Three-parameter approach for elastic–plastic stress field of an embedded elliptical crack

Engineering Fracture Mechanics 76 (2009) 2429–2444

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Three-parameter approach for elastic–plastic stress field of an embedded elliptical crack Junhua Zhao * Institute of Nano Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of Structural Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

a r t i c l e

i n f o

Article history: Received 30 December 2008 Received in revised form 10 April 2009 Accepted 28 June 2009 Available online 2 July 2009 Keywords: Out-of-plane stress constraint factor Tz J-integral Embedded elliptical crack Three-dimensional elastic–plastic finite element

a b s t r a c t A three-parameter approach for an embedded center-elliptical crack under tension is presented by systematic three-dimensional (3D) elastic–plastic finite element. It is shown that even in the well embedded condition, the conventional two-dimensional HRR solution and the extended J–Q description which considers the in-plane constraint modification can hardly provide satisfied description for the crack front fields with the increase of radial distances (r/(J/r0)) and strain hardening exponent n. Thus, a consideration of the out-of-plane constraint and use of the three-parameter description is necessary and efficient to predict the 3D stress fields in the whole plastic zone. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Different three-dimensional (3D) cracks (e.g. through-straight cracks, surface cracks, corner cracks and embedded cracks) frequently initiate and grow at notches, holes or welded joints in structural components. In the past years, many authors made great efforts and studied the fracture problems for these 3D cracks in linear elastic and elastic–plastic materials [1–5]. In the previous work of Guo [6–9] on 3D cracks, it was proven that the out-of-plane stress constraint factor Tz (see Fig. 1) plays an important role in the near-tip fields and the change of Tz may have substantial effects on the crack-tip fields and stress constraints. Furthermore, the stress parameters, such as the equivalent stress re, the hydrostatic stress rm and the stress triaxiality Rr = rm/re, are important 3D stress parameters which have been shown by large amount of evidences [9– 13]. However, much research has proven that these famous two-dimensional theories (e.g. HRR solution [14,15], K–T solution [16], J–A2 [17] and J–Q approach [18,19], etc.) can hardly describe these important stress parameters very well [13,20]. Combining the in-plane constraint theory of K–T or J–Q with the out-of-plane constraint theory of K–Tz [21–23] or J–Tz, K–T– Tz [24–26] or J–QT–Tz [13] the 3D constraint theory with in- and out-of-plane constraints being considered simultaneously can be formed, which is the fundament to solve the widely existed 3D fracture and fatigue problems in engineering structures [27]. However, much attention has been paid to through cracks, surface cracks and corner cracks rather than embedded cracks. In fact, 3D embedded cracks often occur in piezoelectric ceramics and other brittle and ductile materials and result in catastrophic accidents [28–32]. Some research on 3D embedded cracks was presented in linear elastic materials [21,25,32], while no much work was carried out on them in elastic–plastic materials. It is an important reason that the stress field around the embedded elliptical crack front is used to be approximately considered as in-plane strain state. Nevertheless, * Address: Institute of Nano Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. Tel.: +47 735 91 499. E-mail addresses: [email protected], [email protected] 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.06.013

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

y

r

in-plane constraint

θ x

y o

σzz φ

z

z out-of-plane constraint Tz

o x

Fig. 1. The coordinate system and a normal sheet element of an embedded elliptical crack.

