Evaluation of the T-stress for interacting cracks

Computational Materials Science 45 (2009) 349–357

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Evaluation of the T-stress for interacting cracks Y.Z. Chen *, Z.X. Wang, X.Y. Lin Division of Engineering Mechanics, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 10 August 2008 Received in revised form 27 September 2008 Accepted 10 October 2008 Available online 20 November 2008

This paper investigates the T-stress for interacting cracks in general case. The cracks are located in an arbitrary position with the remote tension. If one uses the traction along crack face as unknown function, the problem for interacting cracks can be reduced to a Fredholm integral equation. The solution of integral equation provides necessary knowledge for evaluating the T-stress at a crack tip. The T-stress at the crack tip is generally composed of three parts, one from the uniform field and other two from the perturbation field. Explicit equations for the three parts are proposed. The case for two collinear cracks is taken as an example to examine the accuracy of the suggested numerical method. Many results for T-stress for interacting cracks are presented. It is shown that the interaction is significant for the closer cracks. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: T-stress Stress intensity factor Interaction Fredholm integral equation method Numerical solution

1. Introduction Williams investigated the stress distribution near a crack tip [1]. In the cylindrical coordinates (r, h), the stress components rij can be expressed by







K1 rx rxy f ðhÞ f12 ðhÞ ffi 11 rxy ry ¼ pffiffiffiffiffiffiffiffi 2pr f12 ðhÞ f22 ðhÞ



   K2 T g ðhÞ g 12 ðhÞ þ pffiffiffiffiffiffiffiffiffi 11 þ ðhÞ g ðhÞ 0 g 2pr 12 22

0 0

 ð1Þ

where the first two terms in the expansion form are singular at the crack tip, K1, K2 denote the mode I and mode II stress intensity factors, respectively. The third term in Eq. (1) is bounded and is denoted as the T-stress [2]. The T-stress plays an important role in directional stability for the crack growth path [2–5]. In addition, the T-stress can considerably influence the plastic zone near crack tip in the case of small scale yielding [6,7], and it is also an important factor for onset of fracture [8]. Therefore, evaluation of the T-stress becomes an important topic in fracture analysis. The weight function method was suggested to evaluate the T-stress at the crack tip [9,10]. The Green’s function method in conjunction with the reference solution was proposed to evaluate SIF (stress intensity factor) and T-stress in the crack problem [11–14]. However, most of those studies were devoted to the single crack case.

* Corresponding author. Tel.: +86 511 88780780. E-mail address: [email protected] (Y.Z. Chen). 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.10.006

Direct use of finite element method for evaluating T-stress was summarized [15]. Using the HCE (hybrid crack element), the higher order terms including the T-stress in a variety of cracked problems were evaluated [16,17]. A quarter-point crack tip element in boundary element method was suggested to evaluate T-stress for some cracked plates [18]. From a limit value for the difference of two stress components ahead the crack tip, the T-stress at the crack tip was evaluated [19]. Eigenfunction expansion variational method (EEVM) was developed for evaluating stress intensity factor and T-stress in a circular cracked plate [20]. After assuming a dislocation distribution along the prospective site of the crack, a singular integral equation is obtained. The T-stress at crack tip can be evaluated from the solution of integral equation [21]. The T-stress problem in the case of multiple cracks was addressed [22]. However, the derivation in the paper is incomplete because necessary equations were not provided. For some particular shapes of cracks, the arc crack and the cusp crack, some solutions for T-stress in a closed form were obtained [23]. A detailed derivation for T-stress dependence on the loading on crack was carried out, which has a form of Dirac Delta function [11,22,24]. In addition, a compendium of T-stress solutions in the crack problems was carried out [25]. The traction on the face of a single crack is assumed in the form of the Chebyshev polynomials of the second kind, and the relevant complex potentials are obtained. Starting with the solution for a single crack with loaded crack faces, a system of equation is derived which allows an approximate determination of stress intensity factors of a system of straight cracks [26]. At that time, it was rare to study the T-stress. A simple method of stress analysis in elastic solids with many cracks was proposed in [27]. It is based

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on the superposition technique and the ideas of self-consistency applied to the average tractions on individual cracks. The method is applicable to two-dimensional crack arrays of arbitrary geometry. In addition, numerical solution proves that the method is useful for evaluating the SIFs in the dilute positions of cracks [28]. To determine the stress intensity factors of a microcrack embedded in a solid containing numerous or even countless microcracks, the solid is divided into two regions. The interaction of microcracks in a circular or elliptical region around the considered microcrack is calculated directly by using Kachanov’s micromechanics method, while the influence of all other microcracks is reflected by modifying the stress applied in the far field. Some conventional micromechanics methods for estimating the effective moduli of microcracked materials are evaluated by comparing with the numerical results [29]. This paper investigates T-stress for interacting cracks in general case. The cracks are located in an arbitrary position with the remote tension r1 y ¼ P o . For evaluating the interaction among many cracks, the multiple crack problem is proposed. The multiple crack problem can be solved by using the superposition method, and it can be reduced to a Fredholm integral equation. After the integral equation is solved, the T-stress at the crack tip can be evaluated. The T-stress is composed of three parts. The first part is coming from the uniform field, or the field caused by remote tension for an infinite plate. The second part and the third part are coming from the perturbation field. The second part is defined as the influence at a crack tip from the crack that possesses the crack tip. The third part is defined as the influence at a crack tip from cracks that do not possess the crack tip. Necessary equations for evaluating three parts are clearly indicated. For the two collinear cracks, numerical solution is presented, which fully coincides with the solution in a closed form. Many results for T-stress for interacting cracks are presented. It is shown that the interaction is significant for the case of closer cracks.

