Ding and Li SpringerPlus (2016)5:1490 DOI 10.1186/s40064-016-3105-5

Open Access

RESEARCH

Interface crack between different orthotropic media under uniform heat flow Sheng‑Hu Ding* and Xing Li *Correspondence: [email protected] School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China

Abstract  In this paper, plane thermo-elastic solutions are presented for the problem of a crack in two bonded homogeneous orthotropic media with a graded interfacial zone. The graded interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermo-elastic moduli between dissimilar, homogeneous orthotropic halfplanes, which is assumed to vary exponentially in the direction perpendicular to the crack surface. Using singular integral equation method, the mixed boundary value con‑ ditions with respect to the temperature field and those with respect to the stress field are reduced to a system of singular integral equations and solved numerically. Numeri‑ cal results are obtained to show the influence of non-homogeneity parameters of the material thermo-elastic properties, the orthotropy parameters and the dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors. Keywords:  Functionally graded orthotropic media, Interface zone, Crack, Singular integral equation, Stress intensity factors

Background Functionally graded materials (FGMs) are designed as the special materials which have changed micro-structure and mechanical/thermal properties in the space to meet the required functional performance (Niino et  al. 1987; Suresh and Mortensen 1998). The advantages of FGMs are that the magnitude of residual and thermal stresses can be reduced, and the bonding strength and fracture toughness of such materials can be improved. From both the phenomenological and mechanistic viewpoints, the tailoring capability to produce gradual changes of thermo-physical properties in the spatial domain is the key point for the impressive progress in the areas of functionally graded materials (Miyamoto et al. 1999). By introducing the concept of the FGMs, extensive research on all aspects of fracture of isotropic and orthotropic FGMs under mechanical or thermal loads has been considered (Choi et  al. 1998; Choi 2003; Wang et  al. 2004; El-Borgi and Hidri 2006; Han and Wang 2006; Cheng et al. 2010; Ding and Li 2014; Kim and Paulino 2002; Dag 2006; Zhou et al. 2007). By considering changes in both elastic and thermal properties, Jin and Noda (1991) studied the transient thermo-elastic problems of functionally graded material with a crack. Fujimoto and Noda (2001) investigated the thermal cracking under a

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transient-temperature field in a ceramic/metal functionally graded plate. In addition, assuming the surfaces of the crack are insulated, thermal stresses around a crack in the interfacial layer between two dissimilar elastic half-planes are studied by Itou (2004). With the introduction of the thermal resistance concept, the thermal stress intensity factors for the interface crack between functionally graded layered structures under the thermal loading are investigated by Ding and Li (2015). Zhou and Lee (2011) studied the thermal fracture problem of a functionally graded coating-substrate structure of finite thickness with a partially insulated interface crack subjected to thermal–mechanical supply. Chen (2005) obtained the thermal stress intensity factors (TSIFS) of a graded orthotropic coating-substrate structure with an interface crack. Zhou et al. (2010) considered the thermal response of an orthotropic functionally graded coating-substrate structure with a partially insulated interface crack. Using mesh-free model, Dai et al. (2005) studied the active shape control as well as the dynamic response repression of the functionally graded material (FGM) plate containing distributed piezoelectric sensors and actuators. Natarajan et  al. (2011) considered the linear free flexural vibration of cracked functionally graded material plates by using the extended finite element method. Using extended finite element method, fatigue crack growth simulations of bi-material interfacial cracks have been considered under thermoelastic loading (Pathak et  al. 2013). Using element free Galerkin method, Pathak et  al. (2014) studied quasi-static fatigue crack growth simulations of homogeneous and bimaterial interfacial cracks under mechanical as well as thermo-elastic load. Layered FGM structure are very import in practical engineering (Sofiyev and Avcar 2010; Sofiyev et  al. 2012; Ding et  al. 2014; Ding et  al. 2015). The research of thermal elastic crack problem in layered structure is helpful for the design and application of functionally graded materials. This paper explores the thermal–mechanical response of layered and graded structures using the integral equation approach. The analytical results of the cracked layered material systems with the material properties in the graded coating varying as an exponential function has been obtained by using the integral transform technique. The surface of the crack is assumed to be part of the thermal insulation. The temperature distributions along the crack line are presented. The TSIFS under thermo-mechanical loadings are obtained, which is very important for the designing of layered orthotropic media.

