Global Continuation in Displacement Problems of Nonlinear Elastostatics via the Leray-Schauder Degree

Abstract

We obtain global solution continua of forced displacement problems of nonlinear elastostatics via a Leray-Schauder scheme. We adopt strong ellipticity as a bsic constitutive hypothesis. The usual Leray-Schauder approach, based in part upon the reduction of the boundary value problem to an operator equation for the zeros of a compact vector field, apparently fails. On the one hand, strong ellipticity alone is not enough to insure “invertibility/rdquo; of the principal, quasilinear part of the differential operator. More importantly, the physical requirement of local injectivity of the deformation and the associated growth of the stored-energy function dictate that ellipticity is not uniform. We demonstrate how to overcome these difficulties. Finally, with additional, physically reasonable restrictions on the stored-energy function, we obtain unbounded branches of classical, globally injective solutions.