First-passage fracture probability in stochastic phase-field models

Abstract Predicting fracture probability from an evolving damage field requires an explicit event definition. In the underlying probability space, we define fracture as the set of realizations whose damage trajectories first reach a critical surface by a given time, establishing a first-passage theory within a finite-element stochastic AT2 phase-field model. Mesoscopic fluctuations follow a reflected Langevin process: reflecting boundaries enforce admissible damage bounds, while an absorbing boundary imparts event irreversibility. The corresponding Fokker–Planck equation establishes a mass–flux closure where lost survival mass gives cumulative fracture probability, outward absorption flux gives its evolution rate, and the flux-to-survival ratio defines the effective hazard rate. Conditioning and averaging over exogenous material, geometric, and loading uncertainties yield a survival-weighted population hazard. Seven numerical cases verify probability conservation, delineate quasi-stationary and Kramers regimes, demonstrate one- and two-dimensional mechanical coupling, and jointly calibrate mobility and noise intensity from first-passage and intact-state relaxation data. In a loading–unloading example, cumulative fracture probability remains at 0.3809 despite a nearly zero post-unloading tail probability. This framework unifies fracture as a load-driven probability evolution, intrinsically linking spatial damage dynamics to path-dependent fracture probability.