Mode III crack problems of two bonded functionally graded strips with internal cracks

International Journal of Solids and Structures 46 (2009) 331–343

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Mode III crack problems of two bonded functionally graded strips with internal cracks Yen-Ji Chen, Ching-Hwei Chue * Department of Mechanical Engineering, National Cheng Kung University, No.1, Da-Syue Rd, 70101 Tainan, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 13 May 2008 Received in revised form 8 August 2008 Available online 6 September 2008 Keywords: Functionally graded material Stress intensity factor Singular integral equation Gauss–Chebyshev technique

a b s t r a c t This paper deals with the anti-plane problem of two bonded functionally graded finite strips. Each strip contains an internal crack normal to the interface. The material properties of two strips are assumed to vary along the direction of the crack lines. A system of singular integral equations is derived and then solved numerically by using Gauss–Chebyshev integration formula. The influences of nonhomogeneous parameters, crack interactions and two edge conditions on the mode III stress intensity factors are investigated. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction To get rid of the abrupt change of material properties in laminate structures, functionally graded materials (FGMs) can be used to smooth the stress distribution. The crack problems of FGM layered elastic structures become attractive in the field of fracture mechanics. The internal crack or cracks may be parallel or perpendicular to the interface. The studies of Noda and Jin (1993), Fotuhi and Fariborz (2006) and Wang et al. (2003) can be categorized to the former case. Several studies can be referred to the later case. Erdogan et al. (1991) solved the mode III crack problem in bonded two dissimilar homogeneous half planes with a nonhomogeneous interfacial zone. Choi (1996) studied the bonded dissimilar strips with a crack perpendicular to the functionally graded interface under the mode I loading. Erdogan and Wu (1997) solved the plane crack problem for a nonhomogeneous layer containing a crack. Ueda and Mukai (2002) studied the in-plane crack problem of a functionally graded nonhomogeneous interfacial layer. Surface layer with an internal crack is bonded to an interfacial layer and a substrate. Gao et al. (2004) studied the mode I crack problem in a functionally graded orthotropic strip. For the general case, Long and Delale (2004) presented a general problem for an arbitrarily oriented crack in a FGM layer. In these papers, the influence of the nonhomogeneous material properties, the geometry parameters on the stress intensity factors are discussed in detail. The studies of FGM layer structures have also extended to include the transient, viscoelastic and piezoelectric effects, such as Jin and Paulino (2002); Jin et al. (2003); Ueda (2005). In this study, the stress field of two bonded functionally graded material strips is obtained. Each strip contains an internal crack perpendicular to the bonding surface. The material properties vary exponentially along crack line. Fourier transform is used to formulate the mode III crack problem into a system of singular integral equations, which is then solved by using shev Gauss–Chebyshev integration formula. Numerical results are graphically presented to illustrate the effects of

* Corresponding author. Tel.: +886 6 2757575x62165; fax: +886 6 2363950. E-mail address: [email protected] (C.-H. Chue). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.08.031

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the nonhomogeneous parameters, the crack interaction, strip geometry parameters, and the edge boundary conditions on the stress intensity factors. 2. Formulations Fig. 1 shows the geometry of the structure. Two strips with width h1 and h2, respectively, are bonded along the interface x = 0. Each strip contains an internal crack with crack length 2ai0 perpendicular to the interface. The subscript i indicates the FGM strips 1 and 2. The shear loads applied at the crack surfaces are s1(x) and s2(x), respectively. The shear loadss1(x) and s2(x) can be obtained by using the principle of superposition from the external loads applied at infinity y ? ± 1. The shear moduli of both strips are assumed to vary along the x-axis. Under anti-plane deformation, the constitutive and equilibrium equations are as follows:

sjzðiÞ ¼ lðiÞ ðxÞwðiÞ;j

ð1Þ

sjzðiÞ;j ¼ 0

ð2Þ

where j = x, y. The index i in the parentheses stand for the strips 1 and 2. The quantities w(i) and sjz(i) are the anti-plane displacements of strip i, and anti-plane shear stresses, respectively. The shear moduli l(i)(x) are assumed in the following exponential forms:

lð1Þ ðxÞ ¼ l0 expðbxÞ ðx > 0Þ

ð3aÞ

lð2Þ ðxÞ ¼ l0 expðcxÞ ðx < 0Þ

ð3bÞ

where b and c are the nonhomogeneous parameters of strips 1 and 2, respectively. The shear modulus l0 is assigned at the interface. Using Eqs. (3) and (1), the equilibrium Eq. (2) can be rewritten as

o2 wð1Þ o2 wð1Þ þ ox2 oy2 o2 wð2Þ o2 wð2Þ þ ox2 oy2

! þb

owð1Þ ¼0 ox

ð4aÞ

þc

owð2Þ ¼0 ox

ð4bÞ

!

Employing the Fourier transform on Eqs. (4a) and (4b), the solutions for w1 and w2 become

wð1Þ ðx; yÞ ¼ wð2Þ ðx; yÞ ¼

1 2p 1 2p

Z

1

1 Z 1 1

f11 ða; yÞeiax da þ f21 ða; yÞeiax da þ

2

p 2

p

Z

1

0

Z

0

g 11 ðx; aÞ sinðayÞda

ð5aÞ

g 21 ðx; aÞ sinðayÞda

ð5bÞ

1

Since the problem is symmetric with respect to x-axis, only the upper half-plane y > 0 is considered here. From Eqs. (1), (2), (4) and (5) the unknown functions can be obtained as

Fig. 1. Configuration of two bonded cracked FGM strips.

