Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level‐set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase‐field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time‐dependent crack propagation problems. In line with diffusive damage of the phase‐field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction‐diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant‐Friedrichs‐Lewy condition. In this article, three numerical examples, namely, a L‐shaped section, a rectangular slab with predefined cracks, and an all‐ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase‐field fracture context.

中文翻译：

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使用相场断裂模型的水平集拓扑优化，以使脆性材料的断裂强度最大化

断裂是脆性材料中最常见的破坏模式之一。它会大大降低材料的完整性和结构强度。为解决此问题，我们提出了一种基于水平集（LS）的拓扑优化程序，该优化程序可在受体积约束的情况下优化基体材料中增强夹杂物的分布，以使结构的抗断裂性最大化。本文建立了一个相场断裂模型来模拟裂纹的萌生和扩展，其中开发了一种交错算法来解决此类随时间变化的裂纹扩展问题。符合相场方法对裂缝的扩散性破坏；为断裂力学问题导出了拓扑导数，这些拓扑导数为LS框架中的拓扑优化提供了梯度信息。采用反应扩散方程来更新有限元框架内的LS函数。这样可以通过使用Courant-Friedrichs-Lewy条件克服时间步的限制来避免重新初始化。在本文中，给出了三个数值示例，分别是L形截面，具有预定裂纹的矩形平板和全瓷镶嵌牙桥（即固定局部义齿），以证明基于LS的提议的有效性拓扑优化可增强相场断裂环境下多材料复合结构的抗断裂性能。