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 statistic metrics for quantities of interest, such as the fracture toughness. To depict the material responses and naturally describe the nucleation and growth of fractures, 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 vector, or a combination of random variables truncating the Karhunen-Lo\`{e}ve decomposition or the principle component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field representing the micromechanical parameters. For each sample, the deterministic peridynamic problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate its convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random space as 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 growth for glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are calculated. The numerical estimates of fracture toughness show good consistency with experimental measurements.
翻译: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个可靠的计算方法,以模拟具有物质微结构变异性的各种材料的骨折变形体,并提供相关数量的统计度,例如骨折坚硬度。为了描述材料反应和自然描述骨折的核素变化和生长,我们考虑的是皮肤动力学模型。特别是,开发了一个基于状态的透视性皮肤动力学模型,在这个模型中,微机械性参数是用一个有限的维度硬性随机矢量,或由随机变量组合来调节Karhunen-Lo ⁇ e}Ve腐蚀性或原则组成部分分析(PCA),提供相关数量的统计度。为了解决这个随机的皮肤过敏性过敏性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过硬性过错。对于每个样本来说,确定性过硬性过硬性过硬性过硬性过硬性过硬性过硬性稳定性地压性地压压压压。