Albinmousa2011.pdf

Engineering Fracture Mechanics 78 (2011) 3300–3307

Contents lists available at SciVerse ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Technical Note

A model for calculating geometry factors for a mixed-mode I–II single edge notched tension specimen Jafar Albinmousa a, Nesar Merah b, Shafique M.A. Khan b,⇑ a b

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Ontario, Canada N2L 3G1 Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 21 August 2011 Accepted 8 September 2011

Keywords: SENT Mixed mode Geometry factors Finite element

a b s t r a c t In the present study, a novel approach is presented to obtain closed-form solutions for the geometry factors, which are used to determine the stress intensity factors for various configurations. A single edge notched tension specimen with an angled-crack is used as an example to demonstrate the applicability, simplicity and flexibility of the new approach. Several values for crack inclination angles, plate widths and crack lengths, including micro-cracks, are considered in the analysis. The new approach is validated through comparison with existing analytical and numerical solutions as well as experimental results. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In linear elastic fracture mechanics (LEFM), stress intensity factors (SIFs) characterize the state of stress in the vicinity of the crack tip and therefore hold a pivotal position in the application of LEFM to further problems such as fatigue life prediction [1,2], crack initiation prediction [3–8], crack tip core region characterization [6–9], crack propagation simulation [10–12] and crack stabilization [13,14]. However, closed-form solutions for the calculation of SIFs are only available for cracks in infinite plates (remotely applied stress) or for a limited number of cases of cracked plates of finite size. In addition, for the cases of cracks in finite size plates, most of the closed-form solutions available are for pure mode I loading only. Freese and Baratta [15] used weight function arguments and a boundary collocation technique to re-examine the geometry factor solutions for a finite plate, with a single edge crack, under different loading configurations. Fett [16], using weight functions and superposition method, presented the stress intensity factor solutions for short plates in tension and bending. Cho et al. [17] investigated mode I and mixed mode fracture of polysilicon for microelectomechanical systems (MEMS). They generated edge cracks of crack lengths ranging from 2.5 to 26 lm at different angles of crack inclination using microindentation on specimen sizes between 50 and 400 lm. Finite element analysis was used to determine SIFs, which were used to predict crack initiation angles. Yoneyama et al. [18] proposed a new method for evaluating the mixed mode I–II SIFs from optically observed displacement fields. Hasebe and Inohara [19] performed stress analysis of a semi-infinite plate with an inclined edge crack using rational mapping function of the sum of expressions and the complex variable method and explored the relationship between the stress intensity factors and the crack inclination angle. Chen and Wang [20] calculated the mixed mode I–II geometry factors for a finite plate with an inclined edge crack using weight function method, which are then used to determine the mixed mode I–II SIFs and T-stress for an offset double edge-cracked plate. Merah and Albinmousa [21] determined the mixed mode I–II crack initiation angle for a cracked polycarbonate plate with a single inclined edge crack. The mixed mode I–II SIFs were determined using photoelasticity and finite element method. Jogdand and Murthy

⇑ Corresponding author. Tel.: +966 3 860 7225; fax: +966 3 860 2949. E-mail address: [email protected] (S.M.A. Khan). 0013-7944/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.09.005

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

3301

Nomenclature K I, II b W a Y a, c d /

r

stress intensity factor subscripts denoting the mode of loading crack inclination angle specimen width crack length geometry factor arbitrary fitting coefficients arbitrary fitting exponent loading angle applied stress

