Global continuation in displacement problems of nonlinear elastostatics via the Leray-Schauder degree

We obtain global solution continua of forced displacement problems of nonli near elastostatics via a Leray-Schauder scheme. We adopt strong ellipticity as a bsic constitutive hypothesis. The usual Leray-Schauder approach, base d in part upon the reduction of the boundary value problem to an operator e quation for the zeros of a compact vector field, apparently fails. On the o ne hand: strong ellipticity alone is not enough to insure "invertibility" o f the principal, quasilinear part of the differential operator. More import antly, the physical requirement of local injectivity of the deformation and the associated growth of the stored-energy function dictate that elliptici ty is not uniform. We demonstrate how to overcome these difficulties. Final ly, with additional, physically reasonable restrictions on the stored-energ y function, we obtain unbounded branches of classical, globally injective s olutions.