8_Kar.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 1, 2009, (125-146) ISSN 1307-5543 – www.ejpam.com GENERALIZED THERMOELASTICITY PROBLEM OF A HOLLOW SPHERE UNDER THERMAL SHOCK Avijit Kar∗ and M. Kanoria Department of Applied Mathematics, University of Calcutta, 92, A. P. C. Road, Kolkata - 700 009, India Abstract. This problem deals with the thermo-elastic interaction due to step input of temperature on the boundaries of a homogeneous isotropic spherical shell in the context of generalized theories of thermo-elasticity. Using the Laplace transformation the fundamental equations have been expressed in the form of vector-matrix differential equation which is then solved by eigen value approach. The inverse of the transform solution is carried out by applying a method of Bellman et al. Stresses, dis- placements and temperature distribution have been computed numerically and presented graphically in a number of figures for copper material. A comparison of the results for different theories (CTE, CCTE, TRDTE(GL), TEWOED(GN-II), TEWED(GN-III)) is presented. When the outer radius of the shell tends to infinity, the corresponding results agree with that of existing literature. Key words: Generalized thermo-elasticity, energy dissipation, Laplace transform, step input tempera- ture, vector-matrix differential equation. 1. Introduction The classical theories of thermo-elasticity involving infinite speed of propagation of ther- mal signals, contradict physical facts. During the last three decades, non-classical theories involving finite speed of heat transportation in elastic solids have been developed to re- move this paradox. In contrast with the conventional coupled thermo-elasticity theory [1], ∗Corresponding author. Email address: akmathemati s�yahoo. om (A. Kar) http://www.ejpam.com 125 c© 2009 EJPAM All rights reserved. A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 126 which involves a parabolic-type heat transport equation, these generalized theories involving a hyperbolic-type heat transport equation are supported by experiments exhibiting the actual occurrence of wave-type heat transport in solids, called second sound effect. The extended thermo-elasticity theory (ETE) proposed by Lord and Shulman [1], incorporates a flux-rate term into Fourier’s law of heat conduction, and formulates a generalized form that involves a hyperbolic-type heat transport equation admitting finite speed of thermal signals. Green and Lindsay [2] developed temperature-rate-dependent thermo-elasticity (TRDTE(GL)) theory by introducing relaxation time factors that does not violate the classical Fourier law of heat con- duction and this theory also predicts a finite speed for heat propagation. The closed-form solutions for thermo-elastic problems in generalized theory of thermo-elasticity have been ob- tained by Hetnarski and Ignaczak [3]. Hetnarski and Ignaczak [4] studied the response of semi-space to a short laser pulse in the context of generalized thermo-elasticity. Sinha and Sinha [5] and Sinha and Elsibai [6], [7] studied the reflections of thermo-elastic waves from the free surface of a solid half-space and at the interface of two semi-infinite media in welded contact in the context of generalized thermo-elasticity. Sharma [8] investigated the reflection of thermo-elastic waves from the stress-free insulated