Methodologies for predicting J-integrals under large plastic deformation—I. further developments for tension loading

Engineering Fr~~fure Mechanics

Pergamon

0013-7944(94)00166-9

Vol. 49.

No. 3. pp. 337-354, 1994

Copyright 0 1994 ElsevierScienceLtd Printed in Gnat Britain. All rights reserved

0013-7944/94 57.00+ 0.00

METHODOLOGIES FOR PREDICTING J-INTEGRALS UNDER LARGE PLASTIC DEFORMATION-I. FURTHER DEVELOPMENTS FOR TENSION LOADING C. L. LAU, M. M. K. LEE? and A. R. LUXMOORE Department of Civil Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP, U.K. Abstract-Based on the results of a parametric finite element study on single edge notch specimens with shallow crack geometries (0.05 d a/W < 0.1) under large plastic deformation, a revised J-estimation method is presented. The revised scheme focused on material laws which can be approximated by a power law representation. The scheme uses the format of an existing method which adopts Turner’s EnJ tripartite approach of J-estimation: Linear Elastic Fracture Mechanics, Net Section Yield and Gross Section Yield. The form of the J vs strain relationship in the gross section yield region was derived from a consideration of the HRR strain field equation, the EPRI plastic J equation and the definition of J in terms of the energy release rate. Comparison with finite element results shows that the revised scheme generally provides safe predictions of the crack driving parameter. For low to medium work hardening materials (power law exponent n 2 5), the accuracy of prediction is within about 5%. The scheme has also been shown to be equally applicable to centre cracked panels with short cracks.

1. INTRODUCTION SEVERAL fracture mechanics based methods have been developed in the past two decades for design against fracture and defect assessments in structures. Methods such as the CTOD (or COD) design curve [l] and the J-integral based methods of EPRI [2], Modified EPRI [3], EnJ [4] and R6 [5] are means by which rapid estimation of elastic-plastic crack tip parameters can be obtained. Lee et al. [6] carried out a study on the defect assessment of plain and welded bars with shallow crack geometries [crack depth to specimen depth ratio (a/W) = 0.11 and they compared solutions from these design methods with crack-tip parameters obtained from experimental tests and numerical analyses. For the plain specimens, their findings show that these methods underestimate the crack driving parameters vs strain relationships, particularly for configurations involving tension loading and materials with low work hardening. Prompted by the findings of Lee et al. [6], Azzabi et al. [7] conducted a series of two-dimensional (2D) elastic-plastic plane strain finite element computations on single edge notch specimens under tension loading (SENT). The specimens had a/W = 0.05, 0.1 and 0.2 and the stress-strain curves used were represented by piecewise power hardening and bilinear approximations. Their results indicate that the J vs strain relationships are extremely sensitive to the variation of the power hardening exponent, n, as well as the a/W ratio. The relationships can be schematically represented by three distinct regions, which occur at different levels of global deformation, as shown in Fig. 1:

(1) Linear Elastic Fracture Mechanics (LEFM): a load controlled region wherein plastic deformation is confined to very near the crack tip; (2) Net Section Yield (NSY): a region in which yielding occurs through the net section of the structure and J increases very rapidly with strain and (3) Gross Section Yield (GSY): a region in which yielding occurs in the gross section of the structure and there is a reduction in the rate at which J increases with strain. Based on the finite element results, Azzabi et al. developed an empirical scheme by which the applied J-integral can be estimated for remote strains up to three times the yield strain of the material. tAuthor to whom correspondence

should be addressed. 331

338

C. L. LAU et al.

NET SECTION

YIELD

&sin

Fig. 1. Schematic relationship between J and strain for edge. cracked specimens loaded in tension.