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

8 f11 ða; yÞ ¼ AðaÞ expðm1 yÞ þ A2 ðaÞ expðm3 yÞ > > > < f ða; yÞ ¼ BðaÞ expðm yÞ þ B ðaÞ expðm yÞ 21 2 2 4 > g 11 ðx; aÞ ¼ C 1 ðaÞ expðp1 xÞ þ C 2 ðaÞ expðp2 xÞ > > : g 21 ðx; aÞ ¼ D1 ðaÞ expðq1 xÞ þ D2 ðaÞ expðq2 xÞ

333

ð6aÞ

where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 ¼ m3 ¼  a2 þ iab qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 ¼ m4 ¼  a2 þ iac p1 ¼ b=2  a1 p2 ¼ b=2 þ a1 q1 ¼ c=2 þ a2 q2 ¼ c=2  a2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ a2 þ b2 =4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ¼ a2 þ c2 =4 The roots mj(j = 1, 2  4) are ordered in such a way that Re (m1) < 0, Re (m2) < 0 as a approach minus and plus infinity. From the regularity conditions at y ? 1, the unknown functions in Eqs. (6a) should be rewritten as follows:

8 f11 ða; yÞ ¼ AðaÞ expðm1 yÞ > > > < f ða; yÞ ¼ BðaÞ expðm yÞ 21 2 > g 11 ðx; aÞ ¼ C 1 ðaÞ expðp1 xÞ þ C 2 ðaÞ expðp2 xÞ > > : g 21 ðx; aÞ ¼ D1 ðaÞ expðq1 xÞ þ D2 ðaÞ expðq2 xÞ

ð6bÞ

and A(a), . . ., D2(a) are unknown functions to be obtained from the continuity and boundary conditions. The continuity conditions on the interface x = 0 are

wð1Þ ð0; yÞ ¼ wð2Þ ð0; yÞ

ð7aÞ

sxzð1Þ ð0; yÞ ¼ sxzð2Þ ð0; yÞ

ð7bÞ

The mixed boundary conditions on the y = 0 are as follows:

syzð1Þ ðx; 0Þ ¼ s1 ðxÞ for a1 < x < b1 syzð2Þ ðx; 0Þ ¼ s2 ðxÞ for a2 < x < b2

ð8bÞ

wð1Þ ðx; 0Þ ¼ 0 for 05x5a1 and b1 5x < h1

ð9aÞ

wð2Þ ðx; 0Þ ¼ 0 for h2 < x5a2 and b2 5x50

ð9bÞ

ð8aÞ

The shear loadss1(x) and s2(x) are known functions and are obtained by using the principle of superposition from the external loads applied at infinity y = ± 1. Two dislocation functions are defined as follows (Erdogan, 1985):

o wð1Þ ðx; 0Þ ox o g 2 ðxÞ ¼ wð2Þ ðx; 0Þ ox

g 1 ðxÞ ¼

ð10aÞ ð10bÞ

which must satisfy the following single-valued conditions:

Z

b1

g 1 ðtÞdt ¼

a1

Z

b2

g 2 ðtÞdt ¼ 0

ð11Þ

a2

The unknown functions A (a) and B (a) in Eq. (6b) can be expressed in the form of dislocation functions as

AðaÞ ¼ BðaÞ ¼

i

a i

a

Z Z

b1

g 1 ðtÞeiat dt

ð12aÞ

g 2 ðtÞeiat dt

ð12bÞ

a1 b2

a2

After employing the continuity conditions Eq. (7) and the Fourier inverse transform, we obtain the following relations among four unknown functions C1 (a), . . ., D2 (a):

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D1 ðaÞ þ D2 ðaÞ  C 1 ðaÞ  C 2 ðaÞ ¼ R1 ðaÞ

ð13aÞ

q1 D1 ðaÞ þ q2 D2 ðaÞ  p1 C 1 ðaÞ  p2 C 2 ðaÞ ¼ R2 ðaÞ

ð13bÞ

with

R1 ðaÞ ¼ R2 ðaÞ ¼

Z

a 2a1 p2 a 2a1

b1

a1

Z

b1

a1

g 1 ðtÞep2 t dt 

g 1 ðtÞep2 t dt 

a 2a2 q2

a 2a2

Z

b2

a2

Z

b2

a2

g 2 ðtÞeq2 t dt

ð14aÞ

g 2 ðtÞeq2 t dt

ð14bÞ

Depending on the boundary conditions at the surfaces x = h1 and x = h2, two cases are discussed in this study to get the functions C1, C2, D1, and D2 in terms of the dislocation functions g1 and g2. Case 1: Two edge surfaces are free of traction The two edge surfaces at x = h1 and x = h2 are assumed to be free of traction as

sxzð1Þ ðh1 ; yÞ ¼ sxzð2Þ ðh2 ; yÞ ¼ 0

ð15Þ

From the conditions (15), the following relations can be obtained after performing the Fourier inverse transform:

C 1 ðaÞp1 ep1 h1 þ C 2 ðaÞp2 ep2 h1 ¼ 

a 2a1

D1 ðaÞq1 eq1 h2 þ D2 ðaÞq2 eq2 h2 ¼ 

a 2a2

Z

b1

a1

Z

b2

g 1 ðtÞep1 ðh1 tÞ dt

ð16aÞ

g 2 ðtÞeq2 ðh2 tÞ dt

ð16bÞ

a2

By solving Eqs. (13) and (16) for functions C1, C2, D1, and D2, we have

C 1 ðaÞ ¼

1 f D

aa2 ep2 h1 ½q1 q2 ðeq2 h2  eq1 h2 Þ þ p2 ðq1 eq1 h2  q2 eq2 h2 Þ

R b1 a1

g 1 ðtÞep2 t dt

Rb aa2 ep1 h1 ½p2 ðq1 eq1 h2  q2 eq2 h2 Þ þ q1 q2 ðeq2 h2  eq1 h2 Þ a11 g 1 ðtÞep1 t dt Rb Rb þaa1 p2 ep2 h1 eq2 h2 ðq2  q1 Þ a22 g 2 ðtÞeq2 t dtg þ aa1 p2 ep2 h1 eq1 h2 ðq1  q2 Þ a22 g 2 ðtÞeq1 t dtg Rb C 2 ðaÞ ¼ p21D faa2 ep1 h1 ½p1 p2 ðq2 eq2 h2  q1 eq1 h2 Þ þ q1 q2 p1 ðeq1 h2  eq2 h2 Þ a11 g 1 ðtÞep2 t dt Rb þaa2 ep1 h1 ½p1 p2 ðq1 eq1 h2  q2 eq2 h2 Þ þ q1 q2 p2 ðeq2 h2  eq1 h2 Þ a11 g 1 ðtÞep1 t dt Rb Rb þaa1 p1 p2 ep1 h1 eq2 h2 ðq1  q2 Þ a22 g 2 ðtÞeq2 t dt þ aa1 p1 p2 ep1 h1 eq1 h2 ðq2  q1 Þ a22 g 2 ðtÞeq1 t dtg Rb Rb D1 ðaÞ ¼ D1 faa2 q2 eq2 h2 ep2 h1 ðp1  p2 Þ a11 g 1 ðtÞep2 t dt  aa2 q2 eq2 h2 ep1 h1 ðp1  p2 Þ a11 g 1 ðtÞep1 t dt Rb þaa1 eq2 h2 ½p1 p2 ðep1 h1  ep2 h1 Þ þ q2 ðp2 ep2 h1  p1 ep1 h1 Þ a22 g 2 ðtÞeq2 t dt Rb þaa1 eq1 h2 ½p1 p2 ðep2 h1  ep1 h1 Þ þ q2 ðp1 ep1 h1  p2 ep2 h1 Þ a22 g 2 ðtÞeq1 t dtg Rb Rb D2 ðaÞ ¼ q21D faa2 q1 q2 eq1 h2 ep2 h1 ðp2  p1 Þ a11 g 1 ðtÞep2 t dt þ aa2 q1 q2 eq1 h2 ep1 h1 ðp1  p2 Þ a11 g 1 ðtÞep1 t dt Rb þaa1 q1 eq1 h2 ½p1 p2 ðep2 h1  ep1 h1 Þ þ q2 ðp1 ep1 h1  p2 ep2 h1 Þ a22 g 2 ðtÞeq2 t dt Rb þaa1 q2 eq1 h2 ½p1 p2 ðep1 h1  ep2 h1 Þ þ q1 ðp2 ep2 h1  p1 ep1 h1 Þ a22 g 2 ðtÞeq1 t dtg