detailed 3D finite element calculations have shown a strong 3D nature in the field around the embedded center-elliptical crack front for linear elastic materials in previous work [21,25]. Similarly, the triaxial stress field along the 3D crack front plays a vital role in elastic–plastic materials, and dominates the initiation and propagation behavior of the crack [13,27]. So it is very important to study the 3D constraint effects and the stress state around the 3D elastic–plastic embedded crack front. However, the distributions of J, Q and Tz are so complicated that no theoretical solution is available in the vicinity of the 3D crack border, so the numerical method will be the necessary means. In this paper, detailed 3D elastic–plastic finite element (FE) computations have been carried out for an embedded elliptical crack in plates under tension. The numerical solutions for J, Q and Tz have been obtained along the crack front, and farfield tension and the effect of n are considered. By matching the FE results and theoretical investigations, the empirical formula of Tz and the corresponding stress components are provided to predict the stress state parameters effectively. Focusing on the effects of the in-plane constraint and out-of-plane constraint, a J–QT–Tz approach is proposed which can characterize the 3D stress–strain fields well under small scale yielding (SSY) condition. 2. Numerical analysis 2.1. Finite element modeling 3D elastic–plastic FE analyses are performed to evaluate J-integral, Q term and Tz for an embedded center-elliptical crack in a plate under remote tension. The geometry and coordinate are shown in Fig. 2. Due to the symmetry of Mode I loading plate, only 1/8 of the plate is considered in the analysis. The solutions are performed by use of the commercial code ANSYS with 20-noded isoparametric elements. A typical finite element mesh is illustrated in Fig. 3. In the FE models, a/t, c/w and a/h are less than 0.05 so that the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front. For finite deformation analysis, the crack tip is assigned a finite root radius. The initial notch radius is about 106 times the distance to the boundary. In order to model the 3D stress field accurately, the thickness of the successive element layers gradually reduces toward the free surface along the crack front, and the element size gradually increases with increasing r from the crack tip. The radial size of the elements around the crack tip equals about 0.00005 a. Total 20 elements with orthogonal intersecting lines are used along the crack front, and about 35,000–40,000 elements are used in every model of different aspect ratios. Because of the symmetry of the model and boundary condition, only the uncracked planes in the xoz plane, yoz plane and xoy are symmetrically constrained, respectively. The uniform tension stress is set on the top plane and keeps the same value for different n and a/c. In this paper, the strain hardening exponent n is set at 3 and 13, Poisson’s ratio m for elastic strains is 0.3, the yield stress r0 = 1 MPa and e0 = r0/E = 0.002.

Fig. 2. Geometry of an elastic–plastic plate with an embedded elliptical crack under uniform tension [21]. (a) The 3D model and the rectangular coordinate system. (b) 1/2 Cracks with different a/c (0.2–1.0).

J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

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Fig. 3. The finite element model of whole 1/4 plate. (a) Typical finite element meshes of the 1/4 plate, a/c = 0.5. (b) Meshes of the crack tip in (a). (c) Meshes near the crack front.

2.2. Verifications of the finite element model Because the commercial FE package ANSYS is only used to calculate the J-integral for the straight crack directly, the J-integral of embedded elliptical crack will be obtained by modifying the procedure in the FE package. In the current calculations, the average value of J-integral from the two contours is used. In order to verify the finite element model and the procedure, an embedded elliptical crack model is constructed under tension in linear elastic materials where the geometry and FE meshes have been given [32]. In an isotropic linear elastic body, the relation between stress intensity factor K- and J-integral can be expressed as

J¼

K 2I E0



E0 ¼ E

plane stress

E0 ¼ E=ð1  mÞ2

plane strain

;

ð1Þ

where KI is the stress intensity factor K under the Mode I condition. As shown in Fig. 4, comparisons are made among the present procedure, the 1/4-point displacement method [21] and the results of Newman and Raju’s [33]. It is found that all present results are in very good agreement (<5%) with those given by other two methods in the range of 0° 6 / 6 90°. 3. Definition of the triaxial stress parameters and fundamental equations 3.1. Triaxial stress parameters For the convenience of study, the coordinate system shown in Fig. 1 is adopted, where o is a point of the curve defining the crack front, and x, y and z are the normal, binomial and tangent components at o, respectively. The plane defined by x and y is the normal plane, and r and h are polar coordinates in this plane. In this paper, the out-of-plane stress constraint factor Tz is defined as

Tz ¼

r33 ð1; 2; 3Þ ¼ ðx; y; zÞ or ðr; h; zÞ: r11 þ r22

ð2Þ

In the state of plane stress, Tz = 0. In the state of plane strain, Tz may change from the elastic Poisson’s ratio m of the material to 0.5 [6–8]. Therefore, Tz ranges from 0 to 0.5 in an elastic–plastic body. We define the triaxiality of stress as

Rr ¼ rm =re ; where rm is the hydrostatic stress and re is the Von Mises equivalent stress, that is [13]

ð3Þ

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

1.5

a/c=0.5, a/t<0.1, c/w<0.1, a/h<0.1 1/4-point displacement method [33] Newman's formula 3D J-integral (Present method)

1.4

π K/2σ (π a)

1/2

1.3 1.2 1.1 1.0 0.9 0.8 0

15

30

45

60

75

90

φ (Degree) Fig. 4. Comparisons of present normalized K results with the results of 1/4-point displacement method and Newman and Raju’s.

