A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source

European Journal of Mechanics A/Solids 27 (2008) 607–621

A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source Sadek Hossain Mallik ∗ , M. Kanoria Department of Applied Mathematics, University of Calcutta, 92 A. P. C. Road, Kolkata 700009, India Received 3 July 2007; accepted 14 September 2007 Available online 22 September 2007

Abstract This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique. Numerical computations have been done for magnesium (Mg) and the results are presented graphically. The results for an isotropic material (Cu) have been deduced numerically and presented graphically to compare with those of transversely isotropic material (Mg). © 2007 Elsevier Masson SAS. All rights reserved. MSC: 74F Keywords: Generalized thermoelasticity; Green–Naghdi model; Transversely isotropic material; Spatially varying heat source

1. Introduction Thermoelasticity theories which admit a finite speed for thermal signals (second sound) have aroused much interest in the last three decades. In contrast to the conventional coupled thermoelasticity theory based on a parabolic heat equation (Chadwick, 1960; Nowacki, 1962, 1975), which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. Among these generalized theories, the theory proposed by Lord and Shulman (1967) involving one relaxation time and the one developed by Green and Lindsay (1972) involving two relaxation times have been subjected to a large number of investigations. In Lord–Shulman theory a modified Fourier law of heat conduction including both the heat and its time derivative replaces the conventional Fourier’s law whereas in G-L model Fourier’s law of heat conduction is left unchanged but the classical energy equation and stress–strain temperature relations are modified. In view of some experimental evidences available in favour of finiteness of heat propagation speed, generalized thermoelasticity * Corresponding author.

E-mail addresses: [email protected] (S.H. Mallik), [email protected] (M. Kanoria). 0997-7538/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2007.09.002

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theories are considered to be more realistic than the conventional thermoelasticity theory in dealing with practical problems involving very large heat fluxes and short time intervals, like those occurring in laser units and energy channels. The relevant literature can be found in Chandrasekharaiah (1986), Sherief (1987) and Ignaczak (1979, 1982, 1989). On the experimental side, available evidence in support of the existence of finite thermal wave speed in solids is rather sparse, although an experimental study for second sound propagation in dielectric solids and some related experimental observations were reported nearly four decades ago (Ackerman and Guyer, 1968; Jackson et al., 1970; Jackson and Walker, 1971; Rogers, 1971; Narayanmurti and Dynes, 1972). Most engineering materials such as metals possess a relatively high rate of thermal damping and thus are not suitable for use in experiments concerning second sound propagation. But, given the state of recent advances in material science, it may be possible in the foreseeable future to identify (or even manufacture for laboratory purposes) an idealized material for the purpose of studying the propagation of thermal waves at finite speed. Relevant theoretical developments on the subject were made by Green and Naghdi (1992a). They developed three models for generalized thermoelasticity of homogeneous isotropic materials which are labelled as model I, II and III. The nature of these theories are such that when the respective theories are linearized, model I (Green and Naghdi, 1993) reduces to the classical heat conduction theory based on Fourier’s law. The linearized versions of model II and III permit propagation of thermal waves at finite speed. Model II, in particular, exhibits a feature that is not present in the other established thermoelastic models as it does not sustain dissipation of thermal energy (Green and Naghdi, 1993). In this model the constitutive equations are derived by starting with the reduced energy equation and by including the thermal displacement gradient among other constitutive variables. The Green–Naghdi’s third model (Green and Naghdi, 1992b) admits the dissipation of energy. In this model the constitutive equations are derived by starting with the reduced energy equation, where the thermal displacement gradient in addition to temperature gradient, are among the constitutive variables. Ref. (Green and Naghdi, 1993) includes the derivation of complete set of governing equations of linearized version of the theory for homogeneous and isotropic materials in terms of displacement and temperature fields and a proof of the uniqueness of solution for the corresponding initial-boundary value problem. In the context of linearized version of this theory (Green and Naghdi, 1993), theorem on uniqueness of solutions has been established by Chandrasekharaiah (1996a, 1996b). Chandrasekharaiah (1996c) studied the one dimensional thermal wave propagation in a half space based on the GN model II due to sudden inputs of temperature and stress/strain on the boundary. Chandrasekharaiah and Srinath (1998) studied the thermoelastic interaction caused by a continuous point heat source in a homogeneous isotropic unbounded thermoelastic body by employing linear theory of thermoelasticity without energy dissipation (TEWOED). Mallik and Kanoria (2006, 2007a) have studied the thermoelastic interaction in an infinite rotating elastic medium in presence of heat source in generalized thermoelasticity (Green and Naghdi, 1992b, 1993). The problems have been solved applying eigenvalue approach. Taheri et al. (2005) have employed Green–Naghdi theories of type II and type III to study the thermal and mechanical waves in an annulus domain. Roychoudhuri and Dutta (2005) studied thermoelastic interactions in an isotropic homogeneous thermoelastic solid containing time-dependent distributed heat sources which vary periodically for a finite time interval in the context of TEWOED. Problems concerning generalized thermoelasticity proposed by Green and Naghdi (1992b, 1993) have been studied by many other authors (Mallik and Kanoria, 2007b; Banik et al., 2007; Bandyopadhyay and Roychoudhuri, 2005). Verma and Hasebe (2002) studied wave propagation in transversely isotropic plates in the context of generalized thermoelasticity proposed by Lord and Shulman (1967). In spite of these, relatively less attention has been paid to find thermal stress and displacement in an anisotropic thick plate by considering the equations of generalized thermoelasticity (Green and Naghdi, 1992b, 1993), which has motivated the authors to carry out the present work. The present work deals with a two dimensional problem for a transversely isotropic thick plate containing heat sources. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate is laid down on a rigid foundation and is thermally insulated. The problem has been solved using generalized thermoelasticity theory proposed by Green and Naghdi (1992b, 1993). The governing equations for displacement and temperature fields are solved in Laplace–Fourier double transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier series expansion technique (Honig and Hirdes, 1984). Numerical computations have been done for magnesium (Mg) and the results are presented graphically for both the