boundary of a transversely isotropic half-space in the light of the theory developed by Dhaliwal and Singh [9] in which the theory developed in [1] is extended to anisotropic (transversely isotropic) thermo-elastic materials. Kar and Kanoria [10] investigated thermo-elastic stress wave propagation in an unbounded body with a spherical hole following the theories developed in [2], [15]. Roychoudhury and Bhatta [12] and Roychoudhury and Chatterjee [13] studied thermo-elastic wave propagation in an infinitely extended thin plate following the theory developed in [2] for short-time ap- proximation using the Laplace transform technique. Most engineering materials such as metals possess a relatively high rate of thermal damp- ing 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 foresee- able 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. The relevant theoretical developments on the subject are due to Green and Naghdi[14-16] and provide A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 127 sufficient basic modifications in the constitutive equations that permit treatment of a much wider class of heat flow problems, labelled as types I, II, III. The natures of these three types of constitutive equations are such that when the respective theories are linearized, type-I is the same as the classical heat equation (based on Fourier’s law) whereas the linearized ver- sions of type-II and type-III theories permit propagation of thermal waves at finite speed. The entropy flux vector in type II and III ( i.e. thermo-elasticity without energy dissipation (TEWOED) and thermo-elasticity with energy dissipation (TEWED)) models are determined in terms of the potential that also determines stresses. When Fourier conductivity is dominant the temperature equation reduces to classical Fourier law of heat conduction and when the effect of conductivity is negligible the equation has undamped thermal wave solutions without energy dissipation. Several investigations relating to thermo-elasticity without energy dissi- pation theory (TEWOED) have been presented by Roychoudhuri and Datta [17], Sharma and Chouhan [18], Roychoudhury and Bandyopadhyay [19], Chandrasekharaiah [20], [21]. The main object of the present paper is to study the thermo-elastic stresses, displacement and temperature distribution in an isotropic homogeneous spherical shell due to step input of temperature on the boundaries of the shell in the context of TRDTE [2] and TEWED [16] of type-III of generalized thermo-elasticity. Using the Laplace transformation the fundamen- tal equations have been expressed in the form of vector-matrix differential equation which is then solved by eigen value approach. The inversion of the Laplace transform is done following Bellman et al. [22]. The results obtained theoretically have been computed numerically and are presented graphically for Copper material. A complete and comprehensive analysis and comparison of results of the above theories are presented. 