ð17aÞ

ð17bÞ

ð17cÞ

ð17dÞ

where

  D ¼ 2a1 a2 a2 ðep1 h1  ep2 h1 Þðq1 eq1 h2  q2 eq2 h2 Þ þ ðeq2 h2  eq1 h2 Þðp1 ep1 h1  p2 ep2 h1 Þ From the load conditions (8) on the crack surfaces, the following equations can be obtained:

8 Rb 9 < p1 a 1 ½k1 ðx; tÞ þ k2 ðx; tÞ þ k3 ðx; tÞ þ k4 ðx; tÞ þ k5 ðx; tÞg 1 ðtÞdt = 1 syzð1Þ ðx; 0Þ ¼ s1 ðxÞ ¼ l0 ebx : þ 1 R b2 ½k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞg ðtÞdt ; 6 7 8 9 2 p a2 8 Rb 9 < p1 a 2 ½k10 ðx; tÞ þ k11 ðx; tÞ þ k12 ðx; tÞ þ k13 ðx; tÞ þ k14 ðx; tÞg 2 ðtÞdt = 2 c x syzð2Þ ðx; 0Þ ¼ s2 ðxÞ ¼ l0 e : þ 1 R b1 ½k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞg ðtÞdt ; 15 16 17 18 1 p a1

ð18aÞ

ð18bÞ

where the kernels ki (x, t), (i = 1, . . ., 18) are given in Appendix A. Separating the singular term of the kernels k1(x, t) and k10(x, t), Eq. (18) may be rewritten as follows:

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

8 R  9  < 1 b1 1 þ h1 ðx; tÞ þ k2 ðx; tÞ þ k3 ðx; tÞ þ k4 ðx; tÞ þ k5 ðx; tÞ g 1 ðtÞdt = p a1 tx syzð1Þ ðx; 0Þ ¼ l0 e : þ 1 R b2 ½k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞg ðtÞdt ; 6 7 8 9 2 p a 8 R 2 9  < 1 b2 1 þ h10 ðx; tÞ þ k11 ðx; tÞ þ k12 ðx; tÞ þ k13 ðx; tÞ þ k14 ðx; tÞ g 2 ðtÞdt = a tx p 2 syzð2Þ ðx; 0Þ ¼ l0 ecx : þ 1 R b1 ½k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞ þ k ðx; tÞg ðtÞdt ; 15 16 17 18 1 p a1 bx

335

ð19aÞ

ð19bÞ

where

h1 ðx; tÞ ¼ Im

(Z

1

0

h10 ðx; tÞ ¼ Im

(Z

) rffiffiffiffiffiffiffiffiffiffiffiffiffi ib iaðtxÞ ð 1 þ  1Þe da

a

) rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ic ð 1 þ  1ÞeiaðtxÞ da

0

a

These two equations are called the singular integral equations of the first kind with simple Cauchy-type singularities. All kernels in Eq. (19) are bounded except the term 1=ðt  xÞ, which contribute the singular effects. The dislocation functions g1 and g2 can be solved numerically by using Gauss–Chebyshev integration formula. Case 2: Two edge surfaces are fixed The two surfaces at x = h1 and x = h2 are assumed to be fixed as follows:

wð1Þ ðh1 ; yÞ ¼ wð2Þ ðh2 ; yÞ ¼ 0

ð20Þ

Following the deriving procedures of Case 1, the following relations can be obtained:

D1 ðaÞeq1 h2 þ D2 ðaÞeq2 h2

Z

b1 a g ðtÞep1 ðh1 tÞ dt 2a1 p1 a1 1 Z b2 a ¼ g ðtÞeq2 ðh2 tÞ dt 2a2 q1 a2 2

C 1 ðaÞep1 h1 þ C 2 ðaÞep2 h1 ¼ 

ð21aÞ ð21bÞ

By solving Eqs. (13) and (21) for functions C1, C2, D1, and D2, we have:

