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Ceramics and glasses

The strength of a ceramic follows the Weibull distribution Pf = 1 − exp[−(σ/σ₀) to the power m] with a scale parameter σ₀ = 400 MPa and a Weibull modulus m = 10. Which value is closest to the probability of failure under a load of 300 MPa?

  1. AAbout 5.5%
  2. BAbout 53%
  3. CAbout 95%
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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.

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