On the decay of solutions for porous-elastic systems with history ✩

Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. By using the Lagrange identity, we prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients.

/pdf/nonsimple-material-problems-addressed-by-the-lagrange-s-4l7gk5uqlz.pdf

Nonsimple material problems addressed by the Lagrange’s identity

Thermoelasticity without energy dissipation of materials with microstructure

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy's law for materials with double porosity. This theory is a straightforward generalization of the earlier proposed dynamical theory of elasticity for materials with double porosity. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: through a Kelvin-Voigt material with double porosity three longitudinal and two transverse plane harmonic attenuated waves propagate.

https://www.aimsciences.org/article/exportPdf?id=ffd3a0b5-0d37-4e3f-baf4-b9401093b62e

On the theory of viscoelasticity for materials with double porosity

This article deals with the diffusion model of the linear theory of binary mixtures of the thermoelastic solids. A wide class of external (with respect to an unbounded domain) boundary value problems of steady oscillations is investigated using the potential method. Sommerfeld-Kupradze-type radiation conditions are established. The uniqueness and existence theorems of classical solutions of the aforementioned problems are proved.

Potential method in the linear theory of binary mixtures of thermoelastic solids

Existence, uniqueness and asymptotic behaviour of solutions to the equations of viscoelasticity with voids☆

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 obtai a result on existence, uniqueness, and continuous dependence from initial parameters and supply terms.

On the Incremental Problem in Thermoelasticity of Nonsimple Materials

Existence, uniqueness, and continuous dependence of solutions to the evolution equations that govern small thermoelastic deformations superposed on a general nonlinear thermomechanical deformation with nonuniform temperature is obtained. In our case, the domain occupied by the body may be unbounded and the elasticity tensor may be strongly elliptic. The problem is solved by means of the semigroup theory.

Existence of solutions of the equations of incremental thermoelasticity for unbounded domains

It is usually accepted that the work of Cosserat and Cosserat [1] can be seen as the starting point of the study of solids with microstructure In the sixties an important effort of several authors gave a relevant development of some theories In this sense, we recall the monographs by Truesdell and Toupin [2], Eringen and Kafadar [3], Kunin [4, 5] and Capriz [6] In this paper we restrict our attention to the theory developed by Mindlin [7] In the book, Ciarletta and Iesan [8] have given a current presentation of this theory We study the small deformations superposed on a nonlinear general deformation The current point of view of this problem was given by Iesan [9] Existence results for the equations of the incremental problem have been presented in [10, 11] for the classical thermoelasticity, in [12] for the thermoelasticity of porous materials and in [13] for the viscoelasticity Our main aim is to extend these results to the thermoelastic theory of materials with microstructure

On the Incremental Problem in Thermoelasticity with Microstructure

In this paper we study the linear thermodynamical problem of mixtures of thermoelastic solids. We use some results of the semigroup theory to obtain an existence theorem for the initial value problem with homogeneous Dirichlet boundary conditions. Continuous dependence of solutions upon the initial data and body forces is also established. We finish with a study of the asymptotic behavior of solutions of the homogeneous problem.

Some qualitative results for the linear theory of binary mixtures of thermoelastic solids

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