Rb faa2 p1 q1 q2 ep2 h1 ½p2 ðeq2 h2  eq1 h2 Þ þ ðq2 eq1 h2  q1 eq2 h2 Þ a11 g 1 ðtÞep2 t dt Rb aa2 p2 q1 q2 ep1 h1 ½p2 ðeq2 h2  eq1 h2 Þ þ ðq2 eq1 h2  q1 eq2 h2 Þ a11 g 1 ðtÞep1 t dt Rb Rb þaa1 p1 p2 q1 ep2 h1 eq2 h2 ðq1  q2 Þ a22 g 2 ðtÞeq2 t dtg þ aa1 p1 p2 q2 ep2 h1 eq1 h2 ðq2  q1 Þ a22 g 2 ðtÞeq1 t dtg Rb C 2 ðaÞ ¼ D11 faa2 p1 q1 q2 ep1 h1 ½p2 ðeq1 h2  eq2 h2 Þ þ ðq1 eq2 h2  q2 eq1 h2 Þ a11 g 1 ðtÞep2 t dt Rb þaa2 p2 q1 q2 ep1 h1 ½p1 ðeq2 h2  eq1 h2 Þ þ ðq2 eq1 h2  q1 eq2 h2 Þ a11 g 1 ðtÞep1 t dt Rb Rb þaa1 p1 p2 q1 ep1 h1 eq2 h2 ðq2  q1 Þ a22 g 2 ðtÞeq2 t dt þ aa1 p1 p2 q2 ep1 h1 eq1 h2 ðq1  q2 Þ a22 g 2 ðtÞeq1 t dtg Rb Rb D1 ðaÞ ¼ D11 faa2 p1 q1 q2 eq2 h2 ep2 h1 ðp2  p1 Þ a11 g 1 ðtÞep2 t dt  aa2 p2 q1 q2 eq2 h2 ep1 h1 ðp2  p1 Þ a11 g 1 ðtÞep1 t dt Rb þaa1 p1 p2 q1 eq2 h2 ½q2 ðep1 h1  ep2 h1 Þ þ ðp1 ep2 h1  p2 ep1 h1 Þ a22 g 2 ðtÞeq2 t dt Rb þaa1 p1 p2 q2 eq1 h2 ½q2 ðep2 h1  ep1 h1 Þ þ ðp2 ep1 h1  p1 ep2 h1 Þ a22 g 2 ðtÞeq1 t dtg Rb Rb D2 ðaÞ ¼ D11 faa2 p1 q1 q2 eq1 h2 ep2 h1 ðp1  p2 Þ a11 g 1 ðtÞep2 t dt þ aa2 p2 q1 q2 eq1 h2 ep1 h1 ðp2  p1 Þ a11 g 1 ðtÞep1 t dt Rb þaa1 p1 p2 q1 eq1 h2 ½q2 ðep2 h1  ep1 h1 Þ þ ðp2 ep1 h1  p1 ep2 h1 Þ a22 g 2 ðtÞeq2 t dt Rb þaa1 p1 p2 q2 eq1 h2 ½q1 ðep1 h1  ep2 h1 Þ þ ðp1 ep2 h1  p2 ep1 h1 Þ a22 g 2 ðtÞeq1 t dtg C 1 ðaÞ ¼

1 D1

ð22aÞ

ð22bÞ

ð22cÞ

ð22dÞ

where

  D1 ¼ 2p1 p2 q1 q2 a1 a2 ðeq1 h2  eq2 h2 Þðp1 ep2 h1  p2 ep1 h1 Þ þ ðep1 h1  ep2 h1 Þðq2 eq1 h2  q1 eq2 h2 Þ From the load conditions (8) on the crack surfaces, the governing equations for dislocation functions g1 and g2 are the same as Eq. (19) of Case 1 with different kernels given in Appendix B. 3. Degenerated problems The crack problem governed by Eq. (19) can be degenerated to some simple problems by changing the geometric parameters. Three degenerated problems reduced from Case 1 are discussed.

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3.1. Degenerated problem 1 A FGM strip bounded to a FGM medium (-h2 ? 1) If the edge boundary with x = h2 moves to minus infinity (-h2 ? 1), the kernels k8(x, t), k9(x, t), k12(x, t), k13(x, t), k14(x, t), k17(x, t) and k18(x, t) in Eq. (19) go to zero, and the results become:

8 R 9 < 1 b1 ½k1 ðx; tÞ þ k2 ðx; tÞ þ k3 ðx; tÞ þ k4 ðx; tÞ þ k5 ðx; tÞg 1 ðtÞdt = a p 1 syzð1Þ ðx; 0Þ ¼ l0 ebx : þ 1 R b2 ½k ðx; tÞ þ k ðx; tÞg ðtÞdt ; 6 7 2 p a2 ( Z ) Z 1 b2 1 b1 syzð2Þ ðx; 0Þ ¼ l0 ecx ½k10 ðx; tÞ þ k11 ðx; tÞg 2 ðtÞdt þ ½k15 ðx; tÞ þ k16 ðx; tÞg 1 ðtÞdt

p

p

a2

ð23aÞ

ð23bÞ

a1

where the remaining kernels ki (x,t), (i = 1–7, 10, 11, 15, 16) should be modified and are shown in Appendix C. 3.2. Degenerated problem 2 Two bonded cracked FGM half planes (h1 ? 1, h2 ? 1). If the edge boundaries h1 and h2 move to infinity (h1 ? 1, h2 ? 1), the kernels k3(x, t), k4(x, t), k5(x, t), k7(x, t), k8(x, t), k9(x, t), k12(x, t), k13(x, t), k14(x, t), k16(x, t), k17(x, t) and k18(x, t) in Eq. (19) will disappear, and the results become:

(

)  Z 1 1 b2 syzð1Þ ðx; 0Þ ¼ l0 e ½k ðx; tÞg 2 ðtÞdt þ h ðx; tÞ þ k2 ðx; tÞ g 1 ðtÞdt þ p a1 t  x 1 p a2 6 ( Z  )  Z b2 1 1 b1 cx 1 syzð2Þ ðx; 0Þ ¼ l0 e ½k ðx; tÞg 1 ðtÞdt þ h ðx; tÞ þ k11 ðx; tÞ g 2 ðtÞdt þ p a2 t  x 10 p a1 15 bx

Z

1



b1

ð24aÞ ð24bÞ

The kernels ki (x, t), (i = 2, 6, 11, 15) in the above equations are as follows:

Z

1

a2 ðq1  p2 Þ a1 ðtþxÞ da e p2 a1 ðp1  q1 Þ 0 Z 1 2 c b a ðq2  q1 Þ ða2 ta1 xÞ k6 ðx; tÞ ¼ eð2t2xÞ da e q2 a2 ðp1  q1 Þ 0 Z 1 2 c a ðq2  p1 Þ a2 ðtþxÞ k11 ðx; tÞ ¼ e2ðtxÞ da e a2 q2 ðp1  q1 Þ 0 Z 1 2 c b a ðp1  p2 Þ ða1 tþa2 xÞ k15 ðx; tÞ ¼ eð2t2xÞ da e a1 p2 ðp1  q1 Þ 0 b

k2 ðx; tÞ ¼ e2ðtxÞ

It can be seen that the existence of kernels k3(x, t), k4(x, t), k5(x, t), k7(x, t), k8(x, t), k9(x, t), k12(x, t), k13(x, t), k14(x, t), k16(x, t), k17(x, t) and k18(x, t) in Eq. (19) come from the effects of the boundary surfaces (x = h1, h2) acted on the two cracks. 3.3. Degenerated problem 3 (h1 ? 1, h2 ? 1, 2a20 = 0) Consider the case that the boundary surfaces move to infinity and the crack in material 2 vanishes. It becomes a cracked FGM half plane bonded to a FGM half plane. The governing equation Eq. (24a) for g1is reduced to the following form:

syzð1Þ ðx; 0Þ ¼ l0 ebx

1

p

Z

b1

a1



 1 þ h1 ðx; tÞ þ k2 ðx; tÞ g 1 ðtÞdt tx

ð24cÞ

It agrees with Eq. (20) of Erdogan (1985).

