Scenarios of evolution of some types of simple waves in nonlinear elastic materials

Abstract

Four differing from each other scenarios of evolution of the plane and close to them cylindrical solitary (simple) waves are established and commented. The material of propagation of waves is assumed to be elastic, the nonlinear deformation of which is described by the five-constant Murnaghan model (potential). The initial wave profile is given by the cosinusoidal, Gauss, Whittaker, Macdonald functions. The scenarios for the cosinusoidal (harmonic) profile are used as the classical and reference one. The methods of successive approximations and restriction on the gradient of displacement are utilized. A novelty in the description of scenarios is that the first three approximations are taken into account. This changes essentially the known (based on the first two approximations) scenarios and exposes some new wave effects. The similarities and distinctions of the presented four scenarios are commented. For example, the initial stages and effect of forming the asymmetric profile are quite similar to both the asymmetric and symmetric initial profiles. But also the scenarios show the transition from the one-hump profile to the two-hump or three-hump ones. Some of the scenarios show the drift of the profile relative to the horizontal axis. A row of other features of scenarios is commented.