Thermoelastic instability of a functionally graded layer interacting with a homogeneous layer

International Journal of Mechanical Sciences 99 (2015) 218–227

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Thermoelastic instability of a functionally graded layer interacting with a homogeneous layer Jia-Jia Mao, Liao-Liang Ke n, Yue-Sheng Wang Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 23 January 2015 Received in revised form 16 May 2015 Accepted 21 May 2015 Available online 29 May 2015

The interaction between the thermoelastic distortion and pressure-dependent thermal contact resistance can cause the themoelastic instability when the conductive heat transfers between two elastic bodies in the static frictionless contact. Using the perturbation method, this paper investigates the thermoelastic instability of a system consisting of a functionally graded material (FGM) layer and a homogeneous layer under the plane strain state. The two layers are pressed together by a uniform pressure, and transmit a uniform heat flux at their interface. The material properties of the FGM layer are assumed to be of exponential variation along the thickness direction. The thermoelastic stability behavior of five types of material combinations depending on the ratio of their material properties are discussed in details. The results imply that the FGMs can be used to improve the thermoelastic contact stability of the systems. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Thermoelastic instability Functionally graded materials Contact mechanics Critical heat flux

1. Introduction The thermoelastic deformations caused by the conductive heat transfer between two dissimilar contacting bodies could affect the contact pressure distribution and the extent of the contact area, which will generally affect the boundary conditions of the heat conduction problem in return. As a result, the thermoelastic instability may occur between the contacting bodies. Generally, the thermoelastic instability problem can be identified as two distinct sorts, i.e., the frictionally-excited thermoelastic instability [1] and the static thermoelastic instability [2,3], either because of the frictional heating or because of the pressure-dependent thermal contact resistance. The frictionally-excited thermoelastic instability implies that the sliding contact system involves a frictional heating. Instabilities due to the frictional heating are found in energy dissipating systems, such as brakes and crutches. Dow and Burton [4] investigated the effect of material properties on the frictionallyexcited thermoelastic instability for the case of a scraper or blade sliding normal to its line of contact on a thermally conductive semi-infinite body. Burton et al. [5] analyzed the problem of two sliding flat plates contacting on a straight common edge under a pressure perturbation on the interface. Using the perturbation

n

Corresponding author. Tel.: þ 86 10 51685755; fax: þ 86 10 51682094. E-mail address: [email protected] (L.-L. Ke).

http://dx.doi.org/10.1016/j.ijmecsci.2015.05.018 0020-7403/& 2015 Elsevier Ltd. All rights reserved.

method, Lee and Barber [6] evaluated the influence of finite disk thickness on the stability behavior of an automotive disk brake through a model of a finite thickness layer sliding between two half-plane. Yi et al. [7] developed the finite element method to reduce the thermoelastic instability problem for a brake disk to an eigenvalue problem with the critical speed. They explored the effect of the geometric complexity on the critical speed and the associated mode shape. Yi et al. [8] further determined the critical sliding speed for the thermoelastic instability of an axisymmetric clutch or brake. Lee [9] used a finite layer model with one-sided frictional heating to study the instability in automotive drum brake systems. He also performed some vehicle tests to observe the critical speeds of the drum brake systems with aluminum drum materials. Decuzzi et al. [10] studied thermoelastic instability of a two-dimensional model of a clutch/brake with an infinite number of disks. They analyzed the influence of the disk thickness ratio on the critical speed, critical wave parameter and migration speed of the system. In the static thermoelastic contact, if the heat conducts across the interface between two bodies, the thermoelastic instability could be caused by the interaction of the thermoelastic distortion and pressure-dependent thermal contact resistance. The static thermoelastic instability is visual for many industrial settings, i.e., castings, molding, valves, pistons, thermostats, thermal expansion of railways, cylinder heads, etc. [11]. Barber [12] analyzed the unstable of two half-planes under a sinusoidal contact pressure perturbation on a nominally uniform pressure for sufficiently large

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

heat fluxes. Zhang and Barber [13] investigated the influence of material properties on the stability criterion for the thermoelastic contact between two half-planes. They classified the material combinations into five categories depending upon the ratios of the thermal conductivities, diffusivities and distortivities. Using Zhang and Barber's classification [13], Yeo and Barber [14] considered the effect of a finite geometry on the stability in the system of a layer and a half-plane. Li and Barber [15] discussed the thermoelastic stability of a system consisting of two layers in contact using the perturbation method. Furthermore, the static thermoelastic instability problem was also investigated by Schade et al. [16] for two bonded half-plane and Schade and Karr [17] for a layer bonded to a half-plane. Specially, Ciavarella and his co-authors presented comprehensive studies on the thermoelastic instability problems by considering the combined effects of the pressure-dependent thermal contact resistance and frictional heating. Ciavarella et al. [18] considered thesliding of a one-dimensional rod against a rigid plane with frictional heating and a pressure-dependent thermal contact resistance. Afferrante and Ciavarella [19,20] studied an elastic conducting half-plane sliding against a rigid perfect conductor wall or two half-planes sliding out-of-plane by considering the combined effects. Afferrante and Ciavarella [21] analyzed the Aldo model by introducing the effect of frictional heating. Ciavarella and Barber [22] concerned the stability boundary for the thermoelastic contact of a rectangular elastic block sliding against a rigid wall in the presence of the thermal contact resistance. Functionally graded materials (FGMs) are usually a mixture of two distinct material phases with continuously varying volume fractions of constituent materials, hence their effective material properties change in a continuous and smooth manner. FGMs used as coatings or interfacial zones can reduce the magnitude of residual and thermal stresses, mitigate stress concentration, increase fracture toughness, and resist the contact damage [23–31]. Recently, the thermoelastic instability problems of FGMs were also concerned by many investigators due to their potential application to improve thermolastic stability behaviors in the brake disk system. For the frictionally-excited thermoelastic instability, Jang and his co-authors presented comprehensive works, such as, a stationary FGM layer between two sliding homogeneous layers [32], an FGM half-plane sliding against a homogeneous half-plane [33] and an FGM layer sliding against two homogeneous half-plane [34]. Their studies showed that FGMs could improve the contact stability in the frictional sliding system through an optimal gradient index of FGMs. Hernik [35] dealt with the study of the global thermoelastic instability of a brake disk made of either the isotropic homogeneous metal matrix composite or FGMs. For the static thermoelastic instability, Mao et al. [36] investigated the stability behaviors for three types of material combinations between an FGM layer and a homogeneous half-plane. The results showed that the finite geometry and the gradient index of the FGMs have a great influence on the static thermoelastic instability behavior of them systems. In this paper, we discuss the static thermoelastic instability between an FGM layer and a homogeneous layer under the plane strain state by using the perturbation method. The two layers are pressed together by a uniform pressure, and transmit a uniform heat flux at their interface. The thermoelastic properties of the FGM layer, such as the shear modulus, thermal conductivity coefficient, thermal expansion coefficient, specific heat, density, are assumed to be exponential along the thickness direction. Because of the imperfect contact between these two layers, a pressure-dependent thermal contact resistance is considered at the interface, which depends on the ratios of the material properties of two layers. The effects of the gradient index and layer thickness on the critical heat flux are discussed for five types of material combinations.

