(materials Science Forum Volume 726) Edited By Dariusz Skibicki - Fatigue Failure And Fracture Mechanics-trans Tech (2012).pdf

Fatigue Failure and Fracture Mechanics

Edited by Dariusz Skibicki

Fatigue Failure and Fracture Mechanics

Selected, peer reviewed papers from the Conference on XXIV Symposium on Fatigue Failure and Fracture Mechanics, May 22-25, 2012, Bydgoszcz-Pieczyska, Poland

Edited by

Dariusz Skibicki

Copyright  2012 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Kreuzstrasse 10 CH-8635 Durnten-Zurich Switzerland http://www.ttp.net

Volume 726 of Materials Science Forum ISSN 1662-9760 Full text available online at http://www.scientific.net

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                                        Preface       The conference, one of the most important Polish conferences in the field of materials science and engineering, is organized by the Polish Academy of Sciences and the University of Technology and Life Sciences in Bydgoszcz. The scientific meetings have been held every second year since 1971. The founder of the conference is the author of many monographs, articles and manuals in the field of Fatigue, Failure and Fracture Mechanics – Prof. Stanisław Kocańda.                        The subjects covered in the conference include fatigue life of elements of structures, research on low and high-cyclic fatigue, fatigue in conditions of complex states of stress and strain, analysis of fatigue loads, design taking fatigue into consideration, analysis of random fatigue loads, experimental methods in mechanics of cracking, application of fracture mechanics in engineering issues, the impact of structural, technological and functional factors on fatigue of elements of structures, research methods and equipment.

Prof. Janusz Sempruch

Table of Contents Preface

Chapter 1: Fatigue Life Prediction Low Cycle Fatigue Life of Martensitic Cast Steel after Ageing G. Golański, S. Mroziński and K. Werner Experimental Verification of the Analytical Method for Estimated S-N Curve in Limited Fatigue Life P. Strzelecki and J. Sempruch Fatigue Life Calculation in Conditions of Wide Spectrum Random Loadings – The Experimental Verification of a Calculation Algorithm on the Example of 41Cr4 Steel B. Ligaj and G. Szala The Fictitious Radius as a Tool for Fatigue Life Estimation of Notched Elements G. Robak, M. Szymaniec and T. Łagoda Determination of Fatigue Life on the Basis of Experimental Fatigue Diagrams under Constant Amplitude Load with Mean Stress A. Niesłony and M. Böhm Applying a Stepwise Load for Calculation of the S-N Curve for Trabecular Bone Based on the Linear Hypothesis for Fatigue Damage Accumulation T. Topoliński, A. Cichański, A. Mazurkiewicz and K. Nowicki Concept of Fatigue for Determining Characteristics of Materials with Strengthening E. Marcisz, A. Niesłony and T. Łagoda

3 11 17 27 33 39 43

Chapter 2: Fatigue Properties of Materials Material Properties Investigations With the Use of Microspecimen D. Boroński Effect of Microstructure on Rolling Contact Fatigue of Bearings T.Z. Woźniak, J. Jelenkowski, K. Rozniatowski and Z. Ranachowski Determination of the Fatigue Properties of Aluminum Alloy Using Mini Specimen T. Tomaszewski and J. Sempruch Description of Cyclic Properties of Steel in Variability Conditions of Mean Values and Amplitudes of Loading Cycles G. Szala and B. Ligaj The Comparison of Cyclic Properties of X5CrNi18-10 Steel in the Range of Low-Cycle Fatigue in Conditions of Stress and Strain Control B. Ligaj and G. Szala Method of Determining the Initial Stiffness Modulus for Trabecular Bone under Stepwise Load T. Topoliński, A. Cichański, A. Mazurkiewicz and K. Nowicki

51 55 63 69 77 84

Chapter 3: Fatigue of Welded Structures Fatigue Test Welded Joints Steel S960QL C. Goss and P. Marecki Influence of the Notch Rounding Radius on Estimating the Elastic Notch Stress Concentration Factor in a Laser Welded Tee Joint K. Niklas and J. Kozak Fatigue Life Tests of Explosively Cladded Steel-Titanium Bimetal A. Kurek and A. Niesłony Simulation of Tensile Test of the 1/2Y Welded Joint Made of Ultra High Strength Steel J. Gałkiewicz

93 100 106 110

b

Fatigue Failure and Fracture Mechanics

Identification of Efficient Material S-N Curve for Steel Welded Joints Ł. Blacha and A. Karolczuk Residual Stresses in Steel-Titanium Composite Manufactured by Explosive Welding A. Karolczuk, K. Kluger, M. Kowalski, F. Żok and G. Robak The Impact of the Laser Welding Speed on the Mechanical Properties of Joints in Multilayer Pipes S. Mroziński and M. Piotrowski

118 125 133

Chapter 4: Temperature and Thermo-Mechanical Fatigue Model of the Deformation Process under Thermo-Mechanical Fatigue Conditions J. Okrajni and G. Junak Influence of Temperature on the Cyclic Properties of Martensitic Cast Steel S. Mrozinski and R. Skocki Use of Thermography for the Analysis of Strength Properties of Mini-Specimens A. Lipski and D. Boroński Variations of the Specimen Temperature Depending on the Pattern of the Multiaxial Load – Preliminary Research A. Lipski and D. Skibicki

143 150 156 162

Chapter 5: Multiaxial Fatigue Steel X2CrNiMo17-12-2 Testing for Uniaxial, Proportional and Non-Proportional Loads as Delivered and in the Annealed Condition D. Skibicki, J. Sempruch and Ł. Pejkowski Estimation of Fatigue Life of Materials with Out-of-Parallel Fatigue Characteristics under Block Loading M. Kurek and T. Łagoda Criteria Evaluation for Fatigue Life Estimation under Proportional and Non-Proportional Loadings Ł. Pejkowski and D. Skibicki

171 181 189

Chapter 6: Fatigue Crack Growth Fracture Toughness of Structural Members A. Neimitz Fatigue Crack Growth Rates of S235 and S355 Steels after Friction Stir Processing D. Kocańda, V. Hutsaylyuk, T. Slezak, J. Torzewski, H. Nykyforchyn and V. Kyryliv An Experimental Investigation on Crack Initiation and Growth in Aircraft Fuselage Riveted Lap Joints A. Skorupa, M. Skorupa, T. Machniewicz and A. Korbel Application of Digital Image Correlation in Fatigue Crack Analysis T. Marciniak, Z. Lutowski, S. Bujnowski, D. Boroński and T. Giesko Dual-Band Experimental System For Subsurface Cracks Testing T. Marciniak, Z. Lutowski, S. Bujnowski, D. Boroński and P. Czajka Dual-Camera Vision System for Fatigue Monitoring T. Giesko Modeling of Crack Growth in Steels J. Jackiewicz

195 203 211 218 222 226 233

CHAPTER 1: Fatigue Life Prediction

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.3

Low cycle fatigue life of martensitic cast steel after ageing Grzegorz Golański1,a, Stanisław Mroziński2, b, Krzysztof Werner3,c 1

Czestochowa University of Technology, Armii Krajowej 19, 42 – 200 Czestochowa, Poland

2

University of Technology and Life Sciences in Bydgoszcz, Kaliskiego 7, 85-791 Bydgoszcz, Poland 3

Czestochowa University of Technology, Akademicka 3, 42 – 200 Czestochowa, Poland a

[email protected], [email protected], [email protected]

Keywords: low cycle fatigue, ageing, high – chromium cast steel, cyclic properties

Abstract. The paper presents the results of research on fatigue life of GX12CrMoVNbN9 – 1 (GP91) cast steel in delivery state and after ageing at the temperature of 600 oC. The fatigue of low cycle scope was performed at the temperature of 600 oC at the strain amplitude level of εac = 0.25 ÷ 0.60%. Analysis of the tests carried out has shown that the examined cast steel is characterized by strong cyclic softening without an evident period of stabilization. Ageing of GP91 cast steel leads to the fatigue life reduction by ca. 7 to ca. 35%, depending on the level of strain amplitude εac in comparison with the delivery state. Introduction The GX12CrMoVNbN9 – 1 cast steel belongs to a new group of high-chromium martensitic materials introduced to the power industry in relation to raising the service parameters of power units. This cast steel, whose functional properties are higher than those of previously used low alloy cast steels, was developed on the basis of chemical composition of X10CrMoVNbN9 – 1 steel [1, 2]. During service, the power units components made of steel castings are exposed to the effect of changeable thermal stresses due to start-ups and shut-downs of power units, as welll as mechanical stresses which often exceed the value of yield strength. Repeated cyclic effect of temperature and load contributes to the occurrence of deformations and fractures of fatigue character after a certain number of cycles. Damages, and in extreme cases - complete failures of the castings developing this way, are connected with low cycle fatigue. Damages of massive multi-ton steel castings resulting from low cycle fatigue constitute ca. 65% of all damages in steam turbines [3]. The basic requirement put for high-temperature creep resisting steels/cast steels used in the power industry is retaining stable microstructure for long service time, and thus maintaining certain mechanical properties. The influence of temperature and time, and also stress in the creep conditions, is a cause of changes in microstructure of high-chromium cast steels, which also entails changes in their properties, including cyclic properties [4, 5]. Nowadays the tendency, both: home and abroad, is to aim for creating comprehensive characteristics which determine the usefulness and potential possibility of using modern hightemperature creep resisting steels and cast steels. For this purpose, proper characteristics are necessary, such as fatigue characteristics of new grades of steels and cast steels which determine gradual reduction of properties during the service [6 – 8]. The paper is to present and compare the results of research on fatigue life within low cycle scope for GX12CrMoVNbN9 – 1 cast steel in delivery state and after 8000 hours of ageing at the temperature of 600 oC. Material and methodology of research The material under research was high-chromium martensitic GX12CrMoVNb9-1 (GP91) cast steel of the following chemical composition (%mass): 0.12C; 0.47Mn; 0.31Si; 0.014P; 0.004S; 8.22Cr; 0.90Mo; 0.12V; 0.07Nb; 0.04N. The tests were carried out on test pieces in delivery state, i.e. after

4

Fatigue Failure and Fracture Mechanics

heat treatment (1040 oC/12h/oil + 760 oC/12h/air + 750 oC/8h/furnace) and after 8000 hours of ageing at the temperature of 600 oC. The test pieces were exposed to low cycle fatigue at the temperature of 600 oC at five levels of total strain amplitude εac = 0.25; 0.30; 0.35; 0.50; 0.60 %. Description of methodology that was followed during the research on fatigue life, within low cycle scope at elevated temperature, is presented inter alia in the works [9, 10]. The examined cast steel in delivery state, as well as after the process of ageing, was characterized by a microstructure typical for this grade of materials, that is the microstructure of high-temperature tempered martensite of elongated subgrains whose shape was inherited from lath martensite with numerous precipitates. On grain boundaries of prior austenite and on subgrain boundaries, the M23C6 carbides were precipitated. Inside the subgrains, numerous precipitations of the MX type were observed. Examples of microstructure of the examined cast steel in delivery state and after the ageing process are illustrated in Fig 1. a)

b)

Fig 1. Microstructure of the cast steel: a) delivery state, b) after ageing process Table 1 shows the characteristics of dislocation microstructure of the examined cast steel (dislocation density, mean diameter of subgrains and shape coefficient) in delivery state and after 8000 hours of ageing at the temperature of 600 oC. The analysis carried out by means of transmission electron microscope have revealed that as a result of the ageing process at the temperature of 600 oC, in GP91 cast steel there is a decrease of dislocation density and an increase in the subgrain diameters. Changes in the dislocation microstructure are connected with the processes of recovery and polygonization of the matrix running at elevated temperature, as a result of aiming to obtain dislocation microstructure of lower energy. Table 1. Quantitative parameters of dislocation microstructure of GP91 cast steel Ageing Mean diameter Dislocation Temperature, Shape time, of subgrains, density, [oC] coefficient [h] [µm] [1014 m-2] delivery state --0.708 ± 0.258 0.701 ± 0.151 2.95 ± 2.02 600 8000 0.868 ± 0.249 0.660 ± 0.173 1.59 ± 1.58 Research results and their analysis Static tests The fatigue tests of low cycle GP91 cast steel in delivery state and after the process of ageing were preceded by the static tests of tension performed at two temperatures. Mechanical properties of the investigated cast steel obtained at the above-mentioned temperatures are given in Table 2.

Dariusz Skibicki

5

Table 2. Mechanical properties of GP91 cast steel at room temperature and at 600 °C, in delivery state (sw) and after ageing (ps) Temperature, [°C] Parameter 20 600 600 (sw) (sw) (ps) YS, [MPa] 503 303 297 TS, [MPa] 663 338 331 El., [%] 38.3 63.5 63.8 E, [MPa] 206870 150120 142180 Performed tests have shown that the process of ageing of the investigated cast steel during 8000 hours at the temperature of 600 °C does not have a significant influence on the basic parameters determined by the static tensile test. Whilst a growth of the temperature of the cast steel testing results in the occurrence of plastic strains, as the stresses decrease (there is a fall of the values of stresses of yield strength and tensile strength). At the same time, there is an increase in elongation from 38.3% to 63.5% (63.8% after ageing).

εac, [%]

T= 600 °C (sw)

T= 600 °C (ps)

σ, MPa 350

0.25 -0.3

0

-350 σ, MPa 350

0.35 -0.5

0

-350 σ, MPa 350

0.60 -0.8

0

-350

1 2 3

ε, % 0.3

1 2 3

σ, MPa 350

-0.3

0

-350 σ, MPa 350

0

ε, % 0.5

1 2 3

ε, % 0.8

-0.5 -350 σ, MPa 350

-0.8

0

1 2 3

ε, % 0.3

1 2 3

ε, % 0.5

1 2 3

ε, % 0.8

-350

Fig. 2. Examples of hysteresis loops for the cast steel in delivery state (sw) and after ageing process (ps) obtained at the temperature of 600 °C: 1 – loop at the first cycle; 2 – loop at half-life cycle; 3 – loop for the last cycle Fatigue tests Analysis of the cyclic properties of GX12CrMoVNbN9 – 1 cast steel, in delivery state and after the process of ageing in the conditions of changing load, was performed using the most important parameters of hysteresis loop. These parameters include plastic strain amplitude εap and stress amplitude σa. During fatigue tests, the changes in the basic parameters of hysteresis loop in the function of the number of stress cycles were observed. These changes were very similar at all levels of strain. In the courses of hysteresis loop parameters there were three distinguishable stages. Within these stages, a change in the shape of hysteresis loop in the function of the number of stress cycles was noticed. The character of these changes at all levels of strain at the temperature of testing was comparable. Examples of the loops of hysteresis for the strain amplitude εac=0.25%, 0.35% and 0.60% at three different stages of fatigue are shown in Fig. 2.

6

Fatigue Failure and Fracture Mechanics

The process of low cycle fatigue of GP91 cast steel was characterized by strong cyclic softening. It was recognized by the increase in the width of hysteresis loop ∆εap, with the simultaneous intense decreasing in the values of the range of stress changes ∆σa. At the following stages of cyclic strain, there was no period of stabilization of the hysteresis loop parameters observed for the cast steel, neither in delivery state, nor after ageing (Fig. 3). Cyclic softening of the examined cast steel proceeded until the moment of a crack occurring in the test pieces. The above facts prove cyclic exhausting of fatigue life of the investigated cast steel. a) 300

b) σ a, M P a

σa, MPa

300

280

280

260

260

ε a c = 0.60%

240

240

ε a c = 0.50%

220

ε a c = 0.35%

200

220

ε a c = 0.30%

200

180

ε a c = 0.25% 180

160

160

140

140

0

1 0 00

2000

3 0 00

N u m b e r o f c yc le s N

4000

ε ac=0.60% ε ac=0.50% ε ac=0.35%

0

500

1000

ε ac=0.30%

1500

εac=0.25%

2000

2500

Number of cycles N

Fig. 3. Changes in stress σa as the function of the number of cycles (T=600 °C): a) in delivery state; b) after ageing process In order to assess the influence of the strain level on the extent of changes in the loop parameters, the coefficients of cyclic properties were introduced: δσ – for stress description, δε – for strain description. Graphical interpretation of δσ coefficient is presented schematically in Fig. 4 [10]. σa , MPa

δσ =

σamax

σ amax − σ a min ⋅100% σ a min

σamin

∆σa

εac(i)= const

N=1

N

Fig. 4. Graphical interpretation of δσ coefficient The values of δε coefficient were expressed using analogous dependence. Obtained results of calculations of coefficients δσ and δε for particular levels of strain amplitude εac and temperature of testing are presented in Fig. 5. On the basis of analysis of the graphs included in Fig. 5a and 5b it can be concluded that the extent of changes in the hysteresis loop parameters in the function of the number of stress cycles for the cast steel after ageing process is greater than in the case of material as delivered. When comparing the two analyzed parameters of hysteresis loop (σa, εap), it can be noted that the amplitude of plastic strain εap is characterized by smaller changes for most of the load sequences realized in the research. This rule does not apply only for the level of strain εac=0.25% (Fig. 5b). Smaller scope of changes in δε coefficient (the strain description) also proves that, as regards the area of low cycle fatigue, it is justified to make calculations of fatigue life with a strain-based approach.

Dariusz Skibicki

7

a) 50

δσ , %

b) T=60 0 °C ( ps)

50

T=60 0 °C ( sw)

40

δε , % T=600 °C (p s)

40

30

30

20

20

10

10

T=600 °C (sw)

0

0 0.25

0.30

0.35

0.50

0.25

0.60

0.30

εac, %

0.35

εac, %

0.50

0.60

Fig. 5. The δσ and δε coefficients for the cast steel in delivery state (sw) and after ageing (ps): a) δσ , b) δε Analytical dependence between stress σa and strain εap was also described with Morrow's equation. Fig. 6 presents the graphs obtained as a result of approximation of the loop parameters (σa and εap). Because of the lack of a clear stabilization period of cyclic properties, the values of hysteresis loop parameters necessary for the analytical descriptions were determined from the period corresponding to half the fatigue life n/N=0.5. 1000

σa, MPa

lg σ a = lg K '+ n' lg ε ap

2

1

100 0.0001

Nr. 1 2

n’ R2 T, [°C] K’,[MPa] 600 (sw) 496 0.1384 0.9603 600 (ps) 302 0.0631 0.9112

0.001

0.01

Strain εap, mm·mm-1

Fig. 6. Stress – strain curves of the cast steel behaviour at the 600 °C temperature in delivery state (sw) and after ageing process (ps) Analysis of the tests carried out has proved that the process of ageing of GP91 cast steel causes a significant decrease in the values of both: K’ - coefficient, and n’ - cyclic strain hardening exponent. Moreover, graph (2), plotted for the material after ageing process, slopes more gently in comparison with graph (1) for the material in delivery state. This proves the stress amplitude slowly decreasing as a result of changes running in the microstructure of the aged cast steel. Cyclic softening of the examined cast steel in delivery state, as well as after the ageing process, observed during the tests of constant amplitude at the temperature of 600 °C, is also confirmed by the position of curves of cyclic and static strain. Examples of the curves of static and cyclic strain obtained during the tests are illustrated in Fig. 7.

8

Fatigue Failure and Fracture Mechanics

a) σ, MPa

b)

350

1

σ, MPa

2

250

3

150

-0.4

1 250

2

150

3

50

50 -0.8

350

-50 0

0.4

ε, %

0.8

-0.8

-50 0

-0.4

-150

-150

-250

-250

0.4

ε, %

0.8

Fig. 7. Stress–strain curves of static and cyclic strain: a) test pieces in delivery state (sw), b) test pieces after ageing (ps), 1- curve of static tension, 2- graph of cyclic strain, 3- loops of hysteresis Regardless of the temperature of testing, the curves of cyclic strain lie below the curves of static strain. This proves cyclic softening of the cast steel irrespective of the temperature of testing and level of cyclic strain. a)

b) ∆ε

Strain ε

c   

0.01

∆ε

ac = ε + ε = σ f ae ap 2 E

Strain ε

' b ac = ε + ε = σ f  2 N  + ε '  2 N   ae ap f f  f 2 E 

0.01

εap

'

b  2N  + ε '  2N   f f  f  

  

c

εap

εae

εae 0.001

0.001 1000

2NT

10000

Number of strain reversals 2Nf

1000

2NT

10000

Number of strain reversals 2Nf

Fig. 8. Strain–life curves for GP91 cast steel: a) in delivery state (sw); b) after ageing (ps) The results obtained in the research were used for plotting the graphs of fatigue life of GP91 cast steel which was described with the Manson-Coffin-Basquin equation. Fatigue graphs obtained for the examined cast steel for the temperature of 600 ºC are presented in Fig. 8, while Table 3 contains the parameters of Manson-Coffin-Basquin equation. Analysis of the achieved characteristics (Fig. 8) shows that the abscissa 2NT (the point of intersection of two curves εae = f(2Nf) and εap = f(2Nf)) in both cases lies within the range of small number of cycles and amounts to about 5700 cycles for the cast steel in delivery state, and 4558 cycles for the aged cast steel (Table 3). This proves that at the applied levels of strain εac, the process of cyclic strain of the examined cast steel ran with the dominant role of plastic strain component εap. Therefore, it can be assumed that for these strain levels εac, the cyclic strain resistance of the investigated cast steel will mostly depend on its plastic properties. The comparison between fatigue life of GP91 cast steel in delivery state and after ageing is presented in Fig. 9.

Dariusz Skibicki

9

Number of cycles to failure N

Table 3. Parameters of cyclic curves of GP91 cast steel at the temperature of 600°C Temperature, σ’f b c 2NT ε’f [°C] [MPa] 600 (sw) 248 2.21 - 0.0222 -0.8514 5700 600 (ps) 227 9.96 - 0.03014 - 1.0669 4558 1800

sw

1600

sp

1400 1200 1000 800 600 400 200 0 0.25

0.30

0.35

0.50

0.60

Strain ε ac, %

Fig. 9. Fatigue life of GP91 cast steel at the temperature of 600 oC in delivery state (sw) and after ageing (ps) Fatigue life of GP91 cast steel after the process of ageing was lower in comparison with the life of the cast steel in delivery state, and it was dependent on the strain amplitude εac. This decrease was insignificant and amounted to around 7% in the area of the biggest strains realized in the research (εac=0.60%), and it increased along with the fall of strain value (εac=0.25%) to around 35%, compared to delivery state. Table 4. Fatigue characteristics of GX12CrMoVNbN9 cast steel at the temperature of 600°C Temperature 600°C (sw) Temperature 600°C (ps) εac, [%] σa, [MPa], σa,[MPa], N N (n/N=0.5) (n/N=0.5) 0.25 3548 193 2291 198 0.30 2505 206 1593 202 0.35 1945 208 1321 204 0.50 947 237 857 208 0.60 683 236 637 218 Table 4 shows the characteristics of low cycle fatigue if the examined cast steel. Due to the lack of a clear period of stabilization of cyclic properties (Fig. 3), the value of saturation stress σa was determined from the period corresponding to half the fatigue life (n/N=0.5). Acknowledgements Scientific work funded by the Ministry of Education and Science for the years 2010 - 2012 as a research project No. N N507 510 838.

10

Fatigue Failure and Fracture Mechanics

Conclusions 1. The GX12CrMoVNbN9 – 1 cast steel is characterized by strong cyclic softening without a clear stabilization period at both states: the delivery one and after ageing. 2. The ageing process of GX12CrMoVNbN9 – 1 cast steel contributes to a significant decrease in the values of coefficients K’ and n’. It can be a proof of slow decreasing in the level of stress δa of the investigated cast steel, as a result of changes running in the microstructure of the aged cast steel through the process of the matrix softening. 3. Ageing of GP91 cast steel causes a decrease in fatigue life at the temperature of testing, and its level is dependent on the strain amplitude εac. This decrease is slight (around 7%) in the area of the biggest strains covered in the research (εac=0.60%) and increases as the strain value falls (εac=0.25%) – ca. 35%, compared to the material in delivery state. References [1] [2]

G. Golański High chromium cast steels for power, J. Energetyka, (2010) 58 – 61 (in Polish). H. K. Mayer, H. Cerjak, P. Hofer, E. Letofsky, F. Schuster, Evolution of microstructure and properties of 10% Cr steel castings, in Microstructural development and stability in high chromium ferritic power plant steels, A. Strang and D. J. Gooch (Eds.) The Institute of Materials, London, 1997, pp. 105 – 122. [3] R. Viswanathan, Damage mechanisms and life assessment of high temperature components, ASM International, Metals Park Ohio, USA, 1989. [4] G. Golański, J. Kępa, The Effect of Ageing Temperatures on Microstructure and Mechanical Properties of GX12CrMoVNbN9 -1 (GP91) Cast Steel, Archives of Metallurgy and Materials, 2012 (in print). [5] A. Zieliński, J. Dobrzański, G. Golański, Estimation of the residual life of L17HMF cast steel elements after long – term service, JAMME, 34 2 (2009) 137 – 144. [6] M. Cieśla, G. Junak, Low - cycle characteristic of the latest generation of creep resistant martensitic steels and their welded joints, in Materials and Technology for Construction of Supercritical Boilers and Waste Plants, A. Hernas (Eds.), SITPH Publ., Katowice, 2009, pp. 378 – 399. [7] M. Cieśla, J. Dobrzański, G. Junak, Utility characteristics of a superheater outlet chamber material after overrunning the computational service life, J. Energetyka 1 (2012) 28 – 38 (in Polish). [8] J. Okrajni, M. Cieśla, K. Mutwil, Power plant component life assessment, Inżynieria Materiałowa 1 (2005) 15 – 20. [9] G. Golański, K. Werner, S. Mroziński, Low cycle fatigue of GX12CrMoVNbN9-1 cast steel at 600 ºC temperature, Advanced Materials Research 396-398 (2012) 326-329. [10] S. Mroziński, Stabilization of cyclic properties in metals and its influence on fatigue life, Publ. House of the University of Technology and Life Sciences, Bydgoszcz, Poland 128, 2008 (in Polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.11

Experimental Verification of the Analytical Method for Estimated S-N Curve in Limited Fatigue Life STRZELECKI Przemysław1,a, SEMPRUCH Janusz1,b 1

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, ul. Kaliskiego 7, 85-791 Bydgoszcz a

b

[email protected], [email protected]

Keywords: fatigue design, S-N curve, high-cycle fatigue, accelerated method.

Abstract. The paper presents the method of determining the S-N curve based on the static material properties only. To verify the procedure algorithm, an experiment has been carried out to define the reference fatigue characteristics of material C45+C. The research was performed with the use of the rotary-bending test stand made according to own design. The proper operation of the materials testing machine was verified compliant with the ISO 1143 standard. The analytical method was verified by making statistical calculations assuming the null hypothesis of the equality of slope coefficients of the estimated line with the analytical method and the experimental line. The statistics values calculated have shown that there exist no grounds for rejecting the null hypothesis. Introduction

Stress amplitude ,σa (log)

In engineering calculations, fatigue life or strength of a design element is determined by making the calculations based on a specific model dependent on the loading method. The calculations applied mostly refer to the high-cycle fatigue area. To facilitate such calculations, one shall have fatigue characteristics at their disposal. Frequently at the preliminary phase of the project, it is not possible to perform an experiment which would aim at plotting such a curve. To do so, fatigue characteristics are estimated based on the values of static strengths of the material and the assumed technological process, applying analytical methods. Such methods include e.g. the 1 me FITNET method [1], the method reported in ZG the publication [2] and the algorithm 1 proposed by the authors [3]. Verification of mD these methods can be found in [4] and [5]. The aim of this paper is to verify own proposal of the analytical method used to 106 108 determine the S-N curve. It is based on the Life, N, cycles (log) algorithm from FITNET procedures. It has been described in detail in the publication Fig. 1 S-N curve according to the method proposed [3], and in a form of a diagram, see Fig 1. The primary assumptions of the method are based on the principle that the first step should involve determining the actual static properties of the material (ultimate tensile strength Rm and yield stress Re). The next step is to define the equation of the S-N curve according to equations (1), (2) and (3). Additionally an attempt has been made to verify the method similar in nature published in [1] and [2].

12

Fatigue Failure and Fracture Mechanics

σame N  ZGme N0

(1)

 106  log  N Re   me   0,9 Re  log   ZG 

(2)

R  NRe  400 e   Rm 

10

(3)

where: σa – stress amplitude, N – number of cycles completed, ZG –fatigue limit; it has been assumed according to the FITNET guidelines [after 1], N0 – base number of cycles (it has been assumed as 106), me – slope coefficient. One shall mention that the line with inclination mD is plotted for the application of the characteristics for the elements made of austenitic steel, alloys of aluminium and other metals except for steel and cast steels. The value of slope coefficient mD is 15 for normal stresses and 25 for tangential stresses. The FITNET method is differentiated from the method proposed by the authors by the method of defining slope coefficient me the value of which has been determined as 5 for normal stresses and 8 for tangential stresses. The proposal given in the publication [2], on the other hand, as a different one, defines the fatigue curve by determining the line for the range of limited fatigue life based on two points. The points define the estimated strength for the fatigue life of 106 and 103 cycles. Methodology To verify the above proposals, there were applied data characteristic for the fatigue properties of material C45 +C (Rm = 826 MPa, Re = 647 MPa). To receive that data, test has been made using the rotary-bending fatigue test stand. The drawing of the specimen used for the tests is given in Fig. 2. Interestingly, the materials testing machine has been made according to own design the diagram of which is given in Fig. 3. The tests were made according to norms [6] and [7]. Essentially, the test stand had been verified earlier. The test stand verification involved determining the maximum error of the bending moment applied. The calculations of that value were made compliant with the norm [8] and it was 1.15%. The admissible value here was 1.3%. Additionally there was a measurement of the correctness of Fig. 2 Specimen. calculating the number of cycles. To verify the accuracy of the cycle measurement, the Sentry ST723 tachometer was used (the measurement accuracy of 0.01%). The maximum measurement error was 0.24%.

Dariusz Skibicki

13

l

Mg

Mo

t m·g

m – mass [kg], g – gravitational acceleration [m/s2], l – arm of the load [m].

Mg=m·g·l Fig. 3 Testing equipment diagram

Test results The fatigue test results have been presented in Fig. 4. Besides, in the right lower corner (Fig. 4) you can find the equation of the experimental line calculated according to the norm [9] and the coefficient of determination received. The dashed line stands for the confidence interval calculated for the confidence interval of 95%. Fig. 5 presents an estimated S-N curve according to the method presented above (the dashed line) together with the line defined experimentally.

logN=-8.058logσ +26.33 a

2

R =0.952

Fig. 4 S-N curve for C45+C steel

14

Fatigue Failure and Fracture Mechanics

Experimental curve

Estimated curve

Fig. 5 S-N curve C45+C steel and the estimated curve according to the method proposed [3]

Verification method To verify the method proposed, calculations were made to verify the assumption that the estimated curve shows the slope coefficients equal with the regression curve received from the experimental points. The equations used for the statistical calculations are as follows [10]: n

Sa 

ta 

 (Y  Yˆ ) i

i 1

2

i

2  1 n   (n  1) X i2    X i   n  i 1    i 1 n

a  a0 Sa

(5)

n

n

(Y  Yˆ )   X

2

2

Sb 

tb 

i 1

i

i

i 1

i

2  n  n   (n  2)n X i2    X i    i 1    i 1

b  b0 Sb

where: X=logσ, Y=logN,

(4)

(6)

(7)

Dariusz Skibicki

15

n – number of specimens, =a0X+b0 – equation received with the analytical method, Y=aX+b – equation received in the experiment. Verification results There exist no grounds for rejecting the null hypothesis of the equality of the slope coefficients of the estimated equation from the experimental results and the equation received using the analytical method when the condition below is met: |ta|< t(p,n-2) and |tb|< t(p,n-2).

(8)

Table 1. Statistical calculation results. Method FITNET Method reported reported in the method in the publication [3] publication [2] -9.5 -5.0 -11.8 29.9 18.6 36.0 -1.516 -1.953 3.585 1.386 1.810 -3.444 12.6% 6.32% 0.22% 15.0% 8.03% 0.31% 2.093

The results of the calculations are given in Table 1. To compare the method Value of coefficient a0 proposed with other Value of coefficient b0 algorithms, there ta were determined tb curves according to pvalue (97.5%,ta) the FITNET method pvalue (97.5%,tb) and the method t(97.5%,19) reported in the publication [2]. For all those algorithms the equation of curves were determined. The coefficients of those equations are given in Table 1. To make a quality comparison, Fig. 6 presents the characteristics according to the experiment (the solid line) and the analytical methods (the dashed line stands for the method reported in the publication [3], the dotted line stands for the FITNET method, the dot-and-dashed line stands for the method reported in the publication [2]). Statistics

Fig. 6 Diagrams of fatigue curve according to the experimental and analytical method. For the lines details, see the text.

16

Fatigue Failure and Fracture Mechanics

Conclusions The calculations have confirmed the assumption that the slope coefficients of the estimated curve according to own proposal show values non-significantly different from the coefficients of the experimental curve. A similar assumption can be assumed for the original FITNET method, however, in that case the calculated values of statistics are close to the statistical critical interval established at the level of confidence of α=0.05. The result is reflected in the figure presented in Fig. 6. Interestingly, the curve determined according to the method reported in the publication [3] shows the inclination closer to the experimental curve than the curve received according to the FITNET method. In the statistical calculations the method reported in the publication [2] appeared to be worst, which is connected with the fact that the fatigue limit determined according to that algorithm is much greater than the one determined according to the FITNET method. The error in determining that value was 14.3% and 5.3%, respectively. Acknowledgement The work has been co-financed by the European Union Social Fund, the state budget of Poland and the budget of the Kujawsko-Pomorskie Province as part of the project ‘Krok w przyszłość – stypendia dla doktorantów’, the 4th edition. References [1] Neimitz A., Dzioba I., Graba M., Okrajni J., Evaluation of strength, life and safety of structural components contain defects, Politechnika Świętokrzyska, Kielce 2008, pp. 131-183. [2] Lee Yung-Li, Pan Jwo, Hathaway R. B., Barkey M. E., Fatigue testing and analysis, University of Alabama, Elsevier. 2005, pp. 126-140. [3] Strzelecki P., Sempruch J., Modification of selected methods of rapid determination of fatigue characteristics in the range of limited fatigue life, Journal of Polish Cimac, Vol. 6 No 3, Gdańsk 2011, pp. 289-296. [4] Niesłony A., Kurek A., Chalid el Dsoki, Kaufmann H., A study of compatibility two classical fatigue curve models based on some selected structural materials, Inernational Journal of Fatigue, Vol. 39, June 2012, pp. 88-94. [5] Pejkowski Ł., Skibicki D., Analysis accelerated methods for determination of fatigue curves, Journal of Polish Cimac, Vol. 6 No 3, Gdańsk 2011, pp. 199-214. [6] PN-H-04326:1976, Test of metal fatigue – Bending tests [7] PN-H 04325:1976 Test of metal fatigue - Basic terms and general guidelines for preparing of samples and carry out tests [8] ISO 1143:2010 Metallic materials, Rotating bar bending fatigue testing. [9] ASTM E 739-91:2004, Standard Practice for Statistical Analysis of Linear or Linearized StressLife (S-N) and Strain-Life (ε-N) Fatigue Data. [10] Krysicki W. et al, Probability theory and mathematical statistics in problem, PWN, Warszawa 2000, pp. 184-188.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.17

Fatigue life calculation in conditions of wide spectrum random loadings - the experimental verification of a calculation algorithm on the example of 41Cr4 steel Bogdan Ligaj1, a, Grzegorz Szala1, b 1

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Department of Machine Design, ul. Prof. S. Kaliskiego 7, 85-225 Bydgoszcz a

email: [email protected], b email: [email protected],

Keywords: random loading, fatigue life of steel, programmed fatigue testings, algorithm of fatigue calculations.

Abstract. Precision of fatigue life calculations of structural elements in programmed loading conditions is connected with proper elaboration of loading spectrum and assumption of a proper fatigue characteristic. On the base of literature data and own research there has been elaborated an algorithm for fatigue life calculations in random loading conditions with wide spectrum. Calculations were performed with the usage of chosen mathematical models of two-parametric fatigue characteristics. Results calculated with accordance to the described procedure were validated with experimental test results of specimens made of 41Cr4 steel with a method of programmed fatigue life tests. Nomenclature C(0) C(-1) I N Nc exp Nc cal Nij N0 R S Sa Si Sf (-1) Sf (0) Sm Smax Smin ct i j k nc nij m(-1) m(0)

– constant in the formula describing Wöhler fatigue diagram for off-zero pulsating load (R = 0), – constant in the formula describing Wöhler fatigue diagram for oscillating load (R = -1), – coefficient characterizing width of random loading spectrum, – cycle number – general notation (fatigue life), – number of cycles determined as a result of experimental tests, – number of cycles determined as a result of calculations with the usage of two-parametric fatigue life characteristic, – number of cycles to cracking determined on the base of two-parametric fatigue life characteristic for defined Sai and Smj values, – base number of cycles corresponding to fatigue life (N0 = 106), – cycle asymmetry ratio (R = Smin/Smax), – specimen stress – general notation [MPa], – sinusoidal cycle stress amplitude [MPa], – local stress values on the „i” level of loading [MPa], – fatigue limit under oscillating load (R = -1) for N0 cycle number [MPa], – fatigue limit under pulsating load (R = 0) for N0 cycle number [MPa], – mean sinusoidal cycle stress [MPa], – maximum sinusoidal cycle stress [MPa], – minimum sinusoidal cycle stress [MPa], – constant value in formula describing fatigue life curve, – general notation for the loading level (i = 1,2, ...., k), – index of mean value of sinusoidal cycle loading (j = 1,2, ...., p), – exponent in equation ψN = N-k, – total number of cycles in loading spectrum, – number of cycles in loading spectrum with Sa and i Sm j parameters, – exponent in formula describing Wöhler fatigue diagram for oscillating load (R = -1), – exponent in formula describing Wöhler fatigue diagram for pulsating load (R = 0),

18

Fatigue Failure and Fracture Mechanics

mc cal(x) – exponent in formula describing fatigue life curve in service loading conditions, ψ – factor of material sensitivity to cycle asymmetry, for N = N0, ψN – factor of material sensitivity to cycle asymmetry, for N ≠ N0, 1. Introduction There are numerous procedures of fatigue life calculations of structural elements. As an example of such works there can be mentioned procedures published in 2006 by European Fitness-for-service Thematic Network know as FITNET [3]. From the analysis of these procedures results that in a simplified scope there were discussed problems of elaboration of loading spectra and programs of loading characteristics and determination of fatigue characteristics for cases of service loadings. Detailed analysis of the procedures indicates the need of elaboration of a fatigue life calculation procedure of structural elements subjected to random (service) loading with wide spectrum. Set of cycles determined on the way of schematization of such loadings is characterized by a wide range of variability of mean values Sm and amplitude values Sa [6, 15]. That is why the undertaken task requires the elaboration of numerous detailed methods connected among others with the elaboration of two-parametric loading spectra (2D) [5], selection of proper fatigue characteristics (2D) [8, 13] and selection of an appropriate cumulative fatigue damage hypothesis. [14]. The aim of this work is to present the algorithm of fatigue life calculations on the example of 41Cr4 steel in random loading conditions with wide spectrum. The scope of the paper covers detailed review of the algorithm for the calculation on which background there will be calculated fatigue life of 41Cr4 steel in random loading conditions with an application of chosen two-parametric fatigue characteristics. The essential element of the work is an experimental verification of calculated results. 2. Algorithm for the calculation of fatigue life Fatigue life calculations in random loading conditions require to perform numerous operations connected with the elaboration of a model of material properties and a model of service loading. The algorithm for fatigue life calculations in random loading conditions presented in the fig. 1 includes possibility of separation of loadings in wide and narrow ones. As a criterion for an evaluation the coefficient I was assumed. Application of the algorithm was limited for fatigue life calculations to macro-crack initial stage and methods of calculations in a stress approach. The algorithm has two trails. First of them refers to service loadings while the second one to material properties. Proceeding in the range of the first trail is connected with determination of local extrema followed by their transformation to relative values Smax i/Smax and Smin i/Smax. Such a form of loadings means more convenient data because of conversion of values for assumed ranges of maximum values during calculations. The next important step for functioning of the entire algorithm is the evaluation of width of loading spectrum. There was assumed as a criterion a value of I coefficient on the level 0,95. Courses of loading characterized with higher value are included to the group of loadings with narrow spectrum. In case while the value is lower the course of loading is included to the group with wide spectrum. Classification of loadings influences the selection of a schematization method. For loadings with narrow spectrum it is recommended to apply the peak counting method (PCM) whereas for loadings with wide spectrum there are recommended following methods: simple-range counting (RCM), full cycles counting (FCM), rainflow counting (RFM) and range-pair counting (RPM) [5, 15]. Performed schamatization results as a set of sinusoidal cycles with variable Sai and Smi parameters which is a base for determination of loading spectrum [11]. In case of loadings with narrow spectrum calculations are based on block loading spectrum in which the mean value is constant for individual cycles. For loadings with wide spectrum it is recommended to elaborate loading spectrum that includes variability of loading amplitude of cycles Sai and their mean value Smi [5]. A correlation table as Sa-Sm array [12] is the example of such a type of loading spectrum.

Dariusz Skibicki

19

START

Material properties reading in: - static (R m, Re, E, A, Z), - cyclic (Sf (-1), Sf (0), N0, m(-1), m(0)),

Service data reading in

Determination of local extrema

Transformation into relative values Smax i /Smax i Smin i /Smax No I < 0,95

Determination of coefficient value I = Ni/Ne

No

Yes

Selection of fatigue characteristic in accordance with one of models: H, I, IM, II, III, IV, V

Selection of fatigue characteristic 1D

I < 0,95 Yes Schematization with the method: PCM

Schematization with the method: RCM, FCM, RFM, RPM

Selection of cumulative fatigue damage hipothesis Non-liner cumulative fatigue damage hipothesis

Liner cumulative fatigue damage hipothesis No

Hipothesis based of constant fatigue damage lines Yes

I < 0,95

Damage calculation Di = ni/Ni

Damage calculation D ij = nij/N ij

D = ΣDi

k

D=

Calculation of design fatigue life Nc cal = (1/D) nc

Determination of design fatigue life Nproj

l

∑∑ D ij i =1 j=1

No Nc cal > N proj Yes

Stop

Fig. 1 Algorithm for fatigue life calculations of structural elements [6] The second trail is connected with material properties. Possibility of calculations requires knowledge on static and cyclic properties of a material. Depending on the type of service loading it is recommended to apply suitable characteristics. For narrow spectrum loadings in calculations there are applied one-parametric characteristics S-N (1D) while for loadings with coefficient I < 0.95 it is recommended to apply characteristics Sa-Sm-N (2D). Such recommendations are connected with influence of cycles with specific value of cycle asymmetry coefficient R on fatigue life [8]. For calculations of fatigue life apart from discussed above loading spectra and two-parametric fatigue life characteristics there should be assumed an appropriate cumulative fatigue damage hypothesis. In the presented algorithm there was included possibility to select such a hypothesis.

20

Fatigue Failure and Fracture Mechanics

Calculating one can apply: linear cumulative fatigue damage hypothesis, non-linear cumulative fatigue damage hypothesis or hypothesis based on idea of constant damage line [14]. In this paper there was applied Palmgren-Miner linear cumulative fatigue damage hypothesis. After the selection of hypothesis of accumulation of life damage calculations leading to determine the level of fatigue damage are performed [9]. For loadings with narrow spectrum fatigue damage is calculated from the formula D=∑

ni Ni

(1)

whereas for loadings with wide spectrum k

p

D = ∑∑ i =1 j=1

n ij

(2)

N ij

Finally calculating fatigue life Nc cal is determined using the formula N c cal =

1 ⋅ nc D

(3)

The result of fatigue life calculations Nc cal of a structural element has to be compared with assumed design fatigue life Nproj. In case when fatigue life Nc cal > Nproj one can state that fatigue life of the structural element fulfills design assumptions what enables to finish calculations. In the opposite case (Nc cal < Nproj) one has to return to the beginning of calculations assuming new properties for structural material and/or validating service data conditions. Undertaken corrective actions enable to recalculate and reevaluate fatigue life of the structural element. On the base of presented algorithm for calculations fatigue life was calculated what will be presented in the further part of the work.

3. Service loadings Fatigue life calculations with accordance to the elaborated algorithm will be presented on the example of service course of stress changes appearing in a dangerous section of the steering mechanism of a passenger car. A part of the recorded loading course was presented in the fig. 2. Following values were described in relative values Si/Smax.

Stress Si/Smax

1 0.5 0 -0.5 2s

-1

Time, s

Fig. 2 Courses of stress changes in the form of relative values for a soil roads with steering steering maneuvers General analysis of the loading course indicates its complex character in fields of values, time and frequency that character should be found in factors influencing on a vehicle suspension assembly. While steering a car it is going to be subjected to transverse swinging that occurs as loading with the course close to sinusoidal (periodical) one on which loading caused by road surface imperfections is overlaid [7].

Dariusz Skibicki

21

The presented course of loading changes was statistically analyzed. Values of statistical parameters are : the root mean square ψ2 = 0.1053 (MPa/MPa)2, mean µx = 0.0138 (MPa/MPa) and variance σx2 = 0.1051(MPa/MPa)2. There were also determined values of autocorrelation and spectral power density that were presented in the work [6] in the form of diagrams. Their analysis enables to classify loading service to the group of wide loading spectrum [2]. Another method to evaluate width of loading spectrum that does not require the complex statistical analysis is determination of I coefficient value from the formula I=

Ni Ne

(4)

In the above formula Ni jest is the number of intersections of mean values through half cycles, increasing and decreasing ones, whereas Ne is the number of local extrema appearing in the course (the sum of minimum and maximum values). The method was described in the work [15]. For the assumed loading course the value of I coefficient is 0.6892 what classifies it to the group of wide loading spectrum. Classification of the stress course on the base of I coefficient is concurrent with classification performed on the base of statistical functions. Regarding to fatigue life tests there was established a service loading model in the form of the correlation table in Sa-Sm array (2D spectrum). The set of data including following minimum values and consequently maximum ones was the base for the undertaken actions. In tests the method of full cycle accumulation was assumed [5] as a recommended for the wide spectrum loading analysis. R = -1,0

1 1

R=0

R

-

< R < -1

-1 < R < 0

1

1

Smax = 0

Smin = 0

1

B

D 2

1

1 1 1

2 1

1 1

1
1

1 1

3 2

1

1

5

1

1 2

2

3

1

1

1

2 4

1

0
1 2

1

1

1

1

1

0.9

0.8

0.85

0.7

0.75

0.6

0.65

0.5

0.55

0.4

0.45

0.3

0.35

0.2

0.25

0.1

0.15

0

E 0.05

-0.1

-0.05

-0.2

-0.15

-0.3

-0.25

-0.4

-0.35

-0.5

-0.45

-0.6

-0.55

-0.7

-0.65

-0.8

-0.75

F -0.9

A

2

3

2

5

1 1

0.95

1

-0.95

R =1,0

Smin = -1 C

-0.85

Amplitude Sai / S max

Smax = 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Mean value Smi / Smax

Fig. 3 Two-parametric loading spectrum (2D) [6] Schematization of service loading enabled to determine the set of sinusoidal cycles characterized by variable values of maximums, minimums, amplitudes and means. In calculations and tests there was applied 2D spectrum that is the correlation table in Sa-Sm array presented in the fig. 3. Values of parameters of cycles were presented in relatives that refer to maximum value appearing in the loading course. The structure of such spectra was discussed in the work [5]. Correlation tables are applied in fatigue life tests also by other scientists what is indicated by works [4, 10]. The

22

Fatigue Failure and Fracture Mechanics

insignificantly changed coordinate system in relation to the work [5] was introduced in order to unify the description of loading spectra 2D and two-parametric fatigue characteristics. Introduced changes did not cause any changes in properties of tables. In such a case position o fields with constant values of defined parameters describe suitable coordinates [6]. 4. Experimental tests Test samples were made of 41Cr4 steel classified as a steel for quenching and tempering. The mentioned processes were performed with accordance to recommendations for the type. Surface hardness of specimens was limited to 42÷44 HRC. Tests in static loading conditions were conducted applying specimens made in accordance with standard PN-EN 10002-1:2004 with diameter of a measurement part was 10 mm. Specimens for cyclic loading conditions (for loadings with coefficient R = -1 and R = 0) made in accordance with standard PN-84/H-04334. Diameter of a measurement part was 10 mm and its length 18 mm. Properties of 41Cr4 steel in static loading conditions were determined in experimental tests. Mean values for chosen parameters are following: tensile modulus (Young’s modulus) E = 203900 MPa, plasticity limit R0,2 = 1288 MPa, tensile strength Rm = 1395 MPa, elongation A5 = 24.6 % and contraction Z = 41.2 % . Tests of steel in variable loading conditions with the cycle asymmetry coefficient R = -1 and R = 0 enabled to determine following parameters: m(-1) = 8.53, m(0) = 9.95, C(-1) = 4.093⋅1028, C(0) = 3.272⋅1031, Sf (-1) = 447.6 MPa, Sf (0) = 366.7 MPa. Factors of material sensitivity to cycle asymmetry ψ [13] were calculated from the formula ψ = 0.000314 ⋅ Rm – 0.092429 = 0.3456

(5)

whereas coefficient k [13] from the formula log k = 1,973 – log Rm = 0,0674

(6)

Experimental tests in service loading conditions were performer with the usage of a test stand consisting of the Instron 8501 fatigue testing machine controlled with a dedicated software that enables to perform programmed fatigue life tests. During tests there was applied the service loading model in the form of 2D spectrum. In tests of 41Cr4 steel there were applied following levels of maximum stresses in the loading programme Smax = 800 MPa, 900 MPa, 1000 MPa and 1100 MPa. Number of specimens on individual levels was as following: level 800 MPa – 2 pcs., level 900 MPa – 3 pcs., level 1000 MPa – 3 pcs. and level 1100 MPa – 3 pcs., Fatigue test results in service loading conditions were presented as fatigue life curve (fig. 4) that was defined with the equation log Smax = −

1 log N c 7.92

exp

+ 3.6095

(7)

The square of Pearson correlation coefficient for test results is r2 = 0.9143. Moreover there was determined a value of statistic F = 96.1. It is higher than the critical value Fkr = 5.12 read from the Snedecor’F continuous probability distribution with a confidence interval α = 0.05 for degrees of freedom k1 = 1 i k2 = 9. From the comparison of statistic values F with critical value Fkr it results that equation (7) correctly describes test results what confirms its application in further analysis [1].

Dariusz Skibicki

10000

23

Stress Smax, MPa

3000 2000

1 1000 800 600 400

2

200

100 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Number of cycles Nc exp Fig. 4 Gassner fatigue life curve of 41Cr4 steel specimens (1) on the background of Wöhler fatigue life curve (2) [6] 5. Fatigue life calculations Fatigue life calculations were performed on the base of chosen mathematical models of twoparametric fatigue life characteristics included in the work [8]. Below there are listed the final forms of equations describing assumed models of characteristics including experimental test results of 41Cr4 steel. (Model I (model by J. Szala). Two-parametric fatigue characteristic in accordance with the model I has a following form: - for the range -∞ < R ≤ 0 N=

4,0935 ⋅10 28 (Sa + ψ NSm )8,53

(8)

- for the range 0 < R < 1,0

 447,6 ⋅ (1395 + Sa − Sm )  N = 10 ⋅    1395 ⋅ Sa (1 + ψ N ) 

8 , 53

6

(9)

where - for model I according to the formula ψ N = 1.5380 ⋅ N −0,0167 − 1

(10)

Model I-M (model by G. Szala). Fatigue characteristic is described by formulas (8) and (9). Modification is on the process of calculations of the material sensitivity to cycle asymmetry factor ψN, that was described as a following dependence [13] ψN = N-k = N-0,0674 Where coefficient k was defined with the formula (6).

(11)

24

Fatigue Failure and Fracture Mechanics

Model II (model byl J. Szala). Two-parametric fatigue characteristic in accordance with the model II presented in the form applied in calculations for the range -∞ < R ≤ 1,0  447,6  S  N = 10 ⋅  1 − m    Sa  1395 

8 , 53

6

(12)

Model III (model by A. Lipski). Two-parametric fatigue characteristic form in accordance with the model III for the range -∞ < R ≤ 1,0 is defined by the following equation  447,6   S  2  1 −  m    N = 10 ⋅  S  a   1395  

8, 53

6

(13)

Model IV (model by A. Lipski). Mathematical description of two-parametric fatigue characteristic in accordance with the model IV for the range of cycle asymmetry coefficient -∞ < R ≤ 1,0 is presented by the dependence below 2  447,6  Sm   6 N = 10 ⋅  1−     Sa  1395  

8, 53

(14)

Fatigue life calculations were performed for loading spectrum in the form of the correlation table (fig. 3). There was also assumed Palmgren-Miner linear cumulative fatigue damage hypothesis. Fatigue life calculations were performer for the range of maximum stresses in loading spectrum from 500 to 1200 MPa with 100 MPa grading. Obtained test results for assumed two-parametric characteristics were approximated with line in logarithmic scale and described with the equation x) log S(max =−

1 m c cal

(x)

⋅ log N c obl + c t

(15)

Exponent value mc cal(x) and ct for individual modes are following: model I – mc cal(I) = 8.86 and ct = 3.5656, model I-M – mc cal(I-M) = 9.14 and ct = 3.5250, model II – mc cal(II) = 9.63 and ct = 3.4935, model III – mc cal(III) = 8.70 and ct = 3.5949, model IV – mc cal(IV) = 8.61 and ct = 3.6046.

6. Analysis of test results Experimental verification of fatigue life test results was based on comparative analysis of experimental and calculation fatigue life curves that led to determine the ratio Nc cal/Nc exp or Nc exp/Nc cal in the function of maximum stress Smax (fig. 5). Comparison of service fatigue life Nc exp with calculated fatigue life Nc cal, in accordance with assumed models, indicates that the highest conformity of results was obtained for two-parametric fatigue characteristic marked as a model I-M. In the range of maximum stresses from 250 MPa to 940 MPa test results are in the dangerous area while above 940 MPa in the area of secure estimations. It results from the fig. 5 that the ratio value Nc cal/Nc exp depends on the stress level Smax. For the level Smax = 250 MPa there was obtained the highest ratio value Nc cal/Nc exp approx. 4.2 that, as stress Smax decreases, reaches value of 1 at the level 940 MPa. Above Smax = 940 MPa the ratio value Nc exp/Nc cal increases reaching value approx. 1.5 for the maximum stresses 1300 MPa. For the model II there was obtained the similar course of changes of the ratio value between experimental and calculated values to the course characterized by I-M model. Differences among results are basically on values of Nc cal/Nc exp and Nc exp/Nc cal on defined loading levels Sa. Obtained test results for two-parametric fatigue life characteristics marked as model I, III and IV are located in the dangerous calculation area for the entire range of maximum stress Smax. Similar character of changes of values Nc cal/Nc exp characterizes models III and IV. It should be also noticed that the smallest conformity of calculations with the experiment obtained for models IV and III.