 Note that, the function wðzÞ is an analytic function, which is defined  by wðzÞ ¼ wðzÞ. Except for the physical quantities mentioned above, from Eqs. (3) and (4) two derivatives in a specified direction (abbreviated as DISD) are introduced as follows [31]:

d fY þ iXg ¼ /0 ðzÞ þ /0 ðzÞ dz dz þ ðz/00 ðzÞ þ w0 ðzÞÞ ¼ N þ iT dz d J 2 ðzÞ ¼ 2G fu þ ivg ¼ j/0 ðzÞ  /0 ðzÞ dz dz  ðz/00 ðzÞ þ w0 ðzÞÞ ¼ ðj þ 1Þ/0 ðzÞ  J 1 dz J 1 ðzÞ ¼

ð7Þ

ð8Þ

It is easy to verify that J1 = N + iT denotes the normal and shear tractions along the segment z; z þ dz (Fig. 1). Secondly, the J1 and J2 values depend not only on the position of a point ‘‘z”, but also on the direction of the segment ‘‘dz=dz”. The symbol of derivative d{}/dz is always defined as a derivative in a specified direction (DISD) [31]. For evaluating of the T-stress in crack problem, one more physical quantity J 1 ðzÞ is introduced below. After substituting dz by idz and dz by idz in Eq. (7), the following physical quantity is defined:

J 1 ðzÞ ¼ /0 ðzÞ þ /0 ðzÞ 

dz ðz/00 ðzÞ þ w0 ðzÞÞ ¼ N 1 þ iT 1 dz

ð9Þ

where J1 ðzÞ ¼ N 1 þ iT 1 represents the traction applied on the segment EF which has a rotation angle p/2 with respect to the segment FG (or dz) (Fig. 1). 2.2. Solution for a single crack problem It is assumed that the applied tractions on the crack faces possess the same magnitude and opposite direction, and the remote stresses vanish (Fig. 1). In this case, the boundary conditions are as follows:

2. Theoretical analysis

ðry  irxy Þþ ¼ ðry  irxy Þ ¼ PðtÞ  iQ ðtÞ;

2.1. Basic equations in plane elasticity

where P(t) and Q(t) are the normal and shear tractions applied on the crack faces, and (+) (or ()) denotes the upper (or lower) side of crack, respectively. The boundary value problem defined by Eq. (10) can be reduced to a Hilbert problem. After some manipulations, the following solution is obtained [30,31]:

The fundamentals of the complex variable function method, which plays an important role in plane elasticity, are briefly introduced in what follows [30]. In the method, the stresses (rx, ry, rxy), the resultant forces (X, Y) and the displacements (u, v) are expressed in terms of the complex potentials /(z), w(z) and x(z) such that

UðzÞ ¼ XðzÞ ¼

rx þ ry ¼ 4ReUðzÞ ry  irxy ¼ 2ReUðzÞ þ zU0 ðzÞ þ WðzÞ ¼ UðzÞ 0

þ ðz  zÞU ðzÞ þ XðzÞ

Z

1 2piXðzÞ

a

a

ðjtj < aÞ

ð10Þ

ðPðtÞ  iQ ðtÞÞXðtÞdt tz

ð11Þ

 ðzÞ  zU0 ðzÞ  UðzÞ WðzÞ ¼ X

ð12Þ

ð2Þ 0

f ¼ Y þ iX ¼ /ðzÞ þ z/ ðzÞ þ wðzÞ ¼ /ðzÞ þ ðz  zÞ/0 ðzÞ þ xðzÞ

ð3Þ

2Gðu þ ivÞ ¼ j/ðzÞ  z/0 ðzÞ  wðzÞ ¼ j/ðzÞ  ðz  zÞ/0 ðzÞ  xðzÞ

ð4Þ

where z = x + iy denotes complex variable, G is the shear modulus of elasticity, j = (3  m)/(1 + m) is for the plane stress problems, j = 3  4m is for the plane strain problems, and m is the Poisson’s ratio. In the present study, the plane strain condition is assumed throughout. In Eqs. (2)–(4), the functions x(z), U(z), W(z) and X(z) are defined by [30,31]

 xðzÞ ¼ z/ 0 ðzÞ þ wðzÞ

UðzÞ ¼ /0 ðzÞ;

(σ y − iσ xy ) = P( t ) − iQ( t )

T

G: z+dz F:z

N1 T1

y

E: z-idz

t o 2a

B

x

XðzÞ ¼ x0 ðzÞ;

 ðzÞ  UðzÞ  zU0 ðzÞ WðzÞ ¼ w ðzÞ ¼ X 0

ð5Þ

β

N ±

ð6Þ

Fig. 1. A single crack problem for loadings with same magnitude and opposite direction on crack faces.

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Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

In Eq. (11), we define

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XðzÞ ¼ z2  a2 ;

ðtaking the branch Limz!1 XðzÞ=z ¼ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XðtÞ ¼ X ðtÞ ¼ X ðtÞ ¼ i a2  t2 ; ðjtj < aÞ þ

ð13Þ



ð14Þ

where X+(t) (or X(t)) denotes the value of the function X(z) when the point z is approaching to t from the upper (or lower) side of the crack, respectively. The traction N + iT at a point z on the line FG, from z to z + dz in Fig. 1, can be obtained by simply substituting Eqs. (11) and (12) into Eq. (7), and it can be written as

Z a 1 ½PðtÞ  iQ ðtÞ½Gðz; tÞ þ expð2ibÞGðz; tÞXðtÞdt 2pi a Z a 1  ½PðtÞ þ iQ ðtÞ½ð1  expð2ibÞÞGðz; tÞ 2pi a

N þ iT ¼ 

þ expð2ibÞðz  zÞG0 ðz; tÞXðtÞdt;

ðz 2 FGÞ

1 ; XðzÞðz  tÞ

G0 ðz; tÞ ¼

ð16Þ

ðXðzÞÞ3 ðz  tÞ2

In Eq. (15), b denotes the inclined angle of the segment FG (Fig. 1).  Letting b = 0, z ! t þ o or z ! t o (jtoj < a) in Eq. (15), in either case we will find þ



