Mathematical problems in the coupled linear theory of bone poroelasticity

This paper concerns with the quasi-static coupled linear theory of bone poroelasticity for materials with double porosity and some basic results of the classical theories of elasticity and thermoelasticity are generalized. The system of equations of this theory is based on the equilibrium equations, conservation of fluid mass, the effective stress concept and Darcy’s law for material with double porosity. The Green’s formulas in the considered theory are obtained, the formulas of Somigliana type integral representations of regular vector and regular (classical) solutions are presented. The uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. The single-layer, double-layer and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral method) and the theory of singular integral equations.