Dariusz Skibicki

25

16 7

Nc cal/Nc exp

15 6

1

Model I

2

Model I-M

4

Model III

5

Model IV

3

Model II

5 14 1

5

4

13 4 12 3 11 2 2

Nc exp/Nc cal

1 10 29 3

38 200

400

600

800

1000

1200

1400

Stress Smax, MPa Fig. 5 Experimental to calculated fatigue life ratio for 41Cr4 steel [6] Numerous factors influence conformity of calculated and experimental fatigue life results, to name the most important ones: – conformity of mathematical models of two-parametric fatigue characteristics with characteristics experimentally determined, – conformity of cumulative fatigue damage hypothesis with experimental data. Summary a. Presented algorithm of fatigue life calculations applies to the stress approach that can be applied in calculations for high cycle fatigue loading. Assuming I coefficient as the decision criterion enables to chose the appropriate way of proceedings connected with elaboration of loading spectrum and determination of necessary, because of calculations, fatigue life characteristics. Evaluation of loading spectrum width on the base of I coefficient enables to limit the application of complex statistical apparatus that requires detailed knowledge on interpretation of test results as well as specialist software. b. From the comparative analysis of fatigue life test results with calculated ones according to twoparametric fatigue life characteristics it results that differences between calculation results depend on the applied model and essentially on the maximum stress level in the loading programme what has a direct connection with the evaluated fatigue life Nc cal. c. Cumulative fatigue damage hypothesis was the constant factor in the performed fatigue life calculations with application of assumed two-parametric fatigue life characteristics and as a consequence it did not influence on evaluation results of individual models. d. Regarding to the simplicity of IM and II models there are recommended especially in initial calculations realized in design process. The above recommendation is essentials also because of the smallest range of necessary experimental data that appear in descriptions of the mentioned models. e. One of the essential issues in fatigue life tests and calculations of structural materials and elements is the appropriate elaboration of loading spectrum and on its base the proper loading programme. For random loadings with wide spectrum the essential role is played by the assumption of the cycle accumulation method. In the work there was assumed the method of full

26

Fatigue Failure and Fracture Mechanics

cycles. Elaborated with its application loading spectrum was presented in the form of the correlation table including the set of sinusoidal cycles with variable parameters Sa i and Sm i. Such a system of correlation table corresponds with descriptions applied in two-parametric fatigue life characteristics with the form N(Sa, Sm) applied in fatigue life calculations. Note: This work has been elaborated in the frame of the project No. 0715/B/T02/2008/35 financed by Polish Ministry of Sciences and Higher Education. References [1] [2] [3] [4]

[5] [6]

[7] [8]

[9]

[10] [11] [12] [13]

[14] [15]

J.S. Bendat, A.G. Pierdol, Methods of analysis and measurement of random signals, (in Polish), PWN, Warszawa, 1976. J. Čačko, M. Bìlý, J. Bukoveczky, Random processes: measurement, analysis and simulation, ELSEVIER, Amsterdam – Oxford – New York – Tokyo, 1988. FITNET Fitness-for-Service Procedure – Final Draft MK7, 2006. M. Huck, W. Schultz, R. Fischer, G. Kobler, A standard random load sequence of Gaussian type recommeded for general application in fatigue testing, LBF-Report no. 2909, IABGReport no. TF-570, p.21, 1976. S. Kocańda, J. Szala, Fundamentals of fatigue calculations, (in Polish), PWN, Warszawa, 1997. B. Ligaj, Experimental and calculational analysis of Steel fatigue life in random conditions of wide range spectra, (in Polish), Monographs, 2nd part of monograph: Two-parametric fatigue characteristics of steel and their experimental verification, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2011. B. Ligaj, G. Szala, Loading analysis in tests and calculations of fatigue life of constructional elements – on the example of operating loadings of a car, (in Polish), Logistyka nr 6, 2009. B. Ligaj, G. Szala, Experimental Verification of two-parametic models of fatigue characteristics by using the tests of S355J0 steel as an example, Polish Maritime Research 1/2010 (2010), 39-50. B. Ligaj, G. Szala, A fatigue life calculation method or constructional elements with a use of two-parametric fatigue characteristics, Acta Mechanica et Automatica, vol.3 no.2 (2009), 4751. T. Łagoda, K. Walat, Methods of service loading performance in control systems of fatigue machines, Acta Mechanica et Automatica, vol. 5, no. 1 (2011), 47-52, ASTM standard, Standard Practices for cykle counting in fatigue analysis, ASTM Designation: E 1049-85 (Reapproved 1990). Description of a Fighter Aircraft Loading Standard for Fatigue, ICAF, 1976. G. Szala, Theoretical and experimental analysis of two-parametric fatigue life characteristicsm of constr, (in Polish), Monographs, 1nd part of monograph: Two-parametric fatigue characteristics of steel and their experimental verification, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2011. J. Szala, Hypotheses of fatigue damage accumulation, (in Polish), Monographs, University of Technology and Agriculture, Bydgoszcz 1998. J. Szala, Loads and fatigue life of machine elements, (in Polish), University of Technology and Agriculture, Bydgoszcz 1989.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.27

The fictitious radius as a tool for fatigue life estimation of notched elements Grzegorz ROBAK 1, a, Marcel SZYMANIEC 1, b, Tadeusz ŁAGODA 1, c 1

Opole University of Technology, Faculty of Mechanical Engineering, ul. St. Mikołajczyka, 45-271 Opole, Poland a

b

c

[email protected], [email protected], [email protected]

Keywords: fictitions radius, fatigue life,

Abstract. In this paper, the fictitious radius - according to Neuber’s method for determination of stresses at the notch root was used. Next, the fatigue lives of elements of the ring notches were calculated, and then compared with results of experimental tests of S235JR steel samples. However, the obtained fatigue lives did not bring satisfactory results. It has been demonstrated that the fictitious radius strongly depends on the expected fatigue life. Introdaction Estimation of fatigue life of machine elements has been the subject of intensive research for many years. Numerous failures that have been causing costly repairs of elements or even whole structures, have forced designers to perform this type of research in search for improvement. Moreover, in extreme cases, such failures led to life-threatening situations. At present, the problem of lifetime prediction of machine elements and structures has been given a proper recognition as an important and crucial issue addressed in almost every branch of a modern industry. In particular, the problem has been occuring in industrial activities, where the life, reliability and safety requirements are essential. One of the main reasons for conducting research on fatigue and creating complex calculation algorithms is optimization of machine parts [1]. The necessity of carrying out the optimization of machine elements has made the engineers to design structures with complicated geometry while maintaining lifetime and reliability properties. The aim of this paper is to compare the results of experimental tests performed on cylindrical elements with ring notches to the results of lifetime prediction determined by using the fictitious radius [2-4, 8]. Determination of the value and the fictitious radius for ring-notched elements In order to estimate the fatigue life, a concept of the fictional radius by Neuber was used [2]. The fictitious radius at the notch root allowed to calculate the maximum stresses. Determination of this radius results from the mean stresses at the notch root. Fictitious radius at the notch root was described in the following relation:

ρ f = ρ + sρ * ,

(1)

where: s – multiaxial coefficient ρ* - equivalent microstructural length, ρ - real radius at the notch root. The multiaxial coefficient s for round elements under tension was calculated by referring to the Huber-Mises-Hencky’s criterion, according to the equation:

28

Fatigue Failure and Fracture Mechanics

s=

5 − 2ν + 2ν 2 , 2 − 2ν + 2ν 2

(2)

where ν is the Poisson's ratio. The equivalent microstructural length as proposed by Neuber ρ * for the steel is ca. 0.1 mm. Verification of the fictitious radius considerations based on experimental tests Verification of the fictitious radius assumptions was conducted by comparison of the calculated fatigue life with test results for the notched specimens (Fig. 1) made of stainless S235JR steel [5]. In Table 1, the mechanical properties of S235JR steel are presented. The theoretical stress concentration factor was calculated, according to Nody [6] K t = 3.288, and with the calculations obtained from FE analysis K t = 3.330. However, it must be remembered that the notch factor is determined for the fatigue limit. In other cases, its value strongly depends on fatigue life [3, 4]. Table 1. Mechanical properties of steel S235JR Re, Rm, E, GPa ν MPa MPa 299 423 0.29 200 Experimental tests were performed on the fatigue stand type UFP ± 400 – specimens were subjected to cyclic loading under tension-compression [7]. In view of the fact that the real radius ρ = 0.8 mm, and using equations (1) and (2), the value of fictitious radius ρf = 1.088 mm. Based on the obtained radius, a fictitious model of a specimen was created, then used for determination of stresses at the notch root by a finite element method. The specimen model was limited to only a quarter of the entire specimen, which enabled a greater grid density within the notch root. Software “COMSOL” was applied for calculations. The calculations were performed for five levels of nominal stresses in the notch cross-sections, at which the proper experimental tests were carried out: 125, 160, 200, 230, 300 and 330 MPa. The calculations were made for an elastic range. Figure 2 shows distribution of the applied finite elements at the notch root.

Fig. 1. The geometry of specimen with a real ring notch.

Dariusz Skibicki

29

Fig. 2. Distribution of finite elements at the notch root for the geometry of a specimen with the fictitious radius. Based on Neuber’s assumptions, stress values obtained for a modified geometry of the specimen should be compared with fatigue characteristics of smooth specimens. The obtained values of the fatigue life should correspond to the results of experimental tests. Figure 3 shows two fatigue characteristics for notched specimens and smooth specimens made of S235JR steel. Relevant fatigue characteristic for smooth (3) and notched (4) specimens can be written in S-N as log( N f ) = 2.87 − 0.1 log(σ a ) ,

(3)

log( N f ) = 3.33 − 0.213 log(σ an ) ,

(4)

Fig. 3. Fatigue characteristics for smooth and notched specimens made of S235JR steel.

30

Fatigue Failure and Fracture Mechanics

The obtained fatigue characteristics were then compared with the experimental ones - presented in figure 4. As can be seen, the values of calculated fatigue lives are much lower than the experimental ones. Observing the distribution of characteristics in Fig. 3, note that the higher the amplitude of the load, the difference between the calculated and experimental lives increases. In case of fatigue at about 106, the difference between the obtained results decreases. It should be seen, however, that in this case the model proposed by Nueber for the material analysed is not appropriate. This suggests that Neuber’s assumptions are not correct for the steel for stress values at the fatigue limit, and even more to stresses above this level. In figure 3 comparison of experimental lives for nominal stresses equal to 300 and 330 MPa was not included, because the obtained values of calculated fatigue lives took values below one cycle. From figure 3, it is confirmed that distribution of fatigue characteristics for smooth and notched specimens, which coincide to the point below 3000 cycles, where S235JR material ceases to be sensitive to the notch effect. In such a case, the fictitious radius should aim at infinity, and the fatigue coefficient of the notch effect K f = 1.

Fig. 4. Comparison of the experimental fatigue lives with the calculated ones for specimens made of S235JR steel. The fictitious radius as a fatigue life function According to the remarks discussed in the previous chapter, using fatigue characteristics for smooth and notched specimens, the geometry of the specimen was changed by modeling different fictitious radiuses, which resulted in obtaining various equivalent microstructural length, according to the equation (1) we obtain new formula for equivalent microstructural length :

ρ* =

ρf −ρ s

.

(5)

Dariusz Skibicki

31

The geometry of a specimen was modified in such a way so that the stresses for notched specimens corresponded to stresses that occured for the same values as for smooth obtained from the fatigue characteristics. In the result, an ideal compliance of the estimated fatigue characteristics with experimental ones were therefore obtained. Based on the fatigue characteristics for calculations, two additional levels of nominal stresses of 80 and 100 MPa were determined. In figure 5 were estimated the obtained microstructures radiuses, according to the equation (5) as loading cycles function. From the analysis of this figure, it can be seen that the obtained values of the radius of microstructures are arranged according. to a linear function. This relation was described in the following equation: log( ρ * ) = 2.46 − 0.377 log( N ) .

(6)

Fig. 5. The radius ρ * of the microstructure as a fatigue lifetime function. It should be noted that values of mircrostructure radius take quantities of over 1 millimeter. Only in the case of a nominal stress equal to 80 MPa, the obtained value was 0.85 mm. This result means that a ring notch causes much greater changes in microstructure of a material as compared to Neuber’s assumptions. Summary From the obtained calculations, it can be observed that: 1. Application of the fictitious radius, according to Neuber’s solution, for steel in elastic range under loadings above the fatigue limit does not allow to obtain results comparable with the test results. 2. The equivalent radius of microstructure strongly depends on the fatigue lifetime. 3. The microstructure radius are arranged in a linear function depending on the number of cycles.

32

Fatigue Failure and Fracture Mechanics

The project has been finance from the National Center for Science No 2011/01/B/ST8/06850 REFERENCES [1] [2] [3] [4] [5] [6]

[7]

[8]

Gasiak G., Robak G.: Simulation of fatigue life of constructional steels within the mixed modes I and III loading, Vol. 34, 2011, pp. 389-402 Neuber H.: Über die Berücksichtigung der Spannungskonzentration bei Festigkeitberechnungen. Konstruktion 20, 1968, SS. 245-251 Łagoda T.: Lifetime estimation of welded joint, Springer 2008. Biłołus P., Lagoda T.: Structural notch effect in steel welded joints, Materials and Design, Vol. 30, 2009, pp. 4562-4564 Słowik J., Łagoda T.: The fatigue life estimation of elements with circumferential notch under uniaxial sate of loading, Int. J Fatigue, Vol.33, 2011, pp.1304-1312 Noda N. A., Takasa Y.: Stress concentration formula useful any shape of notch in a round test specimen under tension and under bending, Fatigue, Fract. Engng Mater. Structur., Vol. 22, 1999, pp.1071-1082 Blacha Ł., Karolczuk A., Łagoda T.: Assessment of multiaxial fatigue behaviour of welded joint under consideration of plastic strains in fatigue life calculations, Materials Testing, Vol. 53, 2011, No 6, pp. 339-343 Sonsino C. M., Łagoda T., Demofonti G.: Damage accumulation under variable amplitude loading of welded medium and high-strength steels, Int. J. Fatigue, Vol. 26, No 5, 2004, pp. 487-495

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.33

DETERMINATION OF FATIGUE LIFE ON THE BASIS OF EXPERIMENTAL FATIGUE DIAGRAMS UNDER CONSTANT AMPLITUDE LOAD WITH MEAN STRESS Adam Niesłony1,a and Michał Böhm1,b 1

Department of Mechanics and Machine Design, Opole University of Technology, Mikołajczyka 5, 45-271 Opole, Poland a

[email protected], [email protected]

Keywords: fatigue, mean stress, stress ratio, section method.

Abstract. The paper deals with a comparison of fatigue life calculations, obtained on the basis of classical Basquin diagrams and approximated diagrams acquired with the use of the section method, which allows us to fit the diagrams shape to real material properties. While comparing the calculation results, literature data concerning fatigue tests of welded cruciform specimens from high performance low-alloy steel HSLA-80 presented by Kihl and Sarkani as well as of smooth specimens out of the aluminum alloy 75S-T6 by Grover et al. has been used. It has been noticed that the calculations performed with the use of fatigue diagrams approximated using the section method reflect the true behavior of the material. The models by Niesłony-Böhm and Smith-Watson-Topper compensating the influence of the mean stress gave similar results. Introduction Mechanical fatigue of the material is a process resulting from a time varying load and manifested by the loss of cohesion of the material and its ability to carry loads. The mechanism of this phenomenon is complex and depends mainly on the type of material, the nature of changes in load, the surface condition and shape of the element. In engineering practice, this phenomenon is described by the fatigue diagrams, which are the result of statistical processing of test results of samples of material at strictly defined variable loading conditions. We obtain in this way information sufficient to properly describe the phenomenon without going into the macroscopic fatigue mechanisms that are difficult to describe in a deterministic manner. For this purpose most often are used stress fatigue diagrams, which relate the number of cycles till destruction with the stress amplitude. These diagrams are modeled with exponential functions obtaining the corresponding sections of simple graphs with logarithmic axes. One of the most popular types of these diagrams is the Wöhler and Basquin diagram [1,2]. Unfortunately, they describe the real diagram only for a narrow interval of cycle number, in which we observe approximately a straight line. Therefore the focus of this paper is set on the correct description of the experimental results obtained from literature [3,4] with classical Basquin diagrams for the stress ratio R = −1 and R = 0 as well as with the section method, and later on these diagrams have been used to calculate the fatigue life taking into account the mean stress value of the load . This is the simplest practical case in which the particular role is played by the quality of the fit to the actual characteristics of the fatigue properties of the material. The main objective of this work is to investigate whether and to what extent the stress diagram modeling of fatigue by the classical method and sectional influences the fatigue life calculation. Stress Fatigue Diagrams Stress fatigue diagrams represent material resistance to variable loads and are used in the calculation process of fatigue life. Due to the exponential nature of the changes of this type the fatigue diagrams axes are scaled logarithmically. According to the literature review, the true fatigue curves form the shape of a descending wave, which is schematically presented in Fig.1.

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Fatigue Failure and Fracture Mechanics

Figure 1. True shape of a fatigue curve and the approximation with the Wöhler and Basquin diagram [5]. Due to the fact that most fatigue calculations are made for a range of limited or unlimited fatigue life, a number of proposals for the approximation of these ranges with corresponding functions have been presented [6], the most popular is the Wöhler diagram log(σ a ) = A + B log(N ) for N < NG,

(1)

as well as the Basquin relation σ a = CN b .

(2)

Due to the high ease of use of numerical methods, it is becoming increasingly popular to represent a function with straight sections. This method, called section method is very common in FEM calculation programs, and allows the approximation of experimental data showing no fatigue limit or non-linear nature of the stress fatigue diagram [7]. In this case, to define the further sections is enough to provide pairs of stress amplitude σa and the corresponding number of cycles to destruction N, which define a series of points connected by straight lines on the graph. The number of sections is chosen to suit your needs. The obtained diagrams approximated using this method is presented in Fig. 2 and Fig.3 respectively for the fatigue tests of welded cruciform specimens made out of high-strength low alloy steel HSLA-80 performed by Kihl and Sarkani and for smooth specimens out of the aluminum alloy 75S-T6 performed by Grover et al.. 3

10

= = = = = =

-1; experiment -1; Basquin graph -1; section graph 0; experiment 0; Basquin graph 0; section graph

σmax, MPa

R R R R R R

2

10

3

10

4

10

5

6

10

10

7

10

8

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N, cycle

Figure 2. Stress fatigue diagram on the basis of research by Kihl and Sarkani [3] approximated with the Basquin and section method diagrams.

Dariusz Skibicki

35

3

10

-1; experiment -1; Basquin graph -1; section graph 0; experiment 0; Basquin graph 0; section graph

σ max, MPa

R= R= R= R= R= R=

2

10

3

10

4

10

5

10

6

N, cycle

10

7

10

8

10

Figure 3. Stress fatigue diagram on the basis of research by Grover et al. [4] approximated with the Basquin and section method diagrams Table 1. Points defining the stress fatigue diagrams in the section method.

i 1 2 3 4 5 6 7

Steel HSLA-80, Kihl and Sarkani [3] R=0 R = −1 N Ni σai σai i 3 380 275 5⋅10 5⋅103 4 304 205 1,8⋅10 3⋅104 4 200 100 8⋅10 3⋅105 6 92 62 1⋅10 3⋅106 6 77 50 3⋅10 1⋅108 70 1⋅107 60 1⋅108

Aluminum alloy 75S-T6, Grover et al. [4] R=0 R = −1 N Ni σai σai i 3 385 295 5⋅10 5⋅103 4 325 235 1,8⋅10 2⋅104 4 245 165 8⋅10 6⋅104 5 212 140 3⋅10 2⋅105 6 180 122 3⋅10 3⋅106 150 108 1⋅108 1⋅108

Determination of fatigue life on the basis of σa-N diagrams taking into account the influence of mean stress Reliable prediction of fatigue life is inherently bond with taking into account the mean stress value of the load on the endurance. A multitude of combinations of amplitude loads and mean value found in the actual runs is the main reason for the development of models describing these effects in an analytical way. Various equations created over the years present fatigue life as a function of amplitude, mean value and the selected material constants. As an example we can mention the models proposed by Gerber [8], Goodman [9] or Soderberg [10]. In the context of this work a special attention deserves models that also allow the calculation according to the characteristics described by the section method. Two models have been chosen for this reason. The first proposed by Niesłony and Böhm (NB) [11]: σm , σ aT = σ a + [σ afR =−1 ( N ) − σ afR =0 ( N )] σ af , R=0 ( N )

(3)

where: σaf,R=-1(N) and σaf,R=0(N) are values of the amplitudes read from the stress fatigue diagrams for a certain number of cycles N for the stress ratio R = −1 and R = 0. The second one is the well known and popular Smith-Watson-Topper model (SWT) [12] in the form

36

Fatigue Failure and Fracture Mechanics

σ aT = (σ a + σ m )σ a .

(4)

Transformed stress amplitudes obtained on the basis of the equation (3) and (4) have been used to calculate the number of cycles Ncal on the basis of the Basquin stress fatigue characteristics and characteristics described by the section method, which are presented in Fig.2 and Fig.3 and described earlier. Of course for reading the number of cycles, the characteristic with the stress ratio R = −1 has been used. In the case of the NB model, while performing calculations the number of cycles N had to be chosen for the proper values of the σaf,R=-1(N) and σaf,R=0(N), so that it would be equal to the calculated number of cycles Ncal. Since before computing the transformed amplitude according to equation (3) the number of cycles was not known, N was determined by solving the optimization problem in which the objective function was defined as follows N = N cal .

(5)

The fzero function of the MATLAB environment has been used for that reason. The calculated numbers of cycles Ncal were compared with the ones gained from the experimental research Nexp and presented in Fig.4 for the research conducted by Kihl and Sarkani [3] and for Grover et al. [4] in Fig.5. a)

b) 8

8

10

10

R=0 R=1/3 R=2/3 R=-1/3 R=-2 R=-3

Ncal , cycle

6

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7

10

Ncal , cycle

7

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R=0 R=1/3 R=2/3 R=-1/3 R=-2 R=-3

5

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3

Basquin graph NB model

3

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3

Basquin graph SWT model

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c)

10 Nexp , cycle

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d)

8

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R=0 R=1/3 R=2/3 R=-1/3 R=-2 R=-3

6

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R=0 R=1/3 R=2/3 R=-1/3 R=-2 R=-3

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Ncal , cycle

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3 4

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Nexp , cycle

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8

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Figure 4. Comparison of the calculated number of cycles Ncal on the basis of the NB model (a) and (c) and SWT model (b) and (d) with the number of cycles obtained from the experimental research Nexp for different values of the stress ratio R. Results of the cyclic research by Kihl and Sarkani R = 0 and R = −1 approximated with the Basquin graph (a) and (b) as well as with the section method (c) and (d).

Dariusz Skibicki

37

a)

b) 8

8

10

10

R=0.7 R=0.6 R=0.5 R=0.4 R=0.25 R=0.1 R=0.02 R=-0.6 R=-0.8 R=-1.0

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Ncal , cycle

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R=0.7 R=0.6 R=0.5 R=0.4 R=0.25 R=0.1 R=0.02 R=-0.6 R=-0.8 R=-1.0

7

Ncal , cycle

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3

10 Basquin graph NB model

2

10 2 10

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2

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c) 8

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R=0.7 R=0.6 R=0.5 R=0.4 R=0.25 R=0.1 R=0.02 R=-0.6 R=-0.8 R=-1.0

6

Ncal , cycle

10

5

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R=0.7 R=0.6 R=0.5 R=0.4 R=0.25 R=0.1 R=0.02 R=-0.6 R=-0.8 R=-1.0

7

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3 Section graph SWT model

2

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10 N

exp

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, cycle

Figure 5. Comparison of the calculated number of cycles Ncal on the basis of the NB model (a) and (c) and SWT model (b) and (d) with the number of cycles obtained from the experimental research Nexp for different values of the stress ratio R. Results of the cyclic research by Grover et al. R = 0 and R = −1 approximated with the Basquin graph (a) and (b) as well as with the section method (c) and (d).

Observations and conclusions The application of experimental fatigue diagrams approximated with the section method in the process of fatigue determination affected the accuracy of the calculated results. This is particularly visible for the extreme values of the range of the number of cycles, where the classical Basquin diagram doesn’t describe sufficiently accurately the behavior of the material The section method allows with a sufficiently large number of experimental studies, a better fit to the actual fatigue characteristics. NB and SWT models used to determine the transformed amplitudes due to mean stress allow to compensate the effect of the mean stress value on fatigue life in a satisfying way. We can notice that while using the NB model we obtain better results comparing to SWT taking into account the range of research results in the number of cycles. This can be explained by the fact that there is a dependence of the sensitivity of the material on the mean value due to the number of cycles. The NB model accounts for this change by determining the transformed amplitudes on the basis of amplitudes read from the fatigue diagrams for a fixed number of cycles. There is a need to conduct further review of selected models on the basis of fatigue diagrams predicted approximated by sections to demonstrate their usefulness in predicting the fatigue life in a wide range of number of cycles.

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References [1] Kocańda S. Fatigue Failure of Metals. 1st ed. Springer, 1978. [2] Kocańda S, Szala J. Basics of fatigue calculations. Warsaw: PWN (in Polish), 1997. [3] Kihl DP, Sarkani S. Mean stress effects in fatigue of welded steel joints. Probabilistic Engineering Mechanics (1999), 14:97–104. [4] Grover HJ, Bishop SM, Jackson LR. Fatigue strengths of aircraft materials: axial load fatigue tests on unnotched sheet specimens of 24S-T3 and 75S-T6 aluminum alloys and of SAE 4130 steel. (1951). [5] Basquin OH. The Exponential Law of Endurance Tests. Am. Soc. Test. Mater. Proc. (1910), 10:625–30. [6] Schütz W. A history of fatigue. Engineering Fracture Mechanics (1996), 54:263–300. [7] Berger C, Pyttel B, Schwerdt D. Beyond HCF – Is there a fatigue limit? Materialwissenschaft Und Werkstofftechnik (2008), 39:769–76. [8] Gerber WZ. Bestimmung der zulässigen spannungen in eisen-constructionen (Calculation of the allowable stresses in iron structures). Z Bayer Archit. Ing-Ver (1874), 6:101–10. [9] Goodman J. Mechanics applied to engineering. Longmans, Green & Co.; 1899. [10] Soderberg CR, Sweden V. Factor of safety and working stress. ASME Trans, AER-IS 1930;52:13–28. [11] Niesłony A, Böhm M. Fatigue life of cast iron ggg40 under variable amplitude tension with torsion with mean stress, Engineering Modeling (in Polish) (2011), 41:299–306. [12] Smith KN, Watson P, Topper TH. A stress-strain function for the fatigue of metals. Journal Materials (1970), 5:767–76.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.39

Applying a stepwise load for calculation of the S-N curve for trabecular bone based on the linear hypothesis for fatigue damage accumulation Tomasz Topoliński1, a, Artur Cichański1,b, Adam Mazurkiewicz1,c, Krzysztof Nowicki1,d 1

Faculty of Mechanical Engineering, University of Technology and Life Sciences, Kaliskiego 7 Street, 85-789 Bydgoszcz, Poland a

c

[email protected], [email protected], d [email protected], [email protected]

Keywords: trabecular bone, stepwise loading, linear hypothesis of fatigue damage accumulation.

Abstract: In this work were presented calculated fatigue curves based on fatigue tests of trabecular bone under stepwise load with the application of a linear hypothesis accumulation of fatigue damage. The investigation was performed on 61 cylindrical bone samples obtained from the neck of different femur heads. The bone sample fatigue tests were carried out under compression with stepwise increases of the applied load. The fatigue calculation assumed the Palmgren-Miner (P-M) linear hypothesis accumulation of fatigue damage and the associated modified formulae. The obtained mean fatigue curves were based on the modified stress σ/E0 (E0 – initial stiffness) for the assumed rule-determined slope or y-intercept. The highest agreement with the literature was obtained for Σn/N=10. Introduction Due to the complexity of the fatigue processes, physical descriptions are often impossible to define, and thus, in most methods that allow for a calculated evaluation of fatigue life, the fatigue damage accumulation hypothesis is used. Publications [1,2] refer to several hypotheses of fatigue damage accumulation for different materials. The simplest and the oldest hypothesis is the Palmgren hypothesis supported by the results reported by Miner and referred to as the P-M hypothesis. The modifications to this hypothesis involve introducing more complex forms of the fatigue life relationships, including many of the load parameters and environmental factors. There are also nonlinear hypotheses based on the assumption of equi-damage fatigue lines [3]. The aim of the authors is to present a method for the estimation of the S-N curve for the description of fatigue properties of bone using as the base Palmgren–Miner hypothesis (and its modifications), literature data and a single sample. The authors focused on the presentation of this method and calibration its parameters. Methods The paper uses the research results of 61 cylindrical bone samples that were 10mm in diameter and 8.5mm in length and were obtained from the neck of femur heads. The samples were obtained from 21 men and 40 women undergoing hip joint alloplasty. The samples were stored in a 10% formalin solution at room temperature. All of the samples were scanned with a desktop microCT system (µCT-80, SCANCO Medical AG, Bruettiselllen, Switzerland) with a distance of 36µm between. When scanning the values of many bone structure indices were obtained: trabecular number Tb.N, trabecular thickness Tb.Th, trabecular separation Tb.Sp, bone volume fraction BV/TV, surface fraction BS/BV and the number of joints between individual trabeculae per unit volume of specimen Conn.D. The bone sample fatigue tests were carried out under compression with stepwise increases in the load using the testing machine, INSTRON 8874 (Instron, High Wycombe, England). The minimum loading for all of the loading levels was 5N. The maximum loading started at 10N with a gain every

40

Fatigue Failure and Fracture Mechanics

10N at each successive step. At each loading level, 500 cycles were completed under constantamplitude loadings at a frequency of 1Hz. During the fatigue test were recorded values of displacements and forces with a frequency of 100Hz. Fatigue test termination criterion was the step increase of the recorded displacement. The calculations assumed a linear hypothesis of fatigue damage accumulation. k

n

∑  N 

=D

(Eq. 1)

i =1

where: k is the number of executed load levels, D is the damage parameter. In the first case, it was assumed that the sum of the quotient of the executed cycles, n, at a specific load level to the fatigue life, N, at the same load level at the moment of damage occurrence equals D=1. In the second case, the same linear hypothesis was considered, however parameter D can take values different than 1. In both cases, the fatigue life, N, was derived from the S-N curve of the form shown below:

σ E0 = aN b

(Eq. 2)

where: σ is the maximum stress, a is the y-intercept of the obtained straight line, and b is the slope after linearization in bilogarythmic coordinates. Founded in the work method of calculating the S–N curve based on fatigue test results with stepwise load requires the determination of N in equation 1, based on the equation 2. Then: k

∑ i =1

n 1

(σ / aE0 )b

=D

(Eq. 3)

Based on the equation 3, for the assumed values of D and experimental values of σ were calculated coefficients a or b. The calculations were based on constant values of the equations coefficients for two cases. In first case, all of the fatigue curves assumed the same value of the yintercept, a. In second case, assumed that the value of the slope b from did not change in all calculations. The following values from literature [4,5,6,7,8] were assumed for the main calculations in the case of invariability of the y-intercept: 0.0098 - the mean value a for human bones, in case of invariability of the slope: -0.1094 – the mean value b for human bones. Calculations were also made for the following values of the y-intercept 0.0121 – the maximum value for human bone (same as mean value a for bovine bone) and 0.0241 – the maximum value a for bovine bone. Results As an example, the results of calculations, made for D=1, and the changes in the values of the coefficients for the fatigue equation were a=0.00980, 0.0121 and 0.0241 and b=-0.1094 (mean) are shown on the Fig. 1. Fig. 1 presents a set of fatigue curves in a different form; as an example, for the sum of damage D, which equals 1, the curves for all of the samples show the range of variation in the location; the mean curve (for the mean value of a, Fig. 1a-c, or the mean value of the slope b, Fig. 1d); and the area defined by the intervals of confidence for the mean value. In all cases, α=0.05 was assumed. This distribution demonstrates how the locations of the curves change with increasing values of the y-intercept (Fig. 1a-c) when compared with the curves calculated for the mean value of the slope b.

Dariusz Skibicki

a)

b)

c)

σ/Ε 0

0,1

σ/Ε 0

0,1

σ/Ε 0

0,1

d)

σ /Ε 0

0,1

41

0,01

0,01

0,01

0,01

0,001

0,001 10

100000

0,001 10

N cycle

100000

0,001 10

100000

100000 N cycle

N cycle

N cycle

10

Fig. 1. The results of the calculations of the S-N curve: extreme (represented by thin lines), mean (thick line) and their intervals of confidence for human trabecular bone samples (dashed line), assuming a-c) the constant value of the coefficient of the y-intercept and d) the constant value of the slope b The fatigue curve calculation results for all the conditions assumed are given in Table 1. An increase in the absolute value of the slope bm with an increase in the value of the y-intercept and an increase in the value of the damage parameter D was clearly visible. The damage parameter D is well described by the logarithmic function of the mean value of the slope bm and changing in the value of a. The minimal value of the coefficient of determination R2 compounds to 0.97, whereas the relationship between the value of the mean y-intercept and the value of the sum of the damage was a power function (R2=0.99). Table 1. The mean values of the coefficients, bm and am, for the fatigue curve equation determined for three different values of the y-intercept a and the slope b for four values of D, 0.5, 1, 2 and 10. D=0.5

D=1

D=2

D=10

-0.0617 -0.0775 -0.1189

-0.0810 -0.1007 -0.1511

0.0361

0.03026

bm a

0.0098 0.0121 0.0241

-0.0509 -0.0644 -0.1002

-0.0558 -0.0704 -0.1088

b

-0.1094

0.0420

0.0389

am

The results of the experiments for fatigue life, Ns, when exposed to a stepwise load in reference to their structure index BV/TV for all of the samples are given in Fig. 2. This demonstrates the plotted linear approximating function together with the value of the coefficient of determination. The results constitute a basis for the calculations of the fatigue curves. Discussion This paper is based on the results of research conducted on a different samples of the bone, which were clearly confirmed by the results of the investigations made for the structure indices. In paper [5], the scatter BV/TV for 35 samples from 9 donors (relative standard deviation RSD=38.5%) was comparable with the scatter obtained in the current experiments (61 donors and RSD=37.1%). The coefficient of determination for the fatigue curve in the same paper [5] was not particularly high R2=0.54). In paper [4], the scatter for 29 samples from 4 donors was higher than in our experiment (as much as RSD=42.5%); however, introducing the stress modification to the fatigue equation with the use of the volume fraction and the fabric eigenvalue resulted in a high correlation between the stress and the fatigue life at a constant load amplitude (R2=0.95).

42

Fatigue Failure and Fracture Mechanics

0,50

BV/TV

0,40 0,30 0,20 y = 6E-06x + 0,0895 R2 = 0,689

0,10 0,00 0

10000

20000

30000

40000

50000

60000

Ns cycle

Fig. 2. The relationship between the fatigue life Ns and the volume fraction BV/TV for the samples of bone that were loaded with a stepwise increasing amplitude: the determination coefficient was 0.689 and the p-value was 1.32×10-14 The relationship between the fatigue life results and the volume fraction, and thus one of the structure indices, does exist. It was not a strong correlation (R2=0.689); however, it should be noted that this relationship covers not only the bone properties (including the structure with its individual characteristics, damage and the effects of remodeling) but also the specificity of the fatigue damage process for stepwise loading with the dynamics also associated with the bone properties. Reference [1] Manson S.S., Halford G.R.: Practical Implementation of the Double Linear Damage Rule and Damage Curve Approach for Treating Cumulative Fatigue Damage. International Journal of Fatigue 1981, 17: 169-192 [2] Hashin Z., and Rotem A.: A Cumulative Damage Theory of. Fatigue Failure, Mats. Sci and Eng. 1978, 34: 147-160 [3] Subramanyan S.: A cumulative damage rule based on the knee point of the S-N curve. J. Engng Mater. Technol. 1976, 98: 316-21 [4] Rapillard L., Charlebois M., Zysset P.K.: Compressive fatigue behavior of human vertebral trabecular bone, Journal of Biomechanics 2006, 39(11): 2133-2139 [5] Haddock S.M., Yeh O.C., Mummaneni P.V., Rosenberg W.S., Keaveny T.M.: Similarity in the fatigue behavior of trabecular bone across site and species. Journal of Biomechanics 2004, 37(2): 181-187 [6] Kosmopoulos V., Schizas C., Keller T.S.: Modeling the onset and propagation of trabecular bone microdamage during low-cycle fatigue. Journal of Biomechanics 2008, 41: 515-522 [7] Bowman S.M., Guo X.E., Cheng D.W., Keaveny T.M., Gibson L.J., Hayes W.C., McMahon T.A.: Creep contributes to the fatigue behavior of bovine trabecular bone. Journal of Biomechanical Engineering 1998, 120(5): 647-654 [8] Winwood K., Zioupos P., Currey J.D., Cotton J.R., Taylor M.: Strain patterns during tensile, compressive, and shear fatigue of human cortical bone and implications for bone biomechanics. Journal of Biomedical Materials Research 2006, 79A(2): 289-297

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.43

CONCEPT OF FATIGUE FOR DETERMINING CHARACTERISTICS OF MATERIALS WITH STRENGTHENING Ewa MARCISZ 1,a, Adam NIESŁONY 2,b, Tadeusz ŁAGODA 3,c 1,2,3

Technical University of Opole, Faculty of Mechanical Engineering, ul. Mikolajczyk 5, 45-271 Opole a

[email protected], b [email protected], c [email protected]

Keywords: cyclic fatigue, fatigue characteristics

Abstract. The paper presents the concept of division of the total strain amplitudes. Simulations were performed for high-alloy steel X6NiCr3220 for proposing a new curve of cyclic strain based on the best fit to the experimental points and plotted the hysteresis loop. Proposed division on the total strain amplitude of three parts: plastic strain amplitude, amplitude of the linear elastic strain and strain amplitude of the coupled. In order to preserve the forms of popular formula RambergOsgood and Manson-Coffin-Basquin modified them in such a way that added to their member responsible for the description of the coupled strain. Inclusion of additional term leading to closer representation of the actual material properties. Introduction In most engineering constructions, machines and technical equipment we find the problem of fatigue materials [3]. Fatigue in a small number of cycles in uniaxial loading condition can be presented with the two basic characteristics: the curve Manson-Coffin-Basquin strain amplitude describing the dependence of the number of cycles and cyclic strain curve, the relationship between stress amplitude and the amplitude of the strain during the load-amplitude was. Ramberg-Osgood equation describing the curve of cyclic strain, this equation can be divided into two parts of the elastic and plastic. There are different ways to obtain experimental data to determine the cyclic strain curve: you can use the tests performed with the use of multiple samples or one sample at many levels of stress or strain or fatigue on the basis of the uniaxial loading of a sample of a random process [4]. The most popular method, however, is the first of these, which also allows you to get the results needed to make a chart of fatigue plot strain. The paper presents results of high-alloy steel X6NiCr3220, which is taken from the literature [1]. On their basis, was designated strain amplitude dependence of the amplitude of elastic and plastic, and then plotted the hysteresis loop. Through simulations conducted seen some belonging to the plastic strain amplitude, the amplitude of linear-elastic strain and the third part, which is defined in this paper as a coupled and results from the curvature of the branches of the hysteresis loop during the unloading of the material before reaching a point σ= 0. Division the total strain amplitude of the plastic, linear-elastic and coupled The results of fatigue tests for high-alloy steel subjected to uniaxial tensile X6NiCr3220compression at elevated temperatures are presented in paper [1]. In the literature, we find the description of cyclic strain curve Ramberg-Osgood equation in the form ε ac

σa

1

 σ  n' = + a  E  K' ,

where:

εac - total strain amplitude, σa - the amplitude of stress, E - Young's modulus, K’ – cyclic strength coefficient,

(1)

44

Fatigue Failure and Fracture Mechanics

n’ - exponent of the cyclic deformation strengthening. Ramberga-Osgood equation can be separated into two parts, the first of which defines the amplitude of elastic ε as =

σa E

(2)

and the second part defines the plastic strain amplitude 1

ε ap

 σ n' = a   K'

(3)

We can also be written as ε ap = ε ac −

σa E .

(4) In order to model a hysteresis loop we use the formula (1), in which we move from the amplitudes of stress and strain ranges. Equation takes the form 1

∆σ  ∆σ  n ' ∆ε = + 2  E  2K '  .

(5) Strain amplitude dependence of the number of cycles to damage is given by the Manson-CoffinBasquin as ε ac =

σ 'f E

( 2N f )

b

+ ε ' f (2 N f )c ,

(6)

where: σ’f – coefficient of fatigue strength under tension-compression, b – fatigue strength exponent, ε’f – coefficient of the fatigue plastic strain, c – exponent of fatigue plastic strain, Nf – number of cycles to failure. Equation (6) can also be separated into two parts, the first of which shows the amplitude the elastic strain σ 'f

( 2Nf ) E

b

ε ap = ε ' f ( 2N f

)

εas =

, and the other for the plastic strain amplitude

(7)

c

. (8) In order to illustrate the amplitudes of elastic and plastic strain of cyclic strain curve was plotted and modeled on the basis of the hysteresis loop. Figure 1 illustrats by an example of hysteresis loop together with the designated parts of the elastic and plastic. Plastic deformation, as defined, are those strain that are in the material after removing the load, ie when the stress again reaches the zero value [2].

Dariusz Skibicki

45

Fig. 1.The hysteresis loop and the cyclic strain curve with the selected deformation plastic, linearelastic and coupled. The figure shows that if the dual cyclic yield stress (Re’) is less than the amplitude of stresses, i.e. 2 R e' < σ a (9) then we can identify some plastic ɛ ap, linear-elastic ɛas part conjugate and a new ɛsp. It follows that the newly defined amplitude of strain is described in the coupled model ε sp = ε ac − ε as − ε ap (10) Modeling of hysteresis loops for steel In order to illustrate the idea of sharing the total strain amplitude into three parts modeled hysteresis loop [1]. The analysis was performed for steel X6NiCr3220, for which the calculated: K '= 1455 MPa, n' = 0.242 according to (1) and σ 'f = 1295MPa, b = -0,1460, ε'f = 0.4768, c = -0.57 according to the formula (6). Literature results and their analysis allowed us to determination the cyclic strain curve Cyclic strain curve was determined based on the best fit to the experimental points. Obtained in this way, the parameter values K*=1720 MPa and n* = 0.2728 different from those designated by the classical method. Following the analysis suggested different light patterns describing the cyclic strain curve for the material shown in Figure 2 against plastic strain amplitude, amplitude of the linear elastic strain and strain amplitude of the coupled. The curve is plotted according to the formula Ramberg-Osgood (1) calculating the new material constants, Young's modulus was obtained statically on 193GPa From the figure 1 it can be observed non-linear elastic character. Proposed division of the total strain amplitude of the ɛac for some ɛas linearly elastic, plastic and part of the ɛap coupled ɛsp. In order to preserve the forms hitherto used Ramberg-Osgood equation describing the cyclic deformation of the full use of them by adding the member responsible for the description of the coupled strain. The plot takes the form of cyclic strain 1

ε ac = ε as + ε ap + ε sp =

1

σ a  σ a  n '  σ a  n '' E

+  +   K'  K ''  ,

(11)

where the parameter for the studied steels are as follows: K * '= 2846MPa, n *' = 0.2728, K *'' = 1525MPa, and n *'' = 0.2442.

46

Fatigue Failure and Fracture Mechanics

Fig. 2 The curves of cyclic deformation on the background of elastic, plastic and coupled.

Fig. 3 Cyclic strain of the total, linear-elastic, plastic and incorporated in the system (ɛa - σa) Cyclic strain the total, linear-elastic, plastic and coupled are shown in Figure 3 in the system (ɛa σa). In order to preserve the forms hitherto used equation Manson-Coffin-Basquin strain amplitude describing the total used by placing states responsible for the description of the strain coupled ε ac =

σ 'f E

(2N f )

b

(

+ ε ' f 2N f

)

c

(

+ ε '' f 2 N f

)

d

(12)

where the parameter for the tested steels are: σ 'f = 1191MPa, b = -0.1359, ɛ' f = 0.0411, c = 0.4984, ɛ'' f = 0.3369, d = -0.5486 . gdzie parametr dla badanej stali wynoszą: σ’f = 1191MPa, b = -0,1359, ε’f = 0,0411, c = -0,4984, ε’’f = 0,3369, d = -0,5486.

Dariusz Skibicki

47

Fig.4 The amplitude of strain of the total, elastic, plastic and coupled depending on the number of cycles for the damaged steel X6NiCr3220. In Tables 1 and 2 are summarized the calculated coefficients and exponents of the study of cyclic deformation strengthening of steel, used in the tests carried out. Table. 1. The calculated coefficients of cyclic deformation of steel strengthening X6NiCr3220 Strengthening of the cyclic deformation coefficient, MPa K’ K* K*’ K*’’ 1455 1720 2846 1525 Table. 2. The calculated indices of cyclic deformation of steel strengthening X6NiCr3220 The exponent of cyclic deformation strengthening n’ n* n*’ n*’’ 0.242 0.2728 0.2728 0.2442 Conclusinon In this paper we analyzed the plot of cyclic strain on the example of steel X6NiCr3220. Modeled curve of cyclic strain and stable course of the hysteresis loop by dividing the total strain amplitude in the linear elastic, the coupled and the plastic strain. Following conclusions are detailed: 1. Observed for the analyzed material nonlinear elastic character, which led to propose the division the amplitude of elastic strain on the amplitude of linear-elastic and coupled. 2. For materials modeled by equation (11), for which the coupled strain takes negligibly small values, the equation reduced to the well-known Ramberg-Osgood proposal. 3. For further analysis is necessary to perform experiments for different materials and to analyze the results on the basis of experimental and modeled hysteresis loop. 4. The proposed model is valid for materials which have a double cyclic yield strength is less than the applied stress amplitude.

48

Fatigue Failure and Fracture Mechanics

Acknowledgements This work was supported by the National Centre for Science, contract No. 2011/01/B/ST8/06850. References [1] BÄUMEL A., SEEGER T.: Material Data for Cyclic Loading, Materials Science Monographs, Vol. 42A – E, 1987, Supplement 1, 1990, Elsevier Science Publishers. [2] NIESŁONY A., EL DSOKI Ch., KAUFMANN H., KRUG P.: New mathod for ewaluation of the Manson-Coffin-Basquin and Ramberg Osgood equations with respect to compatibility, International Journal of Fatigue, Vol. 30, 2008, pp. 1967-1977. [3] ROZUMEK D., MARCINIAK Z., LACHOWICZ C. T.: The energy approach in the calculation of fatigue lives under non-proportional bending with torsion. Int. J. of Fatigue, Vol. 32, No. 8, 2010, pp. 1343-1350. [4] WALAT K., ŁAGODA T., KAROLCZUK A.: Fatigue life according to cyclic strain characteristics determined from variable amplitude loading, Materials Testing, (Materialprufung), Vol. 51, No. 5, 2009, pp. 286-290.

CHAPTER 2: Fatigue Properties of Materials

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.51

Material Properties Investigations With the Use of Microspecimen BOROŃSKI Dariusz1, a 1

University of Technology and Life Sciences, Faculty of Mechanical Engineering, Department of Machine Design, al. prof. S. Kaliskiego 7, Bydgoszcz, Poland a

[email protected]

Keywords: cyclic material properties, fatigue, microspecimen, fatigue testing system

Abstract. In the paper there are discussed possibilities of performing static and cyclic investigations of a material properties with the use of small size specimens. An original research system developed at the University of Technology and Life Sciences in Bydgoszcz has been used for static and fatigue testing. Thanks to tests performed on microspecimens it is possible, among others, determining the local properties of material in objects with the material discontinuities. This research provides exemplary results of fatigue tests carried out on microspecimens taken from a laser welded joint. Introduction Development of fatigue design methods, based on the local approach, involves an increasing demand for information on local, cyclic properties of structural materials. It refers both to nonhomogeneous and homogeneous materials which can change their basic properties in result of technology used for production of structural parts or under the influence of loads. Joints made using welding techniques constitute a special group of such elements [1]. However, it was the development of nuclear energy that initiated tests on microspecimens, including materials exposed to radiation [2]. There have been developed different methods for testing [3,4] and preparation of specimens. Also the conference entitled: ‘Small Specimen Test Techniques’ was dedicated to tests on ‘small specimen’. Testing of mechanical properties on a small specimen is also necessary in case of materials used in MEMS electromechanical microstructures [5,6,7]. The worked out technical solutions in the field of test equipment as well as specimen preparation methods are discussed more thoroughly in monograph [8]. System for fatigue properties testing of microobjects (MFS) Thorough knowledge of material properties is especially useful for a fatigue analysis of structural parts involving the local approach which provides the possibility to study the fatigue process directly in the zone of fatigue crack initiation. In such a case, the quality of a structure fatigue analysis largely depends on the familiarity with the material local properties in these areas. It refers mainly to two types of fatigue characteristics: fatigue life curves describing the dependence between fatigue life and the level of load (stress or strain) and cyclic stress-strain curves characterizing the relation between strains and stresses for cyclically variable loading. Both characteristics are most often determined on the basis of the analysis of the stress-strain hysteresis loop parameters recorded during cyclically variable loading applied to the material specimen. However, the analysis of local fatigue properties often requires application of very small size specimens which causes that the so far existing test methods and research instruments are insufficient to perform efficient tests in this field. Apart from performing tests of the material local properties, conducting tests on small size objects, is of great significance for microelements and microstructures which in many cases are produced with the use of materials whose characteristic dimension is considerably decreased [9].

52

Fatigue Failure and Fracture Mechanics

In order to perform fatigue tests with the use of microelements there has been developed an original research system MFS. The system includes: primary loading unit ‘nanodrive’, secondary loading unit ‘microdrive’, unit for strain measurement by digital image correlation method (CID), unit for measurement of strain by laser grating interferometry technique (moiré interferometry) (LFI), units for precise alignment and fastening of objects, computerized control an supply system. A scheme of the system is shown in Figure 1a. An overall view of the system is shown in Figure 1b. a)

b) 11 1 9

5

7 2

4

10 4

10

3 6

8 7

12

Fig.1. Scheme of MFS system (a): carrying base (1), primary loading unit „nanodrive” (2), (3), force measurement unit (4), nanoscale displacement measurement unit (5), microscale displacement measurement unit (6), strain measurement unit by the digital image correlation technique (7), strain measurement unit by the laser grating interferometry technique (8), alignment unit (9), fastening unit (10), thermographic analysis (11), control and supply unit (12). The overall view of MFS system (b) Thanks to the worked out solutions, including a doubled loading system basing on piezo actuator and microstepping motor with a precise ball screw, MFS system enables determination of many static and fatigue properties of the material, including static tension curves, cyclic stress-strain curves and different types of fatigue life curves: stress-life, strain-life. Optoelectronic measurement units used in this system enable measurement of strains on measurement bases with micrometric length. Microspecimen preparation

4

base metal

0.2

On the basis of carried out research, including an assessment of the influence of the specimens preparation technology on their microstructure, the method of laser cutting was chosen to prepare them. First, plates with thickness 0,2 ± 0,02 mm were cut from a 4 mm thick joint by WEDM technique (wire thickness 0,2mm.), according to the scheme presented in Figure 2. From so prepared plates, micro-specimens were cut from particular zones of the joint which were localized by micro-etching. A laser with power range from 1 to 20W and frequency 20-80 kHz, equipped with heads moving with velocity 20 000 mm/s was used to cut out microspecimens.

weld heat affected zone (HAZ)

Fig.2. Scheme of specimen preparation

microspecimen

Dariusz Skibicki

53

Two types of specimens were prepared for the tests: dumbbell shape specimen (Fig.4) and specimens with triangular griping part. Specimens were cut with the use of a small power laser in order to avoid the impact of heat on the specimen structure, which was verified by their metallographic analysis (Fig.5).

15.5 mm

Fig.4. Microspecimen shape and dimensions 50 µm

50 µm

Fig.5. Microspecimen microstructure: specimen taken from base metal zone

Exemplary results of fatigue tests Exemplary results of fatigue tests performed on microspecimens taken from a laser welded joint, made of the S355J2G3 steel (low-alloy steel with improved mechanical properties primarily designed for welded structures), have been shown in Figure 6. In Figure 6a there are exemplary hysteresis loops recorded for different phases of constant amplitude loading course, and in Figure 6b, selected courses of hysteresis loops recorded for two zones of the welded joint: base metal and the weld zone. The presented courses show both diversification of the material properties and their variability under the influence of cyclically variable loading. Further analysis of hysteresis loops recorded for different loading levels make possible determination of local cyclic material properties for particular zones of laser welded joints.

Summary The presented system MFS makes it possible to perform fatigue tests with the use of specimens with dimensions accounting for local differences in material properties. The possibility to determine the material local properties with the use of microspecimens allows to significantly extend the range of fatigue analysis methods based on the local approach.

54

Fatigue Failure and Fracture Mechanics

Application of small specimens and test methods enabling performance of tests in micro-scale, opens new grounds for the development of design methods for prevention against fatigue cracks in objects with different material properties. a)

600

1900

3200 cycle

b)

20 force, N

stress, MPa

weld

15

400

10 200

base metal

5

-0.005

-0.003

0 -0.001 -200

0 0.001

0.003

0.005

strain

-25

-15

-5

5 -5

15

25

displacement, µm

-10 -400

-15 -600

-20

Fig.6. Exemplary results of fatigue test of microspecimens: a) hysteresis loops recorded for different phases of constant amplitude loading, b) hysteresis loops recorded for two zones of the welded joint: base metal and the weld zone This research work is financially supported by the Polish state budget for science as a research project References [1] D. Boroński, Cyclic material properties distribution in laser-welded joints, International Journal of Fatigue, Vol 28/4 (2006) 346-35. [2] G.H. Lucas, Review of small specimen test techniques for irradiation testing, Journal Metallurgical and Materials Transactions A, 21, 5 (1990). [3] S. Saito, K. Kikuchi, Y. Onishi, T. Nishino, Development of piezoelectric ceramics driven fatigue. Journal of Nuclear Materials 307-311 (2002) 1609-1612. [4] W.N. Sharpe, Tensile Testing at the Micrometer Scale: Opportunities in Experimental Mechanics, Experimental Mechanics, 43, 3 (2003) 228-237. [5] S.M. Allameh, An introduction to mechanical-properties-related issues in MEMS structures. Journal of Materials Science, 38 (2003) 4115-4123. [6] Ando T., Shikida M., Sato K. Tensile-mode fatigue testing of silikon films as structural materials for MEMS. Sensors and Actuators, A 93 (2001) 70-75. [7] Cho H.S., Hemker K.J., Lian K., Goettert J., Dirras G. Measured mechanical properties of LIGA Ni structures. Sensors and Actuators, A 103 (2003) 59-63. [8] D. Boroński, Local material properties in fatigue analysis, Publishing House of ITeE-PIB, Bydgoszcz-Radom (2009) (in polish). [9] S.M. Allameh, J. Lou, F. Kavishe, T. Buchheit, W.O. Soboyejo, An investigation of fatigue in LIGA Ni MEMS thin films, Materials Science and Engineering, A 371 (2004) 256-266.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.55

THE EFFECT OF MICROSTRUCTURE ON ROLLING CONTACT FATIGUE OF BEARINGS Tadeusz Z. Wozniak1, a, Jerzy Jelenkowski2, b, Krzysztof Rozniatowski3, c, Zbigniew Ranachowski4, d 1

Kazimierz Wielki University, Chodkiewicza 30, 85-064 Bydgoszcz, Poland 2

3

Institute of Precision Mechanics, Duchnicka 3, 01-796 Warsaw, Poland

Faculty of Materials Science and Engineering, Warsaw University of Technology, Woloska 141, 02-507 Warsaw, Poland

4

Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-103 Warsaw, Poland a

c

[email protected], b [email protected], [email protected], d [email protected]

Keywords: Rolling contact fatigue; Bearing steels; Pitting; Isothermal heat treatment; Bainite; Midrib; Acoustic methods

Abstract. There has been proposed an innovative thermal treatment of bearing steel 100CrMnSi64, where the existing standard heat treatment has been replaced by austempering. The structure of low-temperature tempered martensite has been replaced by a microstructure composed of martensite and lower bainite with midrib. The kinetics of bainitic transformation and isothermal martensitic transition at selected austempering temperatures was controlled by acoustic emission. The research on contact strength was made under the conditions of rolling-sliding friction. The microstructure was revealed with the use of a light microscope and the forms of pitting wear were displayed by a scanning electron microscope. It was found that the optimum microstructure providing the best used contact strength of the tested steel is conditioned by the formation of a lower bainite with midrib at the temperatures near MS. A plausible cause of the increased resistance to pitting is bifurcation of fatigue cracks on dispersion bainitic carbides in combination with primary carbides, in bainitic-martensitic matrix. Introduction The main causes of destruction, friction nodes in machinery parts in working conditions of variable contact loads are seizing and pitting [1-3]. That failure takes place mostly when cooperating surfaces touch each other in rotary motion (bearings, gears, etc.). Pitting is a catastrophic destruction and appears suddenly, after many millions cycles of loading. Starting from first spalling, a further process of the destruction of top layers of material is very fast. Pitting depends on the top layer of material structure, endurance material properties, surface roughness and the values of the friction coefficient. Hardening parameters determine micro-structural habitus, a dispersion of carbides, carbon diffusion and internal stresses. In recent years, the issues on improving the mechanical properties of materials by changing their microstructure have been developed. On the one hand, it is related to an increase in knowledge of the relationship: structureproperties and to the development of new technologies that provide a better control of the material microstructure. There exists both an optimal volume fraction and their optimal dispersity for a definite type of matrix and a definite type of carbides [4, 5]. A material with a uniform carbides distribution or in the form of clusters and thin layers shows higher crack resistance than the microstructures with thick bands or random distribution [5]. This research aimed to verify the possibilities of replacing the standard heat treatment of steel 100CrMnSi6-4 at the isothermal heat treatment at the elevated temperature of austenitising, controlled by the method of sound emission [6, 7, 8]. Both the Hertz stress values and the number of cycles up to the first registered damage of sample were adopted as a criterion for the correctness of the modified heat treatment.

56

Fatigue Failure and Fracture Mechanics

Material and Methodology of the Research Steel 100CrMnSi6-4 (ISO EN 683-17:1999) was used in the research. The content of sulphur and phosphorus was accordingly: 0.014% P and max. 0,007% S. For steel used for the research the degree of carbide banding was 7.3 acc. to Stahl-Eisen-Prüfblatt SEP 1520. The tested steel was delivered as rolled products of 46 mm in diameter in a softened state. The analysis of chemical composition was carried out using an instrument made by Spectrolab. The results of the analysis for chemical composition are shown in Table 1. Table l. Chemical composition of the 100CrMnSi6-4 steel in [wt. %].

C

Cr

Mn

Si

Cu

Ni

Al

Mo

P

S

0.95

1.47

1.10

0.57

0.21

0.07

0.02

0.01

0.014

0.007

In order to determine the temperature MS in the tested steel, dilatometric tests were made for six specimens, 12mm long and 2 mm in diameter. Austenitising of specimens was carried out in an oven of dilatometer at temperature 950°C in a time period of 30 minutes, with registering accuracy of ± 5K. The tests of fatigue contact strength were made with the use of series of specimens austempered and then grinded. Specimens used in the tests were austenitised at 950°C during 0.5 h and quenched in hot oil, maintaining constant temperature (30 - 180°C). The tests were carried out using appliances type ULP-2. There were used specimens of size Φ 8 x 40mm cooperating with two clamping rollers: driven and driving of 150mm in diameter, made of steel ŁH15, of hardness 60HRC. The measure of pitting resistance was the number of cycles until the moment when first spallings occur. The construction of the appliance used and the specimen size enable us to carry out 5 measurements. The tests were made with Hertzian contact pressure 3440 MPa, frequency f = 300 Hz to max. testing limit NG = 2 x 107 cycles, acc. to PN-83/H-04324. Specimens for microscopic research were prepared in a standard way: they were etched with Nital and with Vilell's reagent [9]. Microscopic examination was carried out using a metallographic inverted light microscope type EPIPHOT 200 made by NIKON equipped with a digital colour CCD camera, resolution 2 Mp. Surface geometry of rollers after the tests was imaged thanks to laser interference profilometry, by using a laser appliance WYKO 9300 made by Veeco, a scanning microscope TM-1000 Hitachi in an observation mode BSE [10]. The test method EA [10, 11] made it possible to determine precisely the optimum temperature for austempering (Fig. 1). Testing EA effects were carried out by using modern instrumentation that was able to make austempering in the range of temperature of martensite and bainite formation 30-190ºC. For experimental tests, specimens of 45 mm in diameter and 2 mm thick were used [7, 8, 9]. AE signals sent while austempering were recorded using a special instrumentation to which an ultrasonic transducer WD (20 kHz-900 kHz) was connected. The signals were registered by AE Signal Analyser 10/20 kHz-800 kHz and recorded in PC computer memory by a card ADLINK 9812 with the frequency of 1200 kHz. In order to make spectrogram graph, the Short Time Fourier Transform (STFT) algorithm with the Hamming window was used.