2. Basic equations and constitutive relations We consider a homogeneous isotropic spherical shell of inner radius a and outer radius b in an undisturbed state and initially at uniform temperature T0. We introduce spherical polar co-ordinate (r,θ ,φ) with the centre of the cavity as the origin. In the present problem (due to spherical symmetry) the displacement and temperature are A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 128 assumed to be functions of r and time t only. The stress-strain-temperature relations in the present problem are τi j = λ∆δi j + 2µei j − γ(T + δ1kαṪ )δi j (1) and generalized heat conduction equation is � δ1k + δ2k ∂ ∂ t � K∇2T + δ2kK∗∇2T = ρCe � δ1k Ṫ + � δ1kα ∗ + δ2k � T̈ � + γT0 � ζδ1k + δ2k ∂ ∂ t � ∆̇, (2) where τi j(i = r,θ) is the stress tensor, ∆ is the dilatation, T is the temperature increase over the absolute reference temperature T0, γ = (3λ+ 2µ)αt , λ and µ are the Lame’s constants, αt is the coefficient of linear thermal expansion of the material, α and α∗ are the relaxation times, K is the coefficient of thermal conductivity, K∗ is the additional material constant, ρ is the mass density, Ce is the specific heat of the solid at constant strain, δi j is the Kronecker delta. Consider the following partial cases of equations (1) and (2): (i) if α = 0, α∗ = 0, k = 1, and ζ = 0, then the equations (1) and (2) are reduced to the equations of classical theory of thermo-elasticity (CTE). (ii) if α = 0, α∗ = 0, k = 1, and ζ = 1, then the equations (1) and (2) are reduced to the equations of classical coupled theory of thermo-elasticity (CCTE). (iii) if k = 1, and ζ = 1, then the equations (1) and (2) are reduced to the equations of temperature rate dependent thermo-elasticity theory (TRDTE(GL). (iv) if k = 2 and K = 0, then the equations (1) and (2) are reduced to the equations of thermo-elasticity with energy dissipation (TEWOED(GN-II)). (v) if k = 2, then the equations (1) and (2) are reduced to the equations of thermo-elasticity with energy dissipation (TEWED(GN-III)). A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 129 The thermal relaxation times satisfy the inequalities [2] α≥ α∗ > 0 in the case of TRDTE(GL) theory. If u∼ = [u(r, t), 0,0] be the displacement vector, then er r = ∂ u ∂ r , eθθ = eφφ = u r . (3) The stress equation of motion in spherical polar co-ordinates is given by ∂ τr r ∂ r + 2 r (τr r −τθθ ) = ρ ∂ 2u ∂ t2 . (4) Introducing the following dimensionless quantities U = (λ+ 2µ)u aγT0 ; (R,S) = � r a , b a � ; (σR,σθ ) = 1 γT0 (τr r ,τθθ ); Θ = T T0 ; η= Gt a ; G2 = λ+ 2µ ρ ; α ′ = αG a ; α∗ ′ = α∗G a ; equations (1), (2) and (4) become σR = ∂ U ∂ R + 2λ1 U R − � Θ+ δ1kα ′ δΘ δη � , (5) σθ = λ1 ∂ U ∂ R + (λ1+ 1) U R − � Θ+ δ1kα ′ δΘ δη � , (6) �� δ1k + G a ∂ ∂ η δ2k � c2 K + G a c2 Tδ2k �� ∂ 2Θ ∂ R2 + 2 R ∂Θ ∂ R � = ¨ δ1k ∂Θ ∂ η + � α∗ ′ δ1k + G a δ2k � ∂ 2Θ ∂ η2 « + ε � ζδ1k ∂ ∂ η + G a ∂ 2 ∂ η2 δ2k �� ∂ U ∂ R + 2U R � (7) and ∂ 2U ∂ R2 + 2 R ∂ U ∂ R − 2U R2 = � 1+ δ1kα ′ ∂ ∂ η � ∂Θ ∂ R + ∂ 2U ∂ η2 , (8) where λ1 = λ λ+ 2µ , c2 T = K∗ ρCeG 2 c2 K = K aρCeG , and ε = γ2T0 ρCe(λ+ 2µ) are dimension- less constants. ε being the thermo-elastic coupling constant. Here cT is the nondimensional A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 130 thermal wave velocity and cK is the damping co-efficient. In the case cK = 0, we arrive at the thermo-elasticity equations without energy dissipation theory (TEWOED(GN-II)) for Green- Naghdi model. The boundary conditions are given by σR = 0 on R= 1,S η ≥ 0, Θ= χ1H(η), on R= 1, η > 0, = χ2H(η) on R= S, η > 0, where χ1 and χ2 are dimensionless constants and H(η) is the Heaviside unit step function. The above conditions indicate that for time η ≤ 0 there is no temperature (Θ = 0). Thermal shocks are given on the boundaries of the shell (R = 1, S) immediately after time η = 0. Thermal stresses in the elastic medium due to the application of these thermal shocks are calculated. We assume that the medium is at rest and undisturbed initially. The initial and regularity conditions can be written as U = Θ = 0 at η = 0,R≥ 1, ∂ U ∂ η = 0 at η = 0, U = Θ = 0 when R→∞. 