219

2. Formulation of the thermoelastic instability problem Fig. 1 shows the frictionless contact between an FGM layer (0 r yr h1 ) and a homogeneous layer (  h2 r y o 0) at their common surface y ¼ 0 where h1 and h2 are the thickness of the FGM layer and homogeneous layer, respectively. The two layers are pressed together by a uniform pressure p0 . A uniform heat flux qy ¼ q0 is imposed at the top surface y ¼ h1 and bottom surface y ¼ h2 , flowing across their common interface in the positive y-direction. The thermoelastic properties of the FGM layer are assumed to be the exponential forms as μðyÞ ¼ μb eβy ; β ¼ lnðμt =μb Þ=h1 ;

ð1aÞ

kðyÞ ¼ kb eδy ; δ ¼ lnðkt =kb Þ=h1 ;

ð1bÞ

αðyÞ ¼ αb eγy ; γ ¼ lnðαt =αb Þ=h1 ;

ð1cÞ

cðyÞ ¼ cb eεy ; ε ¼ lnðct =cb Þ=h1 ;

ð1dÞ

ρðyÞ ¼ ρb eςy ; ς ¼ lnðρt =ρb Þ=h1 ;

ð1eÞ

where μðyÞ, kðyÞ; αðyÞ, cðyÞ and ρðyÞ are the shear modulus, thermal conductivity coefficient, thermal expansion coefficient, specific heat, density, respectively; β, δ; γ, ε and ς are the gradient indexes; the Poisson's ratio v is assumed as a constant for simplicity; subscripts “b” and “t” refer to the bottom and top of the FGM layer, respectively. It should be pointed that the metal and ceramic phases are mixed in different volume ratios to form the multi-layered structures in the real FGMs. It can be fabricated by powder metallurgy, chemical vapor deposition, plasma spraying process, centrifugal method [37]. The real material profile of FGMs is of step changes across the thickness, which is often modeled as the linear function or power-law function [38–40]. However, the contact and crack problems of FGMs for linear case or power-law case are quite difficult to obtain the analytical solutions. Therefore, most investigators tend to use the exponential model to analyze the contact and crack problems of FGMs because the exponential model can solve the problems analytically. By using this simple model, we can get an insight on the thermoelastic stability behavior of FGMs. The results may have the potential application on improving the contact stability of the systems. That is why the exponential model is used in the present paper. y q0 p0

h1

α y μ y k y c y ρ y x

h2

α

μ k c ρ

p0 q0 Fig. 1. An FGM layer on a homogeneous layer pressed by a uniform pressure and transmitting heat.

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J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

2.1. Temperature perturbation

U y1 ðyÞ ¼

Following Li and Barber [15], we assume the stability of the system under the perturbation in temperature with an exponential growth rate. Hence, the temperature perturbation T j ðx; y; t Þ can be written as T j ðx; y; t Þ ¼ f j ðyÞebt þ imx ; j ¼ 1; 2; ð2Þ pffiffiffiffiffiffiffiffi where i ¼  1; subscripts “1” and “2” indicate the FGM layer and the homogeneous layer, respectively; f j ðyÞ are complex functions; m is the wave number; and the exponential growth rate b could be either real or complex. Note that the instability will occur if b is positive real or complex with a positive real part. Our aim is to find the critical heat flux when b is zero or a pure imaginary number [15]. Refer to Mao et al. [36], the temperature perturbations T 1 ðx; y; tÞ and T 2 ðx; y; tÞ in the two layers could be expressed as   ð3Þ T 1 ðx; y; t Þ ¼ C 11 eη11 y þ C 12 eη12 y ebt þ imx ;   T 2 ðx; y; t Þ ¼ C 21 e  η21 y þ C 22 eη21 y ebt þ imx ;

ð4Þ

with

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "  ffi#  ffi# 1 b 1 b  δ þ δ2 þ4 m2 þ η11 ¼  δ  δ2 þ 4 m2 þ ; η12 ¼ : 2 λ1 2 λ1 "

ð5Þ

4 X

B1l ef 1l y þ B15 eðγ þ η11 Þy þ B16 eðγ þ η12 Þy ;

ð12Þ

l¼1

where A11 ; A12 ; …; A16 and B11 ; B12 ; …; B16 are unknown constants related to the boundary conditions; and 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u u 16 3  Θ1 7 f 11 ¼ 4  β  t4m2 þ β2  4imβ 5; 2 Θ1 þ 1 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 u u 16 3  Θ1 7 f 12 ¼ 4  β  t4m2 þ β2 þ 4imβ 5; 2 Θ1 þ 1

ð13Þ

2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u 16 3  Θ1 7 t 2 2 f 13 ¼ 4  β þ 4m þ β  4imβ 5; 2 Θ1 þ 1

2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u 16 3  Θ1 7 2 t f 14 ¼ 4  β þ 4m2 þ b þ 4imβ 5; 2 Θ1 þ1

2 f 1l þ βf 1l ðΘ1  1Þ  m2 ðΘ1 þ 1Þ  ; l ¼ 1; 2; 3; 4; B1l ¼ isl A1l ; sl ¼ 2f 1l þβðΘ1 1Þ m

ð14Þ



B15 ¼

N 1 A15 N 2 A16 ; B16 ¼ ; imM 1 imM 2

ð15Þ

ð16Þ

h i  1  2Þ P j  Q j þ 2ðΘ Θ1  1 β β þ γ þ η1j h i; i Mj ¼ h 2ðΘ1  2Þ 4Θ1 m2 1 1 2 Pj Θ Θ1 þ 1P j  Θ2  1 þ m Q j Q j þ Θ1  1 β

ð17Þ

where C 11 , C 12 , C 21 and C 22 are unknown constants to be solved; λj ¼ kj =ρj cj (j ¼ 1; 2) are the thermal diffusivity coefficients of the FGM layer and homogeneous layer, respectively; δ ¼ ε þ ς is the gradient index of the thermal diffusivity coefficient of the FGM layer.

i  h  1 4Θ1 m2 2 1 β þ γ þ η1j Θ Θ1 þ 1P j  Θ2  1 þ m Q j 1 h i; i Nj ¼ h 2ðΘ1  2Þ 4Θ1 m2 1 1 2 Pj Θ Θ1 þ 1P j  Θ2  1 þm Q j Q j þ Θ1  1 β

ð18Þ

2.2. Thermoelastic stress and displacement fields

Pj ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b η21 ¼ m2 þ : λ2

ð6Þ

1

1

We assume that the displacement fields of the two layers induced by the temperature perturbation can be written as

Θ1 þ 1 Θ1 þ 1 ðγ þ η1j Þ2 þ β ðγ þη1j Þ  m2 ; Θ1  1 Θ1  1 3  Θ1 2 þ ðγ þ η1j Þ; j ¼ 1; 2; Qj ¼ β Θ1 1 Θ1  1

uxj ðx; y; t Þ ¼ U xj ðyÞebt þ imx ;

ð7Þ

A15 ¼

uyj ðx; y; t Þ ¼ U yj ðyÞebt þ imx ;

ð8Þ

Therefore, the displacement field of the FGM layer can be rewritten as " # 4 X f 1l y ðγ þ η11 Þy ðγ þ η12 Þy A1l e þA15 e þ A16 e ð21Þ ebt þ imx ; ux1 ðx; y; tÞ ¼

where j ¼ 1; 2; uxj ðx; y; tÞ and uyj ðx; y; tÞ are the displacements in the x- and y-directions, respectively; U xj ðyÞ and U yj ðyÞ are complex functions of the real variable y.

