A strain gradient elastodynamic phase field model for dynamic fracture in heterogeneous materials by multiscale asymptotic homogenization

The macroscopic fracture strength and crack propagation behavior of heterogeneous materials are strongly governed by their underlying microstructures. When the characteristic scale of microstructures becomes comparable to that of the macroscopic structure, strain rate and strain gradient effects play a crucial role in dynamic fracture. In this work, we extend our recently proposed quasi-static strain gradient phase field model to the dynamic regime. The framework consistently integrates microstructural size, strain gradient, strain rate, and phase field evolution within a unified variational energy formulation, and provides mathematically consistent definitions of higher-order constitutive tensors without relying on empirical assumptions. A penalty-based computational strategy is adopted to ensure numerical efficiency while avoiding higher-order continuity requirements. Benchmark simulations demonstrate that the model accurately captures key features of dynamic fracture, including crack branching and propagation velocity, with excellent mesh objectivity. Applications to fiber-reinforced composites and porous materials further confirm the model’s capability to reproduce fracture anisotropy, porosity-dependent dynamic fracture behavior, and experimentally observed damage zones and free-surface velocity histories in spallation problems.