A graph-theoretic approach for multiscale modeling and prediction of crack propagation in polycrystalline materials

Abstract

A multiscale graph theory-based approach is introduced here to predict the microscale crack path in polycrystalline materials. The crack path is represented as the boundary of the partition of a geometric graph. The partitioning is carried out by optimizing an Ising-type hamiltonian. The hamiltonian parameters are chosen such that each partition cost is the same as the energy of the corresponding crack. The interplay of the loading conditions on the specimen and the microstructure of the material near the crack tip determines the crack growth angle in polycrystalline materials. Two different length scales of macro and micro is incorporated for the crack path by defining the crack total energy as the summation of macroscopic energy release and microscopic surface energy. The former term represents the macroscopically favorable crack growth angle by directing the crack to propagate from the crack tip along the direction of the maximum energy release. The latter term guides the microscopic crack path along the macroscopically preferred direction. At this scale, the crack path naturally accommodates both the intergranular and transgranular fractures. In the case of intergranular fracture, the crack propagates along the grain boundaries, while in the case of transgranular fracture, the crack propagates along the crystallographic cleavage planes. The mixed-mode fracture in a thin foil specimen is studied, and the effect of the dihedral angle of the 2D crack is included in defining the effective surface energy. The model is validated using the analytical results for mixed-mode fracture in an isotropic medium and mode-I fracture in a medium with a preferred crack direction. The proposed method can be used to design materials microstructure with optimal fracture resistance.