An Asymptotic Model for Long-Wave Symmetric Motion of a Pre-Stressed Incompressible Plate With Fixed Faces

Abstract

The limitations of classical plate theories are well known and may be classified to fall into two categories. Firstly, all of these theories are only suited to describe relatively low-frequency motions, and secondly, classical theories assume that deformations of the plate’s faces are not fixed. However, many modern applications have to deal with plates experiencing high-frequency loads and/or fixed on the faces, which may be illustrated by an example from geophysics of a coal layer contained within a much stiffer rock. Dynamic motion of such struetures is complicated and classical ad hoc hypotheses are clearly not valid in this context, there-fore an asymptotic analysis of three-dimensional equations of elasticity appears to be a natural approach. A total asymptotic analysis of this kind was performed for thin isotropic struetures in [1]. However, in the previously mentioned example from geophysics, coal layers also usually suffer from large initial Stresses, which generally requires use of a non-linear theory of elasticity.