On computational homogenization of gradient crystal inelasticity
Crystal inelasticity is the main modeling technique employed to study the mechanical behavior of polycrystalline metallic materials. This class of models has the capability to represent micromechanical phenomena such as plastic slip, grain boundary interactions, and dislocation pile-up. An extended class of models, known as gradient crystal inelasticity, can also be used to predict size dependence of the grains, which is a property that has been observed experimentally. An increased understanding, in terms of the challenges in modeling polycrystalline materials, could aid in reducing costs and resources needed to determine the properties of these type of materials experimentally.
The length scales that are characteristic for typical engineering applications and the length scale of the underlying microstructure often differ by many orders of magnitude. As a consequence, it is not computationally feasible to fully resolve the model at a fine enough scale to capture the microstructural characteristics. Instead, computational homogenization is a suitable framework for modeling structures exhibiting these scale separations. Homogenization allows for bridging the microstructural to the effective properties that pertain to the (structural) scale of engineering interest.
In this work, different modeling aspects of gradient crystal inelasticity and their modeling capabilities, in a computational homogenization setting, are investigated. In particular, two variational formats are compared, specifically in terms of convergence rate with respect to mesh refinements, and the effect of applying certain boundary conditions. Furthermore, it is shown that certain effective properties (properties for sufficiently large microscopic models, called Representative Volume Elements) can be bounded from above and below based on simulations performed on finite size models (Statistical Volum Elements), that are amenable to simulation. The bounding property can be used towards estimating how large microscopic models that are needed to produce accurate results in the computational homogenization analysis. Several numerical examples, applied to both two and three-dimensional models, are given, demonstrating the validity of the theoretically made predictions.