A general modelling method for functionally graded

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A general modelling method for functionally graded materials with an arbitrarily oriented crack ab

a

Zhihai Wang , Licheng Guo & Li Zhang

a

a

Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, P.R. China b

Institute of Architecture Engineering, Harbin University of Science and Technology, Harbin 150001, P.R. China Published online: 25 Nov 2013.

To cite this article: Zhihai Wang, Licheng Guo & Li Zhang , Philosophical Magazine (2013): A general modelling method for functionally graded materials with an arbitrarily oriented crack, Philosophical Magazine, DOI: 10.1080/14786435.2013.863437 To link to this article: http://dx.doi.org/10.1080/14786435.2013.863437

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Philosophical Magazine, 2013 http://dx.doi.org/10.1080/14786435.2013.863437

A general modelling method for functionally graded materials with an arbitrarily oriented crack

Downloaded by [Harbin Institute of Technology] at 18:39 27 November 2013

Zhihai Wanga,b, Licheng Guoa* and Li Zhanga* a Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, P.R. China; bInstitute of Architecture Engineering, Harbin University of Science and Technology, Harbin 150001, P.R. China

(Received 15 October 2012; accepted 1 November 2013) In order to analytically solve crack problems regarding functionally graded materials (FGMs), some ideal assumptions are often made. They are: (1) the properties of FGMs are usually assumed to be described by very particular functions; (2) the crack is assumed to be vertical to (or parallel to) the gradient of FGMs. However, these assumptions may not be practical for actual FGMs. Since the controlling differential equations with general mechanical properties are very difficult to solve and the arbitrarily oriented crack causes great trouble in the analytical procedure, a general piecewise-exponential model (GPE model) is proposed to investigate the fracture behaviour of FGMs with general mechanical properties and an arbitrarily oriented crack. “General mechanical properties” means that the mechanical properties in the GPE model are not required to be very particularly pre-defined functions but arbitrary functions determined by fitting the experimental results of FGMs. The studied FGMs are divided into some sub-layers with each layer’s properties varying exponentially so that the general mechanical properties can be approximated by a series of exponential functions and hence the stresses and displacements of each layer which may contain a mixed-mode crack can be solved analytically. By use of integral transform methods, principle of superposition, residual theorem and theory of singular integral equations, the mixed-mode crack problem can be turned into solving a group of singular integral equations from which mixed-mode stress intensity factors (SIFs) can be obtained. Finally, the influences of the nonhomogeneous and geometric parameters on the mixed-mode SIFs are analysed. Keywords: functionally graded materials; fracture mechanics; general piecewise-exponential model; crack; general mechanical properties

1. Introduction One way to improve the bonding strength and toughness and reduce the mismatch of material properties in composites is to process materials with continuous mechanical properties. These materials are known as functionally graded materials (FGMs). Many challenging problems have been addressed in the fracture behaviours of FGMs [1]. This is because the actual material properties of the FGMs may be arbitrarily distributed with spatial position and the fracture behaviours of FGMs are affected greatly by geometric *Corresponding authors. Email: [email protected] (L. Guo); [email protected] (L. Zhang) © 2013 Taylor & Francis


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parameters of the crack. This means that the fracture mechanics investigations of FGMs with arbitrary mechanical properties and an arbitrary oriented crack is significant to the design of FGMs structures and manufacture of lighter and tougher FGMs. Many crack problems in FGMs have been studied in the past years. The stress fields at the crack tip in nonhomogenous materials or FGMs have been investigated by many researchers [2–7]. Jin and Noda [5] pointed out that the singularities and the angular distribution of the crack tip field for the nonhomogeneous materials are identical to those in homogeneous materials provided that the properties of the materials are continuous and differentiable and some of them do not vanish at the crack tip. Parameswaran and Shukla [7] found that the crack-tip stress fields retains the inverse square root singularity and only the higher order terms in the expansion are influenced by the material nonhomogeneity when they investigated the dynamic fracture behaviour of FGMs. Crack-tip stress fields for a stationary crack along or inclined to the direction of linearly varying property gradation were obtained by Jain et al. [3]. For actual FGMs with general mechanical properties, the governing elasticity equations may become partial differential equations with variable coefficients. It is very difficult to obtain the analytical solutions of these governing differential equations. Therefore, in order to solve them analytically, it was often assumed that the FGMs’ moduli vary exponentially with coordinates in many papers [2,8–18]. The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate was investigated by Chen [8]. The surface crack problem of the FGMs was studied by Erdogan and Wu [11]. Guo et al. [13] studied the plane crack problem of a functionally graded coating–substrate system under a concentrated load. The effects of combining FGMs with different inhomogeneous property gradients on the mode-3 propagation characteristics of an interfacial crack were numerically investigated by Kubair and Bhanu-Chandar [19]. A functionally graded layered structure with a crack crossing the interface was studied analytically by Guo and Noda [12]. The dynamic anti-plane problem for a functionally graded magneto-electro-elastic strip containing an internal crack perpendicular to the boundary was investigated by Feng and Su [20]. It should be pointed out that the mechanical properties of functionally graded magneto-electro-elastic materials are particular functions in most of the published papers. Thus, a general modelling method may be significant to investigate the fracture behaviour of functionally graded magneto-electro-elastic materials with arbitrarily distributed properties. It should be mentioned that most of the above mentioned analytical investigations assume that the crack is vertical to (or parallel to) the gradient of FGMs. There are only a few analytical investigations of an arbitrarily oriented crack in FGMs. Konda and Erdogan [14] and Long and Delale [16] investigated an arbitrarily oriented crack in an infinite FGM and a FGM strip, respectively. A general methodology was constructed for the fundamental solution of an arbitrarily oriented crack embedded in an infinite nonhomogeneous plate by Shbeeb et al. [17,18]. However, due to the difficulty in deriving the analytical solutions, all their work assumed the moduli to be particular exponential functions so that analytical solutions could be obtained. However, the above assumed particular material properties may not be practical for actual FGMs. How to study a mixed-mode crack problem of FGMs with arbitrary mechanical properties is significant for the fracture mechanics analysis and design of FGMs. For FGMs with arbitrarily distributed properties, the crack problems are very difficult to solve analytically. Up to now, there are mainly three types of multi-layered


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models [1,3,11,12,16,25,26]. Firstly, a homogenous multi-layered model was adopted by Itou [21], Jin and Paulino [22] and Wang et al. [23]. The main idea of the homogenous multi-layered model is that the FGMs strip was divided into many sub-layers with piecewise constant material properties. In this model, the material properties are not continuous at the interfaces of sub-layers. The second model is referred to as linear multi-layered model [24,25] which divides FGMs into some sub-layers and the mechanical properties in each sub-layer are assumed to be a linear function and continuous at the interfaces between sub-layers. The third model (piecewise-exponential model (PE model)) was proposed by Guo and Noda [1] to investigate the crack problem of FGMs with general mechanical properties. Since the mechanical properties of each layer in the PE model are described by exponential functions, it is very easy to obtain analytical solutions for each layer. In the last two models, the continuous material properties of FGMs can be kept at the interfaces between sub-layers and the problem can be solved with high efficiency. It should be pointed out that, to the authors’ knowledge, all the above mentioned multi-layered models have only considered particular crack forms (cracks vertical or parallel to the gradient of FGMs). As for the general modeling method for the arbitrarily oriented crack problem in FGMs with general mechanical properties, due to the complexity of the analytical procedure, all the above mentioned three piecewise models have not been expanded to the crack problem. Some further mathematical difficulty needs to be overcome and new work needs to be done. Based on the above analysis, a general modelling method for FGMs with general mechanical properties and an arbitrarily (generally) oriented crack is needed. In this paper, a general piecewise-exponential model (GPE model) is proposed to investigate the fracture behaviour of FGMs with general mechanical properties and an arbitrarily oriented crack. It may be thought as significant expansion of the authors’ previously published paper [1] in which only a crack (mode-I) parallel to the gradient of the FGMs was considered by a piecewise-exponential model (PE model). A feasible mathematical procedure is developed for the PE model to be expanded to GPE model. In the GPE model, the studied FGMs are divided into some sub-layers with each layer’s properties varying exponentially so that general mechanical properties can be approximated by a series of exponential functions. Then, analytical solutions of the stresses and displacements of each layer which may contain a mixed-mode crack can be solved analytically. According to the principle of superposition, the stresses of each layer can be divided into two parts. Both of them can be solved from the controlling differential equations using the integral transform method. Then, applying boundary and continuity conditions, the mixed-mode crack problem can be turned into solving a group of singular integral equations so that mixed-mode stress intensity factors (SIFs) can be evaluated. Finally, the influences of the nonhomogeneous and geometric parameters on the mixed-mode SIFs are analysed. 2. GPE model The problem under consideration is illustrated in Figure 1. An infinitely long FGM strip with a finite thickness L along x-axis and L0 denote the distance between the upper surface of the strip and the y-axis. There is an arbitrarily oriented crack of length 2c and crack angle θ in the FGM strip. a0 and b0 denote x0-coordinates of both crack-tips,

