Paris\' law - Citizendia

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. Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and Fracture mechanics is the field of Mechanics concerned with the study of the formation of cracks in materials 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.

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:

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

References

Paris' law-http://www.tech.plym.ac.uk/sme/tutorials/FMTut/Fatigue/FatTheory1.htm.

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