Sae Paper Number 2001014072

2001- 01- 4072

A Review on Crack Closure Models Luiz Carlos H. Ricardo EPUSP, University of Sao Paulo

* Paulo de Mattos Pimenta ** Dirceu Spinelli * EPUSP, University of Sao Paulo, ** EESC, University of Sao Paulo Copyright © 2001 Society of Automotive Engineers, Inc.

ABSTRACT The proposal of this paper is to make a review of analytical crack closure models. Christensen discovered the crack closure in 1963 and later defined by Elber in 1968 in his PhD work. This subject is a topic related to short cracks. The first analytical crack closure model was developed by Newman in 1974 based on the Dugdale model. Since Newman, finite element and difference analysis have been conducted to obtain a basic understanding of crack growth and crack closure processes. Simple and complex models were developed based on the plasticity induced crack closure behavior. Since 1970 until today the most finite elements analysis were conducted using two dimensional under both, plane stress and plane strain conditions. In the literature few works covering three-dimensional models can be found. Chermahini did the first work discussing it, in 1986. This paper also discusses the yielding zone, empirical crack closure model, modified Dugdale and design concepts using short crack theory.

method to study the elastic-plastic stress field around a crack. The application of linear elastic fracture mechanics, i.e. the stress intensity factor range, ∆K, to the “small or short” crack growth have been studied for long time to explain the effects of nonlinear crack tip parameters. The key for these nonlinear crack tip parameters is crack closure. Analytical models were developed to predict crack growth and crack closure processes like Dugdale [ 4 ], or strip yield use the plasticity induced approach in the models considering normally plane stress or strain effects. In this paper will shown a review of some cracks closure models.

THE IMPORTANCE OF CRACK CLOSURE Elber measured the plastic deformation in the wake of a growing fatigue crack measuring nonlinear crack opening behavior. The concept of crack closure under nominal tensile stress cycles was not recognized earlier; but today the fatigue crack problem can be explain without considering crack closure and others mechanisms for crack closure have been proposed.

INTRODUCTION The discovery of crack closure mechanisms, such plasticity, roughness, oxide, corrosion, and fretting product debris, and the use of the effective stress intensity factor range, has provided and engineering tool to predict small and large crack growth rate behavior under service loading conditions. These mechanisms have also provided a rationale for developing new, damage tolerant materials. The major links between fatigue and fracture mechanics were done by Christensen [ 1 ] and Elber [ 2 ]. The crack closure concept put crack propagation theories on a firm foundation and allowed the development of practical life prediction for variable and constant amplitude loading, by such as experienced by modern day commercial aircraft. Numerical analysis using finite elements has played a major role in the stress analysis crack problems. Swedlow [ 3 ] was one of the first to use finite element

The technical significance of crack closure is related to the growth of fatigue cracks under services load histories. The ultimate goal of prediction models is to arrive at quantitative results on fatigue crack growth in terms of millimeters per year or some other service period. Such predictions are required for safety and economy reasons, for example, for aircraft and automotive parts. Sometimes the service load time history ( variable amplitude loading ) is much similar to constant amplitude loading, including mean load effects. In both cases quantitative knowledge of crack opening stress level S op is essential for crack growth predictions, because: •

S op is required to define ∆K eff ( Kmax – K min )

•

∆Keff is supposed to be the appropriate field parameter for correlating crack growth rates under different cyclic loading conditions

The correlation of crack growth data starts from the similitude approach, based on the ∆K eff, which predicts that equal ∆K eff cycles will produce the same crack growth increments. The application of ∆K eff to variable amplitude loading require prediction of the variation of Sop, during variable amplitude load history, which for the more advanced prediction models implies a cycle by cycle prediction. The figure 1 shows the definitions of K values.

