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Transformations and Equation Reductions in Finite Elasticity I: Plane Strain Deformations

For a particular finite elastic material, the governing fourth order nonlinear partial differential equations for plane strain deformations are shown to admit a new first integral, which, together with the constraint of incompressibility, gives rise to a second order problem. By an appropriate transformation of variables, the second order problem can be reduced to a single Monge-Ampere equation. Remarkably, this latter equation admits a linearization to the standard Helmholtz equation, so that the possibility arises for the determination of numerous exact finite elastic deformations. Of particular importance is that one of the parameters arising in the linearization turns out to be the physical angle E involved in standard cylindrical polar coordinates, and therefore these exact solutions might be particularly relevant to problems concerned with sectors of cylinders. Known first integrals of the governing fourth order equations are summarized in cylindrical polar coordinates, and new solutions of the form 0 = g(0) are obtained. Finally, an alternative approach is suggested and the procedure is illustrated with two examples. Corresponding results for the two separate topics of the plane stress theory of highly elastic thin sheets and axially symmetric deformations are given in Part II of the paper.