1 3   h    i1=2 ¼ 1  T z þ T 2z r211 þ r222  1 þ 2T z  2T 2z r11 r22 þ 3 r212 þ r213 þ r223 :

rm ¼ ð1 þ T z Þðr11 þ r22 Þ;

ð4Þ

re

ð5Þ

3.2. Fundamental equations For non-linear analysis, the Ramberg–Osgood type stress–strain relationship is used here, which can be expressed as





e r r n ¼ þa ; e0 r0 r0

ð6Þ

where n is the strain hardening exponent, a is the material constant, and e0 is the reference strain given by r0/E in which E is the Young’s modulus. For the convenience of analysis, a = 1 here. By considering the effects of geometry and size on crack-tip constraint, O’Dowd and Shih [18,19] found that the near-tip stress field is governed by the two parameters of J and Q as follows

rij ¼



J

ar0 e0 IðnÞr

1 nþ1

r~ ij ðhÞ þ Q ij dij



r J=r0

k

r0 ;

ð7Þ

in the forward sector (jhj < p=2), where the first term is the HRR solution, Q is a function of the stress triaxiality achieved ahead of the plane strain cracks. In Refs. [18,19], k is set to zero and Qrr = Qhh and Qij can be given as

Q ij ¼

rij  rij jHRR at h ¼ 0 and r ¼ 2J=r0 : r0

ð8Þ

4. Finite element results and corresponding empirical formulae 4.1. The elastic–plastic J-integral and Q term As shown in Fig. 5, the values of J-integral normalized by (ar0) are given along the embedded elliptical crack front with various a/c by present procedure. Under the same loading, the FE results show that the J-integral changes very slightly with n under SSY condition. So we can give a key decision that the J-integral is independent of n under SSY condition when the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front. Furthermore, the normalized J increases in the range of 0° 6 / 6 45° and decreases in the range of / P 45° with the increase of a/c, especially the normalized J tends to a same value when / equals about 45° for different a/ c and n. Fig. 6 shows that the normalized distributions of Q term are complicated. The Q term gradually increases to maximum and then decreases to a steady value with the increase of / for different n and a/c. For n = 3, the normalized Q term gradually increases with the decrease of a/c in the range of / 6 45°. For a given a/c and /, the Q term increases with increasing n and the increment is about equal except for a/c = 0.2. For a/c = 0.2, the Q term decreases with the increase of n in the range of / 6 45° and increases with increasing n for 45° 6 / 6 90° and the Q term tends to equality with different n when / is about

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

n=13 a/c=0.2 a/c=0.4 a/c=0.5

1E-4

n=3 a/c=0.2 a/c=0.4 a/c=0.5

a/c=0.6 a/c=0.8 a/c=1.0

a/c=0.6 a/c=0.8 a/c=1.0

J/(aσ0)

increasing a/c

1E-5 0

15

30

45 φ (Degree)

60

75

90

Fig. 5. The distributions of J-integral which are normalized by (ar0) around the embedded elliptical crack front with different a/c and / for n = 3 and n = 13.

Q/(aσ0)

80

40

0 n=3 a/c=0.2 a/c=0.4 a/c=0.5

-40 0

15

30

a/c=0.6 a/c=0.8 a/c=1.0

45

n=13 a/c=0.2 a/c=0.4 a/c=0.5

60

a/c=0.6 a/c=0.8 a/c=1.0

75

90

φ (Degree) Fig. 6. The distributions of Q/(ar0) around the embedded elliptical crack front with different a/c and / for two various n.