4. Solutions of the singular integral equations The solutions of the singular integral equations Eq. (19) with the Cauchy type kernel are:

Gi ðtÞ g i ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt  ai Þðbi  tÞ

ði ¼ 1; 2Þ

ð25Þ

where Gj(t) are bounded functions. The stress intensity factors at the crack tips are obtained as

k3 ðb1 Þ ¼ limþ x!b1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G1 ðb1 Þ 2ðx  b1 Þsyzð1Þ ðx; 0Þ ¼ l0 ebb1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1  a1 Þ=2

ð26aÞ

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G1 ða1 Þ 2ða1  xÞsyzð1Þ ðx; 0Þ ¼ l0 eba1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1  a1 Þ=2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 ðb2 Þ cb2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k3 ðb2 Þ ¼ limþ 2ðx  b2 Þsyzð2Þ ðx; 0Þ ¼ l0 e x!b2 ðb2  a2 Þ=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 ða2 Þ k3 ða2 Þ ¼ lim 2ða2  xÞsyzð2Þ ðx; 0Þ ¼ l0 eca2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x!a2 ðb2  a2 Þ=2 k3 ða1 Þ ¼ lim

337

ð26bÞ

x!a1

ð26cÞ ð26dÞ

In deriving the above equations, the following relations have been used (Muskhhelishvili, 1953):

1

p 1

p

Z

b1

a1 b2

Z

a2

1

g 1 ðtÞ G1 ða1 Þe2pi G1 ðb1 Þ dt ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi þ Other terms tx b1  a1 x  a1 b1  a1 x  b1 1 g 2 ðtÞ G2 ða2 Þe2pi G2 ðb2 Þ dt ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi þ Other terms tx b2  a2 x  a2 b2  a2 x  b2

ð27aÞ ð27bÞ

In order to obtain the specific functions G1 (a1), G1(b1), G2(a2) and G2(b2), we define dimensionless quantities (Erdogan et al., 1973) as follows:

xi ¼

xi  ci0 ; ai0

t i ¼ t i  ci0 ; ai0

 ¼ hi  ci0 ; h i ai0

f i ðt i Þ ¼ g i ðtÞ;

ði ¼ 1; 2Þ

then Eq. (25) become:

F i ðt i Þ fi ðt i Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð1 þ t i Þð1  t i Þ

ði ¼ 1; 2Þ

ð28Þ

where F 1 ðt1 Þ and F 2 ðt2 Þ are related to G1(t) and G2(t). Eq. (19) can be expressed in Chebyshev polynomials, which are expressed in Appendix D. The results of stress intensity factors become:

pffiffiffiffiffiffi k3 ðbi Þ ¼ l0 ebbi ai0 F i ð1Þ p ffiffiffiffiffiffi k3 ðai Þ ¼ l0 ebai ai0 F i ð1Þ

ð29aÞ ð29bÞ

with i = 1, 2. Based on the quadratic extrapolation technique, the unknown crack tip values of Fi(1) and Fi(1) can be obtained by using the values of Fi at nodes 2, 3, 4 and n  1, n  2, n  3, respectively. Here n is the number of collocation points along crack lines. 5. Results and discussions In the following numerical computations, the shear loads s1 and s2 applied on the crack surfaces are assumed to be equal. Two crack lengths 2a20 and 2a10 are also assumed to be equal. The stress intensity factors at the crack tips are normalized as:

kai ¼ kbi ¼

k3 ðai Þ pffiffiffiffiffiffiffi

ð30aÞ

k3 ðbi Þ pffiffiffiffiffiffiffi

ð30bÞ

s2 a20

s2 a20

with i = 1, 2. 5.1. A FGM thin film bonded to a homogeneous elastic substrate (h2?1, c = 0) Consider a practical case that a FGM thin film is bonded to a homogeneous elastic substrate. The material properties of this thin film can be selected to fit the functional requirement. The left crack is located at c20/h1 = 1/5. Fig. 2(a) and (b) show the variations of normalized intensity factors with normalized length c10/h1 at different values of ba10 when the boundary surface x = h1 is free of traction. In Fig. 2(a), kb1 is greater than ka1 when ba10 is positive and vice versa. Because of the crack interaction, ka1 increases as the crack approaches the interface. Consider the case ba10 = 0. Due to the edge effect, kb1 is greater than ka1 when the crack is close to the film surface. As the right crack moves to the interface x = 0, ka1 becomes larger and finally ka1 is greater than kb1 according to the crack interaction effect. For all values of ba10 in Fig. 2(b), a general tendency can be seen that both factors kb2 and ka2 decrease and approach to a constant as c10/h1 increases. It is due to the gradually decay of the crack interaction effect. 5.2. Two bonded FGM cracked-strips Let us go back to the most general case that two cracked strips are bonded together. In the following discussion, the geometry parameters of the left crack are kept unchanged. The ratios of c10/h and h1/h2 are subjected to change in order to study the effects of crack location and edge conditions, respectively.

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Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

Fig. 2. Variations of normalized intensity factors with c10/h1 when c = 0, h2 ?  1.

5.2.1. Effects of crack location Fig. 3(a) and (b) are the variations of normalized intensity factors with c10 /h when ba10 =  ca20 =  0.25, and ba10 =  ca20 = 0.25, respectively. The thickness of the strips h1 =  h2. The geometry parameters of the left crack are h2/c20 = 5 and h2/(2a20) = 3.75. From the FGM definition Eq. (3), the stiffness at the interface is highest when ba10 =  ca20 =  0.25, while it is weakest for ba10 =  ca20 = 0.25. In Fig. 3(a), ka1 and kb2 are greater than kb1 and ka2because of the crack interaction and material effects. As the right crack moves to the right with increasing c10 /h, ka2 and kb2 decrease and reach to a constant value, respectively. Consider the case in Fig. 3(b). Since the material effect dominates, ka2 is always greater than kb2. The difference between them is reduced smaller when crack distance c10 /h is small and crack interaction becomes prominent. Same conclusion can be made to the factors ka1 and kb1. However, due to the edge effect, these two factors increase for Case 1 and decrease for Case 2 when the right crack approaches the boundary x = h1.

