Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials

A nonlinear intrinsic theory is used to describe the motions of a straight round elastic rod including the influence of radial shear and inertia. Consideration of steady wave motions reduces the two coupled partial differential equations to ordinary differential equations for which two integrals of the motion may be found. For incompressible elastic materials with the restriction of small strain gradients, but arbitrary finite strains, a large variety of exact solutions may be found by quadrature. These include large amplitude periodic waves (which may contain shocks), solitary waves, and in some cases waves that are transitional from one stress level to another. Such solutions may be found for uniform stress strain curves that are concave up or down or that contain inflections, and even for nonmontonic curves, which have been used to represent phase transitions.