Non-equilibrium nature of fracture determines the crack path

Local processes during fracture, such as the selection of crack paths and the formation of facets and kinks along the paths, are non-equilibrium and discrete in nature and cannot be resolved in the framework of continuum thermodynamics. Atomistic models using existing empirical or machine-learning force fields also fail to offer satisfactory solutions. The problem is addressed here by developing a high-fidelity neural network-based force field (NN-F³) following an active-learning approach that covers the space spanned by strain states up to material failure and the non-equilibrium, intermediate states of atomic-level structures during cracking. Atomistic simulations using NN-F³ identify spatial complexities from lattice-scale kinks to sample-scale crack patterns in 2D crystals such as graphene and h-BN. The selection of crack paths is found to be associated with alternation in the stress intensity factors and anisotropy in the fracture toughness, which defines the roughness of cleaved edges. We find that the non-uniform lattice distortion and undercoordination of cleaved edges at the crack front play critical roles in the fracture process, which cannot be described by the energy densities of relaxed edges as widely used in the literature. Instead, the fracture patterns, the critical stress intensity factors corresponding to the lattice kinks, and the energy densities of edges in the intermediate, unrelaxed states offer reasonable measures for the fracture toughness and its anisotropy, which can be determined from experiments or simulations with the atomic-level resolution and well explain the experimental observation.