Generalized Thermoelasticity Plane Waves in Rotating Media with Thermal Relaxation under the Temperature Dependent Properties

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1 Mechanics and Mechanical Engineering Vol. 9, No. 2 (2005) c Technical University of Lodz Generalized Thermoelasticity Plane Waves in Rotating Media with Thermal Relaation under the Temperature Dependent Properties Mohamed I. A. Othman Department of Mathematics, Faculty of Education, P.O. Bo 2801, Salalah-211, Sultanate of Oman m i othman@yahoo.com Received (1 April 2005) Revised (1 May 2005) Accepted (10 June 2005) A two-dimensional coupled problem in generalized thermoelasticity for rotating media under the temperature dependent properties is studied. The problem is in the contet of the Lord-Shulman s theory with one relaation time. The normal mode analysis is used to obtain the epressions for the temperature distribution, displacement components and thermal stresses. The resulting formulation is applied to two different problems. The first concerns the case of a heat punch moving across the surface of a semi-infinite thermoelastic half-space subjected to appropriate boundary conditions. The second deals with a thick plate subject to a time-dependent heat source on each face. Numerical results are illustrated graphically for each problem considered. Comparisons are made with the results obtained predicted by the two theories in case of absence of rotation. Keywords: Generalized thermoelasticity, rotating media, thermal relaation, temperature dependent properties. 1. Introduction In recent years due to the progress in various fields in science and technology the necessity of taking into consideration the real behavior of the material characteristics become actual. We have seen a rapid development of thermoelasticity stimulated by various engineering sciences [1]. Most of investigations were done under the assumption of the temperature-independent material properties, which limit the applicability of the solutions obtained to certain ranges of temperature. At high temperature the material characteristics such as the modulus of elasticity, the Poisson s ratio, the coefficient of thermal epansion and the thermal conductivity are no longer constants. In some investigations they were taken as functions of coordinates [2] and [3].

2 90 Generalized Thermoelasticity Plane Waves in... In this work we consider the modulus of elasticity is the only temperaturedependent material parameter. The eperimental data [4] show that the changes of Poisson s ratio and the coefficient of linear thermal epansion due to the high temperature can be neglected. The classical uncoupled theory of thermoelasticity predicts two phenomena not compatible with physical observations. First, the equation of heat conduction of this theory does not contain any elastic terms contrary to the fact that elastic changes produce heat effects. Second, the heat equation is of parabolic type predicting infinite speeds of propagation for heat waves. Biot [5] introduced the theory of coupled thermoelasticity to overcome the first shortcoming. The governing equations for this theory are coupled, eliminating the first parado of the classical theory. However, both theories share the second shortcoming since heat equation for the coupled theory is also parabolic. Two generalizations to the coupled theory are introduced. The first is due to Lord and Shulman [6] who obtained a wave-type heat equation by postulating a new law of heat conduction to replace the classical Fourier s law. This new law contains the heat flu vector as well as its time derivative. It contains also a new constant that acts as a relaation time. Since the heat equation of this theory is of the wave-type, it automatically ensures finite speeds of propagation for heat and elastic waves. The remaining governing equations for this theory, namely, the equations of motions and constitutive relations remain the same as those for the coupled and the uncoupled theories. Following the Lord-Shulman s theory, several authors including Puri [7], Ezzat et al. [8] studied the plane thermoelastic wave propagations in a medium of perfect conductivity. This theory was etended by Dhaliwal and Sherief [9] to general anisotropic media in the presence of heat sources. Sherief and Dhaliwal [10] solved a thermal shock problem. Nayfeh and Nasser [11] used Lord-Shulman theory to study plane thermo-elastic surface waves in a half-space. Ezzat et al. [12] studied the effect of reference temperature on thermal distribution for onedimensional problem. Recently, Othman [13] established the model of equations of generalized thermo-elasticity based on the Lord-Shulman theory in an isotropic elastic medium under the dependence of the modulus of elasticity on reference temperature. The second generalization to the coupled theory of the thermoelasticity is what known as the theory of thermoelasticity with two relaation times or the theory of temperature-rate-dependent thermoelasticity. Müller [14] in a review of thermodynamic of thermoelastic solids has proposed an entropy production inequality, with the help of which, he considered restrictions on a class of constitutive equations. A generalization of this inequality was proposed by Green and Laws [15]. Green and Lindsay have obtained an eplicit version of the constitutive equations in [16]. These equations were also obtained independently by Erbay and Şuhubi [17]. This theory contains two constants that act as relaation times and modifies all the equations of the coupled theory not the heat equation only. The classical Fourier s law of heat conduction is not violated if the medium under consideration has a center of symmetry. Using this theory, Agarwal [18], [19] considered, respectively, thermoelastic and magneto-thermoelastic plane wave propagation in an infinite elastic medium. Ezzat and Othman [20] studied the electromagneto-thermoelasticity plane wave with two relaation times. Othman [21] applied the Green-Lindsay s theory

