Lie symmetries, closed-form solutions, and conservation laws of a constitutive equation modeling stress in elastic materials

The Lie-point symmetry method is used to find some closed-form solutions for a constitutive equation modeling stress in elastic materials. The partial differential equation (PDE), which involves a power law with arbitrary exponent n, was investigated by Mason and his collaborators (Magan et al., Wave Motion, 77, 156-185, 2018). The Lie algebra for the model is five-dimensional for the shearing exponent n > 0, and it includes translations in time, space, and displacement, as well as time-dependent changes in displacement and a scaling symmetry. Applying Lie's symmetry method, we compute the optimal system of one-dimensional subalgebras. Using the subalgebras, several reductions and closed-form solutions for the model are obtained both for general exponent n and special case n = 1. Furthermore, it is shown that for general n > 0 the model has interesting conservation laws which are computed with symbolic software using the scaling symmetry of the given PDE.