Paris Law Crack Growth Rate
Fatigue crack growth per cycle from the stress-intensity range using the Paris power law.
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The engineering
The Paris law describes the middle, stable regime of fatigue crack growth: on a log-log plot of da/dN versus ΔK it is a straight line, and C and m are its intercept and slope. Reach for it once a crack has been detected and you need to know how many cycles remain before it reaches a critical size. ΔK combines the applied stress range, the current crack length, and a geometry factor Y that captures the specimen or component shape (1.12 is the classic edge-crack value).
The exponent m is unforgiving — for most metals it sits between 3 and 4, so growth rate scales with roughly the cube of ΔK, and ΔK itself grows with √a. That means crack growth accelerates dramatically as the crack lengthens, which is why inspection intervals shrink near end of life. Paris law is only valid in Region II: below the threshold ΔK_th cracks effectively arrest, and above it near K_IC growth runs away, so don't extrapolate this line past either end.
Get C and m from an ASTM E647 test on your material, heat, and R-ratio — literature values scatter by orders of magnitude and are load-ratio dependent.
Where this math comes from
Paul C. Paris was a young engineer who, in the late 1950s, argued that fatigue crack growth should correlate with the stress-intensity factor from linear elastic fracture mechanics rather than nominal stress. Established reviewers rejected the idea; the story goes that his seminal 1961 paper was turned down by the major journals and finally appeared in a small periodical, The Trend in Engineering, after he and coauthors could not get a hearing elsewhere.
The correlation proved to be one of the most useful relationships in engineering, turning George Irwin's stress-intensity factor into a practical fatigue-life prediction tool. It became the backbone of damage-tolerant design mandated for aircraft after the 1970s, and the measurement of C and m was codified in ASTM E647.
- 1957George R. IrwinFormalizes the stress-intensity factor K, the basis for ΔK.
- 1961Paul C. ParisProposes da/dN correlates with ΔK, after rejection by mainstream journals.
- 1963Paris & Fazil ErdoganPublish the power-law form da/dN = C·(ΔK)ᵐ.
- 1971US Air ForceMIL-A-83444 adopts damage-tolerant design built on crack-growth prediction.
- 2015ASTM E647Standard test method for measuring fatigue crack growth rates and Paris constants.
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