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Equation 27 · Part 9 · Why Materials Fail: Fracture, Flaws and the Inspection Interval

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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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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

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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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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