equal to 45°. From the above analysis, the normalized Q term is the function of n, a/c and /. Compared with the stress components, the values of Q term are comparable and not neglected even if under SSY condition when a/c is very small especially. 4.2. Comparison of the results of the J–Q theory and the finite element results From Figs. 7 and 8 some important information can be obtained: (1) It is very surprising that J–Q solution can effectively describe the influence of the in-plane stress parameters as the radial distances (r/(J/r0)) are relatively small, while the approach can hardly characterize it very well with the increase of r/(J/r0) and strain hardening exponent n. When r/(J/r0) > 2, the J–Q solution may overestimate the in-plane stresses, and the higher r/(J/r0) is, the more disparity between J–Q solution and the FE results is. (2) The J–Q solution is not enough to describe the hydrostatic stress rm near the crack tips. In fact, rm is dependent not only on the in-plane constraint, but also on the out-of-plane constraint. (3) Because the J–Q solution seldom considers the out-of-plane stress constraint, it can hardly give a proper description of re and the important parameter of the triaxiality of stress rm/re. Therefore, the J–Q solution should be modified in ductile fracture of materials in which the damage process is sensitive to rm/re [34]. 4.3. Three-parameter J–QT–Tz description on the stress fields of embedded elliptical cracks From the above analyses of limitations of J–Q solution, the three-parameter J–QT–Tz description on the stress fields is provided in which the influence of the in-plane constraint and out-of-plane constraint are both considered. Based on the J–Q solution of Eq. (7), the modification solution is modified as [8,35]

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

(c)

(a)

8

8

n=3 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

6

σm/σ0

σrr/σ0

6

4

4

2

2 0

2

4

6

n=3 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

8

0

10

2

4

(b)

10

6

8

10

r/(J/σ0)

r/(J/σ0)

(d) 8

n=3 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

6

6

n=3 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

σm/σe

σθθ/σ0

8

4 4 2 2 0

2

4

6

8

10

0

r/(J/σ0)

2

4

6

8

10

r/(J/σ0)

Fig. 7. The radial distributions of stress parameters of finite element results, J–Q solution and J–QT–Tz with various a/c and / for n = 3 (a/c = 0.5, / = 47.97°). (a) The radial distributions of rrr/r0; (b) The radial distributions of rhh/r0. (c) The radial distributions of rm/r0. (d) The radial distributions of rm/re.

rij ¼



J ar0 e0 Iðn; T z Þr

1 nþ1

r~ ij ðh; T z Þ þ Q Tij dij ;

where

Iðn; T z Þ ¼

    

Z p ~r ~h n @u @u ~ ~ ~ ~ ~ ~ ~ ~ r~ nþ1 cos h  sin h r  r þ r þ r Þ dh; u u u u   cos h ½ nðs  2Þ þ 1 ð rr h rh r rr r rh h e @h @h p n þ 1

where

! ~ @2U ~ d1 U þ d2 2 ; nðs  2Þ þ 1 @h ! ~ @u ~r 1 @U n1 ~e ~h ¼ ; u 2ds r  nðs  2Þ @h @h " ! !# ~ ~ ~ ~r ~ n1 @u 1 @ðr Þ @2U @U @3U n1 e ~ ~e ; d1 d1 U þ d2 2 þ r ¼ þ d2 3 nðs  2Þ þ 1 @h @h @h @h @h ! 2~ ~h @u ~ þ d4 @ U  u ~r ; ~ n1 d3 U ¼r e @h @h2

~r ¼ u

r~ n1 e

~ þU ~ 00 ; r~ rr ðh; T z Þ ¼ f U ~; r~ hh ðh; T z Þ ¼ f ðf  1ÞU ~ 0; r~ rh ðh; T z Þ ¼ ðf  1ÞU

ð9Þ

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4.0

(c) 4.0

σrr/σ0

3.5

3.0

n=13 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

3.5

n=13 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

3.0

σm/σ0

(a)

2.5

2.5

2.0

2.0

1.5

1.5 0

2

4

6

8

0

10

2

4

r/(J/σ0)

(b)

6

8

10

8

10

(d) 3.5 n=13 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

5

4

3.0

2.5

σm/σe

σθθ/σ0

6

r/(J/σ0)

2.0

n=13 0 a/c=0.5 φ=47.97 FE J-Q HRR J-QT-Tz

3 1.5

2 0

2

4

6

8

r/(J/σ0)

10

1.0 0

2

4

6

r/(J/σ0)

Fig. 8. The radial distributions of stress parameters of finite element results, J–Q solution and J–QT–Tz with various a/c and / for n = 13 (a/c = 0.5, / = 47.97°). (a) The radial distributions of rrr/r0. (b) The radial distributions of rhh/r0. (c) The radial distributions of rm/r0. (d) The radial distributions of rm/re.