3. Method of solution Let ¦ U(R, p),Θ(R, p) © = ∫ ∞ 0 � U(R,η),Θ(R,η) e−pηdη with Re(p) > 0 denote the Laplace transform of U and Θ respectively. Application of the Laplace transform, equations (7) and (8) yields d2Θ dR2 + 2 R dΘ dR = p    (1+α∗ ′ p)δ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k   Θ+ A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 131 εp ¨ ζδ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k «� dU dR + 2U R � (9) and d2U dR2 + 2 R dU dR − 2U R2 = p2U + (1+α ′ pδ1k) dΘ dR . (10) Differentiating equation (9) with respect to R and using equation (10) we get d2 dR2 � dΘ dR � + 2 R d dR � dΘ dR � − 2 R2 � dΘ dR � = εp3 ¨ ζδ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k « U+ p   {(1+α∗′ p) + εζ(1+α′p}δ1k + (1+ ε)pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k   dΘ dR . (11) Equations (10) and (11) can be written in the form L(U) = p2U + (1+α ′ pδ1k) dΘ dR (12) and L � dΘ dR � = εp3 ¨ ζδ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k « U+ p   {(1+α∗′ p) + εζ(1+α′p}δ1k + (1+ ε)pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k   dΘ dR , (13) where ε is the thermo-elastic coupling constant and L ≡ d2 dR2 + 2 R d dR − 2 R2 . (14) From equations (12) and (13) we have the vector-matrix differential equation as follows LeV = eAeV , (15) where eV = � U , dΘ dR �T , eA=   C11 C12 C21 C22   (16) and C11 = p2, C12 = 1+α ′ pδ1k, C21 = εp 3 ¨ ζδ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k « , C22 = p   {(1+α∗′ p) + εζ(1+α′p}δ1k + (1+ ε)pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k   . A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 132 4. Solution of the vector-matrix differential equation Let eV = eX (m)ω(R, m), (17) where m is a scalar, eX is a vector independent of R and ω(R, m) is a non-trivial solution of the scalar differential equation Lω = m2ω. (18) Let ω= R−1/2ω1. Therefore, from equation (18) we have d2ω1 dR2 + 1 R dω1 dR − � 9 4R2 +m2 � ω1 = 0. (19) Therefore, the solution of the equation (18) is ω= [A1 I3/2(mR) +A2K3/2(mR)]/ p R (20) Substituting equations (17) and (18) into equation (15) we get eAeX = m2 eX , (21) where eX (m) is the eigenvector corresponding to the eigenvalue m2. The characteristic equation corresponding to eA can be written as m4 − (C11 + C22)m 2 + (C11C22 − C12c21) = 0. (22) The roots of the characteristic equation (22) are of the form m2 = m2 1 and m2 = m2 2, where m2 1 +m2 2 = C11 + C22, m2 1m2 2 = C11C22 − C12C21. (23) The eigenvectors X (m j), j =1,2 corresponding to eigenvalues m2 j , j =1,2 can be calculated as eX (m j) =   X1(m j) X2(m j)   =   C12 −(C11 −m2 j )   , j = 1,2. (24) Therefore, the equation (17) can be written as eV = eX (m j)[A1I3/2(m1R) + B1K3/2(m1R)]/ p R+ A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 133 eX (m j)[A1I3/2(m2R)+ B1K3/2(m2R)]/ p R, æ= 1,2. (25) Therefore, from equations in (15) we can write U = ∑ i=1,2 C12[Ai I3/2(miR) + BiK3/2(miR)]/ p R (26) and dΘ dR = − ∑ i=1,2 (C11 −m2 i )[Ai I3/2(miR)+ BiK3/2(miR)]/ p R, (27) where I3/2(miR) and K3/2(miR) are the modified Bessel functions of order 3/2 of first and second kind respectively. Ai and Bi ’s i = 1,2 are independent of R but depend on p and are to be determined from the boundary conditions. Using the recurrence relations of modified Bessel functions [23] we obtain from the equation (27) Θ = ∑ i=1,2 (C11 −m2 i ) mi [Ai I1/2(miR) + BiK1/2(miR)]/ p R, (28) since 1 R1/2 P3/2(mR) = − d dR � P1/2(mR) mR1/2 � , (29) where P = I , K . Taking the Laplace transform of the equations (5) and (6) and using equations (26) and (28) we get σR = a1 ∑ i=1,2 Ai � − 4µ λ+ 2µ I3/2(miR)− p2 mi RI1/2(miR) � /R3/2+ a1 ∑ i=1,2 Bi � − 4µ λ+ 2µ K3/2(miR)− p2 mi RK1/2(miR) � /R3/2 (30) and σΘ = a1 ∑ i=1,2 Ai � 2µ λ+ 2µ I3/2(miR)− λm2 i + (λ+ 2µ)(p2−m2 i ) (λ+ 2µ)mi RI1/2(miR) � /R3/2+ a1 ∑ i=1,2 Bi � 2µ λ+ 2µ K3/2(miR)− λm2 i + (λ+ 2µ)(p2 −m2 i ) (λ+ 2µ)mi RK1/2(miR) � /R3/2, (31) A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 134 where a1 = 1+ α ′ pδ1k. Using the boundary conditions σR = 0 on R = 1, R = S and Θ = χ1 p on R= 1, Θ = χ2 p on R= S. and using the recurrence relations [23] we obtain from equations (28) and (29) A1W11 + A2W12 + B1W13 + B2W14 = 0, A1W21 + A2W22 + B1W23 + B2W24 = 0, A1W31 + A2W32 + B1W33 + B2W34 = χ1 p , (32) A1W41 + A2W42 + B1W43 + B2W44 = χ2 p , where W1i = 4µ λ+ 2µ P3/2(m j) + p2 m j P1/2(m j), W2i = 4µ λ+ 2µ P3/2(m jS) + p2 m j SP1/2(m jS), W3i = p2 −m2 j m j P1/2(m j), (33) W4i = p2 −m2 j m jS 1/2 P1/2(m jS), where P = I for i = j =1,2; P = K for i = 3, j = 1 and i = 4, j = 2. From (32) the values of A1, A2, B1 and B2 are given as   A1 A2 B1 B2   =   W11 W12 W13 W14 W21 W22 W23 W24 W31 W32 W33 W34 W41 W42 W43 W44   −1   0 0 χ1 p χ2 p   . (34) Equation (22) can be written as m4 − � p2 + ¨ p c2 K (1+α∗ ′ p+ εζ(1+α ′ p))δ1k + (1+ ε)p2 c2 T + c2 K p δ2k «� m2+ p3    (1+α∗ ′ p)δ1k + pδ2k c2 K(δ1k + pδ2k) + c2 Tδ2k    = 0. (35) A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 135 Therefore, m1, m2 = p p(δ1k + p pδ2k) 2 p c2 K(δ1k + pδ2k) + c2 Tδ2k ( p α±pβ), (36) where α,β = [ Æ 1+α∗ ′ pδ1k ± Æ c2 K p+ c2 Tδ2k] 2 + [(1+α ′ )ζδ1k + δ2k]ε. (37) Therefore, m1 and m2 are real and positive quantities. 5. Special case (Case of infinite body) From (30) and (31) we write σR = σR(I) +σR(K), (38) σθ = σθ (I) +σθ (K), (39) where σR(I) = a1 ∑ i=1,2 Ai � − 4µ λ+ 2µ I3/2(miR)− p2 mi RI1/2(miR) � /R3/2, (40) σR(K) = a1 ∑ i=1,2 Bi � − 4µ λ+ 2µ K3/2(miR)− p2 mi RK1/2(miR) � /R3/2, (41) σθ (I) = a1 ∑ i=1,2 Ai � 2µ λ+ 2µ I3/2(miR)− λm2 i + (λ+ 2µ)(p2−m2 i ) (λ+ 2µ)mi RI1/2(miR) � /R3/2, (42) and σθ (K) = a1 ∑ i=1,2 Bi � 2µ λ+ 2µ K3/2(miR)− λm2 i + (λ+ 2µ)(p2 −m2 i ) (λ+ 2µ)mi RK1/2(miR) � /R3/2. (43) A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 136 For large value of b i.e. for large value of S, K1/2(miS) and K3/2(miS) tend to zero. Thus for large values of b the asymptotic expressions of σR(I) and σθ (I) are given as σR(I) = a1 χ2 p p S R2 e−m2(S−R) � 4µ λ+ 2µ � 1− 1 m1S � + p2 m1 S � × � 4µ λ+ 2µ � 1− 1 m2R � + p2 m2 R � −e−m1(S−R) � 4µ λ+ 2µ � 1− 1 m2S � + p2 m2 S � × � 4µ λ+ 2µ � 1− 1 m1R � + p2 m1 R � p2 −m2 1 m1 � 4µ λ+ 2µ � 1− 1 m2S � + p2 m2 S � − p2 −m2 2 m2 � 4µ λ+ 2µ � 1− 1 m1S � + p2 m1 S � , (44) → 0 as S→∞ and σθ (I) = a1 χ2 p p S R2 e−m1(S−R) � 2µ λ+ 2µ � 1− 1 m1R � − λm2 1 + (λ+ 2µ)(p2−m2 1) (λ+ 