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Fig. 1. Co-ordinate system and geometry of the plate.

models of Green–Naghdi. Finally we have deduced the corresponding results for isotropic material as a particular case and presented them graphically to compare with the results obtained for magnesium for GN model II (TEWOED). 2. Formulation of the problem We consider an infinite homogeneous transversely isotropic thermally conducting thick elastic plate with spatially varying heat source at an uniform reference temperature T0 in the undisturbed state. The upper surface of this medium is taken traction free and subjected to a known temperature distribution. The lower surface of the plate is laid down on a rigid foundation and is thermally insulated. Let the faces of the plate be the planes x = ±h, referred to a rectangular set of Cartesian co-ordinates axes Ox, Oy and Oz as shown in Fig. 1. We shall consider two dimensional deformation of the plate parallel to xy plane. Then the displacement vector u and temperature T can be taken in the following form   u = u(x, y, t), v(x, y, t), 0 , T = T (x, y, t).

(2.1)

Now the constitutive relations in the present case are σxx = c11 exx + c12 eyy − β1 (T − T0 ), σyy = c12 exx + c11 eyy − β1 (T − T0 ), σxy = (c11 − c12 )exy .

(2.2)

The equations of motion along x and y directions can be obtained as follows c11

∂ 2 u c11 − c12 ∂ 2 u c11 + c12 ∂ 2 v ∂T ∂ 2u + + = ρ 2 + β1 , 2 2 2 2 ∂x∂y ∂x ∂x ∂y ∂t

∂ 2v ∂ 2 v c11 + c12 ∂ 2 u ∂T c11 − c12 ∂ 2 v = ρ . + c + + β1 11 2 2 ∂x∂y ∂y ∂x 2 ∂y 2 ∂t 2 Heat equation and Fourier law in the present case for Green–Naghdi theories can be written as  2   2˙    ∂ T ∂ u¨ ∂ v¨ ∂ 2T ∂ 2 T˙  ∂ T ¨ ˙ K + 2 + χK1 + 2 + ρ Q = ρcv T + T0 β1 + , ∂x ∂y ∂x 2 ∂y ∂x 2 ∂y and



