Paris Law Crack Growth Rate

Until the early 1960s a fatigue crack was the end of a component's life. Paul Paris's contribution, published with Fazil Erdogan in 1963 in the Journal of Basic Engineering, was to show that a crack's growth per cycle is a simple power function of the stress intensity range — and therefore that a cracked part's remaining life is calculable. The paper was rejected by three journals first. It launched damage-tolerant design, which is why a cracked aircraft structure today is inspected and flown rather than scrapped.

The insight is that if $K$ governs everything at a crack tip, then the RANGE of $K$ over a load cycle should govern how far the crack advances in that cycle. Plot $\log(da/dN)$ against $\log \Delta K$ for almost any metal and the middle of the data is a straight line. The slope is $m$ and the intercept is $C$.

The exponent is the part to feel. For steels and aluminium alloys $m$ is between 2 and 4, usually near 3; some high-strength alloys and titaniums reach 6. At $m = 3$, doubling the stress intensity range multiplies the growth rate by eight. Nothing else available to a designer has that leverage, which is why halving a stress range is worth more than any material substitution — and why the fatigue fixes that work are the ones that reduce load range or remove stress concentration, not the ones that specify a stronger alloy. Fatigue crack growth rates in steels are remarkably insensitive to strength grade; a mild steel and a quenched-and-tempered one grow cracks at nearly the same rate under the same $\Delta K$.

Region II only, and this matters in both directions. The real curve has three parts. Region I is the threshold: below $\Delta K_{th}$ — typically 2 to 8 MPa·$\sqrt{\text{m}}$ in steels, and lower at high $R$ ratio — the curve turns down almost vertically and cracks effectively do not grow. Region II is the straight line Paris described. Region III is the acceleration as $K_{max}$ approaches $K_{IC}$, where the curve turns up and runs to fracture. Extrapolating the straight line DOWN into Region I predicts creeping growth where there is none: wasteful, but safe. Extrapolating it UP into Region III predicts a crack still on the line when it is accelerating away from it, so the predicted life is longer than the real one. That error is in the unsafe direction, and it is how a remaining-life estimate comes out optimistic. Check $\Delta K$ at both ends of your integration before believing the answer.

$C$ is unit-bound, and this is the shard's worst trap. The Taylor constant in machining has the same problem; this one is worse, because the exponent sits inside the conversion. $C$ has whatever units make $da/dN$ come out right for the units $\Delta K$ was measured in, so it is not a quantity any calculator can convert on its own. This site fixes one convention and states it on every page that uses it: $C$ is in metres per cycle, with $\Delta K$ in MPa·$\sqrt{\text{m}}$. A source quoting inches per cycle against ksi·$\sqrt{\text{in}}$ needs converting first — multiply by 0.0254 to change inches to metres, and divide by $1.0988434^m$ to change ksi·$\sqrt{\text{in}}$ to MPa·$\sqrt{\text{m}}$. Get $m$ wrong in that conversion and the result is off by orders of magnitude while looking entirely reasonable. A $C$ taken from a table without checking the table's convention is the most common way to get a crack growth prediction wrong by a factor of a thousand.

And $C$ and $m$ are a matched PAIR. They are the intercept and slope of one fitted line, and they trade off against each other in the fit. Taking $m$ from one source and $C$ from another produces a line that passes through neither data set. Record them together, with the $R$ ratio, the environment, the temperature and the $\Delta K$ range they were fitted over.

Two refinements worth knowing exist. Mean stress matters: at higher $R = K_{min}/K_{max}$ the crack faces stay open longer and growth is faster, which Walker's and Forman's modifications handle — and interestingly they mostly adjust $C$ and leave $m$ nearly alone. And under variable amplitude, a single overload leaves a large plastic zone whose contraction puts compressive residual stress at the tip, retarding growth for thousands of subsequent cycles. A cycle-by-cycle sum that ignores retardation is conservative; one that ignores the underloads which cancel it may not be.

One last practical use, running the equation backwards. In many aluminium alloys each cycle leaves one striation on the fracture surface, and the striation spacing IS $da/dN$. Measure it under an electron microscope, solve for $\Delta K$, and a failure investigation has recovered the load history from the broken part itself. Steels often show no clear striations, and a spacing below about 0.3 nm — one lattice spacing — cannot be a per-cycle advance at all.