On the quasiconvex hull for a three-well problem in two dimensional linear elasticity

TL;DR

This paper characterizes the symmetric quasiconvex hull for three-well sets in 2D linear elasticity, providing bounds, explicit descriptions, and conditions for when it matches the lamination convex hull.

Contribution

It offers a complete characterization of the symmetric quasiconvex hull for three wells, including explicit formulas and conditions for equality with the lamination convex hull.

Findings

If two wells are rank-one compatible, the quasiconvex hull equals the lamination convex hull.

Explicit characterization of the lamination convex hull in terms of the wells.

Discussion on the optimality of bounds and relation to quadratic polyconvex functions.

Abstract

We provide quantitative inner and outer bounds for the symmetric quasiconvex hull $Q^e(\mathcal{U})$ on linear strains generated by three-well sets $\mathcal{U}$ in $\mathbb{R}^{2\times 2}_{sym}$. In our study, we consider all possible compatible configurations for three wells and prove that if there exist two matrices in $\mathcal{U}$ that are rank-one compatible then $Q^e(\mathcal{U})$ coincides with its symmetric lamination convex hull $L^e(\mathcal{U})$. We complete this result by providing an explicit characterization of $L^e(\mathcal{U})$ in terms of the wells in $\mathcal{U}$. Finally, we discuss the optimality of our outer bound and its relationship with quadratic polyconvex functions.