effect of crack front curvature on cmod compliance and crack length evaluation for single-edge bend...

7/13/2019 Effect of Crack Front Curvature on CMOD Compliance and Crack Length Evaluation for Single-edge Bend Specimens

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1 Copyright 2014 by CSME

Proceedings of The Canadian Society for Mechanical Engineering International Congress 2014

CSME International Congress 2014

June 1-4, 2014, Toronto, Ontario, Canada

Effect of Crack Front Curvature on CMOD Compliance and Crack LengthEvaluation for Single-edge Bend Specimens

Zijian Yan

Department of Civil and Environmental Engineering

The University of Western Ontario

London, Ontario, Canada

Wenxing Zhou

Department of Civil and Environmental Engineering

The University of Western Ontario

London, Ontario, Canada

e-mail: [email protected]

Abstract This paper presents three-dimensional (3D) finiteelement analyses of single-edge bend (SE(B)) specimens to

investigate the impact of the crack front curvature on the crack

mouth opening displacement (CMOD) compliance and the

crack length predicted from the CMODcompliance. Specimens

with the average crack length aave/W of 0.3, 0.5 and 0.7 and

thickness-to-width ratio B/Wof 1, 0.5 and 0.25 are analyzed.

The curved crack front is assumed to be bowed symmetric and

characterized by a power-law expression. The impact of the

elastic modulus used in the equation to predict the crack length

from the CMOD compliance is also investigated. The results

indicate that the crack front curvature has a negligible impact

on the CMODcompliance and the accuracy of the crack length

evaluated from the CMODcompliance.

Keywords- curved crack front; f racture toughness test; SE(B)specimen; CMOD compliance; ASTM E1820-11

I. INTRODUCTIONFracture toughness resistant curve of ductile material, such

as the J-integral or crack tip opening displacement (CTOD)resistance curve, is an important input of the structural integrityanalysis and usually measured on small-scale specimens, e.g.three-point single edge bend (SE(B)) specimens. The unloadingcompliance method proposed by Clarke et al. [1] is widely usedin fracture toughness test standards, e.g. ASTM E1820-11 [2], todevelop the toughness resistance curve from one singlespecimen. The crack length in the toughness resistance curve istypically predicted from the measured crack mouth openingdisplacement (CMOD) compliance of the specimen.

As specified in ASTM E1820-11 [2], all machine notchedspecimens need to be fatigue pre-cracked to simulate naturalcracks before the resistance curve testing. The fatigue pre-cracking often introduces curved as opposed to straight crackfronts, as illustrated in Fig. 1. The shape of the curved initialcrack front is largely affected by specimen dimensions, notchmachining conditions, fatigue pre-cracking conditions andresidual stress distributions [3]. Furthermore, the crack growthduring the test is in general non-uniform across the crack front.The crack generally grows faster at the mid-plane as a result of

the high local stress triaxiality, and grows slower near the freesurfaces due to the near plane stress conditions [3].

Steenkamp [4] investigated the influence of crack front

curvature on the specimen compliance using two-dimensional(2D) plane strain finite element analyses for SE(B) specimenswith the same average crack length but different degree of crackfront curvature. He concluded that for the same average cracklength by increasing the crack front curvature, the specimencompliance would decrease and for the same degrees of crackfront curvature, the effect of curvature on compliance becamemore pronounced with increasing crack length. However, theactual state of stress in the remaining ligament of a three-dimensional (3D) specimen is not plane strain [5], and how thecrack front curvature impacts the evaluated crack length fromCMOD compliance has not been investigated in previousstudies.

ASTM E1820-11 specifies the allowable deviation of acurved crack front from a straight front based on the so-callednine-point measurement method. It requires that none of thenine physical measurements of the initial (final) crack size differby more than 0.05Bfrom the average initial (final) crack lengthaave obtained from the nine measurements, where B is thethickness of the specimen. Test specimens that do not meet thiscriterion are deemed unacceptable and therefore rejected. In thisregard, the other objective of the present study was to examinethe necessity of this crack front straightness criterion in ASTME1820-11 when evaluating the crack length.

In this study, a systematic 3D finite element analyses ofplane-sided SE(B) specimens with a wide range of thickness-to-width ratios, average crack lengths and crack front curvatures

was carried out. The CMODcompliance value for the specimenwith a straight crack front was compared with the value obtainedfrom a specimen with a curved crack front but having the sameaverage crack length. For a given specimen with either a straightor curved crack front, the crack length predicted from the CMODcompliance was compared with its actual average crack length.The impact of the elastic modulus used in the equation relatingCMODcompliance to the crack length was also investigated. Itis observed that the crack front curvature has a negligible impacton the evaluated crack length and the errors in crack length


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evaluation can be reduced using the effective modulus proposedby Wang et al. [6].

