A second-order strain gradient fracture model for the brittle materials with micro-cracks by a multiscale asymptotic homogenization

In this work, applying a second-order multiscale asymptotic homogenization, an effective fracture model is established for the brittle materials with periodic distribution of micro-cracks. The novel second-order strain gradient fracture model based on the multiscale asymptotic technique is rigorously derived without any phenomenological assumptions, and the fourth-, sixth-, and eighth-order effective elastic tensors of the fracture criterions are obtained by the first-order and second-order multiscale unit cell functions. The significant features of the novel model are: (i) the first-order, second-order strain gradient effect and microstructure size ξ included in the fracture criterion and (ii) the strain energy and the Griffith criterion for micro-crack extensions obtained by the high-order multiscale asymptotic homogenization. Finally, the effectiveness of the proposed model is compared with the direct numerical simulations (DNS), experimental data and some typical fracture problems including Mode I crack plate, rectangular plate with two symmetric V-notch and a holed plate are also evaluated. These examples show that the second-order strain gradient fracture model is valid for solving the brittle materials with periodic distribution of micro-cracks.