4imα1 M 1 4imα1 M 2 C 11 ; A16 ¼ C 12 : Θ1  1 Θ1  1

ð19Þ ð20Þ

l¼1

2.2.1. The FGM layer The governing equations of the FGM layer under the plane strain state are expressed as !   2 ∂2 ux1 ∂2 uy1 ∂ux1 ∂uy1 4α1 eγy ∂T 1 ∇2 ux1 þ þ ¼ ; ð9Þ þ þ β Θ1  1 ∂x2 ∂x∂y ∂y ∂x Θ1  1 ∂x !

∂ uy1 ∂2 ux1 2 ∇2 uy1 þ þ Θ1  1 ∂y2 ∂x∂y   4α1 eγy ∂T 1 ðβ þ γ ÞT 1 þ ; ¼ Θ1  1 ∂y 2

  ∂uy1 β ∂ux1 þ ð 3  Θ1 Þ ð 1 þ Θ1 Þ þ Θ1  1 ∂y ∂x ð10Þ

where α1 ¼ αb ð1 þ ν1 Þ; Θ1 ¼ 3  4ν1 ; and ν1 is the Poisson's ratio of the FGM layer. Substituting Eqs. (3), (7) and (8) into Eqs. (9) and (10), we can obtain the solutions as [36] U x1 ðyÞ ¼

4 X l¼1

A1l ef 1l y þA15 eðγ þ η11 Þy þ A16 eðγ þ η12 Þy ;

ð11Þ

" uy1 ðx; y; tÞ ¼

4 X

isl A1l ef 1l y þ

l¼1

# N1 N2 A15 eðγ þ η11 Þy þ A16 eðγ þ η12 Þy ebt þ imx : imM 1 imM 2

ð22Þ Then, the stress field of the FGM layer can be expressed as (

σ y1 ðx; y; tÞ ¼

4  μ1 eβy X is f ðΘ1 þ 1Þ þimð3 Θ1 Þ A1l ef 1l y Θ1  1 l ¼ 1 l 1l     iðΘ1 þ 1Þμ1 N 1 γ þ η11 iμ1 imðΘ1  3Þμ1 þ   A15 eðβ þ γ þ η11 Þy mM1 ðΘ1  1ÞmM 1 Θ1  1    

iðΘ1 þ 1Þμ1 N 2 γ þ η12 iμ1 imðΘ1  3Þμ1 A16 eðβ þ γ þ η12 Þy ebt þ imx ; þ   ðΘ1  1ÞmM 2 mM 2 Θ1  1

ð23Þ "

   N 1 ðγ þ η11 Þy f 1l  msl ef 1l y A1l þ γ þη11 þ A15 e M1 l¼1    N 2 ðγ þ η12 Þy þ γ þ η12 þ A16 ebt þ imx : ð24Þ e M2

σ xy1 ðx; y; tÞ ¼ μ1 eβy

4  X

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

2.2.2. The homogeneous layer The governing equations of the homogeneous layer are expressed as ! 2 ∂2 ux2 ∂2 uy2 4α2 ∂T 2 ; ð25Þ þ ∇2 ux2 þ ¼ Θ2  1 ∂x2 ∂x∂y Θ2  1 ∂x ∇2 uy2 þ

2

∂ uy2 ∂ ux2 2 þ Θ2  1 ∂y2 ∂x∂y 2

! ¼

4α2 ∂T 2 ; Θ2  1 ∂y

ð26Þ

where α2 ¼ α2 ð1 þ ν2 Þ; Θ2 ¼ 3  4ν2 ; and ν2 is the Poisson's ratio of the homogeneous layer. Substituting Eqs. (4), (7), and (8) into Eqs. (25) and (26), the solutions can be solved as [36] U x2 ðyÞ ¼ ðA21 þ A22 yÞemy þ ðA23 þ A24 yÞe  my þ A25 e  η21 y þ A26 eη21 y ; ð27Þ U y2 ðyÞ ¼ ðB21 þ B22 yÞemy þ ðB23 þ B24 yÞe  my þB25 e  η21 y þ B26 eη21 y ; ð28Þ where A21 ; A22 ; …; A26 and B21 ; B22 ; …; B26 are unknowns related to the boundary conditions, B21 ¼  iA21 þ

iΘ2 iΘ2 A22 ; B22 ¼  iA22 ; B23 ¼  iA23 þ A24 ; m m

B24 ¼  iA24 ; B25 ¼ A25 ¼ 

iη21 iη A25 ; B26 ¼  21 A26 ; m m

4imα2 4imα2   C 21 ; A26 ¼  2 C 22 : η221  m2 ðΘ2 þ 1Þ η21  m2 ðΘ2 þ 1Þ

ð29Þ ð30Þ ð31Þ

 ux2 ðx; y; tÞ ¼ ðA21 þ A22 yÞemy þ ðA23 þ A24 yÞe  my þ A25 e  η21 y þ A26 eη21 y ebt þ imx ;

ð32Þ 

        Θ2 Θ2 uy2 ðx; y; tÞ ¼  i A21 þ y  A22 emy þ i A23 þ y þ A24 e  my m m

 iη  þ 21 A25 e  η21 y  A26 eη21 y ebt þ imx : ð33Þ m

Then, the stress field of the homogeneous layer is expressed as  σ y2 ðx; y; tÞ ¼ iμ2 ½ð1 þ Θ2 ÞA22  2mðA21 þ A22 yÞemy  ½ð1 þ Θ2 ÞA24 þ 2mðA23 þ A24 yÞe  my    2m A25 e  η21 y þ A26 eη21 y ebt þ imx ; ð34Þ

ð35Þ

ð39Þ

where ∂T 1 ðx; y; tÞ ∂y   Θ1  1 η11 A15 η11 y η12 A16 η12 y δy bt þ imx e þ e ; e e ¼ ik1 4mα1 M1 M2

q1 ðx; y; t Þ ¼ k1 eδy

ð40Þ

∂T 2 ðx; y; t Þ ∂y  2  η21  m2 ðΘ2 þ 1Þ ð A25 e  η21 y þ A26 eη21 y Þebt þ imx : ¼ ik2 η21 4mα2 ð41Þ

q2 ðx; y; t Þ ¼ k2

Obviously, there are twelve independent unknowns (i.e., A11 ; A12 ; A13 ; A14 ; A15 ; A16 ; A21 ; A22 , A23 , A24 , A25 and A26 ) in the displacement and stress fields of the two layers. Those unknowns can be solved by the above boundary conditions and the following thermal contact resistance simultaneously. It is well-known that a thermal contact resistance exists at the interface even if there is a full contact between the two bodies. That is because nominally flat surface are always rough at the microscopic scale. It is generally agreed that the thermal contact resistance is a monotonically decreasing function of the contact pressure [19]. According to Barber [12], the pressure-dependent thermal contact resistance R can be defined as q¼