½Nðto Þ þ iTðt o Þ ¼ ½Nðto Þ þ iTðto Þ ¼ Pðt o Þ  iQ ðto Þ;

ðjt o j < aÞ ð17Þ

Eq. (17) means that the obtained complex potentials satisfy the boundary value condition shown by Eq. (10). Simply because the segment EF which has a rotation angle p/2 with respect to the segment FG (or dz) (Fig. 1), The traction N1 + iT1 at a point z on the segment EF will be (Fig. 1)

Z a 1 ½PðtÞ  iQ ðtÞ½Gðz; tÞ  expð2ibÞGðz; tÞXðtÞdt 2pi a Z a 1 ½PðtÞ þ iQ ðtÞ½ð1 þ expð2ibÞÞGðz; tÞ  2pi a

N1 þ iT 1 ¼ 

 expð2ibÞðz  zÞG0 ðz; tÞXðtÞdt;

ðz 2 EFÞ

Note that, if the factor exp(2ib) in Eq. (15) is changed into exp(2ib), Eq. (18) is obtainable. Eq. (18) is useful to evaluate the interaction of T-stress among cracks.

y

a

ð19Þ

ðfor the left crack tip AÞ ðfor the right crack tip BÞ

ð20Þ

Sometimes, one only obtains a solution for P(t) at the discrete points tj = a cos[(2j  1)p/2M] (j = 1, 2, . . . , M), and needs to find the values of P(a) and P(a). In this case, if the P(tj) (j = 1, 2, . . . , M) values are known beforehand, the P(a) and P(a) values can be evaluated by the following extrapolation formulae [31]: M 1 X ð1ÞjþM Pðt j Þ tanðð2j  1Þp=4MÞ; M j¼1 M 1 X ð1Þjþ1 Pðt j Þ cotðð2j  1Þp=4MÞ PðaÞ ¼ M j¼1

PðaÞ ¼

ð21Þ

where

tj ¼ a cos

ð2j  1Þp ; 2M

ðj ¼ 1; 2; . . . ; MÞ

ð22Þ

In Eqs. (21) and (22), M denotes an integer number used in the quadrature rule [31]. The above-mentioned derivation provides the necessary knowledge for evaluating the T-stress and stress intensity factor in the interacting cracks as well as in the single crack. 2.3. T-stress evaluation for multiple crack problem

ð18Þ

σ∞y = Po

Z

where the subscript A (or B) is for left (or right) crack tip, respectively (Fig. 1). After some manipulations, the T-stresses at the crack tips A and B in Fig. 1 can be evaluated by [23,31]

T B ¼ PðaÞ;

ð15Þ

a2 þ tz  2z2

rffiffiffiffiffiffiffiffiffiffiffi at dt ðPðtÞ  iQ ðtÞÞ aþt a r ffiffiffiffiffiffiffiffiffiffi ffi Z a 1 aþt ðK 1  iK 2 ÞB ¼  pffiffiffiffiffiffi dt ðPðtÞ  iQ ðtÞÞ at pa a

1 ðK 1  iK 2 ÞA ¼  pffiffiffiffiffiffi pa

T A ¼ PðaÞ;

where

Gðz; tÞ ¼

From the applied boundary traction, or P(t) and Q(t), the SIFs at the crack tips A and B in Fig. 1 can be evaluated by [31]

The original problem is shown by Fig. 2a where the remote stress is r1 y ¼ P o . In addition, multiple cracks are placed in an infinite plate. The problem shown by Fig. 2a can be considered as a superposition of two particular problems shown by Fig. 2b and c. In Fig. 2b, the infinite plate without cracks is under the remote

σ∞y = Po

y Same in magnitude and opposite in direction

yk

sk

z ko

Bk

xk αk

Ak

2a k

No cracks =

yj sj

Bj

o

xj αj

z jo Aj

+

2a j

x

o

x

o

x

Fig. 2. Principle of superposition for the solution of the multiple crack problem, (a) the original problem, (b) the loading condition for the uniform field, or an infinite plate with the remote loading r1 y ¼ P o , and (c) the loading condition for the perturbation field, or a cracked plate with loadings on the crack faces.

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Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

tension r1 y ¼ P o and the relevant field is called the uniform field. In Fig. 2c, the applied tractions on cracks are same in magnitude and opposite in direction with those in the uniform field in Fig. 2b, and the relevant field is called the perturbation field. For solving the problem shown by Fig. 2c, the general formulation for the multiple crack problem is introduced below (Fig. 3). For the multiple crack problem shown in Fig. 3a, the boundary conditions can be expressed by

ðry  irxy Þk ¼ pk ðsk Þ  iqk ðsk Þ;

ðjsk j < ak ;

k ¼ 1; 2; . . . ; NÞ

ð23Þ

where pk(sk) and qk(sk) are the normal and tangential stresses which are expressed in the local coordinates xkokyk. For the problem shown by Fig. 2a with remote tension r1 y ¼ Po , the imposed boundary tractions on cracks in the perturbation field will be

pk ðsk Þ  iqk ðsk Þ ¼ ð cos2 ak þ i sin ak cos ak ÞPo ; ðjsk j < ak ;

k ¼ 1; 2; . . . ; NÞ

ð24Þ

The multiple crack problem shown in Fig. 3a can be considered as a superposition of N single crack problems shown by Fig. 3b, with following undetermined tractions on cracks:

ðry  irxy Þk ¼ Pk ðsk Þ  iQ k ðsk Þ;

ðjsk j < ak ;

k ¼ 1; 2; . . . ; NÞ

ð25Þ

Simply considering the interaction between cracks, and using Eqs. (15)–(17), a system of Fredholm integral equations is obtained as follows [31]:

Pk ðsk Þ  iQ k ðsk Þ þ

N Z X 0

N Z X 0

ðjsk j < ak ;

aj

½Pj ðsj Þ þ iQ j ðsj ÞDjk ðsj ; sk Þdsj ¼ pk ðsk Þ  iqk ðsk Þ; ð26Þ

P0

where the symbol means that the term corresponding to j = k has been excluded in the summation. The kernels in Eq. (26) are defined by