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Figure 1. Schematic diagram of functionally graded strip with an arbitrarily oriented crack.

respectively. Without loss of generality, the x0 axis of the coordinate system (x0 − y0) can be arranged along the crack line. The total displacement components in the x-direction and y-direction are denoted by utotal(x, y) and vtotal(x, y), respectively. According to Hooke’s law, the constitutive relation can be written as [1] 8 lðxÞ @utotal @vtotal > > total > ½1 þ kðxÞ þ ½3 kðxÞ rxx ¼ > > kðxÞ 1 @x @y > > < total total lðxÞ @v @u (1) ½1 þ kðxÞ þ ½3 kðxÞ ¼ rtotal yy > kðxÞ 1 @y @x > > > > @utotal @vtotal > > þ : stotal xy ¼ lðxÞ @y @x where μ(x) is shear modulus, k(x) = 3 − 4υ(x) for plane strain, k(x) = [3 − υ(x)]/[1 + υ(x)] for plane stress and υ(x) is the Poisson’s ratio. The equilibrium equations are 8 total > @stotal > > @rxx þ xy ¼ 0 < @x @y (2) total total > @r @s xy > yy > þ ¼0 : @y @x The boundary conditions of the FGM strip are total rxx ðL0 ; yÞ ¼ 0 stotal xy ðL0 ; yÞ ¼ 0 (

1\y\1

(3)

rtotal xx ðL L0 ; yÞ ¼ 0 stotal xy ðL L0 ; yÞ ¼ 0

1\y\1

(4)

rtotal y0 y0 ðx0 ; 0Þ ¼ f1 ðx0 Þ total sx0 y0 ðx0 ; 0Þ ¼ f2 ðx0 Þ

a0 \X0 \b0

(5)

where f1(x0) and f2(x0) are known functions depicting the loading conditions on the crack face and can be determined from actual loading conditions. For the actual crack problem of FGMs, since the material properties of FGMs may be arbitrarily distributed along the thickness direction and the crack may be arbitrarily oriented, it is very



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Figure 2. The GPE model: (a) geometric schematic diagram of the GPE model and (b) schematic diagram of the approximated material property in the GPE model.

difficult to obtain an analytical solution of Equation (2). In order to solve the problem, the FGMs strip is divided into M layers in the x-direction. Each sub-layer is marked by a subscript n (n = 1, 2, …, M) as shown in Figure 2(a). The symbols hn−1 and hn denote the x-coordinate of the upper and lower surfaces of the nth layer, respectively. Thus, the nth layer is located between the region x = hn−1 and x = hn, and the thickness of the nth layer is hn − hn−1. It should be noted that h0 = −L0 and hM = L − L0. If the modulus of each layer is assumed to be an exponential function, the continuously varied mechanical properties of actual FGMs can be approximated by a series of exponential functions. Figure 2(b) gives the schematic of the approximated material’s property in the piecewise-exponential model. The actual shear modulus which may be obtained by fitting the experimental results of the FGM strip with general mechanical properties can be defined as an arbitrary function lðxÞ ¼ f ðxÞ

h0 x h M

(6)

Thus, according to the above description of the piecewise-exponential model, the shear modulus of each layer can be described by an exponential function as ln ðxÞ ¼ ln0 edn x

h0 x hM

n ¼ 1; 2; . . .; M

(7)

where μn0 and δn (n = 1, 2, …, M) are nonhomogeneous parameters of each layer. In the local coordinates (x0 − y0), we have

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Z. Wang et al. ln ðx0 ; y0 Þ ¼ ln0 edn x ¼ ln0 ebn x0 þcn y0

h0 x hM

n ¼ 1; 2; . . .; M

(8)


where βn = δncos θ and γn = −δnsin θ. In order to approach the actual property accurately, the actual mechanical property of the FGM should be applied on both surfaces of each layer. Thus, we have ln ðhn1 Þ ¼ f ðhn1 Þ ¼ ln0 edn hn1 n ¼ 1; 2; . . .; M (9) ln ðhn Þ ¼ f ðhn Þ ¼ ln0 edn hn Then μn0 and δn can be derived from Equation (9) as 1 f ðhn Þ ln dn ¼ hn hn1 f ðhn1 Þ ln0 ¼ f ðhn Þedn hn

(10) (11)

According to Equations (1) and (7), the constitutive relations for each layer can be expressed as 8 ln ðxÞ @utotal @vtotal > n n > total > ½1 þ k þ ½3 k ¼ ðxÞ ðxÞ r n n > nxx > kn ðxÞ 1 @x @y > > < total total l ðxÞ @v @u n n n n ¼ 1; 2; . . .; M (12) ½1 þ k þ ½3 k ¼ ðxÞ ðxÞ rtotal n n nyy > kn ðxÞ 1 @y @x > > > > @utotal @vtotal > n > þ n : stotal nxy ¼ ln ðxÞ @y @x Since the total variation range of the Poisson’s ratio is usually small and in one sub-layer the variation is even smaller, it can be assumed to be a constant as many researchers have done [9,12,13,17,18]. Therefore, we have kn(x) = k = 3 − 4υ (plane strain) and kn(x) = k = (3 − υ)/(1 + υ) (plane stress). The equilibrium equations for each layer can be written as 8 total > @stotal > > @rnxx þ nxy ¼ 0 < @x @y n ¼ 1; 2; . . .; M (13) total total > @r @s nyy nxy > > þ ¼0 : @y @x Equations (3)–(5) can be expressed as ( rtotal 1xx ðh0 ; yÞ ¼ 0 stotal 1xy ðh0 ; yÞ ¼ 0 (

rtotal Mxx ðhM ; yÞ ¼ 0 stotal Mxy ðhM ; yÞ ¼ 0

(

rtotal ny0 y0 ðx0 ; 0Þ ¼ f1 ðx0 Þ stotal nx0 y0 ðx0 ; 0Þ ¼ f2 ðx0 Þ

1\y\1

(14)

1\y\1

(15)

a0 \x0 \b0

(16)


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On both surfaces of the nth layer in the piecewise-exponential model, the continuity conditions can be expressed as ( total rtotal nxx ðhn ; yÞ ¼ rðnþ1Þxx ðhn ; yÞ n ¼ 1; 2; . . .; M 1 (17) total stotal nxy ðhn ; yÞ ¼ sðnþ1Þxy ðhn ; yÞ


(

total utotal n ðhn ; yÞ ¼ uðnþ1Þ ðhn ; yÞ total vtotal n ðhn ; yÞ ¼ vðnþ1Þ ðhn ; yÞ

n ¼ 1; 2; . . .; M 1

(18)

Based on the above analysis, Equations (1)–(5) have been transformed into Equations (12)–(18). Therefore, the arbitrarily oriented crack problem of FGMs with general mechanical properties can be investigated by analysing Equations (12)–(18). According to the principle of superposition, for each layer, the solutions of Equations (12)–(18) can be divided into two parts: let rInx0 x0 ðx0 ; y0 Þ, rIny0 y0 ðx0 ; y0 Þ and sInx0 y0 ðx0 ; y0 Þ denote the stress components of the first part which will be solved in the local coordinate system (x0 − y0) and let rIInxx ðx; yÞ, rIInyy ðx; yÞ and sIInxy ðx; yÞ denote stress components of the second part which will be solved in global coordinate system (x − y). The corresponding displacement components are uIn ðx0 ; y0 Þ, vIn ðx0 ; y0 and uIIn ðx; yÞ, vIIn ðx; yÞ, respectively. In global coordinate system (x − y), the total stresses of the mixed-mode crack problem for the nth layer can be written as 8 > rtotal ðx; yÞ ¼ rIInxx ðx;yÞ þ rInx0 x0 ðx0 ;y0 Þ cos2 h þ rIny0 y0 ðx0 ; y0 Þsin2 h 2sInx0 y0 ðx0 ; y0 Þsin h cosh > < nxx 2 II I I 2 I rtotal nyy ðx; yÞ ¼ rnyy ðx;yÞ þ rnx0 x0 ðx0 ;y0 Þ sin h þ rny0 y0 ðx0 ; y0 Þcos h þ 2snx0 y0 ðx0 ; y0 Þsin h cosh > > : stotal ðx; yÞ ¼ sII ðx;yÞ þ rI ðx ;y Þ sin hcos h rI ðx ;y Þ sin hcos h þ sI ðx ; y Þðcos2 h sin2 hÞ nxy

nx0 x0

nxy

0

0

ny0 y0

0

0

nx0 y0

0

0

n ¼ 1; 2; .. .; M

ð19Þ

Similarly, in the local coordinate system (x0 − y0), the total stresses can be expressed as

