The loops a crack grows on

Rainflow counting turns a strain-gauge record into a list of closed loops, and nothing so far has taken it further: a list of ranges and counts, ready to be handed to Miner’s rule and reduced to one number. That reduction throws four things away — the small cycles, the cut-off, the order, and the future — and the essay that said so priced only the first of them.

The loops have somewhere else to go. A crack does not sum damage; it grows, at a rate that depends on how long it already is. Feed the same loops into that instead and everything the Miner sum discarded comes back, in a different order and with different consequences.

A band does not have a rate, it has a rate at a crack length #

The growth law is two lines:

$$\frac{da}{dN} = C\,(\Delta K)^m, \qquad \Delta K = Y\,\Delta\sigma\,\sqrt{\pi a},$$

with $C = 5.21\times10^{-13}$ and $m = 3$ for steel in air, in millimetres and N/mm^1.5. Everything in this essay follows from the fact that $a$ appears on both sides.

A stress range does not have a damage; it has a damage at a crack length. The same 41 N/mm² lorry that produces $\Delta K = 58$ at a half-millimetre flaw produces 257 at a ten-millimetre crack, and because the rate goes as the cube, it does 89 times as much per cycle at the second as at the first. Miner’s rule has no way of expressing that: $n/N$ is fixed the moment the spectrum is written down.

And the threshold is where it becomes a different calculation rather than a more accurate one. Below $\Delta K_{\mathrm{th}}$ — about 63 N/mm^1.5 for structural steel — a cycle does not grow the crack at all. Since $\Delta K$ rises with $\sqrt a$, that threshold is not a stress. Each band of the traffic has a crack length at which it wakes up:

$$a_{\mathrm{th}} = \frac{1}{\pi}\left(\frac{\Delta K_{\mathrm{th}}}{Y\,\Delta\sigma}\right)^2 .$$

For the 62 N/mm² train it is 0.26 mm. For the 7 N/mm² car it is 20.6 mm. A car does nothing to a detail with a small flaw and a great deal to one with a large one, and no spectrum reduced to a single equivalent range can say so.

Half the life is in the first half-millimetre #

The shape of that curve is the practical content of the whole subject, and it is not the shape anybody’s intuition supplies.

Ninety-one years to grow half a millimetre. Twenty-three more to grow the next millimetre. Four to grow the last seventy-five. The crack spends the first half of its life at a size no inspection would find and the last few per cent at a size nothing could miss, and there is no useful middle.

Three consequences follow and they are all uncomfortable.

An inspection programme that starts at year fifty finds nothing, and that is not reassuring. The detail is nine tenths of the way through its safe life in crack-length terms and one twentieth of the way through it in observable terms, because the crack is 0.6 mm and the smallest thing anybody can find reliably is about 2.

A structure that has survived a century is not thereby demonstrated safe. It is a structure whose crack has reached about a millimetre, with the steep part still ahead of it.

And the traffic that matters changes during the life. Early on only the train and the heavy lorry are above the threshold at all. By the time the crack is 4 mm the van has woken up; by 21 mm the cars have. A detail’s spectrum is effectively different at different ages.

Which vehicles matter, and which the sum discards #

Three of the five bands do exactly none of the Miner damage. Between them they grow thirty-seven per cent of the crack.

That is not a small correction and it is not an error in either calculation. It is a disagreement about what a cut-off is. The cut-off on an S-N curve is a real experimental fact: a specimen cycled at a constant range below it does not fail, however many cycles it is given. The inference usually drawn from it — that such cycles are harmless in a spectrum — is a different statement and it is false, because a specimen cycled below the cut-off has no crack and a detail in a spectrum does. The large cycles grow one, and once it is there the small ones grow it further.

Codes know this and handle it by bending the curve rather than by cutting it: a fifth-power slope between the constant-amplitude limit and the cut-off is an admission that the small cycles are not free. The crack-growth calculation says the same thing without a slope change in it anywhere, which is an argument for it being the more fundamental of the two.

And the two lives differ by a third: 196 years by crack growth against 300 by Miner, on the same traffic, the same detail and the same century. Which is right is not a question either calculation can answer; they are answers to different questions, and the crack-growth one is the question an inspection regime is built on.

The number nobody measures #

The curve falls off a cliff and the cliff is the threshold.