X j ðsj Þ C jk ðsj ; sk Þ ¼  ½Gj ðt jk ; sj Þ þ expð2iðaj  ak ÞÞGj ðtjk ; sj Þ; 2pi X j ðsj Þ Djk ðsj ; sk Þ ¼  ½ð1  expð2iðaj  ak ÞÞÞGj ðt jk ; sj Þ 2pi þ expð2iðaj  ak ÞÞðt jk  t jk ÞG0j ðt jk ; sj Þ

y

ð27Þ

p k (s k ) − iq k (s k ) yk sk Bk xk αk 2a k

No crack

sk Ak

Aj

o

ð29Þ

ðk ¼ 1; 2; . . . ; NÞ

ð30Þ

Bj

N

Bk

T A;k2 ¼ Pk ðak Þ ¼

sj

Aj

o

ð31Þ

where sk = acos[(2k  1)p/2M], and Pk(sk) (k = 1, 2, . . . , M) are evaluated from the solution of Eq. (26). In Eq. (29), the part TA,k3 is evaluated from the T-stress influence at the left crack tip Ak from the solutions Pj(sj) of jth crack (j = 1, 2, . . . , k  1, k + 1, N, or not including the term j = k) in the perturbation field (Fig. 3b), which is as follows:

ðk ¼ 1; 2; . . . ; NÞ

ð32Þ

N Z X 0 j¼1

aj

aj

½Pj ðsj Þ  iQ j ðsj ÞC jk ðsj Þdsj þ

N Z X 0 j¼1

þ iQ j ðsj ÞDjk ðsj Þdsj

aj

½Pj ðsj Þ aj

ðk ¼ 1; 2; . . . ; NÞ

ð33Þ

P0

Bj

αj

2a j

M 1 X ð1ÞjþM Pk ðsk Þ tanðð2k  1Þp=4MÞ; M k¼1

ðk ¼ 1; 2; . . . ; NÞ

N1 þ iT 1 ¼

Pj (s j ) − iQ j (s j )

xj

x

In Eq. (29), the part TA,k2 is evaluated from the T-stress influence at the left crack tip Ak from the solution Pk(sk) of kth crack itself in the perturbation field (Fig. 3b). From Eq. (21), this part can be evaluated by

2a k

αj

z jo

ðk ¼ 1; 2; . . . ; NÞ

where

j=1

sj

ð28Þ

In Eq. (29), the part TA,k1 is from the uniform field shown in Fig. 2b. In the case of r1 y ¼ P o , this part is as follows:

αk

=∑

p j (s j ) − iq j (s j ) yj

T A;k ¼ T A;k1 þ T A;k2 þ T A;k3 ;

T A;k3 ¼ ReðN1 þ iT 1 Þ ¼ N1 ;

y

Ak

a2j þ sz  2z2

Also, the meaning of ak, zko, sk and ak (k = 1, 2, . . . , N) has been indicated in Fig. 3. Physically, the kernels Cjk(sj, sk) and Djk(sj, sk) represent the traction influence on the kth crack caused by the traction applied on the jth crack. In fact, two integrals in Eq. (26) can be evaluated in the following way. As mentioned previously, the N + iT influence on a point (sk) of the kth crack caused by the traction on the jth crack can be evaluated by using Eqs. (15) and (16). In the derivation, one simply performs the following substitutions, a ? aj, t ? sj, P(t) ? Pj(sj), Q(t) ? Qj(sj), z ? tjk and b ? ak  aj in Eqs. (15) and (16), two integrals in Eq. (26) are obtained. After the solution for Pk(sk)  iQk(sk)(jskj < ak, k = 1, 2, . . . , N) is obtained from the integral equation (26), the SIFs at the tips of kth crack can be evaluated by using Eq. (19). The T-stress at the crack tip can be evaluated in following way. The T-stress, for example, at the left tip Ak (or sk = ak) of kth crack is denoted by TA,k (Fig. 2a), which is generally composed of three parts

T A;k1 ¼ Po sin ak ;

k ¼ 1; 2; . . . ; NÞ

z ko

G0j ðz; sÞ ¼

; ðX j ðzÞÞ3 ðz  sÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tjk ¼ expðiaj Þðzko þ sk expðiak Þ  zjo Þ; X j ðzÞ ¼ z2  a2j

2

½Pj ðsj Þ  iQ j ðsj ÞC jk ðsj ; sk Þdsj

aj

j¼1

1 ; X j ðzÞðz  sÞ

aj

j¼1

þ

aj

Gj ðz; sÞ ¼

2a j

x

Fig. 3. Principle of superposition used for the perturbation field, (a) many cracks with tractions pk(sk)  iqk(sk) (k = 1, 2, . . . , N) on crack faces, (b) superposition of many individual cracks with tractions Pk(sk)  iQk(sk) (k = 1, 2, . . . , N), on crack faces.

In Eq. (33), the symbol means that the term corresponding to j = k has been excluded in the summation. The kernels in Eq. (33) are defined as

X j ðsj Þ ½Gj ðt jk ; sj Þ  expð2iðaj  ak ÞÞGj ðtjk ; sj Þ; 2pi X j ðsj Þ Djk ðsj Þ ¼  ½ð1 þ expð2iðaj  ak ÞÞÞGj ðt jk ; sj Þ 2pi  expð2iðaj  ak ÞÞðtjk  t jk ÞG0j ðtjk ; sj Þ

ð34Þ

tjk ¼ expðiaj Þðzko  ak expðiak Þ  zjo Þ

ð35Þ

C jk ðsj Þ ¼ 

and the functions Gj(z, s), G0j ðz; sÞ and Xj(z) have been defined previously in Eq. (28).

