TY - JOUR

T1 - A Simplified Probabilistic Model for Nanocrack Propagation and Its Implications for Tail Distribution of Structural Strength

AU - Le, J. L.

AU - Xu, Z.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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.

AB - 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.

KW - Fokker-Planck equation

KW - Weibull distribution

KW - failure statistics

KW - quasi-brittle materials

KW - transition rate theory

UR - http://www.scopus.com/inward/record.url?scp=85065033390&partnerID=8YFLogxK

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U2 - 10.1134/S1029959919020012

DO - 10.1134/S1029959919020012

M3 - Article

AN - SCOPUS:85065033390

VL - 22

SP - 85

EP - 95

JO - Physical Mesomechanics

JF - Physical Mesomechanics

SN - 1029-9599

IS - 2

ER -