RANK-ONE PLUS A NULL-LAGRANGIAN IS AN INHERITED PROPERTY OF 2-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 two dimensions). The purpose of t his paper is to prove that the effective compliance tensor has the sam e representation: rank-one plus the null-Lagrangian. This statement ge neralises the well-known result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropi c and shares the same shear modulus. It also generalises the surprisin g discovery of Avellaneda et al. 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 p lacing it in a more general framework and using some elliptic PDE theo ry rather than the translation method. Our results allow us to calcula te the polycrystalline G-closure of the special class of crystals unde r consideration. Our analysis is contrasted with a two-dimensional mod el problem for shape-memory polycrystals. We show that the two problem s can be thought of as 'elastic percolation' problems, one elliptic, o ne hyperbolic.