Solution of one boundary value task of viscoelasticity in a nonlinear formulation, in the case of a cubic stress-strain relation

Abstract

In this paper, the solution of a boundary value task in the nonlinear formulation is considered by the authors [1][2]. In spite of its proximity to linear theory, the nonlinear theory of viscoelasticityhas not yet been fully developed. This issue is far from being fully completed, since the existing calculation methods do not yet provide a complete answer to the many different questions posedby practice. For this reason, in order to obtain a nonlinear law relating the strains σij and deformations εij a number of conditions are formed: (1) The specific work of deformation A must be a function of the entire deformation historyfrom the beginning of deformation to the current time t. (2) The material of a viscoelastic body is homogeneous and isotropic. (3) For very small deformations the nonlinear relation law between σij and εij in the limit should pass to relations in linear approximation. Key words: bulk compression modulus, linear integral operator, kernel of integral operator, nonlinear dependence,quadratic strain intensity, Fourier and Laplace transforms.