∂T ∂ν + K qx = − χK1 ∂x ∂x

(2.3)

(2.4)

 (2.5)

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where χ = 0 for GN model II and χ = 1 for GN model III and ρ is the density, cij are elastic constants for transversely isotropic material, K1 is the thermal conductivity along x and y direction, K  is the additional material constant for GN theories, β1 is the stress temperature coefficient, cv is the specific heat at constant volume, Q is the heat source and ν˙ = T where ν is the thermal displacement. Introducing the following nondimensional variables x  = c1 ηx, y  = c1 ηy, u = c1 ηu, v  = c1 ηv, t  = c12 ηt, σij qx β1 (T − T0 )β1 t0 = c12 ηt0 , qx  = , σij = , θ= , c11 K1 c11 c11 ρβ1 Q K  Q = , K = K1 c12 η2 c11 K1 c12 η

(2.6)

where η = ρcv /K1 , c12 = c11 /ρ and omitting primes, Eqs. (2.2) to (2.5) can be rewritten in nondimensional form as follows ∂u ∂v + (a2 − a1 ) − θ, σxx = ∂x ∂y ∂u ∂v σyy = (a2 − a1 ) + − θ, ∂x ∂y   (1 + a1 − a2 ) ∂u ∂v σxy = , (2.7) + 2 ∂y ∂x ∂ 2u ∂ 2u ∂ 2u ∂ 2v ∂θ = + a + a + β1 , 1 2 2 2 2 ∂x∂y ∂x ∂x ∂y ∂t 2 2 2 2 ∂ v ∂ v ∂ v ∂ u ∂θ a1 2 + 2 + a 2 = 2 + β1 , ∂x∂y ∂y ∂x ∂y ∂t  2   2    2 2 ˙ ˙ ∂ θ ∂ θ ∂ θ ∂ θ ˙ = θ¨ + ε ∂ u¨ + ∂ v¨ , K + χ + Q + + ∂x ∂y ∂x 2 ∂y 2 ∂x 2 ∂y 2   ˙ ∂θ ∂θ q˙x = − χ + K ∂x ∂x

(2.8) (2.9) (2.10)

where a1 =

c11 − c12 , 2c11

a2 =

c11 + c12 , 2c11

ε=

T0 β12 . K1 c11 η

(2.11)

The above equations are solved subjected to the initial conditions ∂θ ∂u ∂v = = = 0 at t = 0. ∂t ∂t ∂t The boundary conditions for the problem may be taken as   θ (h, y, t) = θ0 H g − |y| , θ = u = v = 0;

(2.12)

σxx (h, y, t) = 0, σxy (h, y, t) = 0, u(−h, y, t) = 0, v(−h, y, t) = 0, qx (−h, y, t) = 0.

(2.13)

In the first equation of (2.13) θ0 and g are constants, while H(.)is the Heaviside unit step function. Thus the surface x = h is traction free and it is heated on a band of width 2g around the y-axis on the upper surface, while the rest of the surface is kept at zero temperature. Laplace–Fourier double transform of a function f (x, y, t) is given by

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f¯(x, y, p) =

∞

611

f (x, y, t) e−pt dt, Re(p) > 0

0

1 fˆ¯(x, q, p) = √ 2π

∞

f¯(x, y, p) e−iqy dy.