3 Othman, MIA 91 to study the effect of rotation and relaation time on plane waves in generalized thermoelasticity. Ignaczak [22] studied a strong discontinuity wave and obtained a decomposition theorem for this theory [23]. In the present paper we shall use the normal mode method to problems of generalized thermoelasticity with thermal relaation in an isotropic medium when the modulus of elasticity is taken as a linear function of reference temperature under the effect of rotation. The resulting formulation is applied in case of two dimensional to two concrete problems. The eact epressions for temperature, displacement components and thermal stress are obtained for the two problems considered. 2. Formulation of the problem We shall consider an infinite isotropic, homogeneous, thermally conducting elastic medium. The medium is rotating uniformly with an angular velocity Ω = Ωn, where n is a unit vector representing the direction of the ais of rotation. The displacement equation of motion in the rotating frame of reference has two additional terms [24]; centripetal acceleration Ω ( Ω u) due to the time-varying motion only and the Coriolis acceleration 2Ω u where u is the dynamic displacement vector measured from a steady state deformed position and supposed to be small. These two terms do not appear in the equations for non-rotating media. The constitutive law for the theory of generalized thermo-elasticity with one relaation time is σ ij = λeδ ij + 2µε ij γ (T T 0 )δ ij, (1) The equation of heat conduction ( k 2 T = t + τ 2 ) 0 t 2 (ρc E T + γt 0 e), (2) The strain-displacement constitutive relations ε ij = 1 2 (u i,j + u j,i ) and We assume that ε ii = e = u i,i (3) E = E 0 f(t), λ = E 0 λ 0 f(t), µ = E 0 µ 0 f(t) and γ = E 0 γ 0 f(t). (4) where f(t) is a given non-dimensional function of temperature, in case of temperatureindependent modulus of elasticity f(t) 1, and E = E 0. The equations of motion, in the absence of body forces, are σ ij,j = ρ[ü i + {Ω (Ω u)} i + (2Ω u) i ]. (5) where all the terms have the same significance as in [16]. The displacement equation of motion in the rotating frame of reference as ρ[ü i + {Ω (Ω u)} i + (2Ω u) i ] = E 0 f[(λ 0 + µ 0 )e,i + µ 0 2 u i γ 0 T,i ] (6) +E 0 f,j [λ 0 eδ ij + 2µ 0 ε ij γ 0 (T T 0 )δ ij ].

