Weibull Survival Probability

A ceramic does not have a strength. It has a distribution of strengths, and the difference is not pedantry — it changes what a design calculation even means. Break thirty nominally identical alumina bars from the same batch, machined on the same grinder, fired in the same kiln, and you will get thirty different numbers spread over a factor of one and a half or two. Nothing was done wrong. A brittle material fails from the single worst flaw the specimen happens to contain, that flaw is drawn at random from a population of pores, inclusions and machining scratches, and the strength is whatever that one flaw allows. The scatter is not measurement error sitting on top of a property. The scatter is the property.

Waloddi Weibull, a Swedish engineer with a background in ballistics, published the distribution that handles this in 1951 in the Journal of Applied Mechanics. The paper is famous partly for its abstract, which concedes with unusual honesty that the function "may sometimes render good service" — and then proceeds to fit everything from the fibre strength of Indian cotton to the height of British adult males. Seventy years later it is the backbone of ceramic design, rolling bearing life, wind resource assessment and half of reliability engineering.

Read $\sigma_0$ correctly, because almost nobody does. $\sigma_0$ is the characteristic strength: the stress at which 63.2% of specimens have already failed. It is not the mean, it is not the median, and it is not a typical value. At $\sigma = \sigma_0$ the bracket $(\sigma/\sigma_0)^m$ equals 1 for every modulus at once, so $P_s = e^{-1} = 0.368$ regardless of $m$ — that fixed point is the pivot the whole distribution turns around, and it is where the odd-looking 63.2% comes from. The mean strength is $\bar{\sigma} = \sigma_0\,\Gamma(1 + 1/m)$, which always sits below $\sigma_0$: about 0.95 of it at $m = 20$, 0.89 at $m = 10$, 0.83 at $m = 5$. This site deliberately does not ship that as a calculator, because it needs a gamma function and cannot be inverted for $m$ in closed form — but knowing it exists is what stops you reading $\sigma_0$ as an average.

The modulus $m$ is the shape of the distribution and the only number that describes consistency. A high $m$ means a narrow spread, which means a predictable material. Below about 5 is a lottery — coarse porous bodies, as-drawn glass edges. Five to ten covers a great deal of commercial alumina. Ten to twenty is a well-made engineering ceramic. Above thirty is unusual and hard-won. For comparison, a structural steel behaves as though $m$ were in the hundreds, which is exactly why metals can be designed with a safety factor and ceramics cannot.

Now the price of the model, which is real and cannot be designed around. The two-parameter form has no threshold. At any stress above zero, however small, it returns a nonzero probability of failure. Taken literally it says a teacup on a shelf has some minute chance of exploding, which is nonsense, and it means every very high survival probability you compute out in the tail is an extrapolation with nothing behind it. The three-parameter form adds a threshold stress $\sigma_u$ below which nothing fails, and it is honest about the physics — but fitting three parameters from thirty specimens is far shakier than fitting two, and the threshold usually comes out with a confidence interval wide enough to include zero. Most of the field uses two parameters and lives with the consequence. The way to actually obtain a threshold is proof testing: load every part to a chosen stress, and the survivors have a demonstrated lower bound rather than an assumed one.

This distribution has other lives on this site, and it is worth knowing they are the same thing. Bearing L10 life is a Weibull distribution with a shape parameter of about 1.1, baked into ISO 281 and almost never named as such — "the life ninety percent of bearings exceed" is a survival probability, and the reason a bearing's life is quoted that way rather than as an average is precisely the reason a ceramic's strength is. A wind resource is quoted as a Weibull shape factor $k$, fitted the same way from ranked data, on wind speed instead of stress. And the reliability corner — series and parallel system reliability, availability from mean time between failures — is the same weakest-link algebra with components in place of flaws and time in place of stress. One distribution, four trades, and each of them thinks it is theirs.

ASTM C1239 is the standard for estimating $m$ and $\sigma_0$ from data, and ASTM C1161 for measuring the strengths that feed it. Neither is reproduced here and neither should be. What this page can tell you is what the numbers mean once you have them.