Some asymptotic expansions of the dispersion relation for an incompressible elastic plate

Abstract

This paper is concerned with a general asymptotic analysis of the dispersion relation associated with waves propagating in a pre-stressed, incompressible elastic plate. In the high wave number limit it is well-known that, whenever a real surface wave speed exists, the fundamental modes of both symmetric and anti-symmetric motions tend to this surface wave speed, with all harmonics tending to a single shear wave speed limit. The character of the two dispersion curves in the moderate and high wave number regimes falls into one of two distinct cases, these being dependent on pre-stress. In the first case all the harmonics are monotonic decreasing functions and as such the asymptotic analysis in this case offers a modest generalisation of an earlier study, see Rogerson and Fu (Rogerson, G. A. and Fu, Y. B. (1995) An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate. Acta Mechanica111, 59–77). In contrast, the second case is quite different in character with the passage to the high wave number limit accompanied by sinusoidal behaviour. This behaviour is fully elucidated by obtaining asymptotic expansions which give phase speed as a function of wave number, pre-stress and harmonic number, sinusoidal terms being found to occur at third order. Both these asymptotic expansions and ones obtained for high harmonic number are found to provide excellent agreement with numerical solutions for Varga materials in the appropriate regimes. It is envisaged that the expansions derived in this paper may well find important potential applications in the numerical inversion of the transform solution sometimes used in impact problems.