4 92 Generalized Thermoelasticity Plane Waves in... Now we introduce the following non-dimensional variables = c 0 η 0, y = c 0 η 0 y, u = c 0 η 0 u, v = c 0 η 0 v, t = c 2 0η 0 t, τ 0 = c 2 0η 0 τ 0, θ = γ0e0 (T T ρc 2 0 ), 0 σ ij = σij. ρc 2 0 (7) Omitting the dashes for convenience, we have σ ij = [(2β 1)eδ ij + (1 β)(u i,j + u j,i ) θδ ij ]f(θ), (8) [ü i + {Ω (Ω u)} i + (2Ω u) i ] = [βe,i + (1 β) 2 u i θ,i ]f(θ) (9) + [(2β 1)ef,i + (1 β)(u i,j + u j,i )f,j θf,i ]. The heat conduction Eq. (2) can be rewritten by using Eq. (3) as: 2 θ = ( θ + τ 0 θ) + ε1 f(θ)δ 0 (ė + τ 0 ë). (10) We consider a special case when T T 0 1, 0 δ 0 1 and f(θ) = (1 α T 0 ). The equations of motion in two-dimension take the form [ 2 ] u α t 2 Ω2 u 2Ω v = 2 u + β 2 v y β 2 u y 2 θ, (11) [ 2 ] v α t 2 Ω2 v 2Ω u = 2 v + β 2 u y β 2 v 2 θ y. (12) The equation of heat conduction 2 θ = ( θ + τ 0 θ) + ε(ė + τ0 ë). (13) where α = ασ = u + (2β 1) v y θ, (14) ασ yy = v + (2β 1) u y θ, (15) ασ y = (1 β)( u y + v ), (16) ασ zz = (2β 1)e θ. (17) 1 1 α T 0, ε = ε 1 δ 0 (1 α T 0 ). (18) Differentiating Eq. (11) with respect to, and Eq. (12) with respect to y, then adding, we arrive at [ ( )] 2 2 α t 2 Ω2 e = 2 θ + 2Ω ζ t. (19)

5 Othman, MIA 93 Differentiating Eq. (11) with respect to y, and Eq. (12) with respect to, then subtracting, we arrive at [ ( )] (1 β) 2 2 α t 2 Ω2 ζ = 2αΩ e t. (20) where 2 = y 2 is Laplace s operator in a two-dimensional space and 3. Normal mode analysis ζ = u y v. (21) The solution of the considered physical variables can be decomposed in terms of normal mode as the following form [u,v,e,ζ,θ,σ ij ] (,y,t) = [ u (y),v (y),e (y),ζ (y),θ (y),σij(y) ] ep(ωt + ia). (22) where ω is the (comple) time constant, i is imaginary unit, a is the wave number in the -direction and u (y), v (y), e (y), ζ (y), θ (y) and σij (y) are the amplitudes of the functions. Using Eq. (22), we can obtain the following equations from Eqs. (13), (19) and (20) respectively [ D 2 a 2 ω(1 + τ 0 ω) ] θ (y) = εω(1 + τ 0 ω)e (y), (23) [ D 2 a 2 α(ω 2 Ω 2 ) ] e (y) = (D 2 a 2 )θ (y) + 2Ωωζ, (24) where [ D 2 a 2 β 1 (ω 2 Ω 2 ) ] ζ (y) = 2β 1 ωωe. (25) D = y, β 1 = α 1 β. (26) Eliminating θ (y) and ζ (y) between Eqs. (23) (25), we get the following sithorder partial differential equation satisfied by e (y) (D 6 a 1 D 4 + a 2 D 2 a 3 )e (y) = 0, (27) where, a 1 = 3a 2 + b 1, (28) a 2 = 3a 4 + 2a 2 b 1 + b 2, (29) a 3 = a 6 + a 4 b 1 + a 2 b 2 + b 3, (30) b 1 = (α + β 1 )ω 2 + (ε + 1)ω 1, (31) b 2 = ω 2 [ω 1 (β 1 + α + εβ 1 ) + αβ 1 ω 2 ] + 4ω 2 Ω 2 β 1, (32) b 3 = β 1 ω 1 [αω ω 2 Ω 2 ], (33) ω 1 = ω(1 + τ 0 ω), ω 2 = ω 2 Ω 2. (34)

