Integral Representation for Three-Dimensional Steady-State Couple-Stress Size-Dependent Thermoelasticity

Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. 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 theory. Then, an integral equation formulation for three-dimensional size-dependent thermoelastic isotropic materials is derived, along with the corresponding singular infinite-space fundamental solutions or kernel functions. For isotropic materials, 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 l, while the thermal coupling is defined in terms of the thermal expansion coefficient α, as in the classical theory of steady-state isotropic thermoelasticity.