Abstract
Abstract In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with voids is considered and some basic results of the classical theory of elasticity are generalized. Indeed, the basic properties of plane harmonic waves are established. The explicit expression of fundamental solution of the system of equations of steady vibrations is constructed by means of elementary functions. The Green’s formulas in the considered theory are obtained. The uniqueness theorems of the internal and external basic boundary value problems (BVPs) are proved. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The formulas of integral representations of Somigliana type of regular vector and regular (classical) solution are obtained. The Sommerfeld-Kupradze type radiation conditions are established. The basic properties of elastopotentials and singular integral operators are given. Finally, the existence theorems for classical solutions of the internal and external basic BVPs of steady vibrations are proved by using of the potential method (boundary integral method) and the theory of singular integral equations.
Highlights
The theories of viscoelasticity initiated by Maxwell, Meyer, Boltzmann, and studied by Voigt, Kelvin, Zaremba, Volterra and others
On the basis of the potential method the uniqueness and existence theorems in the classical theories of viscoelasticity and thermoviscoelasticity for Kelvin–Voigt materials without voids are proved by Svanadze [56]
On the other hand the system of nonhomogeneous equations of steady vibrations in the linear theory of viscoelastic materials with voids can be written as follows μ1 u + (λ1 + μ1) grad div u + b1 grad φ + ρω2u = −ρF, (α1 + ξ2)φ − ν1 div u = −ρs, (2.7)
Summary
The theories of viscoelasticity initiated by Maxwell, Meyer, Boltzmann, and studied by Voigt, Kelvin, Zaremba, Volterra and others. In [31], Iesan extends theory of elastic materials with voids (see, Nunziato and Cowin [32, 33]), the basic equations of the nonlinear theory of thermoviscoelasticity for “virgin”, namely in the absence pre-existing stresses (see, Fabrizio and Morro [9], Deseri et al [13]), Kelvin–Voigt materials with voids are established, the linearized version of this theory is derived, a uniqueness result and the continuous dependence of solution upon the initial data and supply terms are proved. On the basis of the potential method the uniqueness and existence theorems in the classical theories of viscoelasticity and thermoviscoelasticity for Kelvin–Voigt materials without voids are proved by Svanadze [56]
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