Thermal stresses in chiral plates

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Africa Review on 17/04/2014, available online:http://www.tandfonline.com/doi/full/10.1080/01495739.2016.1217180This article is concerned with the linear theory of chiral Cosserat thermoelas- tic bodies. We investigate the deformation of chiral plates. First, we present the basic equations which govern the deformation of thin plates. Then, we present reciprocity and uniqueness results. In the next section, we establish the instability of solutions whenever the internal energy is negative. We use a semigroup approach to prove the existence of a solution. The deformation of an in nite plate with a circular hole is investigated.Postprint (author's final draft

Moore–Gibson–Thompson thermoelasticity

We consider a thermoelastic theory where the heat conduction is escribed by the Moore–Gibson–Thompson equation. In fact, this equation can be obtained after the introduction of a relaxation parameter in the Green–Naghdi type III model. We analyse the one- and three-dimensional cases. In three dimensions, we obtain the well-posedness and the stability of solutions. In one dimension, we obtain the exponential decay and the instability of the solutions depending on the conditions over the system of constitutive parameters.We also propose possible extensions for these theoriesPeer ReviewedPostprint (author's final draft

On uniqueness and stability for a thermoelastic theory

In this paper we investigate a thermoelastic theory obtained from the Taylor approximation for the heat flux vector proposed by Choudhuri. This new thermoelastic theory gives rise to interesting mathematical questions. We here prove a uniqueness theorem and instability of solutions under the relaxed assumption that the elasticity tensor can be negative. Later we consider the one-dimensional and homogeneous case and we prove the existence of solutions. We finish the paper by proving the slow decay of the solutions. That means that the solutions do not decay in a uniform exponential way. This last result is relevant if it is compared with other thermoelastic theories where the decay of solutions for the one-dimensional case is of exponential way.Peer ReviewedPostprint (author's final draft

Moore-Gibson-Thompson thermoelasticity with two temperatures

In this note we propose the Moore-Gibson-Thompson heat conduction equation with two temperatures and prove the well posedness and the exponential decay of the solutions under suitable conditions on the constitutive parameters. Later we consider the extension to the Moore-Gibson-Thompson thermoelasticity with two temperatures and prove that we cannot expect for the exponential stability even in the one-dimensional case. This last result contrasts with the one obtained for the Moore-Gibson-Thompson thermoelasticity where the exponential decay was obtained. However we prove the polynomial decay of the solutions. The paper concludes by giving the main ideas to extend the theory for inhomogeneous and anisotropic materials.Peer ReviewedPostprint (published version

Some qualitative results for a modification of the Green–Lindsay thermoelasticity

In this short note we consider a recent modification of the Green–Lindsay thermoelastic theory proposed at Yu et al. (Meccanica 53:2543–2554, 2018). We consider a functional defined on the solutions of the problem. It allows us to obtain the continuous dependence of the solutions with respect to the initial conditions and to the supply terms, the time exponential decay of solutions and an alternative of Phragme´n–Lindelo¨f type for the spatial behaviour.Peer ReviewedPostprint (author's final draft

Numerical resolution of an exact heat conduction model with a delay term

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.Peer ReviewedPostprint (published version

Some remarks on the fast spatial growth/decay in exterior regions

In this paper we investigate the spatial behavior of the solutions to several partial differential equations/systems for exterior or cone-like regions. Under certain conditions for the equations we prove that the growth/decay estimates are faster than any exponential depending linearly on the distance to the origin. This kind of spatial behavior has not been noticed previously for parabolic problems and exterior or cone-like regions. The results obtained in this work apply in particular for the linear case.Peer ReviewedPostprint (author's final draft

On the asymptotic spatial behaviour of the solutions of the nerve system

In this paper we investigate the asymptotic spatial behavior of the solutions for several models for the nerve fibers. First, our analysis deals with the coupling of two parabolic equations. We prove that, under suitable assumptions on the coefficients and the nonlinear function, the decay is similar to the one corresponding to the heat equation. A limit case of this system corresponds to the coupling of a parabolic equation with an ordinary differential equation. In this situation, we see that for suitable boundary conditions the solution ceases to exist for a finite value of the spatial variable. Next two sections correspond to the coupling of a hyperbolic/parabolic and hyperbolic/ordinary differential problems. For the first one we obtain that the decay is like an exponential of a second degree polynomial in the spatial variable. In the second one, we prove a similar behaviour to the one corresponding to the wave equation. In these two sections we use in a relevant way an exponentially weighted Poincaré inequality which has been revealed very useful in several thermal and mechanical problems. This kind of results have relevance to understand the propagation of perturbations for nerve models.Peer ReviewedPostprint (author’s final draft

On the spatial behavior in two-temperature generalized thermoelastic theories

The final publication is available at link.springer.com via https://doi.org/10.1007/s00033-017-0857-xThis paper investigates the spatial behavior of the solutions of two generalized thermoelastic theories with two temperatures. To be more precise, we focus on the Green–Lindsay theory with two temperatures and the Lord–Shulman theory with two temperatures. We prove that a Phragmén–Lindelöf alternative of exponential type can be obtained in both cases. We also describe how to obtain a bound on the amplitude term by means of the boundary conditions for the Green–Lindsay theory with two temperatures.Peer ReviewedPostprint (author's final draft

Qualitative properties in strain gradient thermoelasticity with microtemperatures

This paper is devoted to the strain gradient theory of thermoelastic aterials whose microelements possess microtemperatures. The work is motivated by an increasing use of materials which possess thermal variation at a microstructure level. In the first part of this paper we deduce the system of basic equations of the linear theory and formulate the boundary-initial-value problem. We establish existence, uniqueness, and continuous dependence results by the means of semigroup theory. Then, we study the one-dimensional problem and establish the analyticity of solutions. Exponential stability and impossibility of localization are consequences of this result. In the case of the anti-plane problem we derive uniqueness and instability results without assuming the positivity of the mechanical energy. Finally, we study equilibrium theory and investigate the effects of a concentrated heat source in an unbounded bodyPeer ReviewedPostprint (author's final draft