Exact solutions of one problem of nonlinear theory of elasticity for two potentials of energy of incompressible material

摘要

In the nonlinear theory of elasticity, which is an apparatus for the study of rubber-like materials, at present there is no single equation of state for such materials, similar to Hooke's law in linear theory. The development of new rubber-like materials, the study of various biological materials leads to the need to create new models of response to external influences. In the framework of hyper elasticity, this leads to the appearance of new mathematical expressions for the deformation energy potential. In commercial packages designed to calculate the stress-strain state of structures from elastomers, the expressions for the deformation energy potential are rather rigidly defined. To accommodate new models, you have to create new application packages. Their verification is carried out using known exact solutions. In this paper, two exact solutions of one model problem for deformation energy potentials proposed by Fung~(1) and Gent et al.~(2), for which we could not find solutions in the literature. The practical significance of exact solutions is not limited to their application for the verification of numerical methods; they allow one to investigate effects not described by linear theory. It is shown that the difference in potentials leads not only to quantitative, but also to qualitative difference of solutions.