Restriction on microstructures for n affine wells in two dimensional linear elasticity
Antonio Capella, Lauro Morales · arXiv (Cornell University) · 2020
We study some particular cases of the $n$-well problem in two-dimensional linear elasticity. Assuming that all wells in $\mathcal{U}\subset\mathbb{R}^{2\times 2}_\text{sym}$ belong to the same 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 the subsets of three wells in $\mathcal{U}$. In the case of three wells such that one pair of wells is rank-one compatible, we show that every young measure limit of linear strains supported on $\mathcal{U}$ has its barycenter in the set $L^e(\mathcal{U})$. We extend this result to a family of four wells where two pairs of wells are rank-one compatible. The proofs are constructive and some explicit examples are presented.