(2.14)

−∞

Applying Laplace–Fourier double transform to Eqs. (2.7)–(2.10) we get d uˆ¯ σ¯ˆ xx = + (a2 − a1 )iq vˆ¯ − θˆ¯ , dx d uˆ¯ + iq vˆ¯ − θˆ¯ , σ¯ˆ yy = (a2 − a1 ) dx   1 + a1 − a2 d vˆ¯ + iq uˆ¯ , σ¯ˆ xy = 2 dx

(2.15)

d 2 uˆ¯ d vˆ¯ d θˆ¯ 2 2 ˆ − (p + a q ) u ¯ + iqa = , 1 2 dx dx dx 2 d 2 vˆ¯ d uˆ¯ ˆ¯ = iq θ, a1 2 − (p 2 + q 2 )vˆ¯ + iqa2 dx dx  ˆ   ¯ d 2 θˆ¯   ˆ 2 2 ˆ¯ 2 du ¯ ˆ + iq v¯ , − (K + χ)q + p θ + p Q = εp dx dx 2

(2.17)

d θˆ¯ qˆ¯˙ x = −(pχ + K  ) . dx

(2.19)

(2.16)

(2.18)

Eliminating uˆ¯ and θˆ¯ from Eqs. (2.16)–(2.18) we get A(q, p)

d 6 vˆ¯ d 4 vˆ¯ d 2 vˆ¯ + B(q, p) + C(q, p) + D(q, p)vˆ¯ = F (x, q, p). dx 6 dx 4 dx 2

(2.20)

ˆ¯ vˆ¯ from Eqs. (2.16)–(2.18) we get Again eliminating u, A(q, p)

d 6 θˆ¯ d 4 θˆ¯ d 2 θˆ¯ + B(q, p) + C(q, p) + D(q, p)θˆ¯ = F1 (x, q, p) dx 6 dx 4 dx 2

(2.21)

where A(q, p) = −a1 a2 , 

 B(q, p) = a2 (K  + χ)(1 + a1 ) + a1 (1 + ε) p 2 + 3a1 (K  + χ)q 2 ,    C(q, p) = (1 + a1 )p 2 + 2a1 q 2 a2 (K  + χ)q 2 + (a2 + ε)p 2 − a2 (K  + χ)(p 2 + q 2 )(p 2 + a1 q 2 ),   D(q, p) = (p 2 + q 2 )(p 2 + a1 q 2 ) a2 q 2 (K  + χ) + (a2 + ε)p 2   − εp 2 (p 2 + a1 q 2 ) (p 2 + q 2 ) − iq 3 (K  + χ) ,  ˆ¯ d 2Q ˆ¯ , F (x, q, p) = a2 pq 2 (a2 − 1) 2 + (p 2 + a1 q 2 )Q dx  ˆ¯   2 ˆ¯ d 4Q 2 2 2 2 2 2 d Q 2 2 2 2 ˆ¯ F1 (x, q, p) = a2 iq a1 4 − (p + q ) + (p + a1 q )a1 − a2 q + (p + q )(p + a1 q )Q . (2.22) dx dx 2

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2.1. Spatially varying heat source We take heat source Q(x, y, t) in the following form Q(x, y, t) =

H (t) cosh(bx) y 2 + a2

where a and b are constants. Then we get



1 π F (x, q, p) = a2 q 2 e−a|q| cosh(bx) (a2 − 1)b2 + (p 2 + a1 q 2 ) a 2 and F1 (x, q, p) =

  π a2 iqe−a|q| cosh(bx) a1 b4 − (p 2 + q 2 ) + (p 2 + a1 q 2 )a1 − a22 q 2 b2 2

+ (p 2 + q 2 )(p 2 + a1 q 2 ) .

1 a

(2.23)

(2.24)

Thus the solutions of Eqs. (2.20) and (2.21) can be obtained as ˆ¯ q, p) = v(x,

3 

Bj (q, p)ekj x + B−j (q, p)e−kj x + G(q, p) cosh(bx)

(2.25)

j =1

and θˆ¯ (x, q, p) =

3 

Aj (q, p)ekj x + A−j (q, p)e−kj x + G1 (q, p) cosh(bx)

(2.26)

j =1

respectively, where

π e−a|q| a2 q 2 [(a2 − 1)b2 + (p 2 + a1 q 2 )] , G(q, p) = 2 a(b2 − k12 )(b2 − k22 )(b2 − k32 )

