An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate

Abstract

This paper concerns an asymptotic analysis of the dispersion relation for wave propagation in a pre-stressed incompressible elastic plate. Asymptotic expansions for the wave speed as a function of wavenumber and pre-stress are obtained. These expansions have important potenatial applications to many dynamic problems such as impact problems. It is shown that in the large wavenumber limit the wave speed of the fundamental modes of both symmetric and anti-symmetric motions tends to the associated Rayleigh surface wave speed, on the other hand, the wave speeds of all the harmonics tend to a single limit which is the corresponding body wave speed. It is also shown that, whereas the fundamental modes are very sensitive to changes in the underlying pre-stress, the harmonics are little affected by such changes, espcially in the small and large wavenumber limits.