Asymptotic Behavior in Linear Thermoelasticity (original) (raw)

Reciprocal and variational principles in linear thermoelasticity without energy dissipation

Mechanics Research Communications, 2010

In the present paper we consider the equations which govern the behavior of an anisotropic and inhomogeneous centrosymmetric material within the framework of the linear theory of thermoelasticity without energy dissipation. We establish a reciprocal relation which is based on a characterization of the boundary-initial value problem in which the initial conditions are incorporated into the field equations. Further, a variational principle is presented too.

Loss of Exponential Stability for a Thermoelastic System with Memory

2010

In this article we study a thermoelastic system considering the linearized model proposed by Gurtin and Pipkin [8] instead of the Fourier's law for the heat flux. We use theory of semigroups [9, 11] combining Pruss' Theorem [10] and the idea developed in [5] to show that the system is not exponentially stable.

Energy decay for hyperbolic thermoelastic systems of memory type

Quarterly of Applied Mathematics, 2001

In this paper we study the hyperbolic thermoelastic system, which is obtained when, instead of Fourier’s law for the heat flux relation, we follow the linearized model proposed by Gurtin and Pipkin concerning the memory theory of heat conduction. In this case the thermoelastic model is fully hyperbolic. We show that the linear system is well posed and that the solution decays exponentially to zero as time goes to infinity.

On a two-dimensional model of generalized thermoelasticity with application

Scientific Reports

A 2D first order linear system of partial differential equations of plane strain thermoelasticity within the frame of extended thermodynamics is presented and analyzed. The system is composed of the equations of classical thermoelasticity in which displacements are replaced with velocities, complemented with Cattaneo evolution equation for heat flux. For a particular choice of the characteristic quantities and for positive thermal conductivity, it is shown that this system may be cast in a form that is symmetric t-hyperbolic without further recurrence to entropy principle. While hyperbolicity means a finite speed of propagation of heat waves, it is known that symmetric hyperbolic systems have the desirable property of well-posedness of Cauchy problems. A study of the characteristics of this system is carried out, and an energy integral is derived, that can be used to prove uniqueness of solution under some boundary conditions. A numerical application for a finite slab is considered ...

On dynamic behavior of a hyperbolic thermoelastic system with memory type in terms of eigenfrequencies

2006 American Control Conference, 2006

This paper studies the dynamic behavior of a onedimensional thermoelastic system with memory type in terms of its eigenfrequencies. The asymptotic expansions for eigenvalues and eigenfunctions are derived. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. From this, we deduce the spectrum-determined growth condition, and as a consequence, the exponential stability of the system is concluded.

On the energy decay of the linear thermoelastic plate with memory

Journal of Mathematical Analysis and Applications, 2005

We analyze the longterm properties of a C 0 -semigroup describing the solutions to a linear evolution system that models a thermoelastic thin plate with memory effects in the heat flux law. Under reasonably general assumptions on the memory kernel, all single trajectories are shown to decay to zero. In spite of that, the semigroup is not exponentially stable.  2004 Elsevier Inc. All rights reserved.

A thermoelastic system of memory type in noncylindrical domains

Applied Mathematics and Computation, 2008

In this paper, we consider a hyperbolic thermoelastic system of memory type in domains with moving boundary. The problem models vibrations of an elastic bar under thermal effects according to the heat conduction law of Gurtin and Pipkin. Global existence is proved by using the penalty method of Lions.