Since the early 1970s, numerous finite element and finite difference analysis have been conducted to simulate fatigue crack growth and closure. These analyses were conducted to obtain a basic understanding of the crack growth and closure processes. Parallel to these numerical analyses, simple and complexes models of the fatigue crack growth process were developed. Although the vast majority of these analyses and models were based on the plasticity induced crack closure phenomenon. Will be discussing here some of these models covering plasticity induced crack closure Newman [ 6 ]. Finite Element and Difference Analysis

Figure 1 Definitions of K Values , Schijve [ 5 ] The application of ∆K eff is considerably complicated by two problems ( 1 ) small cracks and ( 2 ) threshold ∆K values. Small cracks can be significant because in many cases a relatively large part of the fatigue life is spent in the small crack length regime. The threshold problem is particularly relevant for fatigue under variable amplitude spectrum, if the spectrum includes many “ small “ cycles. It then is important to know whether the small cycles do exceed a threshold ∆K value, and to which extent it will occur. The application of similitude concept in structures can help so much, but the correlation to satisfy the results cannot happen and the arguments normally are: •

The similarity can be violated because the crack growth mechanism are no longer similar

•

The crack can be too small for adopting K as a unique field parameter

•

∆Keff and others conditions being nominally similar, it is possible that other crack tip aspects also affect crack growth, such as crack tip blunting and strain hardening, Schijve [ 5 ].

The most analyses in finite element and difference analysis were conducted using two-dimensional analysis under plane stress and plane strain. Since 1980 few works were done covering three-dimensional models. The first model covering was done by Chermahini [ 7 ] . Newman and Armen [ 8 –10 ] and Ohji et al. [ 11 ] were the first to conduct the two dimensional, analysis of the crack growth process. Their results under plane stress conditions were in quantitative agreement with experimental results of Elber [ 2 ], and showed that crack opening stresses were a function of R ratio ( Smin / Smax ) and the stress level ( S max / σ0 ). Blom and Holm [ 12 ] and Fleck and Newman [ 13,14 ] studied crack growth and closure under plane-strain conditions and found that cracks did close but the cracks opening levels were much lower than those under plane stress conditions. Sehitoglu et al. [ 15,16 ] found later the residual plastic deformations that cause closure came from flanks of the crack. McClung [17-19 ] performed extensive finite element crack closure calculations on small cracks at holes, and various fatigue crack growth specimens. Newman [ 20 ] found Smax / σ0 could correlate the crack opening stresses for different flow stresses ( σ0 ) for the middle crack tension specimen, McClung found that K analogy, using Kmax / K0 could correlate the crack opening stresses for different crack configurations for small scale yielding conditions. Very little research on three dimensional finite element analyses of crack closure has been conducted as mentioned before. Chermahini [ 7 ] was the first to investigate the three dimensional nature of crack growth and closure. He found that the crack opening stresses were higher near the obtained experimental crack opening stresses, similar to Chemahini’s calculations, along the crack front using Sunder’s striation method [ 21 ], with backface strain-gages and finite element method.

Empirical Crack Closure Models

NUMERICAL ANALYSIS OF CRACK CLOSURE

The Wheeler [ 22 ] and Willenborg et al. [ 23 ] were the first models proposed to explain crack growth retardation after overloads. These models assume that retardation

exists as long as the current crack-tip plastic zone is enclosed within the overload plastic zone. The physical basis for these models, however, is weak because they do not account for crack growth acceleration due under loads or immediately following an overload. Chang [ 24 ] and Hudson [ 25 ] clearly demonstrated that retardation and acceleration are both necessary to have a reliable model. Later models by Gallagher confirmed it [ 26 ].

the effective stress-intensity calculated by :

Chang [ 24 ] and Johnson [ 27 ] included functions to account for the retardation and acceleration. A new generation of models was introduced by Bell and Wolfman [ 28 ], Schijve [ 29 ], de Koning [ 30 ] and Baudin & Robert [ 31 ] were based on the crack closure concept. The simplest model is the one proposed by Schijve, who assumed that the crack opening stress remains constant during each flight in a flight -by-flight sequence. The other models developed empirical equations to account for retardation and acceleration, similar to the yield zone models.

a - half length of the crack,

factor

range

can

∆ K eff = ∆ Seff π a F

be

(2

) Where:

F – boundary correction factor ∆S eff – effective stress range The figure 2 shows the center crack panel that will be used to evaluate the crack propagation.