6 94 Generalized Thermoelasticity Plane Waves in... Equation (27) can be factorized as (D 2 k 2 1)(D 2 k 2 2)(D 2 k 2 3)e (y) = 0. (35) where, k j, j = 1,2,3 are the roots of the following characteristic equation The solution of Eq. (35) is given by: k 6 a 1 k 4 + a 2 k 2 a 3 = 0. (36) e (y) = 3 e j(y). (37) where e j (y) is the solution of the equation (D 2 k 2 j)e j(y) = 0, j = 1,2,3. (38) The solution of Eq. (38), which is bounded as y, is given by e j(y) = R j (a,ω)e kjy. (39) Substituting from Eq.(39) into the Eq. (37), we obtain: In a similar manner, we get e (y) = θ (y) = ζ (y) = 3 B j (a,ω)e kjy. (40) 3 B j(a,ω)e kjy, (41) 3 B j (a,ω)e kjy. (42) where B j (a,ω), B j (a,ω) and B j (a,ω) are parameters depending on a, ω. Substituting from Eqs. (40) (42) into Eqs. (23) and (25), we obtain B j(a,ω) = εω 1 k 2 j a2 ω 1 B j (a,ω), j = 1,2,3, (43) B j 2ωΩβ 1 (a,ω) = kj 2 B j (a,ω), j = 1,2,3. (44) a2 β 1 ω 2 Substituting from Eqs. (43) and (44) into Eqs. (41) and (42), respectively, we obtain 3 θ εω 1 (y) = kj 2 B j (a,ω)e kjy, (45) a2 ω 1

7 Othman, MIA 95 ζ (y) = 3 2ωΩβ 1 k 2 j a2 β 1 ω 2 B j (a,ω)e kjy. (46) Since, e = iau + Dv, (47) ζ = Du iav. (48) In order to obtain the amplitude of displacement components u and v, which are bounded as y, in terms of Eq. (22) from Eqs. (40), (46), (47) and (48) we can obtain [ ] 3 u 1 2ωΩβ (y) = Ce ay 1 k j + kj 2 ia + a2 kj 2 B j (a,ω)e kjy, (49) a2 β 1 ω 2 v (y) = ice ay [ ] 3 1 2iaωΩβ 1 kj 2 k j a2 kj 2 B j (a,ω)e kjy. (50) a2 β 1 ω 2 where C = 0 to make Eqs. (49) and (50) are bounded as y. In terms of Eq. (22), substituting from Eqs. (40), (45), (46), (49) and (50) into Eqs. (14) (17), respectively, we obtain the stress components in the form σ (y) = {[ ] 3 (2β 1)k 2 j a 2 εω 1 kj 2 a2 kj 2 a2 ω 1 } 4ia(β 1)ωΩβ 1 k j (kj 2 a2 )[kj 2 B j e kjy, (51) a2 β 1 ω 2 ] σ yy(y) = {[ ] 3 k 2 j (2β 1)a 2 εω 1 kj 2 a2 kj 2 a2 ω 1 } 4ia(β 1)ωΩβ 1 k j (kj 2 a2 )[kj 2 B j e kjy, (52) a2 β 1 ω 2 ] σ y(y) = (1 β) { 3 2ωΩβ 1 (kj 2 + } a2 ) (kj 2 a2 )[kj 2 a2 β 1 ω 2 ] + 2iak j kj 2 B j e kjy, (53) a2 σ zz(y) = { } 3 εω 1 2β 1 kj 2 B j e kjy. (54) a2 ω 1 The normal mode analysis is, in fact, to look for the solution in Fourier transformed domain. This assumes that all the field quantities are sufficiently smooth on the real line such that the normal mode analysis of these functions eist.