Dariusz Skibicki

A)

57

B)

Figure 1. A) The stand for testing austempering using the method of acoustic emission (AE), B) Exemplary spectrograms of AE signals for temperatures a) 30 ºC, 160 ºC, 190 ºC.

Research Results Examples of spectrogram graphs for AE signals received during the process of austempering are given in Figure 1 B) in the system coordinates: time - amplitude. In the initial stage of transformation, about 4 minutes, the austempering temperature increases and the individual AE events are gradually expanded. Midribs, formed at the start of the process, do not show any large dilatometric effects, but they are very significant as the structure precursors of bainitic transformation [12, 13]. Thin platelets martensite nucleation in the form of midribs during the quench time makes the first stage of the process. The second stage is related to the enrichment of adjacent austenite in carbon, which leads to its further transition into lower bainite. a)

b)

Figure 2. The microstructure of steel after austempering at different temperature: a) bainite plate of lower bainite with midrib at 160°C for 3 min, white areas of residual austenite with athermal martensite revealed with Vilell's reagent, b) martensite at 100°C during 1450 s, etching with Nital. At temperature 30°C, an intensive range of signal emissions lasts up to approx. 60s. The lower bainite with midrib is formed the fastest at temperature 130°C, which is due to the formation of large quantities of midribs in a shorter time. The longest process of the midribs formation occurs at

58

Fatigue Failure and Fracture Mechanics

temperature 160°C (Figure 2a), which is connected with the diffusion of bainite carbon and the enrichment of austenite in carbon. When the temperature is raised to 190°C, i.e. above MS, the rates of midribs formation and bainitic transition are significantly reduced. Midribs are not observed at temperature below MS of a massive character and above MS (Fig. 2b). Temperature MS determined in dilatometric tests as the deviation from linearity is MS =157°C, while the substantial martensitic transformation of massive character starts at 110°C. a)

b)

Figure 3. The impact of the various options of austempering on contact strength: a) the number of load cycles, 1) quenching below MS, 2) 160°C during 3 min., 3) 160°C during 60 min,, (4) 180°C during 3 min,, (5) 180°C during 10 min., (b) the share in % of population in the number of cycles above the experiment limit: 20x 106 cycles, according to the variant of heat treatment. The research results of fatigue contact strength as a function of temperature and austempering time is shown in Figure 3 a). The research results of austempered specimens at 160ºC and 180ºC significantly exceed the results for steels ŁH15 obtained after customary quenching with tempering at 200ºC [10, 14]. This negative impact of customary quenching is probably caused by blocking of displacement movement in the structure with the martensitic phase prevalence, which delays the relaxation of stresses and thus facilitates the nucleation of micro-cracks. If no pitting was observed after completing the tests, i.e. after 20 million cycles, a limit value was taken to calculate the average value of the test results. It made a certain flattening of average results for different variants of the treatment, which can be seen in Figure 3 a). For this reason, Figure 3 b) additionally shows a share of population in % concerning the number of cycles above the experiment limit 20x 106 cycles for each variant of heat treatment. The largest population reaching as much as 83% was obtained at austempering temperature 160°C during 3 minutes. (variant no. 2 of heat treatment) i.e. on the limit near temperature MS. Top Layer Tests The fatigue contact strength is conditioned by the presence of elastic and plastic strains as well as by the factors accompanying rolling-sliding friction in the system of cooperating elements. Intense lubrication causes spalling (delamination) of about 0.1µm thick shells and approx. 3µm in diameter. Larger damages arise in heavier loads and faulty lubrication, when thin metallic layer flakes and hard carbides, nitrites, etc. are revealed (Fig. 4 a). An intensive and variable load may generate pitting, (Fig. 4 b)), with characteristic elongated strips of larger width and depth. The stria layouts suggest the depth of craters, traces of carbides presence and the speeds with which the generated cracks spread towards the surface. Pitting is connected with the appearance of cracks on or under the surface (Fig. 5).

Dariusz Skibicki

a)

59

b)

Figure 4. Examples of rolling wear owing to joining fatigue cracks near the surface: a) spalling from the strengthened top layer with participation of shear stresses, carbides contact directly with the damaged layer, b) the area formed around the carbides induces pitting of particles of a few micrometers in diameter [10]. First cracks of material appear mostly on a certain depth below the surface near the so-called Bielajew point in the largest material effort area [1, 14-16]. The increase in shear stresses causes displacement of the Bielajew point towards the surface. The highest durability occurs for pitting initiating on the surface (Fig. 4 a, Fig. 5). For pitting initiation on a certain depth below the surface, these areas are ovally shaped and the service life of bearing elements is lower (Fig. 4 b). The differentiation in pitting morphology was also found in the research on other bearing steel 100Cr6 [17], similarly like in the tested here steel 100CrMnSi6-4 after austempering.

Figure 5. The surface geometry of the roller after the tests, treatment variant no. 2, Fig. 3. The image achieved by laser interference profilometry. The results of tests of roller surface geometry for five measurements of wear, marking: Ra - the average arithmetic deviation from the roughness profile, Rq - the average square deviation from the roughness profile, Rz - maximum height of the roughness profile elevation, Rt - maximum depth recesses of the roughness profile [10]. While moving the rolling body, the strains migrate wavely and micro-glides occur as a result of elastic and often plastic strains [14, 18]. Additional strains during the rolling body movement also result from transiting austenite into martensite under the load. The development of pitting cracks is also dependent on the material strengthening. The plastic strain and the growth in dislocation density make carbon diffusion in these areas easier, which in turn affects the formation of areas that are not susceptible to etching, the so-called white etching areas (WEA). In the maximum shear strain area, the layers with a modified structure are formed, the thickness of which increases from about 0.1 mm to 0.7mm with the increase in the value of contact stress and load cycles number.

60

Fatigue Failure and Fracture Mechanics

Impact of Carbides on Contact Fatigue Apart from temperature and time, a form of cementite in annealed steel is an important factor, affecting the kinetics of austenitising. The steel structure fulfilling the optimum of various requirements in respect of technological and functional properties of bearing steel is an equally dispersed spheroidal cementite (Fig. 6 a). In carbides (Fe, Cr, Mn) 3 C, carbon and a great proportion of chromium and manganese are bounded in almost all their amounts [1]. The shape and the dispersion degree of cementite activate carbides dissolution processes, thus affecting the related changes in carbon contents and in alloy elements in matrix. After hardening from 850°C (after austenitising time of 30 min.), approx. 7% of carbides remained and the martensitic matrix contained approx. 1% Cr and 0.6% C [14]. In the areas with segregation and carbide banding, chromium or carbon concentrations may differ in certain areas from their average values. The presence of chromium results in reducing carbon solubility in austenite, which consequently affects the increase in the amount of carbides in steel. a)

b)

Figure 6. Microstructure of steel 100CrMnSi6-4 a) in the initial state after spheroidising annealing, the structure of spheroidal cementite dispersed in a ferrite matrix, SEM image. On the etched surface of scarp, the carbides of different dispersion are revealed. b) Microstructure of steel 100CrMnSi6-4 austempered at 130°C, the inhomogeneity caused by carbide banding along rolling direction, banding degree 7.3 acc. to Stahl-Eisen-Prüfblatt SEP 1520. Etching with Nital reagent [10]. In the tested steel 100CrMnSi6-4 after hardening, inhomogeneous dispersion of primary carbides was observed. An additional factor favourable to inhomogeneous distribution of carbides appearing as clusters is structural heterogeneity caused by diverse fraction of bainite and martensite in microareas. Metallographic analyses of steel 100CrMnSi6-4 revealed the banding of microstructure (Fig. 6b). Banding is defined as light bands of almost pure carbide-poor martensitic structures. Light martensitic bands occur next to dark bands of martensitic-bainitic matrix (lower bainite with a midrib) with the increased concentration of carbides. In bands of bearing steel with the elevated content of carbides, the solid solution after austenitising is richer in carbon, chromium and manganese than in the adjacent bands. This is caused by considerable differentiation in matrix hardenability and microhardness of bands, which affects the resistance to fretting, fatigue and contact wear. The bonded structure also shows a favourable influence on a use life of tools for plastic hot treatment [19]. Complex microstructures containing clusters and laminated arrangement of carbides may improve crack resistance [20, 21]. A positive effect of the impact of complex microstructure is a crack direction path deflection from the initial direction, which increases both the length of the crack path and the total area of the crack. This zone develops further through the coalescence of microcracks in carbides.

Dariusz Skibicki

61

Summary 1. It was manifested that one of the factors to reduce the adverse impact of primary carbide banding on contact strength of bearing steel 100CrMnSi6-4 may become properly selected heat treatment. Not eliminating the beneficial effect of martensitic transformation, we suggest the heat treatment with isothermal hardening between MS and Mf with the use of bainitic transformation, which resulted in eliminating the necessity of tempering. 2. A thermal treatment controlled by the method of acoustic emission, which made it possible to control the kinetics of forming midribs, was suggested. This, in turn, allowed us to achieve the microstructure with a complex dispersion of hard particles in the form of clusters in bainiticmartensitic matrix. In many cases, the level of fatigue contact strength was achieved for above 20x 106 cycles with no macroscopic traces of surface damage in the form of cracks or spallings. 3. Local damages in the tested paths can be divided into two main groups: surface destructions of fraction micrometer size, and pittings i.e. failures in the form of spallings of the mm size. With the initiation of pitting at a certain depth below the surface, the life of bearing elements is lower. The highest life occurs with the initiation of damages on the surface. For austempered specimens at 160180°C, cracks did not propagate deep into material. References [1]

H. K. D. H. Bhadeshia, Steels for Barings, Progress in Materials Science. 57 (2012) 268-435.

[2] S. Pytko, M. Szczerek, Pitting – the form of damaging rolling elements, Tribology. 4/5 (1993) 317-334 (in Polish). [3] Z. Gawroński, Technological toplayer in gears and cam mechanisms, Monographies. Lodz University of Technology, 2005 (in Polish). [4] J. Pacyna, Metallurgy of tool steel cracking, Scientific Bulletins, Metallurgy and Foundry Engineering. No. 120, AGH Krakow, 1988 (in Polish). [5] H. Berns, A. Melander, D. Weichert, N. Asnafi, C. Broeckmann, A. GroB-Weege. A new material for cold forging tools, Computational Materials Science. 11 (1998) 166-180. [6] T.Z. Wozniak, K. Rozniatowski, J. Jelenkowski, New Technologies in Heat Treatment of Steel Rolling Bearings 100CrMnSi6-4, Toolmaker (in Polish). 3 (2009) 17-20. [7] T.Z. Wozniak, K. Rozniatowski, Z. Ranachowski, Acoustic Emission in Bearing Steel during Isothermal Formation of Midrib, Metals and Materials International. 17(3) (2011) 365-373. [8] T.Z. Wozniak, K. Rozniatowski, Z. Ranachowski, Application of Acoustic Emission to Monitor Bainitic and Martensitic Transformation, Metallic Materials -Kovove Materialy. 49(5) (2011) 319-331. [9] C.K. Shui, W.T. Reynolds Jr., G.J. Shiflet, H.I. Aaronson, Etchants for Quantitative Metallography of Bainite and Martensite Microstructures in Fe–C–Mo Alloys, Metallography. 21 (1) (1988) 91-102. [10] Report on implementing the Research and Development Project No. R15 010 02 financed by Polish Ministry of Science and Higher Education, System for determinating an optimal austempering temperature by applying the method of acoustic emission. Head of the Project: prof. dr hab. inż. Rożniatowski K. Faculty of Materials Science and Engineering, Warsaw University of Technology, made by: J. Jelenkowski, T. Z. Wozniak (2007-2010), in Polish. [11] T.Z. Wozniak, J. Jelenkowski, K. Rozniatowski, Z. Ranachowski, Station for austempering of steel with the use of acoustic emission, Engineering Quaterly - Metal Treatment, in Polish. 1 (2010) 2-8.

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[12] T.Z. Wozniak, Z. Ranachowski, Acoustic Emission During Austenite Decomposition into Lower Bainite with Midrib, Archives of Acoustics. 31(3) (2006) 1-15. [13] T.Z. Wozniak, Acoustic Phenomena Near MS in Hypereutectoid Steels, Materials Characterization. 59/6 (2008) 708-716. [14] W. Luty, Bearing Steels, WNT, Warszawa, 1969 (in Polish). [15] S. Pytko, Bases of Tribology and Lubricating Engineering, AGH University Course Books (in Polish), No. 1164, 1989. [16] T. Smolnicki, Physical aspects of the coherence of large-size rolling bearings and deformable supporting structures, Monographies 28. Publishing House of the Wroclaw University of Technology, Wroclaw, 2002 (in Polish). [17] F. Akbasoglu, D. Edmonds, Rolling contact fatigue and fatigue crack propagation in 1C-1.5Cr bearing steel in the bainitic condition. Metall. Mater. Trans. A, 21A (1990) 889-893. [18] S. Pytko, Structures in machine construction and their effect on the devalopment of machinery construction in Poland, Polish and Worldwide Achievements of Science, Gliwice, 2010, pp.123–191 (in Polish). [19] M. Hawryluk, M. Zwierzchowski, Structural analysis of dies used for hot forging in the aspect of their reliability, Maintenance and Reliability. 2B (2009) 31-41 (in Polish). [20] L. Mishnaevsky Jr, U. Weber, S. Schmauder, Numerical analysis of the effect of microstructures of particle-reinforced metallic materials on the crack growth and fracture resistance, International Journal of Fracture. 125 (2004) 33-50. [21] K. Rozniatowski, Methods of the arrangement inhomogenity characterization of the structural elements in multiphase materials, Scientific Papers of the Warsaw University of Technology. Materials Engineering. 22, 2008 (in Polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.63

Determination of the fatigue properties of aluminum alloy using mini specimen Tomasz Tomaszewski1, a, Janusz Sempruch1, b 1

Uniwersytet Technologiczno-Przyrodniczy, al. Prof. S. Kaliskiego 7, 85-791 Bydgoszcz, Poland a

[email protected], [email protected]

Keywords: high-cycle fatigue, mini specimen, aluminum alloy, cyclic hardening

Abstract. There are situations where taking normative specimens is impossible due to the dimensions of the objects investigated (e.g. extruded sections) and one of the solutions is to use mini specimens. As for non-standard specimen testing, it is essential to define the effect of size on fatigue strength. The research methodology facilitates the determination of fatigue characteristics (S-N) for EN AW-6063 aluminum alloy. The material is used to manufacture the extruded section in the process of extrusion of the material through the extruding die. The methodology assumes the geometry of the mini specimen and the normative specimen. As for the material strength identification, a static tensile test for the specimens made directly from finished elements and preliminarily strained in cycles was carried out. As a result of the cyclic material reinforcement, an increase in yield strength Re was observed, which, in turn, rejects Re as the upper criterion of the high-cycle fatigue range. The essential fatigue tests were performed based on unilateral cyclic tension (R = 0.1). The effect of size on fatigue strength was defined. Theoretically aluminum alloy non-sensitive to changes in the size of the cross-section showed a different strength in mini and normative specimens. Introduction The size effect is known in the science of fatigue material failure and design elements. It is usually perceived in the way that we determine the testing area in which that effect is unobserved. The results are objective. In the present paper, the starting point is the statement that there exist situations where the normative specimen (located by its dimensions in the area of a lack of effect/ a limited size effect) is uncomfortable or unfeasible. This is true for the case in which the dimensions of the objects investigated make it possible to manufacture mini specimens only rather than specimens with the geometry recommended by norms [1].

mini specimen

Fig. 1. Diagram of specimen sampling from some extruded sections

64

Fatigue Failure and Fracture Mechanics

The tests reported were performed for the aluminum alloy from which the extruded sections are made (Fig. 1). The dimensions of the objects facilitate making the specimens smaller (mini specimens) than normative ones. They are produced in the process of extrusion of the material through the opening of the extruding die, defining the final shape of the cross-section of the product. Due to the high plastic strain accompanying the process, the initial material shows other strength and cyclic properties than the initial material. It is justifiable to perform tests which involve the specimens taken from finished elements. As was demonstrated in an earlier publication [2], tests with mini specimens are a solution reported in a number of papers which are experimental in nature. These have been applied for example in the nuclear industry. The effect of irradiation on fatigue strength of the steel from which the shields for nuclear reactors are made was investigated [3]. The experiments with the use of mini specimens were also performed in the range of gigacycle strength for high-strength steel; such experiments were performed with the use of non-standard test stands with ultrasonic application where the frequencies of the change load were at the level of 20 kHz [4]. A more detailed description of the research methodology and result is covered in a number of other papers [5, 6]. The review presented in [2] points to a lack of firm research methodology concerning mini specimens. As for the performance of tests with mini specimens, it is essential to define the effect of the size of the cross-section on fatigue strength. That effect is accounted for by the probability of failure of the weakest link of the material structure in the cross-section analyzed. The greater the material volume, the greater the probability of material defects triggered by focal points for fatigue cracking. It is assumed that the size effect is characterized by coefficient [7]: Κd =

Ζd Ζ

(1)

where: - Zd – fatigue strength of specimen of any cross-section, - Z – fatigue strength of specimen of the same material, cross-section area 20 ÷ 80 mm2. The following working hypotheses have been formulated: a) it is possible to develop a methodology for a specific group of structural materials to address fatigue tests performed with the use of a mini specimen. The methodology refers to: - defining the specimen development method (the geometry), - defining the range of feasible loads, - identifying groups of materials showing a varied sensitivity to the size effect; b) performing comparative tests (between the normative specimen and the mini specimen) will make it possible to determine the relationships, defined for a given group of materials, between the values characteristic for fatigue properties for both specimen development methods compared; c) there exists a group of structural materials for which the above relationships have a simplified form (a low level of sensitivity to the size effect) and the performance of tests with mini specimens made from those materials is especially justifiable and cost-effective. This paper focuses on the verification of the research methodology by determining the fatigue characteristics in the high-cycle range. Mini specimens made from EN AW-6063 aluminum alloy were used. Additionally, the effect of cross-section size on fatigue strength was defined (between the mini specimen and the normative specimen). Specimens The mini specimen geometry assumed as part of the methodology showed the dimensions defined below as recommended to be applied by norms [1]. The mini specimens were machined in packages in the process of milling, and the post-treatment surface was neither ground nor polished.

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65

The specimens were made from EN AW-6063 aluminum alloy. The material to be tested was provided in the form of flat bars of the specimen thickness. The mini specimen geometry assumed for testing was compared with the specimen for static tests and the normative specimen for fatigue test in Fig. 2. Both specimens for fatigue testing show the same value of theoretical stress concentration factor αk.

a)

7 ± 0,05

14

R25

R25

130

4

38

b)

c) 50

4

3,5 ± 0,05

7 ± 0,05

7

14

R25

R50

100

1

Fig. 2. Geometry of: a) monotonic test specimen [8]; b) fatigue test specimen [1]; c) fatigue test mini specimen Identification of material properties To identify the mechanical properties of material, a static tensile test compliant with the PN-EN ISO 6892-1:2010 norm was performed. The test was made for 5 specimens of the geometry compliant with Fig. 2a. The tension diagram is provided in Fig. 3 (the solid line), while the mechanical properties determined with them are given in Table 1. As for the metals showing a lack of stability of the cyclic properties, it is assumed that the yield strength (Re) is not a credible criterion of the upper range of high-cycle fatigue strength. As for the aluminum alloys, changes in the mechanical properties are connected with the cyclic hardening of the material [9]. Table 1. Mechanical properties of Table 2. Cyclic load levels of specimens EN AW-6063 aluminum alloy Rm, MPa Re, MPa Ru, MPa E, MPa A, % Z, % 200 167 350 61458 16,7 61,8

P1 P2

σa, MPa σm, MPa 75,5 94,5 84,5 105,5

σmax, MPa 170 190

σmin, MPa 18,9 20,9

A percentage change in value Re as a result of preliminarily completed fatigue loads was determined. Specimens preliminarily loaded (the geometry compliant with Fig. 2a) with fatigue cycle tested for static tension were investigated. The experiments were made for two load levels (P1, P2 according to Table 2). A sinusoidal cycle, unilateral cyclic tension with cycle asymmetry coefficient R = 0.1 was applied. Stress constituted the control parameter. The specimens were

Fatigue Failure and Fracture Mechanics

exposed to fatigue loads with number of cycles Nf = 1000 at the frequency of f = 5 Hz. Then a static tensile test was made. Both the monotonic and the fatigue tests were performed at room temperature. An effect of cyclic strain on the mechanical properties of material was observed. Fig. 3 presents stress–strain curves for the specimen without the cyclic prestrain and for the specimen cyclic prestrain at the maximum stress levels of 170 MPa and 190 MPa of fatigue cycle.

250 200 Stress, σ [MPa]

66

150 100 for σmax 190 MPa for σmax 170 MPa As-receive

50 0 0

2

4

6

8

10

12

14

Strain, ɛ [%]

Fig. 3. Stress–strain curves for as-received and prestrained aluminum alloy at two levels 200 σmax 190 MPa

0,4

160

σmax 170 MPa 0,3

Stress, σ [MPa]

Strain increment, ∆ɛ [%]

0,5

0,2 0,1

3 cycle

10 cycle

30 cycle

120 80 40

0 -0,1 1.E+00

0 1.E+01

1.E+02

1.E+03

1.E+04

0

1

2

3

Strain, ɛ [%]

Number of cycles, Nf

Fig. 4. Strain increment with the number of cycle at different stress amplitudes

Fig. 5. Fragments of the hysteresis loop for maximum stress of 190 MPa

Table 3. Mechanical properties of cyclic prestrained material σmax, MPa 170 190

Rm, MPa 197 198

Re, MPa 174 192

Ru, MPa 342 352

E, MPa 63022 57294

A, % 16,5 17,2

Z, % 62,1 64

ԑ, % 0,96 4,1

The material exhibits a change in mechanical properties as a result of cyclic loads. At the initial state, it showed the yield strength at the level of 167 MPa (plastic strain of 0.2 %). As a result of the initial cyclic strain, an increase in the yield strength by about 15 % was observed (Table 3). There occurred a short period of stability (30 – 40 cycles) of cyclic properties. At the first stage of cyclic load, the greatest strain increment occurred which decreased until saturation was achieved (Fig. 4). Fig. 5 presents changes in the shape of the hysteresis loop for single cycles recorded in various fatigue life periods. The loops were plotted at a single level of the maximum stress (190 MPa).

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67

The studies of cyclic prestrain specimens showed a possibility of rejecting the ‘static’ yield strength as the upper one admissible for fatigue test (σa, σm), which widened the scope of high-cycle studies above the yield strength (167 MPa) up to the value of 190 MPa of maximum stresses. Fatigue test Complete fatigue characteristics (S-N) were provided with the use of the mini specimen (Fig. 2c). To define the size effect (at the level of maximum stress of 180 MPa), additionally 3 normative specimens were investigated (Fig. 2b). The experiments were made for the high-cycle fatigue range (104 ÷ 2x106). Drawing on the cyclic prestrain specimen test results, the range of the stress levels applied was defined. Due to the specimens buckling, loads with the tensile mean component of cycle asymmetry coefficient R = 0.1 (sinusoidal cycle) were used. The frequency of the specimen load change was 5 Hz. Both the monotonic and the fatigue tests were made using the Instron 8874 servo-hydraulic material testing machine, and applying the extensometer of the measurement of 25 mm (monotonic tests). Fatigue test results Based on the fatigue tests with a controlled load, it was possible to plot a fatigue curve (S-N) in the high-cycle fatigue range. Fig. 6 breaks down the results recorded for the mini specimen (complete characteristics) and for normative specimens (3 specimens). Only those results where the specimen was destroyed in the smallest cross-section of the test section have been considered. In yet another case, a crack was developing in the area of the material defect decreasing the actual material fatigue life by making the result non-objective. 220

97,9 logσa = -0,078 logN + 2,37 R2 = 0,93

Stress amplitude, σa [MPa]

93,5

210

91,7

200

89 86,7

190

84,5 81,7

180

80 76,7

170

75,5

Maximum stress, σmax [MPa]

96,7

Mini specimen 71,1

71,7

160

Normative specimen

66,7 1.E+04 66,7

1.E+05

1.E+06

150 1.E+07

Number of cycles, Nf

Fig. 6. S-N curve for EN AW-6063 aluminum alloy For the results a regression line with slope coefficient m = 12.8 and the value of the coefficient of determination of 0.93 was plotted. As for the specimens with a smaller cross-section, higher values of fatigue life were reported than the normative specimens. The relationship can be defined with the coefficient of the cross-section size which, for the results reported, was Kd = 1.17.

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Fatigue Failure and Fracture Mechanics

Summary The research suggests the following conclusions: - it was possible to develop fatigue testing methodology for the high-cycle range for the aluminum alloy applied; that development concerned the specimen geometry, the method and the range of the specimen loads; - the results in the form of an S-N curve for the mini specimen, as compared with the literature data [10], shows a satisfactory compliance of the value of coefficient m of the regression line; for the range of the verification, all that points to the credibility of the result received for the evaluation of the EN AW-6063 aluminum alloy; - the preliminary studies of the size effect showed the values of coefficient Kd = 1.17 the value of which, contrary to the literature data, e.g. [11], must be considered essential for strength analysis. References [1] PN-74/H-04327. Badanie metali na zmęczenie. Próba osiowego rozciągania – ściskania przy stałym cyklu obciążeń zewnętrznych. [2] J. Sempruch, T. Tomaszewski, Application of mini specimens to high-cycle fatigue tests, Journal of Polish Cimac (2011) 279-287. [3] T. Hirose, H. Sakasegawa, A. Kohyama, Y. Katoh, H. Tanigawa, Effect of specimen size on fatigue properties of reduced activation ferritic/martensitic steels, Journal of Nuclear Materials (2000) 283-287. [4] Y. Furuya, Notable size effects on very high cycle fatigue properties of high-strength steel, Materials Science and Engineering A (2011) 5234-5240. [5] M.D. Callaghan, S.R. Humphries, M. Law, M. Ho, K. Yan, W.Y. Yeung, Specimen-size dependency and modelling of energy evolution during high-temperature low-cycle fatigue of pressure vessel steel, Scripta Materialia (2011) 308-311. [6] D. Boroński, Lokalne własności materiałowe w analizie zmęczeniowej, 2009. [7] S. Kocańda, J. Szala, Podstawy obliczeń zmęczeniowych, 1997. [8] PN-EN ISO 6892-1:2010. Metale - Próba rozciągania - Część 1: Metoda badania w temperaturze pokojowej. [9] J. Szala, Hipotezy sumowania uszkodzeń zmęczeniowych, 1998. [10] A.A. Luo, R.C. Kubic, J.M. Tartaglia, Microstructure and fatigue properties of hydroformed aluminum alloys 6063 and 5754, Metallurgical and Materials Transactions A (2003) 2549-2557. [11] A. Neimitz, I. Dzioba, M. Graba, J. Okrajni, Ocena wytrzymałości, trwałości i bezpieczeństwa pracy elementów konstrukcyjnych zawierających defekty, 2008.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.69

Description of cyclic properties of steel in variability conditions of mean values and amplitudes of loading cycles Grzegorz Szala1, a, Bogdan Ligaj1, b 1

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Department of Machine Design, ul. Prof. S. Kaliskiego 7, 85-225 Bydgoszcz a

email: [email protected], b email: [email protected],

Keywords: strength fatigue, two-parametric fatigue characteristics.

Abstract. The work includes description of determination of two-parametric fatigue characteristics (TFC) on the base of experimental test results and description of mathematical models of these characteristics. Mathematical models were verified in C45 and 41Cr4 tests that are essentially different from the point of view of mechanical properties. On the base of analysis of verification test results there were determined ranges of applications of TFC models described in the work. Nomenclature C(0) C(-1) N N0 R Re Rm Sf Sf (0) Sf (-1) Sf( T( 0) )

Sf( T( −) 1) m(0) m(-1) ψ ψN

– constant in the formula describing Wöhler fatigue diagram for off-zero pulsating load (R = 0), – constant in the formula describing Wöhler fatigue diagram for oscillating load (R = -1), – cycle number – general notation (fatigue life), – base number of cycles corresponding to fatigue life (N0 = 106), – cycle asymmetry ratio (R = Smin/Smax), – material yield point [MPa], – material tensile strength [MPa], – fatigue limit - general notation [MPa], – fatigue limit under pulsating load (R = 0) for N0 cycle number [MPa], – fatigue limit under oscillating load (R = -1) for N0 cycle number [MPa], – fatigue life for sinusoidal off-zero pulsating loading (R=0) for the number of cycles N (T = logN) [MPa], – fatigue life for sinusoidal oscillating loading (R=-1) for the number of cycles N (T = logN) [MPa], – exponent in formula describing Wöhler fatigue diagram for pulsating load (R = 0), – exponent in formula describing Wöhler fatigue diagram for oscillating load (R = -1), – factor of material sensitivity to cycle asymmetry, for N = N0, – factor of material sensitivity to cycle asymmetry, for N ≠ N0,

Abbreviations: TFC LCF HCF

– Two-parametric fatigue characteristics, – Low-cycle fatigue, – Hight-cycle fatigue.

1. Introduction In general service loadings of structural elements are random loadings with wide spectrum [1, 2, 3]. In fatigue life calculation methods and in programmed fatigue life tests random loadings are replaced as a result of application of accumulation methods of cycles [4, 5] with a set of sinusoidal cycles that is called loading spectrum [6, 7]. In case of random loadings with wide spectrum it is the set of cycles with parameters changeable in wide limits: amplitude and mean value of loading cycles.

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Fatigue Failure and Fracture Mechanics

Variability of amplitudes and mean values in loading requires to apply, in fatigue life calculations, an appropriate fatigue characteristic for description of material cyclic properties. Two-parametric fatigue characteristics (TFC) N(Sa, Sm) [8] or N(Smin, Smax) [9] correspond to the mentioned conditions. Small set of experimental two-parametric fatigue life characteristic N(Smin, Smax) of steels, aluminum alloys and titanium can be found in the work [9]. Mathematical model of characteristics N(Smin, Smax) was published by Bolotin [10] whereas models in the approach N(Sa, Sm) were given by Heywood in his book [11]. New mathematical concepts of TFC models were published in works [12, 13, 14]. The lack of wide experimental verification of these concepts causes that, for special cases connected with the type of materials and loading conditions, there were obtained diversified conformity of test results with calculated ones. In the presented work there was attempted to describe mathematically TFC that fulfilling the condition of description universality and acceptable conformity of test results with calculated ones. From the analysis of data included in reference materials among others in works [8] and [9] and publications of own tests [13], [14] and [15] it results that depending on the type of material, especially its cyclic properties, curves of experimental characteristics TFC have different forms. Schematic illustration of typical forms of TFC was shown in Fig. 1. a)

b)

c)

Fig. 1 Schematic approach of two-parametric fatigue characteristics: a – salient lines of constant fatigue lives (A type), b – reentrant lines of constant fatigue lives (B type), c – mixed case of constant fatigue lives ((C type) reentrant for high values N = const., salient fot small values N = const.). Lines of constant values of fatigue life N = const. are set in a coordinate system Sa, Sm in the area limited by lines: Sm = Rm, Sa = Rm i -Sm = -Rm, fulfilling the condition Sa + Sm ≤ Rm. The mentioned area was divided into 4 fields (numbers in brackets) in dependence on the range of the cycle asymmetry coefficient R = Smin/Smax. The first field (no.1) covers cycles with the coefficient variability 0 ≤ R < 1, second field (no.2) – -1 ≤ R < 0, third field (3) – -∞ < R < -1, while the fourth field (no.4) covers cycles 1 < R < +∞. Mentioned ranges of the R coefficient variability close all

Dariusz Skibicki

71

possible cases of sinusoidal cycle parameters Sm and Sa separated from random loading. From the analysis of reference data [9] it results that most of experimental TFC is determined in fields (no.1) and (no.2). 2. Formulation of the problem In the monograph [19] there was presented description of TFC determination, TFC mathematical models and experimental verification of the analyzed models. Structural steels C45, S355J0 and 41Cr4 that are different from the point of view of static and cyclic mechanical properties were analyzed as well as models formulated by Heywood (H) [11] and models marked with Roman numerals I – simplification of Haigh curve, II – generalized on the range of limited fatigue life (Goodman formula), III – model based on the parabola equation, IV – model described with the equation of an eclipse and V – model described with the extended parabolic equation [14], [15] and [17]. From the analysis of data included in the paper [17] it results that results of calculations in accordance with the model I the most correspond to the experimental ones. In the model good conformity of test results was on the range of variability of the cycle asymmetry coefficient R from the fields (1) and (2), whereas in fields (3) and (4) of R coefficient variability the conformity, for increased mechanical properties (C45) and low-alloy (S355J0) steels, was significantly lower. The mentioned disadvantage was not observed in case of 41Cr4 alloy steel. Analyzing reference data on steel there can be observed similar phenomenon in a slightly smaller scale because classification of TCF type for alloy steels covers variability of the R coefficient only in fields (1) and (2) excluding fields (3) and (4) where one can observe the clearly diversified form of TFC diagrams. Therefore a conception of TFC modification in accordance with the model I was created towards its universality. The conception was illustrated in Fig. 2.

Fig. 2 Diagram of the model I (line ABCD) and model IM (line ABCEF) of TFC Line of constant value (N = const.) in accordance with the model I with reference to description included in works [15] and [17] is extended between points A B C D (Fig. 2) and described with formulas: m

N=

(S

N 0 ⋅ Sf ((−−11)) (T) fa ( R )

(T) + ψ NSfm (R )

)

m( −1)

for

-∞ < R ≤ 0

(1)

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Fatigue Failure and Fracture Mechanics

and (T)  Sf ( −1) (R m + Sfa( T()R ) − Sfm  (R ) ) N = N0   (T ) Sfa ( R ) R m (1 + ψ N )  

m( −1)

for

0 < R ≤ 1,0

(2)

Mathematical model I described with formulas (1) and (2) refers to TFC of the C type for resilient materials. Model IM is more suitable for plastic-resilient materials. For the model lines of constant value are extended between points A B C E F. These lines are described with formulas: (1) and (2) for the range -1,0 < R ≤ 0 and 0 < R < 1,0 (fields (1) and (2) in the fig. 2). Whereas for the range -∞ < R < -1,0 (segment C E – field (3)) and 1,0 < R < +∞ (segment E F – field (4)) formulas are as following: m

N=

(S

N 0 ⋅ Sf ((−−11)) (T) fa ( R )

(T) − ψ NSfm (R )

)

m ( −1 )

for

-∞ < R < -1

(3)

for

1 < R < +∞

(4)

and (T )  Sf ( −1) (R m + Sfa( T()R ) + Sfm  (R ) ) N = N0   Sfa( T()R ) R m (1 + ψ N )  

m ( −1)

From the above results that for fields (1) and (2) mathematical models I and IM have the same form essentially differing in fields (3) and (4). Parameters appearing in formulas from (1) to (4) were explained in the diagram (Fig. 3) whereas the cycle asymmetry coefficient for metals that appear in formulas from (1) to (4) can be determined in accordance with the method described in works [17] and [19]. Line A C E is a curve of limit stresses whereas the line B D E is an exemplary line of constant fatigue lives (N < N0).

Fig. 3 Diagram of parameters with signification of parameters appearing in TFC mathematical models

ψ N = tg Θ N =

Sf( T( −) 1) − Sfa( T()0) Sfa( T()0)

(5)

Dariusz Skibicki

73

3. Experimental verification of I and IM models of two-parametric fatigue life characteristic Experimental verification of analyzed models I and IM described with formulas in the section 2 will be based on results of C45 and 41Cr4 steel described in details in the work [17]. Wöhler fatigue life curves determined for different values of the cycle asymmetry coefficient R are the base of experimental TFC. These curves are described with following formulas: 1 log Saf ( R ) = − log N + b (6) m(R ) List of parameters from Wöhler fatigue life curves of tested steels for variable values of the cycle asymmetry coefficient R were inserted in Table 1. Table 1. Wöhler fatigue life curves and the ones experimentally determined fatigue for variable values of the cycle asymmetry coefficient R Equation describing Wöhler fatigue life curves 1 Fatigue limit Sfa(R) Cycle logSaf ((RT )) = − log N + b ( R ) m(R ) for N0 = 106 asymmetry coefficient R m(R) b(R) C45 41Cr4 C45 41Cr4 C45 41Cr4 0 17,64 9,95 2,6517 3,1671 204,7 366,7 -0,5 12,02 8,97 2,8188 3,3145 208,7 442,4 -1,0 9,80 8,53 2,9611 3,3546 223,5 447,6 -2,0 12,38 9,17 2,8614 3,3612 238,0 509,8 -3,0 19,80 15,87 2,7440 3,1900 276,0 648,5 Conformity of test and calculation results was evaluated by comparison of fatigue life amplitude value Sfa( T()R ) for individual lines of constant fatigue lives N and variable values of the cycle asymmetry coefficient R. Experimental data and test results in accordance with modified mathematical models were inserted in Table 2 and 3. As a measure of conformity there was obtained the relative value of test and calculation results calculated from the formula: Sfa( T()R ) ex − S(faT()R ) obl δ= (7) Sfa( T()R ) ex Calculation results of relative values were presented in the form of diagrams in individual figures: for C45 steel in Fig. 4a and for 41Cr4 steel in Fig. 4b. In these figures individual diagram lines stand for dependence of relative differences δ from N cycle number that characterizes lines of constant fatigue lives TFC. Lines 1, 2, 3 and 4 correspond to the mathematical model I before modification, in sequence for following values of the cycle asymmetry coefficient R = 0; -0,5; -2 and -3. Lines 5, 6, 7 and 8 correspond to the mathematical model IM (modified) for values of the cycle asymmetry coefficient as above. From assumptions of the analyzed model described in the section 2 and illustrated in the fig. 2 results that diagram lines 1 line up with diagram lines 5 and diagram lines 2 line up with diagram lines 6. In the conformity analysis of test and calculation results for both discussed model diagrams for R = -1,0 were excluded because from assumptions of models results that conformity is total because formulas (3) and (4) are based on Wöhler curve parameters for R = -1,0.

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Fatigue Failure and Fracture Mechanics

Table 2. List of data on fatigue life Saf ((RT )) experimentally determined (Ex) and calculated in accordance with the mathematical models I and IM for C45steel R=0 R = -0,5 R = -2,0 R = -3,0 N Ex I IM Ex I IM Ex I IM Ex I IM 2 10 345 345 374 449 469 486 501 732 483 440 852 449 103 303 303 326 371 388 400 416 541 399 391 600 452 4 10 266 266 279 306 321 326 345 404 325 348 432 311 105 233 233 235 253 264 264 287 304 262 310 316 251 106 205 205 195 209 217 213 238 230 211 276 234 206 7 10 180 180 159 172 178 171 198 175 168 246 175 164 Table 3. List of data on fatigue life Saf ((RT )) experimentally determined (Ex) and calculated in accordance with the mathematical models I and IM for 41Cr4steel R=0 R = -0,5 R = -2,0 R = -3,0 N Ex I IM Ex I IM Ex I IM Ex I IM 102 925 925 760 1235 1155 1059 1390 1535 1052 1159 1672 956 103 734 734 618 955 895 832 1082 1148 823 1002 1234 755 104 582 582 499 739 694 651 842 859 767 867 913 602 5 10 426 426 402 571 538 508 645 644 506 750 677 474 6 10 367 367 321 442 417 396 510 483 395 649 503 373 107 291 291 256 342 323 307 396 363 305 561 374 342 a)

Steel C45 – I i IM

b)

Steel 41Cr4 – I i IM

Fig. 4. Diagrams of relative differences between experimental and test results in accordance with models I and IM for: a – C45 steel, b – 41Cr4 steel

4. Analysis of test and calculation results Test and calculation results described in the section 3 enable the qualitative and quantitative analysis of mathematical models based on the conception of Haigh stress limit curves (model I and IM).

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Analysis of the basic model I presented in the work [17] indicated that the form of the line of constant fatigue lives N of TFC essentially differs from that the form of the line of constant fatigue lives N experimentally determined. Mentioned differences were observed for the coefficient R for the range (-1,0; -∞) especially for plastic-resilient C45 steel, less in 41Cr4 alloy steel. The above statement finds its confirmation on the level of relative differences between calculation and test results illustrated in Fig. 4. These curves indicate that range of differences for C45 steel and the model I is from -0,75 to -0,3 for R = -3,0 and from -0,35 to 0 for R = -2,0 (diagram lines 3 and 4 in Fig. 4a). It should be emphasized that higher of presented values concern smaller number of cycles N (high stress values). Implementation of modification of the model described in the section 2, changing the course of lines of constant fatigue lives N, as in Fig. 2, enabled to obtain much more higher conformities what is reported by the course of lines 7 and 8 in Fig. 4a. Relative values of δ for these cases respectively are from 0,06 to 0,12 for R = -2,0 and from 0,08 to 0,3 for R = -3,0 for this model (IM). It should be emphasized that smaller of presented values correspond to small number of cycles N (high stress values) what, from the point of view of fatigue life calculations of structural elements, is a beneficial case. From the analysis of lines of constant experimental fatigue lives TFC for analyzed steels presented in the work [17] and the analysis of criteria for classification of lines to ranges of LCF and HCF results that the type A dominates in the LCF range while the type B of lines of constant fatigue lives N dominates in the HCF range. The above statement indicates the general TFC form of the C type. Practically in the algorithm of fatigue life calculations of structural elements there should be introduced LCF and HCF criterion and in the first range (LCF) assumed following formulas for calculations (1), (2), (3) and (4) (model IM) whereas for the second range (HCF) there should be assumed formulas (1) and (2) (model I). From HCF and LCF criteria analyzed in works [20] and [21] the criterion of number of cycles Nt from Manson-Coffin curve it is recommended as the most efficient one. From above considerations results the following general conclusion on the application of discussed models in fatigue life calculations of steel elements. In case of calculations in the range of HCF it is recommended to apply the basic model I in accordance with the scheme in the fig. 2 that is described with formulas (1) and (2). In case of loadings in the LCF range it is recommended to apply the modified model marked as IM (Fig. 2) and formulas: form the model IM – (1), (2), (3) and (4). In the analysis of service loadings of structural elements there are cases of loadings with both HCF and LCF values. In these cases in the calculation algorithm there has to be implemented the criterion discussed above and for calculations for cycles from the HCF range the model I has to be applied while for cycle from the LCF range the modified model IM should be applied. Note: This work has been elaborated in the frame of the project No. 0715/B/T02/2008/35 financed by Polish Ministry of Sciences and Higher Education. References [1] [2] [3] [4] [5]

O. Andersen, W. Popp, M. Schaffrabek, D. Stenmetz, H. Stenger, Schaetzen und Testen, Springer Verlag, Berlin, 1976. J.S. Bendat, A.G. Pierdol, Methods of analysis and measurement of random signals, (in Polish), PWN, Warszawa, 1976. L. Gajek, M. Kałuszka, Statistical conclusion validity, (in Polish), WNT, Warszawa, 2000. ASTM standard, Standard Practices for cykle counting in fatigue analysis, ASTM Designation: E 1049-85 (Reapproved 1990). S. Kocańda, J. Szala, Fundamentals of fatigue calculations, (in Polish), PWN, Warszawa, 1997.

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[6]

[7] [8] [9] [10] [11] [12] [13] [14]

[15]

[16]

[17]

[18] [19] [20] [21]

Fatigue Failure and Fracture Mechanics

J. Szala, Programmed fatigue tests of materials and reveted structural components in aviation industry – selected problems, Monographs, 4nd part of monograph: Improvement of fatigue life of rivet connections applied in aviation structures – selected problems, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2010. P. Heuler, H. Klätschke, Generation and use of standardized load spectra and load – time historie, International Journal of Fatigue, 27 (2005). E. Haibach, Betriebsfestigkeit – verfahren und daten zur bauteilberechnung, VDI Verlag, 1989. Atlas of Fatigue Curves, ASM International, The Materials Information Society, Publishing House of Howard E. Boyer, 2003. W.W. Bołotin, Applied statistical methods in mechanics of buildings, Publishing House of Arkady, 1968. R.B. Heywood, Designing Against Fatigue, Chapman Hall, Londyn, 1962. J. Szala, G. Szala, Two-parametric fatigue characteristics formulating problem, Maintenance problems, no. 3 (2001), 287-296. J. Szala, G. Szala, Comparative analysis of two-parametric fatigue characteristics and their experimental verification, Maintenance problems, no. 3 (2001), 297-304. J. Szala, A. Lipski, Conception of description of material fatigue properties in calculations of structural elements (in Polish), Problems of Machines Operation and Maintenance, no. 2 (2005). B. Ligaj, G. Szala, Experimental verification of two-parametric models of fatigue characteristics by using the tests of S355J0 steel as an example, Polish Meritime Research, no.1 (2010), 39-50. B. Ligaj, Experimental and calculational analysis of Steel fatigue life in random conditions of wide range spectra, (in Polish), Monographs, 2nd part of monograph: Two-parametric fatigue characteristics of steel and their experimental verification, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2011. G. Szala, Theoretical and experimental analysis of two-parametric fatigue life characteristicsm of constr, (in Polish), Monographs, 1nd part of monograph: Two-parametric fatigue characteristics of steel and their experimental verification, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2011. I.E. Figle, An empirical wquation relating fatigue limit and mean stress, NASA Technical note TND-3883, Washington, 1967. G. Szala, Stress sensivity coefficient of a material in range of high – cycle fatigue, Maintenance problems, no. 4 (2010), 7-16. G. Szala, B. Ligaj, Evaluation criteria of low- and high-cycle fatigue in fatigue life calculations of constructional elements, Logistyka, no.6 (2009). S. Mroziński, J. Szala, Problem of cyclic hardening or softening in metals under programmed loading, Scientific Problems of Machines Operation and Maintenance, no. 4 (164), vol. 45 (2010), 83-96.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.77

The comparison of cyclic properties of X5CrNi18-10 steel in the range of low-cycle fatigue in conditions of stress and strain control Bogdan Ligaj1, a, Grzegorz Szala1, b 1

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Department of Machine Design, ul. Prof. S. Kaliskiego 7, 85-225 Bydgoszcz a

email: [email protected], b email: [email protected],

Keywords: low-cycle fatigue, diagrams of fatigue life, 1.4301 steel.

Abstract. Service loading of constructional elements are caused by mass forces (dynamic loading) or displacement (kinematic loading). There are methods of estimation of cyclic properties of material based on stress or strain range control corresponding to the mentioned loading types. The paper includes an analysis of differences between diagrams determined in controlled stress and strain conditions as well as relations among: Wöhler curves (W), Manson-Coffin curves (M-C) and Ramberg-Osgood curves (R-O). Basing on the performed analysis there have been formulated conclusions that generally are on connecting loading character with conditions of determination of atigue curves. Nomenclature K’ – strength coefficient of cyclic strain curve [MPa], 2Nf – the number of reversals to failure, N(s) – fatigue life read from Wöhler curve determined with stress controlled conditions for defined value of stress Sa, N(ε) – fatigue life read from Wöhler curve determined with strain controlled conditions for defined value of stress Sa, S – specimen stress – general notation [MPa], Sa – sinusoidal cycle stress amplitude [MPa], Sa(ε) – stress aplitude read from Ramberg-Osgood cyclic strain curve determined with strain controlled conditions for defined εac value [MPa], Sa(σ) – stress aplitude read from Ramberg-Osgood cyclic strain curve determined with stress controlled conditions for defined εac value [MPa], b – fatigue strength exponent in Manson-Coffin equation, c – fatigue plastic deformation exponent in Manson-Coffin equation, m(-1) – exponent in formula describing Wöhler fatigue diagram for oscillating load (R = -1), n' – cyclic hardening exponent in Ramberg-Osgood equation, ε – strain – general notation, εac – total strain amplitude, εac(ε) – total strain amplitude amplituda read from Manson-Coffin curve with strain controlled conditions for the defined number of reversals of loading 2Nf, εac(σ) – total strain amplitude amplituda read from Manson-Coffin curve with stress controlled conditions for the defined number of reversals of loading 2Nf, εf’ – coefficient of plastic fatigue deformation, δ(Sa) – relative difference of stress amplitude value for Ramberg-Osgood curves determined with stress and strain controlled conditions, δ(ε) – relative difference of total strain amplitude value for Manson-Coffin curves determined with stress and strain controlled conditions, δ(N) – relative difference of fatigue life for Wöhler curves determined with stress and strain controlled conditions, σf’ – fatigue strength coefficient [MPa].

78

Fatigue Failure and Fracture Mechanics

1. Introduction Design analysis of different types of machines enables to state that constructional elements are subjected to loads appearing due to control: dynamic (stress) or kinematic (strain) ones. The dynamic one appears when a force (moment) acting on a defined section area leads to changes of strain values while stress values remain constant. As examples of design elements subjected to stress control one can define subassemblies working in engineering designs i.e. aircrafts, seagoing vessels, working stands. The kinematic control appears when value of stresses in the defined crosssectional area is connected with a constant value of strain. As an example of such a case one can define off-centre machine shafts connected with a rigid coupling. A lack of alignment causes the constant value of deflection curve to with the corresponding value of stress. While a shaft turns a rotational bending appears characterized by a constant value of strain during the entire service life while stress changes [2]. The presented method of control of constructional elements finds its refection in experimental research procedures. It is assumed that results of fatigue life tests in stress-controlled conditions are presented as Wöhler curves (W) whereas results determined in strain-controlled conditions are presented as following diagrams: Manson-Coffin fatigue life curve (M-C) and Ramberg-Osgood cyclic strain curve (R-O). Detailed description of the mentioned diagrams was included in papers [1, 6]. Fatigue life diagrams W and M-C and R-O cyclic deformation curve are determined on the base of research conducted in constant-amplitude loading characterized by asymmetry coefficient R = -1 [5]. Multiple repetition of sinusoidal loading cycle enables to observe changes of cyclic properties of a material in specified periods of life. In the fig. 1 there are presented exemplary hysteresis loops recorded during tests of X5CrNi18-10 steel. Superficial analysis of shape and parameters of the loops for different periods of life indicates differences resulting from control methods and cyclic change of material properties. The change of the loop parameters influences stabilization curves of strain amplitude Sa (for the strain-controlled tests) and total strain amplitude εac (for the stresscontrolled tests) pinpointing lack of stabilization of the mentioned parameters in the entire life. Discussed issues were presented in details in works [3,7]. a)

b) Stress S, MPa

600

Stress S, MPa

400

0,9N

0,5N

0,5N 300

200

0,1N

0,1N 0 -1

-0.5

0 0

0,9N -300

-600

0.5

1

Strain ε, %

-0.6

-0.3

0

0.3

0.6

Strain ε, % -200

-400

Fig. 1 Exemplary hysteresis loops for X5CrNi18-10 steel in control conditions: a – strain, b – stress Lack of stablilization of loop parameters in the entire life indicates the problem of choosing the life period to determine curves: M-C fatigue and R-O cyclic deformation. It is essential because of application of the above mentioned curves in fatigue life calculations.

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79

With reference to presented problems connected with a method of control and change of cyclic properties the following hypothesis can be formulated: determination of cyclic properties of metals should be applied in dependence on the character of variable loading (for dynamic loading in a stress picture, and for kinematic one in a strain picture). 2. Formulation of the problem The evaluation of kinematic and dynamic control for test results was performed by comparison of W, M-C and R-O curves. The mentioned curves were determined on the base of data obtained from tests realized in constant amplitude sinusoidal loading conditions (R=1) with stess and strain control. Hysteresis loop parameters recorded for half-life were the base to determine M-C and R-O curves Tests in static loading conditions were performed on round cross-section specimens in accordance with EN ISO 6892-1:2010 standard with a diameter of a measurement base 10 mm. Geometrical features of specimens that are applied in fatigue tests were assumed on the base of PN-84/H-04334 standard. Measurement base of the specimen was 18 mm while its diameter was 10 mm. Tests were conducted with an application of austenitic chrome-nickel steel classified as stainless 1.4301 (designation of X5CrNi18-10) with accordance to PN-EN 10088-1:2007 standard. Material for specimens was bought as a round drawn bar with 24 mm diameter. Delivered bars were in the annealed state. 3. Test results and data elaboration X5CrNi18-10 steel was tested ub static and variable constant amplitude loading conditions. Tests in variable conditions were conducted in strain and stress controlled conditions. Tests in static loading conditions was conducted in accordance with PN-EN ISO 6892-1:2010 standard. Average values of chosen parameters of a tensile test for X5CrNi18-10 steel are as following: yield strength R0,2 = 322 MPa, tensile strength Rm = 652 MPa, tensile modulus (Young's static modulus) E = 166632 MPa, elongation A5 = 60.6 %, contraction Z = 59.8 %, ratio Rm/R0.2 ≈ 2.0. Tests of cyclic properties, strain and stress-controlled, were conducted in sinusoidal conditions characterized by asymmetry coefficient R=-1. Obtained fatigue test results enabled to determine Ramberg-Osgood (Sa-εac) cyclic deformation curve described with the formula: 1

S  S  n' ε ac = a +  a  E  K' 

(1)

Manson-Coffin fatigue life curve (εac-2Nf) ∆ε ac σ' c b = ε 'f (2 N f ) + f (2 N f ) 2 E

(2)

and Wöhler fatigue life curve (Sa-N)

log Sa = −

1 log N + b m ( −1)

Values of parameters appearing in equations (1), (2) and (3) were presented in the table 1.

(3)

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Fatigue Failure and Fracture Mechanics

Equation parameters of curves

Table 1. Parameters of equations describing Ramberg-Osgood, Manson-Coffin and Wöhler curves for X5CrNi18-10 steel Test conditions

M-C

R-O W

E c b εf' σf' n' K’ m(-1) b

MPa

MPa MPa

Stress-controlled (Sa = const.)

Strain-controlled (εac = const.)

166632 -0.3106 -0.1560 0.0580 1540 0.3165 2188 9.05 2.9606

166632 -0.3597 -0.0987 0.0931 929 0.2066 1257 11.56 2.9051

4. Analysis of test results Evaluation of obtained test results strain and stress controlled conditions was based on the comparative analysis of Ramberg-Osgood, Manson-Coffin and Wöhler curves that led to determine differences of relative values. Analysis of Ramberg-Osgood cyclic deformation was conducted on the base of relative difference of amplitudes of nominal stress δ(Sa) determined from the equation: δ (Sa ) =

Sa ( σ ) − S a ( ε ) Sa ( ε )

(4)

Results of amplitudes Sa(σ) and Sa(ε) were read from the Ramberg-Osgood curve for εac specific values. Strain-controlled test results were a reference point in the conducted analysis. Comparison of Mason-Coffin fatigue life curves was conducted on the base of the analysis of difference value of relative total strain amplitude δ(ε) determined from the equation δ(ε) =

ε ac ( σ ) − ε ac ( ε ) ε ac ( ε )

(5)

Values of total strain amplitudes εac(σ) and εac(ε) were read from the Mason-Coffin curve for specific values of the number of reversals of loading 2Nf. Strain-controlled test results were a reference point in the conducted analysis. Differences among test results, shown in a form of Wöhler curves for stress and strain-controlled conditions, were analyzed on the base of a value of relative difference of fatigue lives N calculated from the equation δ( N) =

N ( σ) − N (ε ) N (ε)

(6)

Values of fatigue life N(s) and N(ε) were read from Wöhler curves for specific value of stress amplitude Sa. Strain-controlled test results were a reference point in the conducted analysis. In the fig. 2 there are presented test results in the form of R-O, M-C and W diagrams and results of their comparison. Comparison of cyclic strain curves (fig. 2a) determined in stress and straincontrolled conditions indicates essential differences in the range of diagram courses and value of stress amplitude Sa for specific value of total strain amplitude εac. Analysis of changes of Sa values for the range of εac from 0.001 to 0.03 was performed on the base of relative amplitude difference diagram (fig. 2b). The range of changes of δ(Sa) values is from -0.15 to 0.16 that indicates an

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81

intersection of the analyzed diagrams R-O. The intersection point refers to the value εac = 0.009. For values εac < 0.009 there are obtained lower values of Sa, whereas for εac > 0.009 higher values of Sa with dynamic control. a)

b)

Stress Sa, MPa

720

Relative difference δ(Sa)

0.2

0.15 540 0.1

εac = const.

360

0.05 0

Sa = const.

0

0.01

0.02

Strain εac

-0.05

180

0.03

-0.1 0 0

0.01

0.02

0.03

Strain εac

-0.15 -0.2

Fig. 2 Comparison of Ramber-Osgood cyclic hardening curves in stress and strain controlled conditions (a) leading to determination of relative difference of nominal stress amplitude Sa (b) a) 0.1

b)

Strain εac

0.06

εac = const.

Relative difference δ(ε)

0 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

-0.06

0.01

-0.12

Sa = const.

