Paris' law

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Paris' Law relates the stress intensity factor to sub-critical crack growth under a fatigue stress regime. It is commonly known in materials science and fracture mechanics. The accorded formula reads

$\frac{da}{dN} = C \Delta K^m$ ,

where a is the crack length, N is the number of load cycles, C and m are material constants, and $Δ K$ is the range of the stress intensity factor.

[edit] History and Use

The formula was introduced by P.C. Paris in 1961. It relates the crack growth rate during cyclic loading to the amplitude of the stress intensity factor in a way that is linear on a log-log plot. The law quantifies the residual life of a specimen given a particular crack size. Finding the beginning of the initiation of fast crack initiation:

$K=\sigma Y \sqrt{\pi a}$

One can then find the remaining lifetime using the following simple mathematical manipulations:

$\frac{da}{dN} = C \Delta K^m =C(\Delta\sigma Y \sqrt{\pi a})^m$

From here we can integrate over the size of the crack:

$\int^{Y_f}_0 dy=\int^{a_2}_{a_1}\frac{da}{C(\Delta\sigma Y \sqrt{\pi a})^m }$

[edit] References

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