A nonlinear analysis of instability of a pre-stressed incompressible elastic plate

Abstract

For an isotropic, incompressible elastic plate which is pre-stressed into a state of hydrostatic pressure, linear stability analysis predicts that when the value of the hydrostatic pressure p̄ is increased above p̄ + (say) or decreased below p̄ - (say), the plate will become unstable in the sense that certain forms of travelling waves will become standing waves whose amplitude grows exponentially in time. The upper branch neutral value p̄ + is a function of the product of the wave number and plate thickness while the lower branch neutral value p̄ - is a constant. Those waves which occur when the pressure deviates from the neutral values p̄ + or p̄ - by a small amount are termed near-neutral waves. In this paper the nonlinear evolutionary behaviour of upper-branch near-neutral waves are investigated. It is shown that the amplitude A of such near-neutral waves satisfies an evolution equation of the form d 2 A /d ז 2 = – v 0 A – v 1 A | A | 2 , where v 0 and v 1 are constants and ז is a slow time scale. The general properties of the solution of this equation are studied. It is found that when v 1 > 0 nonlinear effects are stabilizing in the sense that any exponential growth will be suppressed by nonlinearity and turned into an oscillation, and when v 1 < 0 they are destabilizing in the sense that they help the wave amplitude grow. The values of v 1 are calculated for a selection of the plate thicknesses (and wavenumbers) using two material models. We find that the sign of v 1 is not definite and thus nonlinear effects are stabilizing only over a certain wavenumber régime while destabilizing over the remaining wavenumber régime. All the relevant factors which affect the stability of the plate are fully discussed.