~ ¼U ~ ðUÞ is assumed to be the same as in the HRR solution, f is the function of Tz [6,7]. Some where di is a function of Tz and n, U results of I(n, Tz) can be found in the articles [6,7]. QTij is obtained by matching the FE elastic–plastic stress field rij at h = 0 and r/(J/r0) = 2, or

Q Tij ¼

rij  rij jJ—T z at h ¼ 0 and r ¼ 2J=r0 : r0

ð10Þ

It is related to Qij by Eq. (8)

Q Tij ¼ Q þ

rij jHRR  rij jJ—T z : r0

ð11Þ

Let QTrr = QThh, then QT can be derived out from the parameter Q [18,19] through the following relation:

QT ¼ Q þ

rij jHRR  rij jJ—T z : r0

ð12Þ

Combining with Eq. (4), the stress components can be predicted. As shown in Figs. 7 and 8, it is assured that the threeparameter J–QT–Tz approach can describe the 3D stress–strain fields more effectively than the J–Q theory. It can be also observed that the J–QT–Tz solution can describe more effectively with increasing a/c for a given n or with decreasing n for a given a/c. In particular, the J–QT–Tz solution of rm is accurate while the difference is greatly remarkable between J–Q solutions and the FE results. Of course, the difference of stress components between J–QT–Tz solution and FE results is also distinct when r/(J/r0) is small. There may be the following reasons. Near the crack tip, the stress components will be softened in the range of a small scale region in the course of FE calculations [36,37]. Therefore, in the small region, the stress components of FE results are in little agreement with those of J–QT–Tz solution, which will also result in the inaccurate prediction of rm, and the phenomenon is usual and also occurs in other work [36].

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4.4. Distributions of the out-of-plane stress constraint factor Tz From the above analysis, we know that Tz is a very important parameter for the 3D stress–strain fields, so its distributions must be given as soon as possible. Figs. 9 and 10 gives the distributions of Tz with r/a and /, and the typical a/c are 0.2, 0.6, and 1.0, respectively, for n = 3 and n = 13. It can be found that Tz gradually decreases with increasing r/a and then becomes to zero. As r/a = 0, Tz is very close to 0.5. With r/a increasing, Tz gradually decreases and then tends to zero. It is interesting to

(a)

0.5

n=3, a/c=0.2

r/a=0,0.00018,0.0037,0.11,0.38,0.85,1.5

0.4

Tz

0.3 0.2 0.1 0.0

increasing r/a -0.1 0

15

30

45

60

75

90

75

90

75

90

φ (Degree)

(b) 0.5 n=3, a/c=0.6 r/a=0,0.0003,0.0012,0.045,0.11,0.21,0.5

0.4

Tz

0.3 0.2 0.1 0.0

increasing r/a -0.1 0

15

30

45

60

φ (Degree)

(c)

0.5

n=3, a/c=1.0 r/a=0,0.00018,0.00074,0.023,0.064,0.123,0.3

0.4

Tz

0.3 0.2 0.1

increasing r/a

0.0 -0.1 0

15

30

45

60

φ (Degree) Fig. 9. The distributions of Tz with different r/a and / in different a/c for n = 3. (a) a/c = 0.2. (b) a/c = 0.6. (c) a/c = 1.0.

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(a) 0.5 n=13, a/c=0.2

0.4

r/a=0,0.00018,0.0037,0.11,0.38,0.85,1.5

Tz

0.3 0.2 0.1 0.0

increasing r/a -0.1 0

15

30

45

60

75

90

75

90

75

90

φ (Degree)

(b) 0.5 n=13, a/c=0.6 r/a=0,0.0003,0.0012,0.045,0.11,0.21,0.5

0.4

Tz

0.3 0.2 0.1 0.0

increasing r/a -0.1 0

15

30

45

60

φ (Degree)

(c)