2µ)m1 R � × � 4µ λ+ 2µ � 1− 1 m2S � + p2 m2 S � −e−m2(S−R) � 2µ λ+ 2µ � 1− 1 m2R � − λm2 2 + (λ+ 2µ)(p2−m2 2) (λ+ 2µ)m2 R � × � 4µ λ+ 2µ � 1− 1 m1S � + p2 m1 S � p2 −m2 1 m1 � 4µ λ+ 2µ � 1− 1 m2S � + p2 m2 S � − p2 −m2 2 m2 � 4µ λ+ 2µ � 1− 1 m1S � + p2 m1 S � , (45) → 0 as S→∞. Therefore, as b→∞ σR = a1 ∑ i=1,2 Bi � − 4µ λ+ 2µ K3/2(miR)− p2 mi RK1/2(miR) � /R3/2, (46) A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 137 σθ = a1 ∑ i=1,2 Bi � 2µ λ+ 2µ K3/2(miR)− λm2 i + (λ+ 2µ)(p2 −m2 i ) (λ+ 2µ)mi RK1/2(miR) � /R3/2, (47) where Bi ’s (i = 1,2) are given as B1 = − χ1 p m1 � 4µm2K3/2(m2) + (λ+ 2µ)p2K1/2(m2) � 4µ � (p2 −m2 2)m1K3/2(m1)K1/2(m2)− � p2 −m2 1)m2K1/2(m1)K3/2(m2) � +(λ+ 2µ)p2(m2 1 −m2 2)K1/2(m1)K1/2(m2) , (48) and B2 = χ1 p m2 � 4µm1K3/2(m1) + (λ+ 2µ)p2K1/2(m1) � 4µ � (p2 −m2 2)m1K3/2(m1)K1/2(m2)− (p2 −m2 1)m2K1/2(m1)K3/2(m2) � +(λ+ 2µ)p2(m2 1 −m2 2)K1/2(m1)K1/2(m2) . (49) The values of m1 and m2 take the same values for this problem and those in [10] though the dimensionless forms are different. Therefore the above results are equivalent to those in [10]. 6. Numerical results and discussions The solution in the space-time domain is obtained numerically by the method of Bellman et al. [22] for fixed value of the space variable and for dimensionless time η = ηi , i = 1(1)7, where ηi ’s are computed from roots of the shifted Legendre polynomial of degree 7 (see Appendix) with S = 4. The computations for the state variables are carried out for different values of ηi , ηi = 0.0257750, 0.138382, 0.352509, 0.693147, 1.21376, 2.04612, 3.67119. The material chosen for numerical evaluation is copper. The physical data for copper are taken as [11] ρ = 8.96gm/cm3, ε= 0.0168, λ = 1.387× 1012d y/cm2, µ = .448× 1012d y/cm2, and the hypothetical values of relaxation time parameters and material parameters cT and cK are taken as α= 2.0× 10−13sec, α∗ = 1.0× 10−13sec, cT = 2 A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 138 and cK = 1.2. The results of the numerical evaluation of the thermo-elastic stress variations, temperature distribution and displacement are illustrated in figs. 1 to 6. The magnitudes of the variation of stresses, temperature and displacement are observed when the step input of temperatures χ1 = 5 and χ2 = 3 are applied on the inner boundary R=1 and outer boundary S=4 of the shell respectively in all theories (CTE, CCTE, TRDTE(GL), TEWOED(GN-II) and TEWED(GN- III)). Figs.1(a) and 1(b) show the radial stress σR against radial distance R for time η=.69 and .026 respectively. In fig.1(a) it is observed that the qualitative behavior of CTE, CCTE, GL and GN-III models are almost the same. But in GN-II model oscillatory nature is observed. This is due to the rapid reflection of stress wave from outer boundary to inner boundary, since there is no energy dissipation in this theory. In fig.1(b), when time is small (η = .026) i,e, at early stage of wave propagation all the models except GN-II model give close results, whereas for advanced time (Fig.1(a)) the wave propagates with different speeds. The boundary conditions on the boundaries of the shell are found to be satisfied numerically. This is consistent with our theoretical results. Figs.2(a) and 2(b) are plotted for hoop stress σθ against radial distance R for time η= .69 and .026 respectively for all the models. It is clear from