The rest of this paper is organized as follows. Section 2describes the characteristics of the curved crack front as reportedin the literature as well as reflected in the actual crack front datafrom our previous experimental study; the 3D FEA models andanalysis procedures are described in Section 3; Section 4 brieflydescribes the equation used to calculate the crack length from

CMOD compliance and presents the analysis results anddiscussions; the summary and concluding remarks are includedin Section 5.

II. CHARACTERISTICS OF CURVED CRACK FRONTPrevious experimental studies [3, 7-9] showed that curved

crack fronts are typically symmetric about the mid-plane.Therefore, only symmetric crack fronts were considered in thisstudy. The following power-law expression was proposed byNikishkov et al. [8] to characterize a typical symmetric bowedcrack front (see Fig. 1):

0

where x is the coordinate in the specimen thickness directionvarying fromB/2 toB/2; W is the specimen width; a(x) is thecrack length as a function ofx; a(0) and a(B/2) denote the cracklengths at the mid-plane and free surfaces of the specimen,respectively; = a(0)/W - a(B/2)/W; and p (p> 1) is a shapeparameter. For a symmetric bowed crack front such as shown inFig. 1, is equal to amax/W- amin/W.

By examining the fatigue pre-crack fronts of a total of 110CT test specimens with different specimen thicknesses but thesame thickness-to-width ratio (B/W= 0.5), Nikishkov et al. [8]pointed out that the shape parameterpin Eq. (1) is insensitive tothe specimen thickness and can be assigned a fixed value of 3.0.

Equation (1) was adopted to characterize the curved crack frontin the present study. A wide range of values ofwere assumed,whereaspwas set to equal 3.0 for the majority of the analysiscases. Sensitivity analyses were carried out for several caseswithp= 2.5.

Figure 1. Schematic illustration of symmetric bowed crack fronts

Because we investigated the impact of the crack frontcurvature on CMODcompliance and the evaluated crack lengthbased on the same average crack length but different crack frontcurvatures, Eq. (1) was recast in terms of the average cracklength, aave, instead of the crack length at the mid-plane, a(0).The value of aavewas calculated in accordance with the nine-point measurement method specified in ASTM E1820-11. Themeasurements should be made at nine equally spaced pointscentered about the mid-plane of the specimen. The two pointsfarthest from the mid-plane are located at 0.005Wfrom the freesurfaces. The value of aaveis then obtained as follows:

+ == where ai (i= 1, 2, , 9) denote the crack lengths at the ninemeasurement points, with a1 and a9being the measured cracklengths at the two points farthest from the mid-plane (see Fig. 1).

If the crack front is characterized by Eq. (1), aiis then givenby

0 [ 5 (0.25 2)]

1,2 9 where = 0.005W.

Substituting Eq. (3) into Eq. (2) and Eq. (1), and thenconsidering a1= a9due to symmetry, we then recast Eq. (1) intothe following format:

{(0.25 2) 18 5

==

2

}

Equation (4) completely defines a curved crack front given theaverage crack length aave obtained from the nine-pointmeasurement method, and the two parameters andp.

To put Eq. (4) in the context of the crack front straightnesscriterion specified in ASTM E1820-11, a parameter , =max(amax9- aave, aave - amin9)/B, was introduced, where amax9andamin9are the maximum and minimum values of the nine physicalmeasurements, respectively. Note that a straight crack frontcorresponds to = 0; the crack front curvature increases with ,and = 0.05 corresponds to the maximum allowable crack front

curvature as specified in ASTM E1820-11. For specimens withsymmetric bowed crack fronts, the values of andare uniquelyrelated as follows:

0.25 2 418 5==


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Figure 2. Geometric and mesh configuration of the finite element model

Then for given aave/W, B/W and p (p > 1), crack fronts withdifferent levels of curvature can be generated from Eq. (4) andEq. (5) by varying .

