Modelling dynamic brittle fracture with fourth-order phase-field integrated Eulerian SPH

Dynamic brittle fracture presents substantial numerical challenges due to the complex nature of crack initiation, propagation, branching, and fragmentation. In this work, we develop a fourth-order phase-field model for brittle fracture within the Eulerian Smoothed Particle Hydrodynamics (ESPH) framework. The use of higher-order spatial derivatives in the phase-field formulation enables enhanced resolution of crack topology, stable interfaces and smoother energy dissipation. The ESPH method, operating in the current configuration, is particularly suited for modelling large deformations and complex fracture behaviours without the need for remeshing, which might be required for mesh-based methods. We validate our model against several benchmark problems, such as dynamic crack branching in notched plates under tensile loading and asymmetric crack propagation in three-point bending tests. The results highlight the capability of the proposed fourth-order ESPH-phase-field model to accurately predict crack paths, branching, and coalescence phenomena with improved interface regularity and numerical robustness.