fig.2(a) that the maximum stress occurs near the inner boundary and almost oscillatory nature are observed for all the models for time η = .69, whereas for small time η = .026 i,e, at the early stage the maximum stress occurs near the boundaries and almost zero in the interior of the shell for CTE, CCTE, GL and GN-III except in the case of GN-II model, where the oscillatory nature becomes more prominent near the boundaries. Figs.3 and 4 depict σR and σθ versus time η for R= 1.5 respectively. It is seen that the stress waves propagate with time and magnitude of both the stresses in GN-II model are large in comparison with other models. Fig.5 shows the graphs of temperature distribution (Θ) with the radial distance R for fixed time η = .69. All the profiles except GN-II model are found to be convex upwards with respect to the R-axis, and the magnitude of temperature distribution in GN-III model is observed to be greater than that of the other models (CTE, CCTE and GL). This is due to the fact that the term A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 139 involving thermal wave velocity (cT ) is present in the GN-III model. The profile associated with GN-II model appears to be oscillatory in the region 1 ≤ R ≤ 4, as in this case there is no damping term (cK) present in the expression of temperature distribution. It is also to be seen that the boundary conditions are satisfied on the boundaries (χ1 = 5 on R=1 and χ2 = 3 on S = 4). Which are in agreement with our theoretical results. Fig.6 is plotted for radial variation of the displacement for fixed time η = .69. Almost similar behaviors are observed in all the theories. Now figs. 7-11 depicts σR versus R for fixed time η = .69 with inner radius R = 1 and different outer radius S = 3, 6, 8, 10 and 12 for GN- III model. It is observed that as structural size increases the radial stress confined near the boundaries of the hollow sphere where the step input of temperatures are applied. Which is physically plausible. It is also observed that oscillations are accompanied by large R. This is due to the fact that stress waves propagate from the inner boundary to the outer boundary and from the outer boundary to the inner boundary at the same time. Similar behavior is observed in figs. 12-16 for hoop stress. For all above numerical calculations FORTRAN-77 programming Language has been used. (a) Radial Stress vs. R for η = .69. (b) Radial Stress vs. R for η = .026. Figure 1: Radial Stress. A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 140 (a) Hoop Stress vs. R for η = .69. (b) Hoop Stress vs. R for η = .026. Figure 2: Hoop Stress. Figure 3: Radial Stress vs. η for R= 1.5 Figure 4: Hoop Stress vs. η for R= 1.5 Figure 5: Variation of Temperature Distribu- tion vs. R for η= .69. Figure 6: Variation of Displacement vs. R for η= .69. A. Kar and M. Kanoria / Eur. J. Pure Appl. Math, 2 (2009), (125-146) 141 Figure 7: Radial Stress vs. R for η = .69 and S = 3. Figure 8: Radial Stress vs. R for η = .69 and S = 6. Figure 9: Radial Stress vs. R for η = .69 and S = 9. Figure 10: Radial Stress vs. R for η = .69 and S = 10. Figure 11: Radial Stress vs. R for η = .69 and S = 12. Figure 12: Hoop Stress vs. R for η = .69 and S = 3. Figure 13: Hoop Stress vs. R for η = .69 and S = 6. Figure 14: Hoop Stress vs. R for η = .69 and S = 8. REFERENCES 142 Figure 15: Hoop Stress