A Potential Representation for Two-Dimensional Waves in Elastic Materials of Harmonic Type

Abstract

In the present note we consider two-dimensional finite dynamical deformations for the class of homogeneous, isotropic elastic materials introduced by F. John in [1] and referred to by him as materials of harmonic type. The theory of such materials, developed in [1] and [2], appears to be simpler in many respects than that of more general elastic materials, and it may offer the possibility of investigating some features of nonlinear elastic behavior more explicitly than is possible in general. For plane motions of such materials, we derive here a representation for the displacements in terms of two potentials which is analogous to the theorem of Lamé in classical linear elasticity (see [3]) for the case of plane strain. The two nonlinear differential equations satisfied by the potentials reduce upon linearization to the wave equations associated with irrotational and equivoluminai waves in the linear theory. In the following section we state without derivation the equations governing two-dimensional waves in an elastic material of harmonic type. The reader is referred to [1] for details. In Sec. 3 we derive the representation in terms of potentials described briefly above.