Brittle fracture in polycrystalline microstructures with the extended finite element method

A two-dimensional numerical model of microstructural effects in brittle fracture is presented, with an aim towards the understanding of toughening mechanisms in polycrystalline materials such as ceramics. Quasi-static crack propagation is modelled using the extended finite element method (X-FEM) and microstructures are simulated within the framework of the Potts model for grain growth. In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Hence, crack propagation can be simulated without any user-intervention or the need to remesh as the crack advances. The microstructural calculations are carried out on a regular lattice using a kinetic Monte Carlo algorithm for grain growth. We present a novel constrained Delaunay triangulation algorithm with grain boundary smoothing to create a finite element mesh of the microstructure. The fracture properties of the microstructure are characterized by assuming that the critical fracture energy of the grain boundary (G^gb_c) is different from that of the grain interior (Gⁱ_c). Numerical crack propagation simulations for varying toughness ratios G^gb_c/Gⁱ_c are presented, to study the transition from the intergranular to the transgranular mode of crack growth. This study has demonstrated the capability of modelling crack propagation through a material microstructure within a finite element framework, which opens-up exciting possibilities for the fracture analysis of functionally graded material systems.