(1 + pt0 )a2 iqe−a|q| π G1 (q, p) = 2 a(b2 − k12 )(b2 − k22 )(b2 − k32 )  

× a1 b4 − (p 2 + q 2 ) + a1 (p 2 + a1 q 2 ) − a22 q 2 b2 + (p 2 + q 2 )(p 2 + a1 q 2 ) ,

(2.27)

(2.28)

and kj ’s, −kj ’s, j = 1, 2, 3, are the roots of the equation A(q, p)k 6 + B(q, p)k 4 + C(q, p)k 2 + D(q, p) = 0.

(2.29)

Now substituting the expression for vˆ¯ and θˆ¯ into Eq. (2.16) we get 3  

   d 2 uˆ¯ 2 2 ˆ Aj (q, p) − iqa2 Bj (q, p) ekj x − A−j (q, p) − iqa2 B−j (q, p) e−kj x kj − (p + a q ) u ¯ = 1 dx 2 j =1

+ b sinh(bx) G1 (q, p) − iqa2 G(q, p) (2.30)

whose solution is obtained as ˆ¯ q, p) = C1 (q, p)ek4 x + C−1 (q, p)e−k4 x + u(x,

3  [{Aj (q, p) − iqa2 Bj (q, p)}ekj x

kj2 − k42

j =1

− where k4 =



{A−j (q, p) − iqa2 B−j (q, p)}e−kj x ]kj kj2

− k42

+

b sinh(bx)[G1 (q, p) − iqa2 G(q, p)] , b2 − k42

ˆ¯ v, ˆ¯ θˆ¯ compatible with Eq. (2.17) can be obtained as p 2 + a1 q 2 . Now the solution for u,

(2.31)

S.H. Mallik, M. Kanoria / European Journal of Mechanics A/Solids 27 (2008) 607–621

ˆ¯ q, p) = u(x,

3  [{1 − iqa2 fj (q, p)}Aj (q, p)ekj x − {1 − iqa2 fj (q, p)}A−j (q, p)e−kj x ]kj

kj2 − k42

j =1

+ ˆ¯ q, p) = v(x,

613

b sinh(bx)[G1 (q, p) − iqa2 G(q, p)] , b2 − k42

3 

fj (q, p) Aj (q, p)ekj x + A−j (q, p)e−kj x + G(q, p) cosh(bx),

j =1

θˆ¯ (x, q, p) =

3 

Aj (q, p)ekj x + A−j (q, p)e−kj x + G1 (q, p) cosh(bx)

(2.32)

j =1

where fj (q, p) =

kj2

iq , − (p 2 + q 2 )

j = 1, 2, 3.

(2.33)

Now σ¯ˆ xx , σ¯ˆ yy and σ¯ˆ xy can be obtained from Eqs. (2.15) and (2.32) as follows σ¯ˆ xx =

3  {1 − iqa f (q, p)}k 2  2 j j j =1



kj2 − k42



+ (a2 − a1 )iqfj (q, p) − 1 Aj (q, p)ekj x + A−j (q, p)e−kj x

{G1 (q, p) − iqa2 G(q, p)}b2 + (a − a )iqG(q, p) − G (q, p) cosh(bx), 2 1 1 b2 − k42 3  (a − a ){1 − iqa f (q, p)}k 2 

2 1 2 j j 2 = − q fj (q, p) − 1 Aj (q, p)ekj x + A−j (q, p)e−kj x 2 2 kj − k4 j =1  (a2 − a1 ){G1 (q, p) − iqa2 G(q, p)}b2 2 + − q G(q, p) − G1 (q, p) cosh(bx), b2 − k42  3 

1 + a1 + a2  {1 − iqa2 fj (q, p)}iq = + f (q, p) kj Aj (q, p)ekj x + A−j (q, p)e−kj x j 2 2 2 kj − k4 j =1   {G1 (q, p) − iqa2 G(q, p)}iq + + G(q, p) b sinh(bx) . b2 − k42 +

