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This paper is concerned with a linear theory of thermoelasticity without energy dissipation, where the second gradient of displacement and the second gradient of the thermal displacement are included in the set of independent constitutive variables. In particular, in the case of rigid heat conductors the present theory leads to a fourth order equation for temperature. First, the basic equations of the second gradient theory of thermoelasticity are presented. The boundary conditions for thermal displacement are derived. The field equations for homogeneous and isotropic solids are established. Then, a uniqueness result for the basic boundary-initial-value problems is presented. An existence theorem is established for the first boundary value problem. The problem of a concentrated heat source is investigated using a solution of Cauchy-Kowalewski-Somigliana type. [ABSTRACT FROM AUTHOR]

In this work, we study a new thermoelastic model with two temperatures from the numerical point of view. The problem is written as a coupled linear system whose unknowns are the displacement field and the conductivity and thermodynamic temperatures. An existence and uniqueness result recently proved is recalled. Then, a fully discrete approximation is introduced by using the finite element method and the classical implicit Euler scheme. A main a priori error estimates result is proved and, under some appropriate regularity conditions on the continuous solution, we obtain the linear convergence. Finally, some numerical simulations are presented to demonstrate the numerical convergence and the discrete energy decay. [ABSTRACT FROM AUTHOR]

A quasi-static approximation is studied of linearised nonhomogeneous anisotropic compressible thermoelasticity on a non-compact region. Differential inequalities are constructed which under appropriate conditions lead to algebraic spatial growth and decay estimates for various cross-sectional and volume measures of the temperature, displacement and their spatial gradients. [ABSTRACT FROM AUTHOR]

In this work, we consider a multi-dimensional dual-phase-lag problem arising in porous-thermoelasticity with microtemperatures. An existence and uniqueness result is proved by applying the semigroup of linear operators theory. Then, by using the finite element method and the Euler scheme, a fully discrete approximation is numerically studied, proving a discrete stability property and a priori error estimates. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximation and the behavior of the solution in one- and two-dimensional problems. [ABSTRACT FROM AUTHOR]

It is well known that the solutions to the elastodynamic problem do not satisfy continuous dependence properties on the initial values, and/or supply terms, when the elastic tensor fails to be positive. In fact, the behavior of the solutions can be very explosive since the elements of the spectrum can go to infinite. Therefore, it is very relevant to identify thermomechanical mechanisms regularizing the behavior of the solutions. So, the main aim of this note is to show, from an analytical point of view, how the non-locality , in the sense of Eringen, is a mechanism satisfying this property of regularization of the solutions. It is worth noting that such system has not been previously studied from an analytical point of view. We firstly obtain the existence of the solutions to this problem, even when we do not assume any positivity on the elastic tensor. This result is proved with the help of the linear semigroups theory; however, even with these regularizing effects, the solutions to this problem are unstable. A particular easy one-dimensional problem is also considered. The extension of the existence and instability results to the thermoelastic case is pointed out later. Finally, we also study the spatial behavior of the solutions to the problem in the case that the region is a semi-infinite cylinder, and we obtain a Phragmen–Lindelöf alternative of the exponential type. This result is also relevant because a similar result, without considering regularizing terms, is unknown if the elastic tensor is not positive definite. • A nonlocal term regularizes the solution. • An existence and uniqueness result is proved. • The instability of the problem is shown. • The extension to the thermoelastic case is considered. • The exponential spatial decay is also obtained. [ABSTRACT FROM AUTHOR]

In this work, we study an approximate problem arising in the incremental thermoelasticity. The existence of a unique solution is proved applying the theory of linear semigroups. The exponential energy decay is also considered. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are shown. Finally, numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the model. In view of our numerical results the approximate problem could be used only when a certain parameter is small if we compare it with other parameters arising in the problem. [ABSTRACT FROM AUTHOR]