Hence, the displacement field of the homogeneous layer is obtained as

 σ xy2 ðx; y; tÞ ¼ μ2 ½2mA21 þ ð2my þ 1 Θ2 ÞA22 emy  ½2mA23 þ ð2my  1 þ Θ2 ÞA24 e  my  2η21 A25 e  η21 y  þ 2η21 A26 eη21 y ebt þ imx :

uy1 ðx; 0; t Þ ¼ uy2 ðx; 0; t Þ; q1 ðx; 0; t Þ ¼ q2 ðx; 0; t Þ;

221

Tn ; RðpÞ

ð42Þ

where T n is the temperature drop across the interface; q is the heat flux; and p is the pressure. For a small perturbation under the steady state, we have [12] R0 Δq þ q0 ΔR ¼ ΔT;

ð43Þ

where

  ΔR ¼ R0 Δp; R0 ¼ dRðpÞ=dp; R0 ¼ R p0 :

ð44Þ

The perturbations of the contact pressure Δp, temperature drop ΔT, and heat flux Δq at the interface can be expressed as ( 4  1 X s f ðΘ1 þ 1Þ þ mð3  Θ1 Þ A1l Δp ¼  σ y1 ðx; 0; tÞ ¼ iμ1 Θ1 1 l ¼ 1 l 1l     ðΘ1 þ1ÞN 1 γ þ η11 1 m ð Θ1  3 Þ þ A15   mM 1 Θ1  1 ðΘ1  1ÞmM 1    

ðΘ1 þ1ÞN 2 γ þ η12 1 m ð Θ1  3 Þ A16 ebt þ imx ; þ   ð45Þ mM 2 Θ1  1 ðΘ1  1ÞmM 2 ΔT ¼ T 2 ðx; 0; t Þ  T 1 ðx; 0; t Þ "   # η221  m2 ðΘ2 þ 1Þ Θ1  1 A15 A16 ðA25 þ A26 Þ  þ ebt þ imx ; ¼ 4imα2 4imα1 M 1 M 2 ð46Þ

2.3. Boundary conditions and thermal contact resistance relation In the present frictionless contact problem, the two layers are brought into contact by a uniform pressure p0, and transmit a uniform heat flux q0 at their interface. Therefore, the perturbation on the heat fluxes and stresses at the upper surface (y ¼ h1 ) of the FGM layer and the bottom surface (y ¼ h2 ) of the homogeneous layer are zero, i.e., σ xy1 ðx; h1 ; t Þ ¼ 0; σ y1 ðx; h1 ; t Þ ¼ 0; q1 ðx; h1 ; t Þ ¼ 0;

ð36Þ

σ xy2 ðx;  h2 ; t Þ ¼ 0; σ y2 ðx;  h2 ; t Þ ¼ 0; q2 ðx;  h2 ; t Þ ¼ 0:

ð37Þ

At the interface y ¼ 0, the contact requires the continuity of the stresses, displacements, and heat fluxes, σ xy1 ðx; 0; t Þ ¼ σ xy2 ðx; 0; t Þ ¼ 0; σ y1 ðx; 0; t Þ ¼ σ y2 ðx; 0; t Þ;

ð38Þ

 2  η  m 2 ð Θ2 þ 1 Þ ð  A25 þA26 Þebt þ imx : Δq ¼ q2 ðx; 0; t Þ ¼ ik2 η21 21 4mα2

ð47Þ

Substituting Eqs. (45)–(47) into Eq. (43), we obtain     4 q0 μb R0 X Θ1  1 Θ1  1 A15 þ q0 μb R0 W 2 þ A16 Ll A1l þ q0 μb R0 W 1 þ Θ1  1 l ¼ 1 4mα1 M1 4mα1 M2

 2    η21 m2 ðΘ2 þ 1Þ 1 þ k2 η21 R0 A25 4mα2  2    2 η m ðΘ2 þ 1Þ 1 þ k2 η21 R0  21 A26 ¼ 0: 4mα2 þ

ð48Þ

From Eq. (48) and boundary conditions (36)–(39), we can construct 12 homogeneous equations for 12 unknowns A11  A16 and A21  A26 . By setting the determinant of coefficient matrix of

222

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

the 12 equations as zero, the characteristic equation for the exponential growth rate b is expressed as             det g 1i ; g 2i ; g 3i ; g 4i ; g 5i ; g 6i ;             T g 7i ; g 8i ; g 9i ; g 10;i ; g 11;i ; g 12;i ¼ 0; ð49Þ where i ¼ 1; 2; …;12; thesuperscript “T” denotes the transposition  of a matrix; and g 1i  g 12;i are given in Appendix A. Introduce the following notations and dimensionless parameters β n δ n γ λ2 α2 k2 μ θ2 ; δ ¼ ; γ ¼ ; r1 ¼ ; r2 ¼ ; r3 ¼ ; r4 ¼ 2 ; r5 ¼ ; m m m λb α1 kb μb θ1

ð50Þ

rffiffiffiffiffiffiffiffiffiffiffiffi f η η η z n f 1l ¼ 1l ; ðl ¼ 1; 2; 3; 4Þ; ηn11 ¼ 11 ; ηn12 ¼ 12 ; ηn21 ¼ 21 ¼ 1 þ ; r1 m m m m

ð51Þ

βn ¼

P nj ¼ z¼

Qj Pj ; Q n ¼ ; N nj ¼ mN j ; M nj ¼ m2 M j ; H j ¼ mhj ðj ¼ 1; 2Þ; m m2 j

b ; Rn ¼ R0 kb =h1 ; Q n ¼  μb α1 q0 R0 ; m2 λ b

ð52Þ

where C 10 ðH 1 ; H 2 ; 0Þ ¼

D10 ðH 1 ; H 2 ; 0Þ D30 ðH 1 ; H 2 ; 0Þ ; C 20 ðH 1 ; H 2 ; 0Þ ¼ ; D20 ðH 1 ; H 2 ; 0Þ D20 ðH 1 ; H 2 ; 0Þ

ð57Þ

and D10 , D20 and D30 are results from D1 , D2 and D3 via the L’Hôpital's rule, respectively. For the complex root instability (i.e., z ¼ iw), Eq. (55) is complex and cannot be solved explicitly. However, we can separate the real and imaginary parts of Eq. (55) to obtain two real equations as   D3 D1 Rn þ Re Q n þ Re ¼ 0; ð58Þ D2 D2   D3 D1 Q n þ Im ¼ 0; Im D2 D2

ð59Þ

By solving Eq. (56) or Eqs. (58) and (59), we can obtain the stability boundary which is determined from the critical heat flux. When the conducting heat flux reaches a certain critical value, the system will change from stable to unstable.

