Quasi-static problems in the coupled linear theory of elasticity for porous materials

Abstract

In this paper, the coupled linear quasi-static theory of elasticity for porous materials is considered. The system of equations of this theory is based on the constitutive equations, Darcy’s law, the equations of equilibrium, and fluid mass conservation. The system of general governing equations is expressed in terms of the displacement vector field, the volume fraction of pores, and the fluid pressure in pore network. The fundamental solution of the system of steady vibration equations in the considered theory is constructed, and its basic properties are established. Green’s formulas are obtained, and the uniqueness theorems of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are constructed, 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.