-0.18

0.001 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Number of reversals of loading 2Nf

-0.24

Number of reversals of loading 2Nf

Fig. 3 Comparison of Mason-Coffin fatigue life curves in stress and strain controlled conditions (a) leading to determination of relative difference of total strain amplitude εac (b) Mutual position of Mason-Coffin fatigue life curves for dynamic and static control was presented in the fig. 3a whereas values of relative difference calculated from the formula (6) in the fig. 3b. For the entire range of the number of reversals 2Nf higher values of total strain amplitude εac are

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Fatigue Failure and Fracture Mechanics

connected with the period of fatigue life determined in strain-controlled conditions. The highest values of relative difference were obtained for 2Nf = 107 that is δ(ε) = -0.2. On the other hand the lowest value of δ(ε) that is 0.02 applies to the number of reversals from the range 2·104. The diagram of changes of relative difference value δ(ε) resembles a parabola in its form. a) 1000

b)

Stress Sa, MPa

Relative difference δ(N)

0.6

800

0.4

εa = const.

600

0.2

500 400

0 180

300

270

360

450

540

630

-0.2 200

-0.4

Sa = const.

Stress Sa, MPa

-0.6 100 1.E+02

-0.8 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Number of cycles N

-1

Fig. 4 Comparison of Wöhler fatigue life curves in stress and strain controlled conditions (a) leading to determination of relative difference of number of cycles N (b) Analysis of mutual position of Wöhler curves (fig. 4a) determined in dynamic and kinematic control indicates essential differences in their courses. Determined values of relative difference δ(N) (fig. 4b) for the analyzed range of stress amplitude changes Sa are limited from δ(N) = -0.9 to δ(N) = 0.55. Fatigue life curves intersect in a point for which Sa = 510 MPa. For values Sa < 510 MPa the diagram determined in strain-controlled conditions is characterized by higher values whereas for Sa > 510 MPa higher values of N were obtained for the diagram determined with stress-controlled conditions. 5. Summary Presented test results in the form of curves: Ramberg-Osgood cyclic strain curve (R-O), MansonCoffin fatigue life curve and Wöhler fatigue life curve determined in stress and strain controlled conditions indicates differences. Value of differences is connected with the type of material [8] and value of stress amplitudes Sa or total strain amplitude εac. Performed analysis emphasizes the need of sensible choice of a testing method suitable for stress or strain control conditions. The comparative analysis of curves: R-O, M-C and W that were determined in stress and strain controlled conditions indicates that differences are connected with properties of the tested material. The range of changes of relative difference are as following: - cyclic strain curve: δ(Sa) = -0.15 ÷ 0.16, - Manson-Coffin fatigue life curve: δ(ε) = -0.20 ÷ -0.02, - Wöhler fatigue life curve : δ(N) = -0.90 ÷ 0.55. Application of the above mentioned characteristics in fatigue life calculations has to be connected with the analysis of loading control conditions. Appearing differences between Manson-Coffin and Wöhler curves can lead to essential differences in fatigue life calculations resulting from multiple addition of errors in the range of fatigue estimations from 106 to 108 cycles.

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Note: This work has been elaborated in the frame of the project No. 2221/B/T02/2010/39 financed by Polish Ministry of Sciences and Higher Education. References [1] [2] [3]

[4] [5]

[6] [7] [8]

S. Kocańda, J. Szala, Fundamentals of fatigue calculations, (in Polish), PWN, Warszawa, 1997. B. Ligaj: Selected problems of service load analysis of machine components, Journal of Polish Cimac, vol.6, no.1 (2011), 125-131. S. Mroziński, Stabilization of cyclic properties in metals and its influence od fatigue life, (in Polish), Monographs no. 128, Publishing House University of Technology and Life Sciences, Bydgoszcz, 2008. J. Nemec, Strength and stifness of steel elements, (in Polish), WNT, Warszawa, 1968. G. Szala, Theoretical and experimental analysis of two-parametric fatigue life characteristicsm of constr, (in Polish), Monographs, 1nd part of monograph: Two-parametric fatigue characteristics of steel and their experimental verification, Publishing House of Operation Technology Institute - State Research Institute, Radom, 2011. J. Szala, Hypotheses of fatigue damage accumulation, (in Polish), Monographs, University of Technology and Agriculture, Bydgoszcz, 1998. J. Szala, S. Mroziński, The problem of cyclic hardening or softening of metals in programmed loading conditions, (in Polish), Acta Mechanica et Automatica, vol.5, no.3 (2011), 99-107. B. Ligaj, G. Szala, The comparison of cyclic properties of metals in the range of low-cycle fatigue in conditions of stress and strain control, (in Polish), Conference proceedings XXIV Sympozjum Zmęczenie i Mechanika Pękania, Bydgoszcz (2012), 89-90.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.84

Method of determining the initial stiffness modulus for trabecular bone under stepwise load Tomasz Topoliński1, a, Artur Cichański1,b, Adam Mazurkiewicz1,c, Krzysztof Nowicki1,d 1

Faculty of Mechanical Engineering, University of Technology and Life Sciences, Kaliskiego 7 Street, 85-789 Bydgoszcz, Poland a

[email protected], [email protected], c [email protected], [email protected] Keywords: trabecular bone, stepwise loading, initial stiffness modulus.

Abstract. In this work was presented method of initial stiffness modulus E0 calculation based on fatigue tests of trabecular bone under stepwise load. The investigation was performed on 61 cylindrical bone samples obtained from the neck of different femur heads. The bone sample fatigue tests were carried out under compression with stepwise increases of the applied load. The obtained values of the initial stiffness modulus E0 were consistent with literature data and can be used to determine the S-N curve for trabecular bone using the hypotheses of fatigue damage accumulation. It was also an unsuccessful attempt to find a statistical relationship between the values of the initial stiffness modulus E0 and indices of bone structure. Introduction Damage can be defined in various ways. According to the study [1], there are four typical definitions for the characterization of damage: ̶ defects at the microscale described by the number/density of the cracks; ̶ changes in the physical properties, such as material density, acoustic emission recordings, electrical resistivity, ultrasonic waves, and micro-hardness measurements; ̶ the remaining lifetime of the material; ̶ variations in the macromechanical behavior, such as changes in the elastic, plastic or viscoplastic properties. When investigating bones, damage is most often defined as D = E/E0, where E is the current value of the stiffness modulus, E0 is the initial value [2, 3], and by convention, D = 1 – E/E0 [4]. The description D = AD/A is also used, where AD is the bone damage area, A is the initial area [2], and D = DV/BV ,where DV is the damaged volume, BV is the undamaged bone volume [4]. The values of the initial stiffness modulus were determined using different methods, as described below. In paper [4], the tangent modulus (initial tangent stiffness) was the maximum slope of an 11piece linear regression of the stress–strain curve between the applied apparent strains of 0.1% and 0.8%. In paper [6], to determine E0 the slope of the stress–strain curve of the tenth cycle for compression at 1500 microstrain was used. As reported in [7], E0 is defined in a manner analogous to paper [6]; however, the strain ranges from 0.6% to 2.1%. The tangent modulus in [8] was determined prior to fatigue testing using a 100 N (10 MPa) load applied at 2 Hz tensile and compressive ramps. According to work [9], the initial modulus was measured by taking the slope of the best linear fit for the final preconditioning loading cycle (ε=0.001 to 0.003). The initial modulus reported in [10] was measured by taking the slope of the best linear fit of the tenth loading cycle from 0.1% to 0.3% strain. In the work [5] tested each sample of the stress–strain curve at the origin. Samples were loaded cyclically at 2 Hz to 100 N for approximately 20 cycles, which was equivalent to a stress of approximately 12 MPa for the samples. For this test, the slope of the stress-strain curve was linear; therefore, the tangent modulus was calculated by dividing the pretest stress by the strains. This strategy was associated with calculating the standardized stress with the quantity E0 used in the creep and fatigue studies of minimized bone scatter. It appeared that most of the effect on the structure and complexity of the process fell within the quantity E0.

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Evaluating the risk of bone cracking for real-life applications is a very complex procedure. The evaluation critically depends on ability to define the fatigue life of bone tissue; thus, there is a significant part of published work in that field [3, 6, 7, 9, 10] The majority of the research was performed in vitro and using only sinusoidally-variable constant-amplitude loads, most often with compression loads [6, 7, 9, 10], particularly for trabecular bone, whereas tension [5] was the more rarely used type of load [1]. The quantity used to describe the results was generally the input stress associated with the initial stiffness, E0. Similarly, our experiment was developed for cylindrical samples of trabecular bone exposed to sinusoidally-variable loads; however a stepwise loads were applied, which has not been applied in bone experiments, but is commonly used when investigating other materials and structures [11, 12, 13, 14, 15]. This kind of loading makes it possible for the fatigue damage accumulation to involve loads that start from the lowest value to those for which the damage will be visible in a few dozen to a few hundred cycles. Such a load model should allow more information to be obtained about the fatigue process than in the one-step load experimental procedure. Methods The paper uses the research results of 61 cylindrical bone samples that were 10mm in diameter and 8.5mm in length and were obtained from the neck of femur heads. The samples were obtained from 21 men and 40 women undergoing hip joint alloplasty. The samples were stored in a 10% formalin solution at room temperature. All of the samples were scanned with a desktop microCT system (µCT-80, SCANCO Medical AG, Bruettiselllen, Switzerland) with a distance of 36µm between. When scanning the values of many bone structure indices were obtained: trabecular number Tb.N, trabecular thickness Tb.Th, trabecular separation Tb.Sp, bone volume fraction BV/TV, surface fraction BS/BV and the number of joints between individual trabeculae per unit volume of specimen Conn.D. The bone sample fatigue tests were carried out under compression with stepwise increases in the load using the testing machine, INSTRON 8874 (Instron, High Wycombe, England). The minimum loading for all of the loading levels was 5N. The maximum loading started at 10N with a gain every 10N at each successive step. At each loading level, 500 cycles were completed under constantamplitude loadings at a frequency of 1Hz. As stated in the literature, we have used the description of the results for the standardized load, where the standardizing quantity is the quantity, E0. Depending on the requirements and the assumptions made in literature, the standardizing quantity value is either the initial tangent stiffness or the initial secant stiffness. In our work, the calculations of the tangent modulus, E0, involved defining the angle of inclination of the line best adjusted to the pattern of the upper branch of the loop in the range from 85% to 100% of the maximum stress (referenced to the absolute value). The tangent modulus was determined for all of the recorded fatigue loops. This paper makes use of the mean value of the tangent modulus defined by the tangent modulus of the hysteresis loop recorded at the first load step. Full information about sample preparation and recorded during test values contains paper [17].

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Fatigue Failure and Fracture Mechanics

stress range

tg(α) - stiffness modulus E

15% of stress range

Fig. 1. The hysteresis loop with marked: stress range, 15% of the stress, line of best fit to the upper branch of the hysteresis loop, stiffness modulus E Results Fig. 2a shows the stiffnes modulus values for the first block of loading. The continuous lines mark the initial modulus. Fig. 2b shows the stiffnes modulus values for all recorded hysteresis loop. a)

b)

Fig. 2. Stiffness modulus calculated for a) the hysteresis loop recorded in the first block of the load, with a line marked the initial modulus E0 b) all recorded hysteresis loop The results of investigating the structure indices are given in Table 1, which presents the mean values of the selected indices, the standard deviation SD and the relative standard deviation (RSD=SD/mean) for all samples. The considerable variation in the structure of the bone samples investigated should be noted. The relative standard deviation ranges from 19.6% to as much as 48.9%, which must affect the scatter of the parameters recorded. The table was supplemented by the results of calculations of E0, according to the method proposed in our work and layout as for indicators of the structure.

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Table 1. The values of the mean, standard deviation (SD) and relative standard deviation (RSD) for selected indices of the structure and E0 of the bone samples investigated. Structure indices Mean SD RSD [%]

BV/TV 0.204 0.076 37.1

BS/BV [1/mm] 11.998 2.747 22.9

Tb.N [1/mm] 1.133 0.222 19.6

Tb.Th [mm] 0.176 0.045 25.6

Tb.Sp Conn.D. [mm] 0.786 2.764 0.385 0.921 48.9 33.3

E0 [MPa] 388,95 211,05 54,3

N 20630 12078 59

Discussion Our results are consistent with the calculation of the initial stiffness modulus E0 for human vertebral trabecular bone that has been shown in [3.16] for samples cut at an angle of 0°. Table 2 contains the summary of our results and literature data. A higher value for the standard deviation of our results is the effect of much larger number of samples from a larger number of donors. This indicates the correctness of the proposed method for determining the value of E0. The initial stiffness of the cancellous bovine bone samples is much higher, regardless of the direction of the cut samples [3, 10, 16]. Stiffness of cortical bone is also much higher [1]. The obtained values of the initial stiffness modulus E0 were consistent with literature data and can be used to determine the S-N curve for trabecular bone using the hypotheses of fatigue damage accumulation [18]. Table 2 Values of E0 values obtained by method presented in the work and data from the literature No.

References

1 2 3 4 5 6 7 8 9 10 11 12

[1] [6] [3] [10]

Bone human cortical bovine trabecular human vertebral trabecular bovine trabecular bovine trabecular

[16]

our resultes

human vertebral trabecular human femoral trabecular human head femoral trabecular

0° 90° 0° 22° 45° 90° 0°

E0 [MPa] mean±SD min÷max 12370÷15470 2690±900 1190÷4150 251±137 44÷497 1466÷2732 2709±548 1227±725 447±117 159±11 111±68 98±78 1031±461 389±211 69÷867

In addition, an attempt was made to determine the relationship between E0 and calculated indices of the bone structure. Fig. 3 shows scatterplots of the relationship between the values of the sample structure indicators (BV/TV, BMD) and the initial values of stiffnes modulus E0. Regression lines in Fig. 3 are omitted because the coefficient of determination values R2 for all structure indicators in Table 1 according to the E0 did not exceed 0.14. Therefore, no relationships were found between the E0 and bone structure. In our opinion, this relationship should take into account characteristics of strength of individual trabeculae.

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a)

Fatigue Failure and Fracture Mechanics

b)

Fig. 3. Scaterrplots of the relationship between the values of the sample structure indicators BV/TV a), BMD b) and the initial values of stiffness modulus E0 References [1] Zioupos P., Casinos A.: Cumulative damage and the response of human bone in two-step loading fatigue, Journal of Biomechanics 1998, 31(9): 825-833 [2] Kosmopoulos V., Keller T.S: Predicting trabecular bone microdamage initiation and accumulation using a non-linear perfect damage model. Medical Engineering and Physics 2008, 30: 725-732 [3] Rapillard L., Charlebois M., Zysset P.K.: Compressive fatigue behavior of human vertebral trabecular bone, Journal of Biomechanics 2006, 39(11): 2133-2139 [4] Tang S.Y., Vashishth D.: A non-invasive in vitro technique for the three-dimensional quantification of microdamage in trabecular bone. Bone 2007, 40: 1259-1264 [5] Cotton J.R., Winwood K., Zioupos P., Taylor M.: Damage rate is a predictor of fatigue life and creep strain rate in tensile fatigue of human cortical bone samples. Journal of Biomechanical Engineering 2005, 127: 213-219 [6] Haddock S.M., Yeh O.C., Mummaneni P.V., Rosenberg W.S., Keaveny T.M.: Similarity in the fatigue behavior of trabecular bone across site and species. Journal of Biomechanics 2004, 37(2): 181-187 [7] Michel M.C., Guo X.E., Gibson L.J., McMahon T.A., Hayes W.C.: Compressive fatigue behavior of bovine trabecular bone. J Biomechanics 1993, 26(4/5): 453-463 [8] Pattin C. A., Caler W. E., Carter D. R.: Cyclic mechanical property degradation during fatigue loading of cortical bone, Journal of Biomechanics 1996, 29(1): 69-79 [9] Ganguly P., Moore T.L. A., Gibson L.J.: Analysis of fatigue damage in bovine trabecular bone. www.asbweb.org [10] Moore T.L., Gibson L.J.: Fatigue of Bovine Trabecular Bone. Journal of Biomechanical Engineering 2003, 125: 761-768 [11] Landgraf R.W., Morrow J., Endo T.: Determination of the cyclic stress-strain curve. J of Materials 1969, 4: 1621-1653 [12] Janzen W., Ehrenstein G.W.: Bemessungsgrenzen von glasfaserverstärktem PBT bei schwingender Beanspruchung. Kunststoffe 1991, 81(3): 231-236 [13] Orth F., Hoffmann L., Zilch-Bremer H., Ehrenstein G.W.: Evaluation of composites under dynamic load. Composite Structure 1993, 24(3): 265-272

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[14] El Fray M.: A long-term mechanical fatigue examination of thermoplastic elastomers. Elastomery 2004, 8(5): 15-19 [15] Casado J.A., Carrascal I., Polanco J.A., Gutiérrez-Solana F.: Fatigue failure of short glass fibre reinforced PA 6.6 structural pieces for railway track fasteners. Engineering Failure Analysis 2006, 13(2): 182-197 [16] Dendorfer S., Maier H.J., Taylor D., Hammer J.: Anisotropy of the fatigue behaviour of cancellous bone. J Biomech. 2008; 41(3): 636-41 [17] Topoliński T., Cichański A., Mazurkiewicz A., Nowicki K., Study of the behavior of the trabecular bone under cyclic compression with stepwise increasing amplitude, Journal of the Mechanical Behavior of Biomedical Materials 2011, Vol. 4, No. 8: 1755-1763 [18] Topoliński T., Cichański A., Mazurkiewicz A., Nowicki K., Applying a stepwise load for calculation of the S-N curve for trabecular bone based on the linear hypothesis for fatigue damage accumulation, Fatigue Failure and Fracture Mechanics 2012, In Press

CHAPTER 3: Fatigue of Welded Structures

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.93

Fatigue Test Welded Joints Steel S960QL GOSS Czesław1a, MARECKI Paweł1b 1

Military University of Technology, Faculty of Mechanical Engineering, Kaliskiego Street 2, 00-908 Warsaw a

b

[email protected], [email protected]

Keywords: welded joints, low cycles fatigue life.

Abstract. Test results of residual stresses in welded butt joints were also presented. Finite Elments Methods and Jewdokimow and Lawrance’a methods of stress concentration factor αk calculation were comparative sumarized. Presents results is a part more considerable research attendant bridge MS-20. Introduction In last year’s, a investigated of the attendant bridge MS-20, constructed in the Department of Mechanical Engineering Military University of Technology. Load tests were carried out successfully. The research confirmed a sufficient fatigue life of the bridge (5000 cycles). Loads with a larger number of cycles showed cracks occurring in welded joints. Therefore, it was further tests of welded joints of steel from which the bridge was produced. Test results In the first place in order to determine the basic material properties the flat specimen static tensile tests were carried out. The specimens were made in accordance with appropriate [1]. Table 1 presents the chemical composition and Figure 1 shows an example chart of steel tensile test. Table 2 shows the typical mechanical properties obtained during testing. C 0,18

R0,2 [MPa] 1000

Table 1. Steel S960QL chemical composition (melting analysis) Si Mn P S Cr Mo Ni 0,50

1,60

0,02

0,01

0,80

0,60

Table 2. Mechanical properties Steel S960QL Rm [MPa] Ru [MPa] A [%] Z [%] 1090

2209

σ [MPa]

14

38

1100 1000 900 800 700 600 500 400 300 200 100 0 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 ε [%]

Figure 1. Chart of steel S960QL tensile test

2,00

V 0,10

E [GPa] 208

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Fatigue Failure and Fracture Mechanics

Fatigue Test Results The base material flat specimens and steel S960QL specimens with welded V butt joints made using TIG method, coalescence X96-IG, in accordance with appropriate [1] were research subject. The tests were carried out on a machine Instron 8802 with maksimum load 250kN. The specimens were testing using cycling loads with a constant maximum stress value. A fatigue tests were carried out in full cycles in order to obtain a complete steel characteristics. This allowed to obtain Wöhler curve including high-cycle range. 1100

σ a [MPa] 1000

95% 900

95% 800

log Nˆ = 8, 5943 − 0,0045σ

POŁĄCZENIESPAWA WELDED JOINT NE MATERIA BASE MATERIAL ŁRODZIMY

log Nˆ = 8,4124 − 0,0053σ

700

600

500

400 1,00E+02

1,00E+03

1,00E+04

1,00E+05

1,00E+06

N f 1,00E+07

Figure 2. Wöhler curve for base material and welded joint specimens The test results presents a significant decrease in fatigue life of welded specimens compared to base material specimens. Fatigue life in low-cycle range decreased up to 90% (figure 2). It follows that the steel S960QL should not be used for fatigue loaded welded structures. Furthermore a tendency to cyclical strengthening of steel S960QL are presented in following figures (figure 3 and figure 4). 1100 σ[MPa] 1000 900 800 700 600 500 400 300 200 100

1

2

3

4

ε[%]

Figure 3. Chart of stresses and strains changes as cycle number function at a constant stress σmax = 1080 MPa for a base material specimen (Nf = 5770)

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1100 σ[MPa] 1000 900 800 700 600 500 400 300 200 100

1

2

3

4

5

6

7

8 ε[%]

Figure 4. Chart of stresses and strains changes as cycle number function at a constant stress σmax = 1080 MPa for a welded butt joint specimen (Nf = 5770) Residual stresses measurements in the welded joints were also performed [3]. X-ray diffraction measurements were performed using the Rigaku diffractometer STRAINFLEX PSF-2M. The chart illustrated the following types of stresses: x - along the specimen longitudinal axis, y – perpendicular to the specimen longitudinal axis and the average stresses of these directions (figure 5). The measuring points were placed in the center of the welded joint (point 1), on the melting line (point 2) and on the border of heat-affected zone (point 3) (Figure 6). 200 100

Stresses [MPa]

3

2

1

0 -100 direction x -200

direction y average value

-300 -400 -500 -600 Tests points 1,2,3

Figure 5. The residual stresses distribution in a welded butt joint (steel S960QL) 3 2

1

Figure 6. Residual stresses measuring points location in a welded butt joint

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Fatigue Failure and Fracture Mechanics

Determination of the Stress Concentration Factor αk

ρ

θ

Calculation of the Shape Factor by Using Lawrence and Jawdokimov Methods The Lawrance and Jewdokimow methods were used to a shape factor determination [2]. The Lawrence method The measurements of a testing welded joint geometry were performed in order to determining of a shape factor αk. The angle weld slope Θ and transition welding radius ρ in base material were measured (figure 7). Measurements were made from the weld cap and from the root of weld. The obtained values are presents in table 3 [4].

Figure 7. Welded joint geometry - general indications Table 3. Angle weld slope Θ and transition welding radius ρ in base material Parameter Value Θ 26° - 27° Weld cap ρ 4,8° - 7,5° Θ 32° - 35° Root of weld ρ 2,3° - 3° The basic equation used in Lawrence method is shown below: C  t  (1) α k = B ⋅ 1 + A ⋅     ρ    where: A, B, C - constants depends on the type of welded joint, the way it loads and the local geometric shape, t - jointed elements thickness (t = 6 mm), ρ - transition welding radius, Θ - angle weld slope. The values of a coefficients A, B, C are shown in following table (Table 4). Table 4. The values of a coefficients A, B, C Coefficient Value A (weld cap) A = 0,27(tg Θ)0,25 = 0,226 – 0,228 A (root of weld) A = 0,27(tg Θ)0,25 = 0,240 – 0,247 B 1 C 0,5 The shape factor values were obtained using equation 1 with appropriate values of coefficients: weld cap - αk = 1,202 – 1,255, root of weld - αk = 1,339 – 1,399. The maximum value was taken for further calculations. The Jewdokimow Method The shape factor value αk in Jewdokimow method was obtained using a following equation [2]: α k = α kc ⋅ α p (2)

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where: α k –

97

shape factor total value,

α kc –

tensile and bending welded butt joint shape factor of circle cap weld with a width equal to the thickness of the joint elements α p – correction factor taking into account the actual weld width [2].

- weld cap, αk = 1,35 · 1,03 = 1,39 αk = 1,46 · 0,98 = 1,53 - root of weld. The following value was taken for further calculations αk = 1,53. Numerical Modelling of the Shape Factor Value Stress distributions were obtained using by finite element modelling. There was obtained a value of a stress concentration factor: αk = 1,08 - weld cap, αk = 1,20 - root of weld. The shape factor values obtained during numerical analysis were lower than shape factor values obtained during calculations. The stresses and strains distributions in steel S890QL welded specimens during tensile were calculated. Analysis Object The analysis object was flat specimen with welded butt joint in the middle (Figure 8a). a)

b)

Figure 8. a) flat specimen, b) part of analyze (¼ section of the specimen) Computational Model Due to longitudinal and transverse plane of symmetry the computational model was ¼ section of the specimen (only work part) (Figure 8b). C

D

Figure 9. FEM model with an indication of the elements, boundary conditions and loads Model consisted of 10 371 spatial elements (3D), including the 3462 tetrahedral 10-nodes elements and 6909 hexahedral 20-nodes elements (Figure 9). Model mesh consisted of 33232 nodes. The boundary conditions indications in Figure 10 are: „C” – no displacement in y direction „D” – no displacement in x direction. The elastic material model with nonlinear strengthening developed on the tensile test results was assumed. The Calculation Results The results are shown in colored contour lines represented von Mises stresses, displacements and charts along selected lines (figure 10, 11, 12).

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Figure 10. Reduced stress at 850 MPa load

Figure 11. Reduced stress at 600 MPa load

Figure 12. Reduced stress at 500 MPa load

Figure 13. Reduced stress along the bottom edge of the model (middle of the specimen) for three load levels: 850, 600 and 500 MPa

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The stress distributions showed in Figures 10-13 were obtained during numerical analysis. The highest stresses occurred on the melting line. Numerical analysis shows that these stresses approach the yield strength of this steel, although the nominal stresses are much lower. The stress concentration factor was determined on that basis: - root of weld - αk = 1,20, - cap weld - αk = 1,08. Therefore, these values are lower than those obtained using computational methods. All specified values of αk will be used in further work to analyze fatigue life of welded joints. Summary Steel S960QL has cyclic strengthening with the tendency for fast stabilization. This applies both to base material specimens and welded butt joint specimens. During testing using cycling loads with a constant maximum stress value σmax a decrease of hysteresis loop width and amplitude of plastic deformation were showed. The test results showed a significant decrease up to 90 % in fatigue life of welded specimens compared to base material specimens. This tendency is maintained also in high-cycle range. The measurements of residual stresses in welded butt joints showed their considerable value. This fact results in summing of residual stresses and stresses from external forces during tension. This has a substantial effect on the durability of tested elements. The results confirmed a great influence of welded joints on mechanical properties of structures made using this steel. In the case of a bridge we are mainly dealing with similar cyclic loads. Therefore a authors attention will be focus on a such kind of loads. This research are continued mainly for the modification of welded bridge nodes to increase the structure durability. References [1] Polish Standart PN-88/M-69710. (in polish) [2] Goss Cz., Kłysz S., Wojnowski W.; The low cycle fatigue life selected steel AT the welded joint; ITWL, Warszawa 2004 (in polish). [3] Lech-Grega M., Raport Nr 3129/N/1/4 – The residual stress measured, Skawina 2011 (in polish). [4] Goss Cz., Marecki P., The durability fatigue problems welded joint with steel S890QL, Biuletyn WAT, Warszawa 2012 (in polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.100

Influence of the notch rounding radius on estimating the elastic notch stress concentration factor in a laser welded tee joint Karol Niklas1, a, Janusz Kozak2, b 1, 2

Gdansk University of Technology, The Faculty of Ocean Engineering and Ship Technology, Narutowicza 11/12 St., Gdansk, Poland a

[email protected], [email protected]

Keywords: laser welds, fatigue, local stress approach, fem, steel sandwich panel

Abstract. In recent years an increased interest of industry in sandwich-type metal structures can be observed. These structures consist of thin plates of 2.5 mm in thickness separated by stiffeners of different shapes and forms. Welds joining the plates and stiffeners are made on the outer side of the plates using laser welding technique. A locally focused source of heat causes the plate to melt creating a very narrow and elongated joint. As a result, sharp geometric notches are formed on the side of the root of a weld – a place which is inaccessible and cannot be checked. Geometries of individual welded joints vary, sometimes considerably, and this makes their analysis even more complicated. Additionally, the use of laser welding technique influences the formation of untypical distribution of changes in material properties in weld zones. The effect is a joint whose behaviour under load is significantly different from the behaviour of a welded tee joint made with the use of classical methods. Fatigue strength calculations for this type of joints can be conducted based on local stress values in notches, which can be determined with the use of Finite Element Method (FEM). This article analyses the influence of the notch rounding radius on the elastic notch stress concentration factor Kt The aim of the analysis is to evaluate the notch stress concentration according to local notch stress approach. Introduction The growth of transport operating costs causes an increase of interest in new materials and structural solutions. There is an on-going search for innovative structural solutions characterised by, among others, a better strength to mass coefficient and a longer service life. One of the more interesting from the many possible new solutions are steel sandwich panels. These are multi-layered structures consisting of cover plates, stiffeners and a core. The panels are prefabricated as large-size structures made of plates with different thickness, stiffeners of various height and form located between the panels, and different forms of filler material. The stiffeners are flat bars perpendicular to plates, most often in a shape of V, C, O Z or X. Standard plate thicknesses are 1–3 mm, stiffener thicknesses equal 4 mm and the distance between plates is from 20 mm to 250 mm. Overall dimensions of a panel reach from 1.5 m to 10 m. An example of a panel with flat bar stiffeners and other most popular forms of stiffeners is shown in Fig. 1.

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Fig. 1 An example of a steel sandwich panel with flat bar stiffeners (I-core) [1] In all practical applications panel structures must comply with a number of rules (regulations of classification societies, standards, etc.). One of the requirements that welded structures must fulfil is an adequate fatigue strength. The analysed elements of a steel sandwich panel structure that are prone to fatigue failure are welds connecting plates with stiffeners. These joints are laser welded in a very untypical way – by melting a plate from the outer side and forming a joint with a stiffener. The joint created this way has a unique geometry and its fatigue strength is hard to estimate. It should be noted that these joints play a key role in terms of the ability of a multi-layered structure to transfer loads. A theoretical analysis of fatigue strength for this type of joints may be carried out with the use of different criteria: - structural stresses and strains criterion - local criteria (local stresses and local strains) - criteria based on fracture mechanics theory - mixed criteria. Due to the specifics of the analysed laser welds, and especially their geometry, the most advisable fatigue approach seems to be the local stresses criterion. The sections below will discuss the geometries of the analysed laser welds, and than numerical modelling of notch stress concentrations. Numerical modelling of laser welds is performed in order to use local notch stresses to evaluate fatigue strength according to local stress criterion. Geometry of laser welds Welds of steel sandwich panels are prefabricated with a rather untypical technique of laser welding. Plates are joined with stiffeners from their outer side creating joints with a very unique shape. The heat source is a 14 kW CO2 laser whose beam is focalized by parabolic mirrors with a spot size of 0.5 mm and directed onto the outer surface of the panel's plate. The focused laser beam of a very high power density (10e9 W/mm2) makes the plate and stiffener material fuse. A very narrow and elongated joint is created, but without a full penetration on the stiffener edges. On both sides of the weld, between the plate and the stiffener there are technological gaps left. It should be noted that the unpenetrated region of a joint often covers more than a half of the stiffener's width. An example of a laser joint created with the use of this technology and its geometry is shown in Fig. 2. This figure includes mean values of microscopic measurements performed on samples collected from the examined panels. Detailed measurements of SAW and laser welded butt joints on 12 mm thick sheets are published in [3, 4, 5]. However, it must be noted that in case of the analysed joints between plates and stiffeners of steel sandwich panels the dispersion of parameter values describing face and root notches is much greater, in proportion to the thickness of the analysed sheets, than in case of laser joints of about 10 mm thick sheets.

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Fig. 2 Geometry of an elementary laser joint between plates Influence of the notch rounding radius on the elastic notch stress concentration factor The aim of numerical modelling of laser welds is to determine the distribution and, as a consequence, to calculate stress concentrations present in geometrical notches. Determination of stress concentrations is crucial for evaluating the fatigue strength of welds. Following the methodology of numerical calculations according to the local stress criterion, notches should be modelled by fictitious roundings or alternative geometries [6, 7]. However, application of a standard rounding radius ρf = 1 mm, recommended in a number of publications for joints of elements that are over 5 mm thick, is not possible in this case since the elements being joined are much thinner and dimensions of the notches are smaller. The few publications describing the way of calculating elastic stress concentration factors for welds joining very thin sheets (of about 1.5 mm) concern spot welds of car bodies [8, 9]. For this type of intermittent welds the proposed rounding radius is 0.05 mm. Considering the technology of spot welding, different geometry and properties of such welds, generalization of the calculation method with a 0.05 mm radius for other welding technologies is very risky. Other authors suggest the radius value of 0.1 mm for lap joints of thin laser welded sheets [10]. As can be seen from a limited number of publications on the subject, small amount of empirical data and a considerable spread of the proposed values, methods for calculating stress and strain concentrations of welds joining thin elements are a poorly explored research area. Therefore, in the conducted calculations values of rounding radii of face and root notches were adopted as a parameter that will be analysed. The assumptions for the FEM model were: plane stress state, loads in the plane of panel plates, weld symmetry axis. The model geometry with loads and boundary conditions is shown in Fig. 3. Linear elastic material model with Young's modulus E = 2e5 MPa and Poisson ratio ν = 0.3 were assumed. 8-node quadrilateral elements with quadratic shape function were used.

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Fig. 3 Geometry of laser weld model with loads and boundary conditions The examination of the influence of the notch alternative geometry radius ρf on the elastic stress concentration factor Kt was conducted for 6 different values of alternative radii (called ‘fictitious’ in the literature and denoted with ρf ): 0.025 mm, 0.05 mm, 0.1 mm, 0.15 mm, 0.2 mm and 0.25 mm. It was assumed that the values of face and root notch rounding are equal in all analysed cases. Model geometries with face notch rounding and root notch alternative circular geometry assumed for calculations are shown in Fig. 4.

Fig. 4. Model geometries with different values of radius ρf for alternative geometry: a) ρf = 0.025 mm, b) ρf = 0.05 mm, c) ρf = 0.1 mm, d) ρf = 0.15 mm, e) ρf = 0.2 mm, f) ρf = 0.25 mm

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The influence of the notch rounding value ρf on the elastic notch stress concentration factor Kt is shown in Fig. 5. Stress concentration values, calculated as the ratio of maximum notch stress to the nominal stress, observed in the root notch are significantly greater than in the face notch. For the value of rounding radius above ρf = 0.15 changes of Kt factor are insignificant. For the face notch stress concentration values decrease in the whole range of the notch rounding radius ρf and for the higher values of this radius the changes are more and more gentle. The differences between concentration curves for face and root notches result from their different alternative geometries. For the face notch the radius ρf is a rounding, and for the root notch it is the radius of an alternative circular geometry. Looking at the greater values of stress concentrations in the root of a weld and the course changes in the value of Kt factor presented in Fig. 5 it can be said that for numerical modelling of stress and strain concentrations with the use of local approaches a value of alternative geometry radius equal to ρf = 0.15 mm can be adopted. This is the smallest radius value for which the concentration level can be uniquely determined. Application of a greater radius value does not influence the result of concentration, but only causes artificial decrease of cross-section due to removal of weld material.

Fig. 5. Influence of the notch rounding radius ρf on the elastic stress concentration factor Kt Summary One of the key issues connected with the ability of sandwich-type structure to transfer loads is the fatigue strength of laser welds joining plates and stiffeners. Due to the difference of laser welds present in the panel construction it is very hard to reliably estimate their fatigue strength. One of the possible methods for evaluating the fatigue strength for this type of joints is the local stress criterion, for which the elastic notch stress concentration factor can be determined with the use of numerical Finite Element Method. However, the methodology for notch stress concentration calculations assumes the use of a notch rounding or an alternative circular geometry with a specified radius. For most joints of over 5 mm thick sheets a fictitious radius of 1 mm is used. However, for

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the analysed laser welds rounding the notch with a 1 mm radius is not possible as dimensions of the joined elements, welds and notches are substantially smaller. Therefore, an attempt has been made to determine the value of fictitious notch rounding radius. The influence of radius value on stresses was analysed. As a result, curves illustrating the influence of radius value on elastic stress concentration factor were determined. Based on the analysis of the received results a rounding radius of 0.15 mm was proposed as recommended for modelling weld notches for steel structures with 2.5 mm thick cover plates. References [1] Kozak J., Fatigue durability estimation problems of all steel sandwich panels, Gdansk University of Technology, ISBN 83-7348-136-2, Gdansk, 2005 [2] Reinert, T., Mühlenplatzabdeckung mit I-core Sandwich Paneelen. Papenburg: Öffentliche Dokumentation, I-core Panels Sales & Production, 2006 [3] Remes, H., Strain-based approach to fatigue strength assessment of laser-welded joints, Doctoral Dissertation. Espoo: Helsinki University of Technology, ISBN 978-951-22-9189-2, 2008 [4] Laitinen, R., Results of the hardness tests no. 580/01, 622/01, 468/02, 583/02. Ruukki: Raahe, 2003 [5] Laitinen, R., Kujala, P., Remes, H. i Nielsen, S., CO2-laser MAG Weldability of Laser Cutting LASER RAEX Steels, Hull Structural Steel Grade A and High Strength Formable Steel OPTIM RAEX 700 MC. In: Halmøy. Trondheim: E. Proceedings 9th Conference on Laser Materials Processing in the Nordic Countries, Norwegian University of Technology, 2003 [6] Radaj, D. i Helmers, K., Bewertung von Schweisverbindungen hinsichtlich Schwingfestigkeit nach dem kerbspannungskonzept. Konstruktion 49, p. 41–27, 1997 [7] Radaj, D., Sonsino, C. M. i Fricke, W., Fatigue assessment of welded joints by local approaches, 2nd Ed. Cambridge, England: Woodhead Publishing Limited, ISBN-13:978-1-85573948-2, 2006 [8] Eibl, M., Sonsino, C., Kaufmann, H. i Zhang, G., Fatigue assessment of laser welded thin sheet aluminium. International Journal of Fatigue 25, p. 719–731, 2003 [9] Sonsino, C., Kueppers, M., Eibl, M. i Zhang, G., Fatigue strength of laser beam welded thin steel structures under multiaxial loading. Elsevier, International Journal of Fatigue 28, 657–662, 2006 [10] Pinho da Cruz, J., Costa, J., Borrego, L. i Ferreira, J., Fatigue life prediction in AlMgSi1 lap joint weldments. International Journal of Fatigue 22, p. 601–610, 2000

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.106

Fatigue Life Tests of Explosively Cladded Steel-Titanium Bimetal Andrzej Kurek1,a, Adam Niesłony1,b 1

Opole University of Technology, Faculty of Mechanical Engineering Department of Mechanics and Machine Design ul. Mikołajczyka 5, 45-271 Opole, POLAND a

b

[email protected], [email protected]

Keywords: Fatigue, fatigue life, explosive cladding, clad.

Abstract. The paper contains a description of fatigue life tests of titan-steel bimetal. The study involved specimens made of bimetal which was a combination of S355J2 steel and SB G1 265 titanium, which was imposed in the material by explosive cladding method. The research shows that the fatigue life of specimens made of native material, derived from cladded plate is less than the life of specimens of titanium-steel bimetal. Introduction Energy of explosion has been applied for peaceful aims for a long time. Explosive materials can be applied for mineral crushing, demolition of buildings, fire suppression (blowing the flame away), stretching the belts and filling the airbags, shooting rackets or rescue cartridges, treatment of metallic and non-metallic materials. Intense energy released in a short time while explosion gives a possibility of realization of technological processes which could not be realized in typical conditions. The mentioned technologies using the explosion force have one common feature – suitable application of extreme values of velocity and pressure accompanying the explosion [1]. Explosive cladding allows joining materials with completely different properties which are difficult to obtain by means of other methods of joining [2]. Clad obtained with this method are materials of strongly gradient properties and they have complex joining zones. The clad materials are often applied in processing apparatus (chemical and power industries). Wide application of titanium and its alloys in power engineering (condensers, steam condensers, heat exchangers and steam turbines in power plants and thermal-electric power stations) causes that the problem of fatigue life of bimetallic clads, for example, those of steel-titanium type, becomes more and more important. As for many applications, fatigue life of clads is the most important parameter. Taking the specify of the problem into account, the tests are performed according to suitable standards, but the tests not included into the standards are also realized. The strength tests presented in this paper concern the metallic composite, so-called clad, obtained during so-called explosive cladding [3]. This paper is a continuation of the previous papers concerning strength of clad materials [4], and special attention is paid to fatigue life of such materials. The test results This paper presents results of the fatigue tests of the clad material being a joint of steel S355J2 with titanium SB265 G1, being the overlaid material, and the steel being a native material. Thickness of the overlaid material (titanium) was 6 mm, so the specimens of untypical, determined with FEM, shape and dimensions were used for fatigue tests. Next, the specimens were prepared. The specimens 100 x 9 x 9 mm were tested at the fatigue test stand existing at Department of Mechanics and Machine Design, Opole University of Technology. In the tested specimens, the bimetal joint was located exactly in a half of their section. Thus, each specimen was divided into two equal parts – steel and titanium (Fig.1a). Three series of tests were performed. The specimens were subjected to a traditional fatigue test, i.e. alternating bending under sinusoidally changing loading with constant amplitude of bending moment.

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TITANIUIM

Fig. 1 Scheme of attachment and loading of specimens (a) and a scheme of the loading method depending on the specimen attachment: perpendicular (b) or in parallel (c) [4] In the first series of tests, the specimen was located at the test stand in a way allowing to realize bending according to the scheme (Fig. 1b) on the plane perpendicular to the bimetal joint. In such a case, three fatigue tests were performed. They were presented also in the paper [4], and next seven more tests were realized. The results of those fatigue tests were presented in Fig. 3. The specimens are shown as loaded perpendicular to the bimetal joint plane. The next series of tests was performed under different configuration of the clade loading. The specimen was mounted at the stand in the way allowing the vector of the bending force to act on the plane of bimetal joint (Fig. 1c). The test results are presented in Fig. 4 where the specimens are loaded in parallel to the bimetal joint plane. In the third series of tests, the specimens were made of the native material (steel S355J2), and they were obtained from the sheets after the explosive welding. The used specimens had the same shapes and dimensions as the specimens tested in the previous series, so it was possible to relate the obtained results to the results of previous tests. These results are presented in Fig. 5. The fatigue test stand MZGS-100 allows to realize tests under a constant value of the bending moment, so the stress amplitudes used for plotting the presented fatigue characteristics have been determined by means of calculations with FEM, and the linearly-elastic model of the material was assumed. Fig. 2 presents the fatigue test results obtained for the steel-titanium bimetal, where σa is the maximum value of the stress amplitude, and Nf, is a number of cycles to the fatigue crack initiation under loading on the plane perpendicular to the bimetal joint (Fig. 1a – the first series of tests) including the stress amplitude used during the tests, occurring in both steel S355J2, and in titanium SB265 G1. Drop of the specimen stiffness by 30% was assumed as the main criterion of the fatigue crack initiation. Fig. 3 shows the fatigue test results for the steel-titanium bimetal for the stress amplitude σa depending on a number of cycles Nf, under loading on the plane parallel to the bimetal joint (Fig. 1c – the second series of tests) including the stress amplitude applied in tests, occurring in both steel S355J2 and titanium SB265 G1. The double-logarithmic curve shown on Fig. 4 presents the fatigue test results showing dependence between the stress amplitude σa (in steel) and

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the fatigue life Nf for two kinds of the clad material loading and the native material together with characteristics. Scatters of results are very important for fatigue tests, they also occur for homogeneous materials. In the case of bimetallic materials clad with the explosive method we have heterogeneous structure of the material, and differences in the test results can be much different even for similar loadings.

Fig. 2 Graph of dependence of the stress amplitude σa = f(Nf) occurring in steel and titanium

Fig. 3 Graph of dependence of the stress amplitude σa=f(Nf) occurring in steel and titanium Analysis of the obtained results From the tests it appears that titanium SB265 G1 was very often the material in which the crack initiation occurs. It concerns the bimetallic specimens loaded on the plane perpendicular to the joint. In spite of great difference in Young moduli of the joined materials, determining a lower stress value, titanium has a lower fatigue life. On the other hand, in the case of specimens subjected to bending on the plane perpendicular to the joint plane, the crack initiation occurred in the steel layer of the bimetal. Thus, we can state that for a given specimen shape, loading by alternating bending and the resulting stress distribution, the fatigue life of both materials is similar, and the final life is influenced by different random factors caused by technology of materials joining (inclusions,

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heterogeneity of the joining cone etc.). Thus, while designing of machine elements made of bimetals a designer should use fatigue characteristics including fatigue properties of the basic material and the overlaid material [5]. The Wöhler’s curve for the considered material (see Fig.4) is an example of such characteristic. The expected fatigue life of both steel and titanium should be checked, and the lower fatigue life should be assumed as the life of the element.

Fig. 4 Graph presenting dependence of the stress amplitude σa=f(Nf) If the tested specimens are loaded in parallel to the joint plane, the difference of stress amplitude occurring in steel and titanium is greater, so the crack initiation process will proceed in the steel layer at first. From the tests it also appears that the fatigue life of the specimens made of the native material obtained from the clad is lower than the fatigue life of steel-titanium specimens. Moreover, it has been proved that a change of a way of the bimetal loading influences its fatigue life. It can be also seen that the specimen loaded in parallel to the joint plane has a little higher fatigue life. References [1]

[2] [3]

[4]

[5]

Niesłony A., Kurek A., Bański R., Čižek L. 2010. Static and fatigue tests of explosively cladded materials – titanium-steel, Scientific Papers Opole University of Technology, Series Mechanics z. 97, nr 337/2010 (in polish) Mckenney C.R., Banker J.G. 1971. Explosion-Bonded Metals for Marine Structural Applications, Marine Technology, pp. 285-292 Szulc Z., Gałka A., Bański R., Pocica A. 2007. Explosive cladding with titanium - the development of technologies and areas of industrial applications, XII Scientific and Technical Welding Conference "Progress, innovation and quality requirements of welding processes,", Międzyzdroje 29-31.05.2007, s. 13-14 (in polish) Kurek A., Żok F., Niesłony A., Bański R. 2011. Strength tests of steel-titanium explosively cladded bimetal, XXIV Conference "Problems of Development of Machines", s. 47-49 (in polish) A. Niesłony and A. Kurek, “Influence of the Selected Fatigue Characteristics of the Material on Calculated Fatigue Life under Variable Amplitude Loading,” Applied Mechanics and Materials, vol. 104, pp. 197–205, Sep. 2011.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.110

Simulation of Tensile Test of The 1/2Y Welded Joint Made of Ultra-High Strength Steel GALKIEWICZ Jaroslaw1, a 1

Politechnika Świętokrzyska w Kielcach, Wydział Mechatroniki i Budowy Maszyn, A. 1000-lecia PP 7, 25-314 Kielce, Poland a

[email protected]

Keywords: welds, undermaching, heat affected zone, ARAMIS.

Abstract. The detailed analysis of the tensile properties of the 1/2Y welded joint made of ultra-high strength steel S 960 QC was carried out. The analysis concerned various parts of welded joint and has been carried out with the help of both experiment and numerical simulation. Results were compared with the data measured using the ARAMIS system. The purpose of the analysis was to provide the constitutive relations for detailed analysis of the welded joint by finite element method. Introduction The analysis of the welded joint strength is a seemingly simple task. It is probably so when the conventional structural steels are welded. It is certainly not so when the ultra-high strength steel are welded. It is not easy to find a proper a high-strength binder to make the joint as strong, or stronger than the welded material (BM). One may observe a complex distribution of the mechanical properties as well as the secondary stresses within the weld material (WM) and heat affected zone (HAZ). Classical structural integrity assessment proposed in the procedures such as FITNET [1, 2] or API [3] may not be applicable. Often the question arises which materials should be taken into account to decide if the problem of interest concerns undermaching (M<1), overmatching (M>1) or evenmaching (M=1)situation. M=(σeWM)⁄(σeBM) where σe is a yield strength. Namely, the FITNET does not take into account the HAZ in the analysis. It not necessarily is correct in the case of the ultra-high strength steels. It may happen in the case of ultra high strength steel that the WM is the weakest one. In such a case we face the undermatching situation. This problem will be discussed in the paper. In such a case it turns out that the HAZ influences strongly the behavior of the whole welded joint [4],[5]. The micro-structure and the properties of the HAZ change in transverse direction and through the thickness of the joint. Another problem in the strength analysis is to decide on the HAZ size [6]. In the paper the finite element analysis of the welded joint is presented. The undermatching welded joint is considered. Computations were carried out using the ADINA code. Various configurations (models) of the joint were analyzed and compared from the point of view of the benefits reached. Both two-dimensional and tridimensional models were used. Mechanical properties of the welded joint First the surface of the welded joint was polished and observed under the microscope in order to define roughly different zones in the welded joint. In order to define the HAZ more precisely the hardness measurements were made (Fig.1). Then, the mini-specimens (rectangular section 2x4mm and initial length of extensometer 25mm) for tensile tests were machined from different zones of the welded joint. The results of tensile tests are shown in Fig.2 [7].

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Fig.1. Results of the hardness measurements

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Fig.2. Results of tensile tests. Different curves concern the specimens depicted in Fig.1

In the next step the specimens of the cross-section 6x3mm, and the axis perpendicular to the weld were cut out, loaded and the strains were measured using ARAMIS video-extensometer [8]. Then results recorded by ARAMIS were used to produce the constitutive relations in different zones of the welded joint. Numerical simulation of the welded joint based on the data recorded in the tensile test The purpose of this step of analysis was to simulate numerically the behavior of different zones of the stretched welded joint. Computations were made using 3D models field with the four-nodes elements of the tetra-type. To reduce the number of nodes only a half of the specimen was modeled, due to the assumed symmetry. Geometry of the joint is shown in Fig.3.

Fig.3a. Specimen with the boundary conditions and load

Fig,3b. Three different zones of the welded joint

Boundary conditions (Fig. 3a) are introduced as a displacement applied to the selected nodes at the selected cross-section. Displacements were measured by ARAMIS. The nodes at the opposite side of the specimen are fixed in all directions. First, the simulation was carried out before the necking of the weakest element. Until this moment one can assume that the stresses are relatively uniform in the specimen. The strains measured at different points were later used to estimate the constitutive relationships, locally in different zones.

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However, in the first model numerically tested, the constitutive relationships obtained using the micro-specimens (Fig.2) were used. The BM is characterized by curve A4, the WM is characterize by curve A1 and the HAZ by curve A2. The width of the HAZ was assumed 1.8mm. The tensile properties are summarized in Table 1.

Material BM HAZ WM

Table 1. Tensile properties used in the simulations in Model 1. E [GPa] ε0 σ0 [MPa] 5.62E-03 1090 194 3.13E-03 607 194 2.60E-03 505 194

ν 0.3 0.3 0.3

First, the global responses of the specimen to the external loading measured experimentally and computed numerically are compared. The result is shown in Fig. 4a.

Fig.4a. Comparison of the response of the Fig. 4b. Comparison of the strains measured and specimen to the external loading recorded computed along the selected line within the experimentally and computed numerically; welded joint (horizontal line in Fig. 5a) Model 1 The responses measured and computed are close each other. However, the trend of changes measured by the finite element method is not satisfactory. The curves do not converge. One may expect that when the external loading increases the difference between computed and measured values will also increase. The strain distributions were also compared. Qualitatively the strain maps are shown in Fig.5. They look similar to each other. However, when the strain values are computed and measured along the same line (horizontal line in Fig. 5a) the differences are observed and at certain domains are too large. Next, the stress-strain curves recorded in three different points, shown in Fig.5, were compared. One curve was computed numerically, using constitutive relationships received by testing the micro-specimens, the second curve was recorded experimentally (strains were measured directly by ARAMIS, stresses were computed dividing the external force be the actual cross-section). The results are shown in Fig.6 and they are not satisfactory.

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b.

a. Fig. 5a. Strain map computed by FEM

Fig. 5b. Strain map measured by ARAMIS

a. b. c. Fig. 6. Comparison of the stress-strain curves: a) point close to the weld root, b) point close to the specimen axis, c) point at the weld face. More precise numerical simulation of the welded joint In this Section more precise simulation of the welded joint is presented. The volume of the weld was divided into ten strips of different tensile properties (Fig.7). The tensile properties of the materials within each strip was determined using ARAMIS.

Fig. 7. Different zones within the weld material

Fig.8. Tensile curves of different zones in WM and experimentally measured tensile curves within BM and HAZ

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Numerical procedure remains the same as in Model 1, except constitutive equations within the WM. Tensile curves are shown in Fig.8. The global responses of the material measured and computed are compared in Fig.9a. Strains along the same line as in the Model 1 are compared in Fig, 9b. Force-elongation curves are not as close as in Fig.4a. However, now, they are almost parallel. Computed strains are closer to the measured one than in the Model 1.

Fig. 9a. Comparison of the response of the Fig. 9b. Comparison of the strains measured and specimen to the external loading recorded computed along the selected line within the experimentally and computed numerically; welded joint; Model 2 Model 2 In the Model 2 the local stress-strain curves measured in computed in the selected points converge very well. It can be observed in Fig. 10.

a.

b. c. Fig. 10. Comparison of the local stress-strain curves: a) point close to the weld root, b) point close to the specimen axis, c) point at the weld face

Numerical simulation of the welded joint - tensile properties averaged within the HAZ Model 2 provided better results of simulation than Model 1. Thus, in the next step the HAZ was differently modeled. Instead of tensile characteristics of the HAZ, recorded experimentally, using micro-specimens, the constitutive relationship recorded by ARAMIS was used. Quite arbitrarily the representative point within the HAZ was selected. It was assumed that within this point the constitutive relation is representative (average) for the whole HAZ. Such a curve is shown in Fig. 11.

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Fig. 11. Stress-strain curves used in Simulation according to Model 3. The new curve representing the HAZ is the red curve. The previous curve representing the HAZ is denoted by rhombuses. Numerical simulation according to the Model 3 provided a good agreement between the global responses of the welded joint recorded experimentally and computed by FEM (Fig.12a). Strains are distributed along almost parallel curves. However, these curves do not converge (Fig.12b). The local stress-strain curves recorded and computed in three selected points converge perfectly (stressstrain curves are similar to Figs. 10a-c).

Fig. 12a. Comparison of the response of the Fig. 12b. Comparison of the strains measured specimen to the external loading recorded and computed along the selected line within the experimentally and computed numerically; welded joint; Model 3 Model 3

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Fig. 13a. Comparison of the response of the specimen to the external loading recorded experimentally and computed numerically using plane strain and plane stress models

Fig. 13b. Comparison of the strains measured and computed along the selected line within the welded joint. Computations were made using plane strain and plane stress models of the welded joint

Numerical simulations using plane strain and plane stress models of the welded joint In the next step, the numerical simulation was made using simple plane stress and plane strain models of the analyzed welded joint. The material data tested in Model 3 were used. Results of the analysis along with the experimental results are shown in Figs 13a and 13b. One may notice a very good agreement between experimental and numerical results for the plane stress model of the specimen. Certainly, this observation can be used in the simulation of other welded joints of more complex shape. Conclusions In the paper numerical simulation of the welded joint is presented. The undermatching case was analyzed. It may happen when the ultra high strength steel is welded. The structure of the welded joint is very complex. Modeling of the WM and HAZ is very difficult because of varying microstructure and tensile properties. It was assumed that more detailed map of the tensile properties should be limited to the weakest zone. In the analyzed case it was the WM. In other zones the "average" properties were adopted. They were either measured using the micro-tensile-specimens or estimated using video-extensometer ARAMIS, which was useful in the analysis. The global and local responses of the welded joint received by the numerical simulations were satisfactory if the HAZ tensile properties were estimated by ARAMIS technique at the selected "representative average" point. In this case, the computed numerically strain distribution along selected lines is similar to the experimentally measured strains. However, the computed strains for final stage of loading are always several per cent lower than the strains measured by ARAMIS. The numerical results received by using the plane stress model of the welded joint provide good results, which are close to the measured experimentally. Acknowledgements: The financial support from Polish Ministry of Science and Higher Education under contract N N501 199640 is gratefully acknowledged. The ARAMIS testing system was purchased within the European Union Grants to Kielce University of Technology: Modin II and Molab.

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References [1] FITNET Report, (European Fitness-for-service Network). Edited by M.Kocak, S.Webster, J.J.Janosch, R.A.Ainsworth, R.Koers, Contract No. G1RT-CT-2001-05071, (2006) [2] Neimitz A., Dzioba I., Graba M., Okrajni J.: Ocena wytrzymałości, trwałości i bezpieczeństwa pracy elementów konstrukcyjnych zawierających defekty, Wydawnictwo Politechniki Świętokrzyskiej, 2008 [3] American Petroleum Institute, API 579: Recommended practice for fitness-for-service. Washington DC, 2000. [4] Rodgigues D.M., Menezes L.F., Loureiro A., The influence of the HAZ softening on the mechanical behavior of welded joints containing cracks in the weld metal, Engineering Fracture Mechanics, 71, 2053-2064, 2004 [5] Rodgigues D.M., Menezes L.F., Loureiro A., Fernandes J.V., Numerical study of the plastic bahaviour in tension of welds in high strength steels, International Journal of Plasticity, 20, 1-18, 2004 [6] Zhang Z.L., Hauge M., Thaulow C., Odegard J., A notched cross welds tensile testing method for detrmining thrue stress-strain curves for weldments, Engineering Fracture Mechanics, 69, 353366, 2002 [7] Gałkiewicz J., Dzioba I., Pała R., Właściwości mechaniczne złączy spawanych z ultra wysokowytrzymałych stali ferrytycznych, XXIV Sympozjum Zmęczenie i Mechanika Pękania, Bydgoszcz - Pieczyska, maj 2012 [8] ARAMIS User Manual, GOM mbH

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.118

Identification of efficient material S-N curve for steel welded joints Łukasz BLACHA a, Aleksander KAROLCZUK b Opole University of Technology, Faculty of Mechanical Engineering, Department of Mechanics and Machine Design, ul. Mikołajczyka 5, 45-271 Opole, Poland a

b

[email protected], [email protected]

Keywords: welded joints, fatigue failure, weakest link concept, failure probability analysis efficient material.