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Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

  Po 2 EðkÞ 2 P n ðzÞ ¼ c z  KðkÞ 2

In the derivation of Eqs. (33)–(35), Eq. (18) is used. Clearly, the N1 + iT1 influence at the left crack tip (sk = ak) of the kth crack caused by the traction on the jth crack can be evaluated by using Eq. (18). In this case, one simply performs the following substitutions, a ? aj, t ? sj, P(t) ? Pj(sj), Q(t) ? Qj(sj), z ? tjk and b ? ak  aj in Eq. (18), the right hand term in Eq. (33) can be obtained. Similarly, the T-stress at the right crack tip Bk (or sk = ak) in Fig. 2a is denoted by TB,k, which is also composed of three parts

T B;k ¼ T B;k1 þ T B;k2 þ T B;k3 ;

ðk ¼ 1; 2; . . . ; NÞ

2

d 2 c ¼ b þ a; d ¼ b  a; k ¼ 1  2 c Z p=2 2 2 1=2 KðkÞ ¼ ð1  k sin hÞ dh; 0

2

T B;k2 ¼ Pk ðak Þ ¼

1 M

2

2

ð1  k sin hÞ1=2 dh

ð42Þ

0

ð36Þ

ðk ¼ 1; 2; . . . ; NÞ M X

Z p=2

EðkÞ ¼

In Eq. (42), K(k) (E(k)) is the complete elliptical integral of first (second) kind, respectively. In addition, the SIFs at the crack tips C and D are obtained as follows [21,31]:

where

T B;k1 ¼ Po sin ak ;

ð41Þ

ð37Þ

ð1Þjþ1 Pk ðsk Þ cotðð2k  1Þp=4MÞ;

  pffiffiffiffiffiffi 1 EðkÞ ; pc 1  k KðkÞ ! rffiffiffi pffiffiffiffiffiffi 1 c EðkÞ d2 ¼ Po pc  2 ; k d KðkÞ c

K 1A ¼ K 1D ¼ Po

k¼1

ðk ¼ 1; 2; . . . ; NÞ

ð38Þ K 1B ¼ K 1C

Clearly, only if the factor ak in Eq. (35) is replaced by +ak, the part TB,k3 can be evaluated by using Eqs. (32)–(35). Therefore, evaluation of the T-stress and stress intensity factor in the interacting cracks can be carried out from the above-mentioned derivation.

ð43Þ

The T-stress at the crack tip C can be evaluated in the following manner (Fig. 4). Here T-stress at the crack tip C is denoted by TC. In fact, from Eq. (2) we have

rx þ ry ¼ 4ReUþ ðtÞ ðfor z ¼ t þ i0þ ; t > b  a and t ! b  aÞ

3. Particular behaviors for interaction of SIF and T-stress for two collinear cracks

ð44Þ +

In the mode I of fracture, at the point z = t + i0 , t > b  a and t ? b  a, we have rx = TC and ry = Po [23,31]. In addition, at that place, or at z = t + i0+, t > b  a and t ? b  a, the term Pn(z)/X(z) becomes a pure imaginary value. Substituting the mentioned results and Eq. (40) into Eq. (44) yields the T-stress at the crack tip C

The configuration of two collinear cracks is shown in Fig. 4. It is 1 1 assumed that the remote stresses vanishes, or r1 x ¼ ry ¼ rxy ¼ 0. Along the crack face, the following loading is assumed:

ry ¼ Po ; rxy ¼ 0; ðalong the crack faces; or b  a 6 jxj 6 b þ aÞ

T C ¼ Po

ð39Þ

Similarly, we can obtain the T-stresses at the crack tips A, B and D

After some manipulation, the following closed form solution was obtained [31]:

P ðzÞ P UðzÞ ¼ XðzÞ ¼ n  o XðzÞ 2

T A ¼ T B ¼ T D ¼ Po

ð40Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðz2  c2 Þðz2  d Þ;

ðtaking the branch Limz!1 XðzÞ=z2 ¼ 1Þ

y

ð46Þ

The results shown by Eqs. (45) and (46) have also been obtained by Broberg [21]. The problem for two collinear cracks can be considered as a superposition of two single crack problems (Fig. 4a–c). The problem defined by Eq. (39) or shown in Fig. 4a can be solved numerically by using the Fredholm integral equation (26) and Eqs. (29)–(33). In

where

XðzÞ ¼

ð45Þ

σ y = −Po

y1

y2 2b

A

B C

D

x1

x

x2

2a Crack-AB

=

2a Crack-CD

y

y

σ y = P1 ( x1 ) 2b A

2b

B C

D

x1 2a

y 2 σ y = P2 ( x 2 )

y1

x2 2a

A x

B C

D

x1

+ 2a

x2 x 2a

Fig. 4. Principle of superposition for the solution of the two collinear cracks, (a) two collinear cracks with loading on crack faces ry = Po, (b) the crack-AB with loading ry = P1(x1) on crack face, (c) the crack-CD with loading ry = P2(x2) on crack face.

354

Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

Table 1 Non-dimensional SIFs FA(a/b), fA,ex(a/b), FB(a/b) and fB,ex(a/b) for two collinear cracks (see Fig. 4 and Eqs. (47) and (48)). a/b

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

At crack tip A 1.00120 FA 1.00120 fA,ex

1.00462 1.00462

1.01017 1.01017

1.01787 1.01787

1.02795 1.02795

1.04094 1.04094

1.05786 1.05786

1.08107 1.08107

1.11741 1.11741

At crack tip B FB 1.00132 1.00132 fB,ex

1.00566 1.00566

1.01383 1.01383

1.02717 1.02717

1.04796 1.04796

1.08040 1.08040

1.13326 1.13326

1.22894 1.22894

1.45387 1.45387

FA-from numerical solution, fA,ex-from exact solution in a closed form. FB-from numerical solution, fB,ex-from exact solution in a closed form.

computation, M = 25 is assumed in the quadrature rule [31]. After computation, the solution for traction applied on left crack is denoted by P1(x1)(jx1j < a), and on the right crack by P2(x2) (jx2j < a). Clearly, the property P1 (x1) = P2(x1) holds. The computed results at the crack tip A and B for SIFs are expressed as