Problem formulation As shown in Fig.  1, the problem under consideration consists of a functionally graded orthotropic strip (FGOS) of thickness h bonded to two homogeneous semi-infinite orthotropic media with a partially insulated interface crack of length 2c along the x-axis is considered. The subscript j(j  =  1,  2,  3) indicates the FGOS and two semi-infinite orthotropic media respectively. The remaining thermo-mechanical properties depend on the y-coordinate only and are modeled by an exponential function        kx(1) , ky(1) = kx(2) , ky(2) exp δy c

       kx(3) , ky(3) = kx(2) , ky(2) exp δh c

(1) (2)

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Fig. 1  Geometry of the layered orthotropic media under steady-state heat flows

(2) (2) where kx , ky are the thermal conductivities for the homogeneous orthotropic substrate II, and δ is an arbitrary nonzero constant. The temperature satisfies     ∂ ∂ (j) ∂Tj (j) ∂Tj + = 0 (j = 1 − 3) kx ky (3) ∂x ∂x ∂y ∂y

Substituting Eqs. (1) and (2) into the Eq. (3), the heat equation can be given by

kxy0

∂ 2 T1 ∂T1 ∂ 2 T1 + + δ =0 ∂x2 ∂y ∂y2

kxy0

∂ 2 Tj ∂ 2 Tj + =0 ∂x2 ∂y2 (2)

(4)

0
(5)

(j = 2, 3)

(2)

where kxy0 = kx /ky . The heat flux components are written as

∂T1 (x, y) = −Q0 , y → +∞, |x| < +∞ ∂y ∂T2 (x, y) = −Q0 , y → −∞, |x| < +∞ k2 ∂y k3

(6)

We define the following dimensionless quantities

� � � �� � � �� (2)  x, y, h = x, y, h c, T j = Tj −Q0 c ky ,    � �� � (2) σ jkl = σjkl −E0 Q0 α2 c ky ,  � �  � � �� � �  (2)  uj , vj = uj , vj −Q0 α2 c2 ky � � � (1) (2) (3) � αij ,αij ,αij  (1) (2) (3)   , α ij , α ij , α ij =  α0   � � � � 0 0 Exx ,Eyy 0 0 E xx , E yy = E0     � �    C (1) , G (1) , C (2) , G (2) , C (3) , G (3) = ij 66 ij 66 ij 66

j =1−3

(7)

(k, l = x, y)

(i, j = x, y)

(8) �

(1) (1) (2) (2) (3) (3) Cij ,G66 ,Cij ,G66 ,Cij ,G66

E0

�

,

(i, j = 1, 2)

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where α0 and E0 are the typical values of the coefficient of linear thermal expansion and the Young’s modulus of elasticity for the homogeneous orthotropic substrate, respectively. But for simplicity, in what follows, the bar appearing with the dimensionless quantities is omitted. The Duhamel–Neumann constitutive equations for the plane thermo-elastic problem are given by Nowinski (1978)   ∂u ∂v ∂u ∂v ∂u ∂v σxx = C11 +C12 −θ1 T , σyy = C12 +C22 −θ2 T , σxy = C66 + ∂x ∂y ∂x ∂y ∂y ∂x (9) in which

θ1 = C11 αxx + C12 αyy ,

θ2 = C12 αxx + C22 αyy ,

C66 = Gxy

(10)

The elastic stiffness coefficients and the coefficients of the linear thermal expansion in dimensionless form are modeled to take the following forms � � � � (1) (1) (1) (1) (2) (2) (2) (2)  C11 , C12 , C22 , C66 = C11 , C12 , C22 , C66 exp(βy) � � � � (11)  α (1) , α (1) = α (2) , α (2) exp(γ y) yy yy xx xx

where superscripts 1, 2 refer to the FGOS and the homogeneous orthotropic substrate II, respectively, β and γ are graded parameters. The properties of material 3 can be found in Eq. (11) when y is taken as h. In Eq. (11), elastic stiffness coefficients in dimensionless form can be represented by the Young’s moduli and the Poisson’s ratios as (2)

(2)

C11 =

Exx , 1 − νyx νxy

(2)

C22 =

(2) Eyy , 1 − νyx νxy

(2)

(2)

C12 =

Eyy νxy 1 − νyx νxy

(12)

