Nonsimple material problems addressed by the Lagrange’s identity

Marin I Marin,Sr Mahmoud,Ravi P Agarwal

doi:10.1186/1687-2770-2013-135

Abstract

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.

Highlights

Even classical elasticity does not consider the inner structure, the material response of materials to stimuli depends in a relevant way on its internal structure
The objective of our study is to examine by a new approach the mixed initial-boundary value problem in the context of thermoelasticity of nonsimple materials
4 Concluding remarks The uniqueness theorem and the continuous dependence theorems were proved without recourse to any conservation laws or to any boundedness assumptions on the thermoelastic coefficients

Summary

Introduction

Even classical elasticity does not consider the inner structure, the material response of materials to stimuli depends in a relevant way on its internal structure. It has been needed to develop some new mathematical models for continuum materials where this kind of effects was taken into account. Some of them are nonsimple elastic solids. It is known that from a mathematical point of view, these materials are characterized by the inclusion of higher-order gradients of displacement in the basic postulates. The theory of nonsimple elastic materials was first proposed by Toupin in his famous article [ ]. Among the first studies devoted to this material, we must mention those belonging to Green and Rivlin [ ] and Mindlin [ ]

Objectives

Conclusion