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# Paris' law

In materials science and fracture mechanics, Paris' Law is used to relate the stress intensity factor to subcritical crack growth under a fatigue stress regime. $\frac{da}{dN} = C \Delta K^m$

Where a is crack length, C and m are material constants, and ΔK is the stress intensity factor range.

## History and Use

This formula was generated from P.C. Paris' 1961 realization that on a log-log plot of crack growth rate vs stress intensity factor showed a linear relationship linear plot. Using this law, one can quantitative predictions about 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.