[22] proposed a new method for the estimation of the mixed-mode SIFs based on the formation of over-determined system of equations. The proposed method must be implemented in existing finite element codes. Feng et al. [23] employed a continuous dislocation model with finite element method to evaluate stress intensity factors for an arbitrary number of through thickness cracks. Some of the techniques such as the boundary collocation and conformal mapping techniques [24,25], weight functions [16] and mapping functions [19] and finite element techniques [22] are useful tools for calculating mixed mode I–II SIFs. However, most of these techniques require a certain level of mathematical expertise to provide the ability to determine the stress intensity factors. In addition, these methods take considerable time to setup and run the numerical problem. Therefore, there is a pressing need to develop simpler solutions to determine stress intensity factors for various configurations. In the present study, a novel approach employing geometry (or shape) factors, to determine the stress intensity factors for an inclined (mixed mode I–II) edge crack in a finite size plate is presented. The approach is based on finite element analysis, the results of which are used for curve fitting delivering closed-form solutions for geometry factors and consequently the stress intensity factors. The new approach is validated by comparison with available analytical, numerical and experimental results for various combinations of crack inclination angles, crack lengths and plate widths. 2. Problem configuration The problem considered, as shown in Fig. 1a, is the standard configuration of Single Edge Notched Tension (SENT) specimen modified to provide mixed mode I–II. L is the specimen length, W is the width, a is the crack length and b is the crack inclination angle; b = 0 implies pure mode-I loading (i.e. standard SENT). The mixed mode stress intensity factors KI and KII for such a configuration can be calculated from the following relations:

W L/2

L

Fig. 1. (a) Problem configuration and (b) meshed solid model for finite element analysis.

3302

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

Table 1a Mode I geometry factor YI. b°

0 10 20 30 40 50 60 70 80

a/W 0.1

0.2

0.3

0.4

0.5

0.6

0.7

2.11 2.13 2.18 2.28 2.45 2.75 3.33 4.63 8.82

2.43 2.44 2.47 2.54 2.68 2.94 3.48 4.74 8.90

2.95 2.95 2.95 2.97 3.05 3.25 3.72 4.93 9.03

3.74 3.72 3.67 3.60 3.58 3.67 4.05 5.17 9.20

5.00 4.94 4.76 4.53 4.32 4.24 4.48 5.49 9.42

7.12 6.95 6.50 5.92 5.37 5.00 5.01 5.86 9.66

11.07 10.68 9.54 8.12 6.92 6.02 5.69 6.30 9.96

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 1.28 1.28 1.29 1.31 1.33 1.38 1.49 1.84

0.00 1.45 1.44 1.43 1.42 1.42 1.44 1.53 1.85

0.00 1.71 1.68 1.64 1.59 1.55 1.53 1.58 1.88

0.00 2.09 2.02 1.93 1.83 1.72 1.65 1.66 1.91

0.00 2.62 2.51 2.33 2.14 1.95 1.80 1.74 1.95

0.00 3.50 3.26 2.89 2.56 2.24 1.98 1.84 2.00

0.00 4.99 4.46 3.72 3.16 2.61 2.21 1.96 2.05

Table 1b Mode II geometry factor YII. b°

0 10 20 30 40 50 60 70 80

a/W

pffiffiffi K I ¼ Y I r a cos2 b pffiffiffi K II ¼ Y II r a cos b sin b

ð1Þ

where YI and YII are the geometry factors and must be determined for the problem configuration considered. Rearranging Eq. (1) results in:

KI Y I ¼ pffiffiffi r a cos2 b K II Y II ¼ pffiffiffi r a cos b sin b

ð2Þ

Using finite element analysis, the stress intensity factors can be determined for various values of b (varying the strength of each mode in the mixed mode I–II loading) and a/W (varying the width of the cracked plate), which then can be used to calculate the geometry factors using Eq. (2). The commercial software ANSYS is used to determine the stress intensity factors for the configuration considered. The modified mixed mode I–II SENT specimen does not exhibit any symmetry, therefore full 2D models are developed in ANSYS. The problem is idealized as 2D plane stress and the material is considered to be linear isotropic. The lower face is fixed in the y-direction and to prevent rigid body rotation, the node at x = y = 0 is fixed for all degrees of freedom. A uniform stress is applied at the top face of the specimen. 2D structural triangular elements are used pffiffiffi to mesh the solid model. Skewed mid-side node method is used to achieve the 1= r singular behavior at the crack tip using KSCON command in ANSYS. The meshed model is shown in Fig. 1b. The validation of the finite element model was performed by the Merah and Albinmousa [21] using the results of Wilson [26] for various a/W and b values. Using this validated finite element model, a parametric study is performed; while keeping r and a constant, b and a/W are varied between 0° and 80° with a 10° interval and for each value of b, a/W is varied between 0.1 and 0.7 with an increment of 0.1. The resulting values of the mode I and II geometry factors are listed in Tables 1a and 1b, respectively.

