Jm. Hill et Dj. Arrigo, TRANSFORMATIONS AND EQUATION REDUCTIONS IN FINITE ELASTICITY .1. PLANE-STRAIN DEFORMATIONS, Mathematics and mechanics of solids, 1(2), 1996, pp. 155-175
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.