Abstract. The paper presents results of simulation and experimental tests carried out in order to identify the S-N curve introduced for the evaluation of fatigue life for various types of steel welded joints. Such curve is explicitly defined through the corresponding values of Cf and mf parameters. The identification process involved finite element analysis, undertaken for different discrete models varying in notch radius. The obtained results served as an input into iterative calculations carried for different values of Cf and mf parameters. Such calculations were aimed at minimization of the estimator in the form proposed for the process of identification. Introduction Fatigue of steel welded joints in the as-welded condition is an issue of a complex matter which is being attributed to the presence of geometric and material inhomogeneity. Variability in shape of the weld face, especially the notch radius, implies difficulties during the assessment of stress fields. Changes in material microstructure are due to localised melting of the material, corresponding to the presence of heat affected zone (HAZ), transient between the microstructures of parent material and weld. Within the HAZ the following microstructural areas tend to develop: recrystallization area, normalization area, overheated area and the area of fusion. Widmannstätten microstructural pattern observed in the overheated zone indicate HAZ as the area where the damage process of the entire structure is initiated. Scatter of weld profile parameters and inhomogeneous stress fields resulting from varying material structure substantiate the probabilistic approach to fatigue assessment of steel welded joints, e.g. the weakest link concept. The aims of this paper are to: (i) define the efficient material S-N curve, formulated as a part of probabilistic computational model for wide variety of steel welded joints; (ii) introduce the algorithm for identification of such curve; (iii) present results derived from identification process. Probabilistic computational model A computational model based on extension of the weakest link concept into area of steel welded joints was presented in the literature [1]. The weakest link concept is in the group of non-local stress approaches to fatigue assessment, in the sense that it utilizes the complete stress fields instead of the local values. Computational model defines the structural element as a serial system of smaller, independent elements, having its origin in the reliability theory. In the proposed method division of structural element into smaller elements is performed within the frames of meshing operations during the finite element (FE) model generation. Each element is described by failure probability Pf defined by the Weibull distribution. Once the Pf probability for each element is known, application of the weakest link concept in the volume of the material defines the cumulative failure distribution function Pf on a certain loading level as a function of number of cycles to failure N:

Dariusz Skibicki

−

Pf ( N ) = 1 − e

p 1  log N    dV ∫ V0  H  V

,

119

(1)

where: V – material volume, V0 - reference volume, N - number of cycles to failure, H – scale parameter, p – shape parametr, p = 66.82. This form of distribution (1) is described by two parameters: shape parametr p and scale parameter H. The p parameter is interpreted as an exponent corresponding to the volume effect, defined as decrease in fatigue life with increasing volume of the joint. The shape parametr p was determined on the basis of two series of fatigue tests on welded elements submitted to fully reversed loading of a tension-compression type [1]. Volume of the joint in second serie of tests was set at the level of eight times smaller than in the first serie. Identification of the shape parameter p was based on comparison between S-N curves derived from each serie of tests. Obtained results allowed to estimate the value of p = 66.82. The shape parameter is considered as a constant value within the certain range of number of cycles to failure and under the assumption of being independent from stress range ∆σ. Such assumption appears to be valid for cycles to failure that are not exceeding values close to the fatigue strength. The H parameter normalizes logN variable for a given loading level. For a certain range of cycles the scale parameter can be described by the form of H = logNf , where Nf is the number of cycles to failure of an efficient material, for a certain value of Ps: log N f = log C f − m f log ∆σ ,

(2)

where: Cf, mf - fatigue parameters of efficient material, ∆σ - stress range according to the chosen multiaxial failure criterion. In case of joints in as-welded condition grade of the steel (parent material) is of secondary importance for fatigue strength [2]. This feature arises from the joining process: consumable electrode arc welding in shielding gas atmosphere. Cyclic properties of the overheated material sections dominate over the properties of parent material. Cyclic behaviour of a material with this microstructure is described by the scale parameter: S-N (∆σ - Nf ) curve for efficient material. Identification of efficient material S-N curve The well established approaches to fatigue design of welded structures presented in numerous standards and recommendations [3, 4] are based on a series of classified S-N curves corresponding to each class of joint geometry. S-N curves in the form as in recommendations from International Institute of Welding (IIW) [3] are identified by the stress range ∆σ at 2.106 cycles, known as the FAT class. Each curve corresponds to survival probability Ps equal of 95%. The process of identification of the Cf and mf parameters proceeds through minimization of the E(Cf,mf) estimator for efficient material parameters. It leads through iterative selection of the pair of values that determine Eq. 2 and utilizes FAT curves and the nominal stress approach [3]. The idea of identification requires knowledge on number of cycles to failure of a welded joint, obtained for specific value of survival probability; this assumption can be ideally met by application of FAT curves classified by IIW. The performed process was based on calculations that were carried out according to the algorithm shown in Fig. 1. Identification estimator for the efficient material parameters is described by Eq. 3:

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E (C f , m f ) = 0,95 − e

− (log N FAT



1 log C − m f f log ∆σ 

) p ∫  V

p

  dV  

,

(3)

where: NFAT – number of cycles to failure according to the nominal stress approach; referential volume V0 (Eq. 1) for efficient material is assumed to be equal of 1 mm3. Generation of FE model representative for the chosen welded joint classified in nominal stress approach Determination of through-thickness stress field along the weld line and corresponding to the chosen nominal stress ranges Transformation of stresses according to the chosen multiaxial failure criterion Generation of the pair of values for Cf and mf parameters Determination of E(Cf,mf) identification estimator Comparison of estimator values derived for two levels of nominal loading Identification of Cf and mf parameters according to the results from minimization of E(Cf,mf) identification estimator

Fig. 1. Algorithm for identification of the efficient material S-N curve parameters In the assumptions underlying to the model, grade of the steel elements being joined, as well as their geometry, are not affecting the efficient material characteristic. From the standpoint of validation, the identification process was carried for two welded elements of a different geometry: transverse butt weld (type a joint) and a transverse stiffener (type b joint). Geometry of the analysed elements is shown in Fig. 2. In the nominal stress approach [3] these elements are classified as structural details of no. 213 and 511, respectively. (a)

(b)

Fig. 2. Geometry of the investigated elements: a) transverse butt weld [3], b) transverse stiffener [3] Finite element analysis was carried out for FE models of a geometry being representative for both welded elements shown in Fig. 2. Models for the analyzed elements are shown in Fig. 3.The elements were subjected to axial loading. Analysis was undertaken for the loadings corresponding to the stress ranges at the level of ∆σ = 250 [MPa] and ∆σ = 300 [MPa], imposed for the model of 10 variations in notch radius ρ, located along the weld line (ρ = {0.1-1.0} [mm]). Models were meshed by Hex8 [5] elements. Global edge length of the elements in the notch section (section I, Fig. 3) was set to be equal of 0.1ρ. The linear elastic material properties for steel with Young’s modulus equal of 201 GPa and Poisson’s ratio equal of 0.3 were used throughout the calculation process.

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Fig. 3. Partial FE models, representative for the following welded elements: a) transverse butt weld, b) transverse stiffener Calculated components of ∆σij(x,y,z) stress fields acting in vicinity of the welded joint were transformed according to the maximum principal stress criterion. In order to compare the value of the estimators derived for two loading levels, the following transformation formula was used: E = E(21) + E(22 ) ,

(4)

where: E(1), E(2) – estimator values derived for 1st and 2nd loading level, respectively. Results obtained from the performed calculations were further used to determine the map of values of identification estimator E for the efficient material Cf, mf parameters. Example map is shown in Fig. 4. The derived maps allowed to initially assess the location of minimun of the function (4). Retrieved data served as a starting point for the process of minimization through the downhill simplex optimization technique proposed by Nelder and Mead and implemented in MATLAB environment (R2011b). Identification process involved finite elements within the distance x = t/3 from the root of the notch, where t is the thickness of welded plates (in both cases t = 10 mm). The log(Cf), mf parameters extracted for each variation of FE model and joint type are presented in Table 1.

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Fig. 4. An example map of values of identification estimator E for efficient material Cf, mf parameters Table 1. Efficient material log(Cf), mf parameters for different values of notch radius ρ and joint geometry ρ, [mm] 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1

Type a joint (butt weld, IIW no. 213)

Type b joint (transverse stiffener, IIW no. 511)

E, ·10-7

log(Cf)

mf

E, ·10-7

log(Cf)

mf

6 6 7 2 5 6 7 4 9 3

13.77 13.70 13.68 13.66 13.66 13.65 13.65 13.65 13.65 13.65

2.98 3.04 3.07 3.10 3.12 3.14 3.16 3.17 3.19 3.20

7 6 1 5 2 6 7 4 6 7

13.68 13.71 13.69 13.69 13.68 13.67 13.66 13.66 13.65 13.65

3.11 3.11 3.12 3.13 3.13 3.14 3.14 3.15 3.15 3.16

Influence of proper selection of Cf and mf parameters on the estimated fatigue life was tested through sensitivity analysis. Sensitivity was estimated through total differential of Eq. (2): ∆N f =

1 m (∆σ ) f

∆C f +

C f ⋅ ln (∆σ ) m (∆σ ) f

∆m f ,

(5)

where: ∆Cf – overestimation of Cf parameter, ∆mf – overestimation of mf parameter. Calculations were carried out for stress range ∆σ = 100 MPa and Cf, mf values as referred to the FAT curve for structural details no. 213 and 511 (transverse butt weld and transverse stiffener, respectively). Results were used to generate the map of sensitivity assessment in the form shown in Fig. 5. The mf parameter has proved to have greater influence on the resulting number of cycles to failure; 10% overestimation in the value of mf parameter demonstrated 50% overestimation in fatigue life N.

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Fig. 5. Map of sensitivity assessment, where: ∆mf – relative overestimation of mf parameter; ∆Cf – relative overestimation of Cf parameter; ∆N – relative overestimation of fatigue life Summary Based on the results of fatigue tests and iterative calculations, fatigue parameters were extracted that determined the efficient material ∆σ - N curve, introduced as a representation of the microstructure in overheated area of a welded joint. Calculations were carried for two FE models, in 10 variants each, representative for two axially loaded steel welded joints. From the results, the following conclusions were drawn: - the mf and logCf parameters are changing with increasing value of notch radius ρ; this effect is evident since when the notch rounding becomes smoother stresses in the root of the notch are approaching nominal values when ρ →∝ and mf and logCf parameters will gain their values according to the nominal ∆σ -N curve (in the analysed cases impossible due to the modeled shape of joint); - for the notch radius ρ = {0.1-0.2}[mm] logCf value is identical for both joint types while mf values differ by 1.25% which leads to less than 10% change in fatigue life N; - the undertaken analysis over the values of efficient material parameters yielded the following results for the variant of notch radius being modeled as ρ = 0.1 [mm]: logCf = 13.65 and mf = 3.18 (average value derived from two variants, leading to several percent deviation in fatigue life in the analysed cases). Acknowledgements This paper is realized within the framework of research project No. DEC-2011/01/N/ST8/02566 funded by the National Science Centre in Poland.

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References [1] Ł. Blacha, A. Karolczuk, Application of the weakes link analysis to the area of fatigue design of steel welded joints, Fifth International Conference on Engineering Failure Analysis, Elsevier Publishing, The Hague, The Netherlands, 1-4 July 2012. [2] C. M. Sonsino, H. Kaufmann, G. Demofonti, S. Rifisculi, G. Sedlacek, C. Müller, F. Hanus, H. G. Wegmann, High-Strength steels in welded state for light-weight constructions under high and variable stress peaks, ESCC Steel Research Programme, CSM – Roma, LBF – Darmstadt, Published by the European Commission, Brussels 1999. [3] A. Hobbacher, Recomendations for fatigue design of welded joint and components, IIW document XIII-2151-07/XV-1254-07, Paris, 2007. [4] American Bureau of Shipping, Guide for the fatigue assessment of offshore structures, ABS document, Houston, 2003 - 2010. [5] MSC/PATRAN, MSC. The MacNeal-Schwendler Corporation, ver. 2005.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.125

Residual stresses in steel-titanium composite manufactured by explosive welding Aleksander Karolczuk1,a, Krzysztof Kluger1,b, Mateusz Kowalski1,c, Fabian Żok2,d and Grzegorz Robak Author1,e 1

Opole University of Technology, ul. Mikołajczyka 5, 45-271 Opole, Poland 2

Z.T.W Explomet s.c., ul. Oświęcimska 100H, 45-641 Opole, Poland

a

b

c

d

[email protected], [email protected], [email protected], [email protected], e [email protected]

Keywords: Residual stresses; thermal stresses; explosive welding; fatigue; ratcheting

Abstract. The main aim of the paper is determination of residual stresses in explosively welded steel-titanium bimetal. The analysis considers two bimetallic specimens: before and after the heat treatment. In residual stress determination the hole drilling method along with finite element analysis were applied. The results show different residual stress states depending on the heat treatment. The obtained results are confirmed by thermal stress calculation. Introduction Increasing demands in area of efficiency, strength, reliability and fatigue life of modern engineering structures requires application of non-typical materials. Those non-typical materials must satisfy different mechanical properties such as, e.g. high corrosive and temperature resistance along with high strength. In order to meet all the requirements the bimetallic composite are often taken into account. Explosive welding is one of manufacturing technology that allows joining structurally dissimilar metals such as steel and titanium [1]. The joint of materials is obtained during detonation in which one metal (the flyer) undergoes a large deformation. Typical interface in case of explosive welding has wavy shape although flat shape is also possible. The quality of obtained joint depends on many factors [2]. The most important are as follows: detonation velocity, standoff distance between metals, structural and mechanical properties of bonded materials. The joint is obtained only under proper range of welding parameters (welding window) [3]. Detonation imposes a high velocity collision of metals in which a high velocity jet is formed. This air jet swept away the oxide films and other impurities making the bond possible. The research of joint area shows that explosive welding is a solid state metal joining process [4]. Because of unique advantages the bimetals are used in chemical and energy process apparatus, e.g. for heat exchangers, reactors, pipes and others devices that require high resistance against applied fluids [1, 4, 5]. Sensitivity to inappropriate selection of welding parameters is a disadvantage of the process. The dynamic character of explosive welding still prevents full understanding of its mechanisms. The research of bimetallic composites is focused on inhomogeneous structure and optimization of welding process. The problem of monotonic mechanical properties is also raised [6, 7]. Unfortunately, there is not enough information concerning cyclic properties of bimetallic composites. Some limited information could be found in [8, 9]. However, the tests presented in [8] concern only cyclic three point bending and influence of heat treatment on fatigue life of titanium-steel composite. The cyclic stress-strain relation was not investigated. The specific character of explosive welding process is reflected in micro-structural properties [10]. The metallographic research exposes changes in microstructure of subsequent layers of bimetal [10]. Large deformation that undergoes the cladding metal during explosive could result in high residual stresses [5, 11]. The residual stresses could have detrimental effect on fatigue life. Preliminary fatigue tests [12] performed under tensioncompression loading of explosively welded steel-titanium components demonstrate two unexpected fatigue phenomena: (i) ratcheting (accumulation of strain) and (ii) cyclic instability (fig. 1).

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Fig. 1. Preliminary fatigue test results of steel-titanium bimetal (S355J2+N and Titanium ASTM Grade 1) under push-pull loading with zero mean value of force. Hysteresis loops registered under different ratio of current and total number of cycles, n=N/Nf In or order to explain these effects the elastic-plastic simulations are planed but firstly the initial stress conditions must be known. Thus, the residual stresses must be determined. In the present paper the procedure of measurement of residual stresses is presented. Research setup In the residual stress measurement two specimens were used: S1 and S2 (210x180x46 [mm]) made of bimetal obtained by explosive welding process of steel (S355J2+N) and titanium (ASTM Grade 1) plates. Titanium plate with thickness of 6 [mm] was welded to the base plate made of steel with thickness of 40 [mm]. The specimen S1 and S2 were cut out from one welded plate (4330x3150x46 [mm]) designated for a heat exchanger construction. The place of cutting is presented in figure 2.

Fig. 2. Photograph of explosively welded plate with visible place where the specimens were cut out The specimen marked by S1 was cut directly after the welding process and specimen marked by S2 was cut after the heat treatment and flattening process of the original plate. The heat treatment consisted in heating for 90 minutes at 600[oC] and then cooling down to 300[oC] at cooling velocity 100[oC/hour]. The final cooling was performed in the air outside the furnace. The chemical composition and basic mechanical properties of bonded metals are presented in tables 1 and 2, respectively. Table 1. Chemical composition of S355J2 steel (EN 10025-2:2004) and titanium ASTM Grade 1 Steel S355J2 (EN 10025-2: 2004) Chemical element: C Si Mn P S Cu In weight, %: 0.23 0.60 1.70 0.035 0.035 0.45 Titanium Grade 1 (ASTM Grade 1) Chemical element: C Fe H N O Ti In weight, %: 0.10 0.20 0.015 0.03 0.18 99.5

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Table 2. Mechanical properties of S355J2 steel (EN 10025-2:2004) and titanium ASTM Grade 1 Material Re, MPa Rm, MPa E, MPa G, MPa ν, A5, % * * S355J2 382-395 598-605 220000 84000 0.3 24-34* Grade 1 189-215 308-324* 100000 38000 0.37** 43-56* (R02)* * - according to manufacturer certificates, **- in titanium after welding Where: E, G – Young and Kirchhoff modules , Re (R02) – yield stress, Rm- ultimate strength, ν - Poisson ratio, A5 - elongation Measurement of residual stresses. The residual stresses were measured by incremental holedrilling method. The method consists in strain relaxation measurements on the specimen surface around progressive drilled hole. The strain relaxations were measured by a special type strain gauge rosette [13]. In figure 3a a scheme of applied type A strain gauge rosette is presented. The experimental strain results were obtained by using TFrw-1.5/120 gauge [14] with a gauge circle diameter of D = 5.6 [mm] (fig.3a). The TFrw-1.5/120 gauge is characterized by resistance equal to 120 [Ω] and strain sensitive coefficient k = 2.15. Strain values were registered with SCXI-1600 strain gauge (National Instruments manufactured) and LabView Signal Express application. In a hole drilling process a helix drill BOSH HSS-R with diameter 1,5 [mm] (diameter tolerance h8) was used. In figure 3b an example of specimen during drilling operation is presented. (a)

(b)

Fig.3. (a) Scheme of strain gauges type A, (b) Photo of strain gauge under drilling Constant drilling speed equal to 2850 rpm was used. Drilling depths were measured with the help of dial gauge (±0,01 [mm]) which was coupled with drill chuck. Since the value of calibration constants used in standard procedure described in [13] are unknown the finite element method (FEM) was applied to calculate residual stresses (Comsol 3.5). Values of residual stresses in material are determined based on changes in elastic part of strains around a drilling hole. Because the strain gauge location is shifted from the drilled hole the strain relaxation does not correspond directly (through Hook's law) to residual stresses. This discrepancy depends on size and location of strain gauge, and also on the drilling hole depth. These factors could be taken into account using FEM simulation (correction parameter p). The applied algorithm for residual stresses determination is as follows: - strain measurements in three directions (A, B, C) for drilling hole depth h, - change of strain signs, - calculation of the principal strains ε1,2, - calculation of principal stresses σ1, σ2 from the Hook’s law for a plane stress state, - multiplication of the principal stress values by correction parameter p adequate to the hole depth h. Values of correction parameter p were determined with FEM simulation for different values of the hole depth h. The correction parameter value p is simply calculated by division of simulated value of residual stress σr and calculated value of stress σcal, where σcal is computed using the Hook's law

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for a plane stress state from determined by FEM strain relaxation. The strain relaxation determined by FEM is an averaged value of strain relaxation over the area of virtual strain gage (fig. 4). In order to increase the computations the real shape of drill is modeled (fig. 4). Figure 4 presents an exemplary strain distribution εx in cross section of modeled specimen (solid 3D) for the hole depth of 1.5 [mm] (where x – direction of simulated residual stresses, σx = 1 [MPa]). The position and length of the virtual strain gauge along with computed stress field σx is presented in figure 5a. Figure 5b presents exemplary strain distribution εx over the area of the virtual strain gauge. It is evident that the strain field over the area of the virtual strain gauge must be averaged for practical usage since the strain gradient is meaningful.

Fig. 4. Exemplary strain field εx in cross section of modeled specimen along with virtual strain gauge location (h=1.5 [mm]) (a)

(b)

Fig. 5. (a) Exemplary stress field σx along with virtual strain gauge location; (b) Strain distribution εx on specimen surface along axis symmetry of virtual strain gauge For a given (simulated) value of residual stress (σr = 1 [MPa]) several FEM models were created with different hole depth h. For each h value the correction parameter p was computed (fig.6). Furthermore, the distribution of correction parameter p over h was investigated separately for steel and titanium using appropriate linear elasticity material constants which results with the same relation p(h).

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Fig. 6. Distribution of correction parameter p Measured and computed results. Figure 7 presents results of strain measurements in three directions (A,B,C) and residual principal stress (σ1, σ2) computations for specimen S1 (before the heat treatment) and for specimen S2 (after the heat treatment). The strain measurements were performed in two places (point I and point II) for each specimen. The distance between two points where strain gage rosette were attached is around 55 [mm] (fig. 8). (a)

(c)

(b)

(d)

Fig. 7. Measured strains in three directions (A,B,C) in function of depth of drilled hole and calculated principal stresses: (a) without the heat treatment – point I, (b) without the heat treatment – point II; (c) after the heat treatment – point I, (d) after the heat treatment – point II.

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(a)

(b)

Fig. 8. (a) Photo of the specimen S1 with two strain gauge rosettes; (b) Photo of the specimen S2 with two strain gauge rosettes For the specimen S1 without the heat treatment measurements and calculations show tensile stresses in titanium. Whereas, in the specimen S2 that is after the heat treatment the residual stresses are negative (compressive). The applied algorithm of residual stress calculations assumes homogeneous field of stresses. The evaluated residual stresses depend on the depth h of drilled hole (fig.7) that needs a proper interpretation. The value of stresses σ(h) for given depth h is the mean value of residual stresses in material layer of thickness h. The values of residual stresses stabilizes for all measured points for drilling depth of h = 3 mm. The stabilized values are the most representative regarding to fact of large sensibility of p parameter to the hole depth h up to 2 mm (fig. 6). The applied residual stress calculations do not consider the influence drilling operation on stress state which can be significant because of the low yield stress of titanium (tab. 2). However, the introduction of persistent deformations as a result of drilling process has the same effect for all measured points. The qualitative estimation of the residual stresses is correct, i.e.: (i) tensile stresses for titanium without the heat treatment, (ii) compressive stresses for titanium after the heat treatment. The tensile stresses appearance in titanium without the heat treatment can be explained by detailed analysis of explosive welding (fig. 9). The titanium layer undergoes tension in x axis direction as results of detonation pressure because of standoff distance existence between bonded layers (fig. 9). Whereas, appearance of the compressive stresses after the heat treatment is result of different thermal expansion coefficient for bonded metals: for titanium αTi = 8,6·10-6 [1/K] and for steel αSt = 13,0·10-6 [1/K]. The heat treatment causes titanium recrystallization and stress relaxation but only at temperature 600 [oC]. During cooling process down to temperature 20oC titanium and steel reduces theirs volumes. Because the steel has larger thermal expansion coefficient its change of dimensions is larger that initiates the compressive stress for the titanium and tensile stress for the steel. Assuming: (1) the linear dimensions change range (∆L = L·∆T·α); (2) the elastic strain state; (3) deformation compatibility; (4) a plane stress state; it is calculated that as a result of temperature decrease about 580 [oC] stresses for the titanium are σx=σy= -377 [MPa] and for the steel σx=σy=56 [MPa]. The stresses in the titanium exceed the yield stress which means that the above mentioned assumption of the elastic strain state is not satisfied. The assumption of constant thermal expansion coefficients α (assumption 1) is also only an approximation. However, the calculated signs of thermal stresses correspond to measured-computed signs of residual stresses.

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Fig. 9. Scheme of the explosive welding process Summary Basing on the obtained experimental results and performed theoretical analyses the following conclusions are drawn: (i) The performed heat treatment of bimetal steel - titanium does not remove residual stresses but only changes their values and directions. (ii) After the heat treatment the residual stresses in the titanium are compressive that could be beneficial for fatigue loading but introduces tensile stresses in the steel. Initial tensile stresses are unbeneficial for fatigue loading and could be the reason of ratcheting effect observed in preliminary fatigue tests [12]. (iii) Future tests should take into account appearance of persistent strain change caused by drilling process. References [1] J. Banker E., Reineke, Explosion Welding, ASM Handbook, Vol 6 Welding Brazing and Soldering ASM International, 1993, pp. 303-305. [2] S.A.A. Akbari Mousavi, S.T.S. Al-Hassani, A.G. Atkins, Bond strength of explosively welded specimens, Materials and Design 29 (2008) 1334–1352. [3] S.A.A. Akbari Mousavi, P. Farhadi Sartangi, Experimental investigation of explosive welding of cp-titanium/AISI 304 stainless steel, Materials and Design 30 (2009) 459–468. [4] S.A.A. Akbari Mousavi, P. Farhadi Sartangi, Effect of post-weld heat treatment on the interface microstructure of explosively welded titanium–stainless steel composite, Materials Science and Engineering A 494 (2008) 329-336. [5] L.B. Pervukhin, Y.U. Mal’tsev, A. Konon, B.D. Tsemakhovich, A.D. Chydnovskii, Distribution of internal stresses in bimetal steel 22K+steel Kh18N10T obtained by explosive welding, Metal Science and Heat Treatment 17/11 (1976) 934-937. [6] R. Kacar, M. Acarer, An investigation on the explosive cladding of 316L stainless steel-dinP355GH steel, Journal of Materials Processing Technology 152 (2004) 91–96. [7] Mustafa Acarer, Bilge Demir, An investigation of mechanical and metallurgical properties of explosive welded aluminum–dual phase steel, Materials Letters 62 (2008) 4158–4160. [8] L. Čižek, D. Ostroushko, Z. Szulc, R. Molak, M. Prażmowski, Properties of sandwich metals joined by explosive cladding method, Archives of Materials Science and Engineering, 43(1) (2010) 21-29. [9] D. Ostroushko, E. Mazancová, Chosen properties of sandwich CrNi steel-Ti material after explosive cladding, Conference proceedings of 19th International Conference on Metallurgy and Metals, Metal 2010, Czech Republic.

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[10] S.Król, R. Bański, Z. Szulc, A. Gałka, Practical aspects of structural tests of titanium-steel bonds made by explosive cladding and exposed to thermal process loads, Advances in Material Science, 7 (2007) 50-56. [11] M. Sedighi, M. Honarpisheh, Experimental study of through-depth residual stress in explosive welded Al–Cu–Al multilayer, Materials and Design 37 (2012) 577–581. [12] A. Karolczuk, M. Kowalski, R. Bański, Ż. Żok, Fatigue tension-compression testing of steeltitanium bimetal produced by explosive welding, XI Scientific National Polish Conference "Titanium and its alloys - 2011", Rzeszów University of Technology, p. 4 (in Polish). [13] ASTM E837-08. Standard test method for determining residual stresses by the hole drilling strain-gauge method. West Conshohocken: American Society for Testing and Materials; 2008. [14] Information on http://www.tenmex.pl

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.133

The Impact of the Laser Welding Speed on the Mechanical Properties of Joints in Multilayer Pipes Stanisław Mroziński 1,a, Michał Piotrowski 2,b 1

University of Technology and Life Sciences In Bydgoszcz, Prof. Kaliskiego 7 85-796 Bydgoszcz, Poland

2

University of Technology and Life Sciences In Bydgoszcz, Prof. Kaliskiego 7 85-796 Bydgoszcz, Poland a

[email protected], [email protected]

Keywords: Multilayer pipe, laser welding, strength

Abstract. The paper assessed the impact of the laser welding speed on the strength and fatigue properties of the aluminum layer found in multilayer pipes. The conducted experiment has shown that during the adjustment of the welding speed one has to take into account not only the results of static tests, but also the results of fatigue tests. The impact of the welding speed on fatigue life depends on the level of stress σmax. This level is slight in the area of the biggest stresses and increases along with the decrease in stresses. 1. Introduction A large number of advantages of composite materials causes that their application in many industries rises very quickly. One example of such an application are multilayer pipes used both for the distribution of drinking water, usable water and as well as in the installations of central and underfloor heating systems. The multilayered structure of the pipe consists of a few layers: highdensity, high-temperature resistant polyethylene, aluminum and glue (Fig. 1). Because of the many advantages, including: resistance to low and high temperatures, stone overgrowth resistance and the ability to self-compensate thermal elongations multilayer pipes find wide applications. PE

Glue

Al

Glue

Laser beam

PE Aluminum plate

a)

b)

Fig. 1. Multilayer pipes: a) pipe structure b) welding of the aluminum layer One of the main disadvantages of such materials as polyethylene, cross-linked polyethylene, polypropylene and polybutene in temperatures exceeding 60°C is oxygen permeability [1]. The presence of oxygen in installation water is the main cause of corrosion of metal elements in central heating systems. For the production of plastic pipes the application of non-metal layers is made on the pipes limiting the stream size of diffusing oxygen. The more effective way of eliminating this dangerous occurrence is the application of an aluminum layer in pipes 100% reducing the inflow of oxygen to the installations. During the design process of multilayer pipes it is assumed that the pipes during use are exposed to variable loads and creep deformation. Considering the low thickness of the aluminum layer and the necessary tightness, the aluminum layer on a polyethylene pipe is laser-welded. The welding optimization process requires to appropriately adjust welding parameters, which include welding speed and laser beam power. Subjecting the pipe during a technological process to the influence of strong heat stream causes the occurrences of material

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Fatigue Failure and Fracture Mechanics

heterogeneity (structural notches). These apply to both the polyethylene layer and the aluminum layer. During operation there can appear in these areas the beginnings stages of cracks, which further lead to the damage of the pipes. The researched problem is the impact that laser welding parameters have on the fatigue life of multilayer pipes. The presented paper deals with the assessment of the impact of the laser welding speed on the mechanical properties of the welded joints used in multilayer pipes. 2. Experiment methodology 2.1. Researched materials Test samples were taken out of ⌀50mm multilayer pipes, where during the process of joining the 0,6mm thick aluminum layer seven speeds of laser welding were applied ranging from 3,6m/min to 6m/min (3,6; 4,0; 4,4; 4,8; 5,2; 5,6; 6,0m/min). Constant value of the laser beam power equaled 4kW was applied for these welding speeds. The way of extracting the test samples from the pipe is shown below in Fig. 2.

Weld joint

Weld joint

Test sample Test sample

a)

b)

Fig. 2. Test samples: a) samples without a weld, b) samples with a weld After cutting off 10mm wide rings from the pipe a cut was made alongside the weld (Fig. 2a) and on the opposite side from the weld (Fig. 2b). As a result, test samples with and without a weld were obtained. For the purpose of comparison, other samples from a plate used for the pipe braid were also prepared for the experiment. The dimensions of the plate samples matched those samples shown in Fig. 2. Static and fatigue tests were preceded by a metallographic analysis of the laser welds. In Fig. 3 there are shown three selected microsections of laser joints obtained from three different welding speeds.

v=3,6m/min v=4,2m/min v=6m/min a) b) c) Fig. 3. Welding speed impact on the dimensions of the laser weld: a) v=3,5; b) v=4,4; c) v=6m/min On the basis of the conducted analysis of the microsections it can be stated that the welding speed influences the basic weld dimensions. In Fig. 4 there are diagrammatically shown the characteristic dimensions which were analyzed in detail. Values of the characteristic dimensions of the laser welds in relation to the welding speed were compiled in the form of graphs in Fig. 4.

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A

7

Dimenison, mm

6 C

5

D

B

4 3

A B C D

2 1 0

3,6

4,0

4,4

4,8

5,2

5,6

6,0

Welding speed, m/min

Fig. 4. The impact of the welding speed on the weld dimensions Analysis of the given microsections of the laser welds makes it valid to state that the welding speed has its impact mainly on the weld width. As expected, along with the increase of the speed the width of the weld decreases. At the speed of 6m/min the weld width is approx. 30% smaller compared to the weld width obtained with the speed of 3,6m/min, which simultaneously causes the heat affected zone to shrink, alongside with the possibility of crack initiations characterized to that area. Measurements of microhardness were taken on the microsections (both on the weld itself and in the heat affected zone) using the Hanneman method with the help of a Neophot 2 microscope. The spots which were measured for microhardness are diagrammatically shown on the section below in Fig. 5. A

BC D

Microhardness, uHV0,1

67 62 57 52 47 42 37 32

3,6

4,0

4,4

4,8

5,2

5,6

6,0 1

2

A

B

3

C

4

D

Fig. 5. The impact of the welding speed on microhardness Analyzing the results of the obtained measurements makes it possible to state that hardness increases along with the approaching the center of the weld. It is hard to assess the effect of the welding speed on the weld hardness. 2.2 Static tests During the static tests the samples were being overloaded with the speed of the piston feed of the tester machine, which equaled 0,1mm/s. The elongations of the samples were measured using an extensometer with the base of 12,5mm and the measuring range of ±40%. The measurements were taken at the temperature of 21oC. The tests were being conducted up to the point of the sample split within the measuring range of the extensometer. During the static elongation test the temporary values of load force affecting the sample were registered and its elongation.

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Fatigue Failure and Fracture Mechanics

2.3. Fatigue tests For the purpose of fatigue tests only two test samples were used which were taken out of the pipes that constitute the two ends of the welding speed range (v=3,6m/min and v=6m/min). The test samples were subjected to variable tensile stress. The parameters of the tensile stress were compiled in Table 1. and in Fig. 6. The fatigue tests were conducted using the Instron 8501 tester machine equipped with a force gauge head of the ±10 kN measuring range. The experiment was conducted at the temperature of 21oC. Stress frequency f equaled 15Hz at the time. Table 1. Parameters of variable stress Level Stress No. σmax σmin σa ∆σa MPa MPa MPa MPa 1 70 0 35 7 2 80 0 40 80 3 90 0 45 90 a)

σm

σmax

σa

σ

σmin=0

t

b) Fig. 6. Stress program: a) stress parameters, b) stress graph

The criterion assumed for ending the fatigue tests was a crack of the test sample. The strains of the samples were measured using a dynamic extensometer with the range of ±1mm and the base of 10mm. During the experiment the tests samples were secured in hydraulic clamps with a regulated force of the grip. 3. Experiment results and their analysis 3.1. The impact of the plate strengthening on the static properties All the test samples (with and without a weld) fractured in the area of the extensometer’s measuring base. The results of the static tensile stress test is shown in Fig. 7 in the form of graphs on the coordinate system: test sample elongation ε − stress σ. The stresses in the test sample subjected to tensile load were calculated dividing the temporary values of load force by the initial cross-sectional area of the sample. The results are grouped in Table 2.

Original material

100 Stress, MPa

Tabela 2. Results of the tensile stress test of the sample from the aluminum plate

Test sample no.3.0

120

80

Sample type

60

Plate sample

A12,5mm [%] 29

Rm [MPa] 112

RP0,2 [MPa] 70

Pipe sample

38

117

85

40 20 0 0

5

10

15

20

25

30

35

40

Determined parameters

45

Strain, %

Fig. 7. The effect of the plastic strain (strengthening) on the strength parameters After examining the graphs in Fig. 7 it can be stated that, as expected, the process of wrapping the polyethylene pipe with a layer of an aluminum plate increases the pipe’s strength parameters. The phenomenon of the strength parameters’ increase proceeding the earlier plastic deformations is

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related to the so-called material strengthening phenomenon. The graphs in Fig. 7 show that performing the plastic strain process of the aluminum plate during the wrapping of the polyethylene pipe causes an increase in tensile strength Rm and yield strength Rp02. The quantitive changes of the listed parameters proceeding the wrapping process can be found in Table 2. 3.2. The impact of the welding speed on the static properties of the weld joints Regardless of the welding speed the fracture of all the test samples occurred always in the laser weld area. In Fig. 8 there are illustrated example graphs of tensile tests of samples with the welds obtained at various welding speeds. For the purpose of comparison, there is also another graph of a tensile test of the sample without the weld. The graphs depicted in Fig. 8 were made using the basic strength parameters of the studied material. The results of the calculations assembled in Table 3. 120

4.0

100

3.6

Without the weld

4.4 4.8

Stress, MPa

80

5.2 5.6

60

6.0

40

20

0 0

10

20

30

40

Strain, %

Fig. 8. Static tensile tests’ graphs of the welded samples at various speeds Table 3. Test results summary Item no. 1 2 3 4 5 6 7 8

v m/min Without the weld 3,6 4,0 4,4 4,8 5,2 5,6 6,0

A12,5 [%] 38,1 25,4 23,5 23,1 21,3 20,8 19,3 18,7

Defined parameters Rm [MPa] RP0,2 [MPa] 117 122 122 124 124 123 122 121

85 92 95 97 96 97 95 94

On the basis of the obtained results it can be stated that the weld presence, regardless of the welding speed, always caused a slight increase in strength parameters (Rm and Rp02) compared to the test samples without the weld.

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Fatigue Failure and Fracture Mechanics

without the weld

125

3,6

Stress, MPa

115

4

4,4

4,8

5,2

4,4

4,8

5,2

5,6

6

original material

105 95

without the weld

85

3,6

4

5,6

6

Rp0.2

75

Rm

original material

65

2,8

Welding speed, m/min

Fig. 9. The impact of the welding speed on the strength parameters Comparative analysis of tensile strength Rm and yield strength Rp02 illustrated in Fig.9 allows it to state that the welding speed has little effect on the mentioned parameters. As expected, the presence of the weld has its impact on the total elongation of the test samples up to the point of fracture. Regardless of the pipe’s state, elongation A12,5 of the test samples containing the welds is considerably lower than those samples without the welds or made out of the original material. In Fig. 10 there is shown the impact of the welding speed on elongation A12,5 up to the moment of fracture. without the weld

40

Strain, %

35 30 original material

25 20

3,6

4

4,4

4,8

15 10

5,2

5,6

6

2,8

Welding speed, m/min

Fig. 10. The impact of the welding speed on elongation A12,5 It can be concluded from the graph that the welding speed has little effect on the total elongation. The obtained results of elongation A12,5 are very similar for all the given speeds. 3.3. The impact of the welding speed on fatigue life During the fatigue experiment the test samples always fractured outside of the weld.

No. 1 2 3

Table 4. Number of cycles until fracture Number of cycles until fracture N Speed v=3,6m/min Speed 6 m/min Stress σmax 70 450000 593253 80 301127 481281 90 133217 138756

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After an analysis of the obtained results in Table 4 it can be stated that the laser welding speed has an impact on the fatigue life. In order to make the analysis easier the results from Table 4. are also depicted in Fig. 11 in the form of fatigue graphs on the semi-logarithmic coordinate system: stress σmax – number of cycles N. The fatigue results were approximated using the following equation:

log σ max = a log N + b

(1)

where: a- gradient of the fatigue graph, b- y-intercept of simple regression for the fatigue graph. 100

Stress σmax MPa

95 90

σ max = a log N + b v, m/min 3,6 6

a -0,1973 -0,1053

b 934,1 314,5

85 80

v=6 m/min V=3,6 m/min

75 70 65 60 100000

1000000

10000000

Number of cycles

Fig. 11. The impact of the welding speed on the fatigue life of the weld joints After an analysis of the results it can be stated that the weld joint obtained at the speed of v=6m/min has higher fatigue life compared to that joint performed with the speed of v=3,6 m/min. The degree of the impact of the speed is dependent on the level of stress σmax. It is slight in the areas of the biggest stresses and increases along with the decrease of stresses. On the level of stresses σmax=70MPa the fatigue life of the weld joints obtained at the speed of 6m/min is approx. 100% greater compared to those joints at the speed of 3,6m/min. 4. Conclusions The results obtained during the laboratory experiment of the multilayer pipes, which consist mainly of cross-linked polyethylene and aluminum plate 8006 make it possible to formulate the following general conclusions: - an increase in the welding speed while retaining constant laser power causes a considerable decrease in the crosswise dimensions of the weld, the almost double increase in the speed causes an approx. 30% decrease of the dimensions, which has its effect on the size of the heat affected zone in the welded object, - a decrease in microhardness of approx. 8% can be observed in the weld obtained at the speed of 3,6m/min compared to the weld at 6m/min, which is a result of a smaller amount of energy that is distributed to the weld during the welding and it ultimately lowers the temperature of the welding process itself, - the aluminum plate during the process of its wrapping and pressing to the polyethylene pipe strengthens itself which causes an increase in strength parameters, - the presence of the weld, regardless of the welding speed, always caused an increase in strength parameters (Rm and Rp02) compared to the test samples without the weld,

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Fatigue Failure and Fracture Mechanics

-

regardless of pipe’s state, elongation A12,5 of the test samples containing the weld was significantly lesser than the elongation of the samples without the weld and those from the original material - the welding speed has little impact on tensile strength Rm, yield strength Rp02 and elongation A12,5,, - the impact of the speed on the fatigue life depends on the level of stress σmax, it is slight in the area of the biggest stresses and increases along with a decrease in stresses. The obtained results of the conducted tests of the polyethylene–aluminum multilayer pipes illustrated that the welding speed of the aluminum layer of the polyethylene pipe has little impact on the tensile strength of the joints. The welding speed considerably increases the fatigue strength on the low levels of stresses σmax. References: [1] Rutkiewicz. A, Pomiar szybkości przenikania tlenu przez ścianki rur z tworzyw sztucznych, Prace Instytutu Techniki Budowlanej, R38, (2009). [2] PN-74/H04327- Badanie metali na zmęczenie. Próba osiowego rozciągania ściskania przy stałym cyklu obciążeń zewnętrznych

CHAPTER 4: Temperature and Thermo-Mechanical Fatigue

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.143

Model of the deformation process under thermo-mechanical fatigue conditions Jerzy Okrajni 1, a, Grzegorz Junak 1, b 1

Department of Materials Technology, Silesian University of Technology, Krasinskiego 8, 40-019 Katowice, Poland a

[email protected], b [email protected]

Keywords: thermo-mechanical fatigue, low-cycle fatigue, hysteresis loops, fatigue characteristics.

Abstract. The paper focuses on the development of a mathematical representation of deformation characteristics under the conditions of an elevated changeable temperature and mechanical loads. The method proposed in the paper is based on the use of characteristics determined in low-cycle fatigue tests at constant temperatures. Hysteresis loops reflecting the behaviour of a material under the conditions of low-cycle loads at an elevated temperature were primarily used. The effect of relaxation on the course of the hysteresis loop was taken into account. The steady state of the material is referred to in the proposed representation. A calculation algorithm was developed to enable the determination of the stress value for subsequent increases of strain over time. The results obtained were compared with experimentally determined characteristics. Introduction The course of the deformation process in the conditions of simultaneous action of a variable temperature and of mechanical and thermal loads has been described in many papers concerning solid body mechanics. Some papers attempted to generalise the description of the behaviour of materials through constitutive equations for complex stress states that take into account rheological phenomena and dependencies of material properties on the temperature, including Leimatre and Chaboche models [1-3] which are commonly used to describe the properties of materials applied for operation at an elevated temperature or, for example the model of Kichenin and co-authors [3] or the one developed by Figiel and Günter [4]. Therefore, this paper naturally is not an attempt at such generalisation. At the current stage, it only attempts to describe the behaviour of a material subjected to thermo-mechanical fatigue, while taking into account a number of constraints. This is because it assumes that the deformation process occurs in a steady state and that the reaching of the steady state in low-cycle fatigue conditions is tantamount to reaching such a state in the case of thermo-mechanical fatigue. The discussion is referred to the uniaxial state of stress. Description of the proposed method The proposed method of describing dependences between stress and strain under thermomechanical fatigue conditions is based on the use of characteristics determined during tests of lowcycle fatigue at constant temperatures [5,6]. Basic dependences used to describe the mechanical behaviour of a material in such tests are characteristics in the form of hysteresis loops whose courses depend on the value of temperature. The shape of the loops and their parameters may change in successive load cycles as a result of cyclic strengthening or weakening of the material, usually reaching a steady sate that covers a majority of load cycles until failure of the tested specimen. At the current stage, the discussion refers to a steady state. In the developed model, the course of the deformation process is described based on the stabilised hysteresis loops which had been experimentally determined. Function σ`=f(ε`, T), ε`∈(0, ∆ε) describing the course of a hysteresis loop in the state of saturation (Fig. 1a) was introduced.

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Fatigue Failure and Fracture Mechanics

a)

b)

Fig. 1. The courses of steady hysteresis loops, depending on the temperature. schematic representation –a) and a diagram of a hysteresis loop branch for the method of approximating its course used in the paper –b) Function f(ε`, T) can be adopted in various forms. In papers [5,6], the dependence between stress σ`, deformation ε` and temperature T was adopted in the form of the following equation:

σ` = f( ε`,T) = (A - CT v )arctan(Dε`)

(1)

where constants A, C, D and ν are determined based on isothermal fatigue tests. A number of other methods can be proposed to represent the f`(ε’, T) function. The isolating of elastic and elastic–plastic deformation ranges on the characteristics under consideration seems to be an appropriate method. In this case, the curve describing the dependence between stress, strain and temperature will be comprised of two sections. A linear dependence between strain and stress is assumed in the range of values below the strain corresponding to the transition between the elastic R (T ) and elastic-plastic range (Fig. 1b). Above the value of ε ′ = e , strengthening of the material E (T ) occurs, which can be described for example with an exponential dependence. Consequently, we receive the following function:

R (T )  E( T )ε ' for ε ' < e  E( T )  f ( ε ' ,T ) =  n( T ) Re ( T ) '  R ( T ) + C( T ) ε ' − Re ( T )  , for ε ≥ e   E( T )  E( T ) 

(2)

Re is in this case a conventionally treated “yield point” separating the elastic from the elasticplastic range (Fig. 1b).

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Other dependences describing the course of the f`(ε`, T) curve above the yield point Re(T) can also be adopted in the range of elastic-plastic deformations. One can, for example, apply the previously used arctan(x) function. In this way, the following equation can be obtained:

Re ( T )  ' ' E ( T ) for < ε ε  E( T )  ' f ( ε ,T ) =      Re ( T ) + B( T ) arctan ε ' − Re ( T )  m( T ), for ε ' ≥ Re ( T ) E( T )  E( T )   

(3)

The values B(T), C(T), n(T), m(T), E(T) and Re(T), which are dependent on temperature, are determined based on low-cycle fatigue tests. Figure 2 presents examples of σ`(ε`) characteristics determined experimentally and via approximation with the use of dependences (3). E 200

E 500

E 620

M 200

M 500

M 620

800 700

STRESS, MPa

600 500 400 300 200 100 0 0

0.002

0.004

0.006

0.008

STRAIN

Fig. 2. Experimentally determined (E) dependences between strain ε’ and stress σ’ for P91 steel and courses corresponding to them, determined based on a model presentation (M) After obtaining functions (2) or (3), the method described in papers [5,6] can be used to characterise the course of dependences between stresses and strains in the process of thermo-mechanical fatigue. Functional dependences describing the course of hysteresis loop branches with increasing (4) and decreasing (5) strain were substantiated in these papers:

σ=

1 1 f [(ε − ε R ),T ]− f [(ε − ε R ),TR ]+ σ R 0 2 2

(4)

1 1 f [ε − ε C ,T ]+ f [ε − ε C ,TC ]+ σ C 0 2 2

(5)

σ =−

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Fatigue Failure and Fracture Mechanics

TR is the temperature at the beginning moment of the part of the cycle with increasing strain. TC is the temperature at the beginning moment of the part of the cycle with decreasing strain. Constants ε R ,ε C ,σ RO ,σ CO are coordinates of the points of hysteresis loop (Fig. 3a). Figure 3 presents a diagram of a loop which can be obtained in thermo-mechanical fatigue tests (Fig. 3a) and the characteristics of this type of fatigue process which was experimentally determined for the P91 steel (Fig. 3b). This characteristics was compared to a model presentation which uses equations (3)(5). Compliance of the course of both characteristics was achieved, which seems sufficient for technical applications. MODEL

EXPERIMENT

500 400

STRESS, MPa

300 200 100 0 -100 -200 -300 -0.006

-0.004

-0.002

0

0.002

0.004

0.006

STRAIN

a)

b)

Fig. 3. A schematic representation of a hysteresis loop under thermo-mechanical fatigue conditions – a) and a comparison of an experimentally determined hysteresis loop with a hysteresis loop determined based on a model representation –b); steel P91 (Tmax=650oC, Tmin=200oC) The issue becomes much more complex when periods of a steady value of total mechanical strain occur in the deformation process. Under fatigue conditions at elevated temperatures, relaxation of stresses occurs during deformation induced by changes of external factors, especially in periods with a fixed strain value. After a period of relaxation at a fixed strain value, another period of its increase or decrease occurs, which compensates for the effects of the relaxation process. An effect of the return of the material to the state from before the rheological processes can be observed in metal alloys. After a short return period, the material is deformed once again in accordance with the course of the hysteresis loop characteristic for the steady state (Fig. 4). When describing the course of the hysteresis loop under such conditions, a mathematical representation of the deformation process, both during relaxation and under the conditions of “return” to the state preceding this process, is necessary.

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400 0.006

300 0.004

0.002

100 STRAIN

STRESS, MPa

200

0 -100

0

-0.002

-200 -0.004

-300 -0.006

-400 -0.006

2500

-0.004

-0.002

0

0.002

0.004

0.006

2700

2900

3100

3300

3500

TIME, s

STRAIN

a)

b)

Fig. 4. A hysteresis loop determined in an isothermal low-cycle fatigue test (T=550oC) with the total strain under control –a ) whose course over time is shown in the figure –b); steel P91 Having adopted a power law creep, the creep strain is expressed [7] with the following dependence:

ε c = Kσ n tm

(6)

By applying the ageing theory and assuming a constant creep rate for the relaxation process, the following equation is obtained:

 1 1 1   t= − KE (n − 1)  σ n −1 σ on −1 

(7)

Where E is the elasticity modulus and σo is the initial stress value. Using equation (7), the increase of stress caused by relaxation may be written down as follows:

dσ r = − KEσ n dt

(8)

In this case, constant K and elasticity modulus E should be treated as constants depending on temperature. Due to the sign of stress, equation (8) is valid for odd integer values of exponent n. In remaining cases, σ should be replaced with expression σ n

σ

n−1

or sgn(σ )σ . n

The paper was an attempt at expanding the scope of the application of dependence (8) to the range of elastic-plastic deformations. Product KE was treated as a constant depending on temperature S (T ) = K (T )E (T ) . The overall effect of relaxation in time t can be calculated by adding increases of stresses in individual moments of time. Consequently, we receive:

σ r = −∫t S (T )σ n dt

(9)

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While treating the relaxation process as “reversible”, a function expressing the effects of the “return" in the form of changes of the value of the total effect of relaxation σr caused by the strain was introduced in the paper. It was assumed that the intensity of the “return” measured using the increase of stress to the increase of strain ratio depends on stress value σr and temperature. The increase of stress caused by the effects of the “return” was expressed in the first approximation with the following dependence:

 dε  dσ r = M (T )σ r dε = M [T (t )]σ r (t )  dt  dt 

(10)

Consequently, we receive the following equation:

 dε  dt dt  



σ r = − ∫t S [T (t )]σ n − M [T (t )]σ r (t ) 

(11)

Taking into consideration dependences (4) and (5) and equation (11), a recurrent computational algorithm which allows calculating stress values for next strain increases over time was developed. In this algorithm, stresses in next calculation stages are determined as a sum of the results of strain increase, determined from (4) and (5) dependences, and stress changes caused by relaxation - σr (11), while taking into account the effects of the “relaxation return”. An example of thus determined hysteresis loop is presented in Figure 5a which shows an experimentally determined loop and a loop determined based on the developed model representation. Constants S and M were assumed to be dependent on temperature and were determined by trial and elimination based on conducted experiments. Diagrams shown in Figure 5a correspond to the characteristics of changes in stresses as a function of time, shown in Figure 5b. Hysteresis loops determined experimentally and those determined based on a model representation of the characteristics of changes in stresses as a function of time show the conformity of the course, both in the periods in which no discernible influence of rheological processes was observed and during relaxation. MODE L

E X P E R IME NT

400

E X P E R IME NT

MODE L

400

300 300 200

100 S T R E S S , MP a

S T R E S S , MP a

200

0 -100

100 0 -100 -200

-200

-300

-300 -400

-400

0

-0.006 -0.004 -0.002

0

0.002

0.004

0.006

100

200 TIME , s

300

400

S TR AIN

a)

b)

Fig. 5. A hysteresis loop determined in an isothermal low-cycle fatigue test (T=500oC) with total strain control and a hysteresis loop determined based on a corresponding model representation –a ) for those whose courses of stress over time are shown in the figure –b); steel P91

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Conclusion The aim of the tests was to develop a method of the mathematical representation of deformation characteristics under the conditions of an elevated temperature and mechanical loads. At the current stage, calculation methodology was developed and verified for selected cases. The main focus was on representing the relaxation phenomenon. Characteristics shown in Figure 5 refer to fatigue under the conditions of action of constant temperature and variable in time strain whose cycle contained a time interval during which strain remained constant. Dependences assumed in the paper require further verification, to check in particular whether constants determined for particular cases of relaxation itself and isothermal low-cycle fatigue without the participation of rheological processes may serve as a basis for the generalisation of the description of material behaviour under the conditions of cyclic deformation at variable temperature and mechanical strain, while taking into consideration time intervals of relaxation and creep. Currently this type of tests is performed while taking into account various materials and cycles of thermo-mechanical loads and taking into consideration the specificity of operation of devices used under the conditions of mechanical and thermal loadings [8-10], in particular currently developed power engineering devices with enhances operating parameters in the case of which phenomena of fatigue induced by cyclical changes in temperature become more and more important. The results presented in this paper were obtained from research work co-financed by the National Centre of Research and Development in the framework of Contract SP/E/1/67484/10 - "Strategic Research Programme - Advanced Technologies for obtaining energy: Development of a technology for highly efficient zero-emission coal-fired Power units integrated with CO2 cap” References [1]

J. Lemaitre, J.-L. Chaboche, Mechanics of solid materials, Cambridge University Press, Cambridge 1990. [2] J.-L. Chaboche, Cyclic viscoplastic equations part I: e thermodynamically consistent formulation, Journal of Applied. Mech., 1993, vol. 60, pp. 813-821. [3] J. Kichenin, K. Dang Van, K. Boytard, Finite element simulation of a new two-dissipative mechanisms model for bulk medium-density polyethylene, J. Mater. Sc 31(1996), pp.16531661. [4] Ł. Figiel, B. Günter, Modelling the high-temperature longitudinal fatigue behaviour of metal matrix composites (SiC/Ti-6242): Nonlinear time-dependent matrix behaviour, International Journal of Fatigue, v. 30, issue 2, January 2008, pp. 268-276. [5] J. Okrajni, G. Junak, A. Marek, Modelling of the deformation process under thermomechanical fatigue conditions, International Journal of Fatigue, v. 30, issue 2, January 2008, pp. 324-329. [6] J. Okrajni, G. Junak, A. Marek, Description of the deformation process under thermomechanical fatigue, Journal of Achievements in Materials and Manufacturing Engineering, vol. 21 issue 2, 2007, 15-24. [7] G.A. Webster, R.A. Ainsworth, High Temperature Component Life Assessment, Chapman & Hall, London 1994. [8] J. Okrajni, W. Essler, Computer models of steam pipeline components in the evaluation of their local strength, Journal of Achievements in Materials and Manufacturing Engineering, vol. 39 issue 1, 2010, 71-78. [9] J. Okrajni, Thermo-mechanical conditions of power plant components, Journal of Achievements in Materials and Manufacturing Engineering, vol. 33 issue 1, 2009, 53-61. [10] J. Okrajni, K. Mutwil, M. Cieśla, Steam pipelines’ effort and durability. Journal of Achievements in Materials and Manufacturing Engineering, vol. 22, issues 2, June 2007, pp. 63 – 66.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.150

Influence of temperature on the cyclic properties of martensitic cast steel Stanisław Mroziński1, a, Radosław Skocki2,b 1,2

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, A. Prof. S. Kaliskiego 7, 85-789 Bydgoszcz tel.: 48 52 340-82-64, fax: 48 52 340-82-71 a

b

[email protected], [email protected]

Keywords: low cycle fatigue, cyclic properties, martensitic cast steel.

Abstract. This paper attempts to describe changes of the cyclic properties of martensitic cast steel in the function of the number of loading cycles under temperature of 25 and 600oC. The cyclic properties were described by means of three hysteresis loop parameters: stress amplitude σa, plastic strain amplitude εap, plastic strain energy ∆Wpl. It was stated that martensitic cast steel always undergoes clear softening which is independent of the temperature and level of total strain. Introduction The issue of this paper concerns the analytical description of changes of the martensitic cast steel cyclic properties namely the changes of the hysteresis loop parameters in the function of the number of loading cycles. The problem complicates significantly when the course of the changes aside from the mechanical loading is influenced by the changeable temperature [1,2]. At elevated temperature the range of changes of the hysteresis loop parameters is wider than at room temperature [3,4,5]. Due to this reason more propositions are made considering the changes of cyclic properties influenced by the temperature in calculations of fatigue life [6,7]. Basic aim of this work is to determine the influence of the temperature on changes of hysteresis loop parameters in the function of the number of loading cycles. Descriptions of tests

R10

R10

2

15

2

M 20

φ 16.5 h6

0.32

φ 16

φ 8 h6

Specimens for the tests were made of GX12CrMoVNbN9-1 martensitic cast steel. The shape and dimensions of the specimens were in agreement with the standard [8] (Fig. 1).

