Scaling of Fracture Strength in Disordered Quasi-Brittle Materials

TL;DR

This study shows that in disordered quasi-brittle materials, the fracture strength distribution at peak load is better described by a lognormal distribution rather than Weibull or Gumbel, and the mean strength decreases with system size following a logarithmic scaling.

Contribution

It demonstrates that the fracture strength distribution in disordered materials is lognormal and provides a scaling law for the mean fracture strength as system size increases.

Findings

Lognormal distribution fits fracture strength data better than Weibull or Gumbel.

Mean fracture strength scales as 1/(Log L)^ψ with system size L.

Numerical simulations confirm the theoretical distribution and scaling law.

Abstract

This paper presents two main results. The first result indicates that in materials with broadly distributed microscopic heterogeneities, the fracture strength distribution corresponding to the peak load of the material response does not follow the commonly used Weibull and (modified) Gumbel distributions. Instead, a {\it lognormal} distribution describes more adequately the fracture strengths corresponding to the peak load of the response. Lognormal distribution arises naturally as a consequence of multiplicative nature of large number of random distributions representing the stress scale factors necessary to break the subsequent "primary" bond (by definition, an increase in applied stress is required to break a "primary" bond) leading up to the peak load. Numerical simulations based on two-dimensional triangular and diamond lattice topologies with increasing system sizes substantiate that a {\it lognormal} distribution represents an excellent fit for the fracture strength distribution at the peak load. The second significant result of the present study is that, in materials with broadly distributed microscopic heterogeneities, the mean fracture strength of the lattice system behaves as $\mu_f = \frac{\mu_f^\star}{(Log L)^\psi} + \frac{c}{L}$, and scales as $\mu_f \approx \frac{1}{(Log L)^\psi}$ as the lattice system size, $L$, approaches infinity.