Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenization

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in 2D). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalizes the well-known result of Hill [14, 15] that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalizes the surprising discovery in [1] that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closures of the special class of crystals under consideration. Our analysis is contrasted with a two dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as “elastic percolation” problems, one elliptic, one hyperbolic.