This paper addresses the question of the homogenization of fracture properties for three-dimensional disordered brittle solids. The effective toughness, identified as the minimum elastic energy release rate required to ensure crack growth, is predicted from a semi-analytical framework inspired by both micromechanics and statistical physics, that encompasses the decisive influences of both the material disorder and the mechanisms of interaction between a crack and heterogeneities. Theoretical predictions are compared to numerical values of the effective toughness that are computed with the fracture-mechanics-based semi-analytical method of Lebihain et al. (2020). Based on a perturbative approach of Linear Elastic Fracture Mechanics, this method allows for the efficient computation of crack propagation under tensile Mode I loading in composite materials containing several millions of inclusions, where the crack interacts with them through two mechanisms : crossing, wherein the crack penetrates the inclusion, and by-pass, wherein the crack wanders out-of-plane and follows the inclusion/matrix interface. We show that our homogenization procedure provides an accurate prediction of the homogenized fracture properties for a broad range of microstructural parameters such as the inclusion toughness, density or shape. This original theoretical framework constitutes a powerful mean to connect the microstructural parameters of materials to their crack growth resistance, beyond the particular cases considered in the simulations performed. As a result, it provides new strategies for the rational design of optimized brittle composites with tailored fracture properties.

本文解决了三维无序脆性固体断裂特性均质化的问题。有效韧性被确定为确保裂纹扩展所需的最小弹性能量释放速率，是根据受微观力学和统计物理学启发的半分析框架进行预测的，该框架涵盖了材料紊乱和材料之间相互作用的决定性影响。裂纹和异质性。将理论预测与有效断裂韧度的数值进行比较，该数值是用Lebihain等人的基于断裂力学的半分析方法计算得出的。（2020）。基于线性弹性断裂力学的摄动方法，这种方法可以有效地计算拉伸模式I载荷下包含数百万个夹杂物的复合材料中的裂纹扩展，其中裂纹通过两种机制与它们相互作用：交叉（其中裂纹穿透夹杂物）和绕过（其中，裂纹在平面外徘徊，并遵循包含/矩阵界面。我们表明，我们的均质化程序可为各种微观结构参数（如夹杂物的韧性，密度或形状）提供均质化断裂特性的准确预测。这种原始的理论框架构成了将材料的微结构参数与其抗裂纹扩展性联系起来的有力手段，这超出了所进行的模拟中考虑的特殊情况。因此，