8 96 Generalized Thermoelasticity Plane Waves in Applications 4.1. Problem I A time-dependent heat punch across the surface of semi-infinite thermo-elastic half space [1]. We consider a homogeneous isotropic thermoelastic solid occupying the region G given by G = {(,y,z),z R, y 0}. In the physical problem, we should suppress the positive eponentials that are unbounded at infinity. The constants B j, j = 1,2,3 have to be chosen such that the boundary conditions on the surface y = 0 take the form θ(,y,t) = n(,t) on y = 0, (55) σ yy (,y,t) = P(,t) on y = 0, (56) σ y (,y,t) = 0 on y = 0, (57) where n, P are given functions of and t. Eqs. (55) (57) in the normal mode form together with Eqs. (45), (52) and (53) respectively, give L 1 B 1 + L 2 B 2 + L 3 B 3 = n (a,ω), (58) M 1 B 1 + M 2 B 2 + M 3 B 3 = P (a,ω), (59) N 1 B 1 + N 2 B 2 + N 3 B 3 = 0. (60) Eqs. (58) (60) can be solved for the three unknowns B 1, B 2 and B 3. The solution of these equations can be written as 1 B 1 = [(µ µ 2 2 ) + i(µ 2 1 µ 1 2 )], (61) 2 2 where 1 B 2 = [(µ µ 4 2 ) + i(µ 4 1 µ 3 2 )], 2 2 (62) 1 B 3 = [(µ µ 6 2 ) + i(µ 6 1 µ 5 2 )], 2 2 (63) L j = εω 1 k 2 j a2 ω 1 j = 1,2,3, (64) M j = (α 1j + iβ 1j ), j = 1,2,3, (65) N j = (α 2j + iβ 2j ), j = 1,2,3, (66) [ ] k 2 j (2β 1)a 2 εω 1 α 1j = kj 2 a2 kj 2, j = 1,2,3, (67) a2 ω 1 2ωΩβ 1 (kj 2 α 2j = + a2 ) (kj 2 a2 )[kj 2, j = 1,2,3, (68) a2 β 1 ω 2 ] β 1j = 4a(β 1)ωΩβ 1 k j (kj 2 a2 )[kj 2, j = 1,2,3. (69) a2 β 1 ω 2 ]

9 Othman, MIA Problem II β 2j = 2ak j, j = 1,2,3, (70) a2 k 2 j λ 1 = α 12 α 23 β 12 β 23 α 13 α 22 + β 13 β 22, (71) λ 2 = α 12 β 23 + α 23 β 12 α 13 β 22 α 22 β 13, (72) λ 3 = α 11 α 23 β 11 β 23 α 13 α 21 + β 13 β 21, (73) λ 4 = α 11 β 23 + α 23 β 11 α 13 β 21 α 21 β 13, (74) λ 5 = α 11 α 22 β 11 β 22 α 12 α 21 + β 12 β 21, (75) λ 6 = α 11 β 22 + α 22 β 11 α 12 β 21 α 21 β 12, (76) µ 1 = n λ 1 P (L 2 α 23 L 3 α 22 ), (77) µ 2 = n λ 2 P (L 2 β 23 L 3 β 22 ), (78) µ 3 = n λ 3 + P (L 1 α 23 L 3 α 21 ), (79) µ 4 = n λ 4 + P (L 1 β 23 L 3 β 21 ), (80) µ 5 = n λ 5 P (L 1 α 22 L 2 α 21 ), (81) µ 6 = n λ 6 + P (L 1 β 22 L 2 β 21 ), (82) 1 = L 1 λ 1 L 2 λ 3 + L 3 λ 5, (83) 2 = L 1 λ 2 L 2 λ 4 + L 3 λ 6. (84) A plate subjected to time-dependent heat sources on both sides [25]. We shall consider a homogeneous isotropic thermo-elastic infinite conductivity thick flat plate of a finite thickness 2L occupying the region G given by: G = {(,y,z),z R, L y L} with the middle surface of the plate coinciding with the plane y = 0. The boundary conditions of the problem are taken as: (i) The normal and tangential stress components are zero on both surfaces of the plate; thus, σ yy = 0 on y = ±L, (85) (ii) The thermal boundary condition σ y = 0 on y = ±L. (86) q n + h 0 θ 0 = r(,t) ony = ±L. (87) where q n denotes the normal component of the heat flu vector, h 0 is Biot s number and r(,t) represents the intensity of the applied heat sources.

