Fracture Strength: Stress Concentration, Extreme Value Statistics, and the Fate of the Weibull Distribution

The statistical properties of fracture strength of brittle and quasibrittle materials are often described in terms of the Weibull distribution. However, the weakest-link hypothesis, commonly used to justify it, is expected to fail when fracture occurs after significant damage accumulation. Here we show that this implies that the Weibull distribution is unstable in a renormalization-group sense for a large class of quasibrittle materials. Our theoretical arguments are supported by numerical simulations of disordered fuse networks. We also find that for brittle materials such as ceramics, the common assumption that the strength distribution can be derived from the distribution of preexisting microcracks by using Griffith’s criteria is invalid. We attribute this discrepancy to crack bridging. Our findings raise questions about the applicability of Weibull statistics to most practical cases.

Figure 1

Emergent survival probability for $L=4,\dots ,128$, with $k=25$ for the threshold distribution. If the emergent distribution is Weibull, it will follow the solid black line. Clearly, the distribution flows away from Weibull at long length scales, showing that the Weibull distribution is not stable in a renormalization-group sense.

Figure 2

Survival probability for (a) $k=1.5$ and (b) $k=4$. The main figures show the DLB test, while the insets show the Weibull test; straight lines indicate agreement with the tested form.

Figure 3

Crack-width distributions at peak load for Weibull-distributed fuse strengths with exponents (a) $k=1.5$ and (b) $k=4$. The distribution is a power law with an exponential tail for all values of $k$.

Figure 4

(a) Crack-width distributions at peak load for a system with power-law-distributed cracks. The power-law tail has an exponent ${\gamma }_f$ that is larger than the initial one ${\gamma }_i$. (b) The corresponding survival distribution obeys the Weibull law with $k=2{\gamma }_f$.

Figure 5

Relation of the exponents of the crack-width distribution initially ${\gamma }_{rmi}$ and at peak load ${\gamma }_f$. For linear-grown cracks, the relation depends strongly on $p$ and ${\gamma }_i$, while for random-walk-grown cracks we find ${\gamma }_f\approx c{\gamma }_i+d$ for both investigated dilution parameters $p=0.05$ and $p=0.1$.

Figure 6

Results of extrapolating the fits of the Weibull distribution and the suggested transformation-based method. Fits obtained from small data sets (size 200) by using the $T$ method can be extrapolated with confidence to probabilities about 2 orders of magnitude smaller. The figure shows $\pm 1$ standard deviation results.