This paper presents further developments of the above J-based estimation scheme. Finite element computations using the single edge notch (SENT) specimens and centre cracked panel (CCT) under tension loading were carried out. Attention was focused on materials which follow the power hardening material law and crack geometries having a/W ratios between 0.05 and 0.1, as cracks with ratios within this range are of particular interest to engineers involved in defect assessment. The range of the remote strain had been extended to about 10 times the yield strain. Such extreme deformation can occur as a result of collision or explosion, provided that the structure has residual ductility. The numerical results of the SENT specimens formed the database in developing the estimation scheme. Results obtained from the estimation scheme were then compared with a fully plastic analytical solution by He and Hutchinson [S] and the finite elements results of both the SENT and CCT geometries. 2. BACKGROUND TO THE REVISED J-ESTIMATION SCHEME 2.1 Review of previous work For SENT geometries with material stress-strain curves represented by the power hardening law (la) L -_=

d -

CY

(

n

for 6 >+

(lb)

=Y >

(where .fy, cryare yield strain and stress, respectively), Azzabi et al. [7J,using Turner’s EnJ tripartite approach [4], proposed a series of J-estimation curves which relate the normalized J-integral, JlG,
-em2 for :

CW

6 0.85

CY >

(

(b) Region II-NSY J J

J

Gy

Gy

-=-

%.sJ+

LY

J

Gy I.I2 -q10.a5(~-0.85) (1.2 - 0.85)

for0.85<;<1.2

(2W

Predicting J-integrals-I.

Tension loading

339

(c) Region III-GSY

In both the net section yield and gross section yield regions, J/G,, are linear functions of E~/c~, and the curves are valid for values of c “/c~ up to about 3.0. However, in order to cover a much wider range of c*/c~ in the gross yield region, a non-linear function must be used. Figure 2 compares the above scheme with Turner’s EnJ curve and values obtained using finite elements and it is evident that the scheme becomes increasingly unconservative with larger values of P/$ 2.2. Relationship between the J-integral and remote strain in gross section yield For a cracked structure with material behaviour following eq. (l), the HRR [9, lo] near-tip singularity strain field can be written as ~&Z n).

(34

Rearranging eq. (3a) and expressing J in terms of the other parameters gives J=

u,, and c,, are the yield stress and yield Z, is an integration constant depending r and 8 are polar co-ordinates centred 4 is the dimensionless function of the

P-9

strain, respectively, on n, on the crack tip and angle 0 and the power hardening exponent n.

The HRR singularity is derived within the context of a small strain deformation theory of plasticity: it is not necessarily valid for large strain deformation in incremental plasticity, as J dominance is lost beyond small-scale yielding. However, eq. (3b) serves as a pointer with regard to the possible form of relationship between the applied J and the remote strain and the validity of this form is explored further below.

Fig. 2. Comparison of J-strain curves for finite geometries. EFM 49/3-B

C. L. LAU et al.

340

The EPRI [2] J-estimation equation assumes that the total J is the sum of the elastic and plastic components. The plastic J equation for SENT specimens is written as

+,n)($I+‘,

JN=w,q$h,(

(4)

where b is the length of the untracked section a is the crack length h, factor is a function of a/W and n P is the applied load and PO is a reference load Equation (4) shows that J is a function of P"+ ‘, i.e.

(5)

J cc P”+‘. Relating load to stress and using the power hardening law of eq. (lb), the relationship

can be established. Furthermore,

in laboratory J=-;

determination

of J, the compliance relationship

(2 >

(7)

is often used. In eq. (7), B is the specimen thickness and U is total absorbed energy (or area under the load vs load point displacement curve; therefore (8) Again, using eq. (lb) gives (9) From eqs (7) and (9), the relationship between J and strain can be expected to be It+1

(10)

JUCT.

#)-.

.......... 1’............. “i”’

............

“i

8 a_

.......

.......

..?..

/ m_

........

/

e #)-

... ......

..............

,........... i..............._j._. ... .....

..i

“INIT& GK)I ,p

. .. .

..............

0

.,..........; j ..

