Elsevier

Mechanics Research Communications

Volume 31, Issue 1, January–February 2004, Pages 81-89
Mechanics Research Communications

Thermoelastic interactions without energy dissipation in an unbounded body with a spherical cavity subjected to harmonically varying temperature

https://doi.org/10.1016/S0093-6413(03)00082-XGet rights and content

Abstract

The present work is concerned with the thermally induced vibration in a homogeneous and isotropic unbounded body with a spherical cavity. The Green and Nagdhi model of thermoelasticity without energy dissipation is employed. The closed form solutions for distributions of displacement, temperature and stresses are obtained. The solutions valid in the case of small frequency are deduced and the results are compared with the corresponding results obtained in other generalized thermoelasticity theories. Numerical results applicable to a copper-like material are also presented graphically and the nature of variations of the physical quantities with radial coordinate and with frequency of vibration is analyzed.

Introduction

The generalized theories of thermoelasticity, which admit the finite speed of thermal signal, have been the center of interest of active research during last three decades. These theories remove the paradox of infinite speed of heat propagation inherent in the conventional coupled dynamical theory of thermoelasticity introduced by Biot (1956). Because of the inclusion of the thermal relaxation parameters, the basic governing equations involved in the generalized theories of thermoelasticity are all of hyperbolic type differential equations and these theories are also referred to as hyperbolic thermoelasticity theories (Chandrasekharaiah, 1998). Extended thermoelasticity theory proposed by Lord and Shulman (1967), which is also known as thermoelasticity theory with one relaxation time, and the temperature-rate dependent theory of thermoelasticty developed by Green and Lindsay (1972), which is also known as thermoelasticity theory with two relaxation times, are two important models of generalized theory of thermoelasticity. Several experimental studies (Kaminski, 1990; Mitra et al., 1995; Tzou, 1995) indicate that the thermal relaxation effects can be of relevance in the cases involving a rapidly propagating crack tip, a localized moving heat source with high intensity, shock wave propagation, laser technique etc. Several problems revealing interesting phenomena which characterize the generalized theories of thermoelasticity have been persued by many researchers. Out of them (Erbay et al., 1991; Sherief and Salah, 1998; Mukhopadhyay, 1999; Misra et al., 1996; Chandrasekharaiah and Murthy, 1993) are relevant for the present work.

Recently, Green and Nagdhi, 1991, Green and Nagdhi, 1992, Green and Nagdhi, 1993 have formulated three different models of thermoelasticity in an alternative way. Among these, in one of the models (Green and Nagdhi, 1993) the most significance is that the internal rate of production of entropy is identically zero, i.e., there is no dissipation of thermal energy. This theory (GN theory) is known as thermoelasticity without energy dissipation theory (TEWOEDT). In the development of this theory the thermal displacement gradient is considered as a constitutive variable, whereas in the conventional development of a thermoelasticity theory, the temperature gradient is taken as a constitutive variable. Uniqueness theorem in case of linearized version of this theory has been given by Green and Nagdhi (1993) and Chandrasekharaiah (1996a) independently. Later on, Chandrasekharaiah (1996b) have studied free plane harmonic waves without energy dissipation in an unbounded body. Chandrasekharaiah and Srinath, 1997a, Chandrasekharaiah and Srinath, 1998 have studied cylindrical/spherical waves due to (i) a load applied to the boundary of the cylindrical/spherical cavity in an unbounded body, (ii) a line/point heat source in an unbounded body. Sharma and Chouhan (1996) tackled a problem on thermoelastic interaction without energy dissipation due to body forces and heat sources. Mukhopadhyay (2002) dealt with a problem concerning the thermoelastic interactions without energy dissipation in an unbounded medium with a spherical cavity subjected to thermal shock.

