Integral representation for three-dimensional steady state size-dependent thermoelasticity

TL;DR

This paper develops an integral equation formulation for three-dimensional size-dependent steady state thermoelasticity using couple stress theory, highlighting the unique size effects and thermal-mechanical coupling in anisotropic and isotropic materials.

Contribution

It introduces a boundary element method framework for size-dependent thermoelasticity based on couple stress theory, including new integral equations and fundamental solutions for isotropic materials.

Findings

Size-dependent behavior characterized by a single length scale.

Thermal effects depend solely on thermal strain in isotropic case.

Integral formulations enable boundary-only analysis of complex thermoelastic problems.

Abstract

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response, especially in the linear case for which boundary-only formulations are possible. This paper presents the integral equation formulation for size-dependent linear thermoelastic response of solids under steady state conditions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this general size-dependent thermoelastic theory. Then, an integral equation formulation for the three-dimensional isotropic case is derived, along with the corresponding singular infinite space fundamental solutions or kernel functions. Remarkably, for isotropic materials within this theory, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter, while the thermal coupling is defined in terms of the thermal expansion coefficient, as in the classical theory of steady state isotropic thermoelasticity. This simplification permits the development of the required kernel functions from previously defined fundamental solutions for isotropic media.