Nonlinear dynamic longitudinal deformation of a viscoelastic rod with power-law nonlinearity.

In this paper, by methods of symmetry analysis, we study a model of longitudinal dynamic deformation of a viscoelastic rod with a power-law dependence of stress on strain and strain rate. Symmetry analysis of the models of physics and continuum mechanics is one of the most effective ways to obtain quantitative and qualitative characteristics of the physical processes. Model under study is given by a nonlinear differential equation with partial derivatives of the third order. The main Lie group of transformations of this equation is found. For this equation, a formula for the production of new solutions is obtained and all invariant submodels are found. For all these invariant submodels, the invariant solutions that determine these submodels are obtained either in explicit form, or their search reduces to the solving of 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 the 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. [ABSTRACT FROM AUTHOR]