Linearly polarized finite-amplitude plane waves in pre-strained incompressible elastic materials

Abstract

In this paper, general incompressible non-linear elastic materials are considered and also a sub-class of these called “generalized neo-Hookean materials”. We deal with the propagation of finite-amplitude linearly polarized transverse waves in such materials subjected to an arbitrary static homogeneous deformation. Contrary to the case of waves of infinitesimal amplitude (incremental theory), linearly polarized waves are in general not possible for every propagation direction. In this paper, we present two cases when such finite-amplitude waves may propagate. The first case is that of generalized neo-Hookean materials, for which it is shown that a finite-amplitude linearly polarized wave may propagate in any direction. In general, the polarization direction is uniquely determined. The second case is that of general incompressible elastic materials, but the propagation and polarization directions are both assumed to be in the same principal plane of the static deformation. In both cases, the wave motion is governed by a single scalar non-linear wave equation. For simple wave solutions of this equation, a property relating the energy density and the projection of the energy flux onto the propagation direction is derived.