On the instability of inextensible elastic bodies: nonlinear evolution of non-neutral, neutral and near-neutral modes

Abstract

For inextensible elastic bodies, linear theory predicts that if the reaction stress is compressive and sufficiently large, a transverse progressive wave travelling in the direction of inextensibility may have an imaginary wave speed and grow without bound as a standing wave (Chen & Gurtin 1974). The development of these growing standing waves under the influence of nonlinearity is considered in this paper. Attention is focused on the case for which the negative reaction stress deviates by a small amount from the value corresponding to the zero wave speed, so that the question addressed is how the evolution of the near-neutral waves is (slowly) modulated by nonlinear effects. It is shown, both numerically and analytically, that depending on initial conditions, nonlinearity can make a near-neutral wave grow, decay or have constant amplitude (growth occurs even in the neutral case for which linear theory predicts zero growth), but in every case its main action is to distort the wave profile and make it evolve into a shock within a finite time. It is found that the evolution of some near-neutral waves (corresponding to certain initial conditions) is governed by analytical solutions, with the aid of which we can show that any shock, once it has formed, will eventually decay to zero algebraically. For general initial conditions, the further evolution of the shock cannot be determined from the present analysis, but we may conjecture that the shock thus formed will also decay to zero. Hence nonlinearity stabilizes near-neutral waves through the formation of shocks. However, an important result found for near-neutral waves is that corresponding to some initial conditions, high values of strain (and thus stress) may obtain just before the shock forms, so that there is the possibility that the elastic body may fracture before the decay of shock amplitude occurs. The effects of nonlinearity on non-neutral travelling waves are also studied and it is shown that nonlinearity also makes non-neutral travelling waves evolve into shocks, but in contrast with the situation for near-neutral waves it does not change their amplitudes as time evolves. The present analysis is also applicable to surface waves in pre-stressed materials where zero wave speed may be induced by large enough pre-stresses.