ð53Þ 4. Results and discussion

where θ1 ¼ α1 =kb ; θ2 ¼ α2 =k2 :

ð54Þ

Note that θ2 and θ1 represent the thermoelastic distortivity [41] of the homogeneous layer and the bottom material of the FGM layer, respectively. Generally, a large value of θ indicates the great distortivity of a material. Using the above notation and dimensionless parameters, we can rewrite the characteristic Eq. (49) in the dimensionless form D1 ðH 1 ; H 2 ; zÞ þ D2 ðH 1 ; H 2 ; zÞRn þ D3 ðH 1 ; H 2 ; zÞQ n ¼ 0;

ð55Þ

where D1 ðH 1 ; H 2 ; zÞ; D2 ðH 1 ; H 2 ; zÞ and D3 ðH 1 ; H 2 ; zÞ are given in Appendix B. If we set all gradient indexes as zero, Eq. (55) can be reduced to the characteristic equation for two dissimilar homogeneous layers reported by Li and Barber [15].

3. Stability criterion Instability will occur if the solutions (i.e., the dimensionless exponential growth rate z) of the characteristic Eq. (55) is a positive real or a complex with a positive real part. Then, Eq. (55) can be used to solve the critical heat flux Q n corresponding to real roots or complex roots. Normally, when the instability occurs for real roots, the stability criterion is determined by setting z ¼ 0 which is called as the real root instability. When the instability occurs for complex roots, the stability criterion is determined by setting z ¼ iw (w is real) which is called as the complex root instability. For the real root instability (i.e., z ¼ 0), the relation between Rn and Q n can be obtained immediately from Eq. (55),    n  C 10 ðH 1 ; H 2 ; 0Þ  1 1 þ R þ1 ; Qn ¼  ð56Þ C 20 ðH 1 ; H 2 ; 0Þ C 20 ðH 1 ; H 2 ; 0Þ

In this section, a parametric study is presented to discuss the thermoelastic instability between an FGM layer and a homogeneous layer. To compare the stability boundaries for different material combinations, we use the same classification system as that of Zhang and Barber [13]. Their classification system has five types of material combinations, Types 1, 2, 3, 4 and 5, which are dependent on the dimensionless ratio of the materials' thermomechanical properties, r 1 , r 3 and r 5 . Type 1 is for the material combination satisfying r 1 4 1 and 0 or 5 o 1=r 1 ; Type 2 is for r 1 41 and 1=r 1 or 5 o 1; Type 3 is   4 is for 1 or 1 o r 5 and 3 1 þ 1=r 3 for 1 or5 o r 1 ; Type   r 5 =r 1  1  2ðr 5  1Þ 1 þ1=ðr 1 r 3 Þ 4 0 and Type 5 is for 1 o r 1 o r 5      and 3 1 þ 1=r 3 r 5 =r 1  1  2ðr 5  1Þ 1 þ 1=ðr 1 r 3 Þ r 0. For the sake of convenience, the gradient indexes of the shear modulus, thermal expansion coefficient and thermal diffusivity coefficient are assumed to be the same value in the FGM layer, i.e., βn ¼ γ n ¼ δn ¼ n in the following analysis. Indeed, in order to increase the accuracy of the simulation, we need to consider the gradient indexes βn ; γ n and δn on the thermoelastic stability behavior, respectively. This problem will be discussed in our future work. For the stability analysis of the two layers, the materials at the bottom of the FGM layer are selected as the nodular cast iron for Types 1, 2 and 3, while the material of the homogeneous layer is the SiC sintered for Type 1(θ1 4 θ2), the brass for Type 2 (θ1 4θ2), and the magnesium alloy for Type 3 (θ1 oθ2). As for Types 4 and 5, we select the SiC sintered as the bottom material of the FGM layer, the aluminum alloy homogeneous layer for Type 4 (θ1 oθ2) and the magnesium alloy homogeneous layer for Type 5 (θ1 oθ2). Table 1 shows the thermoelastic properties of the selected materials in the five types of material combinations [13]. Note that the positive/negative gradients of the FGM layer control the material properties increasing/decreasing exponentially from the bottom to the top of the FGM layer.

Table 1 Thermoelastic properties of selected materials. Properties

Aluminum alloy

Copper

Nodular cast iron

SiC sintered

Brass

Magnesium alloy

μ (GPa)

27.273 22.0

45.489 17.0

64.122 13.7

172.414 4.4

38.372 19.0

16.667 26.0

173.0 67.16 0.32

381.0 101.93 0.33

48.9 16.05 0.31

110.0 35.48 0.16

78.0 21.35 0.33

95.0 45.11 0.35

α (1C  110  6 ) k(W/m 1C) λ (mm2/s) ν

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

200

Table 2 Stability behavior for five types of material combinations in terms of the gradient index n with R* ¼1.0 and H 2 ¼ H 1 ¼ 1:50. Material combination

Gradient Index n

Critical heat flux

Root

Type 1

 2:0 r n r 0:6 0:6 o n r 2:0 1:3 o n r 2:0

Q n 40

Real Complex Real

Type 2

 2:0 r n r 0:2  0:5 r n r 0:3 0:3 o n r 2:0

Q n 40 Qn o0

Real Complex Real

Type 3

 2:0 r n o 0:0  2r nr 0:6 0:6 o n r 2

Q n 40 Qn o0

Real Complex Real

Type 4

 2:0 r n r 0:2 0:2 o n r 2

Q n 40

Complex Real

Type 5

 2:0 r n o  0:6 0:6 r n r 2

n

Q 40

*

Q o0

Complex: n = -0.5 n = 0.0 n = 0.5

Real:

150

Q

n

223

n = -0.5 n = 0.0 n = 0.5

100

50

0 0.0

0.2

0.4

0.6

0.8

1.0

*

Complex Real

1/(1+R ) Fig. 3. Stability boundaries as a function of R* for different values of n with H2 ¼ H1 ¼ 1.0 (Type 1).

150

550

Li and Barber's result [15]: Real Complex Present result: Real Complex

440

50

H2 = 0.75 H2 = 1.50 H2 = 4.50

Q

*

Q

*

330

Real: 100

220

-50

110

0 0.0

0

-100

0.2

0.4

0.6

0.8

1.0

-150 -2.0

Mao et al. [36]: Real Complex

Complex: H2 = 0.75 H2 = 1.50 H2 = 4.50 -1.5

-1.0

-0.5

Fig. 2. Critical heat flux Q*as a function of R* for an aluminum alloy layer and a copper layer with equal thickness H2 ¼ H1 ¼0.50: comparisons with existing results.

