Potential method in the coupled theory of elastic double-porosity materials

Abstract

In the present paper the linear coupled model of elastic double-porosity materials is proposed in which the coupled phenomenon of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established, and Green’s identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed, and the basic properties of these potentials are given. The determinants of symbolic matrices of the singular integral operators are calculated explicitly. Then, the BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. 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.