8 > rtotal ðx ; y Þ ¼ rInx0 x0 ðx0 ;y0 Þ þ rIInxx ðx; yÞcos2 h þ rIInyy ðx; yÞ sin2 h þ 2sIInxy ðx; yÞ sinh cos h > < nx0 x0 0 0 2 I II II 2 II rtotal ny0 y0 ðx0 ; y0 Þ ¼ rny0 y0 ðx0 ;y0 Þ þ rnxx ðx; yÞsin h þ rnyy ðx;yÞcos h 2snxy ðx; yÞ sinh cos h > > : stotal ðx ;y Þ ¼ sI ðx ; y Þ rII ðx;yÞ sin h cos h þ rII ðx; yÞ sin hcos h þ sII ðx;yÞðcos2 h sin2 hÞ nx0 y0

0

0

nx0 y0

0

0

nxx

nyy

nxy

n ¼ 1; 2;. . .;M

ð20Þ

where x = x0 cos θ − y0 sin θ and y = x0 sin θ + y0 cos θ. The total displacement components total utotal n ðx; yÞ and vn ðx; yÞ of the nth layer can be written as ( II I I utotal n ðx; yÞ ¼ un ðx; yÞ þ un ðx0 ; y0 Þ cos h vn ðx0 ; y0 Þ sin h (21) II I I vtotal n ðx; yÞ ¼ vn ðx; yÞ þ un ðx0 ; y0 Þ sin h vn ðx0 ; y0 Þ cos h

2.1. Solutions of the stress and displacement fields For the stress fields of the first part, the constitutive relations for each sub-layer in local coordinate system (x0 − y0) can be expressed as

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8 ln ðx0 ; y0 Þ @uIn ðx0 ; y0 Þ @vIn ðx0 ; y0 Þ > > I > r ðx ; y Þ ¼ þ ð3 kÞ ð1 þ kÞ > nx0 x0 0 0 > k1 @x0 @y0 > > < ln ðx0 ; y0 Þ @vIn ðx0 ; y0 Þ @uIn ðx0 ; y0 Þ I rny0 y0 ðx0 ; y0 Þ ¼ þ ð3 kÞ ð1 þ kÞ > k1 @y0 @x0 > > > I I > I @u ðx @v ðx 0 ; y0 Þ 0 ; y0 Þ > n n > þ : snx0 y0 ðx0 ; y0 Þ ¼ ln ðx0 ; y0 Þ @y0 @x0

n ¼ 1; 2; . . .; M


(22)

The governing equations are 8 I @rnx0 x0 ðx0 ; y0 Þ @sInx0 y0 ðx0 ; y0 Þ > > þ ¼0 < @y0 @x0 @rI ðx ; y Þ @sI ðx ; y Þ > > : ny0 y0 0 0 þ nx0 y0 0 0 ¼ 0 @y0 @x0

n ¼ 1; 2; . . .; M

(23)

Substituting the constitutive Equation (22) into the governing Equation (23) and applying Fourier transform to the x0 variable, the characteristic equation can be obtained as 8scn bn i I 2 2k3 I 4 I 3 I 2 2 2 ðkn Þ þ 2cn ðkn Þ þ ðcn Þ 2s 2sbn i þ ðbn Þ ðk Þ 2s cn þ kn kþ1 n kþ1 k3 þ s2 ðbn i þ sÞ2 ðcn Þ2 ¼0 n ¼ 1; 2; . . .; M ð24Þ kþ1 Here, s is the Fourier variable. The characteristic roots kInj (n = 1, 2, …, M; j = 1, …, 4) can be derived from Equation (24) and are given in Appendix 1. Using inverse Fourier transform, the expressions of the displacement components can be written as 8 Z 4 I > 1 1X > I > u ðx ; y Þ ¼ EI ðsÞUInj ðsÞeknj ðsÞy0 isx0 ds 0 0 > n < 2p 1 j¼1 nj n ¼ 1; 2; . . .; M (25) Z 1X 4 > > I I kInj ðsÞy0 isx0 > vI ðx0 ; y0 Þ ¼ 1 F ðsÞU ðsÞe ds > nj : n 2p 1 j¼1 nj I Here, s is the Fourier variable and the expressions Enj ðsÞ and FnjI ðsÞ (n = 1, 2,…,M; j = 1,…,4) are given in Appendix 1. Substituting Equation (25) into the constitutive Equation (22), the stress components can be written as 8 Z 1X 4 > I > > rI ðx0 ; y0 Þ ¼ ebn x0 þcn y0 1 BInj ðsÞUInj ðsÞeknj ðsÞy0 isx0 ds > nx0 x0 > > 2p > 1 j¼1 > > > Z 1X 4 < I 1 CnjI ðsÞUInj ðsÞeknj ðsÞy0 isx0 ds rIny0 y0 ðx0 ; y0 Þ ¼ ebn x0 þcn y0 n ¼ 1; 2; . . .; M 2p 1 j¼1 > > > > Z > 4 > I > 1 1X > > sInx0 y0 ðx0 ; y0 Þ ¼ ebn x0 þcn y0 DInj ðsÞUInj ðsÞeknj ðsÞy0 isx0 ds > : 2p 1 j¼1

(26) where the known expressions BInj ðsÞ, CnjI ðsÞ and DInj ðsÞ (n = 1, 2, …, M; j = 1, …, 4) are given in Appendix 1. Without loss of generality, the above characteristic roots can

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ð1Þ

ð1Þ

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ð1Þ

be arranged as Reðkn1 ; kn2 Þ\0 and Reðkn3 ; kn4 Þ [ 0. To satisfy regularity conditions, uIn and vIn must vanish when x20 þ y20 ! 1. Thus, the unknown functions UInj ðsÞ (n = 1,2, …, M; j = 1, 2) are equal to zero when y0 < 0, and UInj ðsÞ (n = 1,2, …, M; j = 3, 4) are equal to zero when y0 > 0. The unknown functions UInj ðsÞ (n = 1, 2, …, M; j = 1, …, 4) can be determined by introducing new auxiliary functions and applying the stress continuous conditions at both crack faces. In order to satisfy the normal and shear stress continuity conditions at y0 = 0, we have lim rIny0 y0 ðx0 ; y0 Þ ¼ lim rIny0 y0 ðx0 ; y0 Þ

n ¼ 1; 2; . . .; M

(27)

lim sInx0 y0 ðx0 ; y0 Þ ¼ lim sInx0 y0 ðx0 ; y0 Þ

n ¼ 1; 2; . . .; M

(28)


y0 !0þ

y0 !0þ

y0 !0

y0 !0

where 0+ is for y0 > 0 and 0− is for y0 < 0. At the crack faces, the following new auxiliary functions are introduced. 8 @ > I I > > u ðx Þ ¼ lim u ðx ; y Þ lim u ðx ; y Þ 0 0 < 1 0 n 0 0 y0 !0 @x0 y0 !0þ n n ¼ 1; 2; . . .; M (29) > @ > I I > u ðx Þ ¼ lim v ðx ; y Þ lim v ðx ; y Þ : 2 0 0 0 n 0 0 y0 !0 @x0 y0 !0þ n Applying Equation (29) and considering the continuity of the displacements at the x0-axis excluding the crack, the following relation can be obtained 8Z b0 > > > u1 ðx0 Þdx0 ¼ 0 < (30) Za0b0 > > > u2 ðx0 Þdx0 ¼ 0 : a0

Using Equations (25)–(29) with the Fourier integral transform method, the unknown functions UInj ðsÞ (n = 1, 2, …, M; j = 1, …, 4) can be expressed as 0 1 Wn11 Wn12 C 1 B B Wn21 Wn22 C W1 ðsÞ ðUIn1 UIn2 UIn3 UIn4 ÞT ¼ n ¼ 1; 2; . . .; M (31) Wn @ Wn31 Wn32 A W2 ðsÞ Wn41 Wn42 Rb Here, the T denotes transpose, W1 ðsÞ ¼ a00 u1 ðx0 Þeisx0 dx0 , and R b0 superscript W2 ðsÞ ¼ a0 u2 ðx0 Þeisx0 dx0 . The expressions Wn (n = 1, 2, …, M) and Wnjk (n = 1, 2, …, M, k = 1, 2, j = 1, …, 4) are given in Appendix 1. The stress fields of the second part can be solved in the global coordinates system (x − y). The constitutive equations can be written as 8 ln ðxÞ @uIIn ðx; yÞ @vIIn ðx; yÞ > > II > ð1 þ kÞ þ ð3 kÞ rnxx ðx; yÞ ¼ > > k 1 @x @y > > < II II l ðxÞ @v ðx; yÞ @u ðx; yÞ n ¼ 1; 2; . . .; M (32) ð1 þ kÞ n þ ð3 kÞ n rIInyy ðx; yÞ ¼ n > k 1 @y @x > > > > @uIIn ðx; yÞ @vIIn ðx; yÞ > > þ : sIInxy ðx; yÞ ¼ ln ðxÞ @y @x

