Localization in heterogeneous materials: A variational approach and its application to polycrystalline solids

In the case when macroscopic instability in a heterogeneous material is defined as the loss of rank-one convexity of the homogenized tangent modulus (namely, the propensity to develop bands of localization), the stability domain is the one of positive definiteness of a potential function on a representative volume element defining the unit cell of the microsctructure. It is possible to bound this domain by minimizing the functional on a set of kinematically admissible velocity fields which intend to catch the localization of deformation in the microstructure.The approach is applied to polycrystal plasticity in a bidimensional geometry: the plane double-slip model of Asaro is retained for the local behavior. The stability domain of a polycrystalline aggregate, schematized by a bidimensional pavement, is displayed as a function of the heterogeneity of crystal orientations, latent hardening and stress distributions. The localization velocity field is estimated in each case. In particular, the destabilizing effect of the coupling between shear strain and rotation at the microscopic level is exhibited.