III. FINITE ELEMENT ANALYSESThe commercial software package ADINA 8.7.4 [10] was

used to carry out 3D linear elastic finite element analyses (FEA)to evaluate the CMODcompliance of SE(B) specimens. All theSE(B) specimens considered in this study are plane-sided, andhave the same width and the standard span-to-width ratio (S/W= 4), but three different relative average crack lengths aave/W(i.e.aave/W = 0.3, 0.5 and 0.7) and three different specimenthicknesses (i.e.B/W = 1, 0.5 and 0.25) that are consistent withthe range of B/W ratios suggested in ASTM E1820-11 [2].Specimens with straight and curved crack fronts wereconsidered. For specimens with curved crack fronts, the crackfront is characterized by Eq. (4) and Eq. (5) with p= 3 and =

0.01 to 0.10 with an increment of 0.01. In addition, specimenswith curved crack fronts characterized by p= 2.5 and selectedgeometric configurations (i.e. B/W = 0.5, aave/W = 0.3, 0.5 and0.7) were also considered to investigate the impact of the shapeparameterpon this study.

Because of symmetry, only a quarter of the specimen wasmodeled. The geometric and mesh configurations for a typicalspecimen are shown in Fig. 2 together with the fixation andloading conditions. The model was divided into ten layers in thethickness direction with the mesh density increasing from themid-plane to the free surface to capture the high stress gradientsnear the free surface. In the vicinity of the crack tip, the smallestelement has dimensions of about 1/3000W and 1/75B in the

width and thickness directions, respectively. There are about11,000 20-node 3D isoparametric brick elements with fullintegration (3 3 3) [10] included in a typical model. Young'smodulus and Poissons ratio were assumed to be 200 GPa and0.3, respectively. The load was applied based on adisplacement-controlled condition, and the magnitude of theforced displacement was changed from 0.001 to 0.1 mm. Noeffect of this applied load-line displacement (LLD) on themeasured CMOD compliance was found. The specimencompliance was calculated from the CMOD value on the

specimen mid-plane and the applied load corresponding to theload-line displacement of 0.1mm (LLD= 0.1mm).

IV. ANALYSIS AND DISCUSSIONSFor a given specimen with a curved crack front, the CMOD

compliance value, Ccurved, was compared with the value, Cstraight,obtained from the specimen with a straight crack front and thesame aave/W and B/W ratios. The values of Ccurved /Cstraight are

plotted against for specimens with curved or straight crackfronts in Figs. 3(a) through 3(c), and those specimens withcurved crack fronts have the same shape parameter p= 3. Thefigures suggest that given aave/Wand B/W, as the crack frontcurvatures characterized by increases from 0 to 0.1, the valuesof Ccurved /Cstraightfirst increases slightly reaching a peak point ataround = 0.03 to 0.05, and then decreases rapidly. After thatpeak point, the effect of the crack front curvature on thecompliance became more pronounced as crack length increases.Due to the dependence of straightness criteria specified inASTM 1820-11 on the specimen thickness, i.e. being afunction ofB, for specimens with the same average crack lengthand values, the crack front curvature impacts the complianceof thick specimens more than that of thin specimens as shown in

Fig. 3. However, the differences between the calculated Ccurvedand Cstraightcorresponding to = 0.05 (the maximum allowablecrack front curvature in ASTM E1820-11) are all within 1% forall the specimen configurations considered in this study.

(a) B/W= 1

(b) B/W= 0.5Figure 3. Variations of Ccurved/Cstraightagainst

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

SE(B) B/W= 1

a/W = 0.3

a/W = 0.5a/W = 0.7

ASTM E1820-11 Limit

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

SE(B) B/W= 0.5

a/W = 0.3

a/W = 0.5

a/W = 0.7

ASTM E1820-11 Limit


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(c) B/W= 0.25Figure 3. Variations of Ccurved/Cstraightagainst (contd)

The relationship between the CMODcompliance, C, and therelative crack length a/W is usually derived from numericalstudies. The following expression given by Tada et al. [11] iswidely used for SE(B) specimens:

4/ 24 0.76 2.28 3.87 2.04 0.66 1

whereE' is the elastic modulus corresponding to the plane straincondition, i.e.E' =E/(1 - v2). Wu [12] and Joyce [13] proposedthe following two inverted expression of Eq. (6), i.e. Eqs. (7) and(8), respectively, to evaluate crack length from the unloadingCMODcompliance during tests:

/ 0.999748 3.950 2.9821 3.21408 51.5156 113.031 / 1.01878 4.5367 9.0101 27.333 74.400 71.489 where (/ )/+,

Equations (7) and (8) have been adopted in ASTM E1820-11(withE ' replaced byE) to evaluate the crack length for SE(B)specimens with deep (0.45 a/W< 1) and shallow (0.05 a/W

< 0.45) cracks, respectively. Because the actual stress state inthe remaining ligament is neither plane stress nor plane strain inreal 3D specimens [4, 5], the use of eitherEorE' will inevitablyimpact the accuracy of the equations. Therefore, the followingso-called effective modulus of elasticity, Ee, was proposed byWang et al. [6] in a recent study: whereA0,A1,A2,A3andA4are coefficients and listed in Table 1.