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The governing equations are 8 II @r ðx; yÞ @sIInxy ðx; yÞ > > þ ¼0 < nxx @x @y @rII ðx; yÞ @sIInxy ðx; yÞ > > : nyy þ ¼0 @y @x

n ¼ 1; 2; . . .; M

(33)

Using the similar procedure to that for deriving the displacement and stress fields of the first part, the displacement components of the second part can be obtained as 8 Z 4 II > 1 1X > II > u ðx; yÞ ¼ EII ðaÞUIInj ðaÞeknj ðaÞxiay da > n < 2p 1 j¼1 nj n ¼ 1; 2; . . .; M (34) Z 4 > 1 1X > II II kIInj ðaÞxiay II > ðx; yÞ ¼ F ðaÞU ðaÞe da v > nj : n 2p 1 j¼1 nj II Here, α is the Fourier variable, and kIInj , Enj ðaÞ and FnjII ðaÞ (n = 1, 2, …, M; j = 1, …, 4) are known expressions shown in Appendix 1. UIInj ðaÞ (n = 1, 2, …, M; j = 1, …, 4) are unknowns functions to be solved. Substituting expressions (34) into (32) yields 8 Z 1X 4 > II > II dn x 1 > r ðx; yÞ ¼ e BIInj ðaÞUIInj ðaÞeknj ðaÞxiay da > nxx > > 2p > 1 j¼1 > > > Z 1X 4 < II 1 CnjII ðaÞUIInj ðaÞeknj ðaÞxiay da rIInyy ðx; yÞ ¼ edn x n ¼ 1; 2; . . .; M (35) 2p 1 j¼1 > > > > Z > 4 > II > 1 1X > > sIInxy ðx; yÞ ¼ edn x DIInj ðaÞUIInj ðaÞeknj ðaÞxiay da > : 2p 1 j¼1

Here, BIInj , CnjII and DIInj ðaÞ (n = 1, 2, …, M; j = 1, …, 4) are known expressions shown in Appendix 1. 2.2. Derivation of the singular integral equations According to the boundary conditions (14), the following equations can be obtained Z b0 u1 ðuÞ II ½A1 ða; xÞjx¼h0 fU1 ðaÞg ¼ du (36) ½G01 ðu; a; xÞjx¼h0 u2 ðuÞ a0 Here, fUII1 ðaÞg is an unknown vector, and [A1(α, x)] and [G01(u, α, x)] are known matrixes shown in Appendix 2. Using the boundary conditions (15), the following equations can be obtained Z b0 u1 ðuÞ du (37) ½G02 ðu; a; xÞjx¼hM ½AM ða; xÞjx¼hM fUIIM ðaÞg ¼ u2 ðuÞ a0 Here, fUIIM ðaÞg is an unknown vector, and [AM(α, x)] and [G02(u, α, x)] are known matrixes shown in Appendix 2. From the stress continuity conditions (17), we have the following equations

Philosophical Magazine ½An ða; xÞjx¼hn fUIIn ðaÞg eðdnþ1 dn Þx ½Anþ1 ða; xÞx¼hn fUIInþ1 ðaÞg Z b0 h i u1 ðuÞ du ½Tn ðu; a; x eðdnþ1 dn Þx ½Tnþ1 ðu; a; xÞ ¼ x¼hn u2 ðuÞ a0

11

n ¼ 1; 2; . . .; M ð38Þ


where fUIIn ðaÞg are unknown vectors, and [An(α, x)] and [Tn(u, α, x)] are known matrixes given in Appendix 2. Applying the displacement continuity conditions (18), the following equations can be obtained ½Zn ða; xÞjx¼hn fUIIn ðaÞg ½Znþ1 ða; xÞjx¼hn fUIInþ1 ðaÞg Z b0 u1 ðuÞ ½½Rn ðu; a; xÞ ½Rnþ1 ðu; a; xÞjx¼hn ¼ du u2 ðuÞ a0

n ¼ 1; 2; . . .; M 1 ð39Þ

where [Zn(α, x)] and [Rn(u, α, x)] are known matrixes shown in Appendix 2. Based on the above analytical procedure, it can be found that some important expressions Γnm(u, α, x) (n = 1, 2, …, M − 1; m = 1, 2, …,8) given in Appendix 2 include infinite integrals and these expressions are complex. If these infinite integrals are not simplified, the singular term in the integrand may result in low efficiency or divergence when calculating these integral numerically. Therefore, a key step for the present GPE is to derive the explicit analytical expressions of these infinite integrals. Actually, through some mathematical manipulations, they can be written into the following forms. Z 1 1 Cnt ðu; a; xÞ ¼ Pn1t ðu; a; x; sÞds n ¼ 1; 2; . . .; M t ¼ 1; 2 (40) 2p 1 1 Cnð2þtÞ ðu; a; xÞ ¼ 2p 1 Cnð4þtÞ ðu; a; xÞ ¼ 2p 1 Cnð6þtÞ ðu; a; xÞ ¼ 2p

Z

1

1

Z

1

1

Z

1

1

Pn2t ðu; a; x; sÞds

n ¼ 1; 2; . . .; M

t ¼ 1; 2

(41)


n ¼ 1; 2; . . .; M

t ¼ 1; 2

(42)


n ¼ 1; 2; . . .; M

t ¼ 1; 2

(43)

Here, Pnmt(u, α, x, s), snl(l = 1, 2, …, 4), fnj and Tnmj (j ¼ 1; . . .; 4; m ¼ 1; . . .; 4) are known explicit expressions shown in Appendix 2. The singular points of the integrands in the above infinite integrals (40)–(43) can be easily found. Thus, by using the theory of residues, the infinite integrals (40)–(43) can be simplified as 4 X Cnt ðu; a; xÞ ¼ 2pi nRes½Pn1t ðu; a; x; sÞ; snl n ¼ 1; 2; . . .; M t ¼ 1; 2 (44) l¼1

Cnð2þtÞ ðu; a; xÞ ¼ 2pi

4 X

nRes½Pn2t ðu; a; x; sÞ; snl

n ¼ 1; 2; . . .; M

t ¼ 1; 2

(45)


n ¼ 1; 2; . . .; M

t ¼ 1; 2

(46)

l¼1

Cnð4þtÞ ðu; a; xÞ ¼ 2pi

5 X l¼1

12

Z. Wang et al. Cnð6þtÞ ðu; a; xÞ ¼ 2pi

5 X


n ¼ 1; 2; . . .; M

t ¼ 1; 2

(47)

l¼1


Here, ξ are defined by 8 > < n ¼ 1 ðu x sec hÞImðsnl Þ [ 0 ðor; u x sec h ¼ 0 and Imðsnl Þ [ 0Þ 1 Imðsnl Þ ¼ 0 n¼ > 2 : n ¼ 0 ðu x sec hÞImðsnl Þ\0 ðor; u x sec h ¼ 0 and Imðsnl Þ\0Þ

(48)

where Imðsnl Þ denotes the imaginary part of singular points snl (l = 1, 2, …, 5). The function Res½Pnmt ðu; a; x; sÞ; snl represents the residue of Pnmt(u, α, x, s) at the singular point snl. Namely, Res½Pnmt ðu; a; x; sÞ; snl ¼ lim ðs snl ÞPnmt ðu; a; x; sÞ s!snl

n ¼ 1; 2; . . .; M

m ¼ 1; . . .; 4 t ¼ 1; 2

ð49Þ

Through the above procedures, combining Equations (36)–(39), the following set of equations can be obtained Z b u1 ðuÞ II ½vðu; a; x0 Þ4M 2 du (50) ½XðaÞ4M 4M fU ðaÞg4M 1 ¼ u2 ðuÞ a where [Ω(α)]4M×4M, {ΦII(α)}4M×1 and [χ(u, α)]4M×2 are known expressions shown in Appendix 2. Solving the set of linear Equations (50), the unknowns UIInj (n = 1, …, M; j = 1, …, 4) can be expressed by the unknown auxiliary functions u1 ðuÞ and u2 ðuÞ. Z b0 u1 ðuÞ 1 II fU ðaÞg4M 1 ¼ ½XðaÞ4M 4M ½vðu; aÞ4M 2 du (51) u2 ðuÞ a0 Using boundary conditions (16) at the crack faces and the total stress expressions (20), the following equations can be obtained. lim ½rIny0 y0 þ rIInxx sin2 h þ rIInyy cos2 h 2sIInxy sin h cos h ¼ f1 ðx0 Þ

y0 !0þ

a0 \x0 \b0 (52)

lim ½sInx0 y0 rIInxx sin h cos h þ rIInyy cos h sin h 2sIInxy ðcos2 h sin2 hÞ ¼ f2 ðx0 Þ

y0 !0þ

a0 \x0 \b0

ð53Þ

Applying expressions (26) and (35), the expressions (52) and (53) can be written as " Z 1X 4 I bn x0 þcn y0 1 limþ e C I ðsÞUInj ðaÞeknj ðsÞy0 isx0 ds y0 !0 2p 1 j¼1 nj # Z 1X 4 dn x 1 II kIInj ðaÞxiay þe Tn5j Unj ðaÞe da ¼ f1 ðx0 Þ ð54Þ 2p 1 j¼1


