In this paper, a recently-developed phase-field damage model is incorporated into the topology optimization framework to take into account crack initiation and propagation in a path-dependent fashion. The proposed topological design can enhance fracture resistance of structures made of brittle materials such as advanced ceramics. For the first time, a path-dependent shape derivative is developed in a step-wise manner during the nonlinear fracture analysis, which enables to drive the topology optimization properly. To measure the fracture resistance of structure, a p-norm function is formulated to aggregate the phase-field variables into a single constraint. To demonstrate the effectiveness of the presented topology optimization procedure, three 2D benchmark examples with single-phasic material and one 3D biomedical example with biphasic materials are studied here. The comparison with the topological designs based upon conventional linear elastic finite element analysis without the damage model indicates that the proposed method can significantly improve the fracture resistance of structures with more efficient use of materials. The proposed method is anticipated to provide an effective approach for sophisticated path-dependent topological design of structures reducing severe stress concentration and high risks of fracture failure.

在本文中，最近开发的相场损伤模型被合并到拓扑优化框架中，以考虑裂纹萌生和以路径相关的方式传播。拟议的拓扑设计可以增强由脆性材料（如高级陶瓷）制成的结构的抗断裂性。首次在非线性断裂分析过程中逐步开发出与路径相关的形状导数，从而能够正确地驱动拓扑优化。为了测量结构的抗断裂强度，p-norm函数被公式化为将相场变量聚合为一个约束。为了证明所提出的拓扑优化程序的有效性，在此研究了三个使用单相材料的2D基准示例和一个使用双相材料的3D生物医学示例。与基于常规线性弹性有限元分析的拓扑设计（不带损伤模型）的比较表明，所提出的方法可以通过提高材料使用效率来显着提高结构的抗断裂性能。预期所提出的方法将为结构的复杂的路径依赖的拓扑设计提供有效的方法，从而减少严重的应力集中和高的断裂破坏风险。