0.5

n=13, a/c=1.0 0.4 r/a=0,0.00018,0.00074,0.023,0.064,0.123,0.3

Tz

0.3 0.2 0.1

increasing r/a

0.0 -0.1 0

15

30

45

60

φ (Degree) Fig. 10. The distributions of Tz with different r/a and / in different a/c for n = 13. (a) a/c = 0.2. (b) a/c = 0.6. (c) a/c = 1.0.

find that the similar variation of Tz with the variation of r/a and / can be seen in Ref. [21]. From the above analyses, we can see that Tz is a function of r/a, a/c, / and n. Fig. 11 plots the distributions of Tz with r/(J/r0) for n = 3 and n = 13, it can be found that Tz sharply decreases with increasing r/(J/r0). In order to give a function to describe the distributions of Tz, we can obtain the form of the function by use of our previous work [21]. For an elastic embedded elliptical crack, the distributions of Tz have been obtained by Zhao et al. [21]

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

  r B2  T z ¼ v exp B1 ; a

ð13Þ

where B1 and B2 are functions of a/c and /. The expressions of B1 can be referred in Appendix A as B2 = 0.55 (Table 1). For an elastic–plastic plane strain crack, Tz can be expressed as [9]

T z ¼ v ep ¼

n1   n1   2:3nþ1 1 1 r0 1 1 r ¼   v v rp 2 2 2 2 re

for small h;

ð14Þ

where mep is defined the secant Poisson’s ratio or the Poisson’s ratio of the elastic–plastic strains. m is the elastic Poisson’s ratio. r p is the crack-tip plastic zone size for an plane strain crack. Attentively, mep will be equal to m because the stress field will be in elastic state as Tz < m. For an elastic–plastic embedded elliptical crack, numerical analyses show a similar variation of Tz with the variation of r/a   B  and / (see Figs. 11 and 12). Therefore, a similar factor exp B1 ar 2 can be assumed to describe the change of Tz [21]. Because the plane strain constraint is the superior limit of that in an embedded elliptical crack, m in Eq. (13) should be replaced by mep and it can be proposed that

  r B2  : T z ¼ v ep exp B1 a

ð15Þ

Consequently, Tz may be predicted by Eq. (15) for the embedded center-elliptical crack when the plastic zone size rp is obtained. As shown in Figs. 11 and 12, it is surprise to find that the Eq. (15) can predict Tz very well (with maximum error <8%) for n = 3 and n = 13 as m 6 Tz < 0.5, where B1 and B2 are the same as those in Eq. (13). When Tz < m, it is obvious in elastic state and then mep = m. Here B2 changes from 0.55 to 0.6 because B2 is a function of a/c and / in fact [21], so the prediction is very well for different n, / and a/c. 4.5. Distributions of the equivalent stress and the equivalent strain In ductile fracture, the equivalent strain ee and equivalent stress re are important parameters which play an important role in the near-tip fields [8]. However, without an exact description of Tz, it is impossible to give a proper prediction of them. For Eq. (14), it is right in-plane strain state. Here we also make use of it to predict the equivalent stress and equivalent strain for the embedded elliptical crack. The equivalent stress and equivalent strain can be expressed as

 

1

 re r 2:3nþ1 ¼ for small h; r p r0 n  2:3nþ1 ee r ¼ for small h: r p e0

ð16Þ ð17Þ

Obviously, once the plastic zone size rp is obtained, re and ee can be predicted by Eqs. (16) and (17). As shown in Figs. 13 and 14, it is surprise to find that the formulae can predict the equivalent stress and equivalent strain well with various / for different n and a/c except for very small r/(J/r0). 0.5

Tz

0.4

0.3

0.2 0.01

n=3 0 a/c=1.0, φ=5.45 0 a/c=0.5, φ=8.33 0 a/c=0.2, φ=90 n=13 0 a/c=1.0, φ=5.45 0 a/c=0.5, φ=8.33 0 a/c=0.2, φ=90 0.1

1

10

100

r/(J/σ0) Fig. 11. The distributions of Tz in a logarithmic (r/(J/r0)) abscissa for n = 3 and n = 13.