Modified Dugdale Model There are many modified Dugdale models for example [ 32 – 35], After Elber [ 2 ] defined the crack closure, the research community began to develop analytical or numerical models to simulate fatigue crack growth and closure. These models were designed to calculate the growth and closure behavior instead of assuming such behavior as in the empirical models. Seeger [ 32 ] and Newman [ 8 ] were the first to develop two type of models. Seeger modified the Dugdale model and Newman developed a ligament or strip yield model. Later a large group of similar models were also developed using the Dugdale framework.

Figure 2 Center Crack Panel, Newman [ 20 ] The figure 3 shows the panel idealized to finite element method.

Budiansky & Hutchinson [ 34 ] studied closure using an analytical model, while Dill & Saff [ 33 ], Fuhring & Seeger [ 36 ], and Newman [ 37 ] modified the Dugdale model. Some models used the analytical functions to model the plastic zone, while others divided the plastic zone into a number of elements. The model by Wang & Blom [ 38 ] is a modification of Newman’s model [ 37 ] but their model was the first to include weight functions to analyses other crack configuration.

Crack Propagation by Finite Element Method The experiments of crack closure from Elber [ 2 ] with constant amplitude loading that was proposed the following equation for fatigue crack propagation rates:

∆a = C (∆K eff ) n ∆N

(1

) Where C and n are constants of the material and ∆Keff is the effective stress intensity factor range.He proposed that

Figure 3 Finite Element Model of Center Crack Panel Newman [ 20 ]

The panel material was assumed to be elastic perfect plastic with a tensile ( and compressive ) yield stress, σ0, of 350.0 MN / m2 and a modulus of elasticity of 70000 MN / m 2 these properties are of aluminum alloy. The released nodes will be done from node A to node F. Of course, the accuracy of the calculated crack opening stresses would be affected by the mesh size chosen to model the crack tip region. A finer element mesh size would give more accurate results. Newman [ 20 ] evaluated three kind of mesh as shows in table 1 Table 1 Meshes at Crack Tip mesh

KT

∆ a ( mm )

elements

Nodes

I

7.2

0.64

398

226

II

14.4

0.16

533

300

III

20.9

0.08

639

358

Figure 5 Crack Surface Displacements and Stress Along Crack Line, Newman [ 37 ] The crack surface displacements, which are used to calculate contact (or closure) stresses during unloading, are influenced by plastic yielding at the crack tip and residual deformations left in the wake of the advancing crack. Upon reloading, the applied stress level at which the crack surfaces become fully open is directly related to contact stresses. This stress is called the “crack opening stress”. Because they are no closed form solutions for elastic plastic cracked bodies, simple approximations must be used. In next will showed the equations that governing the stress and deformations of the analytical crack closure model. Because of symmetry, only one quarter of the plate will be analyzed as showed in the figure 6.

W = 460.0 mm and a ≅ 28.0 mm

Figure 4 Constant Amplitude Crack Extension R = 0, Newman [ 20 ]

with

The figure 4 shows how was stabilized during crack propagation. The mesh that shows the best agreement with experimental results was the mesh II, but the mesh III provides good results too. With the facilities in terms of computer today, normally the time to evaluate this mesh is almost nothing, being size of element and the increment most used today to evaluate a crack closure analysis or propagation. Newman [ 37 ] introduced a model that is possible evaluate crack closure and crack propagation analysis until failure the model received the name FASTRAN. The formulation of FASTRAN is shows in next.

Figure 6 System Used in the Analytical Closure Model Newman [ 37 ] The plate had a fictitious crack of half-length d and was subjected to a uniform stress S. The rigid-plastic bar element connected to point J was subjected to a compressive stress σJ. this element is in contact when the length of the element ( Lj ) is make V j = L j.

