Rigidity and lack of rigidity for solenoidal matrix fields

created on 11 Jun 2002
modified by garroni on 24 Jan 2006

[BibTeX]

Published Paper

Inserted: 11 jun 2002
Last Updated: 24 jan 2006

Journal: Proc R Soc London A
Volume: 460
Number: 2046
Pages: 1789-1806
Year: 2004

Abstract:

We study the problem of identifying conditions under which a divergence free matrix field takes values in some prescribed sets of matrices ${\cal K}$. We treat in detail the case when ${\cal K}$ is made of two or three matrices. Our results are parallel to those on curl free matrices. In that case Ball and James showed rigidity when ${\cal K}$ is made of two matrices and Tartar proved lack of rigidity when ${\cal K}$ is made of four matrices. For our problem we prove rigidity when ${\cal K}$ is made of two matrices and lack of rigidity when is made of three.

We give examples when the differential constraints are yet of a different type and present some applications to composites.

Keywords: microstructures