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：脆性固体常通过添加第二相材料进行增韧。这种做法通常会导致复合材料中存在中尺度（介于过程区尺度与应用尺度之间）的材料非均质性。这种中尺度非均质性的具体构型（包括几何与力学特性）普遍认为在决定裂纹扩展路径及复合材料（有效）韧度方面具有关键作用。本文系统研究了以夹杂物阵列形式存在的中尺度非均质性对动态裂纹扩展的影响。采用变分相场方法，计算了表观裂纹扩展速度与断裂能耗散率，以比较I型加载条件下不同夹杂物构型中的裂纹扩展行为。当夹杂物体积分数固定时，使夹杂物尺寸与K主导区尺寸匹配可获得最佳增韧效果；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时，较低体积分数的构型才能实现更优增韧。由于K主导区尺寸可根据应用场景与材料可得性进行先验估算，原则上可利用该估算值设计材料的中尺度非均质性，从而最优地平衡强度与韧度之间的权衡关系。这为实现极端环境下抗裂纹扩展的功能性（超）材料开辟了新途径。