TABLE I. THE COEFFICIENTS USED IN EQ.(9)

B/W A0 A1 A2 A3 A4

1 1.0773 0.2685 -6.9063 28.9474 -32.3795

0.5 1.0698 -0.0709 -4.7352 22.4058 -24.9463

0.25 1.0414 -0.0706 -2.7256 12.2094 -12.147

In the present study, all of the three elastic moduli, i.e. E,E 'andEe, were used in Eqs. (7) and (8) to predict the average crack

length (denoted as ap) for specimens containing curved or

straight crack fronts from the CMODcompliance obtained from

FEA. It is noted that Eqs. (6) to (9) are all derived based on

specimens with straight crack fronts. For specimens with curved

crack fronts, the predicted crack length apusing those equations

can be assumed as the equivalent straight crack length, which is

generally not the same as the nine point measured average crack

length aaveeven if above equations are perfectly accurate. As the

main concern of this study is to investigate the effect of using

those equations to predict the crack length of specimens with

curved crack fronts, the error ea, where ea= (ap- aave)/aave, was

calculated and plotted against in Figs. 4(a) to 4(i).

(a)

(b)

Figure 4. Variations of the error eaagainst

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

SE(B) B/W= 0.25

a/W = 0.3

a/W = 0.5

a/W = 0.7

ASTM E1820-11 Limit

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.3,B/W= 1, p = 3

ASTM E1820-11 Limit

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.5,B/W= 1,p = 3

ASTM E1820-11 Limit


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(c)

(e)

(g)

(d)

(f)

(h)

Figure 4. Variations of the errorea

against (contd)

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-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.7,B/W= 1,p = 3

ASTM E1820-11 Limit

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress, p = 3 E - plane stress, p = 2.5

E' - plane strain, p = 3 E' - plane strain, p = 2.5

Ef fect ive modulus, p = 3 Effect ive modulus , p = 2.5

/W= 0.5,B/W= 0.5,p = 3 & 2.5

ASTM E1820-11 Limit

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.3, B/W= 0.25, p = 3

ASTM E1820-11 Limit

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress, p = 3

E - plane stress, p = 2.5

E' - plane strain, p = 3

E' - plane strain, p = 2.5

Effective modulus, p = 3

Effective modulus, p = 2.5

/W= 0.3,B/W= 0.5,p = 3 & 2.5

ASTM E1820-11 Limit

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress, p = 3

E - plane stress, p = 2.5

E' - plane strain, p = 3

E' - plane strain, p = 2.5

Effective modulus, p = 3

Effective modulus, p = 2.5

/W= 0.7,B/W= 0.5,p = 3 & 2.5

ASTM E1820-11 Limit

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.5,B/W= 0.25, p = 3

ASTM E1820-11 Limit


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(i)

Figure 4. Variations of the error eaagainst (contd)

It is observed that the magnitude of eais governed by the choice

of the elastic modulus, and insensitive to the crack front

curvature regardless of the aave/WandB/Wratios. Except for

specimens with aave/W= 0.7 andB/W= 1, the use ofEleads to

more accurate predictions of aavethan the use ofE '. The errorfor the use ofEis within 1% for specimens withB/W= 0.5 and

0.25, and 3% for specimens withB/W= 1. Furthermore, the use

of Ee leads to the most accurate prediction of aave for all the

specimens considered. The impact of the shape parameter p

used in Eq. (4) on ea was investigated based on SE(B)

specimens with aave/W= 0.3, 0.5 and 0.7, B/W = 0.5 and = 0

to 0.10. The values of eacorresponding top= 3 and 2.5 for the

specimens considered are depicted in Figs. 4(d), 4(e) and 4(f),

which indicate thatphas a negligible impact on ea.

V. SUMMARY AND CONCLUDING REMARKSSystematic three-dimensional FEA of plane-sided SE(B)

specimens with straight and curved crack fronts and a wide rangeof aave/W(aave/W= 0.3, 0.5 and 0.7) andB/Wratios (B/W= 1, 0.5and 0.25) were performed to investigate the impact of the crackfront curvature on the CMOD compliance and the evaluatedcrack length. Symmetric bowed crack fronts with differentcurvatures were considered in the analysis. The power-lawexpression proposed by Nikishkov et al. [8] was adopted tocharacterize the curved crack front. The level of the crack frontcurvature was characterized by a parameter , which isconsistent with the crack front straightness criterion specified inASTM E1820-11. The CMOD compliance value for thespecimen with a curved crack front was compared with the valueobtained from a specimen with a straight crack front and havingthe same average crack length and thickness. For a givenspecimen with either a straight or curved crack front, the cracklength predicted from the CMODcompliance was examined byits actual average crack length.

