Stability and vibration of pre-stressed compressible elastic plates

Abstract

In a recent paper [1], Ogden and Roxburgh Int. J. Engng Sci. have given a detailed account of plane incremental vibrations of a rectangular plate of incompressible isotropic elastic material subject to an underlying homogeneous pure strain. In this paper corresponding results in respect of a compressible elastic material are derived. Specifically, for a general form of strain-energy function, equations governing the frequency of symmetric and antisymmetric modes of vibration are obtained. Depending on the form of strain-energy function, the underlying state of deformation and the aspect ratio of the plate, nine distinct cases arise together with several subcases. This is similar to the situation in the incompressible theory, although, for compressible materials, additional subcases appear. The emergence of quasi-static modes of deformation, corresponding to zero frequency, is accorded special attention since the frequency equations then reduce to bifurcation equations, which provide information about the boundary of stability of the underlying configuration in deformation space. Stability criteria are analysed in detail for a general form of strain-energy function. The dynamic and static results are illustrated by numerical calculations for a number of simple strain-energy functions.