An Asymptotically Consistent Model for Long-Wave High-Frequency Motion in a Pre-Stressed Elastic Plate

Abstract

A one-dimensional asymptotic model is derived to elucidate the effect of pre-stress on long-wave high-frequency two-dimensional motion in an incompressible elastic plate. Solutions for the leading-order displacements and pressure increment are derived in terms of the long-wave amplitude; a governing equation for which is derived from the second-order problem. This equation is shown to become elliptic for certain states of pre-stress. Loss of hyperbolicity is shown to be synonymous with the existence of negative group velocity at low wavenumber. A higher-order theory is constructed, with solutions obtained in terms of both the long-wave amplitude and its second-order correction. An equation relating these is obtained from the third-order problem. The dispersion relations derived from the one-dimensional governing equations are also obtained by expansion of the corresponding exact two-dimensional relations, indicating asymptotic consistency. The model is highly relevant for stationary thickness vibration of, or transient response to high-frequency shock loading in, thin-walled bodies and also fluid-structure interaction. These are areas for which the effects of pre-stress have previously largely been ignored.