Nonlinear Elastodynamics with Radial Symmetry, Model and Qualitative Investigation

This article deduces a model, stated as an integral equation, for any nonlinear elastic isotropic material undergoing a radially symmetric deformation. Such a model is useful in the study of an explosion, or a spherically symmetric impact. Determining the effects of nonlinear wave propagation, in relation to linear propagation, can be truly challenging in 3D dimensions. By reducing the system to a 1D radial partial integral equation numerical simulations are more accurate and manageable. Also, understanding the radially symmetric model sheds light on the qualitative behaviour of the full 3D nonlinear system. An emphasis is given on an intuitive understanding of the dynamics. After deducing the general integral model we present discontinuous jump conditions, and then discuss and substitute the Mooney-Rivlin approximation for the material. We point out how the model for the linearised material can approximate a Mooney-Rivlin material, and subsequentially present the analytical solution to some important cases of the linearised material. The appendix attempts to be a rather complete exposition which departs from first principles, where the theoretical basis follows the axiomatic treatment of elasticity and the integral formulation of balance principles.