the Weibull modulus
/ VY-bull /
If Weibull statistics describes the whole spread of a ceramic's strengths, the Weibull modulus is the single number that captures how wide that spread is, in other words how reliable the material is. A high modulus means the strengths cluster tightly, so nearly every part is about as strong as every other and you can trust them. A low modulus means the strengths are all over the place, so some parts are much weaker than the average and you dare not push the material near its mean. It is, in short, a consistency score.
The modulus, written m, is the exponent in the Weibull survival law P_s = exp(-(sigma/sigma_0)^m), and on a plot of ln(ln(1/(1-P_f))) against ln(sigma) it is simply the slope of the line through the broken specimens. Roughly, the scatter in strength is about 1/m, so a small m means a broad spread. Typical values tell the story: an ordinary structural ceramic sits around m = 5 to 15, a well-processed engineering ceramic can reach 20 or more, whereas a ductile metal behaves as if m were above 50 or even 100, its strength almost perfectly repeatable. A low m traces back to a broad range of flaw sizes, so improving processing to make flaws uniform is what raises the modulus.
The modulus matters more for design than the average strength does, and this catches people out. Because you must derate to a low failure probability, the safe design stress falls far below the mean when m is small, so a ceramic with a high mean strength but a low modulus can be less useful than a weaker but more consistent one. The modulus also governs how strongly strength falls as parts grow larger, through the size effect. Two honest caveats: a modulus estimated from only a handful of specimens carries wide uncertainty and should come with confidence bounds, and a modulus quoted as a single number hides any bimodal, two-population behaviour, where a single m simply does not describe the data.
Two silicon-carbide grades both average 400 MPa. Grade A has m = 20, grade B only m = 6. For a one-in-a-million design, grade A can be used near 180 MPa but grade B must be derated below 60 MPa. Despite the equal mean, the consistent grade is worth three times as much in service.
Equal average strength, very different reliability: the modulus, not the mean, sets the usable design stress.
A higher mean strength is nearly worthless if the modulus is low, because the low-failure-probability design stress depends more on m than on the mean. And an m from fewer than about thirty specimens is very uncertain, so treat single-number moduli with suspicion unless the sample size is given.