Some remarks on growth and uniqueness in thermoelasticity

R Quintanilla

doi:10.1155/s0161171203108319

Abstract

We use the Lagrange identity method and the logarithmic convexity to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the first contribution of this kind in this theory, it is worth remarking that the assumptions we use here are different from those used in other previous contributions. We assume that the elasticity tensor is positive semidefinite, but we allow that the constitutive tensor of the entropy flux vector (kij), which is a characteristic tensor in this theory, is not sign‐definite. The Lagrange identity method is used to obtain uniqueness in the context of the thermoelasticity of type III. The fundamental key to obtain exponential growth in the thermoelasticity without energy dissipation is the use of a new functional. This functional is inspired in that it is used when the elasticity tensor is not sign‐definite, but (kij) is positive definite.

Highlights

The usual theory of heat conduction based on the Fourier law allows the phenomena of the infinite diffusion velocity which is not well accepted from a physical or engineering point of view
They report instances where the phenomena of second sound has been observed in several kind of materials
Extensive reviews on the second sound theories are the work of Chandrasekharaiah [3] and the books of Jou et al [12], Müller and Ruggeri [14]

Summary

Introduction

The usual theory of heat conduction based on the Fourier law allows the phenomena of the infinite diffusion velocity which is not well accepted from a physical or engineering point of view. The relevant equations of anisotropic inhomogeneous thermoelasticity of type II with a center of symmetry are ρui = aijkhuh,k ,j − aij θ ,j + ρfi, cθ = −aij ui,j + kij θ,j ,i + ρr , (1.1) (1.2)

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Conclusion