Moncef Aouadi

GENERALIZED THEORY OF THERMOELASTIC DIFFUSION FOR ANISOTROPIC MEDIA

The equations of generalized thermoelastic diffusion, based on the theory of Lord and Shulman with one relaxation time, are given in anisotropic media. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases that can be deduced from our generalized model.

Uniqueness and Reciprocity Theorems in the Theory of Generalized Thermoelastic Diffusion

The equations of generalized thermoelastic diffusion, based on the theory of Lord and Shulman, are given. Using Laplace transforms, a uniqueness theorem for these equations is proved. Also, a reciprocity theorem is obtained.

A generalized thermoelastic diffusion problem for an infinitely long solid cylinder

The theory of generalized thermoelastic diffusion, based on the theory of Lord and Shulman, is used to study the thermoelastic-diffusion interactions in an infinitely long solid cylinder subjected to a thermal shock on its surface which is in contact with a permeating substance. By means of the Laplace transform and numerical Laplace inversion the problem is solved. Numerical results predict finite speeds of propagation for thermoelastic and diffusive waves and the presence of a tensile stress region close to the cylinder surface. The problem of generalized thermoelasticity has been reduced as a special case of our problem.

Some Theorems in the Isotropic Theory of Microstretch Thermoelasticity with Microtemperatures

This article is concerned with the linear theory of microstretch thermoelastic bodies with microtemperatures. It is shown that there is exists the coupling of microtation vector field with the microtemperatures for isotropic bodies. The existence of a generalized solutions is proved by means of the semigroup of linear operators theory and the asymptotic behavior of the solutions is studied.

Qualitative Aspects in the Coupled Theory of Thermoelastic Diffusion

In this paper we derive some qualitative results of the coupled theory of thermoelastic diffusion for anisotropic media. We establish a reciprocity relation, which involves two thermoelastic diffusion processes at different instants. We show that this relation can be used to obtain reciprocity, uniqueness and continuous dependence theorems. The reciprocity theorem avoids both the use of the Laplace transform and the incorporation of initial conditions into the equations of motion. The uniqueness theorem is derived without the positive definiteness assumption on the elastic, conductivity and diffusion tensors. We prove also that the reciprocal relation leads to a continuous dependence theorem studied on external body loads. Finally, we prove the existence of a generalized solution by means of the semigroup of linear operators theory.

Theory of Generalized Micropolar Thermoelastic Diffusion Under Lord–Shulman Model

The general equations of motion and constitutive equations, based on the theory of Lord–Shulman with one relaxation time, are derived for a general homogeneous anisotropic medium with a microstructure, taking into account the effects of heat and diffusion. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases which can be deduced from our generalized model.

Stability in thermoelastic diffusion theory with voids

We analyse the longterm properties of a C 0-semigroup describing the solutions to a linear evolution system of thermoelastic diffusion with porosity recently developed by Aouadi [M. Aouadi, A theory of thermoelastic diffusion materials with voids, Z. Angew. Math. Phys. 61 (2009), pp. 357–379]. It is shown that there exists coupling of elasticity with porosity, temperature and diffusion for isotropic bodies. Our main result is to prove the lack of exponential stability and that the time decay of the solutions can be controlled only by a polynomial. We conclude by showing the impossibility of location in time of the solutions.

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C 0 and c 0 exist which satisfy E(t) ≤ C 0 E(0)e −c 0 t .

Qualitative Results in the Theory of Thermoelastic Diffusion Mixtures

The general equations of motion and constitutive equations are derived in a mixture of two interacting continua, taking into account the effects of heat and diffusion. First, the basic equations of the nonlinear theory of heat and diffusion conducting elastic mixtures are derived in lagrangian description. A frame independent nonlinear constitutive relation is derived. Then, the theory is linearized and field equations are given for both anisotropic and isotropic solids. The continuous dependence of solutions on initial data and body sources, and a uniqueness result are presented. An exponential decay estimate of the solutions is established. Finally, we prove the impossibility of the localization in time of the solutions.

Discontinuities in an axisymmetric generalized thermoelastic problem

This paper deals with discontinuities analysis in the temperature, displacement, and stress fields of a thick plate whose lower and upper surfaces are traction-free and subjected to a given axisymmetric temperature distribution. The analysis is carried out under three thermoelastic theories. Potential functions together with Laplace and Hankel transform techniques are used to derive the solution in the transformed domain. Exact expressions for the magnitude of discontinuities are computed by using an exact method developed by Boley (1962). It is found that there exist two coupled waves, one of which is elastic and the other is thermal, both propagating with finite speeds with exponential attenuation, and a third which is called shear wave, propagating with constant speed but with no exponential attenuation. The Hankel transforms are inverted analytically. The inversion of the Laplace transforms is carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical results are presented graphically along with a comparison of the three theories of thermoelasticity.