An initial flaw of 0.25 mm gives a detail an unbounded life; one of 0.30 mm gives it 341 years; one of a millimetre gives it 105. That is a factor of three from a tenth of a millimetre, and a step from finite to infinite across a boundary of a quarter of a millimetre — which is smaller than the undercut at a weld toe, smaller than a slag inclusion, smaller than anything an inspection of a new structure would record.

So the quantity that decides the answer is the one no design calculation contains. The stress range is computed to three figures. The detail category is looked up. The traffic is forecast. And the flaw the detail starts life with — which moves the answer by a factor of ten across the range of sizes a sound weld might plausibly contain — is not measured, not specified, and not on any drawing.

That is the honest reason a fatigue design is done by S-N curve and not by crack growth. The S-N curve has the flaw size in it, averaged over the population of details the tests were done on, which is a statement about a manufacturing standard rather than about a particular weld. Crack growth makes the flaw explicit and then has to guess it.

What the curve is for #

Read the same integration from the other end and it stops being a life and becomes a schedule.

A damage-tolerant design does not promise that a detail will not crack. It promises that the crack will be found, and the promise is arithmetic: the interval between inspections has to be short enough that a crack which was just too small to see at one inspection is still smaller than critical at the next. On this detail a 2 mm crack has 66 years left, so an inspection every thirty years sees it at least twice before it matters, and every ten years sees it six times.

The interval is set by the detectable length and not by the design life, which is why the two numbers have no relationship. A better inspection method — one that finds 1 mm rather than 2 — buys forty more years of interval on this detail, 105 against 65, and the same money spent on a better detail category buys none at all, because the category does not change the curve’s shape.

Where the two calculations have to agree, and where they need not #

It is worth being precise about the relationship between the S-N calculation and this one, because “crack growth is the more fundamental” is the kind of claim that sounds like a dismissal and is not.

They agree about a constant-amplitude test, by construction. Integrate crack growth at one range from a fixed initial flaw to a critical one and the result is $N \propto \Delta\sigma^{-3}$ — which is the S-N curve’s slope of three, arrived at rather than assumed. The detail category is then the initial flaw size in disguise: a category 36 detail and a category 112 detail in the same steel differ by a factor of $(112/36)^3 \approx 30$ in life at the same range, which is a factor of about ten in $a_0$.

They cannot agree about a spectrum, and the reason is exactly the cut-off. An S-N curve carries a cut-off because a constant-amplitude specimen below it never starts a crack; a spectrum on a real detail has a crack from the first week, and after that the cut-off’s experimental basis is gone. This is the one place where a curve fitted to tests is being used outside the conditions the tests were done in.

And they use different information. The S-N route needs the spectrum and the category and nothing else. The crack-growth route needs the spectrum, the initial flaw, the toughness, the geometry factor and two Paris constants — five quantities in place of one, each with its own uncertainty. So the more fundamental calculation is also the less reliable one in practice, and an engineer who replaces a Miner sum with a crack integration has not bought accuracy; they have bought a different set of assumptions, one of which — the flaw size — is the largest single uncertainty in the subject.

The right reading is that each answers its own question. The S-N route answers “how long until this detail probably cracks”; the crack-growth route answers “given that it has, how long until somebody has to know”. A design uses the first and an inspection regime uses the second, and a structure that is designed with one and maintained with the other is being treated correctly.

Where the traffic goes after the crack #

One more consequence of the rate depending on the crack, and it is the one that changes what an owner should worry about.

A detail’s spectrum is effectively different at different ages. At 0.5 mm only two of the five bands are above the threshold at all, so the detail is carrying the traffic of a quiet road. At 5 mm four bands are active. At 21 mm all five are, and the cars — sixty-two thousand of them a day at the far end of the distribution — have joined in.

So a bridge whose traffic composition changes is not simply a bridge with a different Miner sum. An increase in car traffic is worth nothing to a sound detail and something real to a cracked one, and the increase that matters most is the one in the band just below the current threshold, because that band is waiting.

And it inverts the usual advice about weight limits. Restricting the heaviest vehicles has its largest effect early, while the crack is small and only the heavy bands are above the threshold — the phase in which nothing appears to be happening. By the time a crack is found and a weight restriction is the natural response, the heavy bands are a third of the growth rather than all of it, and the restriction buys proportionately less.

