Statistical estimation of damage
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Mechanical damage is calculated using Miner's linear accumulation rule.

The damage :math:`D` caused by :math:`N` cycles of half amplitude :math:`S` is expressed by :math:`D\text{=}\frac{N}{\text{Nr}(S)}`

where :math:`\text{Nr}(S)` is the allowable number of cycles determined by the material's endurance curve.

Mechanical damage is a random variable whose average is determined.




Average damage
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The average damage is written in the form of the mathematical expectation:

:math:`E(D)\text{=}T\underset{{S}_{\text{min}}}{\overset{{S}_{\text{max}}}{\mathrm{\int }}}\frac{N(S)}{{N}_{r}(S)}\text{dS}` where :math:`T` is the duration of the signal

The two proposed counting methods calculate the number of cycles :math:`N(S)` from positive amplitude constraints from where :math:`{S}_{\text{min}}\text{=}0` (except when the Wöhler curve is given in the form of "current zone", in which case :math:`{S}_{\text{min}}\text{=}{S}_{l}` with :math:`{S}_{l}` material endurance limit).

Moreover, the distribution laws used being continuous, :math:`{S}_{\text{max}}\text{=}\mathrm{\infty }`. However, experience shows that the expression to be integrated attenuates quickly and we therefore take :math:`{S}_{\text{max}}\mathrm{=}10{\sigma }_{S}`. where :math:`{\sigma }_{S}` is the standard deviation of the signal.

The calculation of :math:`E(D)` is carried out by numerical integration, by the trapezium method using :math:`\frac{{S}_{\text{max}}\text{-}{S}_{\text{min}}}{300.}` as an integration step.

**Note:**

*In the case of the method for counting level breaches and for a Wöhler curve expressed in the mathematical form proposed by Basquin, the* *average damage per unit of time to an analytical expression (this expression is not used in the command* *POST_FATI_ALEA) .*