Homogenization towards a grain-size dependent plasticity theory for single slip

Size dependent behavior is observed in polycrystalline metals where the yield stress follows an inverse power relation with the grain size. This phenomenon, commonly known as the Hall–Petch effect, is attributed to the resistance at grain boundaries constraining dislocation motion. Classical continuum models cannot capture this phenomenon. A remedy is to adopt higher-order crystal plasticity formulations that model the interfacial behavior with non-standard boundary terms. However, such a fine-scale approach is computationally expensive for large problems. In this paper, a homogenization theory is proposed showing how a crystal plasticity model with one slip system translates consistently into the macroscopic scale. For simplicity, we consider only uniform macroscopic shear and show that the microstructural properties (intrinsic length scale, characteristic grain size and surface modulus) manifest themselves at the macroscopic scale, thus capturing the grain mechanics in an efficient manner.