/

/

0

TRY

1

9

10

Fig. 3. Comparison of J-strain curves for finite and semi-infinite geometries.

.

_j

1

11

Fbdicting J-integrals--I. Tension loading

341

Equations (6) and (10) give the relationship between J and the applied strain. In order to establish if it was applicable to remote strain in gross section yielding in finite bodies, finite element calculations were carried out to obtain J-strain curves for a finite geometry and a semi-infinite geometry. The two models were identical apart from the boundary conditions imposed. The resulting curves, shown in Fig. 3, illustrate that values of the J-integral in the gross section yield region of the two geometries can be correlated by a multiplication factor. It should be noted that the applied strain is equal to the remote strain in semi-inlinite bodies. Hence, for finite geometries in gross section yielding, the remote strain is proportional to the applied strain and the relationship between J and E”/c,, can be described as n+l JGC

;T.

(

>

Equation (11) forms the basis of the revised J-estimation scheme in the GSY region.

aiW

= 0.05

Fig. 4. A typical finite element mesh.

(11)

342

C. L. LAU et al,

3. FINITE ELEMENT

ANALYSIS

All specimens were modelled as 2D edge or centre cracked geometries and a/W ratios of 0.05, 0.075 and 0.1 were used. The aspect ratio, L/W (L being the length of the finite element model or half length of the specimen) of each specimen was set to 4, and the effect of geometry, i.e. dependency of results on L/W ratios, was eliminated by using the remote strain (strain measured at the undisturbed strain field away from the crack tip) [7]. Specimens were loaded by uniformly applied displacements, The numerical procedure adopted was the same as that used in refs [6,7] and it had been proved to produce valid and accurate results [a]. The finite element program ABAQUS [l I] was used to carry out the numerical analyses. Symmetry conditions enabled the modelling of only one-half of the SENT specimen and a quarter of the CCT. The six-noded plain strain isoparametric element CPE6 in the ABAQUS element library was used. The side nodes of the elements at the crack tip were retained in the midpoint position. Rigid body motion for the SENT specimens was prevented by restraining the two degrees of freedom of the corner node opposite the cracked face. All analyses were carried out using incremental plasticity and the von Mises yield criterion with the associated flow rule was adopted. Values of the J-integral were obtained using a domain integral approach. The meshes were generated with the aid of the interactive mesh generation program of ANSYS [12] and a typical mesh is shown in Fig. 4. The different meshes had an average of about 550 elements. The input stress-strain curves were represented by a piecewise linear approximation of eq. (I) with n = 2,3,5,7,10,15,20,25 and 30. Other properties adopted in the computations were: power hardening material constant CI= I, Poisson’s ratio v = 0.3,o; = 700 N/mm’ and c,,= 0.0035. 4. REVISED J-ESTIMATION

SCHEME

The revised ~~stimation scheme proposed here is based on the finite element results of the SENT geometries and follows the same format of the existing procedure due to Azzabi et al. [7]. The scheme is again divided into three distinct regions: LEFM, NSY and GSY regions. The curve for the NSY region will be discussed after those for the LEFM and GSY regions. 4.1. LEFT region

This equation is found to be independent of the type of stress-strain idealization.

0.0

0.25

0.5

0.75

d-it:, Fig. 5s

1.0

1.25

1.S

Predicting J-integrals--I.

343

Tension loading

2.5

0.0

0.25

0.5

0.75

1.0

1.25

1.5

C-k I

0.0

025

0.5

0.75

1.0

1.25

1.5

c-k, Fig. 5. Relationship between J/G,, and P’/cy for (a) a/W = 0.05; (b) a/W = 0.075 and (c) o/W = 0.1.

