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Fatigue Life distribution

What it describes

Fatigue Life is the Birnbaum–Saunders distribution, constructed for the number of cycles until a dominant crack causes failure. Its skewness comes from accumulating damage increments around a critical level.

The support is positive after loc. scale sets a characteristic life and gamma controls relative variability; the curve can have a substantial right tail.

History and names

Birnbaum and Saunders introduced it in 1969 for materials under cyclic loading. The name “fatigue life” preserves that fracture-mechanics motivation.

How it connects to other distributions

Its derivation is connected with a Wiener process and the inverse Gaussian distribution. It also relates to first-passage approximations, but it is not identical to Inverse Gaussian.

Where it appears

  • lifetimes of metal components, welds, and structures under cyclic load
  • reliability tests with censored time-to-failure data

Fitting cautions

Do not read its tail as evidence for a power law: its skewness comes from the damage model and can differ strongly from Pareto.

The crack that turned damage into time

Birnbaum and Saunders imagined a crack exposed to repeated loading cycles. Small increments of damage accumulate until a critical level is crossed; translating that crossing into a cycle count produces the Fatigue Life distribution. The Birnbaum Saunders name preserves its authors and remains the usual term in statistical literature.

Its construction retains a link with the Normal law while producing positive, skewed lifetimes. A useful feature is its reciprocal relationship around the scale parameter, reflecting symmetry in the generating transformation. The crack story does not prove that every fatigued component follows this distribution. Censoring, competing failure modes, and variable loads can substantially change the observed law.

Decision guide

A good candidate when: failure time arises from accumulated damage and the characteristic Birnbaum–Saunders reciprocal transformation is meaningful.

Compare it with: Lognormal and Weibull using probability and hazard plots. All three can look similar in the body but imply different mechanisms and extrapolations.

References

  • SciPy reference: scipy.stats.fatiguelife — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Birnbaum, Z. W. & Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6, 319–327.
  • Owen, W. J. & Ng, H. K. T. (2015). Revisit of relationships and models for the Birnbaum–Saunders and inverse-Gaussian distributions. Journal of Statistical Distributions and Applications, 2, 11.

Fatigue Life Distribution: equations and calculator

Distribution defintion

X∼FatigueLife(γ,Loc,Sc)X\sim\mathrm{FatigueLife}\left(\gamma,\text{Loc},\text{Sc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

γ∈R+,Loc∈R,Sc∈R+\gamma\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(z(x)−1z(x)γ)F_{X}\left(x\right)=\Phi\left(\frac{\sqrt{z(x)}-\sqrt{\frac{1}{z(x)}}}{\gamma}\right)

Probability density function

fX(x)=z(x)+1z(x)2γz(x)ϕ(z(x)−1z(x)γ)f_{X}\left(x\right)=\frac{\sqrt{z(x)}+\sqrt{\frac{1}{z(x)}}}{2\gamma z(x)}\phi\left(\frac{\sqrt{z(x)}-\sqrt{\frac{1}{z(x)}}}{\gamma}\right)

Percent point function/Sample

FX−1(u)=Loc+Sc14[γΦ−1(u)+4+(γΦ−1(u))2]2F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\frac{1}{4}\left[\gamma\Phi^{-1}(u)+\sqrt{4+\left(\gamma\Phi^{-1}(u)\right)^2}\right]^2

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=Loc+Sc⋅μ~1′=Loc+Sc(1+γ22)\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\left(1+\frac{\gamma^2}{2}\right)

Parametric variance

Variance(X)=Sc2⋅(μ~2′−μ~1′2)=Sc2γ2(1+5γ24)\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\gamma^{2}\left(1+\frac{5\gamma^{2}}{4}\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=4γ(6+11γ2)(4+5γ2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{4\gamma(6+11\gamma^2)}{(4+5\gamma^2)^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+6γ2(93γ2+40)(5γ2+4)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3+\frac{ 6 \gamma^2 ( 93 \gamma^2+40 ) }{ ( 5 \gamma^2+4 )^2 }

Parametric median

Median(X)=Loc+Sc14[γΦ−1(1/2)+4+(γΦ−1(1/2))2]2\mathrm{Median}(X)=\text{Loc}+\text{Sc}\frac{1}{4}\left[\gamma\Phi^{-1}\left(1/2\right)+\sqrt{4+\left(\gamma\Phi^{-1}\left(1/2\right)\right)^2}\right]^2

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
ϕ(x):PDF normal standard distribution\phi\left(x\right):\text{PDF normal standard distribution}