A strain gradient phase field model for heterogeneous materials based on two-scale asymptotic homogenization

Due to the inherent microstructural heterogeneity of heterogeneous materials, their macroscopic fracture behavior differs significantly from that of homogeneous materials, exhibiting phenomena such as anisotropic fracture energy and strain gradient effects. To investigate the effect of microstructure on macroscopic fracture behavior, this study proposes a novel multiscale phase field model. Based on the theory of two-scale asymptotic expansion, the model constructs an equivalent multi-field coupled boundary value framework, which includes both a strain gradient elasticity submodel and a homogenized phase field submodel. Through rigorous mathematical derivation, homogenized tensors that characterize the elastic constitutive relations and fracture properties are obtained without relying on any additional assumptions. Moreover, to distinguish the contributions of load components to crack propagation, energy decomposition strategies based on orthogonal projection are introduced for stress and higher-order stress. Compared to full-scale simulations, the proposed model significantly reduces computational cost while maintaining accuracy. Numerical simulations show that the model accurately captures the influence on crack propagation direction induced by microstructure. Additionally, the model effectively demonstrates the hindering effect of strain gradients on crack propagation, offering new insights into the size effect in the fracture of heterogeneous materials. This work provides a new framework for studying the multiscale fracture behavior of heterogeneous materials.