Search

Abstract Our study is dedicated to a mixture composed of a dipolar elastic medium and a viscous Moore–Gibson–Thompson (MGT) material. The mixed problem with initial and boundary data, considered in this context, is approached from the perspective of the existence of a solution to this problem as well as the uniqueness of the solution. Considering that the mixed problem is very complex, both from the point of view of the basic equations and that of the initial conditions and the boundary data, the classical methods become difficult. That is why we preferred to transform it into a problem of Cauchy type on a conveniently constructed Hilbert space. In this way, we immediately proved both the existence and uniqueness of the solution, with techniques from the theory of semigroups of linear operators.

In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

Abstract The goal of this paper is to obtain an extension of the relaxed Saint–Venant principle in order to cover the thermoelasticity of dipolar porous bodies. According to this principle, for a finite time t>0 $t>0$, we identify a bounded domain Dt $\mathcal{D}_{t}$ so that outside of this domain the displacement field ui $u_{i}$, the dipolar displacement field φjk $\varphi_{jk}$, the temperature θ, and the change in volume fraction ϕ do not generate disturbances.

The aim of our paper is the study of the spatial evolution of vibrations in the context of Thermoelasticity without energy dissipation for dipolar bodies. Once we get an a priori estimate for the amplitude of the vibration, which are assumed being harmonic in time, it is possible to predict some spatial decay or growth properties for the amplitude, provided the frequency of vibration is greater than a certain critical value.

The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, in the general case of an anisotropic thermoelastic body. Furthermore, in this general context we have proven a result regarding the uniqueness of the solution of the mixed problem in the dynamical case. We must emphasize that these fundamental results are obtained under conditions that are not very restrictive.

We consider the mixed initial‐boundary value problem in the context of the Moore‐Gibson‐Thompson theory of thermoelasticity for dipolar bodies. We consider the case of heat conduction with dissipation. Even if the elasticity tensors are not supposed to be positively defined, we have proven both, the uniqueness and the instability of the solution of the mixed problem. In the case that the mass density and the thermal conductivity tensor are positive, we obtain the uniqueness of the solution using some Lagrange type identities. [ABSTRACT FROM AUTHOR]

In our study, we formulate the boundary value problem in the context of the nonlinear theory of dipolar, porous, and elastic materials. For this problem, some existence and uniqueness results are proven. The results are natural generalizations of the results obtained by Langenbach for the classical elastic bodies. [ABSTRACT FROM AUTHOR]

In our paper we formulate a theory for thermoelastic porous dipolar bodies in which we consider a new independent variable, namely the heat-flux vector. Furthermore, we add, to the differential equations that describe the behavior of the body, a new differential equation which is an equation of evolution which is satisfied by the components of the heat-flux vector. The basic system of the mixed initial-boundary value problem in this context consists of equations of the hyperbolic type. In order to ensure the consistency of the constructed theory, we formulate and prove an uniqueness result, with regards to the solution of the mixed problem. [ABSTRACT FROM AUTHOR]

We consider a time differential model for dual-phase-lag thermoelastic dipolar material. For the heat flux with dual-phase-lag we introduce a time differential equation to describe the thermal effects, and this is coupled with elastic deformation. Some qualitative results on solutions of the mixed initial boundary value problem, associated with this model, are obtained under mild constitutive restrictions. We will deduce an identity of Lagrange type and will use a conservation law in order to prove the uniqueness of solution, whereas the result of continuous dependence of solution is established by using a Gronwall type inequality and a conservation law. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we generalize the results obtained by Iesan and Quintanilla for microstretch elastic bodies in order to cover dipolar elastic materials with stretch. For the boundary value problem considered in this context, we prove a generalized existence result and, also, an extension of the principle of minimum potential energy. [Copyright &y& Elsevier]