Non-classical problems for viscoelastic solids with microstructure

Abstract

In the present thesis the linear theories of viscoelasticity and thermoviscoelasticity for isotropic and homogeneous Kelvin-Voigt materials with voids are considered and some basic results of the classical theories of elasticity and thermoelasticity are generalized. Indeed, the basic properties of plane harmonic waves in the linear theory of viscoelasticity for Kelvin-Voigt materials with voids are established. There are two longitudinal and two transverse attenuated plane waves in the Kelvin-Voigt material with voids. In the considered theories the fundamental solutions of the systems of equations of steady vibrations are constructed by means of elementary functions and their basic properties are established. The representations of Galerkin type solutions of the systems of equations of steady vibrations are obtained. The Green’s formulas and integral representations of Somigliana type of regular vector and classical solutions are obtained. The formulas of representations of the general solution for the system of homogeneous equations of steady vibrations are established. The completeness of these representations of solutions is proved. The uniqueness theorems of the internal and external boundary value problems (BVPs) of steady vibrations in the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials with voids are proved. The basic properties of surface (single-layer and double-layer) and volume potentials are studied. On the basis of these potentials the BVPs are reduced to the singular integral equations. The corresponding singular integral operators are of the normal type with an index equal to zero. The Fredholm’s theorems are valid for these singular integral operators. Finally, the existence theorems of classical solutions of the above mentioned BVPs of the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials with voids are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations.