Transformations and Equation Reductions in Finite Elasticity II: Plane Stress and Axially Symmetric Deformations

In Part I of this article, the problem of determining plane deformations of a particular perfectly elastic material is shown to give rise to three second order problems. These are evidently considerably easier to deduce exact finite elastic deformations than the full fourth order problem. All three second order problems can be reduced to a single Monge-Ampere equation, which can be linearized. Two of the linearizations involve the physical angle e (or 0) arising in cylindrical polar coordinates (R, 0) and are therefore potentially useful for practical problems. In Part II of the paper, we obtain corresponding results for the two distinct areas of the plane stress theory of thin sheets and for axially symmetric deformations. For the first area we deduce two second order problems, one of which admits a linearization involving two parameters, which turn out to be the polar angle e and the principal stretch X(R, 0) in the transverse direction. This is an important result because it is very unusual to linearize a problem in terms of two parameters, which are both physically meaningful. For axially symmetric deformations we again deduce a second order problem, but for which we are unable to provide a linearization. However, we examine a number of simple solutions of the second order problem.