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Ps=exp⁡[−V(σσ0)m]P_s = \exp\left[-V\left(\frac{\sigma}{\sigma_0}\right)^{m}\right]

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If the strength of a brittle body is set by its largest flaw, and flaw populations are random, then strength is a random variable and reporting a mean is an error of category. The standard treatment is the Weibull weakest-link model, in which the probability that a body of volume V survives a uniform stress σ\sigma is Ps=exp⁡[−V(σσ0)m]P_s = \exp\left[-V\left(\frac{\sigma}{\sigma_0}\right)^{m}\right]. with m the Weibull modulus and σ0\sigma_0 a scale parameter. A larger m means tighter scatter. A review of Weibull analysis across ceramics reports moduli of roughly 10 or below for many advanced ceramics such as alumina, hydroxyapatite and silicon carbide, below about 7 for the glass-ceramics and zirconias surveyed, and above 40 for one magnesium-based glass,…

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Ps=exp⁡[−V(σσ0)m],P_s = \exp\left[-V\left(\frac{\sigma}{\sigma_0}\right)^{m}\right],

Equation 27 · Materials Science

Why Materials Fail: Fracture, Flaws and the Inspection Interval

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

If the strength of a brittle body is set by its largest flaw, and flaw populations are random, then strength is a random variable and reporting a mean is an error of category. The standard treatment is the Weibull weakest-link model, in which the probability that a body of volume V survives a uniform stress σ\sigma is Ps=exp⁡[−V(σσ0)m]P_s = \exp\left[-V\left(\frac{\sigma}{\sigma_0}\right)^{m}\right]. with m the Weibull modulus and σ0\sigma_0 a scale parameter. A larger m means tighter scatter. A review of Weibull analysis across ceramics reports moduli of roughly 10 or below for many advanced ceramics such as alumina, hydroxyapatite and silicon carbide, below about 7 for the glass-ceramics and zirconias surveyed, and above 40 for one magnesium-based glass,…

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