This chapter studies nonlinear dispersive waves in a Mooney-Rivlin elastic rod. We first derive an approximate one-dimensional rod equation, and then show that traveling wave solutions are determined by a dynamical system of ordinary differential equations. A distinct feature of this dynamical system is that the vector field is discontinuous at a point. The technique of phase planes is used to study this singularity (there is a vertical singular line in the phase plane). By considering the relative positions of equilibrium points, we establish the existence conditions under which a phase plane contains physically acceptable solutions. In total, we find ten types of traveling waves. Some of the waves have certain distinguished eatures. For instance, we may have solitary cusp waves which are localized with a discontinuity in the shear strain at the wave peak. Analytical expressions for most of these types of traveling waves are obtained and graphical results are presented. The physical existence conditions for these waves are discussed in detail.

Introduction

Traveling waves in rods have been the subject of many studies. The study of plane flexure waves has formed one focus. See, e.g., Coleman and Dill (1992), and Coleman et al. (1995). Another focus is the study of nonlinear axisymmetric waves that propagate axial-radial deformation in circular cylindrical rods composed of a homogeneous isotropic material. This chapter is concerned with the latter aspect for incompressible Mooney-Rivlin materials. We mention in particular three related works by Wright (1982, 1985) and Coleman and Newman (1990).