4.2. GSY region Figure 5 shows the J/G,, vs P/c,, plots obtained from the finite element computations. The curves have been curtailed in order to illustrate the points of transition from net section yield to gross section yield. It can be seen that the transition occurs at values of P/c,, between 1.05 and 1.2, depending on the values for n and a/W. For simplicity and in keeping with the existing scheme, the start of the gross section yield region is taken to occur at E“/E,, = 1.2 throughout. In Section 2.2, J has been found to be directly proportional to the normalized remote strain raised to the power (n + 1)/n. Thus the estimation curve can be represented by (13) where C and JNGare functions of n and a/W.

344

C. L. LAU

6

10

er al.

15

20

25

30

n Fig. 6. Relationship between JNo and n.

4.2.1. Evaluation of JNo . JNois the value of J at which transition from net section yield to gross yield occurs. The numerical values of JNGwere obtained from Fig. 5 and they are plotted against n for different values of a/W in Fig. 6. It can be seen that the curves converge to the point {n = 1, JNG= 1.85). A trial function can thus be written as J No= l.SS+(n

- l)[Eln(

g)+F].

The constants E and F can be determined by plotting E vs F for diierent values of Jnro,n and a/W, giving a total of 27 straight lines which can be represented by

J’=JNG-1’85_E~n . (n - 1)

Fig. 7. Determination of the coefficients E and F.

(15)

Predicting J-integr&+I.

Tension loading

34s

It is desirable to have only one or an average line for each a/W ratio, and hence only one value of F. It was found that the variation of F for each a/W ratio could be minimized by rewriting eq. (15) as

F=JNG-1’85-Eln . (n - l)‘.’

(16)

The power 1.1 for the term (n - 1) was obtained by trial and error. Figure 7 shows that the three straight lines, represented by eq. (16), meet approximately at the point (E = 0.26, F = 0.95) and thus J NG

1

2

=

1.85+(n - 1)‘.‘[0.26ln( +)+0.95].

a

4

5

6 C-It,

Fig. 8a and b

7

9

(17)

9

19

11

..... ...... !~ C. L. LAU et al.

346

w

80

................

go

................

.....

................ 40 ................

3 $0 % 6 20

....................

,o

......

... .._......... .. .._..........

0~

Fig. 8. The ~lations~p

between C and t w&y for (a) a/W = 0.05; (b) a/W = 0.075 and (c) a/W = 0.1.

42.2. Evaluation of C. Rearranging eq. (13), C can be expressed as --J c=

G

J

‘“-

1.2

y

(

NO

(18)

ll+1’

LY

-7 >

C is plotted against normalized remote strain, Em/+ for the three a/W ratios and the different power hardening exponents n. Figure 8 shows that C takes on a more or less constant value when the remote strain is more than three times the yield strain. This feature also confirms the J-strain relationship described in Section 2.2. It should be noted that, for c”/cY less than 2, the term [(6“I$) - 1.21’”+ ‘Mlin eq. (13) is small and the initial part of these curves is relatively insignificant. Hence, for each n, an average value of C can be taken over the entire gross section yield region without too much loss of accuracy, and C becomes a fiction of a/W and n only. Table 1 gives the values of C calculated from the finite element analyses for different a/W ratios and n. Similar to the evaluation of JNG, C is plotted against n for each a/W ratio in Fig. 9. A trial function for C is written as (19) Since C is known, a relation between AC and n can be established. In Fig. 10, they are shown to be almost linearly correlated, i.e. AC=A+Rn.

(20)

Table 1. Values of C for different a/W and n

a/W 0.05 0.075 0.1

n

2

3

5

7

10

15

20

25

30

2.13

2.43 2.5% 2.80

3.02 3.31 3.84

3.62 4.18 5.13

4.59 5.67 6.78

6.29 8.17 11.24

8.20 11.86 17.59

10.49 16.05 27.01

13.24 22.51 39.94

Predicting J-integrals-I.

0

5

10

341

Tension loading

15

20

25

20

n

Fig. 9. Relationship between C and n.

