On spatial behaviour in linear thermoelasticity

Abstract

In the present paper we consider the linear theory of thermoelasticity of type III for anisotropic and inhomogeneous bodies as developed by Green and Naghdi (1992, 1995). We consider a right prismatic cylinder of either finite or semi‐infinite length and with a plane base, made of an anisotropic and inhomogeneous thermoelastic material. The cylinder is subject to zero body supplies and prescribed tractions and prescribed heat flux on its base and the lateral boundary is heat flux free and traction free. We establish results describing properties of spatial behaviour for the corresponding transient solutions. In fact, for a finite cylinder, we establish some spatial decay estimates for a certain energetic measure of the solution describing the spatial decay of the end effects with respect to the distance to the loaded and heated base, provided the density mass is strictly positive, the specific internal energy is assumed to be a positive definite quadratic form and the heat conductivity tensor is positive definite. In the limiting case when a semi‐infinite cylinder is considered we are able to establish an alternative of Phragmén‐Lindelöf type. We outline that in the case of a thermoelastic material for which the axial component q3 of the heat flux is independent of the time derivative of the thermal displacement gradient, we are able to establish a theorem of influence domain. When the lateral boundary surface is maintained at zero thermal displacement, we obtain spatial estimates by means of a comparison principle involving solutions of the classical one‐dimensional heat equation.