Classical Injective Solutions in the Large in Incompressible Nonlinear Elasticity

TL;DR

This paper proves the existence of large, classical injective solutions in incompressible nonlinear elasticity under various loadings, using a global implicit function theorem and nonlinear Fredholm degree.

Contribution

It establishes a sharp global implicit function theorem for incompressible elasticity, demonstrating solutions in the large without relying on traditional degree theories.

Findings

Existence of unbounded solution branches from the reference configuration.

Solutions exist for any applied loading.

Solutions of arbitrarily large norm are possible under finite loadings.

Abstract

We consider a general class of parametrized displacement boundary value problems in incompressible nonlinear elasticity. We prove the existence of an unbounded solution branch of classical injective solutions emanating from the unforced stress-free reference configuration, which constitutes a sharp global implicit function theorem. The ramifications of this are: There exists at least one solution for any given applied loading, and or there exists solutions of arbitrarily large norm for some finite loading. We refer to this as solutions in the large. The nonlinear constraint equation enforcing incompressibility obviates the direct use of either the Leray-Schauder degree or an oriented degree previously developed for compressible problems. Instead we employ the nonlinear Fredholm degree for proper maps developed by Fitzpatrick, et. al. We establish the requisite admissibility properties in the presence of the constraint equation before proving the main results.