Characterizations of symmetric polyconvexity

TL;DR

This paper characterizes symmetric polyconvex functions in 2D and 3D, explores subclasses, and provides a counterexample to symmetric rank-one convexity implying symmetric polyconvexity, with implications for elasticity theory.

Contribution

It offers a new characterization of symmetric polyconvex functions and investigates subclasses, including a notable counterexample in 3D.

Findings

New characterization of symmetric polyconvex functions in 2D and 3D

Identification of symmetric polyconvex quadratic forms

Counterexample of symmetric rank-one convex quadratic form in 3D

Abstract

Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are easier to handle. This article focuses on symmetric polyconvexity, which is a sufficient condition. We prove a new characterization of symmetric polyconvex functions in the two- and three-dimensional setting, and use it to investigate relevant subclasses like symmetric polyaffine functions and symmetric polyconvex quadratic forms. In particular, we provide an example of a symmetric rank-one convex quadratic form in 3d that is not symmetric polyconvex. The construction takes the famous work by Serre from 1983 on the classical situation without symmetry as inspiration. Beyond their theoretical interest, these findings may turn out useful for computational relaxation and homogenization.