K 1A ¼ F A ða=bÞP o

pffiffiffiffiffiffi pa;

K 1B ¼ F B ða=bÞPo

pffiffiffiffiffiffi pa

ð47Þ

In addition, the relevant exact solution from Eq. (43) is denoted by

K 1A ¼ fA;ex ða=bÞP o

pffiffiffiffiffiffi pa;

K 1B ¼ fB;ex ða=bÞP o

pffiffiffiffiffiffi pa

ð48Þ

The computed results for FA(a/b), FB(a/b) and those from the exact solution fA,ex(a/b), fB,ex(a/b) are listed in Table 1. In addition, the computed results at the crack tip A and B for Tstresses are expressed as

T A ¼ ðGA1 ða=bÞ þ GA2 ða=bÞÞPo ¼ GA ða=bÞPo ;

ðwith GA ¼ GA1 þ GA2 Þ

T B ¼ ðGB1 ða=bÞ þ GB2 ða=bÞÞPo ¼ GB ða=bÞPo ;

ðwith GB ¼ GB1 þ GB2 Þ

the crack-AB (or P1(x1), jx1j 6 a), and it does not depend on the solution for the crack-CD (or P2(x2), jx2j 6 a). In the meantime, the T-stress at the crack tip A is composed of two parts GA1(a/b) and GA2(a/b). The value GA1(a/b) is equal to P1(a) (see Eq. (20)). However, The value GA2(a/b) depends on the solution for the crack-CD (or P2(x2), jx2j < a). From Table 1, it is found that FA(a/b) = fA,ex(a/b) and FB(a/ b) = fB,ex(a/b), where FA, FB are non-dimensional SIF from numerical solution, and fA,ex, fB,ex are from the exact solution in a closed form. From Table 2 for T-stress, it is found that GA(a/b) = gA,ex(a/b) and GB(a/b) = gB,ex(a/b), where GA, GB are non-dimensional T-stress from numerical solution, and gA,ex, gB,ex are from the exact solution in a closed form. Therefore, the suggested numerical technique provides accurate results for the SIF and T-stress evaluations.

4. Numerical examples

ð49Þ where, for example at the crack tip A, GA1 is the influence derived from the applied loading on the crack-AB, and GA2 is the influence derived from the applied loading on the crack-CD. In addition, the relevant exact solution from Eqs. (45) and (46) is denoted by

T A ¼ g A;ex ða=bÞPo ;

T B ¼ g B;ex ða=bÞP o

ðwith g A;ex ða=bÞ ¼ g B;ex ða=bÞ ¼ 1Þ

ð50Þ

The computed results for GA1(a/b), GA2(a/b), GA(a/b), GB1(a/b), GB2(a/ b), GB(a/b) and those from the exact solution gA,ex(a/b), gB,ex(a/b) are listed in Table 2. From above-mentioned analysis, we can see following particular characters for interaction of SIFs and T-stresses in the two collinear cracks. Clearly, both the SIF evaluation and T-stress evaluation depend on the usage of principle of superposition and the integral equation (26). However, once the integral equation is solved, the SIF at the crack tip A only depend on the solution for

Some numerical examples are given to illustrate the efficiency of the presented method. Particular attention is paid to the interaction between cracks. Example 1. In the first example, two cracks are in a series position with a remote tension r1 y ¼ P o (Fig. 5a). The cracks with length 2a have an inclined angle a with respect to the horizontal line. The spacing between two cracks is denoted by 2c. In computation, M = 25 is assumed in the quadrature rule for the numerical solution of the integral equation [31]. In the condition of c/a = 0.2, 1.0 and 5, and a from 5, 10, . . . to 90 (degree), the calculated results for the T-stresses at the crack tip ‘‘A” and ‘‘B” are expressed as

T A ¼ F A ðc=a; aÞPo ;

T B ¼ F B ðc=a; aÞPo

ð51Þ

In addition, in the case of a single crack without interaction, the Tstresses at the crack tip ‘‘A” and ‘‘B” are expressed as

Table 2 Non-dimensional T-stresses GA1(a/b), GA2(a/b), GA(a/b), gA,ex(a/b), GB1(a/b), GB2(a/b), GB(a/b) and gB,ex(a/b) for two collinear cracks (see Fig. 4 and Eqs. (49) and (50)). 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

At crack tip A GA1 1.00114 0.00114 GA2 1.00000 GA 1.00000 gA,ex

a/b

0.1

1.00418 0.00418 1.00000 1.00000

1.00872 0.00872 1.00000 1.00000

1.01450 0.01450 1.00000 1.00000

1.02138 0.02138 1.00000 1.00000

1.02935 0.02935 1.00000 1.00000

1.03853 0.03853 1.00000 1.00000

1.04941 0.04941 1.00000 1.00000

1.06350 0.06350 1.00000 1.00000

At crack tip B GB1 1.00139 0.00139 GB2 GB 1.00000 1.00000 gB,ex

1.00626 0.00626 1.00000 1.00000

1.01613 0.01613 1.00000 1.00000

1.03353 0.03353 1.00000 1.00000

1.06294 0.06294 1.00000 1.00000

1.11321 0.11321 1.00000 1.00000

1.20441 0.20441 1.00000 1.00000

1.39440 0.39440 1.00000 1.00000

1.95356 0.95356 1.00000 1.00000

GA1 – T-stress influence from crack-AB, GA2 – T-stress influence from crack-CD, GA(=GA1 + GA2) – from numerical solution, gA,ex – from exact solution in a closed form. GB1 – T-stress influence from crack-AB, GB2 – T-stress influence from crack-CD, GB(=GB1 + GB2) – from numerical solution, gB,ex – from exact solution in a closed form.