In the present problem an attempt has been made to understand the thermoelastic interactions without energy dissipation in an isotropic elastic medium with a spherical cavity which is subjected to harmonically varying temperature. The closed form solutions for the distributions of displacement, temperature and stresses are obtained analytically. The solutions for small frequency values are also found out. The numerical results are also performed for analyzing the nature of variations of the field variables with radial distance and with frequency.

Section snippets

Governing equations

The basic equations due to Green and Nagdhi (1993) in absence of body force and heat source for isotropic elastic medium are given by

  • (i)

    Displacement equation of motion:μ2u+(λ+μ)(divu)−γθ=ρü

  • (ii)

    Heat conduction equation:cθ̈+γθ0divü=k2θ

  • (iii)

    Stress displacement and temperature relation:σ=λ(divu)I+μ(u+uT)−γθI


where, λ and μ are Lame’ elastic constants, u is the displacement vector, θ is the temperature above uniform reference temperature θ0, ρ is the mass density, c=ρcv is the specific heat at constant

Problem formulation; boundary conditions

We consider an unbounded homogeneous and isotropic elastic medium which possesses a spherical cavity of radius a. It is assumed that there is no body force or heat source and the cavity surface is stress-free and is subjected to a harmonically varying temperature. We assume that the center of the cavity is at the origin of the spherical polar coordinates. Then according to our assumption the boundary conditions areθ=θ0eiΩtatr=aσrr=0atr=awhere r=a is the boundary of the cavity, σrr is radial

Solution of the problem

In view of the nature of boundary conditions, we assume the solutions for U, T, σRR, σϕϕ in the formsU=Ueiωη,T=Teiωη,σRRRReiωη,σφφφφeiωηwhere, the complex amplitudes, U, T, σRR, σϕϕ are independent of η.

Eqs. , then reduce toΔRTR=−ω2cp2UcT22T2T=−εω2Δwhere, Δeiωη.

Decoupling of Eqs. , yields(2+m12)(2+m22)(Δ,T)=0where,m12+m22=1+εcT2+1cp2ω2m12m22=ω4cT2cp2Solutions of equations (14) assuming the regularity conditions that U=T=σRR=σϕϕ=0 for R→∞, η⩾0 can be obtained

Solutions for small frequency

Solving Eqs. , we getm1,2=ωvj,j=1,2where,vj=12(1+ε)cp2+cT2+(−1)j+1Δ1/2withΔ=(1+ε)cp2−cT22+4εcp2cT21/2Thus substituting the values of m1 and m2 in the solutions for U, T, σRR, σϕϕ from , , , we can obtain the closed form solutions for the physical field variables.

Now assuming ω to be small so that neglecting the higher order terms of ω we obtain the expressions for small frequency values in the formsU=U(1)+iωU(2),T=T(1)+iωT(2)σRRRR(1)+iωσRR2,σφφφφ(1)+iωσφφ2where,U(1)=1−1R2U(2)=(R−1)

Discussions

We observe from the approximated solutions obtained above that in cases of small frequencies, the solutions for displacement, temperature and stresses are influenced by the material parameters cp, cT and the thermoelastic coupling constant ε. In case of other thermoelasticity theories (Chandrasekharaiah and Murthy, 1993), the solutions for small frequency are influenced only by the thermoelastic coupling constant and not by any other material parameter. The solutions indicate that at R=1, T=1

Numerical results

With an aim to illustrate the problem we will present some numerical results. The material chosen for the purpose of numerical computation is copper, the physical data for which is given as:ε=0.0168,λ=1.387×1012dyne/cm2,μ=0.448×1012dyne/cm2.In case of TEWOEDT, k is not the thermal conductivity, but a material constant, characteristic of the theory. We have chosen hypothetical values of the material parameters as cp=1.0, cT=0.578 (Chandrasekharaiah and Srinath, 1997b). Using this data the

Acknowledgements

The author would like to thank Professor R.N. Mukherjee for useful discussion and acknowledge the financial support from Council of Scientific and Industrial Research (CSIR), India.

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