Abstract Two-dimensional materials with self-similar second-order hierarchy are considered. The microstructure with fourfold cubic symmetry is generated by a periodic system of voids in brittle parent material, in one case, macrovoids are filled by a microvoided phase, and in the second one, this phase surrounds hollow macrovoids. A crack is embedded in a rectangular periodic domain subjected to the K-field boundary conditions, and the fracture toughness is determined by the analysis of stresses in the crack-tip vicinity. A novel approach based on the discrete Fourier transform reduces the computational cost of the problem. The analysis of the domain, consisting of several hundreds repeating patterns, is reduced to the multiple analysis of a single representative cell in the Fourier transform space. A parametric study is carried out and, in particular, the influence of parent material redistribution between the hierarchical levels is examined. It is found that for the material with filled macrovoids increasing the relative stiffness at the second hierarchical level diminishes the fracture toughness, while for the second layout the effect is opposite. A comparison with non-hierarchical voided materials of the same relative density emphasizes the role of the length scale parameter of a layout.

中文翻译：

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自相似分级材料的断裂韧性

摘要 考虑具有自相似二阶层次的二维材料。具有四重立方对称性的微观结构是由脆性母材中的空隙周期系统产生的，在一种情况下，大空隙被微空隙相填充，而在第二种情况下，该相围绕中空的大空隙。裂纹嵌入在受 K 场边界条件影响的矩形周期域中，断裂韧性由裂纹尖端附近的应力分析确定。一种基于离散傅立叶变换的新方法降低了问题的计算成本。由数百个重复模式组成的域分析被简化为傅立叶变换空间中单个代表性单元的多重分析。进行了参数研究，并且，特别是，检查了层级之间母材再分配的影响。发现对于填充大孔洞的材料，在第二层次增加相对刚度会降低断裂韧性，而对于第二个布局，效果相反。与具有相同相对密度的非分层空隙材料的比较强调了布局的长度尺度参数的作用。