ON THE PROPAGATION OF NONLINEAR TRAVELING WAVES IN AN INCOMPRESSIBLE ELASTIC PLATE

The propagation and stability of travelling waves in an isotropic incompressible elastic plate is considered in this paper. The plate is assumed to be in a state of plane strain and the waves propagate parallel to the surfaces. First, the propagation of a typical monochromatic wave is considered. It is shown that if the plate is initially unstrained, the effects of nonlinearity are to induce a small correction to the wave speed over a long time scale or long length scale without changing the wave amplitude and so the nonlinear solution is a Stokes wave. Next, the stability of such a Stokes wave is considered. It is shown that when the wave is perturbed by long-wavelength modulational waves, the resultant wave field is governed by a cubic Schrodinger equation. The coefficients in this equation are calculated for both a Mooney and a Neo-Hookean material for a selection of plate thicknesses and for the first three modes of the dispersion curve. It is found that the Stokes wave can either be stable or unstable depending on the plate thickness, the mode number and the material. It is deduced that the above conclusions can be generalized to (i) a prestrained isotropic elastic plate in which one principal axis of stretch is perpendicular to the plate surfaces and another one aligned with the direction of wave propagation and (ii) an orthotropic elastic plate which has the three symmetry directions playing the same roles as the three Principal axes Of stretch in (i). In all other cases, nonlinearity is expected to affect the propagation of a monochromatic wave in quite a different manner, and we conjecture that nonlinearity not only induces a small correction to the wave speed but also makes the wave amplitude decay algebraically and so Stokes wave solutions will not exist. The relevance of the above results to a solid-fluid interaction problem is also discussed.