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

339

5.2.2. Effects of edge boundary conditions Let us consider the case that the configuration of Fig. 1 remain unchanged except the right boundary surface x = h1. This surface is extended to the right. Under the conditions h2/(2a20) = 3.75 and h2/c20 = h2/c10 = 2, the variations of normalized stress intensity factor kb1 shown in Fig. 4 are examined to study the edge effects come from edge conditions and the normalized length parameter h1/h2.

Fig. 3. Variations of normalized intensity factors with c10/h when (a) ba10 =  ca20 =  0.25, and (b) ba10 =  ca20 = 0.25.

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Fig. 4. Variations of normalized stress intensity factor kb1 with h1 h2.

Fig. 5. Variations of normalized stress intensity factors with ba10 for a crack in an infinite medium.

In both case ba10 =  ca20 =  0.25 and ba10 =  ca20 = 0.25, the normalized stress intensity factor kb1 is higher when the boundary surface is traction-free than that when the boundary surface is fixed. In addition, the factor kb1 decreases when the traction-free boundary surface is moved to the right and vice versa for the fixed boundary surface case.

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

341

5.3. Numerical validation of degenerated problem 3: a crack in a FGM (h1 ? 1, h2 ? 1, 2a20 = 0, c = b) In this section, we will compare the numerical results of degenerated problem 3 with those of Erdogan (1985). Degenerated problem 3 is an infinite FGM containing a crack. Fig. 5 shows the results of the variations of normalized intensity factors with nonhomogeneous parameter ba10. It agrees well with Fig. 2 in the paper Erdogan (1985). 6. Conclusions The fracture behavior of two bonded cracked FGM strips has been studied. The effects of the nonhomogeneous material parameters, crack locations and edge boundary conditions on the stress intensity factors have been emphasized. From the results, it shows that (1) the stress intensity factors decrease with increase in the distance between two cracks; and (2) the stress intensity factors are larger when the crack tip is located in the stiffer elastic medium and is near the free boundary surface. The kernels ki (x, t) (i = 1, 2, . . ., 18) of Case 1 in Eq. (18) are:

Appendix A.

k1 ðx; tÞ ¼

Z

i 2

1

m1

1

b

k2 ðx; tÞ ¼ e2ðtxÞ

Z

a

eiaðtxÞ da

ðA:1Þ

2a2 a2 p1 ep2 h1 ½q1 q2 ðeq2 h2  eq1 h2 Þ þ p2 ðq1 eq1 h2  q2 eq2 h2 Þ a1 ðtþxÞ e da D

ðA:2Þ

2a2 a2 ep1 h1 ½q1 q2 ðeq1 h2  eq2 h2 Þ þ p2 ðq2 eq2 h2  q1 eq1 h2 Þ a1 ðtxÞ e da D

ðA:3Þ

2a2 a2 p1 ep1 h1 ½ðq2 eq2 h2  q1 eq1 h2 Þ þ p1 ðeq1 h2  eq2 h2 Þ a1 ðtxÞ e da D

ðA:4Þ

2a2 a2 p1 ep1 h1 ½ðq1 eq1 h2  q2 eq2 h2 Þ þ p2 ðeq2 h2  eq1 h2 Þ a1 ðtþxÞ e da D

ðA:5Þ

2a2 a1 p2 ep2 h1 eq2 h2 ðq2  q1 Þ ða2 ta1 xÞ e da D

ðA:6Þ

2a2 a1 p2 ep2 h1 eq1 h2 ðq1  q2 Þ ða2 ta1 xÞ e da D

ðA:7Þ

2a2 a1 p1 ep1 h1 eq2 h2 ðq1  q2 Þ ða2 tþa1 xÞ e da D

ðA:8Þ

2a2 a1 p1 ep1 h1 eq2 h2 ðq2  q1 Þ ða2 tþa1 xÞ e da D

ðA:9Þ

1

0

b

k3 ðx; tÞ ¼ e2ðtxÞ

Z

1

0

b

Z

b

Z

k4 ðx; tÞ ¼ e2ðtxÞ

1

0

k5 ðx; tÞ ¼ e2ðtxÞ

1

0

c

b

Z

b

Z

k6 ðx; tÞ ¼ eð2t2xÞ c

k7 ðx; tÞ ¼ eð2t2xÞ c

b

Z

b

Z

k8 ðx; tÞ ¼ eð2t2xÞ c

k9 ðx; tÞ ¼ eð2t2xÞ k10 ðx; tÞ ¼

Z

i 2

1 0 1 0 1 0 1 0

1

m2

1

c

Z

c

Z

k11 ðx; tÞ ¼ e2ðtxÞ

a

eiaðtxÞ da

ðA:10Þ

2a2 a1 eq2 h2 ½p1 p2 ðep1 h1  ep2 h1 Þ þ q2 ðp2 ep2 h1  p1 ep1 h1 Þ a2 ðtþxÞ e da D

ðA:11Þ

2a2 a1 eq1 h2 ½p1 p2 ðep2 h1  ep1 h1 Þ þ q2 ðp1 ep1 h1  p2 ep2 h1 Þ a2 ðtþxÞ e da D

ðA:12Þ

2a2 a1 q1 eq1 h2 ½q1 ðep2 h1  ep1 h1 Þ þ ðp1 ep1 h1  p2 ep2 h1 Þ a2 ðtþxÞ e da D

ðA:13Þ

2a2 a1 eq1 h2 ½p1 p2 ðep1 h1  ep2 h1 Þ þ q1 ðp2 ep2 h1  p1 ep1 h1 Þ a2 ðtþxÞ e da D

ðA:14Þ

2a2 a2 q2 ep2 h1 eq2 h2 ðp1  p2 Þ ða1 tþa2 xÞ e da D

ðA:15Þ

2a2 a2 q2 ep1 h1 eq2 h2 ðp2  p1 Þ ða1 tþa2 xÞ e da D

ðA:16Þ

2a2 a2 q1 ep2 h1 eq1 h2 ðp2  p1 Þ ða1 ta2 xÞ e da D

ðA:17Þ

2a2 a2 q1 ep1 h1 eq1 h2 ðp1  p2 Þ ða1 ta2 xÞ e da D

ðA:18Þ

1

0

k12 ðx; tÞ ¼ e2ðtxÞ

1

0

c

Z

c

Z

k13 ðx; tÞ ¼ e2ðtxÞ

1

0

k14 ðx; tÞ ¼ e2ðtxÞ

1

0

b

c

Z

c

Z

k15 ðx; tÞ ¼ eð2t2xÞ

1

0

b

k16 ðx; tÞ ¼ eð2t2xÞ

1

0

b

c

k17 ðx; tÞ ¼ eð2t2xÞ

Z

1

0

b

c

k18 ðx; tÞ ¼ eð2t2xÞ

Z

0

where

1

342

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  D ¼ 2a1 a2 a2 ðep1 h1  ep2 h1 Þðq1 eq1 h2  q2 eq2 h2 Þ þ ðeq2 h2  eq1 h2 Þðp1 ep1 h1  p2 ep2 h1 Þ Appendix B.