 πd   W 

Where: ( K )s = Smax πd sec

The equations that govern the response of the complete system were obtained by requiring that compatibility be met between the elastic plate and all of the bar elements along the crack surface and plastic zone boundary. The displacement at point i is :

and

( K )σ 0 n

Vi = Sf ( x i ) − ∑ σ j g ( xi , x j )

(3)

j =1

g( xi , x j ) = G( xi , x j ) + G( − xi , x j )

(4 )

(5)

   2  −1 d − b2 x i − (b 2 − x i ) cosh       d b2 − x i    2 2   2(1 − η )   −1 d − b1 x i  G( x i , x j ) = +  − (b1 − xi )cosh    πE  d b − x 1 i       b b       + d 2 − xi2 sen −1  2  − sen −1  1      d   d    

   sen −1 B − sen − 1 B  2 1   sec  πd   − 1 b 2  − 1  b1   W   sen  d  − sen  d    

(πbK ) W for K = 1 or 2. Where B = K sen(πd ) W

(6)

(7)

b1 = x j − w j ; b2 = x j + w j . The compatibility equation V j = L j. Is expressed as subject to various constraints: n

ij

g ( x i , x j ) = Sf ( x i ) − Li for i = 1 to n

(8)

j =1

One type of constraint is caused by tensile or compressive yielding of the bar elements and the other is caused separation ( Vj ≥ Lj ) along the crack surface. The plastic zone size ( ρ ) for a crack in a finite width specimen was determined by requiring that the finiteness condition of Dugdale be satisfied. This condition states that the stress intensity factor at the tip of the plastic zone is zero and is given by:

(K ) s + ( K )σ 0 = 0

W   πc   πS   ρ = c sen−1 sen  sec max  − 1 πc   W   2ασ0  

( 12 )

In the plastic zone was arbitrarily divided into ten graduated bar elements. The aspect ratios ( 2w i / ρ ) : 0.01; 0.02; 0.04; 0.06; 0.09; 0.12; 0.5; 0.2 e 0.3. The smallest elements were located near the crack tip ( x = c ). Doubling the number of elements in the plastic zone has less than a 1 percent effect on calculated crack opening stresses. At the maximum applied stress, the plastic zone size was calculated from equation ( 12 ). The length ( Li ) of the bar elements in the plastic zone was calculated from equation ( 3 ) as: 10

Li = Vi = S max f ( xi ) − ∑ ασ 0 g ( xi , x j )

( 13 )

j =1

sen

∑σ

 πc     2  sen W    πd  −1 = −ασ 0 1 − sen    πd sec  ( 11 ) π d W   π  sen     W  

solving the ( 21 ) for d and nothing that ρ = d – c gives :

Where f( xi ) e g( xi , xj ) are influence functions given by :

2(1 − η 2 ) πd f ( xi ) = ( d 2 − x i2 ) sec E W

( 10 )

(9)

f ( xi )

Where

and

g ( xi , x j ) are given by

equations ( 4 ) and ( 5 ), respectively. The bar elements act as rigid wedges. The plastic deformation ( Li ) changes only when an element yields in tension ( σj ≥ ασ0 ) or compression (σj ≤ -σ0 ). The division of the plastic zone into a number of finite elements would allow for the eventual use of a nonlinear stress-strain curve with kinematic hardening instead of the rigid perfectly plastic assumptions used here. Crack Opening Stresses The applied stress level at which the crack surfaces are fully open ( no surfaces contact), denoted as S0, was calculated from the contact stresses at Smin. To have no surface contact, the stress-intensity factor due to applied stress increment ( S0 - S min ) is set equal to the stress intensity factor due to the contact stresses. Solving for S0 gives : n −1

2σ j

j =11

π

S0 = S min − ∑

[sen

−1

B2 − sen −1 B1

]

( 14 )