13

"

Z 4 I 1 1X limþ e DInj ðsÞUInj ðaÞeknj ðsÞy0 isx0 ds y0 !0 2p 1 j¼1 # Z 1X 4 dn x 1 II kIInj ðaÞxiay Tn6j Unj ðaÞe da ¼ f2 ðx0 Þ þe 2p 1 j¼1 bn x0 þcn y0

ð55Þ

where Tn5j and Tn6j are known expressions shown in Appendix 2. According to Equation (51), the expressions UIInj ðaÞ (j = 1, 2, …, 4) can be written as Z b0 Z b0 UIInj ðaÞ ¼ Njn21 ðu; aÞu1 ðuÞdu þ Njn22 ðu; aÞu2 ðuÞdu j ¼ 1; 2; . . .; 4 (56)


a0

a0

Njn21 ðu; aÞ

Njn22 ðu; aÞ)

where (or can be obtained using Equations (51). Applying Equations (31) and (56), Equations (54) and (55) can be written as Z b0 ½jn11 ðx0 ; uÞ þ Hn11 ðx0 ; uÞu1 ðuÞdu a0

Z þ

b0

jn12 ðx0 ; uÞ þ Hn12 ðx0 ; uÞ

a0

4ln0 u ðuÞdu ¼ 2pebn x0 f1 ðx0 Þ ð1 þ kÞðu x0 Þ 2 n ¼ 1; 2; . . .; M

Z

b0

jn21 ðx0 ; uÞ þ Hn21 ðx0 ; uÞ

a0

Z þ

b0

a0 \x0 \b0

ð57Þ

4ln0 u ðuÞdu ð1 þ kÞðu x0 Þ 1

½jn22 ðx0 ; uÞ þ Hn22 ðx0 ; uÞu2 ðuÞdu ¼ 2pebn x0 f2 ðx0 Þ

a0

n ¼ 1; 2; . . .; M

a0 \x0 \b0

ð58Þ

where κn11(x0, u), κn12(x0, u), κn21(x0, u), κn22(x0, u), Hn11(x0, u), Hn12(x0, u), Hn21(x0, u) and Hn22(x0, u) are given in Appendix 2. 2.3. Numerical solutions The singular integral Equations (57) and (58) can be solved numerically. Here, the method by Erdogan and Gupta [27] is chosen. The unknown functions u1 ðuÞ and u2 ðuÞ in Equations (57) and (58) can be written as ðb0 a0 Þgn ðuÞ un ðuÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðu a0 Þðb0 uÞ

n ¼ 1; 2

Finally, the mixed-mode SIFs can be expressed as (Shbeeb et al. 1999b) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 2l0n bn a0 b0 a0 e KI ða0 Þ ¼ lim 2ða0 x0 Þry0 y0 ðx0 ; 0Þ ¼ g2 ða0 Þ x!a0 1þk 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l0n bn b0 e KI ðb0 Þ ¼ lim 2ðx0 b0 ÞrTy0 y0 ðx0 ; 0Þ ¼ x!b0 1þk

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 a0 g2 ðb0 Þ 2

(59)

(60)

(61)


14

Z. Wang et al.

Figure 3. Comparison between the normalized SIFs for an arbitrarily oriented crack in an infinite FGM: (a) δ(a0 − b0)/2 = 0.5 and (b) δ(a0 − b0)/2 = 0.25.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l KII ða0 Þ ¼ lim 2ða0 x0 ÞsTx0 y0 ðx0 ; 0Þ ¼ 0n ebn a0 x!a0 1þk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l0n bn b0 KII ðb0 Þ ¼ lim 2ðx0 b0 ÞsTx0 y0 ðx0 ; 0Þ ¼ e x!b0 1þk

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 0 a0 g1 ða0 Þ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0 a0 g1 ðb0 Þ 2

(62)

(63)

More details on the numerical method can be found in Erdogan and Gupta (1972) and Guo et al. [13]. 3. Results and discussion In order to verify the present GPE model, one typical example considered in previously published papers is analysed. An infinite nonhomogenous elastic medium containing an



15

Figure 4. Comparison between the present results and the those of Delale and Erdogan [9] when θ/π = 0.

Figure 5. Comparisons between the actual mechanical properties and the approximated properties.

arbitrary oriented crack with respect to the direction of property gradient is considered. The shear modulus of the nonhomogenous elastic medium was assumed to be an exponential function and the problem was solved under plane strain conditions by Konda and Erdogan [14]. In order to be consistent with the reference, we let L0 =L ¼ 0:5, v = 0.3, μ(x) = μ0eδx, (b0 − a0)/L = 15 (so that the strip is large enough 8l0 compared with the crack length), f2 ðx0 Þ ¼ 1þk e0 edx0 cos h cos sin h and phffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8l0 8l0 dx0 cos h 2 f1 ðx0 Þ ¼ 1þk e0 e cos h. The SIFs are normalized by K0 ¼ 1þk e0 ðb0 a0 Þ=2. Figure 3(a) and (b) show the comparison between the results of the normalized SIFs as δ(b0 − a0)/2 = 0.5 and δ(b0 − a0)/2 = 0.25, respectively. It can be seen that the present results agree well with those of Konda and Erdogan [14].

Z. Wang et al.


16

Figure 6. Variation of the normalized SIFs with θ/π: (a) exponential function, (b) linear function and (c) power-law function.



17

Figure 7. Comparison between the variations of the normalized SIFs with θ/π for different types of properties: (a) KI ða0 Þ=K0 , (b) KII ða0 Þ=K0 , (c) KI ðb0 Þ=K0 and (d) KII ðb0 Þ=K0 .

Next, a limiting case for a crack parallel to the gradient direction [9] will be considered to verify the GPE model. In order to be consistent with the reference, we let (b0 − a0)/(2L) = 40 (so that the strip is large enough compared with the crack length), v = 0.3 and θ/π = 0. The Young’s modulus is assumed to be E(x) = E0exp(δx). The load applied on the crack faces can be written as f1(x0p ) =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −σ0 (σ0 is a positive constant) and f2(x0) = 0. The SIFs are normalized by K0 ¼ r0 ðb0 a0 Þ=2. As shown in Figure 4, the results by the present GPE model agree well with those of Delale and Erdogan [9]. The error is less than 1%. In the following analyses, all the cases are considered under plane strain conditions. For convenience, some parameters are specified for the following calculations. We let μ0 denote the value of shear modulus when x = h0 = −L0 and μM denote the value of shear modulus when x = hM = L − L0. In the following studies, the influence of material parameters on the SIFs will be investigated. To accomplish this, some typical moduli defined by different functions (including, power-law functions, linear functions, exponential functions) will be considered. Thus, different types of shear moduli μ(x) can be written as lðxÞ ¼ l0 þ ðlM l0 Þ½ðx h0 Þ=Lp lðxÞ ¼ l0 e

ðxh0 Þd L

For exponential functions

For exponential functions

(64) (65)

18

Z. Wang et al.


lðxÞ ¼ l0 þ ðlM l0 Þðx h0 Þ=L

For linear functions

(66)

In the following examples, the power p is assumed to be equal to 1/2. Figure 5 shows the comparison between the actual properties which may be obtained by fitting the experimentally tested results and the approximate ones. It can be observed that: when the actual modulus is defined by exponential functions, the problem can be solved directly without dividing layers; when the actual modulus is defined by linear functions, the actual modulus can be approximated very well using ten layers with the same thickness. Moreover, for power-law functions, the number of layers which is sufficient to give a good fit is thirteen. It should be mentioned that the thickness of each layer may be different. For power law functions, the thickness ratio (hn+1 − hn)/(hn − hn−1) (n = 1, 2, …, 12) is 1.5 in Figure 5. As shown in Figure 5, for different functions, the curves of the above actual shear moduli can be approximated by a series of exponential functions. It is observed that the actual properties can be approximated well. In the following study, the crack face tractions f1(x0) and f2(x0) can be determined from actual external loading conditions. One practical case that may be considered is a “uniform strain” applied to the medium away from the crack region. Namely,

Figure 8. Variation of the normalized SIFs with crack length when the actual material property is defined by power-law function: (a) KI ða0 Þ=K0 , (b) KI ðb0 Þ=K0 , (c) KII ða0 Þ=K0 and (d) KII ðb0 Þ=K0 .



