Nonlinear Hyperbolic Waves in Hyperelastic Solids

This paper considers hyperbolic, one spatial dimension nonlinear wave propagation in a hyperelastic solid, and a discussion of the basic theory is presented. Constitutive relations for compressible rubberlike materials, whose internal energies can be expressed as the sum of a function of specific volume only and a function of temperature only, are discussed. These relations are assumed for the analysis of a class of plane wave problems and similarity solutions are obtained. Thermal effects, including the effect of the jump in entropy across a shock for a problem of uncoupled longitudinal wave propagation, are taken into account, however heat conduction is neglected. Solutions for a piezotropic model, which is a model for which mechanical and thermal effects are uncoupled, are obtained for comparison purposes. An axisymmetric problem is also discussed.