Numerical implementation of crack propagation remains an open question, being the subject of numerous previous studies and proposed numerical methods. The numerical manifold method (NMM) is one approach enabling modelling of displacement discontinuities across crack surfaces by employing two independent grid systems (or 'covers'). In this study, building on NMM, particle-grid mapping in the material point method (MPM) is interpreted in a manifold-consistent manner. Distinct from the conventional one-way information transfer from the mathematical cover to the physical cover within the Eulerian view in NMM, the MPM framework for crack propagation is constructed under a hybrid Eulerian-Lagrangian view and features bidirectional information transfer between particles and grids. On this basis, rocks containing closed flaws are discretized by material points, and a two-stage contact algorithm is used to capture the interfacial contact behaviour. Modelling crack propagation is realized through a conversion mechanism governed by the Drucker-Prager yield condition.Comparisons between numerical results and experimental observations indicate that the proposed method effectively predicts the development of cracks. This study provides a rigorous mathematical interpretation for the background grid and material points in MPM from the perspective of NMM and offers a novel strategy for the numerical modelling of crack propagation within NMM.
Wu et al. (2026) studied this question.