On the symmetric lamination convex and quasiconvex hull for the coplanar n-well problem in two dimensions

TL;DR

This paper characterizes the symmetric lamination convex hull for specific well configurations in 2D elasticity, revealing when it coincides with the quasiconvex hull and providing explicit examples.

Contribution

It provides a characterization of the symmetric lamination convex hull for certain well sets in 2D elasticity, including explicit examples and conditions for hull equivalences.

Findings

Symmetric lamination convex hull characterized for n-well sets in 2D.

For certain four-well configurations, lamination convex and quasiconvex hulls coincide.

Explicit examples illustrating the hull structures and their relations.

Abstract

We study some particular cases of the $n$-well problem in two-dimensional linear elasticity. Assuming that every well in $\mathcal{U}\subset\mathbb{R}^{2\times 2}_\text{sym}$ belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull $L^e(\mathcal{U})$ for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in $\mathcal{U}$. For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly contained in its convex hull $C(\mathcal{U})$. We extend this result to some particular configurations of $n$ wells. Most of the proofs are constructive, and we also present explicit examples.