(2) (2) where νij are the Poisson’s ratios and assumed to be constant. Exx and Eyy are Young’s moduli for the homogeneous orthotropic substrate II, respectively. Substituting Eq.  (9) into the equations of equilibrium for the forces reduces these equations to the forms

 � � � � (2) ∂ 2 v1 (2) 2 (2) (2) (2) (2) ∂ 2 u1 ∂v1  1 1  C11 + C66 ∂∂yu21 + C12 + C66 ∂x∂y = θ1 eγ y ∂T + βC66 ∂u ∂y + ∂x ∂x ∂x2 � � 2 � � � 2 2 (2) (2) (2) (2) ∂ v1 (2) (2) 1 (2) ∂v1  u1  C22 + C66 ∂∂xv21 + C12 + C66 ∂∂x∂y = θ2 eγ y (β + γ )T1 + + β C12 ∂u ∂x + C22 ∂y ∂y2

 � � 2 2 2  (2) ∂ u2 (2) (2) ∂T2 (2) ∂ v2 (2) ∂ u2   C11 = θ1 + C + C + C  66 12 66 ∂x2 ∂y2 ∂x∂y ∂x � � 2 2 2  ∂T2  (2) ∂ v2 (2) ∂ v2 (2) (2) ∂ u2  = θ2(2) + C + C + C  C22 66 12 66 2 2 ∂y ∂x ∂x∂y ∂y

∂T1 ∂y

�

(13)

(14)

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 � � 2 2 2  (2) ∂ u3 (2) (2) γ h ∂T3 (2) ∂ v3 (2) ∂ u3    C11 ∂x2 + C66 ∂y2 + C12 + C66 ∂x∂y = θ1 e ∂x � � 2 2 2  ∂T3  (2) ∂ v3 (2) ∂ v3 (2) (2) ∂ u3  = θ2(2) eγ h + C + C + C  C22 66 12 66 ∂y2 ∂x2 ∂x∂y ∂y

(15)

T1 (x, y) = T3 (x, y) |x| < +∞, y = h

(16)

Boundary conditions The temperature filed can be provided using the following boundary condition

� �  −Bi T1 (x, y) − T2 (x, y) |x| ≤ c, y=0     ∂T (x,y)   2 |x| < +∞, y = 0 ∂y

∂T1 (x, y) =  ∂y     

∂T3 (x,y) ∂y

−Q0 /ky(3)

(17)

|x| < +∞, y = h

y → +∞, |x| < +∞

(1)

where Bi = 1/ky (0)/Rc is dimensionless thermal resistance through the crack region. Rc is the thermal resistance through the crack region. The boundary conditions of the stress and displacement field can be given by   0 y = 0, |x| ≤ c 0 y = 0, |x| ≤ c σ1xy (x, y) = σ3xy (x, y) (x, y) = , , σ 1yy y=h σ3xy (x, y) y=h (18)

u1 (x, h− ) = u3 (x, h+ )

v1 (x, h− ) = v3 (x, h+ )

(19)

|x| < ∞

Heat conduction problem By using Fourier transform, the solutions of Eqs. (4) and (5) are given by  � +∞ −δy T1 (x, y) = −∞ (M1 (ω) exp(s1 y)+M2 (ω) exp(s2 y)) exp(−iωx)dω + 1−eδ ,     � +∞ T2 (x, y) = −∞ (M3 (ω) exp(p1 y)+M4 (ω) exp(p2 y)) exp(−iωx)dω + y,    � +∞  T3 (x, y) = −∞ (M5 (ω) exp(o1 y)+M6 (ω) exp(o2 y)) exp(−iωx)dω + e−δh y + 1/δ −

0
y≥h

(20)

where Mk (ω)(k = 1 − 6) can be found in “Appendix 1”. sk , pk and ok are the roots of the characteristic polynomials, which can be given by      1 2 2 −δ ± δ + 4kxy0 ω , p1,2 = ± kxy0 |ω|, o1,2 = ± kxy0 |ω| s1,2 = (21) 2 Introducing the unknown density function

φ(x) =

 ∂  T1 (x, 0+ ) − T2 (x, 0− ) ∂x

(22)

From (17), we obtain



+1 

−1

 1 −2π + H (x, u) φ(u)du =  u−x kxy0

where the kernel H (x, u) can be found in “Appendix 1”.