k1 ðx; tÞ ¼

The kernels kj (x, t) (j = 1, 2, . . ., 18) of Case 2 are:

Z

i 2

1

m1

1

b

Z

b

Z

k2 ðx; tÞ ¼ e2ðtxÞ k3 ðx; tÞ ¼ e2ðtxÞ b

k4 ðx; tÞ ¼ e2ðtxÞ b

k5 ðx; tÞ ¼ e2ðtxÞ

Z Z

a

eiaðtxÞ da

ðB:1Þ

2a2 a2 p1 q1 q2 ep2 h1 ½p2 ðeq2 h2  eq1 h2 Þ þ ðq2 eq1 h2  q1 eq2 h2 Þ a1 ðtþxÞ e da D1

ðB:2Þ

2a2 a2 p2 q1 q2 ep1 h1 ½p2 ðeq1 h2  eq2 h2 Þ þ ðq1 eq2 h2  q2 eq1 h2 Þ a1 ðtxÞ e da D1

ðB:3Þ

2a2 a2 p1 q1 q2 ep1 h1 ½p2 ðeq1 h2  eq2 h2 Þ þ ðq1 eq2 h2  q2 eq1 h2 Þ a1 ðtxÞ e da D1

ðB:4Þ

2a2 a2 p2 q1 q2 ep1 h1 ½p1 ðeq2 h2  eq1 h2 Þ þ ðq2 eq1 h2  q1 eq2 h2 Þ a1 ðtþxÞ e da D1

ðB:5Þ

2a2 a1 p1 p2 q1 ðq1  q2 Þep2 h1 eq2 h2 ða2 ta1 xÞ e da D1

ðB:6Þ

2a2 a1 p1 p2 q2 ðq2  q1 Þep2 h1 eq1 h2 ða2 ta1 xÞ e da D1

ðB:7Þ

2a2 a1 p1 p2 q1 ðq2  q1 Þep1 h1 eq2 h2 ða2 tþa1 xÞ e da D1

ðB:8Þ

2a2 a1 p1 p2 q2 ðq1  q2 Þep1 h1 eq1 h2 ða2 tþa1 xÞ e da D1

ðB:9Þ

1 0 1 0 1 0 1 0

c

b

Z

b

Z

k6 ðx; tÞ ¼ eð2t2xÞ

1

0

c

k7 ðx; tÞ ¼ eð2t2xÞ

1

0

c

b

k8 ðx; tÞ ¼ eð2t2xÞ

Z

1

0

c

b

Z

1

k9 ðx; tÞ ¼ eð2t2xÞ

Z

1

0

im2 iaðtxÞ e da 1 2a Z 1 c 2a2 a1 p1 p2 q1 eq2 h2 ½q2 ðep1 h1  ep2 h1 Þ þ ðp1 ep2 h1 k11 ðx; tÞ ¼ e2ðtxÞ D1 0 Z 1 2 q h p h 2 1 1 2 c 2a a1 p1 p2 q2 e ½q2 ðe  ep1 h1 Þ þ ðp1 ep1 h1 k12 ðx; tÞ ¼ e2ðtxÞ D1 0 Z 1 c 2a2 a1 p1 p2 q1 eq1 h2 ½q2 ðep2 h1  ep1 h1 Þ þ ðp2 ep1 h1 ðtxÞ k13 ðx; tÞ ¼ e2 D1 0 Z 1 2 q1 h2 p1 h1 c 2 a a p p q e ½q ðe  ep2 h1 Þ þ ðp1 ep2 h1 1 1 2 2 1 k14 ðx; tÞ ¼ e2ðtxÞ D1 0 Z 1 2 p2 h1 q2 h2 c b 2 a a p q q ðp  p Þe e 2 1 1 2 2 1 k15 ðx; tÞ ¼ eð2t2xÞ eða1 tþa2 xÞ da D 1 0 Z 1 c b 2a2 a2 p2 q1 q2 ðp1  p2 Þep1 h1 eq2 h2 ða1 tþa2 xÞ k16 ðx; tÞ ¼ eð2t2xÞ e da D1 0 Z 1 c b 2a2 a2 p1 q1 q2 ðp1  p2 Þep2 h1 eq1 h2 ða1 ta2 xÞ k17 ðx; tÞ ¼ eð2t2xÞ e da D1 0 Z 1 c b 2a2 a2 p2 q1 q2 ðp2  p1 Þep1 h1 eq1 h2 ða1 ta2 xÞ k18 ðx; tÞ ¼ eð2t2xÞ e da D1 0

k10 ðx; tÞ ¼

ðB:10Þ  p2 ep1 h1 Þ  p2 ep2 h1 Þ  p1 ep2 h1 Þ  p2 ep1 h1 Þ

ea2 ðtþxÞ da

ðB:11Þ

ea2 ðtxÞ da

ðB:12Þ

ea2 ðtxÞ da

ðB:13Þ

ea2 ðtþxÞ da

ðB:14Þ ðB:15Þ ðB:16Þ ðB:17Þ ðB:18Þ

where

  D1 ¼ 2p1 p2 q1 q2 a1 a2 ðeq1 h2  eq2 h2 Þðp1 ep2 h1  p2 ep1 h1 Þ þ ðep1 h1  ep2 h1 Þðq2 eq1 h2  q1 eq2 h2 Þ Appendix C.

k1 ðx; tÞ ¼

Degenerated the edge boundary h2 to minus infinity

i 2 b

Z

1

m1

1

k2 ðx; tÞ ¼ e2ðtxÞ

a

Z

1 0

eiaðtxÞ da q2 ðp2  q1 Þea1 h1 ea1 ðtþxÞ da a1 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1 

ðC:1Þ

Y.-J. Chen, C.-H. Chue / International Journal of Solids and Structures 46 (2009) 331–343

