Transformations and Equation Reductions in Finite Elasticity III: A General Integral for Plane Strain Deformations

Abstract

In parts I and II of this work, for plane strain, plane stress, and axially symmetric deformations, a number of first integrals are obtained for the so-called perfectly elastic Mirga materials. These first integrals, together with the constraint of material incompressibility, give rise to various second-order nonlinear partial differential equations, some of which are shown in I and II to admit linearization by means of contact transformations. In this paper, for the same perfectly elastic NVrga materials, a general integral involving seven arbitrary real constants is deduced that includes as special cases the various integrals discussed in part I. Moreover, this general integral, in conjunction with the condition of incompressibility, is also shown to be linearized to the standard Helmholtz equation. In part IV of this work, the integral is illustrated with reference to a plane strain similarity deformation.