A Simplified Probabilistic Model for Nanocrack Propagation and Its Implications for Tail Distribution of Structural Strength
A Simplified Probabilistic Model for Nanocrack Propagation and Its Implications for Tail Distribution of Structural Strength
复制标题
纳米裂纹扩展的简化概率模型及其对结构强度尾部分布的影响
作者：
J. Le;Zhifeng Xu
本文提出了一种简化的热激活纳米裂纹扩展的概率模型。在连续极限下，纳米裂纹尖端的概率运动在数学上由Fokker-Planck方程描述。在该模型中，漂移速度显式相关的能量释放率在裂纹尖端通过过渡率理论。该模型被应用于分析的边缘裂纹在纳米级元件的扩展。当纳米裂纹扩展到一个临界长度时，该元件被认为达到失效。Fokker-Planck方程的解表明，纳米级元素的强度和寿命分布都表现出幂律尾部行为，但具有不同的指数。同时，该模型还得到了纳米级元件的平均应力-寿命曲线。当施加的应力足够大时，平均应力-寿命曲线类似于疲劳失效的纳斯昆定律。基于最近发展的有限最弱链模型和准脆性结构失效统计的水平偏移分析，认为模拟的纳米尺度单元强度分布的幂律尾部对宏观结构强度分布的尾部行为具有重要意义。为大尺度准脆性结构强度统计的双参数威布尔分布提供了物理依据。
This paper presents a simplified probabilistic model for thermally activated nanocrack propagation. In the continuum limit, the probabilistic motion of the nanocrack tip is mathematically described by the Fokker-Planck equation. In the model, the drift velocity is explicitly related to the energy release rate at the crack tip through the transition rate theory. The model is applied to analyze the propagation of an edge crack in a nanoscale element. The element is considered to reach failure when the nanocrack propagates to a critical length. The solution of the Fokker-Planck equation indicates that both the strength and lifetime distributions of the nanoscale element exhibit a power-law tail behavior but with different exponents. Meanwhile, the model also yields a mean stress-life curve of the nanoscale element. When the applied stress is sufficiently large, the mean stress-life curve resembles the nasquin law for fatigue failure. nased on a recently developed finite weakest-link model as well as level excursion analysis of the failure statistics of quasi-brittle structures, it is argued that the simulated power-law tail of strength distribution of the nanoscale element has important implications for the tail behavior of the strength distribution of macroscopic structures. It provides a physical justification for the two-parameter Weibull distribution for strength statistics of large-scale quasi-brittle structures.