7 27

105

Fig. 1. Shape and dimensions of specimens used in tests The fatigue tests were preceded by carrying out the static tensile tests. The specimens used in the tests are shown in Fig. 1. The specimens underwent increasing loading with the rate of machine displacement speed of 0.05 mm/s. Specimen’s elongation was measured by a 12.5 mm gauge length

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axial extensometer with measuring range of 3.75 mm. The static tensile tests were carried out under the temperature of 25oC and 600 oC. During these tests momentary loading forces and elongation of the specimen were recorded. After analysing the static tensile tests five levels of total strain εac were accepted in low cycle tests mainly: εac=0,25; 0,30; 0,35; 0,50; 0,60. LCF tests were conducted under controlled total strain εac =const. The same procedure of measuring strain was employed in static tensile test. Test temperature of 25oC and 600 oC and frequency of 0.2 Hz were employed. Accepted sampling frequency of force signal and strain signal allowed to describe loading cycles with set of 200 points. As the end criterion of the fatigue test, the deformation of hysteresis loop (during semi cycle of compression) is accepted. During the tests momentary values of loading force and strain for selected loading cycles were recorded. Results and discussions The examined material cyclically softened independently of the temperature. To analyse this process three hysteresis loop parameters were assumed namely: stress amplitude σa, plastic strain amplitude εap, plastic strain energy ∆Wpl (Fig. 2).

σ +σ

2σa=∆σ

∆Wpl −εa

+ εa ε

-σ ∆εap 2εac= ∆εac

εae

Fig. 2. Hysteresis loop with its basic parameters Fig. 3 presents three graphs of the analysed parameters in the function of the number of loading cycles for the two levels of total strain εac=0,25 and εac=0,60%. According to the results it was stated that the temperature clearly influences fatigue life and the course of the analysed parameters.

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550 σa, MPa

εac=0,25%

550

500

500

450

450

400

400

εac=0,6%

T=25 ºC

350

T=25 ºC

350

σa, MPa

300

300

250

250

T=600 ºC

200

200

T=600 ºC

150

150 0

0

10000 20000 30000 40000

200

400

n a)

600

800

1000

n b)

0,0014

εap, mm/mm

0,0050

εac=0,25%

εap , mm/mm

εac=0,6%

0,0045

0,0012

T=600 ºC

0,0040

T=600 ºC

0,001

0,0035

0,0008

0,0030

0,0006

0,0025 0,0020

T=25 ºC

0,0004

T=25 ºC

0,0015

0,0002

0,0010 0,0005

0 0

0

10000 20000 30000 40000

200

400

n d)

ε ac =0,25 %

0, 8

800 1000

n

c)

∆ Wpl, MJ/m3

600

3

6

∆ Wpl, MJ/m

εac=0,6 %

5

T=600 ºC

T=25 ºC

0, 7

4 0, 6

T=25 ºC

3

0, 5 0

10000

20000

30000

40000

2 0

n

e)

T=600 ºC

200

400

600

800

1000

n f)

Fig. 3. Changes of the hysteresis loop parameters (σa, εap, ∆Wpl) depending on the total strain level and the temperature: a, c, e) εac=0,25%; b, d, f) εac =0,6% On the basis of the presented graphs it can be stated that for each level of total strain there can be distinguished stages with a different softening speed. Fig. 4 presents the course of changes of stress amplitude σa in the function of loading cycles for the level of total strain of εac =0,6% and the temperature of T=20 oC. Three distinctive stages can be observed (A, B, C).

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530 σa , MPa 510

σa =aσ⋅ n +b

490 470 450 430 A

410 390

C

B

A – changeable softening speed B – constant softening speed C – changeable softening speed

370 350 0 Fig. 4.

500

n

1000

Consecutive stages martensitic cast steel softening

In stage “A” cast steel clearly softens. The speed of softening decreases with the increase of the number of loading cycles. In stage “B” the speed of softening is constant. In case of tested cast steel it was the longest stage, independently on the level of total strain and the temperature of the test. In the last stage “C” the speed of softening increases. Crack initiation takes place here and eventually leads to the fatigue failure. It should be noted that the stages mentioned above can be clearly observed for the stress amplitude σa curve. It is more difficult to distinguish these stages for the plastic strain amplitude εap and plastic strain energy ∆Wpl curve, particularly in the temperature of 600oC. This paper attempts to analyse the influence of the temperature on the speed of softening only for the one stage (the longest stage “B”). The hysteresis loop parameters were approximated by the linear regression line with regression equations presented in Fig. 4. To illustrate this approximation in Fig. 5 there were shown the courses of changes of the stress amplitude σa in the temperature of 600oC with all stages and also the courses of this parameter constrained only to the second stage “B”.

σa, MPa

300

300 σa, MPa

280

280

260

260

εac=0,6% εac=0,5% εac=0,35% εac=0,3%

240 220 200

220 200

180

180

160

160

2000

4000

10

100

1000

10000

log n

n a)

εac=0,6% εac=0,5% εac=0,35% εac=0,3% εac=0,25%

240

εac=0,25%

0

σa=aσ⋅n +b

b)

Fig. 5. Changes of stress amplitude σa in the function of the number of loading cycles in the temperature of 600o: a) all stages, b) stage “B” – constant softening speed

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Fatigue Failure and Fracture Mechanics

In Fig. 6 there were collected slopes of the regression lines aσ , aε , a∆W according to the level of total strain εac in the temperature of a) T=25 ºC b) T=600 ºC 0,1

- aε - a∆W - aσ

aσ, aε, a∆W,

aσ, aε, a∆W,

0,1 0,05 0

-0,05 -0,1 a)

0,25 0,3 0,35

0,5

εac, %

0,6

- aε - a∆W - aσ

0,05 0 -0,05 0,25 0,3 0,35

-0,1

0,5

0,6

εac, %

b)

Fig. 6. Slopes of the regression line describing hysteresis loop parameters according to the level of total strain a) T=25 ºC b) T=600 ºC The analysis of the slopes indicates that the course of tested cyclic properties (stress amplitude σa, plastic strain amplitude εap, plastic strain energy ∆Wpl ) is significantly influenced by the level of total strain εac and the temperature. Increasing the speed of softening depending on the temperature can be particularly noticed in the stress amplitude σa curve. Obtained tests results concerning the larger changes in cyclic properties at elevated temperatures confirm reports shown in literature [9,10,11]. Conclusions On the basis of carried out LCF tests it was stated that martensitic cast steel is a material which significantly softens independently of the temperature. The softening process occurs in all realized levels of total strain. Three characteristic stages with different speed of softening can be determined. Independently of the total strain level there is always a stage with a constant softening speed (Fig. 4 stage “B”). In this stage changes of the cyclic properties of the martensitic cast steel (σa, εap, ∆Wpl) in the function of loading cycles can be approximated by the linear regression lines. The analysis of the linear regression line slopes aσ, aε, aW leads to the conclusion that the speed of softening for all stages of softening process (A, B, C) is significantly influenced by the temperature and level o total strain. It can be stated that plastic strain energy ∆Wpl is the least sensitive hysteresis loop parameter to changes of the cyclic properties independently of the temperature and level of total strain. References [1]

Collins J.A., Failure of Materials in Mechanical Design, Analysis, Prediction, Prevention. John Wiley & Sons, New York 1993.

[2]

Kocańda S., Kocańda A., Niskocyklowa wytrzymałość zmęczeniowa metali. PWN Warszawa 1989.

[3]

Vani Shankar, Valerij Bauer, R. Sandya, M.D. Mathew, H.-J. Christ.: Low Cycle fatigue and thermo-mechanical fatigue behavior of modified 9Cr-1Mo ferritic steel at elevated temperatures, Journal of Nuclear Materials vol. 420 (2012), pp. 23-30.

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[4]

Mroziński S., Skocki R., Softening of Martensitic Cast Steel, Journal of Polish CIMAC. Volume 5 No 3, 2011, pp. 173-180.

[5]

Mroziński S., Golański G.: Low cycle fatigue of GX12CrMoVNbN9-1 cast steel at elevated temperature. Journal of Achievements in Materials and Manufacturing Engineering. Vol 49 ISSUE 1, November 2011, pp. 7-16.

[6]

Nagode M., Hack M.: An online algorithm for temperature influenced fatigue life estimation: stress-life approach, International Journal of Fatigue 26 (2004), pp.163-171.

[7]

Nagode M., Zingsheim M.: An online algorithm for temperature influenced fatigue life estimation: strain-life approach, International Journal of Fatigue 26 (2004), pp. 155-161.

[8]

ASTM E606-92: Standard Practice for Strain -Controlled Fatigue Testing.

[9]

Mroziński S., Golański G.: Fatigue life of GX12CrMoVNbN9-1 cast steel in the energy-based approach. Advanced Materials Research. Vol 396-398 (2012), pp. 446-449.

[10] Kaae J.L.: High-temperature low-cycle fatigue of Alloy 800H. International Journal of Fatigue 31 (2009), pp. 332–340. [11] Okrajni J., M. Cieśla M., Mutwil K.: Power plant component life assessment, Inżynieria Materiałowa 1 (2005), 15 – 20.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.156

Use of Thermography for the Analysis of Strength Properties of Mini-Specimens Adam Lipski1,a, Dariusz Boroński1,b 1

The University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Al. Prof. S. Kaliskiego 7, 85-789 Bydgoszcz, Poland a

[email protected] (corresponding author), [email protected]

Keywords: strength properties, mini-specimen, infrared thermography.

Abstract. This paper presents sample applications of passive infrared thermography for research on temperature changes of mini-specimens resulting from monotonously increasing or cyclically variable mechanical load. The MFS system developed in the Department of Machine Design at the University of Technology and Life Sciences in Bydgoszcz (Poland) and designed for testing mechanical properties of microelements were used for tests. The MFS system ensures nanometric measurement accuracy of many static and fatigue-related material properties, including, i.a., static tension curves, cyclic strain curves, fatigue life curves as a function of force, stress and strain. Measurements of the mini-specimens temperature were performed using thermographic camera equipped with microscope lens. The tests have shown that research on the passive infrared thermography may be successfully applied for determining strength properties of materials in micro scale. The used research instrumentation is characterized by sufficient sensitivity and resolution (camera with the microscope lens), while the MFS system ensures accurate load and position control. Introduction The fast development of micro- and nanotechnology as well as material engineering and welding engineering that has recently been observed allows to use completely new design solutions based on advanced structural materials and new methods of their joining. However, those possibilities are often limited by lack of knowledge on specific properties of those materials such as mechanical properties in their broad meaning. One of the main properties of technical objects that may have significant effect on the reliability and safety of the object operation is its ability to operate under variable load, including, in particular, cyclically variable load [1, 2, 3, 4]. Thanks to fast development of material engineering and new manufacturing methods, technical objects are more and more often developed using solutions requiring use of new methods of fatigue analysis, including methods based on local approach. A significant limitation here is often insufficient knowledge on local material properties, including, in particular, fatigue properties. One of the reasons for such situation is high variability resulting mainly from applied material technologies or the object manufacturing processes, generating areas of different mechanical properties located in the close vicinity. Good examples of such structures are e.g. laser-welded joints or friction stir welded (FSW) joints, for which size of zones with different properties often does not exceed several decimal fractions of millimetre. Thus it is necessary to perform fatigue tests using very small specimens, i.e. so called mini-specimens. Strength properties tests (both monotonic and cyclic) can be successfully performed using infrared thermography. Examples of its application are provided e.g. in papers [5, 6, 7]. However, they concern macro scale tests in the broad sense. The professional literature seems not to include any articles concerning that problem as regards micro-scale specimens or structural components. This paper presents sample applications of passive infrared thermography for research on temperature changes of mini-specimens resulting from monotonously increasing or cyclically variable mechanical load. The assessment of the possibility to perform research in that scope is the precondition for further activities concerning passive thermography applied mainly for quick determining of strength properties of materials under cyclic load in micro scale.

Dariusz Skibicki

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Test description Test station. For those tests, the researchers used the MFS system designed for testing mechanical properties of microelements. The system was developed in the Department of Machine Design at the University of Technology and Life Sciences in Bydgoszcz as part of the research and development project funded by The Polish National Centre for Research and Development. The MFS system consists, among others, of such components like: nanodrive load system, microdrive load system, strain measurement system using digital image correlation method (DIC), strain measurement system using laser grating interferometry (LFI) method, precise object alignment and fixing systems, computerized power supply and control system. Overview of the system is shown in Figure 1 next to the drawing of the fatigue test specimen. Thanks to the applied solutions including double load system based on piezoelectric and microstepper actuator equipped with the precise ball screw, the MFS system ensures nanometric measurement accuracy of many static and fatigue-related material properties, including, i.a., static tension curves, cyclic strain curves, fatigue life curves as a function of force, stress and strain. As the system is equipped with optical-electronic measuring instrumentation, it provides micrometric accuracy of strain measurement at measuring points. a)

b)

Fig. 1. Overview of MFS system (a) and a mini-specimen compared with the smallest Polish coin (b) Continuous measurement of the mini-specimen temperature (Fig. 2) was performed during the research, using CEDIP Silver 420M thermographic camera equipped with G3 microscope lens (viewing area of about 3.2 mm x 2.6 mm – image pixel size of about 0.01 mm x 0.01 mm) with the frequency of 100 Hz. Camera images were transmitted via USB 2.0 interface to PC with appropriate software, where they were digitally recorded directly on HDD in form of PTW format files. The following software was installed in the PC computer: – VirtualCAM allowing two-way communication between PC and the thermographic camera via USB 2.0 interface, – CIRRUS Front End which was the user interface of the thermographic camera, – ALTAIR allowing downloading, storage and advanced processing of thermographic images. Main parameters of CEDIP Silver 420M thermographic camera include: – resolution: 320×256 pixels, – pixel size: 25 µm, – pixel matrix pitch: 30 µm, – spectral range: 3.6÷5.0 µm, – sensitivity: below 20 mK (available: 8 mK), – recording frequency at maximum resolution and digital image transfer via USB 2.0 interface: up to 140 Hz (up to 25 kHz at the resolution of 64x8 pixels), – programmable integration time in the range from 10 µs to 10 ms.

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Fatigue Failure and Fracture Mechanics

camera microscope lens

specimen

Fig. 2. Overview of MFS system with the microscope lens

with

installed

thermographic

camera

equipped

Specimens. Samples for strength properties tests at monotonically and cyclically variable load (Fig. 3) were cut with laser from 0.2 mm thick sheet plate made of S355J2G3 (18G2A) steel. b)

0,3

0,2

2,35

60°

R0 ,5

a)

2 4,3

Fig. 3. Dimensions of mini-specimens used in the research (thickness 0.2 mm) (a) and a sample thermogram of a mini-specimen tested under monotonically variable load (b) Sample test results Monotonic tensile test. Figure 4 shows sample tensile force curve and average temperature curve of the mini-specimen at the measuring part as the function of time, recorded during the monotonic tensile test. The analysis of the obtained average temperature curve of the specimen shows gradual temperature decrease at the first phase of loading (by about 0,5°C), due to the thermal-elastic effect. Minimum temperature defines so called thermal-elastic-and-plastic strength σθ. Thermal-elastic cooling effect prevails up to the value of the yield strength, while once the yield strength is exceeded, thermal-plastic heating of the material becomes more and more significant. Those two effects are balanced at the point corresponding to the thermal-elastic-and-plastic strength, i.e. the heating balances the cooling resulted from the thermal-elastic effect. The shift from temperature decrease to increase during the test is smooth and corresponds to properties of metals characterized by absence of explicit yield strength, which is confirmed by the shape of the tensile force curve. The average specimen temperature change recorded during the test amounted to about 2,81°C.

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Fig. 4. Sample curve of the force and temperature change in the mini-specimen as the function of time, recorded during the monotonic tensile test No significant temperature change was observed at the moment of the specimen failure, as compared to curves recorded for specimens of standard size. This may result from the small size of the mini-specimen and, as a consequence, low energy released at the moment of sample failure. Cyclic test. Figure 5 shows fragment (seconds 345 to 355 of the test) of the load force curve and the average temperature curve of the mini-specimen measuring part as the function of time, recorded at variable load.

Fig. 5. Sample fragment of the load force curve and temperature change curve of the mini-specimen as the function of time, recorded during the variable load test

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Fatigue Failure and Fracture Mechanics

Cyclic variation of the mini-specimen load force with average value of about 6.5 N and amplitude value of about 10.8 N was assisted by cyclic temperature variation with the amplitude of about 0.39°C and gradually increasing average temperature value which had not stabilized until the sample failure moment, when the variation achieved its maximum value of about 0.86°C. Gradual increase of the average temperature value during the test resulted from plastic strain causing energy dissipation and heat generation in the specimen, which could be noticed on the basis of the hysteresis loop (Figure 6).

Fig. 6. Sample hysteresis loops recorded during cyclic load test The frequency of the recorded sample temperature curve corresponds to the frequency of constant-amplitude variable load of f = 0.5 Hz. The specimen temperature variation curve is phase shifted with regard to the load curve, i.e. maximum force value in the load cycle corresponds to the minimum temperature value, while the minimum force value in the load cycle corresponds to the maximum temperature value. 331 load cycles were performed during the fatigue test until the specimen failure moment. Summary The tests discussed above have shown that research on the passive infrared thermography may be successfully applied for determining strength properties of materials in micro scale. The described research instrumentation is characterized by sufficient sensitivity and resolution (the thermographic camera equipped with the microscope lens), while the MFS system ensures accurate load control and readout of the selected parameters. The load frequency typical for low-cycle fatigue tests was used in this work. The low load frequency (f = 0.5 Hz) made possible plastic strain development. The presented method allows realising load with nanometric measurement accuracy in range of displacement (minimal extension about 1.7 nm). The measured temperature change for this extension was ca. 0.4°C. This means that it is possible to determining strength properties of objects in micro scale with measure base even less than 200 µm by using coupled mechanical and temperature fields. For example the strain resolution is 0.001 % for 0.17 mm measure base. Thus it is possible to determine the local material characteristics, e.g. for different zones of welded joints.

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Results of further research on that area will be presented in future conferences dedicated to fatigue and experimental mechanics. This research work is financially supported by the Polish state budget for science in 2010-2013 as a research project. References [1] [2] [3] [4] [5] [6] [7]

S. Kocańda, J. Szala, Basics of the fatigue calculations, PWN, Warsaw, 1997 (in Polish). J. Szala, Fatigue damage accumulation hypothesis, University of Technology and Agriculture in Bydgoszcz, 1998 (in Polish). S. Kocanda, Fatigue Failure of metals (Fatigue and Fracture), Springer, 1978. J. Schijve, Fatigue of Structures and Materials, Kluwer Academic Publishers, DordrechtBoston-London, 2001. N. Harwood, W.M. Cummings (eds), Thermoelastic Stress Analysis, IOP Publishing Ltd., Bristol, 1991. G. La Rosa, A. Risitano, Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components, Int. J. Fatigue, 22 (2000), pp. 65–73. A. Lipski, The use of passive infrared thermography for tests of materials and riveted joints used in aviation industry- selected problems. Part II of the collection of monographs (edited by J. Szala): Experimental methods in studies of materials and riveted joints used in aviation industry - selected problems. Institute For Sustainable Technologies –National Research Institute in Radom, Bydgoszcz-Radom, 2010 (in Polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.162

Variations Of The Specimen Temperature Depending On The Pattern Of The Multiaxial Load - Preliminary Research Adam Lipski1,a, Dariusz Skibicki1,b 1

University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Al. Prof. S. Kaliskiego 7, 85-789 Bydgoszcz, Poland a

[email protected] (corresponding author), [email protected]

Keywords: multiaxial fatigue, nonproportional load, infrared thermography.

Abstract. This paper provides the results of research on temperature changes of cylindrical specimens depending on the pattern of the multiaxial load. The research were made by using passive infrared thermography. It was found out that the average temperature value is significantly dependent on the plastic strain energy and that the temperature change amplitude depends on the nominal normal stress (except for torsion). Introduction Many physical phenomena related to mechanical load cause temperature changes. Those changes can be observed using passive infrared thermography. Many of those phenomena have already been subject to research (e.g. temperature changes caused by monotonically increasing mechanical load, discussed in the papers [1] and [2]. However, temperature measuring instrumentation that is usually used is designed for spot temperature measurement and, moreover, it requires direct contact with the surface of the tested object, which can influence the temperature of the object and the test result. Many of those phenomena are also related to surface, hence determining temperature at selected points of a specimen or a structure is not sufficient. It is necessary to determine distribution of temperature over the entire surface instead. Temperature distribution recording rate together with the accuracy of the temperature measuring system is of key importance, particularly for variable load tests. The above specified conditions for surface temperature measurement using contactless method and the temperature distribution recording rate and accuracy are met by modern scientific matrix thermographic cameras. Their application in passive infrared thermography research on general strength of materials and structures was presented in the papers [3] and [4]. Thanks to modern research instrumentation such as thermographic cameras, new information can be obtained on processes that have already been thoroughly analysed and described from another point of view. Such instrumentation also often allows to reveal completely new phenomena. Such processes include multiaxial fatigue, for which it is hard to find any papers in the available professional literature, providing examples of thermography use in the analysis of, for example, how the extent of load nonproportionality influences dissipation processes associated with the degree of the specimen or the structural component effort, manifesting in the temperature changes. This paper provides the results of research on temperature changes of cylindrical specimens depending on the pattern of the multiaxial load. That research should provide grounds for broader analysis of the effect of nonproportional load on fatigue properties, which manifests in the temperature changes. Test description Test station. The research was carried out using biaxial testing machine Instron 8874 with the load range ±25 kN for tension-and-compression and ±100 Nm for torsion (fig. 1). The extensometric research was performed using 8-channel, versatile extensometric bridge NI SCXI-1520 manufactured by National Instruments – and NI SCXI-1600 USB module for data acquisition (fig. 2). 1-RY85-0.6/120#-3-3m strain rosettes manufactured by HBM (fig. 3), with measuring base of 0.6 mm and quick-drying adhesive 1-Z70 were also used.

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Fig. 1. CEDIP Silver 420M thermographic camera aiming at the specimen installed in the handle of the testing machine Instron 8874

Fig. 2. The extensometric bridge and the computer controlling Instron 8874 machine as well as computers retrieving data from the extensometric bridge and the thermographic camera. a)

b)

c) y x

Fig. 3. The specimen installed in the handle of the testing machine Instron 8874 (a), arrangement of extensometers over the rosette (b) and the assumed reference coordinate system (c)

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Fatigue Failure and Fracture Mechanics

Signals from three extensometric channels of the rosette were recorded during tests together with force and torque signals from Instron machine. Data was recorded using NI LabVIEW Signal Express software. The surface temperature distribution of the specimens was additionally recorded with the frequency of 50 Hz using thermographic camera CEDIP Silver 420M (fig. 1). Camera images were transmitted via USB 2.0 interface to PC with appropriate software, where they were digitally recorded directly on HDD in form of PTW format files. The following software was installed in the PC computer: – VirtualCAM allowing two-way communication between PC and the thermographic camera via USB 2.0 interface, – CIRRUS Front End which was the user interface used to control the thermographic camera, – ALTAIR allowing downloading, storage and advanced processing of thermographic images. Main parameters of CEDIP Silver 420M thermographic camera include: – resolution: 320×256 pixels, – pixel size: 25 µm, – pixel matrix pitch: 30 µm, – spectral range: 3.6÷5.0 µm, – sensitivity: below 20 mK (available: 8 mK), – recording frequency at maximum resolution and digital image transfer via USB 2.0 interface: up to 140 Hz (up to 25 kHz at the resolution of 64x8 pixels), – programmable integration time in the range from 10 µs to 10 ms. Specimen. As nonproportional load was planned to be applied in the research, specimens made of austenitic steel X2CrNiMo17-12-2 characterized by high sensitivity to nonproportional fatigue load were used. The results of the chemical composition test were presented in table 1. Basic mechanical properties of the tested steel were also determined with the following results: Ultimate Tensile Strength UTS = 416 MPa, Tensile Yield Strength TYS = 210 MPa and Vickers Hardness HV10 = 234. Table 1. Chemical composition of X2CrNiMo17-12-2 steel grade and its comparison with the requirements of the standard PN-EN 10088-1

specimen min max

Fe

C

Si

Mn

63.9 -

0.028 0 0.03

0.25 0 1

1.48 0 2

P [%] 0.033 0 0.045

S

Cr

Mo

Ni

<0.005 0 0.015

18.03 16.5 18.5

2.22 2.0 2.5

12.3 10 13

Test specimens were prepared in accordance with the standard ASTM E2207-02 (fig. 4). Ra

Fig. 4. Fatigue test specimen

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Loading conditions. The following types of oscillatory, sinusoidal load (see tab. 2 and fig. 5) were used for the discussed tests: tension-and-compression (1), torsion (2), combined proportional load including tension-and-compression and tension with the amplitude ratio of τa/σa = 0.5(3) as well as combined nonproportional load including tension-and-compression and torsion with the amplitude ratio of τa/σa = 0.5(4) or 0.826 (5) and the phase angle of ϕ = 90°. The amplitude ratio values were assumed based on desired features of the nonproportional load. For τa/σa = 0.5 and ϕ = 90° the value of the maximum tangent stress vector does not change over the entire fatigue cycle. This pattern of load is very popular in many studies available in the professional literature thus results of individual studies can be easily compared. Whereas due to the amplitude ratio τa/σa = 0.826, the value of the equivalent stress according to Zenner formula applied in this research, is constant over the entire fatigue cycle. Thus, this is the most nonproportional load and one should expect that the effect of the nonproportionality shall be most evident for that load. Amplitudes of normal and tangent stress were selected so as to ensure that all load patterns are characterized by equal equivalent stress value amounting to σeq = 330 MPa. In case of that equivalent stress value, average lives for uniaxial and proportional loads amount to about 150 thousand cycles, while for the nonproportional load (4) about 55 thousand cycles, whereas for the most destructive load (5) – about 25 thousand cycles. The tests were performed for four levels of the equivalent load σeq = 300, 310, 320 and 330 MPa at the load frequency of 1 Hz. Table 2. Main load parameters assumed in the research for the equivalent stress level of σeq = 330 MPa

σeq

load type

σa

τa

[MPa]

1

tension-and-compression

2

torsion

3

proportional 0.5

4 5

ϕ

N

[°]

[cycles]

330.0

0.0

-

150 000

0.0

273.1

-

150 000

282.5

141.2

0

150 000

nonproportional 0.5

330.0

165.0

90

55 000

nonproportional 0.8

330.0

272.6

90

25 000

1)

330

2) MY PY

3) MY PY

4) MY PY

5) MY PY

MY PY

Fig. 5. Load patterns used in the research: tension-and-compression (1), torsion (2), proportional 0.5 (3), nonproportional 0.5 (4) and nonproportional 0.8 (5) Test results. Figure 6 shows the curve of the average specimen temperature T in the measuring part for following load patterns: nonproportional 0.8 (fig. 6a), proportional (fig. 6b), tension-andcompression (fig. 6c) and torsion (fig. 6d). Performed analysis of the amplitude and average temperature recorded during tests suggests that the temperature significantly depends on the pattern of the load applied to the specimen. The highest change of the average temperature over time was observed for nonproportional load, where the temperature change stabilized at the level of about 1.15°C after 60 seconds of load application. While in case of proportional load, the average

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temperature change value stabilized at the level of about 0.48℃, for tension-and-compression – at the level of about 0.55°C and for torsion - at the level of about 0.40°C. Moreover, it was found out that the nature of the average temperature value change corresponds to the nature of the change of the highest principal strain. The highest value of the temperature amplitude was recorded for the nonproportional load. While the value of the recorded temperature amplitude for torsion was negligible. a)

0,25 ε% 0

ε1

ε2 t, s

60

63

-0,25

b) ε1

0,2 ε% 0

ε2

60

t, s 63

-0,2

c) ε2

0,1 ε% 0

t, s 63

60 ε1

-0,1

d) ε1

0,2 ε% 0

ε2 t, s

60

63

-0,2

Fig. 6. Sample curves of the specimen temperature change during first 70 seconds of variable load for the equivalent stress level of σeq = 330 MPa for nonproportional load 0.8 (a), for proportional load (b), for tension-and-compression (c) and for torsion (d)

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Analysis of test results The following parameters were calculated for the values constituting curves presented in test results paragraph: amplitudes and average temperature values, principal strain vectors εI and corresponding normal stress vectors σε, plastic strain values σPL and the plastic strain energy EPL. Authors analysed recorded hysteresis loops, to determine plastic strain value and plastic strain energy. Sample loops are presented in figure 7. It is worth to analyse the way of the hysteresis loop change with the increase of load nonproportionality (1) – zero value of nonproportionality, (4) – low nonproportional load and (5) – the load of the maximum nonproportionality. 400

σε, MPa 300

200

(1) 100

εI, % 0 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-100

(4) -200

(5) -300

-400

Fig. 7. Sample hysteresis loops for loads (1), (4) and (5) The authors analysed the possibility of correlation between the mean value of the average temperature curve of the measured specimen part surface TM and the maximum average principle strain εI (fig. 8a), plastic strain εPL (fig. 8b), plastic strain energy EPL (fig. 8c), as well as between the amplitude TA of the average temperature curve of the measured specimen part surface and the amplitude of the minimum normal stress σY (fig. 8d). It was found out that the average temperature value TM is significantly dependent on the plastic strain energy EPL and that the temperature change amplitude TA depends on the nominal normal stress σY (except for torsion). No TA change for torsion is the subject of further work. Summary The preliminary research results provided in the paper herein showed that it is possible to perform research on temperature changes of variably loaded specimens, depending on the nature of the multiaxial load. Cyclic temperature changes corresponding to the load were observed. That phenomenon currently is subject to further research, the results of which shall be presented in future conferences dedicated to fatigue.

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Fatigue Failure and Fracture Mechanics

a)

b)

c)

d)

Fig. 8. Graphical presentation of the correlation between parameters determined for the following individual multiaxial load patters (1-5): average temperature TM and maximum average principle strain εI (a), average temperature TM and plastic strain εPL (b), average temperature TM and plastic strain energy EPL (c) as well as amplitude of temperature TA and amplitude of nominal normal stress σY (d) The scientific research financed by funds of the Polish National Centre for Science. References [1] [2] [3]

[4]

N. Harwood, W.M. Cummings (eds), Thermoelastic Stress Analysis, IOP Publishing Ltd., Bristol 1991. G. Rudowski, Infrared and its application, Wydawnictwo Komunikacji i Łączności, Warszawa 1978 (in Polish). R. Litwinko, W. Oliferuk, Yield Point Determination Based On Thermomechanical Behaviour Of Polycrystalline Material Under Uniaxial Loading, Acta Mechanica et Automatica, Vol. 3, No.4 (2009), pp. 49-51. A. Lipski, The use of passive infrared thermography for tests of materials and riveted joints used in aviation industry- selected problems. Part II of the collection of monographs (edited by J. Szala): Experimental methods in studies of materials and riveted joints used in aviation industry - selected problems, Institute For Sustainable Technologies –National Research Institute in Radom, Bydgoszcz-Radom, 2010 (in Polish).

CHAPTER 5: Multiaxial Fatigue

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.171

Steel X2CrNiMo17-12-2 Testing for Uniaxial, Proportional and Non-Proportional Loads as delivered and in the Annealed Condition Dariusz Skibicki1, a, Janusz Sempruch1,b and Łukasz Pejkowski1,c 1

University of Technology and Life Science, Faculty of Mechanical Engineering, Kaliskiego 7, 85-796, Bydgoszcz, Poland a

[email protected], b [email protected], c [email protected]

Keywords: multiaxial fatigue, fatigue life, fatigue criteria, fractography.

Abstract. The article presents the results of fatigue life and fractographic testing of steel X2CrNiMo17-12-2 exposed to proportional and non-proportional fatigue loads. The following load types were applied: tension-compressive strength, torsion, proportional combined/complex loads produced by tension-compressive strength and torsion as well as non-proportional combined load – by tension-compressive strength and torsion by the phase shift angle φ=90°. The paper analyses the effect of the load method on the fatigue life and fractography of fatigue fractures recorded, and especially the effect of non-proportional load. Introduction The results presented in the present article refer to the first stage of the research project which aims at developing the methodology of research programmed for multiaxial load with non-proportionally changing components by the adaptation of uniaxial ideas of programmed fatigue research. The first stage of the project programme was planned to include the plotting of reference fatigue life plots for the uniaxial, proportional and non-proportional loading. The article also presents the fractographic analysis of the fatigue specimen fractures produced in the above-mentioned fatigue specimen types. Testing conditions Loads. The research involved the application of the following types of fully-revered, sinusoidallyvariable loads: tension-compressive strength, torsion, proportional combined loads produced by tension-compressive strength and torsion with the amplitude ratios of τa ⁄σa =0.5 and 0.826 as well as non-proportional combined loads produced by tension-compressive strength and torsion with the amplitude ratios of τa ⁄σa =0.5 and 0.826 and the phase shift angle of φ=90°. The values of the quotient of amplitudes were assumed by determining the desired non-proportional load features. For τa ⁄σa =0.5 and φ=90° the rotating vector of maximum tangent stress has the same value throughout the fatigue cycle. It is a very frequently applied type of load in many literature reports. With that in mind, it offers a possibility of a comparison of the research results. On the other hand, the ratio of amplitudes τa ⁄σa =0.826 results in the value of equivalent stress according to the Zenner formula applied in this research is constant throughout the fatigue cycle. It is therefore the load of the highest degree of non-proportionality and one must expect that for that type of load the effect of non-proportionality on the fatigue life will be more visible. Material. The research material was selected according to its sensitivity to non-proportionality. Out of 6 preliminarily selected materials (Table 1.), there was chosen the steel of the highest sensitivity to non-proportionality; X2CrNiMo17-12-2 as delivered and complete annealing. The material selection was made based on the results of specimens of monotonic tension using the relationships proposed by Borodii and Shukaeva [1]. In this paper the authors report in the relationship between the quotient of ultimate tensile strength and the yield strength σu ⁄σy , and coefficient DN expressing the change in the fatigue life under non-proportional load (the higher the value of the fraction, the lower the fatigue life under non-proportional load in relation to fatigue life under proportional load).

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Fatigue Failure and Fracture Mechanics

Complete annealing was performed in electric vacuum oven with argon protection atmosphere. The samples were heated up to the temperature of 1050°C, annealed in this temperature for 2 hours, after which it was cooled to the temperature of 620°C. Then, having been taken out from the oven/furnace, the specimen were cooled in the open air. Table 1. Calculation results for the evaluation of the effect of non-proportionality on fatigue life σu Nn β = -1 Material ln|α| = 0.705β-1.22 DN = p -1 σy N C45 0.3302 0.1030 0.5 Cu-ETP 0.0958 0.0704 0.5 MO58 0.7800 0.2137 0.55 S355J2G3 0.1679 0.0791 0.5 X2CrNiMo17-12-2 0.5924 0.1576 0.58 X2CrNiMo17-12-2 1.4570 0.6414 0.75 (annealed) The selected steel was identified in terms of its monotonic properties, chemical properties and microstructure. For the material non-annealed and annealed, the following were reported: σu = 416 MPa, σy = 210 MPa, HV10 = 234 as well as σu = 589 MPa, y = 240 MPa, HV10= 153. The results of research of the chemical composition are given in Table 2. Table 2. Chemical composition recorded for steel X2CrNiMo17-12-2 and the comparison with norm PN-EN 10088-1 Fe C Si Mn P S Cr Mo Ni 63.9 0.028 0.25 1.48 0.033 <0.005 18.03 2.22 12.3 Min 0 0 0 0 0 16.5 2.0 10 Max 0.03 1 2 0.045 0.015 18.5 2.5 13 The steel microstructure is given in Fig. 1. The metallographic specimens were digested with a water solution of the mixture of hydrofluoric and nitric acids (Mi16Fe) - 1 volumetric part of nitric acid. 2 volumetric parts of hydrofluoric acid. 3 volumetric parts of glycerine. The enlargement applied during the registration of microstructure was identical and it was 250x. The composition of a visible austenitic structure includes austenite grains with visible twins and grain borders. The austenite grain size differs considerably. According to norm PN-84/H04507-01, the grain size was determined at the enlargement of 100x. As for the specimen as delivered the grain size is g=8, while for the specimen after complete annealing g=5.

a) b) Fig. 1. Microstructure of steel X2CrNiMo17-12-2 as delivered (a) and after complete annealing (b) Specimen. The research was made using biaxial strength machine Instron 8874. Performing tension-compressive strength in the range of +/-25 kN and torsion – 100 N·m. The specimens for research were designed according to norm ASTM E2207-02 (Fig. 2).

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Fig. 2. Fatigue testing specimen Fatigue Life Analysis Uniaxial Loads. For the steel selected there were made plots of fatigue life for tension and torsion (Figs 3 and 4). The results were approximated in a double-logarithm design using linear functions. The forms of the functions are given in Table 3. Table 3. Approximating functions of fatigue life under tension-compressive strength and torsion for steel X2CrNiMo17-12-2 as delivered and in the annealed condition. Tension-Compressive σ=S·Nb Torsion strength log(σ)=b·log N +log (S) S=478, b=-0.0504, S=561, b=-0.0477, Steel as delivered R2=0.88 R2=0.91 S=833, b=-0.1119, S=3518, b=-0.2757, Annealed steel R2=0.91 R2=0.71 The fatigue plots for torsion in both cases are located under fatigue plots of tension-compressive strength. As for the material as delivered both plots are approximately parallel (close to the value of coefficient b). As for annealed material, the torsion plot is more inclined than the tension plot. The problem of out-of-parallelism of the fatigue characteristics is widely discussed by Kurek and Łagoda [2]. A comparison of the plots for the specimens from the samples of the same load, however, for the material as delivered and in the annealed condition, one can observe that in both cases of load of the plots for annealed steel are found below the plots for steel as delivered Figs 5 and 6). The plots also show a greater inclination.

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Fatigue Failure and Fracture Mechanics

Fig. 3. Fatigue life for tension and torsion specimens for nominal stresses as delivered. Symbol ‘x’ stands for non-cracked specimens.

Fig. 4. Fatigue life for tension and torsion specimens for nominal stresses for annealed material. Symbol ‘x’ stands for non-cracked specimens. ‘A’ stands for material annealed.

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175

Fig. 5. Fatigue life for specimens exposed to tension for nominal stresses as delivered and in the annealed condition

Fig. 6. Fatigue life for specimens torsion for nominal stresses as delivered and in the annealed condition Proportional Load. Based on the results of tests of tension-compressive strength and torsion, the analysis of multiaxial fatigue criteria was made in terms of the accuracy of fatigue life evaluation [3]. The best criterion was the Zenner criterion [4]. Applying that criterion, there have been proposed the values of amplitudes for proportional combined loads – tension-compressive strength with torsion with two different ratios of amplitudes τa ⁄σa =0.5 and 0.826. The results of fatigue life for uniaxial and proportional loads have been compiled in Figs 7 and 8 for steel as delivered and in Fig 9 for material after annealing. In the first case almost all the results fall within the scatter bound of factor 2, while in the other one – in the scatter bound of factor 3.

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Fatigue Failure and Fracture Mechanics

Fig. 7. Fatigue life for material as delivered for the specimen exposed to tension, torsion and proportional specimens τ ⁄σ = 0.5 and 0.826, for equivalent stresses according to Zenner. Symbol ‘x’ stands for non-cracked specimens.

Fig. 8. Comparison of calculated fatigue life with the experimental ones for the specimen of tension, torsion and proportional specimens τ ⁄σ = 0.5 and 0.826, for equivalent stresses according to Zenner, for the material as delivered.

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Fig. 9. Comparison of calculated fatigue life with the experimental ones for the specimen of tension, torsion and proportional load τa ⁄σa =0.5 and 0.826, for equivalent stresses according to Zenner, for the material after annealing. Non-proportional load. Then the fatigue life was tested under non-proportional load. The research was performed for tension with torsion with the nominal amplitude ratios of τa ⁄σa =0.5 and 0.826 and for the phase shift angle φ=90°. The results are presented in Fig. 10. The fatigue life reported under non-proportional load are much lower than the uniaxial and complex proportional loads and are decreasing with an increase in the degree of non-proportionality – for loads ⁄ = 0.826 life times are lower than for the ratio of amplitudes 0.5. Fig. 11 demonstrates fatigue life for τa ⁄σa =0.5 are located in the scatter bound of factor 3, and for τa ⁄σa =0.5 they are found beyond the bound.

Fig. 10. Fatigue life for non-proportional loads τa ⁄σa =0.5 and 0.826 and the phase angle shift between components φ=90°.

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Fatigue Failure and Fracture Mechanics

Fig. 11. Comparison of calculated fatigue life with the experimental ones for non-proportional load τa ⁄σa =0.5 and 0.826 and phase angle shift between components φ=90°. Fractographic analysis As for tension-compressive strength, the fatigue fracture is perpendicular to the direction of the effect of load (Fig. 12). Cracks were initiated from a single focus found on the specimen surface. As for torsion, for high values of stresses the direction of macro-cracking coincided with the direction of the effect of maximal tangent stresses (Fig. 13). A decrease in the amplitude of stress resulted in a change in the direction of macro-cracking by 45°, which means that they were perpendicular to the directions of principal stresses. The occurrence of cracks in both planes was characteristic. For an even lower amplitude of stress, the crack maintained the direction perpendicular to the direction of the effect of principal stresses, however, it occurred only in one of the planes of the effect of that stress. As for the proportional load with the amplitude ratios of τa ⁄σa =0.5 the cracks were perpendicular to the specimen axis (Fig. 14), while in case of load of a greater share of tangent stress, namely for the ratio of amplitudes of τa ⁄σa =0.826, the cracks were turned by about 20° as compared with the specimen axis. The direction is perpendicular to the direction of the effect of principal stresses. In both cases fatigue fractures were similar to those reported for tension-compressive strength. Non-proportional loads resulted in the pattern of the trace of the crack on the specimen surface being very irregular. As for the load with the amplitude ratios of τa ⁄σa =0.5, the directions of cracks were perpendicular to the direction of the effect of principal stresses, whereas for the load with the amplitude ratios of τa ⁄σa =0.826 the crack directions were perpendicular to the specimen axis. Conclusions The Zenner criterion allows for a satisfactory forecast of fatigue life exposed to proportional multiaxial load. This criterion, however, cannot be applied under non-proportional load. According to the forecasts, steel X2CrNiMo17-12-2 is a material which is sensitive to nonproportionality. The material sensitivity to non-proportionality is high enough to allow for revealing the effect of non-proportional load of a varied degree of non-proportionality. The fractography of fatigue fractures for uniaxial and proportional loads complies with the forecasts. The directions of cracks, on the other hand, for non-proportional load require further analyses.

Dariusz Skibicki

Fig. 12. Specimen fractography for tension-compressive load.

Fig. 13. Specimen fractography for torsion loads.

179

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Fatigue Failure and Fracture Mechanics

Fig. 14. Specimen fractography for complex, proportional loads.

Fig. 15. Specimen fractography for complex, non-proportional loads. Acknowledgements The project has been financed from the National Centre for Science References [1]

[2] [3]

[4]

M.V. Borodii, S.M. Shukaev. Additional cyclic strain hardening and its relation to material structure, mechanical characteristics and lifetime. International Journal of Fatigue. 29 (2007) 1184–1191. M. Kurek, T. Łagoda. Fatigue life estimation under cyclic loading including out-of-parallelism of the characteristics. Applied Mechanics and Materials. 104 (2012) 125-132. D. Skibicki, Ł. Pejkowski. Integrals fatigue criteria evaluation for life estimation under uniaxial, combined proportional and non-proportional loadings, Journal of Theoretical and Applied Mechanics. 4 (2012) H. Zenner, A. Simbürger, J. Liu. On the fatigue limit of ductile metals under complex multiaxial loading, International Journal of Fatigue. 2 (2000) 137-145.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.181

Estimation of fatigue life of materials with out-of-parallel fatigue characteristics under block loading Marta Kurek1, a, Tadeusz Łagoda1,b 1

Department of Mechanics and Machine Design, Faculty of Mechanical Engineering , Opole University of Technology, ul. Mikołajczyka 5, 45-271 Opole, POLAND a

[email protected], b [email protected]

Keywords: fatigue life, out-of-parallelism, fatigue characteristics, block loading

Abstract. The paper presents the algorithm of fatigue life estimation for materials with out-ofparallel fatigue characteristics under block loading. Brass CuZn40Pb2, medium-alloy steel 30CrNiMo8 and high-alloy steel 35NCD16 belong to such materials. Brass CuZn40Pb2 was used for analysis. The experimental results were compared with those calculated according to the assumed model, and satisfactory results were obtained. Introduction All machinery, equipment and structures must be characterized by the greatest durability and reliability while maintaining safety. However, there are difficulties in determining the fatigue life resulting from the inability to determine the stability using a single equation. In [1] presents an analysis of the relationship between fatigue strength of pure bending and pure torsion of selected construction materials, which amounts to determining the relative value of normal stress to shear stress σ B =

τ

af

,

(1)

af

where: σaf – fatigue limit for bending, τaf – fatigue limit for torsion. Result of work [1] to review and distribution of construction materials due to the variability of the parameter (1) depending on the number of cycles to destruction. For most materials, this relationship shows constancy. However, there is a group of materials, for which relation (1) is variable and dependent on the number of cycles. These are materials which are characterized by a lack of parallelism of the characteristics of fatigue in pure bending and pure torsion such as brass CuZn40Pb2, medium-alloy steel 30CrNiMo8 and high-alloy steel 35NCD16. For such materials, estimation of fatigue life is complicated because it must take into account the variability of the parameter B in the multiaxial fatigue criteria which take into account this parameter. They include: the criteria of Gough and Pollard [2], Nisihary and Kawamoto [3], Lee [4], Findley [5], Carpinteria, and Spagnoli [6], Ogonowski [7,8] and Walat [9]. Aim of this work is the presentation an algorithm for estimating the fatigue life under loads block for materials characterized by a lack of parallelism of the mutual characteristics of fatigue. All calculations were done for brass.

Experimental research CuZn40Pb2 brass was used for the analysis, the results of fatigue presented in [10]. Block loads are increasingly used in fatigue tests as a kind of random loads [11]. Figure 1 shows the fatigue graph for pure bending and pure torsion material analyzed.

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Fatigue Failure and Fracture Mechanics

500 400

logNf=19.9-5.86logσa

σa, τa, MPa

300

200

logNf=45.31-17.18logτa

100 4 10

10

5

10

6

10

7

N , cycles f

Fig. 1. Fatigue curves for pure bending and pure torsion for CuZn40Pb2 The authors of [10] tested 36 samples at various load levels and different order of these levels. Fig. 2 shows the geometry of the specimens used for testing.

Fig. 2. Geometry of the tested specimens Figure 3 shows an example fragment of the course stresses.

Fig. 3. Part of the course of stress Four specimens were used in the experiment for each combination of alternating blocks of torsion and bending. The block was made up of nτ cycles of torsion and nσ cycles of bending, each sinusoidal. Periods that have been adopted to study accounted for 10% durability at this level charge for NI, NII and NIII (Fig. 4). Therefore, the moments causing bending and torsion were alternately and lasted respectively, nI = 104, nII = 3 ·104 or nIII = 105 cycles, until the destruction of the sample, where: nI=0.1·NI .

(2)

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Fig.4. Applied load blocks Table 4 contains detailed characteristics of levels and amplitudes, which were calculated by using the characteristics of fatigue. Tab. 1. Levels of research alternating bending and torsion and the corresponding number of cycles in the blocks [10] number of cycles type of load

bending

nσ

nτ

σ a , MPa

τ a , MPa

Ig

10 4

-

364

-

II g

3·10 4

-

302

-

245

-

-

222

-

209

-

194

III g Is torsion

stress amplitude

load level

II s III s

10 -

5

10

4

-

3·10

-

5

10

4

The algorithm of fatigue life assessment Stages of the algorithm of fatigue life determination under block loading are shown in Fig.5. The initial number of cycles taken for the calculation of the first cycle algorithm is Ni = 106. The first stage of proceedings is to record the input data, which are the stress courses coming from bending and torsion in their blocks: σ xx (t ) = σ a sin(ωt ) ,

τ

xy

(t ) = τ sin(ωt ) a ,

(3)

(4)

where: σa – amplitude of normal stress coming from bending, τa – amplitude of shear stress coming from torsion, ω – angular frequency, t – time.

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Fatigue Failure and Fracture Mechanics

Fig. 5. Fatigue life determination under block loading The next important step in fatigue life determination is determination of the critical plane orientation angle corresponding to the maximum effort of the material. In the paper, the critical plane position was determined with the method of damage accumulation. For loads of block accumulation was calculated respectively for bending: j  1  Sσ (α ) = ∑  n(σ ) i =1  N f   

(5)

and for torsion: j  1 Sτ ( α ) = ∑  i =1  N f 

 n( τ ),  

(6)

where Nf is the number of cycles corresponding to the shear stress from the bending, respectively: 1

τ ηs (t , α ) = − σ xx (t ) sin 2α 2

(7)

and for torsion: τ ηs (t , α ) = τ xy (t ) cos 2α ,

(8)

according to the standard of ASTM [12]. The course of stress (7) and (8) set the amplitude. Then, in the case of bending fatigue life are determined from the equation (7) log N f (α ) = Aσ − mσ log τ ηsa (α ) ,

(9)

Dariusz Skibicki

185

where: Aσ, mσ - coefficients of the equation for bending, In the case of torsion τηsa substitute for the value of the formula (8) and fatigue life are determined from the equation: log N f (α ) = Aσ − mσ log τ ηsa (α ) .

(10)

After summing up the cumulative (5) and (6) search angle, for which the cumulative value was the maximum: S (α ) = Sσ (α ) + Sτ (α ) ,

(11)

α = max(S (α )) .

(12)

The criterion on the plane of maximum shear stresses [7] was applied in analysis. According to this criterion, the history of equivalent stress can be written as: σ eqa = (2 − B) ⋅ σ ηa + B ⋅τ ηsa ,

(13)

where: σ ηa = σ xxa cos 2 α + τ xya sin 2α ,

(14)

1

τ ηsa = − σ xxa sin 2α + τ xya cos 2α . 2

(15)

In the case of bending equation (14) and (15) reduces to: σ ηa = σ xxa cos 2 α ,

(16)

τ ηsa = σ xxa sin 2α .

(17)

For the torsion: σ ηa = sin 2α ⋅τ xya ,

(18)

τ ηsa = τ xya cos 2α .

(19)

Variability parameter B (Nf) is included in the multiaxial fatigue criterion (13). To calculate the fatigue life (Ncal) the method of the average amplitude of the damage [13]. Modified amplitude was calculated according to the expression: σ md =

n(σ ) ⋅ σ σeqa (σ ) mσ +1 + n(τ ) ⋅ σ τeqa (τ ) mσ +1 n(σ ) ⋅ σ σeqa (σ ) mσ + n(τ ) ⋅ σ τeqa (τ ) mσ

.

(20)

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Fatigue Failure and Fracture Mechanics

Then, looking for fatigue life are determined from the formula [13]:  σ σeqa N f = n(σ ) ⋅   σ md

   

mσ

 σ τeqa + n(τ ) ⋅   σ md

   

mσ

.

(21)

The presented algorithm is based on the method of iterations, so the rest of the work to calculate the ratio between constant obtained by initial: ∆=

N i +1 Ni

.

(22)

It should be noted that Ncal calculated from equation (21) is the sum of two blocks of fatigue life originating from the bending and torsion which represents 20% of the total according to equation (2). This procedure is repeated for successive calculated fatigue lives to the moment when the following condition is satisfied: 0.99<∆<1.01.

(23)

Thus, the error at the level 1% was assumed, what is satisfactory in the case of fatigue calculations of machine element and structures. If the condition (23) is fulfilled, the obtained fatigue life is the searched quantity. Comparison of the calculated and experimental fatigue lives Fig. 6. presents comparison of the calculated and experimental fatigue lives if the calculated fatigue life was calculated according to the expression: N f = 5 ⋅ (n(σ ) + n(τ )) .

(24) 10

10

10

7

IIt-Ib IIt-IIb IIt-IIIb It-IIb IIIt-Ib IIIt-IIb It-IIIb IIIt-IIIb It-Ib

6

5

2.3 4

10 4 10

10

5

10

6

10

7

Fig.6. Comparison of the calculated fatigue life Ncal according to the criterion in the plane of maximum shear stresses and the experimental fatigue life Nexp for CuZn40Pb2 when B=const

Dariusz Skibicki

187

Fig.7 shows comparison of calculated and experimental fatigue lives for brass CuZn40Pb2 when B depends on (1).

Ncal, cycles

10

10

10

7

IIt-Ib IIt-IIb IIt-IIIb It-IIb IIIt-Ib IIIt-IIb It-IIIb IIIt-IIIb It-Ib

6

5

2.3 4

10 4 10

10

5

10

6

10

7

Nexp, cycles

Fig.7. Comparison of the calculated fatigue life Ncal according to the criterion on the plane of maximum shear stresses with the experimental fatigue life Nexp for CuZn40Pb2 when B(Nf) From the figures it appears that in the case of including variability of the coefficient B(Nf) all the results are included into the scatter band of the coefficient 2.3. If B is constant, the results are worse. Conclusion 1. From analysis of the obtained results it appears that the proposed algorithm can be applied for calculations of the fatigue life under block loadings for the materials with out-of-parallel characteristics for pure bending and pure torsion. The performed verification gave satisfactory results. 2. From analysis of simulation tests it appears that the best results are obtained when B is a function of a number of cycles. 3. The proposed model should be a subject of further analyses for other materials and loadings. "Project financed from the funds of the National Center of Science, awarded according to the decision No. DEC-2011/01/N/ST8/06900" References [1] Kurek M., Łagoda T., Comparison of fatigue characteristics for some selected structural materials under bending and torsion, Materials Science, Vol. 47, No. 3, November, 2011, pp. 334-344. [2] Gough H.J., Some Experiments on the Resistance of Metals to Fatique under Combined Stresses, London: His Majesty’s Stationery Office, 1951. [3] Nishihara T, Kawamoto M., The strength of Metals under Combined Alternating Bending and Torsion with Phase Difference. Memoirs of the College of Engineering, Kyoto Imperial University, Vol.X, No. 6, 1941. [4] Lee S.B., A criterion for fully reversed out–of–phase torsion and bending, Multiaxial fatigue ASTM STP 853, Philadelphia1985, pp.553–568. [5] Findley W.N., A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending, Journal of Engineering for Industry, 1959pp.301–306.

188

Fatigue Failure and Fracture Mechanics

[6] Carpinteri A., Spagnoli A., Multiaxial high–cycle fatigue criterion for hard metals, Int J Fatigue 23, 2001, pp.135–145. [7] Łagoda T., Ogonowski P., Criteria of multiaxial random fatigue based on stress, strain and energy parameters of damage in the critical plane, Mat.-wiss. u. Werkstofftech, Vol.36, No 9, 2005 pp. 429-437. [8] Walat K., Kurek M., Ogonowski P., Łagoda T.: The multiaxial random fatigue criteria based on strain and energy damage parameters on the critical plane for the low-cycle range, International Journal of Fatigue, 37, 2012 ss. 100-111 [9] Walat K., Łagoda T., Application of the covariance on the critical plane for determination of fatigue life under cyclic loading, Procedia Engineering, Vol. 2, 2010, pp. 1211–1218 [10] Kohut M., Łagoda T., Trwałość zmęczeniowa elementów wykonywanych z mosiądzu MO58 w warunkach blokowych obciążeń skręcających i zginających, Problemy rozwoju maszyn roboczych Konferencja Naukowa Zakopane, 2007, pp.165-167 (in Polish). [11]Skibicki D. Experimental verification of fatigue loading nonproportionality model, Journal of theoretical and applied mechanics, 2007, 45, 2, 337-348. [12] ASTM E 739-91, Standard practice for statistical analysis of linearized stress–life (S–N) and strain life (ε–N) fatigue data, in: Annual Book of ASTM Standards, Vol. 03.01, Philadelphia, 1998, pp.614–620. [13] Łagoda T., Sonsino C.M., Comparison of different methods for presenting variable amplitude loading fatigue results, Matt. – wiss. u. Werkstofftech, 2004, 35, No.1 pp. 13-19.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.189

Criteria evaluation for fatigue life estimation under proportional and non-proportional loadings Łukasz Pejkowski1,a, Dariusz Skibicki1,b 1

Faculty of Mechanical Engineering, University of Technology and Life Sciences in Bydgoszcz, Poland a

[email protected] (corresponding author), [email protected]

Keywords: multiaxial fatigue, non-proportional loading, fatigue criteria, fatigue life estimation

Abstract. The paper presents the fatigue criteria, representing the integral criteria, most frequently reported in literature. They were verified for uniaxial loadings and for combined: tensioncompression with torsion both proportional and non-proportional. The verification involved a comparison of the fatigue life reported based on the criterion with the experimental fatigue life. Introduction A special case of a varied state of stress is the state causing a change in the directions of the principal axes. The loading triggering such a state is referred to as non-proportional. The experiments show that this type of loading results in a considerable decrease in fatigue strength and fatigue life. The paper analyses the criteria representing the group of integral criteria in terms of the applicability for the fatigue calculations in the cases of proportional and non-proportional loadings. Criteria selected for the analysis Integral approach to the Huber-Mises-Hencky hypothesis. The idea of integral criteria was initiated by Novoshilov [1] who proved that both the second invariant of the deviator of the stress state, [2] as well as the equivalent stress according to the HMH hypothesis [3] can be interpreted as the mean squared value from stresses tangential for all the physical planes passing through the material point considered: =

15 8

sin

. (1)

Symbols and in the above notations denote the angles of the polar system in which the location of the physical plane is described [1, 2]. The Zenner Criterion. The integral approach was developed by Zenner. In his papers [2,3], he claims that the HMH hypothesis assumes a constant ratio of the fatigue limits for tension and torsion . According to HMH, it is 1⁄√3, and in real life it falls within the range 0.5 < ( ⁄ ) < 0.8 for ductile materials. The HMH hypothesis provided as in the formula (2) suggests that the value of tangential stress acting on the physical plane is the only one which affects the material fatigue behaviour. In the real life also the changing value of normal stress affects fatigue. The author of the criterion claims that those facts should be reflected in the calculations. The effect of those considerations is the criterion which is expressed in the following formula: #

=

15 8

($

,&

'1 + )

,*

+ +,

,& '1

−.