 πb  sen K  W  Where BK = for K = 1 or 2  π c0  sen   W 

( 15 )

and c 0 is the current crack length minus ∆c*. The increment the width of element n, and its significance is discussed in the next section. If σj = 0 for j = 11 to n – 1 at the minimum applied stress, then the crack is already open, and S0 cannot be determined from equation ( 14 ). The stress σj at the crack tip changes from compression to tension when the applied stress level reaches S0. Crack Extension and Approximations The crack growth equation proposed by Elber [ 2 ] states that the crack growth rate is a power function of the effective stress intensity factor range only. Later , Hardraht et al. [ 9 ] showed that the power law was inadequate at high growth rates approaching fracture. The results presented herein show that is also inadequate at low growth rates approaching threshold. To account for these effects, the power law was modified to :

dc = C1∆K effC2 dN

 ∆K 0 1−   ∆K eff 

   

2

K max = S max πc F

( 16 )

2

K  1 −  max   C5   S0 where : ∆ K 0 = C 3 1 − C 4 S max 

   

element at the crack tip. The length while the crack was grown under cyclic loading ( cycle-by-cycle ) over the length ∆c*. The number of load (∆N ) required to grow the crack an increment ∆c* was calculated from equation ( 16 ) and the cyclic load history. When the sum of the crack growth increments (∆c ) equaled or exceeded ∆c*, the analytical closure model was exercised. If ∆N reached 300 cycles, the model was exercised whether or not ∆c* was reached. This limits the number of cycles that can be applied before the model is exercised. The increment ∆c* was set equal to summation of ∆c’s. Thus, ∆c* was less than or equal to that computed from equation ( 19 ), and the number of cycles ranged from 1 to 300. During the cyclic growth computations, the cyclic stress history was monitored to find the lowest applied stress before ( S minb ) and after ( Smina ) the higest applied stress level ( S maxh ). The application of the analytical closure model consisted of : • • • • • • •

Applying minimum stress Sminb at crack length c Applying maximum stress Sminh at crack length c Extending crack and increment ∆c* Applying minimum stress Smina at crack length c + ∆c* Calculating cyclic load history Calculating new ∆c* from equation 19 Repeating process when crack extension reaches new ∆c* or reaches 300 cycles.

( 17 )

( 18 )

and

∆ K eff = (S max − S 0 ) π c F The constants C1 to C5 are determined by experimental test under constant amplitude loading. The factor F is the boundary correction factor on stress intensity. The analytical closure model provides extending the crack an incremental value at he moment of maximum applied stress. The amount of crack extension (∆c* ) was arbitrarily defined ∆c* = 0.05ρ max

( 19 )

Where ρ max is the plastic zone caused by the maximum applied stress occurring during the ∆c* was calculated from equation ( 16 ) and the cyclic load history. Typical values of ∆c* ranged between 0.004 and 1.0 mm, depending upon the applied stress level and crack length. The simulated crack extension (∆c* ) creates a new bar

Figure 7 Crack Surface Displace under Constant Amplitude Loading, Newman [ 37 ]

5.

Schijve, J.” Fatigue Crack Growth Closure: Observations and Technical Significance”, ASTM STP 982, PP. 5-34, USA, 1988

6.

Fracture Mechanics Concepts a Historical Perspective”, Progress in Aerospace Sciences, nº 34, pp. 347-390, USA, 1998

7.

Chermahini, R.G. “ Three Dimensional Elastic – Plastic Finite Element Analysis of Fatigue Crack Growth and Closure”, PhD Thesis, Old Dominion University, Norfolk, USA, 1986

8.

Newman, J.C. Jr., “ Finite Element Analysis of Fatigue Crack Propagation Including The Effects of Crack Closure”, PhD Thesis, VPI & SU, Blacksburg, USA, 1974

9.

Newman, J.C. Jr.; Armen, H. Jr. “ Elastic-Plastic Analysis of Fatigue Crack Under Cyclic Loading”, AIAA Journal, nº 13, pp. 1017-1023, USA, 1975

Figure 8 Calculated Crack Opening Stresses as Function of Crack Length under Constant Amplitude Loading R = 0, Newman [ 37 ] The figures 7 and 8 show an example of application of the use of FASTRAN in a 2219-T851 aluminum alloy.

CONCLUSION

10. Newman, J.C. Jr., “ A Finite Element Analysis of Fatigue Crack Closure ”, ASTM 490, pp. 281-301, USA, 1976

The paper did a review of some analytical crack closure models, giving a special attention for Newman’s models that are the most used in the aerospace industry. The paper shows also that the methodology can be used at the autoparts and carmakers in short time, as one of the criterion to design the structures and systems.

11. Ohji K, Ogura K, Ohkubo Y. “ Cyclic Analysis of a Propagating Crack and Its Correlation with Fatigue Crack Growth” Eng. Fracture Mechanics, nº 7, pp.457-464, England, 1975

ACKNOWLEDGMENTS Dr. Wolf Elber and Dr. Jim C. Newman Jr. from NASA ( National Aeronautics and Space Administration ) at Langley Research Center, Virginia, 23665, USA

REFERENCES 1.

Christensen, R. H. “ Fatigue Crack Growth Affected by Metal Fragments Wedged Between Opening Closing Crack Surface”, Appl. Mater. Res., nº 2,Ocotber, pp. 207-210, USA, 1963

12. Blom, A.F. & Holm, D.K. “ An Experimental and Numerical study of Crack Closure”, Eng. Fracture Mechanics, nº 22, pp. 997-1011, England, 1985 13. Fleck, N.A., “ Finite Element Analysis of Plasticity Induced Crack Closure Under Plane Strain Conditions” , Eng. Fracture Mechanics, nº 25, pp. 441-449, England, 1986 14. Fleck, N.A. & Newman, J.C. Jr, “ Analysis of Crack Closure Under Plane Strain Conditions ”, ASTM STP 982, pp. 319-341, USA, 1988

2.

Elber, W. “ Fatigue Crack Propagation”, PhD Thesis, University of New South Wales, Australia, 1968

15. Lalor, P., Sehitoglu H. & McGlung, R.C, “ Mech. Aspects of Small Crack Growth from Notches – The Role of Crack Closure. The Behavior of Shorts Fatigue Cracks, EFG1”. London: Mechanical Engineering Publications, pp. 369-386, England, 1985

3.

Swedlow, J. L. “ The Effect and Plastic Flow in Cracked Plates “, PhD Thesis, California Institute of Technology, Pasadena, USA, 1965

16. Sehitoglu, H.;Gall, K. & Garcia, A.M. “ Recent Advances in Fatigue Crcak Growth Modeling”, Int. Journal Fracture, nº 80,2,pp. 165-192, USA, 1996

4.

Dugdale, D.S. “ Yielding of Steel Sheets Containing Slits ”, J. Mech. Phys. Solids, pp. 100-104, n.0 8, USA, 1960

17. McGlung, R.C., & Sehitoglu, H. “ Finite Element Analysis of Fatigue Crack Closure – Numerical Results”, Eng. Fracture Mechanics, nº 33, pp. 253 – 272, England, 1989

18. McGlung, R.C “ Finite Element Modeling of Fatigue Crack Growth” – Theoretical Concepts and Numerical Analysis of Fatigue ” In : Blom A, Beevers, C. Editors, West Midlands, EMAS, Ltda, pp. 153-172, England, 1992 19. McGlung, R.C. “ Finite Element Analysis of Specimen Geometry Effects on Fatigue Crack Closure”, Fatigue Fract. Eng. Mater. Structures, nº 17 , pp. 861-872, USA, 1994

31. Baudin G. & Robert, M. “ Crack Growth Life Time Prediction Under Aeronautic Type Loading “ , ECF5, Lisbon, Portugal, pp. 779-792, 1984 32. Seeger, T.” Ein Breitag zur Berechnung von Statisch und Zyklisch Belasteten Reisscheiben nach dem Dugdale-Barenblatt Model” , Institut fur Statik und Stahlbau, Report nº 21, Darmstadt, Germany , 1973