Obviously, by setting the gradient indexes of FGM layer as zero, i.e., βn ¼ γ n ¼ δn ¼ n¼0.0, the present problem can be reduced to the thermoelastic instability of two dissimilar homogeneous layers discussed by Li and Barber [15]. Fig. 2 presents the critical heat flux Qn as a function of Rn for an aluminum alloy layer and a copper layer with equal thickness H2¼H1¼0.50. For a direct comparison, Fig. 2 also provided Li and Barber’s results including both real and complex root instabilities. It can be seen that the present results are in accordance with Li and Barber’s results. In order to show the effect of the gradient index n on the stability behavior, Table 2 presents the stability behavior for the five types of material combinations in terms of the gradient index n with Rn ¼ 1.0 and H 2 ¼ H 1 ¼ 1:50. Note that “Real” and “Complex” indicate that the system exhibits the instability with a real root and a complex root, respectively. Obviously, the stability behavior is very sensitive to the gradient index of FGMs. For all five types of material combinations, the system can exhibit either complex root instability or real root instability for a certain range of the gradient index n. These results indicate that controlled gradients in thermoelastic properties of FGM layer can change the critical heat flux and stability boundaries, and hence modify the thermoelastic stability behavior of systems.

0.5

1.0

1.5

2.0

n

*

1/(1+R )

0.0

Fig. 4. The effect of H2 on the critical heat flux Q* versus gradient index n curves with H1 ¼1.50 and R* ¼ 1.0 (Type 1).

4.1. Type 1 material combination Fig. 3 shows the stability boundaries as a function of the thermal contact resistance Rn for different values of the gradient index n with H2 ¼ H1 ¼1.0. It is showed that the stability boundary is determined from the complex root for the positive gradient index (n ¼ 0:5), while it is determined from the real root for the negative gradient index (n ¼  0:5). However, for the contact between two homogeneous layers, i.e., n ¼ 0.0, there is a critical value of Rn, below which the critical heat flux is dependent on the complex root and above which it is dependent on the real root. The similar phenomenon was also observed by Mao et al. [36] and Li and Barber [15]. Obviously, the value of the critical heat flux Qn increases with the increase of the contact resistance Rn. That is to say, the increase of the contact resistance Rn can postpone the static thermoelastic instability. Fig. 4 presents the effect of H2 on the critical heat flux Qn versus gradient index n curves with H1 ¼1.50 and Rn ¼ 1.0. Note that Mao et al. [36] have studied the thermoelastic instability between an FGM layer and a homogeneous half-plane. Their results are also plotted in these figures for a comparison. When the heat flux transmits into the more distortive material (i.e. the FGM layer,

224

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

800

Real: n = -0.5 n = 0.0 n = 0.5

600 400

600

Complex: n = -0.5 n = 0.0 n = 0.5

Real:

200 *

0

Q

*

200

Q

n = -0.5 n = 0.5

400

0

-200

-200

-400 -600

-400

-800 0.0

0.2

0.4

0.6

0.8

1.0

-600 0.0

*

1/(1+R )

400

Real: 300 200

600

0.8

Mao et al. [36]: Real Complex

400 200 *

Q

*

Q

0.6

Mao et al. [36]: Real Complex

0 -200 -400

-300 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-600 -2.0

Real: H2 = 0.75 H2 = 1.50 H2 = 4.50

Complex: H2 = 0.75 H2 = 1.50 H2 = 4.50

-200

-400 -2.0

1.0

Fig. 7. Stability boundaries as a function of R* for different values of n with H2 ¼ H1 ¼ 1.0 (Type 3).

100

-100

0.4

1/(1+R )

Complex: H2 = 0.75 H2 = 1.50 H2 = 4.50

H2 = 0.75 H2 = 1.50 H2 = 4.50

0.2

*

Fig. 5. Stability boundaries as a function of R* for different values of n with H2 ¼ H1 ¼ 1.0 (Type 2).

0

Complex: n = -0.5 n = 0.0 n = 0.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

n

n Fig. 6. The effect of H2 on the critical heat flux Q* versus gradient index n curves with H1 ¼ 1.50 and R* ¼ 1.0 (Type 2).

Fig. 8. The effect of H2 on the critical heat flux Q* versus gradient index n curves with H1 ¼1.50 and R* ¼ 1.0 (Type 3).

Q n 4 0), the stability boundary is determined by both real and complex roots, and the critical heat flux is decreasing by the increased thickness of the homogenous layer. For the given thickness of the homogeneous layer, the critical heat flux first increases, and then decreases with the gradient index n increasing from  2.0 to 2.0. However, when the heat flux transmits into the less distortive material (i.e. the homogeneous layer, Q n o 0), the system is unstable only for the positive gradient index, and the critical heat flux is determined by the real root instability. For a thin homogeneous layer, the present results have a significant different with those given by Mao et al. [36]. However, for a thicker layer (H 2 Z 4:50), the present results are close to the results for a homogeneous half-plane in Mao et al. [36].

instabilities with the positive heat flux or the complex root instability with the negative heat flux. Specially, the latter case for n ¼  0:5 can only happens with the small Rn. Fig. 6 examines the effect of H2 on the critical heat flux Qn versus gradient index n curves with H1 ¼1.50 and Rn ¼1.0. For the thin homogeneous layer (H 2 r 1:50), the system will be unstable only for the real root with a certain range of gradient index under the positive heat flux, but exhibit both complex root instability and real root instability under the negative heat flux. With the increase of H 2 from 1.50 to 4.50, the complex root instability occurs for the positive heat flux, and the critical heat flux Qn decreases. Obviously, for a thicker homogenous layer (such as H 2 ¼ 4:50), the present results are close to the results given by Mao et al. [36].

4.2. Type 2 material combination

4.3. Type 3 material combination

Fig. 5 plots the stability boundaries as a function of the thermal contact resistance Rn for different values of the gradient index n with H2 ¼ H1 ¼1.0. When n ¼ 0:5, the system exhibits both real and complex root instabilities with the negative heat flux, but the critical heat flux is determined only by the real root instability; when n ¼ 0:0 and n ¼  0:5, the system exhibits real root

Fig. 7 depicts stability boundaries as a function of Rn for different values of the gradient index n with H2 ¼ H1 ¼1.0. It shows that the gradient index n does not have a significant effect on the real root stability boundaries, but it has considerable influence on the complex ones. For the negative gradient index (n ¼  0.5), the stability boundary is determined from the real root with the

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

0

-30

Q

*

-60

-90

Complex: n = -1.0 n = 0.0 n = 1.0

-120

-150 0.0

0.2

0.4

Real: n = -1.0 n = 0.0 n = 1.0

0.6

0.8

1.0

*

1/(1+R )

0

-50

-100

Q

*

Mao et al. [36]: Real Complex

-150

-200

-250 -2.0

-1.5

-1.0

-0.5

H2 = 0.75 H2 = 1.50 H2 = 4.50

0.0 n

0.5

Complex: H2 = 0.75 H2 = 1.50 H2 = 4.50 1.0

1.5

2.0

Fig. 10. The effect of H2 on the critical heat flux Q* versus gradient index n curves with H1 ¼ 1.50 and R* ¼ 1.0 (Type 4).