σ¯ˆ yy

σ¯ˆ xy

(2.34)

The boundary conditions in Laplace–Fourier double transform domain take the form

ˆθ¯ (h, q, p) = 2 θ0 sin(qg) , π pq σ¯ˆ xx (h, q, p) = 0, σ¯ˆ xy (h, q, p) = 0, ˆ¯ u(−h, q, p) = 0, ˆ¯ v(−h, q, p) = 0, qˆ¯ x (−h, q, p) = 0. Now using the boundary conditions (2.35) we get

3 

2 θ0 sin(qg) kj h −kj h − G1 (q, p) cosh(bh), Aj (q, p)e + A−j (q, p)e = π pq j =1

(2.35)

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 3 

k42 − {a1 kj2 + (a2 − a1 )k42 }fj (q, p) Aj (q, p)ekj h + A−j (q, p)e−kj h kj2 − k42 j =1  2 k G1 (q, p) − {(a2 − a1 )k42 + a1 b2 }iqG(q, p) cosh(bh), =− 4 b2 − k42  3 

iq + {kj2 − p 2 + (a2 − a1 )q 2 }fj (q, p) kj Aj (q, p)ekj h + A−j (q, p)e−kj h kj2 − k42 j =1  iqG1 (q, p) + {b2 − p 2 + (a2 − a1 )q 2 }G(q, p) b sinh(bh), = b2 − k42 3 

kj2 {1 − a2 iqfj (q, p)} b sinh(bh)[G1 (q, p) − a2 iqG(q, p)] = , Aj (q, p)e−kj h − A−j (q, p)ekj h kj2 − k42 b2 − k42 j =1 3 

Aj (q, p)e−kj h + A−j (q, p)ekj h fj (q, p) = −G(q, p) cosh(bh), j =1 3 

Aj (q, p)e−kj h − A−j (q, p)ekj h kj = b sinh(bh)G1 (q, p).

(2.36)

j =1

Solution of the above system of linear equations gives the unknown parameters Aj (q, p), A−j (q, p), j = 1, 2, 3. This completes the solution of the problem in the transformed domain. The results for isotropic material can be deduced from our problem by simply replacing a1 = λ/(λ + 2μ), a2 = (λ + μ)/(λ + 2μ),  = γ 2 T0 /(ρcv (λ + 2μ)), γ = (3λ + 2μ)αt in our calculations, where λ, μ are Lame’s constant, and ρ, cv and αt are respectively density, specific heat at constant volume and coefficient of linear thermal expansion for isotropic material. 3. Inversion of double transform The expression for function fˆ¯(x, q, p) in Laplace transform domain can be given as 1 f¯(x, y, p) = √ 2π

=

2 π

∞

eiqy fˆ¯(x, q, p) dq

−∞

∞

cos(qy)fˆ¯e (x, q, p) + i sin(qy)fˆ¯o (x, q, p) dq

(3.1)

0

where fˆ¯e and fˆ¯o denote the even and odd part of fˆ¯(x, q, p) respectively. To get the solution in space time domain we have to apply Laplace inversion to equation (3.1) which have been done numerically using a method based on Fourier series expansion technique. The outline of the numerical inversion method used is given below. Let f¯(x, y, p) be the Laplace transform of a function f (x, y, t). Then the inversion formula for Laplace transform can be written as 1 f (x, y, t) = 2πi

d+i∞ 

ept f¯(x, y, p) dp

d−i∞

where d is an arbitrary real number greater than real part of all the singularities of f¯(x, y, p).

(3.2)

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615

Taking p = d + iw, the preceding integral takes the form f (x, y, t) =

edt 2π

∞

eitw f¯(x, y, d + iw) dw.

