Exponential decay in one-dimensional type III thermoelasticity with voids

In this paper we consider the one-dimensional type III thermoelastic theory with voids. We prove that generically we have exponential stability of the solutions. This is a striking fact if one compares it with the behavior in the case of the thermoelastic theory based on the classical Fourier law for which the decayis generically slower.Peer ReviewedPostprint (author's final draft

Exponential stability in three-dimensional type III thermo-porous-elasticity with microtemperatures

We study the time decay of the solutions for the type III thermoelastic theory with microtemperatures and voids. We prove that, under suitable conditions for the constitutive tensors, the solutions decay exponentially. This fact is in somehow striking because it differs from the behaviour of the solutions in the classical model of thermoelasticity with microtemperatures and voids, where the exponential decay is not expected in the general case.Peer ReviewedPostprint (author's final draft

Exponential decay in one-dimensional Type II/III thermoelasticity with two porosities

In this paper we consider the theory of thermoelasticity with a double porosity structure in the context of the Green-Naghdi types II and III heat conduction models. For the type II, the problem is given by four hyperbolic equations and it is conservative (there is no energy dissipation). We introduce in the system a couple of dissipation mechanisms in order to obtain the exponential de- cay of the solutions. To be precise, we introduce a pair of the following damping mechanisms: viscoelasticity, viscoporosities and thermal dissipation. We prove that the system is exponentially stable in three different scenarios: viscoporosity in one structure jointly with thermal dissipation, viscoporosity in each structure, and viscoporosity in one structure jointly with viscoelasticity. However, if viscoelasticity and thermal dissipation are considered together, undamped solutions can be obtained.Peer ReviewedPostprint (author's final draft

Exponential decay in one-dimensional porous-thermo-elasticity

This paper concerns the one dimensional problem of the porous-thermo-elasticity. Two kinds of dissipation process are considered: the viscosity type in the porous structure and the thermal dissipation. It is known that when only thermal damping is considered or when only porous damping is considered we have the slow decay of the solutions. Here we prove that when both kinds of dissipation terms are taken into account in the evolution equations the solutions are exponentially stable.Peer Reviewe

Lord–Shulman thermoelasticity with microtemperatures

In this paper we consider the Lord–Shulman thermoelastic theory with porosity and microtemperatures. The new aspect we propose here is to introduce a relaxation param- eter in the microtemperatures. Then we obtain an existence theorem for the solutions. In the case that a certain symmetry is satisfied by the constitutive tensors, we prove that the semigroup is dissipative. In fact, an exponential decay of solutions can be shown for the one-dimensional case. In the last section, we restrict our attention to the case where we have an isotropic and homogeneous material without porosity effects and assuming that two of the constitutive parameters have the same sign. We see that the semigroup is dissipative.Peer ReviewedPostprint (author's final draft

On the time decay for an elastic problem with three porous structures

In this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature concerning the time decay of solutions in porous elasticity, can appear in the system of field equations. The new couplings provided in this situation show a strong relationship between the elastic and the porous components of the material. In this situation, we obtain an existence and uniqueness result for the solutions to the problem using the Lumer-Phillips corollary to the Hille-Yosida theorem. Later, assuming a certain condition determining a “very strong” coupling between the material components, we can use the well-known arguments for dissipative semigroups to prove the exponential stability of the solutions to the problem. It is worth emphasizing that the proposed condition allows bringing the decay of the dissipative porous structure of the problem to the macroscopic elastic structurePeer ReviewedPostprint (published version

Energy decay in thermoelastic bodies with radial symmetry

In this paper, we consider the energy decay of some problems involving domains with radial symmetry. Three different settings are studied: a strong porous dissipation and heat conduction, a weak porous dissipation and heat conduction and poro-thermoelasticity with microtemperatures. In all the three problems, the exponential energy decay is shown. Moreover, for each of them some finite element simulations are presented to numerically demonstrate this behaviorPeer ReviewedPostprint (published version

Thermoelasticity with temperature and microtemperatures with fading memory

In this paper, we investigate a model of poro-thermoelasticity with microtemperatures, where the behavior of the body is influenced by the history of both temperature and microtemperatures. Mathematically, this translates into a system of partial integro-differential equations. Under suitable condition on the tensors appearing in the model, we prove that the resulting system is well posed. In the one-dimensional case, the exponential decay of the energy is provePeer ReviewedPostprint (author's final draft

Decay of quasi-static porous-thermo-elastic waves

We study the behavior in time of the solutions to several systems of equations for porous-thermo-elastic problems when one of the variables is considered to be quasi-static or, in other words, whose second time derivative can be neglected. We analyze three different situations using the classical Fourier law and also the type II or type III Green–Naghdi heat conduction modelsPeer ReviewedPostprint (author's final draft

An a priori error analysis of a Lord–Shulman poro-thermoelastic problem with microtemperatures

In this paper, we deal with the numerical analysis of the Lord–Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature and the microtemperatures. An existence and uniqueness result is stated. Then, a fully discrete approximation is introduced using the finite element method and the implicit Euler scheme. A discrete stability property is shown, and an a priori error analysis is provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation, the comparison with the classical Fourier theory and the behavior of the solution in two-dimensional examples.Peer ReviewedPostprint (author's final draft