Asymptotic homogenization of phase-field fracture model: An efficient multiscale finite element framework for anisotropic fracture

The intractable multiscale constitutives and the high computational cost in direct numerical simulations are the bottlenecks in fracture analysis of heterogeneous materials. In an attempt to achieve a balance between accuracy and efficiency, we propose a mathematically rigorous phase-field model for multiscale fracture. Leveraging the phase-field theory, the difficulty of discrete-continuous coupling in conventional cross-scale crack propagation analysis is resolved by constructing a continuum description of the crack. Based on the asymptotic expansion, an equivalent two-field coupled boundary-value problem is well-defined, from which we rigorously derive the macroscopic equivalent parameters, including the equivalent elasticity tensor and the equivalent fracture toughness tensor. In our approach, both the displacement field and the phase-field are simultaneously expanded, allowing us to obtain a fracture toughness tensor with diagonal elements of the corresponding matrix controlling anisotropic fracture behavior and non-diagonal elements governing crack deflection. This enables multiscale finite element homogenization procedure to accurately reproduce microstructural information, and capture the crack deflection angle in anisotropic materials without any a priori knowledge. From the numerical results, the proposed multiscale phase-field method demonstrates a significant reduction in computation time with respect to full-field simulations. Moreover, the method accurately reproduces physical consistent anisotropic fracture of non-centrosymmetric porous media, and the experimentally consistent damage response of fiber-reinforced composites. This work fuses well-established mathematical homogenization theory with the cutting-edge fracture phase-field method, sparking a fresh perspective for the fracture of heterogeneous media.