The three straight lines, obtained by the least square method, can be generalized into one equatidn by using Lagrange interpolation to fit the following points: a/w

0.05

B (slope)

0.1559

0.075 0.1835

::;485

The interpolation procedure yields the slope of these straight lines as a function of a/W. In addition, these lines meet at a point of n = 3.55 when AC = 0.25. Thus eq. (20) becomes a AC = 0.25 + (n - 3.55) [ 29 .92 (,I-2.636(;)+0.2129].

(21)

6-

B-

Y’ 2-

O-

;,

li

lb

tb

Al

A5

II

Fig. 10. Relationship between AC and R for different crack length.

A

348

C. L. LAU et ai.

............

~......._.__.....................

:.

i _.L._._

,

..:

:.

i

I IN

.~

.............+..............

..._.____._.. t’

0

1

2

3

4

6

6

7

.

.

8

.

j

.

.

8

. j

10

c”lc, Fig. Il. Comparison of J-integral predictions with the fully plastic solutions by He and Hutchinson [8].

Substituting eq. (21) into eq. (19) gives C=2(1~~~+,-3.,,,[29.92(~~-2.~3~(~)+0.2129]+0.2~.

(22)

4.3. J-estimation for NSY In engineering practice, a straight line is adequate in predicting the complicated NSY behaviour. For the linear representation, this straight line is obtained by joining the last point of the linear-elastic curve at ~“/a = 0.85 and the first point of the GSY curve at cm/c,,= 1.2. Since

Therefore

J=J~~+

(JNG12

GY

0722y $-0.85). . - is5 .

(23)

Equation (23) is valid for 0.85 < ~“/Iz,,6 1.2. 5. VALIDATION OF THE REVISED EZWIMATION SCHEME The revised J-estimation scheme represented by eqs (12), (13) and (23) is based on 2D elastic+plastic plane strain finite element analyses of a variety of configurations. In this section, the validity of the estimation scheme will be established by comparison with the fully plastic solutions due to He and Hutchinson [8] and the finite element results of both the SENT and CCT geometries. Table 2. Values of C for different n-comparison between the estimate scheme and the fully plastic solution of He and Hutchinson He and Hutchinson Estimation scheme

1.5

2

3

4

5

7

1.539 1.814

1.715 1.920

2.019 2.133

2.280 2.346

2.514 2.559

2.929 2.985

Predicting J-integral+I. 5.1.

Tension loading

349

Comparison with analytical solutions for fully plastic cracked problems

He and Hutchinson [8], using modified principles of complementary potential energy and potential energy, obtained bounds to the J-integral for edge-cracks in semi-infinite bodies. The form of the solution for J, in terms of the effective stress and strain, for the pure power hardening material, eq. (lb), is J = aa,“Qh

(n),

(24

where h (n) is a numerical value depending on the power hardening exponent n. For the comparison shown in this section, the values of h (n) used are the upper bound values. It has been shown by Kaliske et al. [13] that eq. (24) can be expressed as (25)

(4

Fig. 12. Comparison of J-integral predictions with finite element results for SENT with a/W = 0.05 for (a)n=2and3and(b)n=S-30.

C. L. LAU et al.

350

Non-dimensionalizing

eq. (25) by G,, gives h(n),

where the elastic shape factor is taken to 1.12. Values of h (n) for 1.0 < n < 7.0 are given in ref. [8]. The foregoing fully plastic solutions for edge cracks in semi-infinite bodies can be compared with the values obtained from the proposed estimation scheme using eq. (13), with slight modifications. As there is no net section yield in a semi-infinite body with work hardening material behaviour, JNc is set to zero. In addition, the LEFM region is no longer applicable giving (27)

(4

m_

.._

.,

.,........

.,.............

G...

-----E8-f TION go_ ____. _..._......_.._______,____ ._._._....._... .. ..______ ‘MA. ... ..

/’

no .......

n&o

,..’ ___ ___,......

5. ” g-

8 #

..-20 -76’

4---

10 +, @>&_....__... .

