Leath P L, Duxbury P M
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849.
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-1116.
J Res Natl Inst Stand Technol. 1994 Jul-Aug;99(4):337-344. doi: 10.6028/jres.099.031.
Simple models play a key role in the microstructural analysis of mechanical failure in composites and other materials with complex and often disordered microstructures. Although equal load-sharing-models are amenable to rigorous statistical analysis, problems with local load enhancements near failed regions of the material have so far resisted exact analysis. Here we show for the first time, that some of the simpler of these - can be solved exactly using a sub-stochastic matrix method. For diluted fiber bundles with local load sharing, it is possible to find a compact expression for the characteristic equation of the sub-stochastic matrix, and from it obtain an asymptotic expansion for the largest eigenvalue of the matrix. This in turn gives the asymptotic behavior of the size effect and statistics of the fiber-bundle models. We summarize these results, and show that the important features of the exact result can be obtained from a single scaling analysis we had developed previously. We also compare the statistics of fracture with the appropriate limiting extreme-value survival distribution (a Gumbel distribution), and, as previously indicated by Harlow and Phoenix, note that the Gumbel distribution performs quite poorly in this problem. We comment on the physical origin of this discrepancy.
简单模型在复合材料及其他具有复杂且往往无序微观结构的材料的机械失效微观结构分析中起着关键作用。尽管等载荷分担模型适用于严格的统计分析，但材料失效区域附近局部载荷增强的问题迄今仍无法进行精确分析。在此，我们首次表明，其中一些较为简单的模型——可以使用次随机矩阵方法精确求解。对于具有局部载荷分担的稀释纤维束，可以找到次随机矩阵特征方程的紧凑表达式，并由此获得该矩阵最大特征值的渐近展开式。这进而给出了纤维束模型尺寸效应和统计量的渐近行为。我们总结了这些结果，并表明可以从我们之前开发的单一标度分析中获得精确结果的重要特征。我们还将断裂统计与适当的极限极值生存分布（耿贝尔分布）进行了比较，并且正如哈洛和菲尼克斯之前所指出的，注意到耿贝尔分布在这个问题上表现相当差。我们对这种差异的物理根源进行了评论。
