Morphology of fracture profiles and toughness: competition between inter and transgranular fracture in two dimensional brittle solids

Two dimensional intergranular brittle cracks propagating through a microstructured material produce fracture profiles which, at scales larger than the microstructural length scale, are anti-persistent and close to directed random walks with Hurst exponent similar to 0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 0.5$$\end{document}. The extent of intergranularity is controlled by the ratio of the toughness of the grain boundaries to that of the grain interior. However, experiments suggest [e.g. Ponson et al. (Phys Rev Lett 97(12), 2006)] that even when transgranular crack propagation is possible, the fracture profile is still close to a random walk. In this work, generating fracture profiles in a material with an idealised honeycomb microstructure using a phase field based model of crack propagation, we show that the competition between inter and transgranular fracture manifests in a manner that is more nuanced than what the experiments suggest. While the fracture profile is indeed always anti-persistent, transgranularity resulting from toughening the grains leads to profiles that can have roughness exponents much lower than 0.5. Moreover, in such cases, the overall toughness of the specimen scales with the Hurst exponent. On the other hand, transgranularity resulting from weakening the grain boundaries, without changing the toughness of the grain interior, always lead to fracture profiles close to the random walk.