19

Figure 9. Comparison between the variations of the normalized SIFs with crack length for different types of properties: (a) KI =K0 and (b) KII =K0 .

enyy ðx; 1Þ ¼ e0

n ¼ 1; . . .; M

(67)

According to the superposition principle, the unknown functions f1(x0) and f2(x0) can be expressed as f1 ðx0 Þ ¼ 2ð1 þ mÞlðx0 cos hÞe0 cos2 h f2 ðx0 Þ ¼ 2ð1 þ mÞlðx0 cos hÞe0 cos h sin h

a0 \x0 \b0 a0 \x0 \b0

(68) (69)

For all pthe following cases, the SIFs will be normalized by K0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ tÞe0 ðb0 a0 Þ=2. The Poisson’s ratio is assumed to be equal to 0.3 and L0 =L ¼ 0:5. (1) In order to study the influences of the crack angle on the SIFs, let us consider the arbitrarily oriented crack with its centre located at origin of the local coordinate system (x0 − y0) (namely, (b0 + a0)/2 = 0). The calculations were carried


20

Z. Wang et al. out for different types of properties with the angle θ varying from 0 to 0.5π. Figure 6(a)–(c) shows the variations of the SIFs with the direction of crack when (b0 − a0)/L = 0.25 and μM/μ0 = 5. It can be observed that the mode I SIFs decrease with increasing θ, while the mode II SIFs, increase first and then decrease with increasing θ. Particularly, it can be found that: the curves of the mode-II SIFs usually attain their extremum when θ/π = 0.25. In Figure 6(a)–(c), the values of the mode-I SIFs are always bigger than those of the mode-II SIFs at the beginning when the FGM strip is mostly under mode I loading. The values of the mode-I SIFs become smaller than those of the modeII SIFs, when the crack angle θ increases to some degree and the mode II SIF starts to dominate. This trend is not affected by different types of properties. (2) The comparison of the varying characteristics of the SIFs with the crack angle for different types of properties is shown in Figure 7(a)–(d). It can be observed that the values of the SIFs are obviously affected by the types of properties. It indicates that it is not sufficient to consider a single type of function describing material properties for practical FGMs. (3) For different modulus ratio μM/μ0, the influence of various crack lengths on the SIFs are considered when (b0 − a0)/L varies between 0.1 and 0.5. In the case of power-law functions, Figsure 8(a)–(d) depict the varying characteristics of the SIFs with crack length with the modulus ratio μM/μ0 varying from 2 to 5. In Figure 8(a), KI ða0 Þ=K0 increases with increasing crack length. KII ða0 Þ=K0 varies slightly with increasing crack length in Figure 8(c). From Figure 8(b) and (d), it can be observed that KI ðb0 Þ=K0 and KII ðb0 Þ=K0 increase with increasing crack length. The varying characteristics of the SIFs with the crack lengths for different types of properties are compared in Figure 9 when μM/μ0 = 5. From Figure 9, it can be observed that the SIFs are obviously affected by the types of properties.

4. Conclusions In the past decades, many researchers have tried to develop analytical models for the fracture mechanics of FGMs. However, in order to solve the problem analytically, most of the published papers assumed the material properties to vary according to specific functions and only a few analytical models have been developed for FGMs with general mechanical properties. Differently from previous works, a GPE model is developed for FGMs with general mechanical properties and an arbitrarily oriented crack. The significance of the GPE model lies in: (1)

The present GPE model generalizes the PE model of Guo and Noda [1]. A feasible mathematical procedure is developed for the PE model to be expanded to GPE model. Since the GPE model is based on the analytical model of a single layer with exponential properties, with the concept of the GPE model, hundreds of analytical models published in the past 30 years on the elastic crack problems of FGMs assuming exponential properties can be unified as they form a strong base for establishing a general analytical model of FGMs with arbitray mechanical properties and arbitrarily oriented cracks.


21

(2) The concept of FGMs has been applied in many fields, such as thermal barrier materials and piezoelectric materials. In these fields, the material properties in analytical fracture mechanics models are also usually assumed to have very particular functions (mainly exponential functions).The establishment of the present GPE model indicates that the idea of piecewise-exponential model has great potential in studying the crack problems of FGMs or nonhomogeneous materials. With the GPE model, it is found that:


(a)

(b)

The mixed-mode SIFs may be affected greatly by the crack angle in FGMs. This implies that it is not sufficient to study only the crack vertical to or parallel to the gradient direction of FGMs and an arbitrarily (generally) oriented crack should be considered. The variations of the mixed-mode SIFs with the crack angle are affected greatly by the types of material properties. It indicates that it is not sufficient to assume material properties to be only described by one type of particular functions for practical FGMs.

Acknowledgements This work is supported by NSFC (10872056), NCET(08-0151), SRF for ROCS (SEM), Heilongjiang Science Fund for ROCS, Doctoral Programs Foundation of Ministry of Education of China, Science Funds for Distinguished Young Scholar of Heilongjiang Province and Program for Science and Technology Innovation Team in University of Heilongjiang Province, China.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

L.C. Guo and N. Noda, Int. J. Solids Struct. 44 (2007) p.6768. Y.F. Chen and F. Erdogan, J. Mech. Phys. Solids 44 (1996) p.771. N. Jain, C.E. Rousseau and A. Shukla, Theor. Appl. Fract. Mech. 42 (2004) p.155. Z.H. Jin and R.C. Batra, J. Mech. Phys. Solids 44 (1996) p.1221. Z.H. Jin and N. Noda, J. Appl. Mech. Trans. ASME 61 (1994) p.738. N. Noda and Z.H. Jin, JSME Int J., Ser. A 38 (1995) p.364. V. Parameswaran and A. Shukla, Mech. Mater. 31 (1999) p.579. J. Chen, Int. J. Fract. 133 (2005) p.303. F. Delale and F. Erdogan, J. Appl. Mech. 50 (1983) p.609. F. Erdogan, Functionally Graded Mater. 1996 (1997) p.105. F. Erdogan and B.H. Wu, J. Appl. Mech. Trans. ASME 64 (1997) p.449. L.C. Guo and N. Noda, Mech. Mater. 40 (2008) p.81. L.C. Guo, L.Z. Wu and L. Ma, Compos. Struct. 63 (2004a) p.397. N. Konda and F. Erdogan, Eng. Fract. Mech. 47 (1994) p.533. C.Y. Li, G.J. Weng and Z.P. Duan, Int. J. Solids Struct. 38 (2001) p.7473. X. Long and F. Delale, Int. J. Fract. 129 (2004) p.221. N.I. Shbeeb and W.K. Binienda, J. Appl. Mech. Trans. ASME 66 (1999) p.492. N.I. Shbeeb and W.K. Binienda, J. Appl. Mech. Trans. ASME 66 (1999) p.501. D.V. Kubair and B. Bhanu-Chandar, J. Mech. Phys. Solids 55 (2007) p.1145.

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[20] [21] [22] [23] [24] [25] [26] [27]

W.J. Feng and R.K.L. Su, Int. J. Solids Struct. 43 (2006) p.5196. S. Itou, Int. J. Fract. 110 (2001) p.123. Z.H. Jin and G.H. Paulino, Int. J. Fract. 107 (2001) p.73. B.L. Wang, Y.W. Mai and N. Noda, Int. J. Fract. 116 (2002) p.161. Z.Q. Cheng, D.Y. Gao and Zheng Zhong, Int. J. Mech. Sci. 54 (2012) p.287. G.Y. Huang, Y.S. Wang and D. Gross, Eur. J. Mech. A. Solids 22 (2003) p.535. Z. Zhong and Z.Q. Cheng, Int. J. Solids Struct. 45 (2008) p.3711. F. Erdogan, G.D. Gupta and Q. Appl, Math. 30 (1972) p.525.


