R. Quintanilla

Growth and uniqueness in thermoelasticity

A uniqueness theorem is proved for two theories of thermoelasticity capable of admitting finite speed thermal waves, the theories having been proposed by Green & Naghdi. Uniqueness is proved under the weak assumption of requiring only major symmetry of the elasticity tensor; no definiteness whatsoever is postulated. It is shown how to demonstrate uniqueness by a Lagrange identity method and also by producing a novel functional to which to apply the technique of logarithmic convexity. It is remarked on how to extend the result to an unbounded spatial domain without requiring decay restrictions on the solution at infinity. Finally, conditions are derived which show how a suitable measure of the solution will grow at least exponentially in time if the initial ‘energy’ satisfies appropriate conditions. This complements the fundamental work of Knops & Payne, who produced corresponding growth results in the isothermal elasticity case.

A note on discontinuity waves in type III thermoelasticity

Two recent nonlinear theories of thermoelasticity, developed by Green and Naghdi, are examined. It is shown that in type II theory, second sound is permissible and both mechanical and temperature waves may propagate. In type III theory we show that the situation is more analogous to that in classical nonlinear thermoelasticity: one wave propagates and a homothermal temperature wave is allowed.

Slow decay for one-dimensional porous dissipation elasticity

This paper concerns the one-dimensional linear theory of porous elastic solids. We prove the slow decay for the solutions of two initial-boundary value problems determined by several boundary conditions.

ON A THEORY OF THERMOELASTICITY WITH MICROTEMPERATURES

The article is concerned with a linear theory for elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. In the first part of the article we derive an existence result for the dynamical theory and establish the continuous dependence of solutions on the initial data and body loads. Then we consider the equilibrium theory and present a uniqueness result and a solution for the field equations. The theory is illustrated with the solution of the problem of thermal stresses in an elastic space with a spherical cavity.

Damping of end effects in a thermoelastic theory

The aim of this paper is to present a spatial decay estimate in the thermoelasticity of Type III. We prove that the rate of decay is bounded below by an exponential of a second degree polynomial of the distance.

Exponential Stability in the Dual-Phase-Lag Heat Conduction Theory

Abstract In this paper we obtain the existence of solutions and continuous dependence on the supply terms and initial conditions for the equation of the dual-phase-lag heat conduction theory. When the phase-lag constants satisfy a certain condition we can prove exponential stability with respect to time and spatial variables. When this condition is not satisfied we can prove the instability of solutions.

On Saint-Venant's principle in linear elastodynamics

We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x3>ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x3<ct, an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaKazaaiacaGGOaGccaaIXaGaaeiiaiabgkHiTiaabccadaWcaaqa% amXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hha4naaBaaaleaacaqGZa% aabeaaaOqaaiaabogacaqG0baaaKazaakacaGGPaaaaa!4BB0!\[(1{\text{ }} - {\text{ }}\frac{{x_{\text{3}} }}{{{\text{ct}}}})\]. Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragmén-Lindelöf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end.

A well-posed problem for the three-dual-phase-lag heat conduction

The three-dual-phase-lag theory based on the constitutive law q(P, t + τ q ) = −(k∇T(P, t + τ T ) + k*∇ν(P, t + τν)), was proposed by Roy Choudhuri as an extension of the theory by Tzou where the recent theories of Greeen and Naghdi could be recover. Although it proposes a law that could be compatible with our intuition, when we adjoin it with the energy equation −∇q(x, t) = cṪ(x, t), we obtain an ill-posed problem, that is, a problem which has a sequence of eigenvalues such that their real part are positive (and go to infinity). A consequence of this fact is that the problem is always unstable and we cannot obtain continuous dependence on the initial data. In this note we combine this constitutive equation with two-temperature heat conduction theory and we show in this new context that the problem is well-posed.

Qualitative aspects in dual-phase-lag heat conduction

We investigate the equation of the dual-phase-lag heat conduction proposed by Tzou. To describe this equation, we use the phase lag of the heat flux and the phase lag of the gradient of the temperature. We analyse the basic properties of the solutions of this problem. First, we prove that when both parameters are positive, the problem is well posed and the spatial decay of solutions is controlled by an exponential of the distance. When the phase lag of the gradient of the temperature is bigger than the phase lag of the heat flux, the problem is exponentially stable (which is a natural property to expect for a heat equation) and the spatial behaviour is controlled by an exponential of the square of the distance. Also, a uniqueness result for unbounded solutions is proved in this case.

EXISTENCE IN THERMOELASTICITY WITHOUT ENERGY DISSIPATION

In this article we investigate some qualitative aspects of the solutions of the equations of thermoelasticity without energy dissipation. In this sense we establish the suitable framework in which the problem of the linear anisotropic thermoelasticity without energy dissipation is well posed. The fundamental tool with which to obtain the framework is the semigroup of linear operators theory.