Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.

提出了一种新的矩修正多项式维数分解（PDD）方法，用于在任意边界条件下对三维颗粒 - 基体功能梯度材料（FGMs）进行随机多尺度断裂分析。该方法包括通过正交多项式基对分量函数进行傅里叶 - 多项式展开，结合蒙特卡罗模拟的附加控制变量用于计算展开系数，以及一个矩修正随机输出以考虑颗粒位置和几何形状的影响。对二维FGM进行的数值验证表明，新方法，特别是单变量PDD方法，产生相同的粗略蒙特卡罗结果，但计算量减少了五倍。对一个在混合模式变形下的三维边缘裂纹FGM试样的数值结果表明，单变量PDD方法可以有效地生成裂纹驱动力的统计矩或概率分布以及断裂起始的条件概率。沿裂纹前沿，应力强度因子和断裂起始概率的概率特征存在显著变化。此外，结果对并发多尺度分析中的子域大小不敏感，如果明智地选择子域大小，将能高效地计算概率解。