(3.3)

−∞

Expanding e−dt f (x, y, t) in a Fourier series in the interval [0, 2T ] we obtain the approximate formula (Honig and Hirdes, 1984) f (x, y, t) = f∞ (x, y, t) + ED

(3.4)

where ∞

 1 f∞ (x, y, t) = c0 + ck for 0  t  2T , 2 k=1    edt ikπt ¯ ikπt T e f x, y, d + ck = T T

(3.5)

(3.6)

and ED is the discretization error. The error ED can be made arbitrary small by choosing d large enough (Honig and Hirdes, 1984). Since the infinite series in Eq. (3.5) can be summed up to a finite number N of terms, the approximate value of f (x, y, t) becomes  1 fN (x, y, t) = c0 + ck 2 N

for 0  t  2T .

(3.7)

k=1

Using the formula to evaluate f (x, y, t) we introduce a truncation error ET that must be added to the discretization error. Two methods are used to reduce the total error. First the ‘Korrecktur’ method is used to reduce the discretization error. Next the ε-algorithm is used to accelerate convergence. The Korrecktur method uses the following formula to evaluate the function f (x, y, t) 

f (x, y, t) = f∞ (x, y, t) − e−2dT f∞ (x, y, 2T + η) + ED

(3.8)



where the discretization error |ED |  |ED |. Thus, the approximate value of f (x, y, t) becomes fN K (x, y, t) = fN (x, y, t) − e−2dT fN  (x, y, 2T + η).

(3.9)

We shall now describe the ε-algorithm that is used  to accelerate the convergence of the series in Eq. (3.5). Let N = 2q + 1, where q is a natural number and sm = m k=1 ck is the sequence of partial sum of series in (3.7). We define the ε-sequence by ε0,m = 0,

ε1,m = sm

and εp+1,m = εp−1,m+1 +

1 , εp,m+1 − εp,m

p = 1, 2, 3, . . . .

It can be shown that (Honig and Hirdes, 1984) the sequence ε1,1 , ε3,1 , ε5,1 , . . . , εN,1 converges to f (x, y, t) + ED − c20 faster than the sequence of partial sums sm , m = 1, 2, 3, . . . . The actual procedure used to invert the Laplace transform consists of using Eq. (3.6) together with the ε-algorithm. The values of d and T are chosen according to the criterion outlined in Honig and Hirdes (1984).

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4. Numerical results and discussions With the view of illustrating the numerical results, the material chosen for the plate is magnesium (Mg), the physical data for which is given by as follows (Dhaliwal and Singh, 1980) c11 = 5.974 × 1010 N m−2 , c13 = 2.17 × 1010 N m−2 , c44 = 1.510 × 1010 N m−2 , cv = 1.04 × 103 J K g−1 deg−1 ,  = 2.02 × 10−2 ,

c12 = 2.624 × 1010 N m−2 , c33 = 6.17 × 1010 N m−2 , β1 = 2.68 × 106 N m−2 deg−1 , K1 = 1.7 × 102 W m−1 deg−1 ,

ρ = 1.74 × 103 K g m−3 ,

T0 = 298 K.

To get the roots of the polynomial equation (2.29) in complex domain we have used Laguerre’s method. For numerical calculation we have taken K  = 200. Here the numerical inversion of Laplace transform has been done using a method based on Fourier series expansion technique (Honig and Hirdes, 1984). Figs. 2–7 are drawn to give comparison of the results obtained for displacements, temperature and stresses for two models of Green and Naghdi, namely GN model II and GN model III, against the thickness x for y = 0 and t = 0.25. Here the graphs are plotted to show the variation of displacements, temperature and stresses against x, i.e against the thickness for y = 0 and t = 0.25. Figs. 8–13 are drawn to give a comparison of the results for displacements, temperature and stresses with that of an isotropic material, which is taken to be copper whose material properties are given as follows (El-Maghraby, 2004)

Fig. 2. Variation of displacement component u against x for t = 0.25 and y = 0.

Fig. 3. Variation of displacement component v against x for t = 0.25 and y = 0.

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Fig. 4. Variation of temperature θ for t = 0.25 and y = 0.

Fig. 5. Variation of stress component σxx for t = 0.25 and y = 0.