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

(a)

0.5 0.4

Tz

0.3 0.2 0.1

n=3 a/c=0.2 0

φ=14.90 FE 0 φ=90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 5

1E- 4

1E- 3

0.0 1

0.1

1

r/a

(b)

0.5 0.4

Tz

0.3 0.2

n=3 a/c=0.5 0

0.1

φ=8.33 FE 0 φ=90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 6

1E- 5

1E- 4

1E- 3

0.0 1

0.1

1

0.1

1

r/a

(c) 0.5 0.4

Tz

0.3 0.2

n=3 a/c=1.0 0

0.1

φ=5.45 FE 0 φ=90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 6

1E- 5

1E- 4

1E- 3

0.0 1

r/a Fig. 12. The finite element results of Tz and corresponding curves of Eq. (15) with various a/c at h = 0° in a logarithmic (r/a) abscissa for n = 3. (a) a/c = 0.2. (b) a/c = 0.5. (c) a/c = 1.0.

5. Discussions For a straight-through crack under SSY condition, it has been shown by Ref. [8] that r p can be predicted by

rp ¼

p n



KI

8 1 þ n ar0

2 ;

for h ¼ 0;

ð18Þ

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

(a)

0.5 0.4

Tz

0.3 0.2 0.1

n=13 a/c=0.2 0

φ =14.90 FE 0 φ =90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 5

1E- 4

1E- 3

0.0 1

0.1

1

r/a

(b)

0.5 0.4

Tz

0.3 0.2

n=13 a/c=0.5 0

0.1

φ =8.33 FE 0 φ =90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 6

1E- 5

1E- 4

1E- 3

0.0 1

0.1

1

0.1

1

r/a

(c)

0.5 0.4

Tz

0.3 0.2

n=13 a/c=0.1 0

0.1

φ =5.45 FE 0 φ =90.00 FE

Eq. (15) Eq. (15)

0.0 1E- 6

1E- 5

1E- 4

1E- 3

0.0 1

r/a Fig. 13. The finite element results of Tz and corresponding curves of Eq. (15) with various a/c at h = 0° in a logarithmic (r/a) abscissa for n = 3. (a) a/c = 0.2. (b) a/c = 0.5. (c) a/c = 1.0.

a¼

1 þ r po =B ; 1 þ r po =B  2v

ð19Þ

where rpo ¼ rp ja¼1 , B is the thickness of the plate in Ref. [9] and r p is the average size of plastic zone through the thickness of the plate.

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

(a)

4 n=3 0 0 0 a/c=0.2 φ =90 a/c=0.5 φ =47.97 a/c=1.0 φ =5.45 FE FE FE Eq. (16) Eq. (16) Eq. (16) J-Q J-Q J-Q

σe/σ0

3

2

1

0 0

5

10

15

20

r/(J/σ0)

(b)

2.0 n=13 0 0 0 a/c=0.2 φ =90 a/c=0.5 φ =47.97 a/c=1.0 φ =5.45 FE FE FE Eq. (16) Eq. (16) Eq. (16) J-Q J-Q J-Q

σe/σ0

1.5

1.0

0.5 0

5

10

15

20

r/(J/σ0) Fig. 14. The radial distributions of equivalent stress of finite element results, J–Q solution and Eq. (16) with different a/c and / for (a) n = 3 and (b) n = 13.

For an elliptical surface crack, r p is can be also obtained by Eqs. (18) and (20) when B is replaced by equivalent Beq [26]. As shown in Fig. 15, the equivalent thickness Beq is defined as Beq = 2 min{B1, B2}, where B1 and B2 are the distances from the analyzed point P on the crack border to the boundary of the cracked bodies along the tangential line of the crack front line at P. Substituting Eq. (1) into Eq. (18) gives (See Fig. 16)



JE0

p n

E0 ¼ E

plane stress 8 1 þ n ða r E0 ¼ E=ð1  m2 Þ plane strain 1 þ r p =Beq a¼ : 1 þ r p =Beq  2v

rp ¼

2

2 0Þ

for h ¼ 0;

ð20Þ ð21Þ

In the above analysis, only the constitutive relationship Eq. (6) is considered as a = 1. Once the constitutive relationship is changed, the distribution of the stress triaxial constraint and its effects on the fields may change as well. Different type of constitutive relationship is discussed for a plane strain crack in the Refs. [8,9]. For the widely used Ramberg and Osgood stress–strain relation





r ar r n e¼ þ 0 : E E r0

ð22Þ

It can be obtained that

v eq ¼

  1 1  v 2 2

1  n1 :

