TRANSFORMATIONS AND EQUATION REDUCTIONS IN FINITE ELASTICITY .2. PLANE-STRESS AND AXIALLY-SYMMETRICAL DEFORMATIONS

In Part I of this article, the problem of determining plane deformatio ns 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 M onge-Ampere equation, which can be linearized. Two of the linearizatio ns involve the physical angle Theta (or theta) arising in cylindrical polar coordinates (R, Theta) and are therefore potentially useful for practical problems. In Part II of the paper, we obtain corresponding r esults for the two distinct areas of the plane stress theory of thin s heets and for axially symmetric deformations. For the first area we de duce two second order problems, one of which admits a linearization in volving two parameters, which turn out to be the polar angle Theta and the principal stretch lambda(R, Theta) in the transverse direction. T his is an important result because it is very unusual to linearize a p roblem in terms of two parameters, which are both physically meaningfu l. For axially symmetric deformations we again deduce a second order p roblem, but for which we are unable to provide a linearization. Howeve r, we examine a number of simple solutions of the second order problem .