Brittle strength varies from specimen to specimen because failure is often controlled by the most severe flaw sampled by the stressed volume or surface.

A common model is the Weibull distribution. For a uniformly stressed volume in a simple two-parameter form, the probability of failure by stress $\sigma$ is

$$P_f=1-\exp!\left[-\left(\frac{\sigma}{\sigma_0}\right)^m\right],$$

where $m$ is the Weibull modulus and $\sigma_0$ is a scale parameter.

A large $m$ produces a narrow strength distribution; a small $m$ indicates large specimen-to-specimen scatter.

Example

If $m=10$, $\sigma_0=500,\mathrm{MPa}$ and the applied stress is $400,\mathrm{MPa}$,

$$P_f=1-\exp[-(0.8)^{10}]\approx0.102.$$

So roughly ten percent of nominally identical specimens would be expected to fail by that stress under the model assumptions.

The weakest-link idea also creates a size effect: a larger highly stressed volume samples more potential critical flaws and can therefore have a lower survival probability than a smaller specimen at the same stress.

This statistical view is especially useful for ceramics and glasses, where little crack-tip plasticity exists to reduce sensitivity to flaw populations. A single deterministic 'strength' value can therefore be inadequate for reliability design.