Critical Manifolds in Polycrystalline Grain Structures

With the development of new, fully three-dimensional metallographic techniques, there is considerable interest in characterizing three-dimensional microstructures in ways that go beyond twodimensional stereology. One characteristic of grain structures is the surface of lowest energy across the microstructure, termed the critical manifold (CM). When the grain boundaries are sufficiently weak, the CM lies entirely on grain boundaries, while when the grain boundaries are strong, cleavage occurs. A scaling theory for the cleavage to intergranular transition of CMs is developed. We find that a critical length scale exists, so that on short length scales cleavage is observed, while at long length scales the CM is rough. CMs for realistic polycrystalline grain structures, determined by a network optimization algorithm, are used to verify the analysis.