Restrictions on microstructure
Kaushik Bhattacharya, Nikan B. Firoozye, Richard D. James, Robert V. Kohn · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1994
We consider the following question: given a set of matrices with no rank-one connections, does it support a nontrivial Young measure limit of gradients? Our main results are these: (a) a Young measure can be supported on four incompatible matrices; (b) in two space dimensions, a Young measure cannot be supported on finitely many incompatible elastic wells; (c) in three or more space dimensions, a Young measure can be supported on three incompatible elastic wells; and (d) if supports a nontrivial Young measure with mean value 0, then the linear span of must contain a matrix of rank one.