Ip

rs

s 0

1

2

3

10

4

11

&“lC, Fig. 13. Comparison of J-integral predictions with finite element results for SENT with a/W = 0.075 for (a)n=2and3and(b)n=S30.

Predicting J-integrals-I.

351

Tension loading

Values of the parameter C are obtained from eq. (22) with a/W = 0, as the dimension of the crack becomes insignificant in a semi-infinite body. Equation (27) thus becomes n+l $=(1.494+0.213n)

z (

Y

?

(28)

>

Results obtained from eqs (26) and (28) are compared in Fig. 11 for remote strains up to about 10 times the yield strain. In addition, Table 2 compares the values of C obtained by fully plastic solutions of He and Hutchinson and the J-estimation scheme. The proposed scheme can be seen to be able to predict the applied J to a high degree of accuracy, particularly for lower work hardening materials (high n values). The increasing divergence for the high work hardening cases of n = 1.5 and 2 can be attributed to the difference between deformation and incremental plasticity theories. The elastic component of the applied J, which is proportional to loadz, makes a not insignificant contribution to the total

@)

500.

n-m 400,

/ iTION

aoo, 25

g 200.

4 . . .... . ..c---- 5 .. .. .. HV @

100.

s

0.

....... . 20 15 ‘-lr?

s

I

0

1

2

a

5

Fig. 14. Comparison of J-integral predictions with finite element results for SENT with o/W = 0. I for (a) n = 2 and 3 and (b) n = S-30.

352

C. L. LAU et al.

J for high work hardening materials [ 141.This component accounts for the discrepancies between the J-estimation scheme and the fully plastic solution. 5.2. Comparison with finite element results 5.2.1. SENT geometries. Figures 12-14 compare the finite element results for the SENT geometries with predictions obtained from the J-estimation equations for the three a/W ratios of 0.05, 0.075 and 0.1, respectively. For the medium to low work hardening cases, i.e. n 2 5, the scheme provides accurate and, in general, conservative estimations of the applied J-integrals. A slight underestimation (maximum of about 10%) occurs only in the cases of a/W = 0.1 for n = 25 and 30 for the range of f?/c,, between 1.2 and 4.0. The accuracy of prediction is, in general, to within 5%. For the high work hardening cases of n = 2 and 3, accurate and slightly unsafe predictions are obtained for values of &“1%up to about 4. Beyond this range, the scheme becomes increasingly conservative, producing errors of up to about 20% at the highest normalized strain. 5.2.2. CCT geometries. For a certain normalized remote strain, a CCT geometry gives only a marginally different applied J than a SENT geometry for the same values of a/W and n. Hence, the proposed scheme can also be applied to CCT geometries. Figure 15 compares the finite element results for the CCT geometries with the predicted values for the medium to low work hardening cases. Un~n~~ative estimates of up to about 10% are obtained for the low work hardening cases of n = 25 and 30 at low remote strain range. For the other cases, the agreement is good.

6. CONCLUSIONS In the present study, a revised estimation scheme based on the results of a parametric finite element study and the J-strain relationship for single edge notch specimens with shallow crack geometries is developed. The accuracy of the proposed scheme has been established by comparison with a fully plastic analytical solution by He and Hutchinson. The procedure can be used for rapid prediction of the crack driving parameter J-integral in both SENT and CCT geometries under large plastic deformation as a result of excessive localized loading. It has been demonstrated that the accuracy of prediction is very good for low to medium work hardening materials and within acceptable limits for the high work hardening cases. In Part II, the development of an estimation methodology for single edge notch specimens in pure bending (SENB) will be presented.

.......... j ..‘.....

“4

..“...... i

0

12

a

4

5

6 c"lcy

Fig. 1Sa

7

8

0

lo

11

predicting J-integmls-I.

Tension loading

w

100-j .......!