1 þ a rr0

Consequently, it can be obtained for a plane strain crack based on the Eq. (23) and the discussions in Refs. [8,9]

ð23Þ

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

(a)

8

n=3 0 0 0 a/c=0.2 φ =90 a/c=0.5 φ =47.97 a/c=1.0 φ =5.45 FE FE FE Eq. (17) Eq. (17) Eq. (17)

εe/ε0

6

4

2

0 0

5

10

15

20

r/(J/σ0)

(b) 8

n=13 0 0 0 a/c=0.2 φ =90 a/c=0.5 φ =47.97 a/c=1.0 φ =5.45 FE FE FE Eq. (17) Eq. (17) Eq. (17)

εe/ε0

6

4

2

0 0

5

10

15

20

r/(J/σ0) Fig. 15. The radial distributions of equivalent strain of finite element results and Eq. (17) with different a/c and / for (a) n = 3 and (b) n = 13.

Fig. 16. Equivalent thickness Beq (Beq = 2 min{B1, B2}) of 3D embedded crack.

T z ¼ v eq ¼

  1 1  v 2 2

  1 1 1 v  n1 ¼  2 2 re

1 þ a r0

Submitting Eq. (24) into Eq. (11) gives

1 : n1  2:3nþ1

1þa

r p r

ð24Þ

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J. Zhao / Engineering Fracture Mechanics 76 (2009) 2429–2444

Tz ¼

  1 1  v 2 2

1þa

  r B2  1 exp B1 : n1  2:3nþ1 a

ð25Þ

r p r

Similar to the above investigation, we can give the corresponding formulae in order to predict the stress parameters for an embedded elliptical crack.In addition, the J–QT–Tz solution can describe the stress–strain fields for a plane strain crack not only under SSY condition but also under LSY condition in Ref. [35]. Therefore, we can speculate that the J–QT–Tz solution could also describe the stress–strain fields for a corner crack or other shape crack, which will be our following work. 6. Conclusions The 3D stress–strain fields near the embedded elliptical crack front are investigated under small scale yielding condition based on the FE results and the analytical methods. The conclusions are in the following: (1) The angular distributions and amplitude of the stress parameters near the crack tip gradually deviate from the HRR solution and J–Q solution seriously with increasing r/(J/r0), so a consideration of the out-of-plane constraint and use of the three-parameter description of J–QT–Tz is necessary and efficient to predict the 3D stress fields in the whole plastic zone. (2) The out-of-plane stress constraint factor Tz is strongly coupled with the stress field. For an embedded elliptical crack, Tz decreases rapidly from 0.5 to 0 at some r/a as it departs from the crack tip. The region of Tz ? 1/2 is very small in usual materials, when m is obviously lower than 0.5. (3) The J-integral is independent of n under the same loading when the stress-free boundary conditions at the side and back surfaces of the plate have negligible effect on the stress state along the crack front. Acknowledgement The work is supported by NSF of China, Aviation science foundation (Nos. 2007ZA52011 and 2008ZF52062), Scientific Research Innovation Project of Jiangsu Province (No. xm06-42). Appendix A The parameters of Eq. (13) are expressed in Table 1.

Table A1 Values of B1 as expressed as Eq. (13) [21]. a/c

B1

0.2 0.4 0.5 0.6 0.8 1.0

pffiffiffiffi 0:34394 þ 1:34051= / pffiffiffiffi 0:81166 þ 1:2357= / pffiffiffiffi 1:31941 þ 0:92127= / pffiffiffiffi 1:83679 þ 0:59022= / pffiffiffiffi 2:53128 þ 0:26236= / 3.09572

where m is Poisson’s ratio, B2 is the function of a/c and / in fact (here we take B2 = 0.55), B1 is functions of a/c and / which can be given as pffiffiffiffi B1 ¼ 0:11414ða=cÞ2 þ 3:75902a=c  0:48969 þ ½0:19318ða=cÞ2 þ 1:4979ða=cÞ1  1:31755= /.

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