,& +

) sin

≤

. (2)

190

Fatigue Failure and Fracture Mechanics

Coefficients a, b, n and m are determined based on the material constants [3]. Papadopoulos Criterion 1 (1997). The first Papadopoulos criterion was formulated based on the considerations on the crystalline microstructure of metals. He claims that the main reason for the initiation of the fatigue crack are plastic slips in metal grains [4,5]. They can be triggered even in the case of slight external loadings, e.g. during high-cycle fatigue. It happens so since the grains are provided with the systems of easy slip. According to the author of the criterion, the tangential stresses acting in the directions of slips are the main quantities with the effect on the initiation of the crack. The criterion proposed is expressed in the relationship [4,5,6]: 1〈3& 〉 + $ )$56 〈7〉 ≤

. (3)

In the above formula 〈3& 〉 stands for the mean squared value of the generalised amplitude of tangential stress 3& , )$56 〈7〉 stands for the maximum value the mean normal stress reaches during the loading cycle and coefficient $ is calculated based on the material constants.

Papadopoulos Criterion 2 (2001). The second of the Papadopoulos criteria assumes widening of the term of uniaxial fatigue limit by the cases of mulitaxial loading [7]. At the same time the author claims that the criterion of functionality in the engineering term should divide the stress levels into safe and unsafe and to estimate the levels based on simple material data without going into microstructural aspects. The criterion has been written below: max 3& + ;<

, *&=

≤

, (4)

where 3& stands for the generalised amplitude of tangential stress, , *&= is as maximum hydrostatic stress and ;< - the coefficient determined based on material constants. Analysis of the criteria Criteria analysis method. The paper compares the experimental fatigue life values with the calculated ones. The equivalent stress values, calculated based on the criteria, have been related with fatigue life values based on the Basquin equation [8]. Coefficients of the equation were determined drawing on the adequate uniaxial tests [9]. Then there were defined the statistical parameters in a form of mean dispersion of fatigue life 3? and mean squared error fatigue life estimation 3@ A , calculated according to formula [10]: 7K=L,M 1 3? = 10BC ; EC = F log J P ; 3@ . 7N&O,M Q

M R

A

= 10BSTU ; E@

A

=

7 ∑QM R WXY Z K=L,M [ 7N&O,M . (5) .

Figures providing a comparison of the values of mean squared error. The figures below present a comparison of the values of mean squared error 3@ A . If, for a given type of loading, the mean dispersion of fatigue life assumed the negative value, which points to the overestimation of fatigue life, indicator showing the value of mean squared error was directed towards the negative side, which can be reflected with the relationship: \]Y.(3? − 1)3@ A .

Dariusz Skibicki

6 4 2 0 -2 -4 -6 -8 -10

191

T-C T P HMH INT

Zenner

Papadopoulos 1

N

Papadopoulos 2

Fig. 1. Comparison of the mean squared error of fatigue life estimation for aluminium alloy 7075T651 [11]. 10 7,5

> 6000

T-C

5

T

2,5 0 -2,5

P2 HMH INT

Zenner

Papadopoulos 1

-5

Papadopoulos 2

< -130

-7,5

N2 N0,5

-10

Fig. 2. Comparison of the mean squared error of fatigue life estimation for 1045 steel [12]. 50

T-C

>100

40

T

30

P0,5

20

P1

10

P10 P2

0 -10

HMH INT

< -200 Zenner

Papadopoulos 1

Papadopoulos 2

P4

Fig. 3. Comparison of the mean squared error of fatigue life estimation for steel 1045 [13]. 100 90 80 70 60 50 40 30 20 10 0 -10 -20 -30

T-C

> 1900

T P 0,5 P 0,8 N 0,5

< -1000 HMH INT

Zenner

Papadopoulos 1

N 0,8

Papadopoulos 2

Fig. 4. Comparison of mean squared error of fatigue life estimation for X2CrNiMo17-12-2 steel [14].

192

Fatigue Failure and Fracture Mechanics

Conclusions To use the experimental fatigue life data, one can state that: 1. Relatively the best results were reported by applying the second Papadopoulos criterion and the Zenner criterion, and the worst – by applying the integral approach according to the Huber-Mises-Hencky hypothesis. 2. None of the criteria analysed can be applied to estimate the fatigue life under nonproportional loadings for materials that are sensitive to non-proportional loads. 3. The integral approach can be effective under non-proportional loadings, however the range of use should be proven by comparision to wide number of experimental results.. References [1] [2]

[3] [4] [5] [6] [7] [8] [9]

[10]

[11] [12]

[13] [14]

H. Zenner, I. Richter, Eine Festigkeitshypothese für die Dauerfestigkeit bei beliebigen Beanspruchungskombinationen, Konstruktion. 29 (1977) 11-18. H. Zenner, R. Heidenreich, I. Richter, Schubspannunsintensitätshypothese – Erweiterung und experimentelle Abstützung einer neuen Festigkeitshypothese für schwingende Beanspruchung, Konstruktion. 32 (1980) 143-152. H. Zenner, A. Simbürger, J. Liu, On the fatigue limit of ductile metals under complex multiaxial loading, International Journal of Fatigue. 22 (2000) 137-145. I. V. Papadopoulos, A high-cycle fatigue criterion applied in biaxial and triaxial out-of-phase stress conditions, Fatigue & Fracture of Engineering Materials & structures. 18 (1995) 79-91. I. V. Papadopoulos, A new criterion of fatigue strength for out-of-phase bending and torsion of hard metals, Fatigue. 16 (1994) 377-384. I. V. Papadopoulos et al., A comparative study of multiaxial high cycle fatigue criteria for metals, International Journal of Fatigue. 19 (1997) 219-235. I. V. Papadopoulos, Long life fatigue under multiaxial loading, International Journal of Fatigue. 23 (2001) 839-849. R. I. Stephens, A. Fatemi, R. R. Stephens, H. O. Fuchs, Metal fatigue in engineering, Willey Interscience, New York, 2001. D. Skibicki, Ł. Pejkowski, Integrals fatigue criteria evaluation for life estimation under uniaxial, combined proportional and non-proportional loadings, Journal of Theoretical and Applied Mechanics. 50 no 4 (2012). K. Walat, T. Łagoda, Fatigue life of machine elements on the critical plane determined by the stress covariance extremum (in Polish), Oficyna wydawnicza Politechniki Opolskiej, Opole, 2011. E. N. Mamiya, F. C. Castro, R. D. Algarte, J. A. Araujo, Multiaxial fatigue life estimation based on a piecewise ruled S - N surface, International Journal of Fatigue. 33 (2011) 529-540. Y. Verreman, H. Guo, High-cycle fatigue mechanisms in 1045 steel under non-proportional axial-torsional loading, Fatigue & Fracture of Engineering Materials & Structures. 30 (2007) 932-946. D. L. McDiarmid, Multiaxial fatigue life prediction using a shear stress based critical plane failure criterion, Fatigue design Vol. 1, Technical Research Center of Finland. (1992) 21-33. D. Skibicki, J. Sempruch, Ł. Pejkowski, Steel X2CrNiMo17-12-2 Testing for Uniaxial, Proportional and Non-Proportional Loads as delivered and in the Annealed Condition, Materials Science Forum. (2012).

CHAPTER 6: Fatigue Crack Growth

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.195

Fracture toughness of structural members Neimitz Andrzej Kielce University of Technology, Al. 1000 lecia P.P.,25-314 Kielce [email protected] Keywords: fracture toughness, in-plane constraint, out-of-plane constraint

Abstract. In the paper several formulae to compute the fracture toughness are presented. The formulae include either parameter characterizing the in-plane constraint or out-of-plane constraint or both. The formulae are based on different assumptions and approaches to fracture mechanics. Namely, small or finite strains were assumed, global or local approach was adopted. In all cases the standard, plain strain fracture toughness was used as a reference state. Introduction It is well known that fracture toughness is not a material property. It depends on the shape and size of the structural element. Fracture toughness measured according to the proper standards [1],[2],[3] (KIC, JIC and δTC) is most often used in fracture criteria. It is so, because such measured a critical value belongs to the smallest from the all other fracture toughness’s measured using specimens with a shorter crack and a smaller thickness than defined in the standards. It was shown in Fig.1, where, the fracture toughness’s measured according to the standards requirements are those values for the relative crack length, a/W, from the range (0.45 – 0.65). However, using KIC, 40H SEN(B) T TEMP=680 0C B=16mm, W=25mm

2000

potential drop compliance change

JQ [kN/m]

1600 1200 800 400 0 0

0.2

0.4 a0/W

0.6

0.8

Figure. 1. The influence of the relative crack length on fracture toughness. Triangles denote results obtained using potential drop method to measure the crack extension. Rhombuses denote results received using the compliance change technique. JIC or δTC one receives conservative; thus, save results; they are not always economically acceptable. In this paper several models are presented to compute the “real” fracture toughness of the structural elements. They will be based on the parameters which characterize the stress field in front of the crack, namely Q – introduced by O’Dowd and Shih [4], and = introduced by Guo

[5]. The former one represents the influence of relative crack length; the later one represents element’s thickness. Models concern ductile materials, which are the most often used for structural elements. Thus, not only results concerning the stress fields received using the assumption of the small strains were utilized. Also, the stress analysis using finite strains, more realistic one, was used.

196

Fatigue Failure and Fracture Mechanics

Model based on the experimental results Recent engineering procedures, SINTAP [6], and later FITNET [7] adopted the formula, received by approximation of experimental results [8]

[

c K mat = K mat 1 + α (−Q) k

]

(1)

where Kmat is the critical stress intensity factor (SIF) computed from the formula:

=

(

,

)

and coefficients α, k were given for selected steels in SINTAP or they can be found in the look-up tables after the time consuming calibration procedure which must be performed to estimate the parameters of the Weibull distribution [9],[10], J is J-integral, E is Young modulus, ν is Poisson ratio. Analytical models based on the assumption of small strains Classical Hutchinson [11], Rice and Rosengren [12] solution (HRR) for stress distribution in front of the crack was corrected by O’Dowd and Shih [4] to take into account the in-plane constrains: 1

 EJ  1+n σ ij = σ 0  2  σ~ij (θ , n ) + Qσ 0σˆ ij (θ )  ασ 0 I n r 

(2)

where σ0 is yield stress, α and n are Ramberg-Osgood coefficients, r and θ are polar coordinates, and are functions which can be found using computer program [13]. Guo [5] derived approximate formula for the stress and strain fields in front of the crack in tridimensional geometry. Guo’s formula can be extended to the form similar to Eq. (2) [14]. Functions and depend now on Tz parameter and they can be also found using program [13]. Function Q should be replaced by Q* and the later one can be computed similarly to Q [4] but the reference HRR state must be replaced by the Guo state. Tz parameter may characterize the out-ofplane constraint. O’Dowd [15] assumed that the critical state in front of the crack is met when the opening stress σ22, riches the critical value, σc, which was assumed to be a material constant, at certain distance rc, which also was assumed to be a material constant. Selecting the specimen geometry, thickness and crack length, according to standards [2] and assuming that for this reference state the Q parameter is equal to zero, O’Dowd received a very simple formula:  Q   J C = J IC 1 −   σC /σ0 

n +1

(3)

When in the reference state Q≠0, Eq.(3) can be written in the form:  σ − Qσ 0   J C = J IC  C  σ C − Qref σ 0   

n +1

(4)

Using Guo’s approach without the second term correction, assuming that the Q*=0, the appropriate formula is as follows: J C = J IC

I n (Tm , n)  σ~22 (Tm = 0.5, n)    I n (Tm = 0.5, n)  σ~22 (Tm , n) 

(1+ n )

(5)

In Eq.(5) it was assumed that at the reference state the average through the thickness value of Tz function, Tm is equal to 0.5. When one wishes to take into account both the in-plane and out-ofplane constraints may use another form

Dariusz Skibicki

J C = J IC

197

I n (Tm , n)  σ~22 (Tm = 0.5, n)  Qm* (n, Tm ,σ 0 , a W )  1−   I n (Tm = 0.5, n)  σ~22 (Tm , n)  σC σ0 

(1+ n )

(6)

In Eq.(6) the reference state (standard specimen) assumes that Q*=0 and Tm=0.5, If in the standard specimen ∗ ( ) ≠ 0 the appropriate formula is as follows:

J C = J IC

 σ~ (T = 0.5, n)  Q* (n, T , σ , a W )   m 0 1 − m  ⋅  22~ m   σ σ σ ( T , n ) 22 m C 0   I n (Tm , n)    I n (Tm = 0.5, n)  σ − Q* σ   C m 0     *  − σ Q σ m ( ref ) 0   C 

(1+ n )

(7)

Analytical models based on the assumption of finite strains Usually, structural steels are ductile at the service temperature. In such a case the strains within plastic domain in front of the crack are large, on the order of several or more than 20 per cent. Thus, the assumption, that the strains are finite seems to be quite obvious. However, there are no closed form analytical solutions for the stress-strains fields in front of the crack in the case of finite strains. Nevertheless, the finite element solutions provide an interesting picture concerning the stress fields. Stresses are finite and the maxima of all the normal components are observed at certain small distance from the crack tip. When the external loading increases, the opening stress maximum moves out from the crack tip. If plasticity is constrained, it means that the Q-parameter does not drop much below zero and the average value of Tz parameter at the maximum location does not drop much below 0.4, the stress maximum remains constant. In such a case the following formula can be derived [16]   Q  J C = J IC 1 − max  σ  22 / σ 0  

n +1

(8)

where "" # is the maximum opening stress. It should be computed numerically or approximated using formula [16]: ""

$

#

= %&

' $ -('⁄(, $ ⁄* ) , ++ ( * /"

(9)

where coefficients c, and d can be found in [16]. If plasticity is not constrained, the plastic zones are large – even the total yielding is observed, the stress maximum drops slightly (4%-5%) when external loading increases. In such a case one can use the following formula. 01 = 021 31 −

$

− 6(

""

(

#)

78$ − ( "" # "" )79$

#)

79$ :

(

;

)

(10)

Numerical and experimental investigations allowed to formulate the fracture criterion [16] similar but not identical to the famous Ritchie, Knot and Rice [17] postulate. It was assumed that cleavage fracture may take place if the opening stress in front of the crack assumes higher value than the critical stress σc, over the critical length lc. The critical length was assumed to be a material constant, independent of the temperature, also the critical stress σc, was assumed to be a material constant; however, this quantity depends on the temperature. Example values of the critical stress and the critical distance are listed in Table 1.

198

Fatigue Failure and Fracture Mechanics

Table 1. An example values of the critical stress and critical distance [18]

Temperature -200C -500C

Ferritic-bainitic material (13HMF steel) critical stress critical distance (MPa) (µm) 1300-1350 ~250 ~1575

~250

Ferritic material (13HMF steel) critical stress critical distance (MPa) (µm) 970 ~270 1020

~270

In Fig.2 the stress field is shown for two different specimens made of the same steel. Specimens contain cracks of different relative length a/W=0.5 and a/W=0.2. Rhombs denote the state right before stable crack growth, triangles show the state right before the unstable crack growth for a/W=0.5 and the state during stable, ductile crack propagation for a/W=0.2. For the long crack the stress maximum moves from the crack tip until stresses are higher than the critical value over the critical length lc. If it happens, the unstable crack jump is observed. For the short crack situation changes because the plasticity is free to expand and the Q-parameter drops down. As a result the opening stresses decrease below the critical value and cleavage is not observed. However, if plasticity evolution is limited, e.g. by microstructure and/or temperature the stress maximum remains constant when external loading increases (Fig.3).

Fig.2. Material: ferritic bainitic, Temp. -20oC, a/W=0.5, solid symbols, a/W=0.2 empty symbols, rhombuses – crack before the stable growth, triangles – crack before the cleavage jump. Critical stress, σcrit=1300 MPa, lcrit=200 µm.

Dariusz Skibicki

199

Fig.3 Material: martensitic, Temp. -80oC, a/W=0.2, rhombuses: opening stress, circles: effective stress, triangles: accumulated effective plastic strains. The characteristic features of the stress field in front of the crack, measured numerically using the assumption of finite strains, allow for a simple geometric model, which in turn allows formulating the fracture criterion (Fig.4).

Fig.4. Material: ferritic, Temp. -20oC, a/W=0.5. Broken line: beginning of the stable crack growth. Solid line: before the cleavage crack

200

Fatigue Failure and Fracture Mechanics

Fig.5. The scheme, helpful to derive the failure criterion. Symbols in Figs 4 and 5 denote: lcrit - critical length (material constant), lact - actual length where opening stresses are higher than the critical value, ψ - coefficient, computed numerically which is used to define the distance of the stress maximum location from the crack tip. If extension of the plastic zone is large this coefficient is usually constant and it is close to 1. For low in-plane constraint it can slightly decrease when the external loading increases. Simple analysis of the similar triangles in Fig.5 leads to the following formulae: J crit =

lcσ 0 (σ max − σ 0 ) 2ψ (σ max − σ c )

(11)

or using the relationship between the external loading and the J-integral [19] 1

 l  n +1 Pcrit = Pact  crit   lact 

(12)

where Pcrit, and Pact are critical and actual external loadings respectively. Next we compare two states; the critical one reached for standard specimen and the critical one for the specimen containing shorter crack than a/W=0.45. To simplify computations we replace the σmax value with the opening stress value computed using HRR formula at the location of the stress maximum, (σ 22 )HRR(ψ ) .   Qσ 0 1 +  ψ  (σ 22 )HRR(ψ ) − σ 0  J c = J IC ψ*  Qσ 0 1 +   (σ 22 )HRR(ψ ) − σ c 

(13)

where ψ* is the value computed for shorter crack. Usually both ψ* and ψ are close each other and the ratio ψ/ ψ* can be neglected. In deriving Eq.13 the self-similarity of the HRR fields was utilized, ( "" )<==(>) = ( "" )<==(>∗ .

Dariusz Skibicki

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Using the same arguments as above the formula can be derived for the situation in which nonstandard specimen is not dominated by plane strain state.

[ [

][ ][

ψ (σ 22 )ψpstrain − σ c Qσ 0 + {(σ 22 )m }GUOψ * − σ 0 J c = J IC ψ * (σ 22 )ψpstrain − σ 0 Qσ 0 + {(σ 22 )m }GUOψ * − σ C

] ]

(14)

In Eq.14, ( "" >?@ is computed from the HRR formula at the stress maximum, A( "" BCDE>∗ is computed from the Guo formula at the stress maximum, and average value of this maximum is computed. Summary In the paper several different approaches to determine the real fracture toughness of the structural element were presented. They take into account both the in- and out-of plane constraint measures. All approaches require numerical computation to determine both the Q parameter and Tz (Tm) parameters. Finite strains analysis requires, in addition, computation of the maximum opening stress in front of the crack and location of this maximum. Acknowledgements: The financial support from Polish Ministry of Science and Higher Education under contract N N501 199640 is gratefully acknowledged. Literature [1] E 1820-05 Standard Test Method for Measurement of Fracture Toughness. ASTM, 2005. [2] ASTM (2002), ASTM E 399: Standard Test Method for Plain-Strain Fracture Toughness of Metallic Materials, ASTM International. [3] BS 5762:1979, Method of crack opening displacement (COD) Testing, (1979) [4] O’Dowd, N.P., Shih, C.F., „Family of crack-tip fields characterised by a triaxiality parameter-I. Structure of fields”, Journal of the Mechanics and Physics of Solids, Tom 39, str. 898-1015 (1991 [5] Guo W. Elastoplastic three dimensional crack border field - I. Singular structure of the field. Engineering Fracture Mechanics 1993; 46(1):93-104. [6] FITNET, Report (European Fitness-for-service Network). Edited by Kocak M., Webster S., Janosch J.J., Ainsworth R.A., Koers R., Contract No. G1RT-CT-2001-05071, 2006 [7] SINTAP: Structural Assessment Procedure for European Industry, Final Procedure, 1999, BriteEuram Project No. BE95-1426, British Steel. [8] Ainsworth R.A., "A constraint based failure assessment diagram for fracture assessment". International Journal of Pressure Vessels and Piping, 64, 277-285,(1995) [9] Sherry, A.H., France C.C., Goldthorpe M.R., „Compendium of T-stress solutions for two and three dimensional cracked geometries”, Fatigue and Fracture of Materials and Structures, 1995, Vol. 18, No. 1, str. 141-155. [10] Sherry, A.H., Wilkes M.A., Beardsmore D.W., Lidbury D.P.G. (2005), “Material constraint parameters for the assessment of shallow defects in structural componenets – Part I: Parameter solutions”, Engineering Fracture Mechanics, &@, str. 2373-2395 [11] Hutchinson, J.W., „Singular behaviour at the end of a tensile crack tip in a hardening material”, Journal of the Mechanics and Physics of Solids, Tom 16, str. 13-31 (1968). [12] Rice, J.R., Rosengren, (1968), G.F., Plane Strain Deformation Near a Crack Tip in a Powerlaw Hardening Material, Journal of the Mechanics and Physics of Solids, 16, pp.1-12,

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Fatigue Failure and Fracture Mechanics

[13] Graba–Gałkiewicz, http://php.tu.kielce.pl/~pgfm/html/hrr.htm see also: Gałkiewicz J., Graba M., Algorithm for Determination of σ~ij (n,θ ) , ~εij (n,θ ) , u~i (n,θ ) , d n (n) , I n (n) Functions in HutchinsonRice-Rosengren Solution and its 3d Generalization, Journal of Theoretical and Applied Mechanics, 44, 1:19-30, (2006). [14] Neimitz A., Graba W., Analytical-Numerical Hybrid Method to Determine the Stress Field in Front of the Crack in 3D Elastic-Plastic Structural Elements, Proceedings of XVII European Fracture Conference. Brno, Electronic version, Book of Abstracts p.85 (2008) [15] O’Dowd N.P. (1995), “Applications of two parameter approaches in elastic-plastic fracture mechanics”, Engineering Fracture Mechanics, Vol. 52, No. 3, 445-465. [16] Neimitz A., Graba, M., Gałkiewicz J. „An alternative formulation of the Ritchie-Knott-Rice local fracture criterion”, Engineering Fracture Mechanics, 74, 8, str. 1308-1322, (2007) [17] Ritchie, R.O., Knott, J.F., Rice, J.R., (1973), “On the Relationship Between Tensile Stress and Fracture Toughness in Mild Steels”, Journal of the Mechanics and Physics of Solids, 21, pp. 395-410. [18] Neimitz A., Gałkiewicz J., Dzioba I., “The ductile to cleavage transition in ferritic Cr-Mo-V steel: A detailed microscopic and numerical analysis”, Engineering Fracture Mechanics, vol.77, pp.2504-2526, 2010 [19] Kumar V., German M.D., Shih C.F., “An Engineering Approach for Elastic-Plastic Fracture Mechanics”, Electric Power Research Institute. Palo Alto, Ca (1981) EPRI Report NP-[67]1931

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.203

Fatigue crack growth rates of S235 and S355 steels after friction stir processing KOCANDA Dorota1,a, HUTSAYLYUK Volodymyr1, SLEZAK Tomasz1, TORZEWSKI Janusz1, NYKYFORCHYN Hryhorij2,b , KYRYLIV Volodymyr2 1

Military University of Technology, 2 gen. S. Kaliski Str., 00-908 Warsaw, Poland

2

G.V.Karpenko Physico-Mechanical Institute of NASU, 5 Naukova Str., 79060 Lviv, Ukraine a

b

[email protected], [email protected]

Keywords: carbon steel, fatigue crack growth rate, friction stir processing

Abstract. In the study, there were investigated the effects of friction stir processing (FSP) which was applied in order to improve the surfaces of notched specimens made of S235JR and S355J2 carbon steels, on their fatigue crack growth rates in the air. There were presented the results of comparative fatigue tests conducted at asymmetric tension (R= -0.2) for these steels treated by means of FSP and for the ones in the delivery state. The method of successive etched material layers used revealed the presence of internal tensile stresses in the surface layers of treated specimens. Crack growth rates were described on the basis of non-linear fracture mechanics, taking the effects of internal stresses into account. 1. Introduction Over the last decade, there has been observed strong interest in friction treatment as a new technology used mainly to bond metallic materials (Friction Stir Welding, abbr. FSW) that are difficult to join by means of traditional methods without reducing their fatigue properties. Another prospective application of friction treatment is the possibility of modifying the surfaces of components and forming multi-layer structures of various fatigue strength and operating properties by means of combining the friction method with mechanical stirring of substrates. This modified form of FSW treatment is called as Friction Stir Processing (FSP) [1]. The two mentioned methods are characterized by completely different ways of thermal activation of material bonding processes or modification of component surfaces when compared to traditional methods. In the FSW and FSP technologies, microstructural changes in materials do not take place for the reason that temperatures reached in friction nodes are significantly lower. The effectiveness of the mentioned methods depends on the selection of treatment parameters such as tool pressure force applied on a component surface, the tool's rotational and linear speeds as well as the tool's inclination angle to the surfaces being bonded or the components being modified. The shape and type of a welding tool have a considerable effect on the method's effectiveness as well [2]. On an industrial scale, FSW is used mainly as an advanced technology for welding of light alloys of Al, Mg, Cu and Ti. Welding of the mentioned materials with the use of conventional methods is significantly limited considering the formation of undesirable structural changes in the material as well as defects as a result of supplying large amounts of heat to the welding area. In aircraft structures, the quality of bonding between sheets created by means of the FSW method as well as the fatigue strength of a joint are of priority significance. Comprehensive research conducted by Airbus [3] and Bombardier [4] has shown a significant improvement of static and fatigue properties of joints between components made with the use of the FSW method in relation to traditional rivet bonding. The use of friction technology to bond steel sheets meets huge difficulties. Considering high plasticization temperatures of steel, creating a FSW joint requires using high-strength materials for welding tools and machines (turning lathes and milling machines) that can produce high pressure

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Fatigue Failure and Fracture Mechanics

forces and high rotational speeds of the tool in a friction node. When the said difficulties concerning steels are overcome, instant implementation of the FSW method may be expected in various engineering applications (building engineering, power engineering, bridges, ship building, etc.). Good weld quality and the structure's grain size reduction in the area of the weld were obtained for low and medium carbon steels [5-7]. The joint's mechanical properties slightly improved in relation to the base material. At the same time, the presence of complex structural phases in the weld was observed. The joint's microhardness increased significantly, namely to 900 µHV, in relation to the parent material's microhardness, which was 250 µHV. Its profile depended considerably on the tool's rotational speed. Repeatable quality of friction welded joints is closely connected with providing invariable conditions of the welding process. To this end, primary process parameters are constantly monitored. Apart from the main application of the mentioned methods to create lasting metallic joints and multilayer structures [8], it is of particular interest to develop technologies concerning thin anticorrosive layers with properly modified structures on structural carrier materials [9]. The findings of numerous studies that have been published confirm the prospects of the said technologies from the point of view of studies that are being conducted on new methods for increasing the fatigue strength and corrosion resistance of structural steels. The study presents the results of fatigue tests on crack propagation in air of two structural carbon steels (S235JR and S355J2), which were surface treated by means of FSP method at asymmetric tension (R= -0.2). Crack growth rates in the said specimens were compared with crack growth rates of the specimens in the delivery state. The examined steels are used in building structures, bridges and ships. 2. Materials. Research methodology Two structural carbon steels of ordinary (S235JR) and of higher strength (S355J2) were selected for fatigue tests. In shipbuilding industry, they are used as hull steels. These ones are characterized by good welding properties and cold crack resistance. The steels' chemical compositions and mechanical properties are presented in Table 1. Tension tests on fatigue crack growth rate in the air of FSP treated and untreated specimens at asymmetric tension (R= -0.2) were performed for flat double edge-cracked specimens (DET). The specimens were cut out from 12-mm thick S235JR and S355J2 steel sheets. Specimen geometry is illustrated in Figure 1. In the initial state, both steels (S235JR and S355J2) show a ferritic-pearlitic structure with average grain size of 15-25 µm, however, in S235JR steel, the ferritic structure prevails. Table 1. Chemical composition and mechanical properties of S235JR and S355J2 steels C Mn Si P S Cr Ni A5 σY σUT [%] [%] [%] [%] [%] [%] [%] [%] MPa MPa S235JR 0.08 0.85 0.4 0.006 0.010 0.02 0.02 262 396 36.1 S355J2

0.19

1.67

0.48

0.013 0.006

0.02

0.01

360

534

18.5

Friction-mechanical treatment of both steel specimens was conducted on two surfaces with the use of a tool in the form of a cylindrical disc, which rotated at the speed of 50m/sec. Plastic deformation of the steel surface layers was obtained after the tool had run two times along the specimens’ axes at linear speeds that were 0.5 m/min and 10 m/min, respectively. The measure of the tool's pressure force to specimen surfaces was the depth of the tool's penetration into the material. The depth was 0.25 mm when the tool run along the specimen for the first time and 0.1 mm for the second time. Surface treatment was performed at the Physico-Mechanical Institute in Lviv, Ukraine.

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Figure 1. Geometry of specimens for fatigue tests It should be emphasized that the physical aspect of friction-mechanical treatment differs from treatment with the use of a stirring mandrel in FSW and FSP methods. As a result of contact friction, the component's surface layers heat up to a temperature that is higher than the temperature of the material's phase transformations (approximately 1100-1300°C). Simultaneously, there take place plastic deformation of the component surface, intense cooling in the air of the near-to-surface layers of the tool as well as the component and the tribological medium at the speed of the order of 103-104 degrees C/sec. Such temperature and force conditions occur during forced friction between a rotating metal disc, which plays the role as a strengthening tool, and a component being treated. High local heat-up temperature in the metallic micro-area, high contact pressure in a friction node and dynamic tool action lead to the hardening of the surface and grain refining to nano-sizes in subsurface layers. A metallographic analysis of specimen cross-section structures after friction treatment revealed grain size reduction in steel to the size of 20 – 50 nm. As a result of friction treatment, in steel components there was formed a plastically deformed surface layer with size-reduced grains of a total thickness up to 150-250 µm. It was confirmed by microhardness tests conducted on longitudinal and transverse cross-sections of steel specimens that were etched in 4% nital. Vickers microhardness in the near-to-surface layer of plastically deformed S235 steel varied from 300 to 170 µHV0.1 in the parent material, whereas for S355J2 steel, it varied from 360 to 180 µHV0.1 in the parent material. Figures 2 and 3 show the images of specimen cross-sections and the structures of the surface layers of S235JR and S355J2 steels, respectively. a)

b)

c)

d)

Figure 2. Plastically deformed surface layer in S235JR steel observed on a longitudinal crosssection (a), on a transverse cross-section (b), the martensitic-austenitic structure of the second layer (c), surface layer with size-reduced grains (d) at higher magnification The surface layer of both treated steels consists of three layers. The first one, which is a near-tospecimen surface, is a white layer (the colour is a result of etching in nital) of a thickness from several to a dozen or so micrometers. It was formed from remelted metal, which solidified again very quickly. The said layer is characterized by a very fine-grained structure of a hardness that is slightly higher than the parent material hardness. The second layer, which shows a martensitic-austenitic structure, is a strongly heat affected layer. Its hardness is considerably higher than the parent material hardness. An increase hardness in this layer results, among other things, from the diffusion of carbon into steel. The carbon comes from the decomposition of the tribologic liquid in the friction node, a very high speed of steel cooling in micro-areas as well as the tool and the tribologic liquid.

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Fatigue Failure and Fracture Mechanics

The structure of the third layer is a result of heat effect as well. Heat absorbed by the steel causes the tempering of the layer. However, the speed of cooling in this layer is too low to lead to its rehardening. The hardness of this layer, which is called a layer of tempered material, is significantly lower than the hardness of the second layer and may be even slightly lower than the parent material hardness. a)

b)

c)

Figure 3. Plastically deformed surface layer in S355J2 steel seen on a longitudinal section (a nearto-surface layer at the bottom of the image) (a), on a transverse section with size-reduced grains in the surface layer (b), at higher mag. (c) Results of an analysis of the mechanical properties of steels in the delivery state and after friction- mechanical treatment revealed slight changes (10-15%) in primary material quantities for both steels, namely yield stress σY and ultimate tensile strength σUT after friction treatment. Moreover, the plasticity of S235 steel decreased by 10%, i.e. σUT - σY interval, after surface treatment. In the case of S355J2 steel, plasticity decreased by about 4%. The results of the above-described tests prove that the surface treatment changed the static properties of steels to a little degree. However, it led to a decrease in the plasticity of both steels to a greater degree. Internal stress measurements with the use of the method of successive etched material layers were performed on both surfaces, A and B, of treated steels. Direction "x" of the component of normal stress tensor, σxx, agreed with the direction of the specimen longitudinal axis and, at the same time, with the direction of the tool action along specimen surfaces. Direction "y" and normal stress measurement, σyy, corresponded with transverse stresses in specimens. By means of the method of successive etched material layers, the level of stresses in the surface layer was assessed to the depth of 0.1 mm. The results of the mentioned tests prove the operation of high longitudinal tensile stresses σxx (200 MPa) and transverse tensile stresses σyy (400 MPa) on surface A in the S235 steel specimen. Whereas on surface B in the same specimen, there were present compressive stresses σxx (170 MPa) and tensile stresses σyy (160 MPa). For the S355 steel specimen, on both surfaces, there were measured compressive longitudinal σxx and tensile transverse stresses σyy, which reached 120-300 MPa and 300 MPa, respectively. At the depth of 0.1 mm, there was observed the presence of compressive internal stresses σxx and σyy of the order of 160-180 MPa. Such a heterogeneous distribution of internal stresses as well as residual stresses left in sheets after the manufacturing process had a significant effect on the process of fatigue cracking of steel specimens. 3. Results of fatigue tests on S235 and S355 steels In order to assess the effects of surface friction treatment of S235 and S355 steels on their fatigue behavior, there were conducted crack propagation tests in DET untreated and surface treated specimens at asymmetric tension (R= -0.2). Crack length measurements were performed optically with an accuracy of 0.01 mm on a stand equipped with a telescope. At each stress level, two specimens were tested. For both investigated steels, a characteristic feature of crack propagation in either untreated or treated specimens was different crack propagation from edge notches. In the initial stage, cracks propagated identically from both notches. Then, the crack propagating from one of the notches arrested and did not grow until an ultimate specimen fracture. From the other notch in the specimen, the crack propagated during the whole period of the specimen's life. This latter crack con-

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207

tributed to specimen fracture. Sample surface fractures in S235 and S355 steel specimens with noticeable non-identical crack propagations from notches are illustrated in Figures 4a and 4b, respectively. In the Figures, an arrow denoted as PSO indicates a range of specimen cracking in a plane strain state, whereas an arrow denoted as PSN indicates cracking in a plane stress state. PSO

PSN

PSN

a)

PSO

b)

Figure 4. S355 fracture surface (σmax = 115 MPa) (a) and S235 fracture surface (σmax = 85 MPa) (b), that indicate fatigue crack propagation mainly from one side of the notch specimen The above pictures prove that, in the cracking zone of the plates, a substantial area of the fracture surface corresponds with a stable cracking in plane strain state. After a transition to cracking in plane stress state, the crack growth rate increases. It is a range of subcritical crack growth until an ultimate plate fracture. Sample curves of fatigue crack growth rates in the air of untreated specimens (the blue curve) and treated ones (the red curve) of investigated steels at the stress of σmax= 85 MPa (R= -0.2) are presented in Figures 5a and 5b, respectively. a)

b)

Figure 5. Experimental curves of crack growth rates in the air for S235 steel (a) and S355 steel (b) that were surface treated and untreated at σmax= 85 MPa Experimental results indicate a higher crack growth rate in the air of treated steel than of steel in the delivery state, particularly in the case of cracking in a plane strain state. The said stage of specimen cracking constituted a major part of their fatigue lives. Cracking in a plane stress state was faster in untreated specimens. Similar behavior of crack growth rates in specimens was also observed at other stress levels. Such behavior of specimens with treated surfaces was a result of less stable, in thermodynamic sense, surface after friction treatment. However, the fatigue life of specimens that were subjected to friction treatment was higher by 10-15% for both steels in relation to the fatigue life of specimens in the delivery state. 4. Description of DET specimen cracking It follows from the analysis of fracture surfaces of S235JR and S355J2 steel DET specimens that for the most part of their fatigue lives, the cracking proceeded stably in a plane strain state. Cracking in a plane stress state proceeded fast and led to an ultimate specimen failure.

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Fatigue Failure and Fracture Mechanics

Nevertheless, the cracking process in the surface treated S235 and S355 steels develops in tensile internal stress field σS (secondary stresses). Either in the white layer or in heat-affected layer for both steels there was found an intergranular brittle cracking mechanism up to the depth of about 150 µm (Figures 6a and 6b). In accordance with the FITNET procedures [10] the influence of primary stress σp and secondary stress σS fields on specimen’ cracking is expressed in the following way: K I ,eff (σ , a ) = K I  σ p , a  + K I  σ s , a  + ρ (a )    

(1)

where: Keff is an effective stress concentration factor, a - crack length, σP and σS mark respectively primary (external ) and secondary (internal ) stresses, ρ(a) is the correction factor to account enlarged plasticity of the material under the influence of tensile internal stresses. a)

b)

Figure 6. Mechanism of intergranular brittle cracking observed in the surface layer of the S235 (a) and the S355 (b) FSP treated steels. The distribution of secondary stresses σP in the depth g of a sample with tensile internal stresses acted in near-to-surface layer is described by Equation (2). In this equation the denotation c means the depth of a specimen at which the internal stresses change their sign. Hence, the stress intensity factor K Is (σ , a ) suitable a flat DET sample is defined by Equation (3) [10]:

[(

)(

σ s = σ max ⋅ 1 − ( g / c )2 / 1 + ( g / c )2

(

)]

(2)

)

(

)

  K IS (σ , a ) = σ max ⋅ π ⋅ a ⋅   1 + (a c )4 − (a c )2  / 1 + (a c )4       

1/ 2

(3)

and the correction factor ρ(x) is estimated by formula (4):

ρ (x ) = 0.1 ⋅ ( x )0.714 − 0.007 ⋅ ( x )2 + 3 ⋅ 10 −5 ⋅ (x )5 ; where

x = Lr ⋅  K IS K IP   

(4)

In order to describe fatigue cracking in an elastic-plastic material, there was used the J-integral. Compliance function Y(a/W) for a flat DET specimen of a width of 2W with a notch cut along length a has the following form: Y=

π ⋅ a / 2W 

2

3

a a a a ⋅ 1.122 − 0.561⋅   ⋅ −0.205 ⋅   + 0.471⋅   + 0.190 ⋅   1 − a / W  W  W  W  W 

4

(5)

  

For a stable crack growth in a plane strain state the J-integral range ∆J has the form: ∆J = ∆Je +∆J pl =

and

π ⋅Y 2(a /W) ⋅ ∆σ 2 ⋅ a

( )

E / 1−ν

2

H⋅ ∆σ (n+1) ⋅ a + F

( (

( )n '

; H = W − a ⋅ Y  a  ; F = K '

∆J e = ∆K I2, eff (σ , a ) E / 1 − ν 2

W

W 

(6)

))

where: a is the sum of a physical crack length and the depth of a notch, n’ is hardening exponent determined from a cyclic stress-strain curve of the material; K’ is the fatigue strength coefficient. K‘ and n’ values were determined experimentally.

Dariusz Skibicki

209

The condition for the occurrence of cracking and a transition from a plane strain state to a plane stress state is a situation where ∆J=JR , where JR is the critical value of the plastic strain energy that is necessary to initiate cracking by tearing in a component: ( n +1) H ⋅ σ cr ⋅a E ⋅R where (7) JR = σ cr = 2 F Y ⋅π ⋅a Material cracking resistance R is related to the energy release rate G by the following relation: π ⋅ a ⋅ σ cr2 (8) R = G IC ⋅ 1 − ν 2 = ⋅ 1 −ν 2 E It is well known, that in the case of plasticization of a short distance, JR and R quantities are identical; therefore, it is enough to use one of the above parameters for cracking process description. However, in order to allow for material plasticization of a longer distance, both parameters are necessary to characterize this process properly. With the use of the above relations (5-8), there were developed diagrams of crack growth rates da/dN=f(∆J), which are presented in Figure 7. In calculations of estimated crack growth rates in steels, the following values were adopted for the quantities that appear in the abovementioned equations: S235 untreated: n’ = 0.1473, K’= 581.34 MPa and for treated steel n’ = 0.129, K’ = 501.5 MPa, S355 untreated: n’ = 0.156, K’= 798.2 MPa and for treated steel n’ = 0.136, K’ = 608.2 MPa

(

)

(

)

a)

b)

Figure 7. Crack growth rate curves against the J-integral and the JR-integral in the air for S235 steel (a) and S355 steel (b) that were untreated and surface FSP treated at σmax= 85 MPa In Figures 7a and 7b that present crack growth rate curves in examined steels in the air in the function of the J-integral and the JR-integral, there were marked the stages of specimen cracking. Stage I is related to cracking in a plane strain state. Stage II is related to cracking plane change and a transition from a plane strain state to a plane stress state. Stage III is related to the critical crack growth. It was described by the JR-integral, taking internal tensile stresses operating in the surface layers of analyzed specimens into account. 5. Conclusions The study discussed characteristic features of fatigue cracking in the air of two series of DET specimens made of S235JR and S355J2 carbon steels under asymmetric tension (R= -0.2). One series was used in tests on untreated steel specimens, whereas the other series was used in tests on surface treated by FSP treatment specimens. The results of the fatigue tests indicated faster cracking of treated specimens in a plane strain state in the phase of stable crack growth than in specimens in the delivery state. It is connected with a thermodynamically less stable surface layer in specimens after friction treatment, which can also be proved by the presence of high internal tensile stresses on specimen surfaces and a heterogeneous structure of surface layers. However, cracking of the said specimens in a plane stress state was slower than in untreated specimens. As a result, the fatigue life of treated specimens was higher by 10-15% compared to the specimens in the delivery state. It

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Fatigue Failure and Fracture Mechanics

should be stressed that the surface modification of a 12-mm thick steel by means of friction treatment was carried out on a non-professional stand. From an analysis of world-wide literature, it follows that few studies in this field concern modification of steel surfaces with the use of the FSP method. The research was financed by The Polish Ministry of Science and Higher Education within the scope of the project N N501 0097 33. References [1] R. S. Mishra, Friction Stir Processing technologies, Advanced Mater. Processes, October 2003. [2] H. Fuji, L. Cui, M. Maeda, K. Nogi, Effect of tool shape on mechanical properties and microstructure of friction stir welded aluminium alloys, Mater. Sci. Eng. A 419, 2006, pp. 25-31. [3] H. J. Schmidt, B. Schmidt-Brandecker, Advanced materials and manufacturing technologies for aircraft application. 2nd Int. Conf. Material and component performance under variable amplitude loading. 2009, Germany, pp. 67-87. [4] L.J.J. Kok, K. Poston, G. Moore, Bombardier aerospace FSW demonstrator. ICAF 2011 Structural Integrity: Influence of Efficiency and Green Imperatives. Springer, New York, 2011, Proc. 26th ICAF Symposium, Canada, 2011, pp.73-81. [5] L.Cui, H. Fuji, N. Tsuji, K. Nogi, Friction stir welding of a high carbon steel, Scripta Mater., Vol. 56, 2007, pp. 637-640. [6] Cho Hoon-Hwe, Heung Nam Han, Sung-Tae Hong et al. Microstructural analysis of friction stir welded ferritic stainless steel. Mater. Sci. Eng. A 528 (211) pp. 2889-2894 [7] A.P. Reynolds, Wei Tang, T. Gnaupel-Herpld, H. Prask, Structure, properties and residual stress of 304L stainless steel friction stir welds, Scripta Mater. 48 (2003), pp. 1289-1294 [8] J. Gandra, R. Miranda, P. Vilica, J.P. Teixeira, Functionally graded materials produced by friction stir processing. J. Mater. Proc. Tech. 211 (2011), pp. 1659-1668. [9] D. Kocańda, A. Górka, Formation of a metal coating by means of friction stir processing, ICAF 2011 Structural Integrity: Influence of Efficiency and Green Imperatives. Springer, New York, 2011, Proc. 26th ICAF Symposium, Canada, 2011, pp.167-178. [10] A. Neimitz, Fracture mechanics, PWN SA, Warsaw, 1998 (in Polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.211

An experimental investigation on crack initiation and growth in aircraft fuselage riveted lap joints Andrzej Skorupa1,a, Malgorzata Skorupa1,b, Tomasz Machniewicz1,c, Adam Korbel 1,d 1

AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, A. Mickiewicza Av. 30, 30-059 Kraków, Poland

a

[email protected], [email protected], [email protected], [email protected]

Keywords: riveted joints, fatigue tests, squeeze force, thickness effect, fatigue life.

Abstract. Effects of variables related to design and production of riveted lap joints representative of longitudinal sheet connections for a pressurized transport aircraft fuselage were experimentally investigated. The specimens from an aircraft Al alloy D16 Alclad sheets of three different thicknesses (1.9, 1.2 and 0.8 mm) were assembled under load control using round head rivets and rivets with the compensator from a P24 Al alloy. For the joints from 1.9 mm thick sheets fatigue tests indicated a dependency of the crack initiation site and crack path on the squeeze force level and on the rivet type. At the same time, increasing the squeeze force led to improved fatigue properties of the joints, specimens assembled using the rivets with the compensator showing fatigue lives consistently longer than joints with the round head rivets. All observed trends have been explained based on hole expansion and load transfer measurements. For thin sheets connected using the round head rivets, local deformations and indentations under the driven rivet head promoted crack initiation and failure in the adjacent sheet. Fatigue test results indicated that the detrimental effect of this type imperfections could outweigh the benefits associated with a decrease in secondary bending due to thinning the sheets. The rivets with the compensator were observed to cause significant local imperfections beneath the manufactured head, which adversely affected the joint fatigue performance. Introduction Riveting remains a preferred method for connecting elements of an aircraft structure, though adhesive-bonded and riveted-bonded joints are also applied. A typical design solution for joining sheets of a pressurized transport aircraft fuselage in the longitudinal direction is a riveted lap joint, usually comprising three rivet rows, as shown in Fig. 1. Due to eccentricities occurring in the overlap region for this type of a joint, the so-called secondary bending is induced under nominally axial loading on the sheets. The phenomenon of secondary bending can lead to considerably elevated stresses in the sheets and affects the mode of failure of the joint [1]. The fatigue crack nucleation location, crack path geometry and fatigue properties of a riveted lap joint depend on the integrated effect of a number of factors related to joint design and production as well as loading conditions. This paper focuses on the influence of the squeeze force, sheet thickness and rivet type. Specimens and Testing Equipment Configuration of three-row riveted lap joint specimens used in the fatigue tests is shown in Fig. 2 and the specimens’ dimensions are specified in Table 1. The rivet row spacing s=5d (d – rivet diameter) and the rivet pitch in row p=5d are typical for fuselage skin connections. The rivet holes were drilled according to the process specification of the Polish aircraft industry. The total length L of the specimens was chosen to eliminate the effect of specimen fixture in the fatigue machine on stress conditions in the overlap region [2].

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Fig. 1. Typical fuselage longitudinal riveted lap splice joint

Fig. 2. Specimen for fatigue tests Table 1. Characteristic dimensions of riveted specimens Sheet thickness Rivet diameter Hole diameter Specimen length t [mm] d [mm] do [mm] L [mm] 0.8 3.5 208 +0.12 1.2 4.0 260 (d+0.05) 0 1.9 5.0 345 The sheet material was a Russian Al alloy D16CzATWH in the Alclad condition. The mechanical properties (0.2% yield stress = 291 MPa, ultimate strength = 433 MPa, elongation = 13%) and the fatigue crack growth behaviour of this material are similar to those of the western Al 2024-T3 alloy [3]. Two types of protruding head rivets differing in the manufactured head geometry, namely with a round head and with the so-called compensator were used to assemble the sheets, Fig. 3. The compensator, which is a small protrusion on the mushroom rivet head, causes increased rivet hole expansion. The rivet material was the P24 Al alloy equivalent to the western 2117-T3 material used for the AD rivets. Force controlled riveting was applied using a squeezer

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mounted in the grips of a MTS 810 fatigue machine [4]. The same machine was utilized in the fatigue tests carried out under constant amplitude loading at a stress ratio of 0.1. This type of loading simulates variations of the hoop stress in the fuselage skin generated due to the cabin pressurization. Crack growth on the sheet surface was monitored using a travelling microscope. The testing equipment is shown in Fig. 4

Fig. 3. Rivet types used in experiments: (a) round head rivet; (b) rivet with the compensator

Fig. 4. Fatigue testing equipment Effect of Squeeze Force In production practice, the squeeze force is represented by a ratio of the rivet driven head diameter (D) to the rivet shank diameter (d) which increases with the squeeze force level. The D/dvalue is, therefore, a first indicator of the riveting process quality. Typical D/d ratios range from 1.3 to 1.5, the latter value being considered as optimal [2]. The rivet installation causes rivet hole expansion, which generates compressive residual tangential stresses in the hole vicinity. The higher the squeeze force level, the larger the compressive tangential stress area, which affects the initiation location and path of fatigue cracks at rivet holes and the joint fatigue life. Increasing squeeze force yields also a higher residual clamping between the sheets beneath the rivet heads. This leads to transmitting a portion of the applied load by friction, which again can influence a mode of joint failure. As an example, Table 2 gives fatigue test results observed under an applied maximum cyclic stress Smax=120 MPa for specimens from 1.9 mm thick sheets with the round head rivets installed using four different squeeze force levels, resulting in four different D/d-values.

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Table 2. Fatigue lives and crack behaviour for specimens riveted with different squeeze forces (1.9 mm thick sheets, round head rivets) D/d

Fatigue life* kcycles

1.3

81.6

Under driven head; Quarter elliptical

1.4

160.0

Under driven head; Quarter elliptical

1.5

235.5

Under manufactured head; Quarter/semielliptical

1.6

298.2

Under manufactured head; Semi-elliptical

Crack initiation site; Crack shape

Crack path Net cross section Above net section, through rivet hole Outside rivet hole

*

Average from three tests

A trend of increasing the fatigue life with the squeeze force, demonstrated in Table 2, was also exhibited at Smax of 100 and 80 MPa. No impact of the stress level on the location of crack nuclei and crack path was found. An illustration of the results from Table 2 are fractographic observation results shown in Fig. 5. It is seen that fatigue cracks always initiate on the faying surface in one of end rivet rows, which results from the influence of secondary bending [1]. For a limited squeeze force, the cracks initiate at the edge of the rivet hole and propagate in the net cross section, Fig. 5a. A more intense squeezing of the rivet leads to crack initiation outside the hole, but propagation through the hole, usually shifted above the net cross section, Fig. 5b. For a relatively high squeeze force fatigue cracks nucleation occurs above the hole, near the edge of the clamping area beneath the rivet head, and the crack propagates outside the hole, Fig. 5c. The latter behaviour is partly contributed by fretting [5].

Fig. 5. Effect of squeeze force on fatigue crack initiation and path (explanation in text) It is seen in Table 2 that the specimens with D/d of 1.5 and 1.6 always failed in the sheet adjacent to the rivet manufactured head, while in the case of D/d≤1.4 the crack nucleation and failure occurred in the sheet under the driven head. The above behaviour can be explained based on rivet hole expansion measurements shown in Fig. 6 and load transfer measurements shown in Fig. 7. In Fig. 6a, hole expansion is defined as he=(de – do)/ do, where de is the expanded hole diameter. As shown in Fig. 6a, for D/d of 1.5 and, especially, 1.6, he in the sheet next to the rivet driven head considerably exceeds that in the sheet next to the manufactured head. At the same time, Fig. 7 demonstrates that loads transferred by the end rivet rows are almost equal. Consequently failure occurs in the sheet with smaller hole expansion, i.e. under the manufactured head. For D/d≤1.4 he in both sheets is relatively small and only slightly larger under the driven head (Fig. 6a). In that case, the negative influence of a much higher transfer load in the sheet adjacent to the driven head (Fig. 7)

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dominates and determines the failure location. A more uniform load transmission distribution for D/d of 1.5 compared to D/d of 1.3 shown in Fig. 7 stems from lower flexibility of rivets installed with a higher squeeze force [1].

Fig. 6. Hole expansion measurement results for sheet thickness t=1.9 mm: (a) round head rivet; (b) rivet with the compensator

Fig. 7. Load transfer distribution in riveted joint for two squeeze force values: round head rivet, sheet thickness t=1.9 mm Fig. 6b shows measurements results on he for the rivet with the compensator for two D/d-values. From a comparisons with Fig. 6a is seen that due to the compensator he in the sheet next to the manufactured head becomes considerably larger than for the standard, round head geometry. Fig. 6b indicates that he below the manufactured head of the rivet with the compensator is larger than below its driven head, which explains why in all fatigue tests on specimens assembled using this type rivets fatigue failure occurred in the sheet adjacent to the rivet driven head. Similarly as in the case of specimens with the round head rivets, a higher squeeze force yielded an increase in the fatigue life. For a given D/d ratio, fatigue lives of specimens assembled using the rivets with the compensator observed at Smax=120 and 100 MPa were by 40 to 90% higher than for specimens with the round head rivets. Effect of Sheet Thickness In order to assess the effect of sheet thickness on the mode of failure and fatigue properties of the joint, specimens from 0.8 mm and 1.2 mm thick sheets were fatigue tested in addition to the specimens from 1.9 thick sheet considered in the previous section. The sheets were connected using the round head rivets applying two different squeeze force values leading to D/d of 1.3 and 1.5 for either specimen series. The fatigue tests were carried out at three Smax stress values, namely 120, 100

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and 90 or 80 MPa. In the case of the D/d=1.3 specimens, the crack path for both sheet thicknesses and at all load levels was through the rivet holes, slightly above the net section, Fig. 8a. With the D/d=1.5 specimens, the cracks initiated and propagated above the rivet holes, Fig. 8b. In all cases failure took place in the sheet adjacent to the rivet driven head. It can be concluded from confronting the above observations with information in Table 2 that the mode of failure for joints from the thin sheets (0.8 and 1.2 mm) is different than in the case of joints from the thicker sheets (1.9 mm). The reason behind the above differences can be local deformations and indentations under the rivet driven head that occur during the rivet installation in thin sheets due to their low stiffness. Note that the driven head diameter is smaller than the manufactured head diameter (about 2d).

