Study on the Method for a Solution to Some Class of Quasi-Static Problems in Linear Viscoelasticity Theory, as Applied to Problems of Linear Torsion of a Prismatic Solid

Abstract

One possible method is proposed for reducing the problem of isotropic hereditary elasticity to solving a set of similar quasi-static problems in the theory of elasticity and thermoelasticity. The representability of the solution of the problem of linear hereditary elasticity in the form of the sum of solutions of three problems is substantiated: the linear theory of elasticity for imaginary bodies-incompressible and having a zero Poisson’s ratio and stationary uncoupled thermoelasticity for a body whose properties do not depend on temperature. The shear and bulk relaxation kernels are considered independent; the viscoelastic Poisso ratio is time dependent. Two theorems that reduce solutions of the general quasi-static problem of linear viscoelasticity theory to a solution of the corresponding problem of elasticity theory are proved. These theorems hold if one of the following conditions is satisfied: 1) the material is close to a mechanically uncompressible material; 2) the mean stress is zero; 3) the shift and volume hereditary functions are equal. The theorems provide free direct and inverse transforms between solutions of viscoelasticity and elasticity problems, which make them convenient in applications. They have been applied to solutions of problems on the pure torsion of a prismatic viscoelastic solid with an arbitrary simply connected cross section. Some examples describing the obtained results have been considered.