20. Newman, J.C. Jr, “ A Finite Element Analysis of Fatigue Crack Closure”, ASTM STP 590, pp. 281301, USA, 1976

33. Dill, H.D & Staff, C. R. “ Spectrum Crack Growth Prediction Method Based on Crack Surface Displacement and Contact Analyses”, ASTM, STP 595, pp. 306-319, USA, 1976

21. Sunder, R. & Dash, P.K. “ Measurement of Fatigue Crack Closure Through Electron Microscopy”, Int. Journal of Fatigue, nº 4, pp. 97-105, USA, 1982

34. Budianski, B. & Hutchinson, J.W “ Analysis of Closure in Fatigue Crack Growth ”, J. Appl. Mech., nº45, pp. 267-276, USA, 1978

22. Wheeler, O.E. “ Spectrum Loading and Crack Growth”, Journal of Basic Engineering, nº 94, 1, pp. 181-186, USA, 1972

35. Hardrath H.F, Newman, J.C. Jr, Elber, W., Poe, C.C. Jr.; “ Recent Developments in analysis of Crack Propagation and Fracture of Practical Materials”, In: Perrone, N.; Liebowitz, H. Mulville, D., Pilkey W., editors, Fracture Mechanics, University of Virginia Press, pp. 347-364, 1978

23. Willenborg, J.D., Engle, R.M. & Wood, H.A. “ A Crack Growth Retardation Model Using an Effective Stress Concept”, AFFDL-TM71-1FBR, Dayton, USA, 1971 24. Chang, J.B. ; Engle, R.M & Stolpestad, J. “ Fatigue Crack Growth Behavior and Life Predictions for 2219T851 Aluminum Subjected to Variable Amplitude Loadings” , ASTM STP 743, pp. 3-27, USA, 1981 25. Chang, J.B. & Hudson, C.M. “ Methods and Models for Predicting Fatigue Crack Growth Under Random Loading”, ASTM STP 748, 1981 26. Gallager, J.P. “ A generalized devolpment of Yield Zone Models” , AFFDL-TM-74-28-FBR, USA, 1974 27. Johnson, W.S. “ Multi-Parameter Yield Zone Model for Predicting Spectrum Crack Growth”, ASTM STP 748, pp. 85-102, USA, 1981 28. Bell, P.D & Wolfman A. “ Mathematical Modeling of Crack Growth Interaction Effects”, ASTM STP 595, pp. 157-171, USA, 1981 29. Schijve, J. “ Prediction Methods for Fatigue Crack Growth in Aircraft Material”, ASTM STP 700, pp. 3 -34, USA, 1980

36. Fuhring, H. & Seeger, T. “ Dugdale Crack Closure Analysis of Fatigue Cracks under Constant amplitude Loading”, Eng. Fracture Mechanics, nº 11, pp. 99-122, England, 1979 37. Newman, J.C. Jr. “ A Crack Closure Model for Predicting Fatigue Crack Growth Under Aircraft Spectrum Loading”, ASTM STP 748, pp. 53-84, USA, 1981 38. Wang, G.S. & Blom, A.F. “ A Modified DugdaleBarenblatt Model for Fatigue Crack Growth Predictions Under General Load Predictions”, The Aeronautic Research Institute of Sweden, Report FFA TN 1987-79, 1987 39. Newman, J.C. Jr. “ FASTRAN II – A Fatigue Crack Growth Structural analysis Program”, NASA TM-10459, 1992

CONTACT Luiz Carlos H. Ricardo, EPUSP, University of Sao Paulo email: [email protected] Paulo de Mattos Pimenta, EPUSP, University of Sao Paulo email : [email protected]

30. de Koning, A.U., “ A simple Crack Closure Model for Predictions of Fatigue Crack Growth Rates Under Variable Amplitude Loading”, ASTM STP 743, pp. 63-85 , USA, 1981

Dirceu Spinelli , EESC, University of Sao Paulo email : [email protected]