0

i...............i ......f_ &?&.L

1

2

8

4

8

g

7

8

e

10

11

E-k,

Fig. 15. Comparisons of J-integral predictions with Bnite element results for CCT (a) a/W = 0.05; (b) a/W=O.O?S and (c) a/W=O.l.

Rc&nowl&ement+-The work described in this paper was carried out with the 6nancial support from the Defence Research Agency. The authors are grateful to Dr J. D. 0. Sumpter for initiating the work and valuable comments on the manuscript.

REFERENCE on methods for assessing the acceptability of flaws in fusion welded structures. British Standard Institution 111Guidance PD6493: 1991(1991).

PI V. Kumar,M. D. German and C. F. Shih, An engineering approach for elastic-plastic fractum analysis. Electric Power

Research Iustitute (1981). M R. A. Ainsworth, The assessment of defects in structures of strain hardening material. &ng Frucrnre Me&. 19, 633642 (E&4). [41C. E. Turner, Further developmenta of a J-based design curve and its relationship with other procedures. Second Symposium Elastic-Plastic Fracture, Vol. II. Fracture Resistance Curves and Engineering Applications, ASTM STP gtI3,80-102 (1983). PI I. Milne et a%.Assessment of the integrity of structures containing defects-R6. Central Electricity Generating Board (1986).

C. L. LAU et al.

354

@IM. M. K. Lee, A. R. Luxmoore and J. D. G. Sumpter, A numerical and experimental study on defect assessment of nlain and welded bars with shallow crack aeometries. In& J. Fruc~ure62. 245-268. (1993).

n K. 1. Ax&i, A. R. Luxmoore and hi. hi. k. Lee, J-integral estimation for single ‘edge notch specimens in

tension.

hf. J. Fracture 63, 75-87, (1993).

181M. Y. He and J. W. Hutchinson, Bounds for fully plastic crack problems for intinite bodies, Second Symposium Elastic-Plastic Fracture, Vol. 1, Inelastic Crack Analysis, ASTM STf %ej, 177-290 (1983).

191J. W. Hutchinson, Singular behaviour at the end of a tensile crack tip in a hardening material. J. Mech. Phys. Solti 16, 13-31 (1968).

IlO1J. R. Rice and 0. F. Rosengren, Plain strain deformation near a crack tip in a power-law hardening material. J. Me&

Phys. solids 16, l-12 (1968). ABAQUS User’s Manual, Version 4.8, Hibbit, Karlsson & Sorenson Inc., U.S.A. (1989). Iti; ANSYS, Version 4.4, Swanson Analysis System, Inc., P.O. Box 65, Johnson Road, Houston, U.S.A. (1989). u31 M. Kaliske, A. R. Luxmoore and J. D. G. Sumpter, A comparison of analytical and finite element J-integral for shallow cracks, Proc. 5th Int. Conf. Numerical Methoak in Fracture Mechanics, pp. 501-514, Freiburg. Germany (1990). v41 J. D. G. Sumpter, Private communication (1993). (Receiued 19 September 1993)

APPENDIX This Appendix gives a summary of the main equations of the J-estimation procedure presented in this paper so that users can have rapid access to the scheme without referring to the main body of the paper. The basic form of the equations follows Turner’s EnJ approach with the applied J-integral normalii by the energy release rate at yield (G,). Accurate prediction can be achieved provided that 0.05 < a/W < 0.1 and 2 6 n 6 30. There are three parts to the estimation scheme: Linear EIastic Fracture Meckanics, Net Section YKId and Gross &&on Yieid. 1. Linear EIartic Fracture Mechanics J -= (7,

Cm = for F g 0.85 ( cy )

(Al)

G,, = Y 2noysa

Y = 1.12 for shallow edge cracks. 2. Net Section Yield

JNG= 1.85;t(n - l)fO.26ln(

;)+0.95].

(A3)

3. Gross Section Yield (A4) where (As) with the value of JNo obtained from eq. (A3).