Quasi-static problems in the coupled linear theory of thermoporoelasticity

Abstract

This article is concerned with the coupled linear quasi-static theory of thermoporoelasticity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy’s law of the flow of a fluid through a porous medium, Fourie’s law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of general governing equations is expressed in terms of the displacement vector field, the changes of the volume fraction of pores, the fluid pressure in pore network and temperature. The fundamental solution of the system of quasi-static equations in the considered theory is constructed and its basic properties are established. On the basis of Green’s formulas the uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.