0

Fig. 9 shows the stability boundaries as a function of Rn for different values of the gradient index n with H2 ¼ H1 ¼1.0. Type 4 material combination permits the instability only for the negative heat flux. The critical heat flux is determined from the complex root for n ¼  1:0, while it is determined from the complex root with the larger Rn, and the real root with the smaller Rn for n ¼ 0:0 and n ¼ 1:0. Note that for the contact between two homogeneous layers (i.e., n¼0), its stability behavior is different from that of two homogeneous half-planes as described by Zhang and Barber [13] from which the instability occurs only for the real root. Fig. 10 discusses the effect of H2 on the critical heat flux Qn versus gradient index n curves with H1 ¼ 1.50 and Rn ¼ 1.0. It is shown that increasing the thickness of the homogeneous layer accelerates the instability of the system. The thick homogenous layer (i.e., H 2 ¼ 4:50) have the same unstable tendency with the half-plane reported by Mao et al. [36]. Furthermore, the critical heat flux is determined from the complex root for the negative n or positive n with a small value, and determined from the real root for positive n with large value. For example, the complex root instability occurs for  2:0 r n o  0:6, and the real root instability occurs for 0:6 rn r 2:0 when H 2 ¼ H 1 ¼ 1:50. For these cases, the absolute value of the critical heat flux decreases slightly as the gradient index increases from  2.0 to 2.0. 4.5. Type 5 material combination

-30

Fig. 11depicts the stability boundaries as a function of Rn for different values of the gradient index n with H2 ¼H1 ¼ 1.0. Only the negative heat flux can cause the instability for Type 5 material combination which is quite similar with Type 4 material combination. For Type 5 material combination, the effects of gradient index n and dimensionless thickness H2 on the critical heat flux Qn are also similar to those for Type 4 material combination, and therefore are not shown for brevity.

Q

*

-60

Real: -90

n = -1.0 n = 0.0 n = 1.0

Complex: n = -1.0 n = 0.0 n = 1.0

-120

-150 0.0

between the real root instability and complex root instability. Obviously, for the curve of two homogeneous layer, i.e., n ¼ 0.0, only the complex root instability occurs with the negative heat flux. Fig. 8 discusses the effect of H2 on the critical heat flux Qn versus gradient index n curves with H1 ¼1.50 and Rn ¼1.0. Similar to Types 1 and 2 material combinations, Type 3 material combination permits the instability for both positive and negative heat flux. For the positive heat flux, only the real root instability can occur with the negative gradient index. The thicker the homogeneous layer is, the more unstable the system is. However, for the negative heat flux, both real and complex root instabilities can occur. The critical heat flux changes rapidly for the thick homogeneous layer whereas it changes slowly for the thin homogeneous layer when n r 0. 4.4. Type 4 material combination

Fig. 9. Stability boundaries as a function of R* for different values of n with H2 ¼ H1 ¼ 1.0 (Type 4).

Real:

225

0.2

0.4

0.6

0.8

1.0

*

1/(1+R ) Fig. 11. Stability boundaries as a function of R* for different values of n with H2 ¼ H1 ¼ 1.0 (Type 5).

positive heat flux and complex root with the negative heat flux. For the positive gradient index (n ¼0.5), the stability boundary is determined from both real and complex roots with the negative heat flux. There is a critical value of Rn which is the boundary

5. Conclusions By the perturbation method, this paper studies the thermoelastic instability of a two-layered structure with an FGM layer and a homogeneous layer under the plane strain state. The two layers are pressed together by a uniform pressure, and transmit a uniform heat flux in the thickness direction. The characteristic equation is obtained to determine the stability boundary for five types of material combinations. The effects of the gradient index, layer thickness and material combination on the critical heat flux and stability boundaries are discussed in detail. The results of the

226

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

present analysis are validated by reducing the problem to the contact between two homogeneous layers. It is found that: (1) For Types 1, 2 and 3 material combinations, the systems can exhibit instability at both directions of the heat flux; (2) For Types 4 and 5, only the negative heat flux can cause the instability of systems; (3) The gradient index has a great effect on the critical heat flux for Types 1, 2, and 3, while the effect is less pronounced for Types 4 and 5; (4) For all five types, the absolute value of the critical heat flux Qn increases significantly with the increase of the thermal contact resistance; (5) The system with a thin homogeneous layer is generally more stable than that with a thick one; (6) For a thicker homogeneous layer (H 2 Z4:50), the present results are close to the results given by Mao et al. [36].

Acknowledgments The work described in this paper is supported by National Natural Science Foundation of China under Grant numbers 11272040 and 11322218, Program for New Century Excellent Talents in University under Grant number NCET-13-0656, and Beijing Higher Education Young Elite Teacher Project under Grant number YETP0562.

Appendix A



T   η eh1 η11 η12 eh1 η12 g 8i ¼ 0; 0; 0; 0; 11 ; ; 0; 0; 0; 0; 0; 0 ; M1 M2

ðA8Þ

  n g 9i ¼ 0; 0; 0; 0; 0; 0; 2meη21 h2 ;  ð1 þ R2 þ 2mh2 Þeη21 h2 ;

o 2meη21 h2 þ 2mh2 ; ð1 þ R2  2mh2 Þeη21 h2 þ 2mh2 ; 2me2η21 h2 þ mh2 ; 2memh2 ; ðA9Þ n

  g 10;i ¼ 0; 0; 0; 0; 0; 0; 2meη21 h2 ;  ð1 þ R2 þ 2mh2 Þeη21 h2 ;  2meη21 h2 þ 2mh2 ; ð1  R2 þ2mh2 Þeη21 h2 þ 2mh2 ; o  2η21 e2η21 h2 þ mh2 ; 2η21 emh2 ;

ðA10Þ

o   n g 11;i ¼ 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; eη21 h2 ; e  η21 h2 ;

ðA11Þ

   q0 μb R0 q0 μb R0 q0 μb R0 q0 μb R0 L1 ; L2 ; L3 ; L4 ; q0 μb R0 W 1 g 12;i ¼ Θ1  1 Θ1  1 Θ1  1 Θ1  1 Θ1  1 Θ1  1 ;q μ R0 W 2 þ ; þ 4α1 mM 1 0 b 4α1 mM 2   2  ðΘ2 þ 1Þ η21 k2 R0  1 η21 m2 ; 0; 0; 0; 0; 4mα2 )   2  ðΘ2 þ1Þ η21 k2 R0 þ 1 η21  m2  : 4mα2

ðA12Þ

  ðΘ1 þ 1ÞN 1 γ þ η11 1 mðΘ1  3Þ ;   mðΘ1  1ÞM 1 mM 1 Θ1  1   ðΘ1 þ 1ÞN 2 γ þ η12 1 mðΘ1  3Þ ; W2 ¼   mM 2 Θ1  1 mðΘ1  1ÞM 2 W1 ¼

