Transformations and Equation Reductions in Finite Elasticity IV: Illustration of the General Integral for a Plane Strain Similarity Deformation

摘要

In three previous parts of this work, for plane strain, plane stress, and axially symmetric deformations, a number of new first integrals are deduced for the so-called perfectly elastic Varga materials. These results constitute a considerable advance in the theory of finite elastic deformations, there being no results similar to these in existing theory. The new integrals, together with the constraint of incompressibility, mean that certain highly nonlinear fourth-order partial differential equations admit second-order systems, every solution of which is a solution of the corresponding fourth order problem. Morevoer, many of the second-order partial differential equations admit linearization to either the harmonic equation or the Helmholtz equation, thus giving rise to the posssibility of generating quite general solutions. However, it is not immediately clear how such solutions relate to solutions of the full system.This part of the work (IV) attempts to discover the extent to which the solutions of these new first integrals span the solutions of the full space. In this paper, the general integral for plane strain deformations, given in Part III, is illustrated with reference to a specific similarity deformation, which maps one wedge-shaped region into another such region and for which the general solution of the full system can be obtained in closed form.