Nonlinear travelling waves in a rod composed of a modified Mooney–Rivlin material. I. Bifurcation of critical points and the non–singular case

Abstract

This paper studies finite axial–radial deformations in a rod composed of a modified Mooney–Rivlin material. Our focus is on travelling wave solutions, for which the governing partial differential equation can be reduced to a system of ordinary differential equations. Two important features of this system are: (a) the phase plane has no or one or two singular lines; (b) the critical points are governed by a sixth–order polynomial equation involving four physical parameters, whose roots cannot be found by direct calculations. In the first part of this series of two papers, we consider the non–singular case. Since the bifurcation equation is of sixth order, how to determine its roots and their dependence on the physical parameters imposes a major difficulty. By using the singularity theory, this is successfully resolved. We are able to classify precisely the number of critical points and their types in different ranges of the physical parameters. In total, we find there are seven types of qualitatively different phase planes, among which three types contain paths representing physically acceptable solutions. These paths are then studied one by one. We find in total seven types of travelling waves. These are: solitary waves of radial contraction, solitary waves of radial expansion, kink waves, antikink waves and three types of periodic solutions. The solution expressions in terms of integrals are obtained. We also examine in detail the physical conditions under which each type of wave can arise. The existence of kink waves in hyperelastic rods with no external source is conjectured in the literature. Here, it is proved by vigorous mathematics that such waves can indeed arise in a rod composed of a modified Mooney–Rivlin material.