10 98 Generalized Thermoelasticity Plane Waves in... We now make use of the generalized Fourier s law of heat conduction in the non-dimensional form, namely, Eq. (88) in the normal mode form q n + τ 0 q n t = θ y. (88) qn 1 = (1 + τ 0 ω) Combining Eqs. (45), (87) and (89) we arrive at θ y. (89) H 1 B 1 cosh(k 1 L) + H 2 B 2 cosh(k 2 L) + H 3 B 3 cosh(k 3 L) = (1 + τ 0 ω)r. (90) Equations (85) and (86) in the normal mode form together with Eqs. (52) and (53) respectively give: M 1 B 1 cosh(k 1 L) + M 2 B 2 cosh(k 2 L) + M 3 B 3 cosh(k 3 L) = 0, (91) N 1 B 1 sinh(k 1 L) + N 2 B 2 sinh(k 2 L) + N 3 B 3 sinh(k 3 L) = 0. (92) Equation (90), (91) and (92) can be solved for the three unknowns B 1, B 2 and B 3. (1 + τ 0 ω)r B 1 = ( )cosh(k 1L) [(λ λ 8 4 ) + i(λ 7 4 λ 8 3 )], (93) where B 3 = B 2 = (1 + τ 0 ω)r ( )cosh(k 2L) [(λ λ 10 4 ) + i(λ 9 4 λ 10 3 )], (94) (1 + τ 0 ω)r ( )cosh(k 3L) [(λ λ 12 4 ) + i(λ 11 4 λ 12 3 )], (95) H j = εω 1 k 2 j a2 ω 1 [ k j tanh(k j L) + h 0 (1 + τ 0 ω)], j = 1,2,3, (96) λ 7 = (α 21 α 23 β 12 β 23 )tanh(k 3 L) (α 13 α 22 β 13 β 22 )tanh(k 2 L), (97) λ 8 = (α 12 β 23 + α 23 β 12 )tanh(k 3 L) (α 13 β 22 + α 22 β 13 )tanh(k 2 L), (98) λ 9 = (α 11 α 23 β 12 β 23 )tanh(k 3 L) (α 13 α 21 β 13 β 21 )tanh(k 1 L), (99) λ 10 = (α 11 β 23 + α 23 β 11 )tanh(k 3 L) (α 13 β 21 + α 21 β 13 )tanh(k 1 L), (100) λ 11 = (α 11 α 22 β 11 β 22 )tanh(k 2 L) (α 12 α 21 β 12 β 21 )tanh(k 1 L), (101) λ 12 = (α 11 β 22 + α 22 β 11 )tanh(k 2 L) (α 12 β 21 + α 21 β 12 )tanh(k 1 L), (102) 3 = H 1 λ 7 H 2 λ 9 + H 3 λ 11, (103) 4 = H 1 λ 8 H 2 λ 10 + H 3 λ 12. (104)