(23)

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Thermal stress analysis By using the standard Fourier transforms to Eqs. (13)–(15), following results for the displacement fields for the FGOS and two homogeneous orthotropic media are obtained � � � �  4 2 ξ (ω) � +∞ � � �  j  (γ +sj )y mj y −ixω dω + +∞  e e−ixω dω u (x, y) = C (ω)e e 1 j  −∞ −∞ dj (ω)  j=1 j=1 � � � �  4 2   v (x, y) = � +∞ � C (ω)q (ω)emj y e−ixω dω + � +∞ � ξj+2 (ω) e(γ +sj )y e−ixω dω + χ eγ y + χ e(γ −δ)y  j j 1 2  1 −∞ −∞ dj (ω) j=1

j=1

(24)

�  � +∞ � ξ (ω)   u2 (x, y) = −∞ C5 (ω)en1 y + C6 (ω)en2 y + d53 (ω) e(γ +p1 )y e−iωx dω � � +∞ �  (γ +p1 )y e−iωx dω +  v2 (x, y) = −∞ C5 (ω)q5 (ω)en1 y + C6 (ω)q6 (ω)en2 y + dξ63(ω) e (ω)

(2)

θ2

(2)

2C12

y2

(25)

 � � +∞ � (γ +o2 )y e−iωx dω   u3 (x, y) = −∞ C7 (ω)en3 y + C8 (ω)en4 y + dξ74(ω) (ω) e � �  v (x, y) = � +∞ C (ω)q (ω)en3 y + C (ω)q (ω)en4 y + ξ8 (ω) e(γ +o2 )y e−iωx dω + χ eγ h + χ e(γ −δ)h  3 7 7 8 8 1 2 −∞ d4 (ω)

(26)

where Cj (ω)(j = 1 − 8) are unknown function. ξj , qj (j = 1 − 8), dj (j = 1 − 4), χj (j = 1, 2) are given in “Appendix 1”. mj (j = 1 − 4) and nj (j = 1 − 4) are the roots of the characteristic polynomials, which can be given by     1 2 2 −β − β − 2�1 ± 2 (�1 ) − 4�2 m1,2 = (27) 2     1 2 2 −β + β − 2�1 ± 2 (�1 ) − 4�2 m3,4 = (28) 2

n1,2

1 =− 2



 −2�3 ± 2 (�3 )2 − 4�4 ,

n3,4

1 = 2



 −2�3 ± 2 (�3 )2 − 4�4

(29)

where

�1 = ω

2



(2)

(2)

(C12 )2 (2)

(2)

C22 C66

−

(2)

�2 = ω 4

C11

(2) C22

C11

(2)

C66

(2)

+2

C12

(2)

C22



,

(2)

+ ω2 β 2

C12

(2) C22

,

�3 = �1 ,

�4 = ω4

2 C11 2 C22

Solution procedure and near‑tip field intensity factors Introducing the density functions Φ1 (x) =

 ∂  u(x, 0+ ) − u(x, 0− ) , ∂x

Φ2 (x) =

 ∂  v(x, 0+ ) − v(x, 0− ) ∂x

(30)

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Substituting Eqs. (24)–(26) into Eqs. (18)–(19), we obtain 1

� � 1 2π · ωT (x) + K11 (x, u) Φ1 (u) + K12 (x, u)Φ2 (u) du = � 1 u−x kxy0 −1 � � � � 1�  1 2π · ωT (x)   + K22 (x, u) Φ2 (u) du = � 2 K21 (x, u)Φ1 (u) +  u−x kxy0 −1 �    

��

(31)

where Kij (x, u)(i, j = 1, 2), ω1 (x)T , ω2 (x)T are given in “Appendix 2”. The singular integral Eq. (31) are solved numerically with the unknown density functions R1 (u) and R2 (u) having the following form  N �  R1 (u)  R1 (u) = bn Tn (u)  Φ1 (u) = √1−u 2 n=1 (32) N �    Φ2 (u) = √R2 (u)2 R2 (u) = cn Tn (u) 1−u

n=1

Once R1 (u) and R2 (u) have been determined, the thermal stress intensity factors ahead of the crack tip can be defined and calculated as follows