355

Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

σ ∞y = Po σ ∞y = Po

α

D C

2d B

2a α

D

α

2d α

B 2c

A 2c

2a

C

2a

2a A

c = d sin α

c = d cos α

σ∞y = Po

σ ∞y = Po

C A

D

D 2a

α

C

2h 0.5a

B

2a B

A

2a

2a

2d

Fig. 5. Configurations of two cracks: (a) two inclined cracks in series, (b) two inclined cracks in a stacking position, (c) two cracks in horizontal position, (c1) two cracks in a horizontal position with h/a = 0.2 and a/d = 0.4, (c2) two cracks in a horizontal position with h/a = 0.2 and a/d = 1, and (d) one crack (AB) in a horizontal position and another crack (CD) in rotation.

T A ¼ T B ¼ fA ðaÞPo ¼ fB ðaÞPo ; 2

ðwith f A ðaÞ ¼ fB ðaÞ

2

¼ sin a  cos aÞ

ð52Þ

The computed results for FA(c/a,a) and fA(a) are plotted in Fig. 6, and the computed results for FB(c/a,a) and fB(a) are plotted in Fig. 7. From the computed results we can see the following properties for the T-stress interaction. For the crack tip A in the outer location, the interaction is comparatively small. For example, the curve FA (c/ a,a) in the case of c/a = 0.2 has a less deviation with the curve fA(a). However, for the crack tip B in the inner location, the interaction is comparatively large. The curve FB(c/a, a) in the case of c/a = 0.2 has a significant deviation from the curve fB(a). For example, we have FB(c/a,a)jc/a=0.2a=p/12 = 0.3382 and fB(a)ja=p/12 = 0.8860. However, when c/a = 5 the interactions for the T-stress at both tips A and B are vary small (Figs. 6 and 7). In the case of a = 0 and two cracks are in series (Fig. 5a), even a/ d = 0.99, or (d  a)/d = 0.01, the suggested numerical method provides accurate results, which coincide the exact solution [28].

FA c/a=0.2 FA c/a=5

0.5

fA FA c/a=1

0.0

-0.5

FA(c/a,α) 2

2

fA(α)=sin α-cos α

-1.0

0

20

40

60

80

α (degree) Fig. 6. Non-dimensional T-stresses FA(c/a,a), fA(a) at the crack tip A for two inclined cracks in series (see Fig. 5a and Eqs. (51) and (52)).

Non-dimensional T-stress

1.0

1.0

Non-dimensional T-stress

Example 2. In the second example, two cracks are in a stacking position with a remote tension r1 y ¼ P o (Fig. 5b). The cracks with length 2a have an inclined angle a with respect to the horizontal line. The spacing between two cracks is denoted by 2c. In computation, M = 25 is assumed in the quadrature rule for the numerical solution of the integral equation [31]. The computed results for T-stresses at crack tips A and B are expressed using the same expression (51). Similarly, Eq. (52) is used for T-stresses without considering the interaction. The computed results for FA(c/a,a) and fA(a) are plotted in Fig. 8, and the computed results for FB(c/a,a) and fB(a) are plotted in Fig. 9. In this case, the curve FA(c/a,a) in the case of c/a = 0.2 has some derivation from the curve fA(a). For example, we have FA(c/ a,a)jc/a=0.2a=p/18 = 1.3319 and fB (a)ja=p/18 = 0.9397. In addition, the curve FB(c/a, a) in the case of c/a = 0.2 also has some derivation with the curve fB(a). For example, For example, we have FB(c/a, a)jc/a=0.2a=p/3 = 0.6854 and fB (a)ja=p/3 = 0.5000. In the case that a = 0 and two cracks are in stacking position (Fig. 5b), the computation becomes difficult when d/a becomes a

FB c/a=0.2 0.5

FB c/a=1

0.0

fB

FB c/a=5

-0.5

FB(c/a,α)

-1.0

2

2

fB(α)=sin α-cos α

-1.5

0

20

40

60

80

α (degree) Fig. 7. Non-dimensional T-stresses FB(c/a,a), fB(a) at the crack tip B for two inclined cracks in series (see Fig. 5a and Eqs. (51) and (52)).

Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

rather small value, for example d/a < 005. In fact, it is a complicated problem in the case of stacking cracks when d/a ? 0. However, in the case of c/a P 0.2 with c = dcosa, no illness phenomena in computation have been found. Example 3. In the third example, two cracks are in a horizontal position with a remote tension r1 y ¼ P o (Fig. 5c). The cracks with length 2a have a spacing 2h in vertical direction. The dimension in the horizontal direction is denoted by 2d. In computation, M = 25 is assumed in the quadrature rule for the numerical solution of the integral equation [31]. In the condition of h/a = 0.2, 1.0 and 5, and a/d from 0.05, 0.1, 0.15 to 1.0, the calculated results for the T-stresses at the crack tip ‘‘A” and ‘‘B” are expressed as

T A ¼ F A ðh=a; a=dÞPo ;

T B ¼ F B ðh=a; a=dÞPo

1.0

Non-dimensional T-stress

356

F B c/a=0.2 0.5

F B c/a=1 0.0

F B c/a=5 fB

-0.5

F B (c/a,α )

-1.0

2

-1.5

ð53Þ

0

20

40

ð54Þ

The computed results for FA(h/a,a/d) and fA are plotted in Fig. 10, and the computed results for FB(h/a,a/d) and fB are plotted in Fig. 11. It is found from Fig. 11 that for the crack tip B in the inner location, the interaction is comparatively large. The curve FB(h/a,a/ d) in the case of h/a = 0.2 has a significant deviation from the value fB = 1 (not considering interaction), particularly, in the range 0.4 6 a/d 6 1. For example, we have some FB(h/a,a/d) values as follows: (1) 1.1094 (for h/a = 0.2, a/d = 0.4), (2) 1.7232 (for h/ a = 0.2, a/d = 0.5), (3) 0.1431 (for h/a = 0.2, a/d = 0.6), (4) 0.0143 (for h/a = 0.2, a/d = 0.7), (5) 0.0950 (for h/a = 0.2, a/d = 0.8), respectively. For explaining the geometry situation happened, the configurations for two cases: (1) h/a = 0.2 and a/d = 0.4, (2) h/ a = 0.2 and a/d = 1, are plotted in Fig. 5c1 and c2. In fact, if 0.5 6 a/ d 6 1, two cracks are in stacking position.

