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 viscoelasticity has 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 posed by  practice.   For  this  reason,  in  order  to  obtain  a  nonlinear  law  relating  the  strains  σ_ij  and deformations ε_ij a number of conditions are formed:

• The specific work of deformation A must be a function of the entire deformation history from the beginning of deformation to the current time
• The material of a viscoelastic body is homogeneous and
• For very  small  deformations  the  nonlinear  relation  law  between  σ_ij  and  ε_ij  in  the  limit should pass to relations in linear