Thermoelastic interactions without energy dissipation in an unbounded body with a spherical cavity subjected to harmonically varying temperature
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:
- (ii)
Heat conduction equation:
- (iii)
Stress displacement and temperature relation:
where, λ and μ are Lame’ elastic constants, 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 arewhere 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 formswhere, the complex amplitudes, , , , are independent of η.
Eqs. , then reduce towhere, .
Decoupling of Eqs. , yieldswhere,Solutions 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 getwhere,withThus substituting the values of m1 and m2 in the solutions for , , , 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 formswhere,
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,
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:In case of TEWOEDT, 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.
References (22)
Mech. Res. Commun.
(1996)- et al.
Mech. Res. Commun.
(1997) - et al.
J. Mech. Phys. Solids
(1967) - et al.
Int. J. Engng. Sci.
(1996) - et al.
Int. J. Engng. Sci.
(1998) J. Appl. Phys.
(1956)J. Thermal stresses
(1996)Appl. Mech Rev.
(1998)- et al.
J. Thermal Stresses
(1993) - et al.
J. Elasticity
(1997)
J. Elasticity
Cited by (29)
Rotational and voids effect on the reflection of P waves from stress-free surface of an elastic half-space under magnetic field and initial stress without energy dissipation
2013, Applied Mathematical ModellingCitation Excerpt :On the spatial behavior in thermoelasticity without energy dissipation has been illustrated by Quintanilla [20]. An attempt has been made by Mukhopadhyay [21] to investigate the thermoelastic interactions without energy dissipation in an unbounded body with a spherical cavity which is subjected to harmonically varying temperature. Othman and Song [22] have investigated a reflection phenomena of the plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation.
A three-dimensional thermoelastic problem for a half-space without energy dissipation
2012, International Journal of Engineering ScienceCitation Excerpt :When Fourier conductivity is dominant the temperature equation reduces to the 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 the thermo-elasticity without energy dissipation (TEWOED) theory have been presented by Roychoudhuri and Dutta (2005), Sharma and Chouhan (1999), Roychoudhuri and Bandyopadhyay (2004), Chandrasekharaiah and Srinath (1998, 1997), Das, Lahiri, Sarkar, and Basu (2008), Mukhopadhyay (2002, 2004) and Mukhopadhyay and Kumar (2008). Many problems in engineering practice involve determination of stresses and/or displacements in bodies that are three-dimensional.
Thermo-elastodynamic response of a spherical cavity in saturated poroelastic medium
2010, Applied Mathematical ModellingMode of a spherical cavity's thermo-elastodynamic response in a saturated porous medium for non-torsional loads
2010, Computers and GeotechnicsA study of generalized thermoelastic interactions in an unbounded medium with a spherical cavity
2008, Computers and Mathematics with ApplicationsCitation Excerpt :Sharma and Chauhan [21] investigated a problem concerning thermoelastic interactions without energy dissipation due to body forces and heat sources. Mukhopadhyay [22] tackled a problem concerning the thermoelastic interactions without energy dissipation in an unbounded medium with a spherical cavity subjected to harmonically varying temperature. Othman and Song [23] have investigated the effect of rotation on the reflection of magneto-thermoelastic waves under thermoelasticity without energy dissipation.
A numerical solution of magneto-thermoelastic problem in non-homogeneous isotropic cylinder by the finite-difference method
2007, Applied Mathematical Modelling