 √ √ kxy0   K (1) = lim 2(x − 1)σ (x, 0) = −  I yy 2 R2 (1),  +  x→1 √   √ k    KI (−1) = lim 2(−x − 1)σyy (x, 0) = 2xy0 R2 (−1), x→−1− √  K (1) = lim √2(x − 1)σ (x, 0) = − kxy0 R (1),   xy 1 II 2  x→1+  √   √ kxy0   2(−x − 1)σxy (x, 0) = 2 R1 (−1).  KII (−1) = lim

(33)

x→−1−

Numerical results and discussion In this paper, the orthotropy and non-homogeneity parameters of Tyrannohex can be found in Ootao and Tanigawa (2005). The material properties can be given by Exx = 135 GPa, Eyy = 87 GPa, νxy = 0.15, αyy = 0.32 × 10−5 /◦ C,

kx = 2.81 W/m ◦ C,

νyx = 0.09667,

αxx = 0.32 × 10−5 /◦ C,

ky = 3.08 W/m ◦ C

In the presented results the values of the thermal stress intensity factors are normal√ (2) ized by k0 = E2 Q0 α2 c/ky . The crack is located along the interval −1 ≤ x ≤ 1. Figure  2a, b show the effects of the thermal conductivity parameter δ on the crack surface temperature when Bi = 0.1 and Bi = 0.5, respectively. From Fig. 2a, b, it can be found that the temperature jump across the crack surfaces increases with an decrease of the absolute values of δ. At the other hand, for smaller value of Bi, the temperature will become more pronounced. As expected, the temperature jump across the crack becomes more pronounced as the crack surfaces become more insulated, that is, as Bi decreases.

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a

b Fig. 2  Influences of thermal conductivity parameter δ on the normalized crack surfaces and crack extend line (2) y = 0 temperatures T (x, 0+ )/T0 and T (x, 0− )/T0 ,T0 = Q0 c/ky , h = 1.0, kxy0 = 2.0, a Bi = 0.1, b Bi = 0.5

Figure 3a, b show the effects of the thermal conductivity parameter δ and kxy0 on the mode I and kxy0 = 0.5II thermal stress intensity factors. It can be found that the mode I thermal stress intensity factors increases with an increase of the thermal conductivity parameter δ for either or kxy0 = 2.0; while increases with an increase of kxy0 for both δ = −1.0 and δ = 1.0. And the values of mode II thermal stress intensity factors decreases with the increasing of the thermal conductivity parameter δ regardless of the value of kxy0. Meanwhile, the values of mode II thermal stress intensity factors decreases (2) with the increasing of an increase of kxy0 regardless of the value o αxx f δ. (2) Figure 4a, b illustrate the effects of the stiffness parameter β and Exx on the mode I and II thermal stress intensity factors. It can be seen that the mode I thermal stress inten(2) sity factors increases with a decrease of the stiffness parameter β for both Exx = 0.5 and

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a

b Fig. 3  Influences of the thermal conductivity parameter δ and kxy0 on the normalized thermal stress intensity factors, h = 1.0. a mode I . b mode II

(2) (2) = 2.0; while increases with an increase of Exx Exx regardless of the value of the stiffness parameter β. For the mode II thermal stress intensity factors, the contrary is the case. Figure 5a, b show the II effects of the thermal expansion parameter γ and on the mode I and II thermal stress intensity factors. It may be obtained that the absolute values of both mode I and mode II thermal stress intensity factors increases with an increase of the thermal expansion parameter γ for either kxy0 = 0.5 or kxy0 = 2.0; and the absolute values of both mode I and mode II thermal stress intensity factors increases with an (2) increase of αxx .

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a

b (2) Influences of the stiffness parameter β and Exx on the normalized mode thermal stress intensity fac‑

Fig. 4  tors. a I . b mode II

Figure  6a, b illustrate the effects of different thickness of functionally graded orthotropic strip on the mode I and II thermal stress intensity factors when δ = −1.0 and δ = 1.0, respectively. We can see that the mode I and thermal stress intensity factors increase or decrease with the increasing of h, and then reach a steady value.

Conclusions In this paper, thermo-mechanical stress and displacement fields for an interface crack between an orthotropic functionally graded interlayer and two homogeneous orthotropic media are obtained. In addition to the mechanical fields, temperature fields are

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a

b (2)

Fig. 5  Influences of the thermal expansion parameter γ and αxx on the normalized mode thermal stress intensity factors. a I . b II

also developed for exponentially varying thermal properties along the gradation direction. TSIFS are numerically calculated based on a singular integral equation derived from the dislocation density along the crack faces. The variations in temperature distribution and the thermal stress intensity factors due to the change in non-homogeneity parameters of the material thermo-elastic properties, the orthotropy parameters and the dimensionless thermal resistance are investigated.