vs. R for η = .69 and S = 10. Figure 16: Hoop Stress vs. R for η = .69 and S = 12. ACKNOWLEDGEMENTS. We are grateful to Prof. S. C. Bose of the Department of Applied Mathematics, University of Calcutta for his kind help and guidance in the preparation of the paper. We also express our sincere thanks to the referees for their valuable suggestion for the improvement of the paper. Avijit Kar is grateful to the Council of Scientific and Industrial Research (C.S.I.R), New Delhi for the award of a fellowship. References [1] H.W. Lord and Y. Shulman, A generalized dynamic theory of thermoelasticity, J. Mech. Phys. Solids, 15 (1967), 299-309. [2] A.E. Green and K.A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7. [3] R.B. Hetnarski and J. Ignaczak, Generalized Thermoelasticity: closed form solutions, J. Thermal Stresses, 16 (1993), 473-498. [4] R.B. Hetnarski and J. Ignaczak, Generalized Thermoelasticity: Response of semi-space to a short laser pulse, J. Thermal Stresses, 17 (1994), 377-396. [5] A.N. Sinha and S.B. Sinha, Reflexion of Thermoelastic Waves at a Solid Half-space With Thermal Relaxation, J. Phys. 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Co., New York, 1966. [23] J.N. Watson, Theory of Bessel Function(2nd edn.), Cambridge Univ. Press, 1980. REFERENCES 144 Appendix Let the Laplace transform of σi(R,η) be given by σ j(R, p) = ∫ ∞ 0 e−pησ j(R,η)dη. (A.1) We assume that σ j(R,η) is sufficiently smooth to permit the use of the approximate method we apply. Putting x = e−η in equation (A.1) we obtain σ j(R, p) = ∫ 1 0 x p−1 g j(R, x)d x , (A.2) where g j(R, x) = σ j(R,−log x). (A.3) Applying the Gaussian quadrature rule to the equation (A.2) we obtain the approximate rela- tion n∑ i=1 Wi x p−1 i g j(R, x i) = σ j(R, p), (A.4) where x i ’s(i = 1,2, ....n) are the roots of the shifted Legendre polynomial and Wi ’s(i = 1,2, ....n) are the corresponding weights [22] and p = 1(1)n. For p = 1(1)n), the equations (A.4) can be written as W1 g j(R, x1) +W2 g j(R, x2) + ............+Wn g j(R, xn) = σ j(R, 1) W1 x1 g j(R, x1) +W2 x2 g j(R, x2) + ............+Wn xn g j(R, xn) = σ j(R, 2) .................................................................. REFERENCES 145 .................................................................. W1 xn−1 1 g j(R, x1) +W2 xn−1 2 g j(R, x2) + ............+Wn xn−1 n g j(R, xn) = σ j(R, n) Therefore   g j(R, x1) g j(R, x2) .. .. .. g j(R, xn)   =   W1 W2 . . . Wn W1 x1 W2 x2 . . . Wn xn . . . . . . . . . . . . . . . . . . W1 xn−1 1 W2 xn−1 2 . . . Wn xn−1 n   −1  σ j(R, 1) σ j(R, 2) . . . σ j(R, n)   . (A.5) (As the matrix is the product of diag{Wi}multiplied by Vander Monde matrix, it can be shown that the matrix is non-singular.) Hence g j(R, x1), g j(R, x2), ...., g j(R, xn) are known. For n= 7 we have Roots of the shifted Corresponding Weights Legendre Polynomial 2.5446043828620886E-2 6.4742483084434816E-2 1.2923440720030282E-1 1.3985269574463828E-1 2.9707742431130145E-1 1.9091502525255938E-1 5.0000000000000000E-1 2.0897959183673466E-1 7.0292257568869853E-1 1.9091502525255938E-1 8.7076559279969706E-1 1.3985269574463828E-1 9.7455395617137909E-1 6.4742483084434816E-2 REFERENCES 146 From equations in (A.5) we can calculate the discrete values of g j(R, x i) i,e, σ j(R,ηi); (i =1,2,....,7) and finally using interpolation we obtain the stress components σi(R,η); (i = R,θ).