11 Othman, MIA Numerical results The copper material was chosen for the purpose of numerical evaluations. Since we have ω = ω 0 + iζ, e ωt = e ω0t (cos ζt + isin ζt) and for small values of time, we can take ω = ω 0 (real). The numerical constants of the problems were takes as: ε = 0.003; δ 0 = ; ρ = 8954; τ 0 = 0.03; n = 500; P = 100; h 0 = 100; r = 1000; a = 1.2; ω = 1; α = 1.8; α = ; (1/K) and T 0 = 293 K. The computations were carried out for a value of time t = The numerical technique, outlined above, was used for the real part of the thermal temperature θ distribution, the displacement components u, v and the stress components σ, σ yy and σ y for each problem, for Problem I on the plane y = 5 and for Problem II on surface at y = 2 and on the middle plane y = 0, where L = 4 for two different values of Ω = 0 and Ω = The results are shown in Figs The graphs shows the four curves predicted by different theories of thermoelasticity. In these figures, the solid lines represent the solution corresponding to using the theory of heat conduction (τ 0 = 0), the dashed lines represent the solution for the Lord-Shulman s theory (τ 0 = 0.03). It can be seen from these figures that the rotation acts to decrease the magnitude of the real part of the temperature, displacement and the stress components. We notice also that the results for the temperature, displacement and stress components when the relaation time is appeared in the heat equation are distinctly different from those the relaation time is not mentioned in the heat equation. This is due to the fact that thermal waves in the Fourier theory of heat equation travel with an infinite speed of propagation as opposed to finite speed in the non- Fourier case. This demonstrates clearly the difference between the coupled and the generalized theory of thermoelasticity. Figs. 1 6 demonstrate the effect of rotation for Problem I on the plane y = 5. Figs represent the effect of rotation on the surface of Problem II at y = 2 and Figs represent the effect of rotation on the middle plane at y = 0, where L = Conclusion We can obtain the following conclusions according to the analysis above and the illustrated figures 1. It is clear that the rotation has decreasing effect with the modulus of elasticity being dependent on the reference temperature. 2. The horizontal component of displacement is vanishes on the surface and middle plane of problem II when the rotation equal to zero. 3. We note that since the vertical component of displacement v and the stress component σ y are odd function of y there values on the middle plane of Problem II are always zero. References [1] Nowacki, W: Thermoelasticity, Addison-Wesley Pub. Com. Inc. (1962), London, 5.

12 100 Generalized Thermoelasticity Plane Waves in... [2] Tanigawa, Y: Some basic thermoelastic problems for non-homogeneous structural materials. Appl. Mech. Rev., (1995), 48, 115. [3] Ootao, Y, Akai, T, Tanigawa, Y: Three-dimensional transient thermal stress analysis of a non-homogeneous hollow circular cylinder due to moving heat source in the aial direction. J. Thermal Stresses, (1995), 18, 89. [4] Manson, SS: Behavior of material under conditions of thermal stress, (1954), NACA [5] Biot, M: Thermoelasticity and irreversible thermo-dynamics. J. Appl. Phys., (1956), 27, [6] Lord, H, Shulman, Y: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solid, (1967), 15, [7] Puri, P: Plane waves in thermoelasticity and magneto-thermoelasticity, Int. J. Engng Sci., (1972), 10, [8] Ezzat, M, Othman, MIA, El-Karamany, AS: Electromagneto-thermoelasticity plane waves with thermal relaation in a medium of perfect conductivity, J. Thermal Stresses, (2001), 24, [9] Dhaliwal, R, Sherief, HH: Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., (1980), 33, 1-8. [10] Sherief, HH, Dhaliwal, R: Generalized One-dimensional thermal-shock problem for small times, J. Thermal Stresses, (1981), 4, 407. [11] Nayfeh, AH, Nasser, SN: Thermoelastic waves in solids with thermal relaation, Acta Mech., (1971), 12, [12] Ezzat, M, Othman, MIA, El-Karamany, AS: The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity, J. Therm. Stresses, (2001), 24, [13] Othman, MIA: Lord-Shulman theory under the dependence of the modulus of elasticity on the reference temperature in two-dimensional generalized thermo-elasticity, J. Therm. Stresses, (2002), 25, [14] Müller, I: The Coldness, A universal function in thermoelastic solids, Arch. Rat. Mech. Anal., (1971), 41, 319. [15] Green, AE, Laws, N: On the entropy production inequality, Arch. Rat. Mech. Anal., (1972), 45, 47. [16] Green, AE, Lindsay, KA: Thermoelasticity, J. Elasticity, (1972), 2, 1. [17] Erbay, S, Şuhubi, ES: Longitudinal wave propagation in a generalized elastic cylinder, J. Therm. Stresses, (1986), 9, 279. [18] Agarwal, VK: On plane waves in generalized thermoelasticity. Acta Mech. (1979), 13, [19] Agarwal, VK: On electromagneto-thermoelastic plane waves, Acta Mech., (1979), 34, [20] Ezzat, M, Othman, MIA: Electromagneto-thermoelasticity plane waves with two relaation times in a medium of perfect conductivity, Int. J. Engng Sci., (2000), 38, [21] Othman, MIA: Effect of rotation on plane waves in generalized thermoelasticity with two relaation times, Int. J. Solids and Structures, (2004), 41, [22] Ignaczak, J: A strong discontinuity wave in thermoelasticity with relaation times, J. Therm. Stresses, (1985), 8, [23] Ignaczak, J: Decomposition theorem for thermoelasticity with finite wave speeds, J. Therm. Stresses, (1978), 1, 41.

