TRANSFORMATIONS AND EQUATION REDUCTIONS IN FINITE ELASTICITY .1. PLANE-STRAIN DEFORMATIONS

For a particular finite elastic material, the governing fourth order n onlinear partial differential equations for plane strain deformations are shown to admit a new first integral, which, together with the cons traint of incompressibility, gives rise to a second order problem. By an appropriate transformation of variables, the second order problem c an be reduced to a single Monge-Ampere equation. Remarkably, this latt er equation admits a linearization to the standard Helmholtz equation, so that the possibility arises for the determination of numerous exac t finite elastic deformations. Of particular importance is that one of the parameters arising in the linearization turns out to be the physi cal angle Theta involved in standard cylindrical polar coordinates, an d therefore these exact solutions might be particularly relevant to pr oblems concerned with sectors of cylinders. Known first integrals of t he governing fourth order equations are summarized in cylindrical pola r coordinates, and new solutions of the form theta = g(Theta) are obta ined. Finally, an alternative approach is suggested and the procedure is illustrated with two examples. Corresponding results for the two se parate topics of the plane stress theory of highly elastic thin sheets and axially symmetric deformations are given in Part II of the paper.