The parameter m of the Weibull distribution that expresses how small the scatter in strength is. The larger it is, the smaller the scatter: about 5–20 for ceramics and several tens or more for metals.
Japanese name: ワイブル係数
Weibull modulus appears in 2 practice questions across 1 topic, including Ceramics and glasses.
Field: Ceramics and glasses
From the weakest-link idea, in which strength is set by the largest flaw, the strength of brittle materials is often described by the Weibull distribution of failure probability Pf = 1 − exp[−(σ/σ₀)^m]. σ₀ is the scale strength at which the failure probability is about 63.2%, and m is the Weibull modulus, which expresses how small the scatter is. The coefficient of variation of strength (standard deviation/mean) is roughly 1.2/m: about 12% for m = 10 and about 6% for m = 20. Since m reflects how uniform the flaw sizes are, it is also used as an indicator of manufacturing quality.
The strengths of n specimens are ranked from smallest to largest, and the failure probability of the i-th is estimated as (i − 0.5)/n, (i − 0.3)/(n + 0.4) or similar. Plotting ln ln[1/(1 − Pf)] on the vertical axis against ln σ on the horizontal gives a straight line whose slope is m. However, m found by least squares from this plot is easily biased when there are few specimens, so standards such as ASTM C1239 use maximum likelihood estimation. The fewer the specimens, the wider the uncertainty (confidence interval) of m, so a reliable m calls for specimens numbering in the tens.
A large m means the strength is uniform, not that it is high. A material with high mean strength but small m includes pieces that fail at low stress. Also, m and σ₀ depend on the test method; even for the same material, a tensile test, which stresses a larger volume, gives lower strengths than a bending test. In metals, yielding blunts crack tips, so strength scatter is small and m reaches several tens or more; Weibull statistics are therefore rarely used in their design.
For ceramic parts, the allowable stress is set from a target failure probability. For a failure probability of 10⁻⁶, the allowable stress is (10⁻⁶)^(1/m) times σ₀: about 0.25 times for m = 10 and about 0.50 times for m = 20. This shows that raising m is about as effective in design as raising the strength itself. In addition, the difference in effective volume between part and specimen is corrected for, and proof testing, which applies a stress above the service stress before shipment to weed out weak pieces, is combined to ensure reliability.
Answer: A. About 5.5%
σ/σ₀ = 300/400 = 0.75, and its tenth power is 0.75¹⁰ ≈ 0.0563. So Pf = 1 − exp(−0.0563) ≈ 0.055, about 5.5%. About 53% forgets the power m and computes 1 − exp(−0.75). About 95% mistakes exp(−0.0563) ≈ 0.945 for the failure probability; that is the probability of surviving (the reliability). Because the strength of ceramics is set by the size of internal flaws, it scatters more than that of metals, with m typically about 5–20. The larger m, the smaller the scatter, and the faster the failure probability falls at stresses below σ₀.
Answer: C. About 397 MPa
In the weakest-link picture, a larger volume is more likely to contain a large flaw, so strength falls. σ₂ = 500 × (1/10) to the power 0.1 = 500 × 0.794 ≈ 397 MPa. About 50 MPa assumes strength is inversely proportional to volume; in fact it enters through the m-th root, so the drop is gradual. About 500 MPa treats strength as a size-independent material constant, as for metals. Using the bend strength of small specimens directly to design a large component errs on the unsafe side, so the design of ceramic components corrects the strength for differences in effective volume.
Substituting σ = σ₀ into the equation gives Pf = 1 − exp(−1) = 1 − 1/e ≈ 0.632. σ₀ is also called the characteristic strength; it is close to the mean strength but is not the mean itself.
There is no fixed number, but the estimation error grows as the number decreases, so tests on the order of tens of specimens are desirable. With about 10 specimens the estimate of m can deviate greatly from the true value, so a confidence interval should be reported as well.
Plastic deformation blunts crack tips in metals, so strength depends little on flaw size and scatters only slightly. Metals are therefore normally handled with ordinary statistics using the mean and standard deviation.
Perovskite structure Piezoelectric effect Sintering Transformation toughening