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a

b Fig. 6  Influences of the thickness h of the functionally graded orthotropic strip on the normalized mode I and mode II thermal stress intensity factors, a δ = −1.0, b δ = 1.0

Authors’ contributions SHD established the model and completed the derivation and calculation. XL analyzed the numerical results. Both authors read and approved the final manuscript. Acknowledgements Financial support from the National Natural Science Foundation of China (11261045, 11261041, 11472193), the China Scholarship Council (CSC), the Fundamental Research Funds for the Central Universities (1330219162) and Shanghai Pujiang Program (14PJ1409100) are gratefully acknowledged. Competing interests The authors declare that they have no competing interests.

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Appendix 1 The expressions of Mj (w)(j = 1 − 6) are given by  �1 ip1 (o2 −s2 )es2 h  φ(x)eiωx dx M1 (ω) =  s h s h 2 1  2π ω((s1 −o2 )(p1 −s2 )e +(o2 −s2 )(p1 −s1 )e ) −1   �1  ip1 (s1 −o2 )es1 h   M2 (ω) = φ(x)eiωx dx   2π ω((s1 −o2 )(p1 −s2 )es1 h +(o2 −s2 )(p1 −s1 )es2 h ) −1  �1 i[s1 (o2 −s2 )es2 h +s2 (s1 −o2 )es1 h ] iωx dx M3 (ω) = s1 h +(o −s )(p −s )es2 h ) −1 φ(x)e  2π ω((s −o )(p −s )e 1 2 1 2 2 2 1 1     M4 (ω) = M5 (ω) = 0     �1  ip1 (s1 −s2 )e(s1 +s2 −o2 )h  M6 (ω) = φ(x)eiωx dx 2π ω((s −o )(p −s )es1 h +(o −s )(p −s )es2 h ) −1 1

2

1

2

2

2

1

(34)

1

The kernel function H (x, u) is

H(x, u) =



+∞



0

  s1 p1 (o2 − s2 )es2 h + s2 p1 (s1 − o2 )es1 h −2 + Bi − 1 sin[ω(u−x)]dω kxy0 ω (s1 − o2 )(p1 − s2 )es1 h + (o2 − s2 )(p1 − s1 )es2 h

(35)

The expressions of ξj , qj (j = 1 − 8), dj (j = 1 − 4), χj (j = 1, 2) are given by

   2 2 − θ12 C22 γ + sj γ + sj + β θ22 C12   2  2 + C66 ω2 θ12 + γ + sj + β θ22 , j = 1, 2

ξj = iωMj



   2 2 ξj = ω2 Mj−2 − θ22 C11 γ + sj−2 + β θ12 C12 2    2 2 2  j = 3, 4 ω θ1 + γ + sj−2 + β θ22 + γ + sj−2 C66

     2 2 2 ξ5 = iωM3 p12 θ22 C12 − θ12 C22 ω2 θ12 + p12 θ22 + C66

     2 2 2 ξ6 = ω2 M3 p1 θ12 C12 − θ22 C22 ω2 θ12 + p12 θ22 + p1 C66      2 2 2 ξ7 = iωM6 o22 θ22 C12 − θ12 C22 ω2 θ12 + o22 θ22 + C66      2 2 2 ξ8 = ω2 M6 o2 θ12 C12 − θ22 C11 ω2 θ12 + o22 θ22 + o2 C66 qj (ω) =

  2   2 + βC 2 iω mj C12 + C66 12

qj (ω) =

 2  2 iωnj−4 C12 + C66

2 − ω2 C 2 mj (mj + β)C22 66

2 − ω2 C 2 n2j−4 C22 66

,

,

j =1−4

j =5−8

(36)

(37) (38) (39) (40) (41)

(42)

(43)

