Heterogeneous crystal and poly-crystal plasticity modeling from a transformation field analysis within a regularized Schmid law

The regularized Schmid law (RSL) has recently been proposed as a plastic flow criterion for poly-crystals under the crude assumptions of either uniform stress or uniform strain. We first reconsider this law for application to heterogeneous intra-crystalline plasticity, with reference to a Homogeneous Equivalent Super-Crystal. We then extend the modeling to poly-crystals with the goal to account for both stress and strain heterogeneities within as well as between grains. The transformation field analysis (TFA) is used as the homogenization procedure. This TFA is known to be accurate for materials that can be described as assemblies of plastically homogeneous domains. Otherwise, the estimates of the material effective behavior that result from its application are too stiff. Because stress and strain fields are almost everywhere uniform in laminates, we consider crystal slip organizations into multi-laminate structures. It is demonstrated that laminate layers either parallel to slip planes or normal to slip directions do not contribute to the over-stiffness due to the TFA. Thus, hierarchical multi-laminate (HML) structures are introduced where the successive laminate orientations are taken parallel to the crystal slip planes. It is shown that a conveniently weighted superposition of all the possible plane hierarchies cancels out most of the undesirable TFA contributions to the overall stiffness estimates. A relevant extension to poly-crystal plasticity of this (RSL-TFA-HML) modeling is presented.