SOME INTEGRABLE AND EXACT SIMILARITY SOLUTIONS FOR PLANE FINITE ELASTIC DEFORMATIONS

The governing partial differential equations for static deformations of homogeneous isotropic incompressible hyperelastic materials (sometimes referred to simply as perfectly elastic materials) are highly nonlinear and consequently only a few exact solutions are known. In this paper, the general plane similarity deformation for such materials, which includes a number of previously studied deformations and which is a well-defined deformation for every material of this type, is considered. By a variety of ad hoc procedures (transformation of variables, utilization of a known general solution, and direct integration), a number of special situations for the neo-Hookean and Varga materials are compiled which give rise to integrable cases. The solutions so obtained have not been given previously and are exact in the sense that closed-form expressions are obtained. In common with most closed-form solutions of nonlinear ordinary differential equations, the expressions obtained tend to be neither simple nor explicit, because as a general rule they tend to be obtained in parametric forms which are often awkward to utilize in the context of specific problems. However, in view of the scarcity of exact solutions in finite elasticity, these isolated integrable cases would seem worthwhile recording. At the least, such special solutions are valuable as a means of checking numerical schemes. © 1990 Oxford University Press.