ðA13Þ

  n 1 ðΘ1 þ 1ÞN1 γ n þ ηn11 Θ1  3 ;  n n Θ1  1 M1 ðΘ1  1ÞM 1  n  n n 1 ðΘ1 þ 1ÞN2 γ þ η12 Θ1  3 ;  W n2 ¼ n  Θ1  1 M2 ðΘ1  1ÞM n2

W n1 ¼ 



 N1 N2 Θ2 Θ2 η η ; ; 1;  ;  1;  ;  21 ; 21 ; g 1i ¼ s1 ; s2 ; s3 ; s4 ;  mM 1 mM 2 m m m m ðA1Þ

   N1 g 2i ¼ f 11 ms1 ; f 12  ms2 ; f 13  ms3 ; f 14  ms4 ; þ η11 M1

N2 þγ; þ η12 þ γ; 0; 0; 0; 0; 0; 0 ; M2  

g 3i



  ¼ 0; 0; 0; 0; 0; 0; 2m; 1  Θ2 ;  2m; 1  Θ2 ;  2η21 ; 2η21 ;

 g 4i ¼

ðA15Þ ðA2Þ Appendix B ðA3Þ D1 ðH1 ; H2 ; zÞ ¼

g6i

ðB1Þ

L1 eh1 f 11 L2 eh1 f 12 L3 eh1 f 13 L4 eh1 f 14 ¼ ; ; ; ; W 1 eh1 ðη11 þ γÞ ; W 2 eh1 ðη12 þ γ Þ ; 0; 0; 0; 0; 0; 0 ; Θ1  1 Θ1  1 Θ1  1 Θ1  1





n





 g 7i ¼ eh1 f 11 f 11  ms1 ; eh1 f 12 f 12  ms2 ;     eh1 f 13 f 13  ms3 ; eh1 f 14 f 14  ms4 ;    eh1 ðη11 þ γ Þ N 1 þ M 1 η11 þ γ ; M1 )    eh1 ðη12 þ γ Þ N 2 þ M 2 η12 þ γ ; 0; 0; 0; 0; 0; 0 ; M2

n

n

ðB2Þ

D3 ðH 1 ; H 2 ; zÞ ¼ 4ðΘ1  1Þηn12 M n1 W n1 F 1 n n  4ðΘ1  1Þeðη11  η12 ÞH1 ηn11 M n2 W n2 F 1 þ 4ðΘ1 1Þηn12 F 2 V 4

4 X m¼1

ðA6Þ 



D2 ðH 1 ; H 2 ; zÞ ¼  ðΘ1  1Þ2 eH1 ðη11  η12 Þ 1 ηn11 ηn12 H 1 F 1 ;

ðA5Þ 

h n i n 8 9  n < n n = eðη11  η12 ÞH1 1 e2η21 H2 þ 1 eðη11  η12 ÞH1 ηn11 ηn12  ηn11 ηn12 ðΘ1  1Þ2 F 1  2ηn H  n : ; e 21 2 1 η r 3 21

ðA4Þ

   k η ðΘ1  1Þ kb η12 ðΘ1  1Þ ; ; g 5i ¼ 0; 0; 0; 0; b 11 α1 M 1 α1 M 1  2   ) k2 η21 ðΘ2 þ 1Þ η21  m2 k2 η21 ðΘ2 þ 1Þ η221  m2 ; 0; 0; 0; 0; ; α2 α2



n

Ll ¼ ðΘ1 þ 1Þsl f 1l  mðΘ1  3Þ; Lnl ¼ ðΘ1 þ 1Þsl f 1l  ðΘ1  3Þ; l ¼ 1; 2; 3; 4:



L`1 μb L`2 μb L`3 μb L`4 μb ; ; ; ; μ W 1 ; μb W 2 ; 2mμ2 ; Θ1  1 Θ1  1 Θ1  1 Θ1  1 b   ðΘ2 þ1Þμ2 ; 2mμ2 ; ðΘ2 þ 1Þμ2 ; 2mμ2 ; 2mμ2 ;

ðA14Þ

h

Lm

4 X

i¼1 iam

Ei

4 4 X X

n

eijk eH1 f 1j

n

n f 1j  sj

j¼1 k¼1 j a m ka m

 n  n n n eðη11 þ γ ÞH1 ðΘ1 1Þ f 1k  sk Y 1 eH1 f 1j Lnk Y 2  n  o n n  Lnj eðη11 þ γ ÞH1 f 1k  sk Y 3



i ðB3Þ

where F1 ¼ ðA7Þ

4 X m¼1

Em

4 X

4 4 X  n X  n  n n ð  1Þi ef 1i H1 f 1i  si eijk ef 1i H1 f 1k  sk Lnj ;

i¼1

j¼1 k¼1 j am k am

iam

J.-J. Mao et al. / International Journal of Mechanical Sciences 99 (2015) 218–227

F2 ¼

4 X

Lnm

m¼1

4 X

4 4 X  n X  n  n n ef 1i H1 f 1i  si eijk ef 1i H1 f 1k  sk Lnj ;

i¼1 iam

j¼1 k¼1 j am ka m

   Em ¼ ð1 þ Θ2 Þ  1 þ e4H2 þ 4e2H2 H 2 Lnm þ 4sm r 4  1  4e4H2

þ 2e2H2 2 þ 4H 22 ðΘ1  1Þ; m ¼ 1; 2; 3; 4;

n n 2ðΘ1  1Þ 1  eðη11  η12 ÞH1 r 2 r 4 ηn11 ηn11 ðηn  ηn ÞH1 V ¼ G1  n e 11 12 G2 þ ðY 4  Y 5 Þ;  2ηn H  n2  η12 e 21 2  1 η21  1 r 3 ηn21 Y 1 ¼ ηn12 M n1 W n1  ηn11 M n2 W n2 ;     n n Y 2 ¼ γ n þ ηn11 M n1 þ N n1 ηn12  eðη11  η12 ÞH1 γ n þ ηn12 M n2 þ N n2 ηn11 ;  n     Y 3 ¼ γ þ ηn11 M n1 þ N n1 ηn12  γ n þ ηn12 M n2 þ N n2 ηn11 ;     n Y ¼ 2e2H2 2H  ηn þ e4H2 ηn þ 1  2eðη21 þ 1ÞH2 ðH  1Þ 4

2

21

21

2

n  2eðη21 þ 3ÞH2 ðH 2 þ 1Þ þ ηn21  1;   n n Y 5 ¼ 2eðη21 þ 1ÞH2 ðH 2 1Þ 2e2ðη21 þ 1ÞH2 2H 2 þ ηn21     n n n þ 2eðη21 þ 3ÞH2 ðH 2 þ 1Þ þ e2ðη21 þ 2ÞH2 ηn21  1 þ e2η21 H2 ηn21 þ 1 ; h

i Gi ¼ 4r 4 e4H2  2e2H2 1 þ 2H 22 þ 1 Nni þ ðΘ2   þ 1Þ e4H2  4e2H2 H 2 1 M ni W ni ; i ¼ 1; 2:

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