The Backward in Time Problem of Double Porosity Material with Microtemperature

Olivia A. Florea

doi:10.3390/sym11040552

Abstract

In the present study, the theory of thermoelastodynamics is considered in the case of materials with double porosity structure and microtemperature. The novelty of this study consists in the investigation of a backward in time problem associated with double porous thermoelastic materials with microtemperature. In the first part of the paper, in case of the bounded domains the impossibility of time localization of solutions is obtained. This study is equivalent to the uniqueness of solutions for the backward in time problem. In the second part of the paper, a Phragmen-Lindelof alternative in the case of semi-infinite cylinders is obtained.

Highlights

In recent years, many authors have been interested in the study of the thermoelastic bodies with double porosity structure
The theory of thermoelastic solids with double porosity structure consists in the study of two types of porosity: the macro porosity represented by the body’s pores and the micro porosity represented by the fissures that appear in the skeleton
The aim of this paper is to show that in the case of thermoelasticity with double porosity structure and microtemperature, the only solution that vanishes after a finite time is the null solution, when the mechanisms of dissipation are the double porous dissipation, the temperature and the microtemperature

Summary

Introduction

Many authors have been interested in the study of the thermoelastic bodies with double porosity structure. The theory of thermoelastic solids with double porosity structure consists in the study of two types of porosity: the macro porosity represented by the body’s pores and the micro porosity represented by the fissures that appear in the skeleton. According to Barneblatt et al [2], Berryman et al [3] and Khalili et al [4], some interesting applications of materials with double porosity structure are encountered in geophysics and according to Cowin, [5] in mechanics of bone. In the case of elastic materials with double porosity, the basic equations contain the displacement vector field, and two types of pressure associated with the macro and micro porosity [6,7,8]. The fluid pressure is independent of the displacement vector field in the case of equilibrium theory

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Conclusion