2 2 2 4 dj (ω) = C66 C12 [C11 ω + ω2 [(γ + sj )2 + (γ + sj + β)2 ] + (γ + sj )2

2 2 2 2 2 (γ + sj + β)2 C22 ] + ω2 (γ + sj )(γ + sj + β)[(C12 ) − C11 C22 ],

j =1−2

(44)      2 2 2 2 2 2 2 2 2 d3 (ω) = C66 C11 ω + 2ω2 p12 C12 + p14 C22 − C11 C22 + ω2 p12 C12

(45)

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     2 2 2 2 2 2 2 2 2 d4 (ω) = C66 C11 ω + 2ω2 o22 C12 + o42 C22 − C11 C22 + ω2 o22 C12 χ1 =

θ22 2 γ δC22

χ2 = −

θ22 2 δC22 (γ

(46)

(47)

− δ)

Appendix 2 The expressions of Kij (x, u)(i, j = 1, 2),ω1 (x)T , ω2 (x)T are given by 





+∞

K12 (x, u) = lim



+∞

2i  ω kxy0



 A22 D26 n2 y A21 D25 n1 y e − e cos[ω(u − x)]dω D D

(49)

K21 (x, u) = lim



+∞

2i  ω kxy0



 A24 D16 n2 y A23 D15 n1 y e − e cos[ω(u − x)]dω D D

(50)

K22 (x, u) = lim



+∞



K11 (x, u) = lim

y→0−

y→0−

y→0−

y→0−

0

0

0

0

2  ω kxy0

2  ω kxy0





A22 D16 n2 y A21 D15 n1 y e − e D D

A24 D16 n2 y A23 D15 n1 y e − e D D





− 1 sin[ω(u−x)]dω

(48)



− 1 sin[ω(u−x)]dω

(51)

 � � 8  � �  +∞ T  Ij J1j − B23 e−iωx dx   ω1 (x) = −∞ j=1 � � 8  � +∞ �  T  Ij J2j − B21 − B22 e−iωx dx   ω2 (x) = −∞

(52)

j=1

with

I1 = ξ5 /d3 −

2 

ξj /dj

j=1

I3 = e(γ +o2 )h q7 /d4 −

I5 = B23 −

2  j=1

F1j

I2 = ξ6 /d3 −

2 

e(γ +sj )h ξj /dj

j=1

I6 = B21 + B22 −

I7 = e(γ +o2 )h B31 + eo2 h B32 −

2  j=1

2 

(53)

ξj+2 /dj

j=1

I4 = e(γ +o2 )h q8 /d4 − 2 

2 

e(γ +sj )h ξj+2 /dj

j=1

(54)

B1j

(55)

j=1

e(γ +sj )h B1j

I8 = e(γ +o2 )h B33 −

2 

e(γ +sj )h F1j

j=1

(56)

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j+1

J1j = (−1)



A22 Dj6 A21 Dj5 − D D

2 2 A1j = (−iω)C12 + C22 mj qj



j+1

J2j = (−1)



  2 F1j = C66 (γ + sj )ξj − iωξj+2 /dj

2 2 A32 = C22 n4 q8 − iωC12

B32 = −θ22 M6 eγ h

2 E1j = C66 (mj − iωqj )

2 2 A22 = C22 n2 q6 − iωC12

B22 = −θ22 M3

2 A24 = C66 (n2 − iωq6 )

2 B23 = C66 [(γ + p1 )ξ5 − iωξ6 ]/d3

2 2 A31 = C22 n3 q7 − iωC12

  2 2 B31 = C22 (γ + o2 )ξ8 − iωC12 ξ7 /d4

2 A33 = C66 (n3 − iωq7 )

2 A34 = C66 (n4 − iωq8 )

(57)

(58)

j =1−4

  2 2 B21 = C22 (γ + p1 )ξ6 − iωC12 /d3 2 A23 = C66 (n1 − iωq5 )



j =1−4

  (2) 2 2 B1j = (−iω)C12 ξj + C22 (γ + sj )ξj+2 /dj − θ2 Mj

2 2 A21 = C22 n1 q5 − iωC12

A23 Dj5 A24 Dj6 − D D

2 B33 = C66 [(γ + o2 )ξ7 − iωξ8 ]/d4

(59)

(60) (61) (62)

(63) (64) (65)

(66) (67)

Here D is the determinant of the Dij (i, j = 1 − 8). Dij is the sub-determinant of the linear system of Eqs. (24)–(26) corresponding to the elimination of the ith row and jth column. Received: 13 January 2016 Accepted: 18 August 2016

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