A stochastic approach for grain boundary toughness in phase-field fracture modeling

Fracture is fundamentally an atomic scale phenomenon. Grain-boundary structure, solute embrittlement, and grain-boundary defects can all impact crack propagation, but explicitly including atomic scale effects in models designed to study fracture at the micron or higher length scale is challenging. In contemporary phase-field fracture methods, it is common to assume an average critical energy release rate for a material domain; however, this approach has deleterious consequences for accuracy since crack propagation is not determined by local average toughness, but instead by a weakest link. We develop a phase-field fracture methodology where local critical energy release rates are input stochastically at grain boundaries in a polycrystal to implicitly include atomic scale effects. We perform multiple mode I fracture simulations to determine distributions of critical stress intensity K_c, strain energy density ψ, and maximum tensile stress σ_max. The mean, standard deviation, initial crack position, and mesh size were varied to determine their effects. We generally found that the mean and variance of the output distribution of K_c correlated strongly with the mean and variance of the input distribution for critical energy release rate. Ultimately, in 83.8% of simulations, for reasonable expectations of the mean and variance of the critical energy release rate, locally stochastic fracture properties result in lower fracture toughness compared to models with uniform grain-boundary properties due to this weakest-link behavior. This validates the ability of the method to capture the extreme-value nature of fracture and demonstrates the importance of incorporating distributions of the critical energy release rate into simulations, not just the mean.