New nonlinear models of dynamic longitudinal deformation of a viscoelastic rod

Abstract

In this paper we study a general model of dynamic longitudinal deformation of a viscoelastic rod with a nonlinear dependence of the stress on the strain and on the strain rate. This model is given by a nonlinear differential equation with partial derivatives of the third order. A group classification of this model which depends on two arbitrary functions is carried out. Six basic models are obtained, for which their main Lie groups of transformations are found. A model that admits the widest Lie group of transformations is studied in detail. For it, formula for the production of new solutions is obtained and all invariant submodels are found. For all its invariant submodels, the invariant solutions defining these submodels are found either in explicit form, or their search reduces to solving the systems of first-order differential equations. For these systems, boundary value problems that have a physical meaning are studied. Conditions are obtained that ensure the existence and uniqueness of solutions to these boundary value problems. This allows one to correctly solve these boundary value problems numerically. Boundary value problems for some specific values of the parameters included in them are solved numerically. The research carried out is especially relevant in rocket engineering, aircraft engineering, shipbuilding and other fields.