Z

q2 ðq1  p2 Þea1 h1 ea1 ðtxÞ da a1 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  0 Z 1 b p1 ðp1  q2 Þea1 h1 k4 ðx; tÞ ¼ e2ðtxÞ ea1 ðtxÞ da a ½ðp  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  1 0 1 Z 1 b p1 ðp2  q2 Þea1 h1 k5 ðx; tÞ ¼ e2ðtxÞ ea1 ðtþxÞ da a ½ðp  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  1 0 1 Z 1 c b p2 ðq1  q2 Þea1 h1 k6 ðx; tÞ ¼ eð2t2xÞ eða2 ta1 xÞ da a2 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  0 Z 1 c b p1 ðq2  q1 Þea1 h1 k7 ðx; tÞ ¼ eð2t2xÞ eða2 tþa1 xÞ da a2 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  Z 1 0 i m2 iaðtxÞ e da k10 ðx; tÞ ¼ 2 1 a Z 1 a1 h1 c e fp1 p2 ðea1 h1  ea1 h1 Þ  q2 ðp2 ea1 h1  p1 ea1 h1 Þg a2 ðtþxÞ k11 ðx; tÞ ¼ e2ðtxÞ da e a2 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  0 Z 1 a h c b q2 ðp2  p1 Þe 1 1 k15 ðx; tÞ ¼ eð2t2xÞ eða1 tþa2 xÞ da a1 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  0 Z 1 c b q2 ðp1  p2 Þea1 h1 k16 ðx; tÞ ¼ eð2t2xÞ eða1 tþa2 xÞ da a1 ½ðp1  q2 Þea1 h1 þ ðq2  p2 Þea1 h1  0 b

k3 ðx; tÞ ¼ e2ðtxÞ

Appendix D.

1

ðC:2Þ ðC:3Þ ðC:4Þ ðC:5Þ ðC:6Þ ðC:7Þ ðC:8Þ ðC:9Þ ðC:10Þ

Eq. (19) can be solved after reducing them into the following Chebyshev polynomial equations:

8 n 9 8 h i9 P 1 > > 1 > > > > > > F 1 ðt k Þ t x þ p ðh ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t ÞÞ > > 1 r 2 r 4 r 4 r 5 r k k k k k > > = < n r k > > > > k¼1 > > bða x þcÞ r 10 > > s 1 ðxr Þ ¼ l0 e > > n > > P > > > > > > p > > > > þ F 2 ðt k Þ½k6 ðxr ; tk Þ þ k7 ðxr ; tk Þ þ k8 ðxr ; tk Þ þ k9 ðxr ; tk Þ > > ; : > > n > > > > k¼1 > 9> 8 n > > = < h i P > > 1 1 > > F 2 ðt k Þ t x þ p ðh ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t ÞÞ > > 10 r 11 r 12 r 13 r 14 r k k k k k = < n r k > > > > cða20 xr þcÞ k¼1 > > > > > > s2 ðxr Þ ¼ l0 e n P > > > > > > > > p > > > > F ðt Þ½k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ k ðx ; t Þ þ 1 15 r 16 r 17 r 18 r > k k k k k ; : > n > > > > k¼1 > > > > > > > > n n P P > > > > p p ; : F ðt Þ ¼ 0; F ðt Þ ¼ 0 n 1 k n 2 k k¼1

343

ðD1Þ

k¼1

where tk = cos(2k-1)p/2n, (k = 1, 2, . . ., n); xr = cos (rp/n), (r = 1, 2, . . ., n  1) are the nodes satisfy Chebyshev polynomial of the first and second kind respectively, Eq. (D1) involve 2n simultaneous linear equations to solve 2n unknowns Fi(tk) (i = 1, 2 and k = 1, 2, . . ., n). References Choi, H.J., 1996. Bonded dissimilar strips with a crack perpendicular to the functionally graded interface. International Journal of Solid and Structures 33, 4101–4117. Erdogan, F., 1985. The crack problem for bonded nonhomogeneous materials under antiplane shear loading. Transactions of the ASME, Journal of Applied Mechanics 52, 823–828. Erdogan, F., Gupta, G.D., Cook, T.S., 1973. Numerical solution of singular integral equations. In: Sih, G.C. (Ed.), Mechanics of Fracture. 1. Method of Analysis and Solution of Crack Problem. Noordhoff International Publishing, Leyden, The Netherlands. Erdogan, F., Kaya, A.C., Joseph, P.F., 1991. The mode III crack problem in bonded materials with a nonhomogeneous interfacial zone. Transactions of the ASME, Journal of Applied Mechanics 58, 419–427. Chapter 7. Erdogan, F., Wu, B.H., 1997. The surface crack problem for a plate with functionally graded properties. Transactions of the ASME, Journal of Applied Mechanics 64, 449–456. Fotuhi, A.R., Fariborz, S.J., 2006. Anti-plane analysis of a functionally graded strip with multiple cracks. International Journal of Solids and Structures 43, 1239–1252. Gao, L.C., Wu, L.Z., Ma, T.Z., 2004. Mode I crack for a functionally graded orthotropic strip. European Journal of Mechanics 23, 219–234. Jin, Z.H., Paulino, G.H., 2002. A viscoelastic functionally graded strip containing a crack subjected to in-plane loading. Engineering Fracture Mechanics 69, 1769–1790. Jin, B., Soh, A.K., Zhong, Z., 2003. Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip. Archive of Applied Mechanics 73, 252–260. Long, X., Delale, F., 2004. The general problem for an arbitrarily orient crack in a FGM layer. International Journal of Fracture 129, 221–238. Muskhhelishvili, N.I., 1953. Singular Integral Equations. Noordhoff International Publishing, Groningen, The Netherlands. Noda, N., Jin, Z.H., 1993. Thermal stress intensity factors for a crack in a strip of a functionally gradient material. International Journal of Solids and Structures 30, 1039–1056. Ueda, S., 2005. Impact response of a functionally graded piezoelectric plate with a vertical crack. Theoretical and Applied Mechanics 44, 329–342. Ueda, S., Mukai, T., 2002. The surface crack problem for a layered elastic medium with a functionally graded nonhomogeneous interface. JSME International Journal, Series A 45, 371–378. Wang, B.L., Mai, Y.W., Sun, Y.G., 2003. Anti-plane fracture of a functionally graded material strip. European Journal of Mechanics A/Solids 22, 357–368.