The numerical results show that for SE(B) specimen with acurved crack front satisfying the straightness criterion asspecified in ASTM E1820-11, i.e. 0.05, the correspondingCMODcompliance differs by less than 1% compared with thecompliance of the specimen with a straight crack front and the

same crack length. In addition, the crack front curvature has anegligible impact on the crack length predicted for all theconsideredvalues regardless of aave/WandB/Wratios. The useof three different elastic moduli, i.e. E, E ' andEe, in theprediction of the average crack length from the CMODcompliance for SE(B) specimen was investigated. It is observedthat the use ofEeas reported by Wang et al. [6] can lead to highlyaccurate prediction of the average crack length for wide range ofa/W,B/Wand crack front curvatures of SE(B) specimens.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial supportprovided by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada and TransCanada Ltd. through theCollaborative Research and Development (CRD) program.

REFERENCES

[1] G. A. Clarke, W. R. Andrews, P. C. Paris, and D. W. Schmidt, Singlespecimen tests for JIc determination, Mechanics of Crack Growth,ASTM STP 590, 1976, pp. 27-42.

[2] ASTM, ASTM E1820-11: Standard Test Method for Measurement ofFracture Toughness, America Society of Testing and MaterialsInternational, West Conshohocken, PA, 2011.

[3] J. Zhou, and W. O. Soboyejo, An investigation of the effects of crackfront curvature on the crack-tip opening displacement of A707 steel,International Journal of Fracture, vol. 115, No. 3, 2002, pp. 287-305.

[4] P. Steenkamp, JR-curve testing of three-point bend specimen by theunloading compliance method, Fracture Mechanics 18th Symposium,ASTM STP 945, 1985, pp. 583-610.

[5] G. Shen, W. R. Tyson, and J. A. Gianetto, CMOD compliance of BxBsingle edge bend specimens, Proceedings of the 2012 ASME PressureVessel and Piping Division Conference, Toronto, Ontario, Canada, July1519, 2012, ASME, New York.

[6] E. Wang, W. Zhou, and G. Shen, A numerical study on effective modulusof elasticity in crack length evaluation for single-edge bendingspecimens, Journal of Testing and Evaluation, vol. 41, No. 5, 2013, pp.1-8, doi: 10.1520/JTE20120269.

[7] O. L. Towers, Fatigue crack front shape and its effect on fracturetoughness measurements, Journal of Testing and Evaluation, vol. 11, No.1, Jan. 1983, pp. 34-45.

[8] G. P. Nikishkov, J. Heerens, and D. Hellmann, Effect of crack frontcurvature and side grooving on CTOD 5 and J -Integral in CT and 3PBspecimens, Journal of Testing and Evaluation, Vol. 27, No. 5,September1999, pp. 312319.

[9] E. Wang, W. Zhou, G. Shen, and D. Duan, An experimental study onJ(CTOD)-R curves of single edge tension specimens for X80 steel,International Pipeline Conference (IPC2012), Calgary, Alberta, Canada,September 2428, 2012, Paper Number: IPC2012-90323.

[10] ADINA,Theory and Modeling Guide. ADINA R & D Inc., Watertown,MA, 2012.

[11] H. Tada, P. C. Paris, and G. R. Irwin, The Stress Analysis of CracksHandbook, ASME press, New York, 2000.

[12]

S. X. Wu, Crack length calculation formula for three point bendspecimens, International Journal of Fracture, vol. 24, No. 1, 1984, pp.R33-R38.

[13] J. A. Joyce, J-resistance curve testing of short crack bend specimensusing unloading compliance, Fracture Mechanics, Proceedings of the22nd National Symposium, vol. 1, 1992, pp. 904-924.

[14] W. O. Soboyejo, Mechanical Properties of Engineered Materials, MarcelDekker, Inc., New York, 2003.

-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

E - plane stress

E' - plane strain

Effective modulus

/W= 0.7,B/W= 0.25,p = 3

ASTM E1820-11 Limit

effect of crack front curvature on cmod compliance and crack length evaluation for single-edge bend...

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