13 Othman, MIA 101 [24] Schoenberg, M, Censor, D: Elastic waves in rotating media, Quarterly of Applied Mathematics, (1973), 31, [25] Nowacki, W: Dynamic Problem of Thermo-elasticity, Noordhoof International, (1975), The Netherlands.

14 102 Generalized Thermoelasticity Plane Waves in... Nomenclature Symbol Description λ, µ Lamé s constants ρ density C E specific heat at constant strain E(T) Temperature dependent modulus of elasticity α empirical material constant [ ] 1 K η 0 c 2 0 ε 1 = ρce k = (λ0+2µ0)e0 ρ = γ0e0 ρc E E 0 const. modulus of elasticity at α = 0 ν Poisson s ratio t time T absolute temperature α T coefficient of linear thermal epansion δ 0 non-dimensional constant µ 0 = 1 2(1+ν) f(θ) is a given nondimensional function of temperature = ε 1 δ 0 f(θ) ε β = E0(λ0+µ0) ρc 2 0 ν λ 0 = (1+ν)(1 2ν) γ 0 = αt 1 2ν T 0 = δ0ρc2 0 γ 0E 0 = δ0 σ ij ε ij e = ε ii u i k τ 0 = 1 2(1 ν) ( 1 ν α T 1+ν ) reference temperature components of stress tensor components of strain tensor dilatation components of displacement vector thermal conductivity one relaation time

15 Othman, MIA Figure 1 Temperature distribution θ for Problem I at α = u Figure 2 Horizontal displacement distribution u for Problem I at α = 1.8

16 104 Generalized Thermoelasticity Plane Waves in v Figure 3 Vertical displacement distribution v for Problem I at α = Figure 4 The distribution of stress component σ for Problem I at α = 1.8

17 Othman, MIA yy Figure 5 The distribution of stress component σ yy for Problem I at α = yy Figure 6 The distribution of stress component σ y for Problem I at α = 1.8

18 106 Generalized Thermoelasticity Plane Waves in Figure 7 Temperature distribution θ for Problem II at α = u Figure 8 Horizontal displacement distribution u for Problem II on the surface at α = 1.8

19 Othman, MIA v Figure 9 Vertical displacement distribution v for Problem II on the surface at α = Figure 10 The distribution of stress component σ for Problem II on the surface at α = 1.8

20 108 Generalized Thermoelasticity Plane Waves in yy Figure 11 The distribution of stress component σ yy for Problem II on the surface at α = y Figure 12 The distribution of stress component σ y for Problem II on the surface at α = 1.8

21 Othman, MIA Figure 13 Temperature distribution θ for Problem II on the middle plane at α = u Figure 14 Horizontal displacement distribution u for Problem II on the middle plane at α = 1.8

22 110 Generalized Thermoelasticity Plane Waves in Figure 15 The distribution of stress component σ for Problem II on the middle plane at α = yy Figure 16 The distribution of stress component σ yy for Problem II on the middle plane at α = 1.8