In this paper we consider the one-dimensional version of thermoelasticity with two porous structures and porous dissipation on one or both of them. We first give an existence and uniqueness result by means of semigroup theory. Exponential decay of the solutions is obtained when porous dissipation is assumed for each porous structure. Later, we consider dissipation only on one of the porous structures and we prove that, under appropriate conditions on the coefficients, there exists undamped solutions. Therefore, asymptotic stability cannot be expected in general. However, we are able to give suitable sufficient conditions for the constitutive coefficients to guarantee the exponential decay of the solutions. [ABSTRACT FROM AUTHOR]

The validity of the coupled and uncoupled quasi-static approximations is considered for the initial boundary value problem of linear thermoelasticity subject to homogeneous Dirichlet boundary conditions, and for solutions and their derivatives that are mean-square integrable. Essential components in the proof, of independent interest, are conservation laws and associated estimates for the exact and approximate systems. [ABSTRACT FROM AUTHOR]

We study solutions for the one-dimensional problem of the Green–Lindsay and the Lord–Shulman theories with two temperatures. First, existence and uniqueness of weakly regular solutions are obtained. Second, we prove the exponential stability in the Green–Lindsay model, but the non-exponential stability for the Lord–Shulman model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article is concerned with the linear theory of chiral Cosserat thermoelastic 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 infinite plate with a circular hole is investigated. [ABSTRACT FROM PUBLISHER]

In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear strain gradient theory of Cosserat thermoelastic bodies. The theory is capable of predicting a finite speed of heat propagation and leads to a symmetric conductivity tensor. The constitutive equations for isotropic chiral thermoelastic materials are presented. In this case, in contrast with the classical Cosserat thermoelasticity, a thermal field produces a microrotation of the particles. The thermal field is influenced by the displacement and microrotation fields even in the equilibrium theory. Existence and uniqueness results are established. The theory is used to study the effects of a concentrated heat source in an unbounded homogeneous and isotropic chiral solid. [ABSTRACT FROM AUTHOR]

Alternative growth and decay estimates, reminiscent of the classical Phragmén-Lindelöf principle, are derived for a linearised thermoelastic body whose plane cross-sections increase unboundedly with respect to a given direction. The proof uses a modified Poincaré inequality to construct a differential inequality for a weighted linear combination of the cross-sectional mechanical and thermal energy fluxes. Decay estimates are deduced also for the cross-sectional mean square measures of the displacement and temperature. An explicit upper bound in terms of base data is established for the amplitude occurring in the decay estimates. [ABSTRACT FROM AUTHOR]

In this article, we use the Nunziato–Cowin theory of materials with voids to derive a theory of thermoelastic solids, which have a double porosity structure. The new theory is not based on Darcy's law. In the case of equilibrium, in contrast with the classical theory of elastic materials with double porosity, the porosity structure of the body is influenced by the displacement field. We prove the uniqueness of solutions by means of the logarithmic convexity arguments as well as the instability of solutions whenever the internal energy is not positive definite. Later, we use semigroup arguments to prove the existence of solutions in the case that the internal energy is positive. The deformation of an elastic space with a spherical cavity is investigated. [ABSTRACT FROM AUTHOR]

This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations. [ABSTRACT FROM AUTHOR]

Abstract: This note is concerned with the linear (and linearised) type III thermoelastic theory proposed by Green and Naghdi. First, the continuous dependence of the solutions upon the initial data and supply terms is established for noncentrosymmetric bodies. Then a uniqueness result for centrosymmetric materials is established. [Copyright &y& Elsevier]

Abstract: In this paper we study the equations of the nonlinear theory of type II Thermoelasticity proposed by Green and Naghdi. We propose a constitutive framework for the free energy function arising in this theory and study various consequences including the existence and uniqueness of quasistatic solutions and the possibility of thermally activated instability. [Copyright &y& Elsevier]

