Revisiting Andrews method and grain boundary resistivity from a computational multiscale perspective

Abstract

The effective material response as observed at a macro level is a manifestation of the material microstructure and lower scale processes. Due to their distinct atomic arrangement, compared to bulk material, grain boundaries significantly affect the electrical properties of metals. However, a scale-bridging understanding of the associated microstructure–property relation remains elusive so that phenomenological approaches such as the Andrews method are typically applied. In the present contribution we revisit Andrews method from a computational multiscale perspective to analyse its limits and drive concepts to go beyond. By making use of homogenisation techniques we provide a solid theoretical foundation to the Andrews method, discuss its applicability and tacit assumptions involved, and resolve its core limitations. To this end, simplistic analytical examples are discussed in a one-dimensional setting to show the fundamental relation between the Andrews method and homogenisation approaches. Building on this knowledge the importance of the underlying microscale morphology and associated morphology-induced anisotropies are in the focus of investigations based on simplified microstructures. Concluding the analysis, scaling laws for isotropic microstructures are derived and the transferability of the results to realistic, (quasi-)isotropic polycrystals is shown.