3. Proposed model To increase the usefulness of these results and to generalize, following model is proposed to evaluate the geometry factors and thus the stress intensity factors for any combination of crack inclination angle and a/W for the mixed-mode I–II single edge notched tension configuration:

3303

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

h a a idi Y i ¼ ci cos þ ai W W

ð3Þ

where i = I or II. Eq. (3) is based on curve fitting performed in Matlab environment, which also provided the following relations for the fitting parameters:

cI ¼ 1:9½cos b0:921  0:38b2:03 dI ¼

aI ¼

8:53  5:57b b2  0:82b þ 1:37 1:12 ð4Þ

b3  0:73b2 þ 0:8 0:3

cII ¼ 1:2½cos b

 0:15b

3

dII ¼ 2:85b  6:4b2 þ 5:1

aII ¼ 0:8b3  2:53b2 þ 1:66b þ 0:54 where b is in radians. A total of 23 curves were fitted in order to correlate YI and YII with a/W and b considering all values of b and a/W ratios analyzed using finite element method in the fitting process. It should be noted that Eqs. (3) and (4) for b = 0° will yield a

Table 2a Goodness of fit for Eq. (3) for mode I geometry factor YI. b°

SSE

R-square

Adjusted R-square

RMSE

0 10 20 30 40 50 60 70 80

0.0002182 0.0005558 0.0008335 0.0030020 0.0018330 0.0017600 0.0013310 0.0011790 0.0003667

1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9995 0.9997

1.0000 1.0000 1.0000 0.9998 0.9998 0.9997 0.9996 0.9992 0.9995

0.007385 0.011790 0.014430 0.027400 0.021410 0.020980 0.018240 0.017170 0.009574

Table 2b Goodness of fit for Eq. (3) for mode II geometry factor YII. b°

SSE

R-square

Adjusted R-square

RMSE

10 20 30 40 50 60 70 80

0.0004874 0.0000368 0.0002686 0.0002844 0.0002705 0.0001117 0.0000866 0.0000618

1.0000 1.0000 0.9999 0.9999 0.9998 0.9998 0.9995 0.9983

0.9999 1.0000 0.9999 0.9998 0.9997 0.9997 0.9993 0.9975

0.011040 0.003033 0.008195 0.008431 0.008223 0.005284 0.004652 0.003931

Table 3a Goodness of fit for Eq. (4) for mode I geometry factor YI.

cI dI

aI

SSE

R-square

Adjusted R-square

RMSE

0.0005646 0.0016510 0.0085580

1.0000 1.0000 0.9918

1.0000 0.9999 0.9890

0.01063 0.01817 0.03777

Table 3b Goodness of fit for Eq. (4) for mode II geometry factor YII.

cII dII

aII

SSE

R-square

Adjusted R-square

RMSE

0.0008007 0.0199700 0.0044060

0.9977 0.9990 0.9919

0.9968 0.9987 0.9859

0.01266 0.06320 0.03319

3304

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

positive value of YII instead of zero. However, this has no physical meaning since KII is zero for such a condition. The curve fitting quality is a critical issue in this regard. In order to achieve a good fitting quality, the goodness of fit terms is evaluated using four methods: the sum of squares due to error (SSE), R-square, adjusted R-square and root mean squared error (RMSE). The goodness of fit terms is listed in Tables 2a and 2b for Eq. (3) for YI and YII respectively, which are fitted with respect to different a/W ratios. Tables 3a and 3b list goodness of fit terms for the coefficients in Eq. (4) for YI and YII respectively, which are fitted with respect to different crack inclination angles. Failure to achieve good values of the goodness of fit terms will lead to an obvious error in calculating the mixed mode geometry factors YI and YII and thus stress intensity factors KI and KII.