Abstract: Three dual-phase-lag heat conduction theory is based on the constitutive law . It is an extension of the dual-phase-lag which is able to recover the Green and Nagdhi theories when Taylor approximations are considered. If we adjoin this constitutive law with the energy equation , an ill-posed problem is generically obtained. That is, a problem with a sequence of eigenvalues for which the real part goes to infinity. As a consequence, the problem is unstable and, moreover, there is no continuous dependence of solutions with respect initial data. In this note we show that this behavior does not apply when τ ν > τ q = τ T . We prove continuous dependence with respect the initial data and supply terms. We also show how to obtain the solutions of the problem by means of a recurrent scheme. Travelling wave solutions are also obtained. The continuous dependence results are extended to the thermoelastic case and several approximation theories are considered. [Copyright &y& Elsevier]

In correcting a small mistake in [10], we can prove a new result on non-exponential stability for a coupled system arising in resonators. This also gives another (surprising and simple) example for a thermoelastic system changing from exponential stability to non-exponential stability, when changing from Fourier's law to Cattaneo's law in modeling of the heat conduction. [ABSTRACT FROM AUTHOR]

Abstract: In this note we investigate the continuous dependence of the solutions for a theory of heat conduction with a delay term. We use energy arguments to obtain the continuous dependence results and spectral arguments to prove the non-uniqueness result. The extension to the thermoelastic problem is also pointed out. [Copyright &y& Elsevier]

Green and Naghdi proposed three theories of thermoelasticity labelled as type I, II and III, respectively. Here we investigate the spatial behavior of solutions of a problem determined by the type III thermoelasticity. We study the growth/decay alternative in the spatial variable for the linear theory in the case that the thermoelastic deformations at the instant T are constrained to be proportional to the ones at the instant zero. In the case that the solutions decay, an upper bound for the amplitude term by means of the data of the problem can be obtained. We sketch how to obtain it. We finish by noting (through examples) that for several families of cases the solutions can be periodic in the spatial variable and the Phragmen-Lindelof alternative is not satisfied. [ABSTRACT FROM AUTHOR]

THE PROBLEM DETERMINED by thermoelastic deformations when the internal energy is not positive definite, becomes ill-posed. We recall that this kind of situation happens in the study of prestressed thermoelastic solids. Thus, it will be of interest to obtain qualitative properties of solutions in this case. In this note we prove the uniqueness of solutions for the linear thermo-poro-elasticity with microtemperatures theory, when the internal energy is not to be assumed to be positive definite. We use the energy arguments combined with the Lagrange identities. [ABSTRACT FROM AUTHOR]

In this work we study modifications of the non-classical models of thermoelasticity, the one proposed Green and Lindsay and the one stated by Lord and Shulman, to two-temperature setting. We prove uniqueness results for the solutions of the systems of equations that model both theories for isotropicmaterial. We also provide growth estimates for the solutions. [ABSTRACT FROM AUTHOR]

Abstract: In the first part of this paper we present a linear theory of thermoelastic bodies with microstructure and microtemperatures which permits the transmission of heat as thermal waves at finite speed. The theory is based on the entropy balance postulated by Green and Naghdi [A.E. Green, P.M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. London A 432 (1991) 171–194]. We consider bodies with microstructure whose microelements can stretch and contract independently of their translations. Then we establish existence and uniqueness results in the context of the dynamic theory. [Copyright &y& Elsevier]

WE CONSIDER THE SYSTEM of micro-beam resonators in the thermoelastic theory of Lord and Shulmann. First, we prove the uniqueness and instability of solutions when the sign of a parameter is not prescribed. Existence of solutions and uniform bounds for the real part of the spectrum have been found. We finish the paper by proving the impossibility of the time localization of solutions. [ABSTRACT FROM AUTHOR]

In this paper we investigate the asymptotic behavior of the solutions of the one-dimensional generalized porous-thermo-elasticity problem. First, we prove that when only thermal damping is present the decay of solutions is slow (generically). Second, we show that if the porous dissipation is also present the decay of the solutions is controlled by an exponential. [ABSTRACT FROM AUTHOR]

We consider the linear theory of homogeneous and isotropic thermoelastic solids for the types II and III theories. First, we present the basic equations which characterize the bending of thin thermoelastic plates. Then, we establish a uniqueness result with no definiteness assumption on constitutive coefficients. Existence of solutions is proved under the assumption that the internal energy density is positive definite; the asymptotic behavior is then analyzed. Finally, we study the spatial behavior of solutions. [ABSTRACT FROM AUTHOR]

