Broadening global families of anti-plane shear equilibria

TL;DR

This paper develops a global bifurcation theory for nonlinear elastic materials under anti-plane shear, revealing broadening behaviors and ellipticity loss, with implications for failure mechanics.

Contribution

It introduces a rigorous global bifurcation framework for two classes of nonlinear elastic materials, demonstrating broadening of solution supports and ellipticity loss phenomena.

Findings

Solution curves exhibit broadening behavior.

Ellipticity loss indicates potential failure or instability.

The phenomena relate to crack formation and stability issues.

Abstract

We develop a global bifurcation theory for two classes of nonlinear elastic materials. It is supposed that they are subjected to anti-plane shear deformation and occupy an infinite cylinder in the reference configuration. Curves of solutions to the corresponding elastostatic problem are constructed using analytic global bifurcation theory. The curve associated with first class is shown to exhibit broadening behavior, while for the second we find that the governing equation undergoes a loss ellipticity in the limit. A sequence of solutions undergoes broadening when their effective supports grow without bound. This phenomena has received considerable attention in the context of solitary water waves; it has been predicted numerically, yet it remains to be proven rigorously. The breakdown of ellipticity is related to cracks and instability making it an important aspect of the theory of failure mechanics.