On the linear theory of double porosity thermoelasticity under local thermal non-equilibrium

In the present paper, the linear theory of thermoelasticity of double porosity materials under local thermal non-equilibrium is considered and the basic boundary value problems are investigated by using the boundary integral equation method (potential method). Indeed, the fundamental solution of system of steady vibrations equations of the considered theory is constructed explicitly by means of elementary functions and its basic properties are established. A Galerkin-type solution to this system of equations is presented and the completeness of this solution is proved. The basic internal and external boundary value problems of steady vibrations are formulated and the uniqueness theorems for classical solutions of these problems are proved. The basic properties of the surface and volume potentials and singular integral operators are established. Finally, the existence theorems for classical solutions of the above-mentioned boundary value problems are proved by using the potential method and the theory of singular integral equations.