Paris' law

Relevance

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.

<math> \frac{da}{dN} = C \Delta K^m </math>

Where a is crack length, C and m are material constants, and \Delta K 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:

<math> K=\sigma Y \sqrt{\pi a} </math>

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

<math> \frac{da}{dN} = C \Delta K^m =C(\Delta\sigma Y \sqrt{\pi a})^m </math>

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

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

References

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

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