Fundamental solutions in the theory of elasticity for triple porosity materials

Abstract

This paper concerns with the full coupled linear theory of elasticity for triple porosity materials. The system of the governing equations based on the equations of motion, conservation of fluid mass, the constitutive equations and Darcy’s law for materials with triple porosity. The cross-coupled terms are included in the equations of conservation of mass for the fluids of the three levels of porosity and in the Darcy’s law for materials with triple porosity. The system of general governing equations of motion is expressed in terms of the displacement vector field and the pressures in the three pore systems. Five spatial cases of the dynamical equations are considered: equations of steady vibrations, equations in Laplace transform space, equations of quasi-static, equations of equilibrium and equations of steady vibrations for rigid body with triple porosity. The fundamental solutions of the systems of these PDEs are constructed explicitly by means of elementary functions and finally, the basic properties of the fundamental solutions are established.