Appendix 1 kþ1 l kI ðc þ kInj Þ k 1 n0 nj n 3k ln0 sðcn þ kInj Þ FnjI ¼ i ln0 ðs þ ibÞkInj þ k1 I Enj ¼ ln0 sðs þ ibn Þ þ

8 kþ1 3k > > l EI s þ l F I kI BI ¼ i > < nj k 1 n0 nj k 1 n0 nj nj 3k kþ1 I ln0 Enj ln0 FnjI kInj CnjI ¼ i sþ > > k 1 k 1 > : I I I Dnj ¼ ln0 ðiFnjI s þ Enj knj Þ

kIn1

n ¼ 1; . . .; M and j ¼ 1; . . .; 4 (A1) n ¼ 1; . . .; M and j ¼ 1; . . .; 4 (A2)

n ¼ 1; . . .; M and j ¼ 1; . . .; 4

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffiffi! cn bn 3 k 1 u 3k 3k t ¼ þ4si bn cn c n þ bn þ 4s2 1þk 1þk 2 2 1þk 2 n ¼ 1; 2; . . .; M

kIn2

ðA5Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffiffi! cn bn 3 k 1 u 3k 3k t þ ¼ þ4si bn cn c n þ bn þ 4s2 1þk 1þk 2 2 1þk 2 n ¼ 1; 2; . . .; M

kIn4

ðA4Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffiffi! cn bn 3 k 1 u 3k 3k t ¼ þ þ4si bn þ cn c n bn þ 4s2 1þk 1þk 2 2 1þk 2 n ¼ 1; 2; . . .; M

kIn3

(A3)

c b ¼ nþ n 2 2

ðA6Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi!2 rffiffiffiffiffiffiffiffiffiffiffi! 3k 1u 3k 3k t þ þ4si bn þ cn c n bn þ 4s2 1þk 2 1þk 1þk n ¼ 1; 2; . . .; M

ðA7Þ


23

I I I I I I I I I I I I I Wn11 ¼ i Cn4 Dn3 Fn2 Cn3 DIn4 Fn2 Cn4 DIn2 Fn3 þ Cn2 DIn4 Fn3 þ Cn3 DIn2 Fn4 Cn2 DIn3 Fn4 n ¼ 1; . . .; M

ðA8Þ

I I I I I I I I I I I I I Wn12 ¼ i Cn4 Dn3 En2 Cn3 DIn4 En2 Cn4 DIn2 En3 þ Cn2 DIn4 En3 þ Cn3 DIn2 En4 Cn2 DIn3 En4 n ¼ 1; . . .; M ðA9Þ


I I I I I I I I I I I I I Wn22 ¼ i Cn4 Dn3 En1 Cn3 DIn4 En1 Cn4 DIn1 En3 þ Cn1 DIn4 En3 þ Cn3 DIn1 En4 Cn1 DIn3 En4 n ¼ 1; . . .; M ðA10Þ

I I I I I I I I I I I I I Wn21 ¼ i Cn4 Dn3 Fn1 Cn3 DIn4 Fn1 Cn4 DIn1 Fn3 þ Cn1 DIn4 Fn3 þ Cn3 DIn1 Fn4 Cn1 DIn3 Fn4 n ¼ 1; . . .; M

ðA11Þ

I I I I I I I I I I I I I Dn2 Fn1 Cn2 DIn4 Fn1 Cn4 DIn1 Fn2 þ Cn1 DIn4 Fn2 þ Cn2 DIn1 Fn4 Cn1 DIn2 Fn4 Wn31 ¼ i Cn4 n ¼ 1; . . .; M ðA12Þ

I I i I I I I I I I I I I Wn32 ¼ i Cn4 Dn2 En1 Cn2 DIn4 En1 Cn4 DIn1 En2 þ Cn1 DIn4 En2 þ Cn2 DIn1 En4 Cn1 DIn2 En4 n ¼ 1; . . .; M

ðA13Þ

I I I I I I I I I I I I I Wn41 ¼ i Cn3 Dn2 Fn1 Cn2 DIn3 Fn1 Cn3 DIn1 Fn2 þ Cn1 DIn3 Fn2 þ Cn2 DIn1 Fn3 Cn1 DIn2 Fn3 n ¼ 1; . . .; M ðA14Þ

I I I I I I I I I I I I I Wn42 ¼ i Cn3 Dn2 En1 Cn2 DIn3 En1 Cn3 DIn1 En2 þ Cn1 DIn3 En2 þ Cn2 DIn1 En3 Cn1 DIn2 En3 n ¼ 1; . . .; M

ðA15Þ

I I I I I I I I I I I Wn ¼ s Cn1 ðDIn4 En3 Fn2 þ DIn3 En4 Fn2 þ DIn4 En2 Fn3 DIn2 En4 Fn3 DIn3 En2 Fn4 I I I I I I I I I I I Fn4 Þ þ Cn2 ðDIn4 En3 Fn1 DIn3 En4 Fn1 DIn4 En1 Fn3 þ DIn1 En4 Fn3 þ DIn2 En3 I I I I I I I I I I I þ DIn3 En1 Fn4 DIn1 En3 Fn4 Þ þ Cn3 ðDIn4 En2 Fn1 þ DIn2 En4 Fn1 þ DIn4 En1 Fn2 I I I I I I I I I I I DIn1 En4 Fn2 DIn2 En1 Fn4 þ DIn1 En2 Fn4 Þ þ Cn4 ðDIn3 En2 Fn1 DIn2 En3 Fn1

I I I I I I I I DIn3 En1 Fn2 þ DIn1 En3 Fn2 þ DIn2 En1 Fn3 DIn1 En2 Fn3 Þ n ¼ 1; . . .; M ðA16Þ

kþ1 ln0 a2 þ ln0 kIInj dn þ kIInj k1 3k II II II l k þ ln0 dn þ knj Fnj ¼ ia k 1 n0 nj

II Enj ¼

8 3k kþ1 II II II II > > > Bnj ¼ i k 1 ln0 Fnj a þ k 1 ln0 Enj knj > < kþ1 3k II II ln0 FnjII a þ ln0 Enj CnjII ¼ i knj > > k 1 k 1 > > : II II Dnj ¼ ln0 ðiEnj a þ FnjII kIInj Þ

n ¼ 1; . . .; M

and

j ¼ 1; . . .; 4

n ¼ 1; . . .; M and j ¼ 1; . . .; 4

(A17)

(A18)

n ¼ 1; . . .; M and j ¼ 1; . . .; 4 (A19)


24

Z. Wang et al.

kIIn1

dn 1 ¼ 2 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3k 2 4a2 þ ðdn Þ 4dn ai kþ1

kIIn2

dn 1 ¼ 2 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3k 2 4a2 þ ðdn Þ þ 4dn ai kþ1

kIIn3

dn 1 ¼ þ 2 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3k 2 4a2 þ ðdn Þ þ 4dn ai kþ1

kIIn3

dn 1 ¼ þ 2 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi 3k 2 4a2 þ ðdn Þ 4dn ai kþ1

Appendix 2

Z In1j ðxÞ ¼

1

ekn1j y0 isx0 þiay dy

n ¼ 1; 2; . . .; M

(A20)

n ¼ 1; 2; . . .; M

(A21)

n ¼ 1; 2; . . .; M

(A22)

n ¼ 1; 2; . . .; M

(A23)

n ¼ 1; . . .; M

j ¼ 1; 2

(B1)

n ¼ 1; . . .; M

j ¼ 3; 4

(B2)

x tan h

Z In2j ðxÞ ¼

sn1

x tan h

1

ekn1j y0 isx0 þiay dy

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½4a2 ð1 þ kÞ þ 4iadn 3 þ 2k k 2 þ ð1 þ kÞðdn Þ2 cos2 h 1@ pffiffiffiffiffiffiffiffiffiffiffi idn cos h þ ¼ 2 1þk 1 þ 2a sin hA

n ¼ 1; . . .; M

ðB3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½4a2 ð1 þ kÞ 4iadn 3 þ 2k k 2 þ ð1 þ kÞðdn Þ2 cos2 h 1@ pffiffiffiffiffiffiffiffiffiffiffi idn cos h þ ¼ 2 1þk 1 0

sn2

þ 2a sin hA

n ¼ 1; . . .; M

ðB4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½4a2 ð1 þ kÞ þ 4iadn 3 þ 2k k 2 þ ð1 þ kÞðdn Þ2 cos2 h 1@ pffiffiffiffiffiffiffiffiffiffiffi idn cos h ¼ 2 1þk 1 0

sn3

þ 2a sin hA

n ¼ 1; . . .; M

ðB5Þ


sn4

25

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½4a2 ð1 þ kÞ 4iadn 3 þ 2k k 2 þ ð1 þ kÞðdn Þ2 cos2 h 1@ pffiffiffiffiffiffiffiffiffiffiffi idn cos h ¼ 2 1þk 1 þ 2a sin hA

n ¼ 1; . . .; M ðB6Þ


sn5 ¼ 0

n ¼ 1; . . .; M

(B7)

fn1 ¼ ½kn12 cos h þ iða s sin hÞ½kn13 cos h þ iða s sin hÞ½kn14 cos h þ iða s sin hÞ n ¼ 1; . . .; M ðB8Þ fn2 ¼ ½kn11 cos h þ iða s sin hÞ½kn13 cos h þ iða s sin hÞ½kn14 cos h þ iða s sin hÞ n ¼ 1; . . .; M