λ = 7.76 × 1010 N m−2 ,

μ = 3.86 × 1010 N m−2 ,

K = 386 W m−1 deg−1 ,

cv = 383.1 J K g−1 deg−1 ,

T0 = 293 K,  = 0.0168,

ρ = 8954 K g m−3 ,

αt = 1.78 × 10−5 N m−2 deg−1 , θ0 = 1,

h = 0.5,

g = 1,

a = 1,

b = 1.

Figs. 2–3 depict displacement components u & v against the thickness x for y = 0 and t = 0.25 respectively for GN model II (TEWOED) and GN model III (TEWED). It is observed from these figures that on the rigid base at x = −0.5 the displacements are zero which confirms the boundary conditions taken. On the upper surface of the plate i.e. at x = 0.5 the displacements are maximum which supports the physical fact. The displacement for GN model II is larger than that of GN model III. This is due to the fact that the energy dissipation term (T˙ ) is present in GN model III (see Eq. (2.4)). In Fig. 4 the temperature distribution is plotted against x for y = 0 and t = 0.25. On the upper surface the boundary condition is taken as θ = 1 in −1  y  1. So Fig. 4 tallies with the boundary conditions taken and as the lower surface of the plate is thermally insulated and kept on a rigid base, the maximum temperature is observed there and the oscillatory nature is seen near it because of the reflection of thermal wave from there. Again in Fig. 5 the normal stress σxx is plotted against x for the same set of parameters for GN model II and GN model III. As the upper surface is stress free the figure also shows the expected behaviour and the wave nature is observed near the lower surface because of the same reason as mentioned above. Similar observation has been found in Fig. 6 for shearing stress σxy . Fig. 7 shows the variation of normal stress σyy against x. Figs. 8 and 9 are plotted to show the variation of displacements u and v against x for y = 0 and t = 0.25 and make a comparison of the results with isotropic material (Cu) for GN model II. The qualitative behaviour of the

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Fig. 6. Variation of stress component σxy for t = 0.25 and y = 0.

Fig. 7. Variation of stress component σyy for t = 0.25 and y = 0.

Fig. 8. Variation of displacement component u against x for t = 0.25 and y = 0.

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Fig. 9. Variation of displacement component v against x for t = 0.25 and y = 0.

Fig. 10. Variation of temperature θ for t = 0.25 and y = 0.

Fig. 11. Variation of stress component σxx for t = 0.25 and y = 0.

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Fig. 12. Variation of stress component σxy for t = 0.25 and y = 0.

Fig. 13. Variation of stress component σyy for t = 0.25 and y = 0.

displacement u is similar to that of El-Maghraby (2004) (shown in Fig. 3 there) for isotropic material (Cu) where LS and GL model were taken. From Figs. 8 and 9 it is observed that the magnitude of the displacements for transversely isotropic material is larger than that of isotropic material. Similar behaviour is seen for temperature distribution and thermal stress distribution in Figs. 10–13. Acknowledgements We are grateful to Prof. S.C. Bose of the Department of Applied Mathematics, University of Calcutta for his valuable suggestions and guidance in preparation of the paper. We also express our sincere thanks to the reviewers for their valuable suggestions for the improvement of the paper. S.H. Mallik is grateful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India for the award of a fellowship. References Ackerman, C.C., Guyer, R.A., 1968. Temperature pulses in dielectric solids. Ann. Phys. 50, 128–185. Bandyopadhyay, N., Roychoudhuri, S.K., 2005. Thermoelastic wave propagation without energy dissipation in an elastic half space. Bull. Cal. Math. Soc. 97 (6), 489–502. Banik, S., Mallik, S.H., Kanoria, M., 2007. Thermoelastic interaction with energy dissipation in an infinite solid with distributed periodically varying heat sources. Int. J. Pure Appl. Math. 34 (2), 231–245. Chadwick, P., 1960. Thermoelasticity, the dynamic theory. In: Sneddon, I.N., Hill, R. (Eds.), Progress in Solid Mechanics, vol I. North-Holland, Amsterdam, p. 265.

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