Summary. This paper concerns the thermoelasticity with two temperatures. First, we state the suitable frame where the boundary initial value problem of thermoelasticity with two temperatures is well posed. Second we study the structural stability with respect the parameter which is typical in this theory. When this parameter tends to zero we obtain the system of the usual thermoelasticity. We also study the convergence of the solutions in this theory to the classical one. Last section is devoted to study the spatial behavior for the backward-in-time problem. [ABSTRACT FROM AUTHOR]

Investigates the structural stability of thermomechanics and heat conduction theories. Convergence of these theories to the classical version of thermoelasticity and heat conduction; Lord-Shulman theory of thermoelasticity; Structural stability with respect to the relaxation parameter; Convergence of the solutions withe the relaxation parameters tends to zero.

In this article we study a one-dimensional problem of the Chandrasekharaiah-Tzou thermoelastic theory. In this theory two delay parameters τ[sub θ] and τ[sub q] are considered. We prove that when τ[sub θ] > θ[sub q]/2 the solutions decay exponentially and when τ[sub θ] < τ[sub q]/2 there exist unstable solutions. These results suggest that when we want to study a problem in this thermoelastic theory we must assume that the inequality τ[sub θ] > τ[sub q]/2 holds. [ABSTRACT FROM AUTHOR]

In this article we propose a model of the nonsimple thermoelastic theory without energy dissipation. The equations of the linear theory are also obtained. Later, we obtain a uniqueness theorem for materials with a center of symmetry and an existence and uniqueness result in the general case. It is worth noting that we present two uniqueness results that apply to two different families of equations that govern the thermoelasticity without energy dissipation of nonsimple materials. [ABSTRACT FROM AUTHOR]

This paper derives spatial decay bounds in a dynamical problem of incremental thermoelasticity defined on a semi‐infinite cylindrical region. Previous results for isothermal elastodynamics and the parabolic heat equation lead us to suspect that the solution of the problem should tend to zero faster than a decaying exponential of the distance from the finite end of the cylinder. We prove that an energy expression is actually bounded above by a decaying exponential of a quadratic polynomial of the distance. As it has been previously proved for usual linear thermoelasticity, this is reminiscent of the decay rate in second order parabolic problems. [ABSTRACT FROM AUTHOR]

The chiral effects cannot be described by means of the classical theory of elasticity. In this paper we study the thermoelastic deformation of chiral plates in the context of the strain gradient theory of thermoelasticity. The work is motivated by the interest in using chiral continuum as model for some carbon nanotubes, auxetic materials and bones. First, we derive the basic equations which govern the deformation of thin thermoelastic plates. In contrast with the theory of achiral plates, the stretching and flexure cannot be treated independently of each other. A system of Timoshenko–Ehrenfest type is presented and an existence result is established. Then, we consider the dynamic theory of plates and present a uniqueness result with no definiteness assumption on the elastic constitutive coefficients. The effects of a concentrated heat source are investigated. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the incremental problem in the thermoelastic theory of nonsimple elastic materials. After we set down the basic equations of this problem, we obtain a result on existence, uniqueness, and continuous dependence from initial parameters and supply terms. [ABSTRACT FROM AUTHOR]

In this paper, a strain gradient thermoelastic problem is studied from the numerical point of view. The heat conduction is modeled by using the type II thermal law and the second gradient of the thermal displacement is also included in the set of independent constitutive variables. An existence and uniqueness result is recalled. Then, the fully discrete approximations are introduced by using the implicit Euler scheme and the finite element method. A discrete stability property and a main a priori error estimates results are proved. Then, some numerical simulations are performed, including some issues as the numerical convergence of the approximations, the effect of two possible dissipative terms (second- and fourth-order) or a comparison with the type II strain gradient thermoelasticity. [ABSTRACT FROM AUTHOR]