Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians

TL;DR

This paper explores explicit twist maps and shear maps in nonlinear elasticity, revealing conditions under which Jacobians vanish or are positive, and analyzing their impact on the regularity and uniqueness of energy minimizers.

Contribution

It provides explicit examples of twist maps with vanishing Jacobians and characterizes the unique shear map minimizer with mixed Jacobian positivity, highlighting their regularity properties.

Findings

Explicit twist maps with Jacobian vanishing on positive measure sets.

Unique shear map minimizer with mixed Jacobian positivity and discontinuities.

Regularity and uniqueness conditions for energy minimizers in nonlinear elasticity.

Abstract

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian $\det \nabla u$ and the uniqueness and regularity of energy minimizers $u$ that are either twist maps or shear maps. We exhibit \emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma}: \Omega\to \mathbb{R}^2$ in a model, two-dimensional case. The shear map minimizer has the properties that (i) $\det \nabla u_{\sigma}$ is strictly positive on one part of the domain $\Omega$, (ii) $\det \nabla u_{\sigma} = 0$ necessarily holds on the rest of $\Omega$, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma}$ is not continuous on the whole domain.