It is now a well-established fact that even simple topology variations can drastically change the fracture response of structures. With the objective of gaining quantitative insight into this phenomenon, this paper puts forth a density-based topology optimization framework for the fracture response of structures subjected to quasistatic mechanical loads. One of the two key features of the proposed framework is that it makes use of a complete phase-field fracture theory that has been recently shown capable of accurately describing the nucleation and propagation of brittle fracture in a wide range of nominally elastic materials under a wide range of loading conditions. The other key feature is that the framework is based on a multi-objective function that allows optimizing in a weighted manner: ($i$) the initial stiffness of the structure, ($ii$) the first instance at which fracture nucleates, and ($iii$) the energy dissipated by fracture propagation once fracture nucleation has occurred. The focus is on the basic case of structures made of a single homogeneous material featuring an isotropic linear elastic behavior alongside an isotropic strength surface and toughness. Novel interpolation rules are proposed for each of these three types of material properties. As a first effort to gain quantitative insight, the framework is deployed to optimize the fracture response of 2D structures wherein the fracture is bound to nucleate in three different types of regions: within the bulk, from geometric singularities (pre-existing cracks and sharp corners), and from smooth parts of the boundary. The obtained optimized structures are shown to exhibit significantly enhanced fracture behaviors compared to those of structures that are optimized according to conventional stiffness maximization. Furthermore, the results serve to reveal a variety of strengthening and toughening mechanisms. These include the promotion of highly porous structures, the formation of tension-compression asymmetric regions, and the removal of cracks and sharp corners. The particular mechanism that is preferred by a given structure, not surprisingly, correlates directly to the elastic, strength, and toughness properties of the material that is made of.

现在一个公认的事实是，即使是简单的拓扑变化也能极大地改变结构的断裂响应。为了定量了解这一现象，本文提出了一个基于密度的拓扑优化框架，用于准静态机械载荷下结构的断裂响应。所提出的框架的两个关键特征之一是它利用了一个完整的相场断裂理论，该理论最近被证明能够准确地描述在广泛的名义弹性材料中脆性断裂的成核和传播。负载条件范围。另一个关键特征是该框架基于允许以加权方式优化的多目标函数：（$我$) 结构的初始刚度, ($我我$) 断裂成核的第一个实例，和 ($我我我$) 一旦发生断裂成核，断裂传播所耗散的能量。重点介绍由单一均质材料制成的结构的基本情况，该材料具有各向同性的线性弹性行为以及各向同性的强度表面和韧性。针对这三种类型的材料特性中的每一种都提出了新的插值规则。作为获得定量洞察力的第一项努力，该框架被部署以优化二维结构的断裂响应，其中断裂必然在三种不同类型的区域中成核：在体积内，从几何奇异点（预先存在的裂缝和尖角） ), 以及边界的平滑部分。与根据传统刚度最大化优化的结构相比，所获得的优化结构显示出显着增强的断裂行为。此外，结果有助于揭示各种强化和增韧机制。其中包括促进高度多孔结构、形成拉伸-压缩不对称区域以及去除裂纹和尖角。毫不奇怪，给定结构优选的特定机制直接与制成材料的弹性、强度和韧性特性相关。拉压不对称区域的形成，裂纹和尖角的去除。毫不奇怪，给定结构优选的特定机制直接与制成材料的弹性、强度和韧性特性相关。拉压不对称区域的形成，裂纹和尖角的去除。毫不奇怪，给定结构优选的特定机制直接与制成材料的弹性、强度和韧性特性相关。