Finite J-Integral Formulation at the Atomic Scale and Breakdown of Continuum Fracture Mechanics

In this work, a finite energy-based formulation of the J-integral is proposed and applied to atomic-scale defects to investigate the breakdown of continuum fracture mechanics. The J-integral is evaluated through molecular statistics simulations as the potential energy difference between two identically deformed configurations with neighbouring crack lengths, where the finite crack advance corresponds to a single atomic bond break. Singleedge cracked single-crystal silicon specimens are analyzed by progressively reducing the specimen width from the macroscale to approximately 10 nm. The results show that the critical atomistic J-integral remains essentially constant (2.5 J/m2 ) across all considered sizes, demonstrating scale independence and confirming that localized bond-breaking events govern brittle fracture. The spatial extent of the fracture process zone is quantified through atomic displacement fields and is found to be approximately constant (≈0.5 nm), independent of specimen size. In contrast, the conventional continuum J-integral progressively deviates from the atomistic value when the specimen width approaches the characteristic size of the fracture process zone, indicating the breakdown of the infinitesimal crack-extension assumption. The proposed formulation provides a simple and computationally efficient framework for extending fracture mechanics concepts to the atomic scale and clarifies the physical origin of the breakdown of continuum-based fracture mechanics.