ðB9Þ

fn3 ¼ ½kn11 cos h þ iða s sin hÞ½kn12 cos h þ iða s sin hÞ½kn14 cos h þ iða s sin hÞ n ¼ 1; . . .; M

ðB10Þ

fn4 ¼ ½kn11 cos h þ iða s sin hÞ½kn12 cos h þ iða s sin hÞ½kn13 cos h þ iða s sin hÞ n ¼ 1; . . .; M ðB11Þ Tn1j ¼ Bn1j cos2 h þ Cn1j sin2 h 2Dn1j cos h sin h

n ¼ 1; . . .; M j ¼ 1; . . .; 4 (B12)

Tn2j ¼ Bn1j sin h cos h Cn1j sin h cos h þ Dn1j ðcos2 h sin2 hÞ

n ¼ 1; . . .; M

j ¼ 1; . . .; 4

ðB13Þ Tn3j ¼ En1j cos h Fn1j sin h

n ¼ 1; . . .; M

j ¼ 1; . . .; 4

(B14)

Tn4j ¼ En1j sin h þ Fn1j cos h

n ¼ 1; . . .; M

j ¼ 1; . . .; 4

(B15)

Tn5j ¼ Bn2j sin2 h þ Cn2j cos2 h 2Dn2j sin h cos h

n ¼ 1; . . .; M

Tn6j ¼ Bn2j sin h cos h þ Cn2j cos h sin h 2Dn2j ðcos2 h sin2 hÞ

j ¼ 1; . . .; 4 (B16)

n ¼ 1; . . .; M

j ¼ 1; . . .; 4

ðB17Þ C0m ðu; a; xÞ ¼ C1m ðu; a; xÞ C0ð4þmÞ ðu; a; xÞ ¼ CMm ðu; a; xÞ

m ¼ 1; . . .; 4 m ¼ 1; . . .; 4

(B18) (B19)

26

Z. Wang et al.

1 Cnt ðu; a; xÞ ¼ 2p

Z

1

2 X

1

j¼1

Tn1j Wnjt In1j ðxÞ þ

4 X

! Tn1k Wnkt In2k ðxÞ

k¼3

n ¼ 1; . . .; M 1 Cnð2þtÞ ðu; a; xÞ ¼ 2p

Z

1

2 X

1

j¼1


4 X

eisu ds Wn t ¼ 1; 2 !

Tn2k Wnkt In2k ðxÞ

k¼3


n ¼ 1; . . .; M Z

1 Cnð4þtÞ ðu; a; xÞ ¼ s 2p

1

2 X

1

j¼1


4 X

1 Cnð6þtÞ ðu; a; xÞ ¼ 2p

Z

1

2 X

1

j¼1

eisu ds Wn

t ¼ 1; 2 !


k¼3

n ¼ 1; . . .; M Tn4j Wnjt In1j ðxÞ þ

4 X

ðB20Þ

ðB21Þ

eisu ds Wn

t ¼ 1; 2 !


k¼3

n ¼ 1; . . .; M

ðB22Þ

eisu ds Wn

t ¼ 1; 2

ðB23Þ

P4 i½sðux sec hÞþxa tan h 1 j¼1 Tnmj Wnjt fnj e Pnmt ðu; a; x; sÞ ¼ 2p ðs sn1 Þðs sn2 Þðs sn3 Þðs sn4 ÞWn n ¼ 1; . . .; M m ¼ 1; . . .; 4 t ¼ 1; 2 " ½An ða; xÞ ¼

ðB24Þ

BIIn1 ðaÞekn1 ðaÞx




DIIn1 ðaÞekn1 ðaÞx




II

II

II

II

II

fUIIn ðaÞg ¼ f UIIn1 ðaÞ UIIn2 ðaÞ UIIn3 ðaÞ

II

ðB25Þ

T

(B26)

ðn ¼ 1; 2; . . .; M Þ

C02 ðu; a; xÞ C04 ðu; a; xÞ

C05 ðu; a; xÞ ½G02 ðu; a; xÞ ¼ C07 ðu; a; xÞ


Cn1 ðu; a; xÞ ½Tn ðu; a; xÞ ¼ Cn3 ðu; a; xÞ ½Zn ða; xÞ ¼

ðn ¼ 1; 2; . . .; M Þ UIIn4 ðaÞ g

II ðaÞekn1 ðaÞx En1 II II Fn1 ðaÞekn1 ðaÞx II

II En2 ðaÞekn2 ðaÞx II II Fn2 ðaÞekn2 ðaÞx II

#

II


½G01 ðu; a; xÞ ¼

"

II

Cn2 ðu; a; xÞ Cn4 ðu; a; xÞ

II En3 ðaÞekn3 ðaÞx II II Fn3 ðaÞekn3 ðaÞx II

(B27) (B28)

(B29) II En4 ðaÞekn4 ðaÞx II II Fn4 ðaÞekn4 ðaÞx II

# (B30)

Philosophical Magazine ½Rn ðu; a; xÞ ¼ 2

½XðaÞ4M 4M

½A1 ða; h0 Þ 6 ½A1 ða; h1 Þ 6 6 ½Z1 ða; h1 Þ 6 6 ... ¼6 6 6 0 6 4 0 0


0 0 eðd2 d1 Þh1 ½A2 ða; h1 Þ 0 ½Z2 ða; h1 Þ 0 .. .. . . 0 0 0 0 0 0

0 0 0 .. .. . . 0 0 0

27


(B31)

0 0 0 .. .

0 0 0 .. .

3

7 7 7 7 7 7 7 ½AM 1 ða; hM 1 Þ eðdM dM 1 ÞhM 1 ½AM ða; hM 1 Þ 7 7 5 ½ZM1 ða; hM1 Þ ½ZM ða; hM1 Þ 0 ½AM ða; hM Þ


ðB32Þ fUII ðaÞg4M 1 ¼ f fUII1 ðaÞgT

fUIIn ðaÞgT

fUIIM ðaÞgT g

T

3 ½G01 ðu; a; 0Þ 7 6 ½T1 ðu; a; h1 Þ eðd2 d1 Þh1 ½T2 ðu; a; h1 Þ 7 6 7 6 ½R1 ðu; a; h1 Þ ½R2 ðu; a; h1 Þ 7 6 7 6 .. 7 6. 7 6 7 6 ½Tn ðu; a; hn Þ eðd2 d1 Þh1 ½Tnþ1 ðu; a; hn Þ 7 6 ¼6 7 ½R ðu; a; h Þ ½R ðu; a; h Þ n nþ1 n 7 6 n 7 6. 7 6 .. 7 6 6 ½TM 1 ðu; a; hM 1 Þ eðd2 d1 ÞhM 1 ½TM ðu; a; hM 1 Þ 7 7 6 5 4 ½RM 1 ðu; a; hM 1 Þ ½RM ðu; a; hM 1 Þ

(B33)

2

½vðu; aÞ4M 2

(B34)

½G02 ðu; a; hM Þ Z

jn11 ðx0 ; uÞ ¼ Z jn12 ðx0 ; uÞ ¼

1

1

j¼1

"

2 X j¼1

Z jn22 ðx0 ; uÞ ¼

Hn11 ðx0 ; uÞ ¼

1 2p

Hn12 ðx0 ; uÞ ¼

1 2p

2 X

1 j¼1

2 X

1

Z jn21 ðx0 ; uÞ ¼

"

1

1

# Wnj1 2sln0 i Dn1j ðsÞ eisðx0 þuÞ ds ð1 þ kÞjsj Wn 2 X

1 j¼1 1

4 X

1 j¼1

Z

Wnj1 isðx0 þuÞ e ds Wn

# Wnj2 2sln0 i Cn1j ðsÞ eisðx0 þuÞ ds ð1 þ kÞjsj Wn

1

Z

Cn1j ðsÞ

1

4 X

1 j¼1

Dn1j ðsÞ

Wnj2 isðx0 þuÞ e ds Wn

(B35)

(B36)

(B37)

(B38)

Tn5j Njn21 ðu; aÞekn2j ðaÞxiay da

(B39)


(B40)

28

Z. Wang et al. 1 Hn21 ðx0 ; uÞ ¼ 2p


1 Hn22 ðx0 ; uÞ ¼ 2p

Z

1

4 X

1 j¼1

Z

1

4 X

1 j¼1


(B41)


(B42)