4. Results and discussions Figs. 2 and 3 illustrate the results from the proposed model along with results from previous numerical studies. Fig. 2 presents results for the variation of YI with a/W ratio for a pure mode I crack and compares these with Freese and Baratta [15], Benthem and Koiter [26] and Fett [16] for complete range of a/W ratios. Fig. 3 presents results for the variation of YI and YII with a/W ratio for a mixed mode I–II crack and compares these with Chen and Wang [20] and Wilson [27]. As clearly observed, the proposed model compares well with previous studies especially for the values of a/W less than 0.7, which represents the range used for curve fitting. It must be noted here that the previous studies use much more involved (e.g., Wilson

70

σ 60

Fareese & Baratta [15] Benthem and Koiter [26] Fett [16] Proposed Model

w W

50 a

YI

40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a /W Fig. 2. Variation of YI for the complete range of a/W.

4 o

Y I & β = 45

3.5 o

Y II & β = 45

3

YI & YII

2.5

o

Y I & β = 22.5

o

Y II & β = 22.5

σσ Chen & Wang [20] Wilson [27] Proposed Model w W Chen & Wang [20] Wilson [27] a β Proposed Model Wilson [27] Proposed Model Wilson [27] Proposed Model

2 o

Y I & β = 22.5

1.5

o

Y I & β = 45

1 Y II & β = 45

0.5 0

o

o

Y II & β = 22.5

0.3

0.35

0.4

0.45

0.5

a /W Fig. 3. Variation of YI and YII with a/W.

0.55

0.6

3305

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

2.0 294 N

1.8 1.6

196 N

KI (MPa √m)

1.4 98 N

1.2

Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Merah & Albinmousa [21] Proposed Model

{ { {

1.0 0.8

σσ w W a β

Ref. [21] & current

σσ

φ

w W

294 N

0.6 a

196 N

Ref. [18, 28]

0.4 98 N

0.2 0.0

0

10

20

30

40

50

60

70

80

90

β (degrees for Ref. [21] & Current) φ (degrees for Ref. [18, 28]) Fig. 4a. Variation of KI with mixed mode loading.

1.0 σσσ

0.9

σ

294 N

w W

w W

0.8

φ

196 N

a β

0.7

KII (MPa √m)

a

Ref. [21] & current

0.6

98 N

Ref. [18, 28]

Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Yoneyama et al. [18] Richard & Benitz [28] Proposed Model Merah & Albinmousa [21] Proposed Model

{ { {

0.5 0.4 294 N

0.3

196 N

0.2 98 N

0.1 0.0 0

10

20

30

40

50

60

70

80

90

β (degrees for Ref. [21] & Current) φ (degrees for Ref. [18, 28]) Fig. 4b. Variation of KII with mixed mode loading.

[27] and Freese and Baratta [15] used boundary collocation method and Chen and Wang [20] used weight function method) analysis, whereas the present study uses the simple Eq. (3). Figs. 4a and 4b present results for the variation of KI and KII respectively with mixed mode loading and compare the proposed model with experimental studies in the open literature. Yoneyama et al. [18] and Richard and Benitz [28] tested standard SENT (straight crack) specimens and varied the loading angle, / to produce mixed mode I–II loading condition. To compare with these cases, the crack inclination angle is set to zero in the proposed model, i.e., b = 0° and the load is multiplied by cos / and sin / for KI and KII, respectively. In Fig. 4b, the abscissa represents the crack inclination angle for the current study and Merah and Albinmousa [21] and the loading angle / for Yoneyama et al. [18] and Richard and Benitz [28]. The present model is compared for three different values of the applied loads (i.e., 294, 196 and 98 N) as considered in the experiments by Yoneyama et al. [18] and Richard and Benitz [28]. It can be seen that the present model compares well with the experimental studies throughout the entire range of crack inclination angle (i.e. complete range of mixed mode I–II) and for various values of applied loads for both KI and KII.

