Lightweight topology optimization of cracked structures based on the mixed-mode brittle fracture criterion

Engineering structures are prone to microcracking due to the influence of manufacturing processes or environmental conditions, while accurately simulating crack behavior remains complex. Consequently, the topological optimization of cracked structures presents a significant challenge in structural design. This paper proposes a topology optimization method based on the brittle fracture criterion for mixed-mode cracks, aimed at optimizing cracked structures. An optimization model is established to minimize volume while satisfying constraints on strength, stiffness, and crack resistance. The extended finite element method (XFEM) is employed to achieve high-accuracy simulations of crack behavior. Moreover, the sensitivity expressions for the stress intensity factor across different fracture modes are derived, and an innovative solution to the lightweight design problem of structures exhibiting mixed-mode fracture modes under a given load is presented. A comparative study of the lightweight design under different working conditions, including the presence or absence of cracks, is carried out using four benchmark cases and an engineering structure example. The influence of crack parameters such as length, location, and size on the optimized topology is also investigated. The results of numerical examples demonstrate that this proposed method provides an effective framework for generating topological structures that meet diverse requirements for stiffness, strength, and crack resistance.