Representative and statistical volume elements for grain boundary networks: A stereological approach

Abstract

Experimental and computational studies have demonstrated that the structure of grain boundary networks (GBNs), e.g. the connectivity of certain types of grain boundaries (GBs), has a significant impact on macroscopic material properties. Because minor changes in GBN structure can cause large changes in effective properties, it is common to use large microstructures or report averages over many microstructural instantiations with the assumption that results become more representative as the size or number of samples increases. For crystallographic texture, and composite materials, such ideas have been made rigorous through the definition of representative volume elements (RVEs) and statistical volume elements (SVEs). However, for GBNs, the size of an RVE and the cardinality of a set of SVEs have not yet been determined. In this study, we employ stereological methods to evaluate the convergence of triple junction fractions in both idealized and realistic 2D GBNs to quantitatively define RVEs and SVE sets for GBNs. We compare these results to those for crystallographic texture in the same polycrystals and find that the trends for GBNs and texture are remarkably different. Because the vast majority of experimental work currently relies on 2D microstructure characterization, the results obtained for 2D systems have great practical value in and of themselves. However, the stereological approach employed also allows us to make quantitative predictions of RVEs and SVEs for fully 3D microstructures. These results are expected to aid future experimental and computational work in selecting appropriately sized material volumes to achieve robust quantitative results.