3306

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

1.6 1.4

YI & YII

Hasebe & Inohara [19] Proposed Model (a /W = 0.2) Proposed Model (a /W = 0.05) Hasebe & Inohara [19] Proposed Model (a /W = 0.2) Proposed Model (a /W = 0.05)

YI

1.2

Y II

1.0

σσ

YI

0.8 a

β

0.6 0.4 Y II

0.2 0.0

0

10

20

30

40

50

60

70

80

90

β (degrees) Fig. 5. Edge-cracked semi-infinite plate: variation of YI and YII with b.

YI

2.6

Y II

2.4

YI

{ {

β β β β

1.30

o

= 10 o = 40 o = 10 o = 40

1.25

1.20

Y II

1.15

2.2 YI

YII

2.8

σσ

1.10

2.0 w W

a

1.8

β

1.05

1.00

1.6 0

0.02

0.04

0.06

0.08

0.1

a /W Fig. 6. Variation of YI and YII for micro-cracks (a/W < 0.1).

4.1. Flexibility of the proposed model In order to check the flexibility of the proposed model, the results are also compared against a case of semi-infinite plate with an inclined edge crack in uniaxial tension. The solution, using conformal mapping to find mixed mode geometry factors, for such a case is available in the open literature [19]. To investigate a suitable value of a/W, the width W is increased gradually while keeping the crack length a constant. It is found that a value for the ratio of a/W = 0.05 is suitable to simulate the case of semi-infinite plate. A comparison between the proposed model and Hasebe and Inohara [19] is given in Fig. 5 for two different values of a/W, which shows a good agreement for a/W = 0.05. This shows the flexibility of the proposed model and that with a careful manipulation of parameters, Eq. (3) could be applied to different problem configurations. Citing the unavailability of closed-form solutions for geometry factors, Cho et al. [17] used finite element method to determine SIFs for micro-cracks in MEMS applications. Although the range of a/W ratio used and the resulting values of geometry factors are not given in their paper, an analysis of values of a and W used (mentioned in the introduction above) provide a range of 0.00625–0.52 for a/W ratio. Based on this information, the proposed model was evaluated for a/W < 0.1. The results, shown in Fig. 6 for two different values of crack inclination angle, demonstrate that the proposed model is sensitive to small values of a/W ratio and therefore a candidate for micro-cracks or MEMS application.