Fig. 8. Failure mode for specimens from thin sheets with round head rivets: (a) t=1.2 mm, D/d =1.3, Smax=90 MPa; (b) t=0.8 mm, D/d =1.5, Smax=120 MPa Results presented in Table 3 indicate that sheet thickness has an impact on the joint fatigue life. Increasing sheet thickness should yield a lower fatigue life due to the effect of secondary bending. For example, at Smax of 120 MPa the bending factor kb=Sb/Smax, where Sb is the nominal bending stress computed according to Schijve’s model [6], equals 1.1 and 0.85 for t=1.9 and 0.8 mm respectively. For Smax=80 MPa, somewhat higher kb factors of 1.25 and 0.9 are obtained for the above t-values [1]. However, as seen in Table 3, the observed effect of thickness on the fatigue life is not systematic, due to the addressed above imperfections inherent in the joints. Applying the rivets with the compensator to connect thin sheets brings no benefits compared to the round head rivets because, due to a specific shape of the manufactured head bottom surface (cf. Fig. 3b), significant local imperfections of the sheet beneath that rivet head precipitate failure. For the above reason, fatigue cracks develop in the sheet under the manufactured head and can grow outside the rivet hole, Fig. 9. Table 3. Fatigue lives (kcycles) for specimens of round head rivets and different thicknesses Smax, MPa 120 100 90 D/d 1.3 1.5 1.3 1.5 1.3 1.5 t=0.8 mm 288.5 322.2 483.0 1666.1 743.6 1665.0 t=1.2 mm 177.0 396.4 347.7 768.5 586.8 1135.4 t=1.9 mm 81.6 235.5 257.2 355.0 507.3* 1174.5* * Results for Smax=80 MPa

Fig. 9. Typical failure mode for specimens from thin sheets and rivets with the compensator: t=0.8 mm, D/d =1.4, Smax=120 MPa

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Conclusions Experimental observations presented in the paper lead to the following conclusions: 1. The initiation and growth of fatigue cracks in riveted lap joints and the joint fatigue performance depend on rivet hole expansion, and hence on the rivet type and rivet squeeze force, as well as on the sheet thickness. Fatigue cracks initiate always on the faying surface of the sheets in one of the outer rivet rows. 2. Essentially, joint from thicker sheets fail in a sheet with smaller hole expansion, but the distribution of load transfer through the joint can also play a role. For the round head rivet smaller hole expansion occurs in the sheet below the manufactured head, while for the rivet with the compensator smaller expansion is observed in the sheet adjacent to the driven head. For relatively low rivet squeeze forces the crack path is close to the net cross section along one of the outer rivet rows. At high squeeze forces cracks can start and grow outside the rivet hole. The fatigue life increases with the squeeze force value and is always longer for the rivets with the compensator than for the round head rivets. 3. The above observations are not valid for joints from thin sheets. For round head rivets the riveting process can locally introduce imperfections in the sheet adjacent to the rivet driven head, which promotes crack nucleation at this location. In this case, no systematic dependency of the joint fatigue life on the sheet thickness is exhibited. Rivets with the compensator are not suitable for connecting thin sheets because significant local imperfections beneath the manufactured head cause a premature failure at that location. Acknowledgements The financial support from the governmental research funds within the years 2009-2012 is acknowledged. References [1] M. Skorupa, A. Skorupa, Load transmission and secondary bending in lap joints of aircraft fuselage. Institute of Aviation Scientific Publications, Warsaw, 2010. [2] R.P.G. Müller, An experimental and analytical investigation on the fatigue behaviour of fuselage riveted lap joints. The significance of the rivet squeeze force, and a comparison of 2024-T3 and Glare 3, PhD Dissertation, Delft University of Technology, The Netherlands, 1995. [3] J. Schijve, M. Skorupa, A. Skorupa, T. Machniewicz, P. Gruszczyński, Fatigue crack growth in the aluminium alloy D16 under constant and variable amplitude loading, Int. J. Fatigue, 26 (2004) 1–15. [4] M. Skorupa, A. Skorupa, T. Machniewicz, A. Korbel, Effect of production variables on the fatigue behaviour of riveted lap joints, Int. J. Fatigue, 32 (2010) 996–1003. [5] A. Skorupa, M. Skorupa, Riveted lap joints in aircraft fuselage. Design, Analysis and properties, Springer, Dordrecht, 2012. [6] J. Schijve, Some elementary calculations on secondary bending in simple lap joints. Report NLR TR 72036, National Aerospace Laboratory, Amsterdam, 1972.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.218

Application of Digital Image Correlation in Fatigue Crack Analysis MARCINIAK Tomasz 1,a, LUTOWSKI Zbigniew 1,b, BUJNOWSKI Sławomir 1,c, BOROŃSKI Dariusz 2,d, GIESKO Tomasz 3,e 1

University of Technology and Life Sciences, Faculty of Telecommunications and Electrical Engineering, Kaliskiego 7, 85-789 Bydgoszcz, Poland

2

University of Technology and Life Sciences, Faculty of Mechanical Engineering, Kaliskiego 7, 85-789 Bydgoszcz, Poland

3

Institute for Sustainable Technologies - National Research Institute, Pułaskiego 6/10, 26-600 Radom, Poland

a

b

c

[email protected], [email protected], [email protected], d e [email protected], [email protected]

Keywords: digital image correlation, crack growth analysis, multi-processor graphic cards, wholefield displacement analysis

Abstract. In the paper method of displacement analysis in the cracking zone based on digital image correlation and advanced multi-processor graphic cards procedures was presented. The basic assumption for the discussed displacement and strain measurement method under time variable loads was obtaining high measurement sensitivity by simultaneously minimizing the measurement time consumption. The developed digital procedures for correlation of images has been used for an example of displacement analysis in the crack propagation testing in airplane riveted joints. Introduction The possibility to apply digital image correlation method to analyse the displacement and strain has been an exceptionally attractive choice in the mechanics of solids [1,2,3]. However, due to technological limitations, its practical application is far from being satisfactory, especially for an analysis of small strain values, typical for operating loads of structures and materials with high stiffness, for example steel. Measurement of small displacements, which later provides basis for determination of strains, requires the use of very high resolution digital video cameras and this involves the necessity of transmission and processing of huge amount of data in a very short time. The complexity of applying strain determination methods based on analysing the object’s surface image appears to be high for time variable loads. Additionally long lasting cyclical loads involve a significant increase in the amount of data to be processed with further reduction of the analysis time and a considerably higher probability of the method de-correlation. The remainder of this paper presents a method of displacement determination in the fatigue crack zone, illustrated by example of tests carried out on sample from airplanes aluminium structure. Digital Image Correlation Method Based on GPU The image correlation methods, developed in the 80s, involve comparing a sample image before and after exposing the object to strain in the form of lighting with white light or laser light (spot methods) [4,5,6,7]. Displacements of surface characteristic points allow determining the strain values within the analysed area. The sensitivity of this method depends on the parameters obtained using image observation methods such as dimensions of the observation field or the image geometric resolution. Image’s surface points are recorded on a PC before and after exposing the object to strains. This enables correlation of these images on the basis of intensity recorded for each pixel using a CCD matrix or a other type of the image sensor.

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Development of the digital image correlation techniques is connected with the progress in optoelectronics. Constantly increasing resolution of video cameras makes it possible to obtain more and more accurate measurements. An increased image resolution and the usage the digital image transmission systems involves the necessity of transmitting a huge amount of data in a very short time. This significantly limits the possibilities of applying digital image correlation (DIC) methods for time variable loads. The basic assumption for the discussed displacement measurement method with time variable loads was obtaining high measurement sensitivity by simultaneously minimizing the measurement time. For this purpose special computing procedures based on multi-processor graphic cards (GPU) were developed. Analysing displacements based on the assessment of the cross correlation coefficient C is one of the basic solutions used in the image cross correlation methods. The cross correlation coefficient is determined for a chosen point of the environment, P[x,y], as follows: b d  I n ( x i , y j ) I m ( xi , y j )  C (u, v) = ∑∑   f ⋅g i = a j =c  

(1)

where f and g are, respectively: f =

∑∑ [I b

d

i =a j =c

n ( xi , y j )

]

2

, g=

∑∑ [I b

d

m

( xi , y j )

]

2

(2)

i = a j =c

where: a,b - initial and final values of the correlated image coordinate x index, c,d - initial and final values of the correlated image coordinate y index, In(xi,yj) - the image intensity in the point with coordinates x y, recorded in the phase of loading, Im(xi,yj) - the image intensity in the point with coordinates x y, recorded in the phase of loading. The value of the coefficient C can change in the range from 1 to 0. When C=1, the images of point P in the environment, recorded in phases n and m, are entirely consistent and when C=0 they are entirely different. In order to define a displacement of a chosen point P based on images recorded in the two phases of loading n and m, sub-area O covering point P is separated from image n and m is displaced in relation to the analogical area in image n. The value of cross correlation coefficient C is calculated for each location of sub-area On. Due to low efficiency of standard functions to determine the coefficient C, e.g. those available in OpenCV library, an original algorithm of correlative search of the image similar areas has been developed. The implementation was performed by means of CUDA library in version 3.2, facilitated by NVidia Company. Since there are two known ways of obtaining a resultant matrix of the pattern adjustment to the searched image (correlation matrix), it is necessary to make a choice in the initial phase, between the method based on Fourier transforms used by OpenCV library and the method to directly determine the values of the correlation function. Therefore it was decided to carry out our own implementation of the algorithm determining the correlation function according to the version directly defined by equation (1). For this version of the algorithm tests were carried out comparing the performance with its possible maximum optimised equivalent, taken from OpenCV library. Thus, the developed implementation, computing the functions of correlation on GPU was compared with a multi-thread OpenCV implementation on CPU and GPU. The results of comparison tests are shown in Figure 1, where "CPU" is the algorithm with simplest implementation performed on CPU (one thread), "CPU-CV" represents an algorithm using OpenCV library in a multithread manager, "GPU-CV" is utilization of OpenCV library in GPU version. "GPU" and “GPU II” are the tested implementation of algorithms performed on GPU, where GPU II is an effect of additional GPU resources occupancy optimisation. The authors’ implementation on GPU is more efficient than the other solutions for T

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pattern dimensions not exceeding 65x65 points, where T represents size of correlated part of analysed images. For T pattern dimensions 30x30 points the computation speed achieved was ten times faster (Fig.1 - “GPU”). Further optimisation of the algorithm’s speed improve efficiency of image analysis even by 100% (Fig.1 - “GPU II”). The last phase of the algorithm optimisation involved the use of textures memory for storing the pattern and searched images data. By using the memory of textures the algorithm’s efficiency was improved by approximately 25%, regardless of pattern area T. 1000 100 10

time, ms

1

CPU CPU-CV GPU-CV

0.1 0.01

GPU GPU II

0.001 0.0001

0

10

20

30

40 50 60 pattern T dimension, points

70

80

90

100

Fig.1. The results of procedures efficiency comparison tests Example results of displacement analysis in fatigue crack zone The developed digital procedure for correlation of images has been used for an example of displacement analysis in the method of fatigue crack propagation testing in airplane riveted joints. A camera with an image resolution of 2448x2050 pixels and a set of lens with 50 mm area of view was used for the sample image recording. Images was recorded on a PC equipped with GPU card NVIDIA GTX480 by means of GigaEthernet card with a maximum frequency of 17Hz. Measurement sensitivity s=0.0025 mm was obtained for the applied system configuration. In Figure 2a, an images of specimen made of aluminium alloy 2024-T3 with a propagated crack, which was exposed to constant amplitude of nominal stress and the stress ration R=0, has been presented. Determined on their basis, distributions of displacements δ within the crack environment are demonstrated in Figure 2d. a)

c)

d)

b)

Fig.2. Crack images for minimal (a) and maximal (b) value of loading cycle. Crack image with analysis pattern (c) and displacement map against the background of specimen image (d)

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Distribution of displacement δ along lines marked at Figure 3a and displacement gradient (Fig.3b) in considered crack zone area were determined for tested specimen. On their basis crack line and crack tip was determined. Further, the course of the crack length changes were determined on the base of obtained displacement δ distributions in a function of the load cycle number, for the discussed tests example. a)

87

c) -20

b) crack tip

51

20 1 px = 2.5 µm

δ, px -25

-30

fatigue crack line 143 137 124 111

-35

137

20

51

87 111 143 line 124

-40 60

80

100

120 160 point140 number

180

200

220

Fig.3. Displacement distribution in crack zones: a) displacement map with cross-section lines, b) displacement gradient distribution, c) displacement distributions along lines marked at Fig.3a Summary The considered method to analyse displacements with the use of digital image correlation technique supported by GPU technology enables its application in displacement analysis for objects exposed to time variable loads. By applying modern optoelectronic and IT solutions, the image analysis algorithms developed for the equipment configuration used in this work was able to determine the strain distribution in an 2448x2050 pixels image, with the size of a searched patterns matrix of 30x30 elements and the pattern size of 20x20 pixels, in less than 0.2 sec. Further reduction of the analysis time is possible for GPU cards with a higher number of cores. Presented example show possibility of applying the developed method in fatigue crack length and growth testing and analysis. It is possible use of elaborated displacement analysis method for crack tip detection also. References [1] D. Giesko, D. Boroński, A. Zbrowski, P. Czajka, Detection and measurement of fatigue cracks in solid rocket propellants, Maintenance Problems, 74/3 (2009) 75-84. [2] A. Pilch, A. Mahajan, T. Chu, Measurement of whole-field surface displacements and strain using a genetic algorithm based intelligent image correlation method, Journal of Dynamic Systems, Measurement, and Control, v. 126, 3 (2004) 479-488. [3] F. Lagattu, J. Brillaud, M.-C.Lafarie-Frenot, High strain gradient measurements by using digital image correlation technique, Materials Characterization 53 (2004) 17-28. [4] T. Schmidt, J. Tyson, Dynamic Strain Measurement Using Advanced 3D Photogrammetry, Proceedings of IMAC XXI, 2003, Kissimmee. [5] Information on http://www.correlatedsolutions.com [6] Information on http://www.gom.com [7] D. Boroński, Local material properties in fatigue analysis, Publishing House of ITeE-PIB, Bydgoszcz-Radom (2009) (in polish).

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.222

Dual-Band Experimental System For Subsurface Cracks Testing Tomasz Marciniak1, a, Zbigniew Lutowski2,b , Sławomir Bujnowski3,c , Dariusz Boroński4,d, Piotr Czajka5,e 1,2,3

University of Technology and Life Sciences, Faculty of Telecommunications and Electrical Engineering, Kaliskiego 7, 85-796 Bydgoszcz, Poland

4

University of Technology and Life Sciences, Faculty of Mechanical Engineering, Kaliskiego 7, 85-796 Bydgoszcz, Poland

5

Institute for Sustainable Technologies - National Research Institute, Pułaskiego 6/10, 26-600 Radom, Poland a

c

b

[email protected], [email protected], d e [email protected], [email protected], [email protected]

Keywords: cracks, dual-band system, thermography, hybrid method

Abstract. The authors of this article have presented a hybrid method for inspection in the visible and infrared bands. A model of a test stand equipped with a monochromatic CCD and thermovision cameras which enable execution of tests in the field of active and passive thermography has been presented. Application of two vision channels provides the possibility of observing the defects caused by cracks and non-uniformity of a material structure, in the subsurface layer. An analysis of experimental tests performed on selected objects has been presented. Images from both channels, effects of the image overlapping and profiled charts along characteristic lines in thermograms have been shown. Introduction Fast development of optoelectronics has contributed to a wider use of optical measurement methods and rapid growth of a new scientific discipline, that is, machine vision [1]. The main methods of this type employ visible range of the light. Since not all defects that occur in mechanical structures can be seen in the visible band, these systems are often supplemented with a vision channel operating in infrared. Thanks to this, it is possible to monitor the temperature of objects, detect areas of local differences in temperature and zones of heat accumulation, and with the use of active thermography, spot the defects in the subsurface layer [2]. Such possibilities are especially useful for assessment of a material fatigue damage. Images transformation Application of two vision channels causes additional problems, one of which being transformation of one image into the other. It results from the fact that the cameras working in infrared and in the visible light are separate physical objects. Thus, it’s not possible to get images in the visible light or in infrared recorded exactly from the same location. Due to the fact that cameras have different optical axes the images do not match each other (they cannot be directly put on each other). In order to make the two images overlap, one of them needs to undergo perspective transformation which levels the angle of view from the other camera. If cameras viewpoints are close, then most image changes can be modeled through a planar homography - linear mapping relating two corresponding planar points in two views [4]. Planar homography between two views can be determined by finding sufficient constraints to fix the (up to) 8 degrees of freedom of the relation. It can be estimated from the matching of 4 points or lines in general positions in two views. In this case the homography determining matrix H was estimated with the RANSAC algorithm use [3].

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The system has two modes of homographic transformation determination. In the first mode key points are indicated manually on the infrared and visual images respectively, so determined transformation relates directly infrared and visual band images - Figure 1.

Fig.1. Planar homography (matrix H) relation between infrared (IR) and visual band camera pictures (VIS) – manual mode In the second mode the special pattern is placed instead of tested object, and obtained images of the pattern are source of automatically searched key points. This makes possible to calculate two matrixes H describing transformations between images and plane of the pattern. Test stand model The vision module consists of a thermovision camera and a visible band camera, placed on separate rotary stools. The thermovision camera is equipped with a non-cooled microbolometrical matrix with resolution 640x480 pixels. The camera sensor operates in a long-wave (7,5÷14µm) range of infrared radiation. A vision system using a monochromatic 1626x1236 pixels resolution camera was used to record images in the visible band. Two white diode panel illuminators were applied to illuminate the observation area. A system with two infrared radiators, each with power of 500W, was used for heat impulse stimulation. LED illuminators and IR radiators were mounted on articulated arms. Additionally a heating plate with 1000 W power was applied to simulate the heat processes which occur during production and machining or simulation of operating conditions for the tested objects. The heating plate temperature regulation range is 50÷300°C. This device makes it possible to heat the samples through their direct contact with the plate surface. The use of this heating method is limited mainly to highly heat conductive materials such as metal products. Figure 2 shows a tests stand with its components. Observation examples The basic functions of the experimental inspection system, based on two vision channels are: - detection and identification of defects of surface structures in the visible band; - analysis of heat emission from the product surface enabling detection of surface and subsurface defects and areas of heat accumulation. Application of active and passive thermography provides the possibility to identify different kinds of faults, including: fatigue cracks, surface cracks, flaking failures, voids in surface and subsurface material, machining faults, discoloration of the coating and areas of elevated temperature. In Figure 3, there are sample images of a faulty disc obtained in the visible band and infrared active thermography, which has revealed cracks, invisible in the visible band.

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Fatigue Failure and Fracture Mechanics

Thermovision camera Slide Infrared radiators CCD camera

Frame

Panel illuminators

Heating plate

Tested object

Fig.2. Structure of the test stand

a)

b) Fig.3. Image of a faulty disc: a) in the visible band, b) in infrared

Hybrid analysis of images from both bands (the visible and infrared channels) provides more testing possibilities. For this purpose the test stand has been equipped with advanced methods of image treatment enabling planar homographic transformation, thanks to which it is possible to match the images recorded at different points of space and with different resolutions.

a)

b)

c)

Fig.4. Image of a detail with a subsurface defect: a) in the visible band, b) in infrared, c) in the visible band with a defect marked after matching the pictures

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Tests were performed on a sample made of polyamide 6 in which blind holes with diameters 1,2,4 and 6 mm were drilled, leaving 2 mm of material (from the plate head) in one case, and 2 mm of material in the second sample. Samples were exposed to heating impulse with two infrared radiators, each with power 500 W. The impulse lasted 5 seconds. In Figure 4 there is a temperature profile obtained from the tested sample along two lines L1 and L2. Line L1 is a straight line running through the holes centers for which there has been left 2 mm of material (from the hole bottom ), whereas, for line L2 there has been left 1 mm.

Fig.5. Temperature profile along two lines L1 and L2 Summary Despite the widespread use of thermovision in the cracks detection systems (including dual systems - e.g. with ultrasound or ultraviolet) there are still rarely seen thermovision solutions combined with visible light images. The experiments performed at the above discussed test stand prove, that application of a second vision channel in the infrared band is justified and enables detection of a material subsurface defects. Efficiency of detection depends on the material type and depth of the defect occurrence. However, it requires application of more complicated algorithms of the image treatment, hence computers with higher computing capacity are needed. Application of hybrid visual analysis of specimens during fatigue testing make possible detection and tracking of fatigue crack growth process. Research work carried out within the framework of the European Union Strategic Program called "Innovative Systems of Technical support for Sustainable Development of Economy " in the "Innovative Economy Operational Programme". References [1] T. Marciniak, D. Boroński, Z. Lutowski, S. Bujnowski, T. Giesko, Digital Image correlation universal tools versus custom solutions, Wydawnictwo Naukowe ITeE-PIB, 4 (2010) 19-28. [2] W. Minkina, S. Dudzik: Infrared thermography - errors and uncertainties, John Wiley & Sons Ltd, Chichester, 2009. [3] M.A. Fischler, R.C. Bolles, Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography, Communications of the ACM, 24(6) (1981) 381–395. [4] R. Hartley, A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge University Press, 2000.

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.226

Dual-camera vision system for fatigue monitoring Tomasz Giesko1, a 1

Institute for Sustainable Technologies – National Research Institute, ul. Pułaskiego 6/10, 26-600 Radom, Poland a

[email protected]

Keywords: Fatigue monitoring, Vision system, Dual-camera system.

Abstract. The article presents a dual-camera vision system for fatigue monitoring composed of a vision unit, a camera positioning set and a computer unit. Vision modules are mounted onto the four degrees-of-freedom (4DOF) positioning sets, which allows for an easy determination of the position of the camera in relation to the sample. The application of motorized measurement lenses with changeable configuration, thanks to the alteration of the distance of observation and the vision angle, enables the adaptation of the system to different scales of observation of the fatigue processes in the specimen surface. Automatic focus setting is realised with the use of the implemented algorithm. The software developed allows for the analysis of fatigue fracture for two twodimesional (2D) images or the three-dimensional (3D) stereovision image. Introduction Fatigue processes taking place in materials and structural elements are one of the most important issues in the area of knowledge on the maintenance of technical objects. The monitoring of fatigue is facilitated by the application of experimental methods with the use of optoelectronic technologies that are applied both in fatigue diagnostics of materials and structures as well as local deformation analyses. Among the aforementioned, the optical methods are more common. They employ sensor cameras of high resolution, lenses ensuring proper vision angle, and mechatronic systems for the positioning of the vision module in relation to the object. On the basis of images acquired, the crack trajectory is analysed and the increase of the crack length in the function of load and number of cycles is measured [1]. Great accuracy and high frequency of measurements, as well as the possibility of automation of the measurement process are the basic requirements for the apparatus enabling the conduction of the advanced fatigue tests. The primary restrictions to the development of measurement apparatus for fatigue tests employing visual methods stemmed from the incapability to meet these requirements properly. The more complex and advanced equipment solutions and programming tools enabling the fatigue monitoring and the measurement of cracks are the results of the intense development of optoelectronic technologies and image processing and analysis methods. The literature review indicates that experimental systems equipped with single CCD camera have so far been the major type of optoelectronic measurement apparatus used for fatigue monitoring and tests. The optomechatronic systems for the monitoring of fatigue fracture in materials and structural elements developed jointly by the Institute for Sustainable Technologies – National Research Institute (ITeE-PIB) in Radom and the University of Technology and Life Sciences in Bydgoszcz [2,8] constitute good examples of such solutions. These systems are equipped with single camera vision module and mechatronic system for the positioning of camera in relation to the objects tested, and they are adapted to work with fatigue testing machines. The developing imaging systems employing high resolution CCD cameras are being widely used in surface geometry measurements in micro and macro scale [3]. Making use of the possibility of simultaneous monitoring of the surface from the two cameras, the assessment of deformations employs the advanced methods of three-dimensional digital image correlation (3D-DIC), which facilitates the development of experimental mechanics. The application of dual-camera vision systems in the experimental tests for the fatigue monitoring in the specimen under various types of

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load is still a rare phenomenon. The experimental stereovision system for three-dimensional surface deformation measurement and crack length measurement in the tension-torsion loading test is presented in publications [5,7]. In the dual imaging system high speed CCD cameras are used. The supporting structure of the cameras mounted on the frame of the fatigue testing machine enables the positioning of the cameras in the range of 16÷80 [º] between optical axes. The leading commercial solutions of this type are the systems by ARAMIS GOM GmbH [10] and Trilion Quality Systems [11]. In these dual-camera systems, the position of the cameras in relation to the specimen is regulated manually. A similar solution is offered by the system for measuring the shape, displacement and strain of surfaces in three dimensions developed by the Correlated Solutions, Inc. [12]. The system allows for the monitoring of objects within the distance of 1÷10 [mm]. In the aforementioned systems DIC methods are used for the strain and displacement analysis. However, the high price of these solutions impedes their wide application in research organizations. Thus, tasks aiming at the development of alternative solutions of similar technological level but lower price are being undertaken. Publication [6] presents the dual-camera system equipped with the cameras of sensor resolution ca. 1 megapixel enabling the stereovision imaging, whereas publication [4] deals with the topic of the stereovision system enabling 3D measurements at the time of experiments conducted on the fatigue testing machine. In the systems applying analysis of images taken from several cameras, the problem of the calibration of the cameras deciding on the measurement accuracy is of great importance. The calibration processes employs different calibration patterns, frequently developed by the authors themselves. On the basis of the publications available, it can be stated that the scarce multi-camera vision systems that have so far been developed are mainly of experimental character. No structures with vision and mechatronic modules enabling precise positioning of optical paths with the measurement of their position for each degree of freedom have been found. The article presents a dual-camera vision system for the monitoring of fatigue wear in materials and structural elements developed at the Institute for Sustainable Technologies – National Research Institute in Radom. The software of the system was developed in close cooperation with the University of Technology and Life Sciences in Bydgoszcz. SYSTEM CONCEPT The system concept consists in the application of a dual-camera system for the monitoring of the surface of the object based on the epipolar geometry (Fig. 1). In the system presented, the optical axes of both the cameras cross in one point on the surface of the object.

Fig. 1. Epipolar geometry of the dual-camera system The imaging system in such a case ensures the achievement of a wide range of the dimensional observation field, which was one of the basic requirements of the monitoring system. The scale of the observation field is the result of the parameters of the lens used and the size of the sensor of the camera (Fig. 2).

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a)

b)

Fig. 2. View of the model of the vision module: a) degrees of freedom, b) basic geometrical parameters The optical resolution of the vision path is determined by the relation of the real dimensions of the observation field to the pixel size of the camera sensor. The parameters of the optical system were determined with the use of basic formulas in classic optics. As the measurement system was intended to be applied for tests on fatigue testing machines, the functional requirements and structural restrictions stemming from the assumed work conditions were also taken into account. In the positioning system for each of the modules four degrees of freedom were assumed, which is supposed to ensure the easiness of the regulation of the position of the vision path with reference to the sample observed. Within the analysis conducted, the specification of basic parameters of the designed stereovision system was developed (Tab. 1). Table 1. Selected parameters of the stereovision system Parameter Value 80 [mm] ÷ 360 Working distance WD [mm] Field of view FOV (dimension H) 1 [mm] ÷ 50 [mm] Maximum optical resolution 3 [µm] Pan angle range between the 16 [º] ÷ 110 [º] cameras α Positioning range in the O-Y axes 150 [mm] Positioning range in the O-Z axes 300 [mm] Positioning range αx ±45 [º] Positioning range αy ±30 [º] The matrix of design solutions for the structure of the vision module was developed, which constituted a starting point for the determination of its individual components. It is, however, worth underlying that the design of the structure of the vision system is a process of selecting from the limited availability products the following components: the camera, lens, filters, lighting set-up. Using the parameters identified, the components meeting the predefined requirements were selected. In order for the required measurement resolution to be ensured, the CCD camera with 2448x2050 pixel sensors was used. Due to the necessity to adjust optical parameters of the lens to the observation scale, the motorised lens Zoom6000 by Navitar was selected [9], easy to reconfigure with the use replaceable components. The simulation and the analysis of the dependency between the working distance of observation (WD) of the specimen mounted onto the fatigue testing machine and the (α) pan angle between optical axes of cameras are possible (Fig. 3).

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110

80

100

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Pan angle between cameras α [°]

Pan angle between cameras α [°]

90

70 60 50 40 30 Lower limit

20

Upper limit (with grips) Upper limit (without grips)

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0 100

105

110

115

210

120

230

250

270

Working distance WD [mm]

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310

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350

370

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Working distance WD [mm]

Fig. 3. Positioning ranges for the angle between optical axes with mounting grips considered The grips in the fatigue testing machine limit the top range of vision angles for greater working distances. The grips, however, do not limit the bottom ranges for the regulation of vision angles. The optional configurations of lenses ensure that the required measurement ranges and optical resolutions are obtained (Fig. 4 and 5). The realisation of the assumed configuration of the Zoom6000 lens consists in the application of replaceable components (body tubes and accessory optics) that cooperate with the main lens module. 100 90

Field of view FOV [mm]

80 70 60

Upper limit line

50

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40 30 20 10

II

III

Lower limit line

0 0

50

100

150

200

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300

350

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Working distance WD [mm]

Fig. 4. FOV for configurations of lenses I, II, III with reference to the WD working distance (possible solutions in the I, II, III areas between the Lower and Upper limit lines) 30

Optical resolution [µm]

25

I

20 15 10

II III

5 0 0

50

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150 200 250 Working distance WD [mm]

300

350

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Fig. 5. Optical resolution with reference to the WD working distance The Zoom6000 lens is equipped with miniature DC motors that adjust the zoom and focus, which allows for the automation of the image scaling and calibration processes. The developed mechanical structure of the monitoring system is presented in Fig. 6.

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Supporting frame

LED illuminator

Zoom6000 Lens

Specimen CCD camera

Camera positioning set

Fig. 6. General view of the fatigue monitoring system mounted onto the fatigue testing machine Rigid aluminium profiles ensuring high stiffness of the structure at a considerably low weight were used. The frame mounted onto the fatigue testing machine is the main support element. The vision modules are fixed on the 4DOF positioning sets, which allow the adjustment of the camera position in relation to the specimen. The degrees of freedom of the vision module include the following: the linear shift along the optical axis, rotation in the vertical and horizontal planes, as well as rotation in relation to the optical axis. In the lighting module for the scene, LED ring illuminators with the illuminance of ca. 18 [klux] at the distance of ca. 300 [mm] from the sample were used (Fig. 7). The LED illuminators are triggered in the frequency range of 1÷99 [Hz]. In the case of the imaging of high speed processes, when a very high luminous flux density is needed, a halogen illuminator with the illuminance of ca. 500 [klux] at the distance of ca. 300 [mm] from the specimen can be applied. a)

b)

Fig. 7. Lighting systems: a) with the use of LED ring illuminators, b) with the use of halogen illuminator The developed software allows for the analysis of fatigue processes. The method of digital image correlation DIC is used for the detection and identification of the areas characteristic for the images recorded. In order to ensure the proper functioning of the programme modules responsible for the realisation of digital image correlation in the 2x2D mode and the reconstruction of the surface analysed in the stereo 3D mode, the camera calibration process is conducted. The automatic image sharpness setting realised by means of the electronic lens control is also applied.

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Discussion The developed dual-camera vision system presents the alternative concept of the structure and mounting of the system compared with ARAMIS GOM GmbH and Trilion Quality Systems. The direct mounting onto the fatigue testing machine ensures the rigid and stable position of the vision system in relation to the specimen which is a significant advantage in long-term tests. It should be emphasized that every position change of the vision system (intended or unintended) requires the calibration of the cameras to be repeated. Regulation ranges for the camera position in the 4DOF system enable similar observation possibilities as aforementioned commercial solutions. The rare experimental stereovision systems [5,7] are not equipped with the positioning mechanical modules. These systems allow for a very limited positioning of cameras and are definitely inferior to the system presented in the paper. Summary The dual-camera vision system was developed to meet the needs of the advanced fatigue tests in particular. It allows for the observation of the surface of the specimen enabling the analysis of created deformations, detection of the initiated fatigue crack and the monitoring of the crack propagation. The wide range of possible applications is the main advantage of the system, as it allows for the independent positioning of the cameras in the 4DOF systems and the reconfiguration and modification of the components of the vision module, i.e. the camera, lens and illuminator. The system can cooperate with the user’s individual software. Further development of the system can include automation of the function of the positioning of the vision module in relation to the specimen at the stage of calibration as well as during the monitoring of the fatigue process. Scientific work created within the framework of the “Innovative Systems of Technical Support for Sustainable Development of Economy” Strategic Programme within the Innovative Economy Operational Programme. References [1]

[2] [3]

[4]

[5] [6]

[7]

[8]

Boroński D., Szala J., Giesko T.: Automatic measurements of fatigue crack length and trajectory. 19th Danubia-Adria Symposium on Experimental Methods in Solid Mechanics, Polanica-Zdrój 2002 130-131. Giesko T., Boroński D., Zbrowski A., Czajka P., Detection and measurement of fatigue cracks in solid rocket propellants, Maintenance Problems 3 (2009) 75-85. Hügli H., Mure-Dubois J., 3D vision methods and selected experiences in micro and macro applications, Two and Three Dimensional Methods for Inspection and Metrology IV (proc. SPIE) Vol. 6382 Iss. 10 (2006) 209-216. Karpour A., Zarrabi K., A Stereo Machine Vision System for measuring three-dimensional crack-tip displacements when it is subjected to elastic-plastic deformation, E-Leader Conference Singapore 2010, http://www.g-casa.com/E-Leader-Singapore_Program.htm. Sharpe W.N. (ed.), Springer Handbook of Experimental Solid Mechanics, Springer 2008 589597. Tang Z.Z., Liang J., Xiao Z.Z., Guo C., Hu H., Three-dimensional digital image correlation system for deformation measurement in experimental mechanics, Optical Engineering 49 (10) (2010) 103601-103609. Yan J.H., Sutton M.A., Deng X., Wei Z., Mixed-mode crack growth in ductile thin-sheet materials under combined in-plane and out-of-plane loading, International Journal of Fracture, vol. 14 no 4 (2009) 297-321. Zbrowski A., Samborski T., Giesko T., Boroński D.: Opto-mechatronic system for fatigue crack monitoring of riveted joints. Maintenance Problems 4 (2010) 153-162.

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Information on http://www.navitar.com Information on http://www.gom.com/metrology-systems/ Information on http://www.trilion.com/ Information on http://www.correlatedsolutions.com

© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.726.233

MODELING OF CRACK GROWTH IN STEELS Jacek Jackiewicz University of Technology and Life Sciences in Bydgoszcz, Faculty of Mechanical Engineering, Department of Applied Mechanics, 7, Prof. S. Kaliski Street, PL 85-789 Bydgoszcz, POLAND email: [email protected] Keywords: Direct and iterative solvers, Fracture mechanics, BCC and FCC steels, Peierls stress, Strain localization

Abstract. The paper discusses issues related to the damage accumulation and cracking in steels. Special attention is paid to the selection of appropriate methods in the modeling of progressive damage development. In special cases, the damage accumulation and crack propagation may lead to the brutal destruction of machine parts. Hence, some attention is drawn to the conditions that can lead to this brutal destruction. In order to model unstable crack propagation the applied solver for solving systems of resulting equations should be as efficient as possible. Physical description of the cracking of steels Generally, brittle solids contain small microcracks. Such cracks may propagate at a stress that is lower than that required for any slip of dislocations. Brittle fracture in brittle solids can be defined as occurring at very small plastic deformation when the average stress carried by material is less than the yield stress. If preexisting cracks in materials are small, then the stress can reach the level required to initiate slip or twinning. Brittle cracks nucleated by slip or twinning are not able to propagate immediately. The ability of steel to plastic deformation in the vicinity of the crack tip prevents the spread of brittle cracks. Substantial plastic strain or grain boundary sliding can cause the nucleation of larger grain boundary cracks or cause preexisting cracks to grow in a stable manner, until an increased length of one of these cracks coupled with higher stress due to work hardening leads to unstable propagation of the cleavage crack. During the ductile fracture of steel tensile strength is less than the stress required for crack propagation. In this case, steel first deforms plastically, causing the nucleation and growth of voids. Such cracking occurs along the grain boundaries and is called intergranular, when most of the second phase particles, where voids are formed, is located on grain boundaries. If the distribution of second phase particles is relatively uniform, the cracking occurs through grains and is called transgranular. The modeling of dislocations in crystals remains attractive due to reduction of computational effort when compared to, for example, atomistic simulations. In steels, which have the bodycentered cubic (BCC) crystal lattice structure, such as ferritic steels, the Peierls stress for their slip systems is predicted to increase with decreased temperature, thereby increasing the yield stress at low temperatures. The Peierls stress (discovered by Rudolph Peierls and modified by Frank Nabarro) represents the force necessary to move a dislocation within a plane of atoms in the unit cell. The magnitude of the force needed to move the dislocation depends on the distance between planes of atoms in the unit cell and also the size and width of the dislocation according to the following equation [1, 2]: =

exp

‖ ‖

,

(1)

where μ is the shear modulus, ν – Poisson's ratio, ζ ≝ 0.5d ⁄ 1 − ν" – the half-width of the dislocation, d ≝ a⁄√h + k + l – the interplanar spacing between planes of atoms in the cubic crystal with Miller indices h, k and l of cubic planes form a notation system in crystallography for planes and directions in crystal (Bravais) lattices, a – the lattice constant (the edge of the cube), –

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the Burgers vector associated with a dislocation that represents the measure of a lattice distortion caused by the presence of the line defect. The length of the Burgers vector is usually represented by the equation: ‖ ‖ = 0.5a√h + k + l

(2)

but its direction depends on the most likely plane for slip, which is usually the closest-packed plane of the unit cell. In most metallic materials, the length of the Burgers vector for a dislocation is equal to the spacing between atoms in the most closely packed direction on the most closely packed plane. Because of the Peierls stress for the slip system of material is predicted to decrease with increasing planar density of atomic packing (represented by d ⁄‖ ‖), any slip of dislocations is preferred on closely packed planes. The shortest interatomic distances, assessed for materials with face-centered cubic (FCC) structures, are found to be smaller than that assessed for BCC ones. The yield stress dependence on the temperature is low for FCC materials, therefore these materials can be used in cryogenic conditions. However, BCC metals, of which the yield stress has strong dependence not only on the temperature but also on the strain rate, are prone to brittle fracture. Sequential multiscale modeling of crack growth requires the use of the incremental methodology for the input equations of the structural model. If an idealized problem of damage accumulation and crack growth in a steel part is solved by the use of the finite element method (FEM), a system of algebraic equations with the coefficient matrix is formulated. The coefficients of the system reflect relationships between grid nodes specified by the spatial discretization for the assumed mathematical model. Since crack propagation creates new surfaces, the number of finite elements should be increased in the numerical model. If the number of finite elements is increased than it is obvious that the number of system equations should be also increased. Very often, the coefficients of a system of equations are written in the form of a matrix. There is a significant effect of the bandwidth of the coefficient matrix on the time of calculations and the size of used memory. In practice, it is difficult to determine optimal numbers of mesh nodes in the context of the bandwidth reduction problem taking into consideration new surfaces of a propagating crack. For a symmetric sparse matrix of the coefficients, which are determined for a system of equations, the problem is to reduce the matrix bandwidth by permuting rows and columns such as to move all the nonzero elements of the matrix in a band as close as possible to the matrix diagonal. The use of the reverse Cuthill-McKee algorithm [3] for renumbering mesh nodes makes sense only for the initial spatial discretization of the numerical model up to the start of crack propagation. The bottleneck effect in simulations of crack propagation is generally associated with the necessity of solving a large system of equations, which gets expanded every time not only by the creation of new equations but also by the extension of old ones due to the occurrence of crack growth. The extended finite element method (XFEM) [4] may be used to model propagation of various discontinuities: strong (cracks) and weak (material interfaces) without remeshing. A key advantage of XFEM is that discontinuous basis functions are added to standard polynomial basis functions for nodes that belong to elements intersected by a discontinuity. The discontinuous basis functions provide a basis for discontinuity opening displacements using the concept of the partition of unity method. In general, the partition of unity method is a powerful technique to model discontinuities and singularities accomplished through local enrichment within a finite element setting. Direct and iterative methods for solving of linear equations used to model crack growth Modeling of damage accumulation and crack growth in steels by means of the FEM requires solving systems of equations with sparse coefficient matrices, which contain relatively large numbers of zero elements. Both direct as well as iterative solvers may be used to find a numerical solution of a system of linear equations.

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The direct solvers allow to obtain a solution after a finite number of elementary operations. The performance of elementary manipulations on coefficients and constant terms of equations transforms the input original system of linear equations into an equivalent one by adding the two equations from a system previously multiplied or divided by a nonzero number, by switching places of two equations from a system or by multiplying both sides of one equation from a system by a nonzero number. One of the direct methods for solving systems of linear equations, which is equally simple, as well as accurate, is the method of Gaussian elimination. The standard Gaussian elimination algorithm can be used to solve a general system of n linear equations of the form: + - = , with n unknowns included in the vector - and n constant terms included in the vector . The goal of elementary operations of the Gaussian elimination is to transform or rather reduce the original matrix of coefficients, + = ./01 2 , into the upper 3×3

triangular matrix, 5 = .601 2 . Hence, the original system of equations, + - = , can be replaced 3×3 by an equivalent system, 5 - = 7, with the transformed vector of constant terms, 7. In the first step of forward elimination, the first unknown 8 is eliminated from all n − 1 equations of the system by subtracting the multiplied first equation of the system from the 9-th equation in order to zero the coefficient /0 assigned to 8 in the 9-th equation (for 9 = 2, … , n). The subtraction of the relevant multiples of elements / 1 of the first row of + and the term = , which are defined by quotients /0 ⁄/ , from the elements /01 of the 9-th row of + and term =0 can be represented algebraically by the equations: >

"

/01 = /01 − > ?@ / 1 ,

(3a)

=0

(3b)

"

>

@@

= =0 − > ?@ = . @@

In the second step, the second unknown 8 is eliminated from all n − 2 equations of the " transformed system, + " - " = , by means of the same type of manipulations as in the first " step. These manipulations are performed to elements /0 of the coefficient matrix + " (for 9 = 3, … , n), which are assigned to 8 . Continuing this approach recursively the system of equations, 5 - = 7, with the triangular coefficient matrix, 5, is obtained after n − 1 steps. A solution of the equation system, 5 - = 7, can be determined by the inverse procedure: 80 = >BC@ =0 3 ??

"

− ∑31E0F /013

"

81 = G HI0 − ∑31E0F 601 81 J, ??

(4)

where: 9 = n, n − 1, … ,1. The Gaussian elimination method is almost always effective if main diagonal elements /00 of + are not equal to zero. Otherwise, its modifications are used with partial or full choice of pivot entry. In the case of matrix computations, a pivot entry (i.e., an element of a matrix, which is selected first by an algorithm) is usually required to be at least distinct from zero. The pivot element is the first nonzero entry in the row, which does the elimination. The pivots are usually found along the diagonal. Finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. The direct method for solving systems of linear equations based on the Cholesky algorithm for the decomposition of the matrix of coefficients, +, is often used. The Cholesky algorithm is a stepby-step decomposition procedure for the symmetric positive-definite matrix, +, with real entries into the product, + = K KL, of a lower triangular matrix K and its conjugate transpose KL . The Cholesky decomposition is possible only if + is symmetric and positive definite. In this case, the solution of the system of equations, + - = , can be obtained by first setting the vector M by means of the formula: K M = , and then calculate the vector - using the system KL - = M.

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An alternative form of the Cholesky decomposition is the symmetric indefinite factorization [5]: + = KN O KLN . This form eliminates the need to take square roots. When + is positive definite the elements of the diagonal matrix, O, are all positive. However, this factorization may be used for most invertible symmetric matrices. An example of an invertible matrix, for which decomposition is undefined, is one in that the first entry is zero. The KN O KLN and K KL factorizations may be easily related by: @

@

@

@

+ = KN O KLN = KN OP OP KLN = KN OP OP

L

@

@

KLN = KN OP KN OP

L

= K KL.

(5)

However, the main disadvantage of direct methods, which are used to solve a system of linear equations with a sparse matrix, is the creation of new nonzero elements in this matrix during the calculations. The new nonzero elements generated in individual items of the matrix + that have been forced to zero before performing the arithmetic operations are called fill-in matrix elements. The difference between direct and iterative types of algorithms for solving systems of linear equations should be distinguished. Direct methods give the exact solution, - = + , after a finite number of computational steps involving exact arithmetical operations. Iterative methods are based on iterative improvements of an approximate solution until it reaches a satisfactory accuracy. The processing performance of iterations begins with an initial approximation -Q . For each iterative method, it is also necessary to define the criterion for terminating iterations, usually in the form of the maximum permissible limit value for: - the length of the line segment expressed by the norm: R- SF " − - S" R, between successive approximations of a solution obtained in iterations S and S + 1, which, in Euclidean geometry, allows to determine the distance between two points represented by the vectors - S" i - SF " , - the vector norm given by the residual formula: RT SF " R = R − + - SF " R. Iterative methods, which use exact arithmetic operations, does not give the exact solution after a finite number of computational steps. It can be shown on the example of the iterative operatorsplitting method. The iterative operator-splitting method is based on the transformation of the matrix + using mathematical operations defined by the formula: + = U + + − U". The system of equations, + - = , can then be transformed as follows: VU + + − U"W- = , U - = U − +"- + , - = U U − +"- + U

(6a) (6b) (6c)

.

Hence, the iterative method - SF

"

=U U− +" - S" + U ] XYYYZY YY[ E\

E^

= \- S" + ^

(7)

determines the series _- " , - " , - `" , … , - S" , … a, for which limS→e - S" = - = + . Gauss may be counted as adherents of iterative methods for solving systems of linear equations due to the fact that these methods are easier to perform than direct ones. In 1823 he wrote in his native German-language a piece of text [6] that after translation into English sounds as: The method of elimination is difficult to apply directly, at least when you have more than two unknowns. Iterative methods can be carried out half asleep, or thinking about other things. Large systems of equations, which are usually characterized by spare matrices of coefficients, are solved mainly by using iterative methods. The required number of iterations can be reduced by the use of the initial conditions called as preconditioning. Assuming that if in Eq. 7 the positive-definite symmetric matrix, \, which approximates the matrix +, is much easier to reverse than +, then instead of solving the system, + - = , it can be . In particular, if \ = +, then the considered the following system of equations: \ + - = \

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determination of a solution can be carried out only in one iteration because of + + = f. The matrix \ is called a preconditioner for +. Selection of the matrix \ occurs as a result of a compromise of the following requirements: - preconditioner should be a sparse matrix of simple structure, in order to be easily reversed, - preconditioner should have a structure as close as possible to the matrix +, in order to accelerate the convergence of the algorithm as efficient as possible. Very often there are used the following selection methods for the matrix \: - diagonal scaling: g01 = h00 i01 (represented by the main diagonal of the matrix +) for 9 = j, whereas g01 = 0 for 9 ≠ j, - scaling method using the main diagonal and two adjacent minor diagonals: g01 = h01 for |9 − j| ≤ 1, whereas g01 = 0 for |9 − j| > 1, - incomplete Cholesky decomposition: \ = K∗ K∗L . It should be noted that during the implementation of an incomplete Cholesky factorization as a preconditioner \, the sparse matrix K∗ is only a certain approximation of a lower triangular matrix K of the full Cholesky decomposition: + = K KL. At present time, the most commonly used iterative method for symmetric positive definite matrices is the conjugate gradient method. This method can be applied to solve systems of equations with sparse matrices that may be too large to be handled by direct algorithms such as the Cholesky decomposition. In addition, this method can also be used to solve optimization problems without constraints, including determination of the minimum potential energy. The method of conjugate gradients (CGs) was developed independently by Stiefel of the Institute of Applied Mathematics at Zurich and Hestenes with the cooperation of Rosser, Forsythe and Paige of the Institute for Numerical Analysis of the U.S. Bureau of Standards. The well-known form of the algorithm of this method was developed jointly by Hestenes and Stiefel during a stay of the second scientist at the U.S. Bureau of Standards [7]. The CG method can be best described by using the algorithm for minimizing of the following square functional (which takes the quadratic form): p -" = -L +- −

L

-+7

(8)

for a particular set of vectors. If the matrix + is nonsingular, then the functional p -" can be written as: p -" = -L +- − "L +

+- − " −

L

+

+ 7.

(9)

The functional p -" reaches a minimum only for the vector -, which is the solution of the system, +- = . Hence, the designation of the minimum of the functional p -" is equivalent to solve the system, +- = . If in Eq. 8 the matrix + is singular and its inverse + represents the generalized inverse matrix, then, in consequence, the determination of the minimum of the functional p -" is still equivalent to solve the system, +- = , for which minimization is performed in the orthogonal complement of the null space of +. For the algebraic residuum: q Q" = + - Q" − , which corresponds to the initial guess - Q" , the CG method is carried out as the sequential line minimizations in a linear subspace of ℝ3 generated by the images of q Q" under the first S powers of the square matrix + of degree n (starting from +Q = s): q Q" , + q Q" , + q Q" , … , +S q Q" , i.e., tS H+, q Q" J = span_q Q" , + q Q" , + q Q" , … , +S q Q" a. The subspace determined by the formula (10) is known as the order- S Krylov subspace.

(10)

238

Fatigue Failure and Fracture Mechanics

The algorithm for finding a solution consists of looking for next approximate solutions along the direction of the vector v S" according to the formula: - SF

"

= - S" + τS v S" ,

(11)

where for the S-th iteration the parameter τS is chosen in order to minimize the functional pH- S" + τS v S" J given by Eq. 8, i.e., τS =

x v S" q S"

S" x + v S"

v

x v S" H+ - S"

=

v

S" x + v S"

J

.

(12)

Moreover, the gradient of p in the point specified by the vector - SF " , which is equal to the residuum q SF " , should be perpendicular to the direction of search v S" . This fact can be proven as follows: - SF

"

= - S" + τS v S"

v

S" L

q

SF "

⇒ ⇒ =

+ - SF " − = + - S" − + τS + v S" ⇒ q SF " = + - SF " − = q S" + τS + v S" ⇒ v

S" L

q S" + τS v

S" L

(13)

+ v S" = 0

During each iteration of the CG method the relatively simple procedure based on the product of the matrix + by the vector v S" , which does not change the form of +, is used. In comparison to direct methods, the CG method allows to reduce the computer memory requirements and to use more effectively the specific structure of the sparse matrix. There are a number of variants of the CG method. One of them is the biconjugate gradient method for asymmetric matrices [8]. Note that the conjugate residual (CR) method [9, 10] differs to the similar method of CGs primarily in that the CR method is applicable for Hermitian matrices (i.e., symmetric matrices with real coefficients), which should not be positively defined. This makes the CR method can be implemented to deal with the numerical optimization problems using Lagrange multipliers in order to find saddle-points of the objective function instead of minima, and during computer simulations of the damage accumulation and cracking of strain-softening materials by mean of FEM. Having regard to an initial (arbitrary) estimate of the solution of the system of equations, - Q" , the CR method may be characterized by the concise algorithm shown in Example 1. Example 1. Program code of the CR method without preconditioning - Q" = preliminary assessment of the solution (some initial guess) T Q" : = − + - Q" v Q" : = T Q" 9{ℯ6/{}, with S starting at 0: S: = 0 6ℯ~ℯ•€ L T S" + T S" αS = + v S" "L + v S" " - SF = - S" + αS v S" T SF " = T S" − αS + v S" _convergence criterion for - SF " a ‰Š HT SF " is small enoughJ €Œℯ• exit loop ℯ•Ž ‰Š βS =

T SF

"L

L

+ T SF

"

T S" + T S" = T SF " + βS v S" v + v SF " = + T SF " + βS + v S" S: = S + 1 ℯ•Ž •ℯ~ℯ•€ SF "

Dariusz Skibicki

239

Carrying out calculations by the CR method in comparison to the CG method requires the introduction of an additional vector of the upgrading solution, and therefore the implementation of 2n more elementary arithmetic operations. The algorithm of the CR method with preconditioning to improve the convergence of results can be written, as in Example 2. Example 2. Program code of the CR method with preconditioning - Q" = some initial guess T Q" : = \ H − + - Q" J v Q" : = T Q" 9{ℯ6/{}, with S starting at 0: S: = 0 6ℯ~ℯ•€ L T S" + T S" αS = + v S" "L \ + v S" " - SF = - S" + αS v S" T SF " = T S" − αS \ + v S" _convergence criterion for - SF " a ‰Š HT SF " is small enoughJ €Œℯ• exit loop ℯ•Ž ‰Š βS =

T SF

"L

L

+ T SF

"

T S" + T S" SF " v = T SF " + βS v S" + v SF " = + T SF " + βS + v S" S: = S + 1 ℯ•Ž •ℯ~ℯ•€ The preconditioning matrix \

must be symmetrical.

Problems in modeling of damage accumulation leading to the fracture of material Strain localization usually appears in narrow bands of material. It is identified with heterogeneous forms of strain distribution in a place of the local loss of stability of material. The mathematical description of this phenomenon is identified with ambiguity of a solution to the stress equilibrium partial differential equations that results in a loss of ellipticity of these governing differential equations. For strain localization problems, the acoustic tensor method can be applied to only strain softening materials. If the phenomenon of strain localization is considered in fault zones of materials exhibiting strain hardening and strain softening, then the vertex damage-coupled theory developed by Stören and Rice [11] can be applied. The elastoplastic theory for non-associated plastic strains (the yield surface does not coincide with the plastic potential) shows that it is possible to fulfill the localization criterion (vanishing value of the acoustic tensor determinant) before the plastic limit criterion (vanishing value of the determinant of the elastoplastic matrix). For certain stress states steels will exhibit narrow bands of intense deformation called shear bands, since the deformation mode in these bands is usually shear. It is often claimed that shear banding corresponds to a zero value of the determinant of the acoustic tensor. The eigenvalue problem corresponds to the perturbed (i.e., linearized) equation of motion by means of the governing equations for the continuum. Two general types of methods, namely transformation methods and iterative methods, are available for solving eigenvalue problems. The transformation methods, such as Jacobi, Givens and Householder schemes, are preferable when all eigenvalues and eigenvectors are required. The iterative methods, such as power method, are preferable when few and eigenvectors are required.

240

Fatigue Failure and Fracture Mechanics

Conclusions 1. Direct and iterative methods for solving systems of linear equations, which are intended solely for the positive definite matrices of coefficients are unsuitable for modeling crack growth in steels. 2. Parallel iterative solution method for large sparse linear equation systems should be preferred. 3. Checks for path stability need to be included in finite element programs for damage. References [1] J.P. Hirth, J. Lothe, Theory of Dislocations, 2nd Edition. Krieger Publishing Company, 1992. [2] R.W. Hertzberg, R.P. Vinci, J.L. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 5th Edition. Wiley, 2012. [3] E. Cuthill, J. McKee, Reducing the bandwidth of sparse symmetric matrices, in: Proc. 24th ACM National Conference, Association for Computing Machinery, New York, 1969, 157–172. [4] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46 (1999) 131-150. [5] J.A. George, J.W-H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs NJ 1981. [6] C.F. Gauss, Brief an Gerling vom 26 Dec. 1823, Werke, vol. 9, 278–281, in: Mathematical Tables and Other Aids to Computation (a translation by G.E. Forsythe), vol. 5, 1950, 255–258. [7] M.R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards 49 (1952) 409-436. [8] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Section 2.7.6 Conjugate gradient method for a sparse system, in: Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, New York 2007, 87-92. [9] Y. Saad, Iterative methods for sparse linear systems (2nd ed.), Society for Industrial and Applied Mathematics, Philadelphia, PA 2003, 181–182. [10] T. Sogabe, M. Sugihara, S.L. Zhang, An extension of the conjugate residual method to nonsymmetric linear systems, Journal of Computational and Applied Mathematics 226 (2009) 103-113. [11] S. Stören, J.R. Rice, Localized necking in thin sheets, Journal of the Mechanics and Physics of Solids 23 (1975) 421-441.

Keywords Index 1.4301 Steel

77

A Accelerated Method Acoustic Methods Ageing Algorithm of Fatigue Calculations Aluminum Alloy ARAMIS

11 55 3 17 63 110

B Bainite BCC Steel Bearing Steels Block Loading

55 233 55 181

C Carbon Steel Clad Crack Growth Analysis Cracks Cyclic Fatigue Cyclic Hardening Cyclic Material Properties Cyclic Properties

203 106 218 222 43 63 51 3, 150

77 218 233 222 226

118 106 125

F Failure Probability Analysis Fatigue Fatigue Characteristics

H Heat Affected Zone High-Chromium Cast Steel High-Cycle Fatigue Hybrid Method Hysteresis Loop

110 3 11, 63 222 143

In-Plane Constraint Infrared Thermography Initial Stiffness Modulus Isothermal Heat Treatment Iterative Solver

195 156, 162 84 55 233

L

E Efficient Material Explosive Cladding Explosive Welding

Fatigue Life Estimation Fatigue Life of Steel Fatigue Monitoring Fatigue Strength Fatigue Testing System Fatigue Tests FCC Steels Fem Fictitions Radius Fractography Fracture Mechanics Fracture Toughness Friction Stir Processing

203 171, 189 11 118 27, 106, 171, 181, 211 189 17 226 69 51 211 233 100 27 171 233 195 203

I

D Diagrams of Fatigue Life Digital Image Correlation Direct Solvers Dual-Band System Dual-Camera System

Fatigue Crack Growth Rate Fatigue Criteria Fatigue Design Fatigue Failure Fatigue Life

118 33, 51, 100, 106, 125 43, 143, 181

Laser Weld Laser Welding Linear Hypothesis of Fatigue Damage Accumulation Local Stress Approach Low Cycle Fatigue Low Cycles Fatigue Life

100 133 39 100 3, 77, 143, 150 93

242

Fatigue Failure and Fracture Mechanics Trabecular Bone Two-Parametric Fatigue Characteristics

M Martensitic Cast Steel Mean Stress Microspecimen Midrib Mini Specimen Multi-Processor Graphic Cards Multiaxial Fatigue Multilayer Pipe

150 33 51 55 63, 156 218 162, 171, 189 133

39, 84 69

U Undermaching

110

V Vision System

226

W N Non-Proportional Loading Nonproportional Load

189 162

O Out-Of-Parallelism Out-of-Plane Constraint

181 195

P Peierls Stress Pitting Programmed Fatigue Testings

233 55 17

R Random Loading Ratcheting Residual Stress Riveted Joints Rolling Contact Fatigue

17 125 125 211 55

S S-N Curve Section Method Squeeze Force Steel Sandwich Panel Stepwise Loading Strain Localization Strength Strength Properties Stress Ratio

11 33 211 100 39, 84 233 133 156 33

T Thermal Stress Thermo Mechanical Fatigue Thermography Thickness Effect

125 143 222 211

Weakest Link Concept Welded Joint Welds Whole-Field Displacement Analysis

118 93, 118 110 218

Authors Index B Blacha, Ł. Böhm, M. Boroński, D. Bujnowski, S.

118 33 51, 156, 218, 222 218, 222

C Cichański, A. Czajka, P.

39, 84 222

M Machniewicz, T. Marciniak, T. Marcisz, E. Marecki, P. Mazurkiewicz, A. Mrozinski, S. Mroziński, S.

211 218, 222 43 93 39, 84 150 3, 133

N G Gałkiewicz, J. Giesko, T. Golański, G. Goss, C.

110 218, 226 3 93

203

Okrajni, J.

143

P

J Jackiewicz, J. Jelenkowski, J. Junak, G.

195 33, 43, 106 100 39, 84 203

O

H Hutsaylyuk, V.

Neimitz, A. Niesłony, A. Niklas, K. Nowicki, K. Nykyforchyn, H.

233 55 143

Pejkowski, Ł. Piotrowski, M.

171, 189 133

R K Karolczuk, A. Kluger, K. Kocańda, D. Korbel, A. Kowalski, M. Kozak, J. Kurek, A. Kurek, M. Kyryliv, V.

118, 125 125 203 211 125 100 106 181 203

L Łagoda, T. Ligaj, B. Lipski, A. Lutowski, Z.

27, 43, 181 17, 69, 77 156, 162 218, 222

Ranachowski, Z. Robak, G. Rozniatowski, K.

55 27, 125 55

S Sempruch, J. Skibicki, D. Skocki, R. Skorupa, A. Skorupa, M. Slezak, T. Strzelecki, P. Szala, G. Szymaniec, M.

11, 63, 171 162, 171, 189 150 211 211 203 11 17, 69, 77 27

T Tomaszewski, T. Topoliński, T.

63 39, 84

244 Torzewski, J.

Fatigue Failure and Fracture Mechanics 203

W Werner, K. Woźniak, T.Z.

3 55

Z Żok, F.

125