In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in material microstructures, and to provide statistical metrics for quantities of interest, such as the fracture toughness. To describe the material responses such as the nucleation and growth of fractures in a uniform setting, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random field, e.g., a combination of truncated random variables determined by the Karhunen–Loève decomposition or the principal component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field that represents the micromechanical parameters. For each sample, the corresponding peridynamic problem is solved by an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and verify the convergence rate with a number of benchmark problems. The proposed scheme not only possesses the asymptotic compatibility spatially but also achieves an algebraic or sub-exponential convergence rate in the parametric random space when the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and its growth of the glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are also calculated accordingly. The numerical estimates of fracture toughness show good consistency with data from experimental measurements.

在这项工作中，我们的目标是开发一个统一的数学框架和一种可靠的计算方法来模拟具有材料微观结构可变性的异质材料中的脆性断裂，并为感兴趣的数量提供统计指标，例如断裂韧性。为了描述材料响应，例如在均匀环境中的成核和裂缝扩展，我们考虑了近场动力学模型。特别是，开发了一种基于随机状态的近场动力学模型，其中微机械参数由有限维随机场建模，例如，截断的组合由 Karhunen-Loève 分解或主成分分析 (PCA) 确定的随机变量。为了解决这个随机的近场动力学问题，采用概率搭配法（PCM）对代表微机械参数的随机场进行采样。对于每个样本，通过基于优化的无网格正交规则解决相应的近场动力学问题。我们对所提出的方案进行了严格的分析，并通过一些基准问题验证了收敛速度。该方案不仅在空间上具有渐近相容性，而且在参数上实现了代数或次指数的收敛速度。搭配点数增加时的随机空间。最后，为了验证这种方法在实际断裂问题上的适用性，我们考虑了微晶玻璃材料中的结晶增韧问题，其中材料在微观结构尺度上同时包含非晶玻璃和结晶相。所提出的随机近场动力学求解器用于捕获具有不同晶体体积分数的微晶玻璃的裂纹萌生及其生长，并据此计算平均断裂韧性。断裂韧性的数值估计与实验测量的数据具有良好的一致性。