J. Albinmousa et al. / Engineering Fracture Mechanics 78 (2011) 3300–3307

3307

5. Conclusions A novel approach to obtain closed-form solutions for geometry factors in mixed mode I–II loading for different configurations is presented. The approach is based on finite element analysis and parametric curve fitting. The proposed method is used to determine the geometry factors for a single edge-notched tension specimen modified to accommodate mixed mode I–II conditions by using an inclined crack. The generated results compare very well with available analytical, numerical and experimental studies in the open literature. In addition, the flexibility of model is emphasized by simulating an inclined edge crack in a semi-infinite plate and considering its suitability for micro-cracks/MEMS applications. References [1] Mohanty JR, Verma BB, Ray PK. Evaluation of overload-induced fatigue crack growth retardation parameters using an exponential model. Eng Fract Mech 2008;75:3941–51. [2] Mohanty JR, Verma BB, Ray PK. Prediction of fatigue life with interspersed mode-I and mixed-mode (I and II) overloads by an exponential model: extensions and improvements. Eng Fract Mech 2009;7 6:454–68. [3] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Eng 1963;85:519–27. [4] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974;10(3):305–21. [5] Kong XM, Schluter N, Dahl W. Effect of triaxial stress on mixed-mode fracture. Eng Fract Mech 1995;52(2):379–88. [6] Theocaris PS, Andrianopoulos NP. The T-criterion applied to ductile fracture. Int J Fract 1982;20:R125–30. [7] Ukadgaonker VG, Awasare PJ. A new criterion for fracture initiation. Eng Fract Mech 1995;51(2):265–74. [8] Khan SMA, Khraisheh MK. A new criterion for mixed mode fracture initiation based on the crack tip plastic core region. Int J Plast 2004;20:55–84. [9] Theocaris PS, Andrianopoulos NP. The mises elastic–plastic boundary as the core region in fracture criteria. Eng Fract Mech 1982;16(3):425–32. [10] Plank R, Kuhn G. Fatigue crack propagation under non-proportional mixed mode loading. Eng Fract Mech 1999;62:203–29. [11] Bhattachary A. Analysis of crack growth rate based on rotation at the crack tip plastic zone. Int J Fract 1997;83:159–65. [12] Weber W, Willner K, Kuhn G. Numerical analysis of the influence of crack surface roughness on the crack path. Eng Fract Mech; in press. (doi: 10.1016/ j.engfracmech.2010.03.024). [13] Joyce JA, Link RE, Roe C, Sobotka JC. Dynamic and static characterization of compact crack arrest tests of navy and nuclear steels. Eng Fract Mech 2010;77:337–47. [14] Zaikin AD. Crack stabilization in a brittle body using stiffeners. J Appl Mech Technol Phys 2006;47(6):886–91. [15] Freese CE, Baratta FI. Single edge-crack stress intensity factor solutions. Eng Fract Mech 2006;73:616–25. [16] Fett T. Stress intensity factors for edge-cracked plates under arbitrary loading. Fatigue Fract Eng Mater Struct 1998;22:301–5. [17] Cho SW, Jonnalagadda K, Chasiotis I. Mode I and mixed mode fracture of polysilicon for MEMS. Fatigue Fract Eng Mater Struct 2007;30:21–31. [18] Yoneyama S, Ogawa T, Kobayashi Y. Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods. Eng Fract Mech 2007;74:1399–412. [19] Hasebe H, Inohara S. Stress analysis of a semi-infinite plate with an oblique edge crack. Ing Arch 1980;49:51–62. [20] Chen CH, Wang CL. Stress intensity factors and T-stresses for offset double edge-cracked plates under mixed-mode loadings. Int J Fract 2008;152:149–62. [21] Merah N, Albinmousa J. Experimental and numerical determination of mixed mode extension angle. ASTM J Test Eval 2008;37(2):95–107. [22] Jogdand PV, Murthy KSRK. A finite element based interior collocation method for the computation of stress intensity factors and T-stresses. Eng Fract Mech 2010;77:1116–27. [23] Feng XQ, Shi YF, Wang XY, Li B, Yu SW, Yang Q. Dislocation-based semi-analytical method for calculating stress intensity factors of cracks: Twodimensional cases. Eng Fract Mech 2010;77:3521–31. [24] Bowie OL, Neal DM. A modified mapping-collocation technique for accurate calculation of stress intensity factors. Int J Fract Mech 1970;6(2):199–206. [25] Bowie OL. Solutions of plane crack problems by mapping techniques. In: Sih GC, editor. Mech Fract. Leyden: Noordhoff International Publishing; 1973. [26] Benthem JP, Koiter WT. Asymptotic approximations to crack problems. In: Sih GC, editor. Methods of analysis and solutions of crack problems. Noordhoff Internal Publishing; 1973. p. 131–78. [27] Wilson WK. Research Report 69-IE7-FMECH-RI.1969; Westinghouse Research Laboratory. Pittsburg. [28] Richard HA, Benitz K. A loading device for the creation of mixed mode in fracture mechanics. Int J Fract 1983;22(2):R55–8.