Potential method in the theory of thermoviscoelastic mixtures

In this article, the linear theory of binary thermoviscoelastic mixtures is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Namely, the fundamental solution of the system of equations of steady vibrations is constructed explicitly and its basic properties are established. Green’s second and third identities are obtained and the uniqueness theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved. The surface and volume potentials are constructed and their basic properties are given. The determinants of symbolic matrices are calculated explicitly. 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 internal and external BVPs of steady vibrations are proved by the potential method and the theory of singular integral equations.