Brittle solids are often toughened by adding a second-phase material. This practice often results in composites with material heterogeneities on the meso scale: large compared to the scale of the process zone but small compared to that of the application. The specific configuration (both geometrical and mechanical) of this mesoscale heterogeneity is generally recognized as important in determining crack propagation and, subsequently, the (effective) toughness of the composite. Here, we systematically investigate how dynamic crack propagation is affected by mesoscale heterogeneities taking the form of an array of inclusions. Using a variational phase-field approach, we compute the apparent crack speed and fracture energy dissipation rate to compare crack propagation under Mode-I loading across different configurations of these inclusions. If fixing the volume fraction of inclusions, matching the inclusion size to the K-dominance zone size gives rise to the best toughening outcome. Conversely, if varying the volume fraction of inclusions, a lower volume fraction configuration can lead to a better toughening outcome if and only if the inclusion size approaches from above the size of the K-dominance zone. Since the size of the K-dominance zone can be estimated \textit{a priori} given an understanding of the application scenario and material availability, we can, in principle, exploit this estimation to design a material's mesoscale heterogeneity that optimally balances the tradeoff between strength and toughness. This paves the way for realizing functional (meta-)materials against crack propagation in extreme environments.

翻译：脆性固体通常通过添加第二相材料来增韧。这一做法常导致复合材料在介观尺度上产生非均匀性：该尺度虽远大于过程区尺寸，但远小于应用尺度。这种中尺度非均匀性的具体构型（包含几何构型与力学特性）被普遍认为是决定裂纹扩展行为进而影响复合材料（等效）韧性的关键因素。本文系统研究了以夹杂物阵列形式存在的中尺度非均匀性对动态裂纹扩展的影响规律。通过变分相场方法，我们计算了表观裂纹扩展速度与断裂能量耗散率，以比较I型加载条件下不同夹杂物构型对裂纹扩展的影响。研究表明：当保持夹杂物体积分数恒定时，使夹杂物尺寸与K主导区尺寸相匹配可获得最优增韧效果；反之，当改变夹杂物体积分数时，仅当夹杂物尺寸从高于K主导区尺寸的方向逼近该特征尺寸时，低体积分数构型才能产生更优增韧效果。由于K主导区尺寸可根据应用场景与材料特性的先验认知进行预估，原则上可利用该预估来设计材料的中尺度非均匀性，从而最优地权衡强度与韧性。这为实现极端环境下抗裂纹扩展的功能性（超）材料开辟了新路径。