80

Fig. 9. Non-dimensional T-stresses FB(c/a,a), fB (a) at the crack tip B for two inclined cracks in a stacking position (see Fig. 5b and Eqs. (51) and (52)).

-0.8

Non-dimensional T-stress

ðwith f A ¼ fB ¼ 1Þ

60

α (degree )

In addition, in the case of a single crack without interaction, the Tstresses at the crack tip ‘‘A” and ‘‘B” are expressed as

T A ¼ T B ¼ fA P o ¼ fB Po ;

2

fB (α )=sin α -cos α

Example 4. In the forth example, two cracks with a remote tension r1 y ¼ P o have the same length 2a (Fig. 5d). The first crack (crackAB) is in horizontal position, and the second crack (crack-CD) has an inclined angle a with respect to the horizontal. In computation, M = 25 is assumed in the quadrature rule for the numerical solution of the integral equation [31].

FA h/a=5 -1.0

fA=-1 FA h/a=1

-1.2

FA h/a=0.2 FA(h/a,a/d)

-1.4

fA=-1 -1.6 0.0

0.2

0.4

0.6

0.8

1.0

a/d Fig. 10. Non-dimensional T-stresses FA(h/a,a/d), fA = 1 at the crack tip A for two cracks in a horizontal position (see Fig. 5c and Eqs. (53) and (54)).

0.4

FB h/a=0.2

0.2

Non-dimensional T-stress

Non-dimensional T-stress

1.0

FA c/a=5

0.5

fA 0.0

FA c/a=1 FA c/a=0.2

-0.5

-1.0

FA(c/a,α)

2

2

fA(α)=sin α-cos α

-1.5

0.0

FB(h/a,a/d)

-0.2

fB = -1

-0.4

FB h/a=1

-0.6 -0.8

FB h/a=5

-1.0

fB = -1

-1.2 -1.4 -1.6 -1.8

0

20

40

60

80

α (degree) Fig. 8. Non-dimensional T-stresses FA(c/a,a), fA (a) at the crack tip A for two inclined cracks in a stacking position (see Fig. 5b and Eqs. (51) and (52)).

0.0

0.2

0.4

0.6

0.8

1.0

a/d Fig. 11. Non-dimensional T-stresses FB(h/a,a/d), fB = 1 at the crack tip B for two cracks in a horizontal position (see Fig. 5c and Eqs. (53) and (54)).

Y.Z. Chen et al. / Computational Materials Science 45 (2009) 349–357

357

5. Conclusions

1.2

Non-dimensional T-stress

1.0 0.8 0.6

fC

0.4

fD FD

0.2 0.0

FC

-0.2 -0.4 -0.6

FB

fA fB

-0.8 -1.0

FA

-1.2 -1.4 0

20

40

60

80

α (degree) Fig. 12. Non-dimensional T-stresses FA(a), FB(a), FC(a), FD(a), fA(a) = fB(a) and fC(a) = fD(a) at the crack tips A, B, C and D for two cracks (see Fig. 5d and Eqs. (55) and (56)).

In the condition of a from 0, 5, 10, . . . to 90 (degree), the calculated results for the T-stresses at the crack tips A, B, C and D are expressed as

T A ¼ F A ðaÞPo ;

T B ¼ F B ðaÞPo ;

T C ¼ F C ðaÞPo ;

T D ¼ F D ðaÞPo ð55Þ

In addition, in the case of a single crack without interaction, the Tstresses at the crack tips A, B, C and D are expressed as

T A ¼ T B ¼ fA P o ¼ fB Po ; ðwith f A ¼ fB ¼ 1Þ T C ¼ T D ¼ fC ðaÞPo ¼ fD ðaÞPo ; 2

ðwith f C ðaÞ ¼ fD ðaÞ ¼ sin a  cos2 aÞ

ð56Þ

The computed results for FA(a), FB(a), FC(a), FD(a), fA = fB = 1 and fC (a) = fD(a) are plotted in Fig. 12. Since the two cracks are in a close location, interactions for Tstress in this example are significant. Except for the component FA(a) and its counterpart fA = 1, the components FB(a), FC(a), and FD(a) have much difference with their counterparts fB = 1 and fC(a) = fD(a). In this case, the iteration for T-stress at the crack tips should not be neglected. In the computation for third and forth examples (Fig. 5c and d), if the two cracks have an extremely closer distance, the computation becomes difficult. For example, the distance 0.5a is replaced by 0.05a in Fig. 5d. However, in the case of configurations used in the examples, no illness phenomena in computation have been found.

In the available literatures for evaluating T-stress, most of them are devoted to the single crack case. However, it is seen from the present study that the interaction for T-stress among cracks is generally significant, particularly, in the case of closer cracks. Therefore, it is important to study the T-stress interaction in the multiple crack case. The formulation for a single crack problem, or the content in Section 2.2, becomes a fundamental in the study. Once the boundary tractions (ry  irxy)± = P(t)  iQ(t) along the crack face is known beforehand, the N + iT influence and the N1 + iT1 influence can be determined by Eqs. (15), (17), (18), respectively. This is an essential step in the present study. After using this fundamental solution, the multiple crack problem can be reduced to a Fredholm integral equation with the boundary tractions as unknown functions. The solution of the integral equation will provide the results for SIFs and T-stresses at the crack tips. It is found in the present study that the interaction for T-stress among cracks is very complicated in general. The interaction has to evaluate by using numerical method. In the previous studies, researches paid attention to the one edge crack or two edge cracks in symmetric position, for example, in [11,14]. Therefore, those suggested method for evaluating the T-stress might not be useful in the present case that is devoted to the interaction of cracks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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