The whole of it, once, by hand #

The detail starts with a flaw of 0.5 mm, $Y = 1.12$, and carries the 62 N/mm² train band, 2 per cent of 800 crossings a day — 16 cycles a day, 5,844 a year.

At $a = 0.5$ mm the train gives

$$\Delta K = 1.12 \times 62 \times \sqrt{\pi \times 0.5} = 87\ \text{N/mm}^{1.5},$$

above the threshold of 63, so it grows the crack at

$$\frac{da}{dN} = 5.21\times10^{-13} \times 87^3 = 3.4\times10^{-7}\ \text{mm per cycle},$$

which is 0.002 mm a year. Five hundred years at that rate, from the train alone, to add a millimetre — except that the rate is not constant, and by $a = 1$ mm it has grown by $2^{1.5} = 2.8$ times.

At the other end, $a = 100$ mm:

$$\Delta K = 1.12 \times 62 \times \sqrt{\pi \times 100} = 1{,}231, \qquad \frac{da}{dN} = 9.7\times10^{-4}\ \text{mm per cycle},$$

5.7 mm a year from the train alone. Three thousand times the rate, from the same vehicle, on the same bridge. The crack does not fail because the traffic got worse; it fails because the crack got longer, and the traffic never changed at all.

Integrating that properly is what the figures do, band by band, with each band switched on only where its $\Delta K$ clears the threshold. The answer is 195.8 years and a critical crack of 125.7 mm.

Which free body produced the number #

The free body is a through-thickness crack in a wide plate, with $\Delta K = Y\Delta\sigma\sqrt{\pi a}$ and $Y = 1.12$ — the standard edge-crack factor. The critical length is where the peak stress-intensity reaches the toughness, which is the flaw that sets the strength read as a length rather than as a stress: $K_{\max} = Y\sigma_{\max}\sqrt{\pi a_c} = K_c$, with $\sigma_{\max}$ the steady stress of 80 N/mm² plus the largest range, giving 125.7 mm for a toughness of 3,160 N/mm^1.5.

The integration is over the same five-band spectrum and the same 800 crossings a day the S-N calculation used, so the two lives on this page are lives of the same bridge. The check is that the Miner sum computed alongside it reproduces that essay’s answer.

What the picture cannot show #

Initiation. Every cycle here grows a crack that is assumed already to exist. In a plain, unwelded member the larger part of the life is spent making one, and crack growth has nothing to say about that half — which is why the calculation applies to welded details, where the flaw is there from the first day, and badly to anything else — a plain rolled edge is governed by the notch it does not feel in full rather than by anything here.

The crack’s shape. $Y = 1.12$ is a through crack in a wide plate. Real fatigue cracks at a weld toe start as semi-elliptical thumbnails, grow through the thickness first, break through, and only then behave like this. A proper calculation integrates two dimensions with two geometry factors.

Residual stress. A welded detail sits in a field at yield, so every applied cycle is fully tensile whatever its mean, and the crack never closes. That assumption is why mean stress does not appear anywhere — and it is also why the threshold used here is the low-$R$ one rather than the higher value a stress-relieved detail would have.

Any scatter at all. $C$ and $m$ are mean values fitted to a cloud of test data two orders of magnitude wide. A life computed from them is a central estimate and the distribution around it is not narrow.

The assumption that decides everything #

That $m = 3$. The exponent is the whole of the behaviour: it is why the rate rises a thousandfold as the crack grows, why the last few per cent of the life is short, why the order of the cycles can matter, and why the cube-weighted equivalent range of the S-N calculation has the same three in it.

It is measured rather than derived, it varies between 2.5 and 4 across steels and environments, and the answer is roughly as sensitive to it as it is to everything else combined. At $m = 2.5$ this detail lives longer and the small vehicles matter more; at $m = 3.5$ the reverse. The convergence of the two calculations on the same number three — Paris’s exponent and the S-N slope — is not a coincidence and it is not a derivation either: the S-N slope is what the crack-growth exponent produces when a fixed initial flaw is integrated to failure, which is the sense in which the crack-growth calculation is the more fundamental.

Still open: the loops in the order they arrived #

Everything above integrates the spectrum band by band and repeats it, which is Miner’s own assumption wearing different clothes: the bands are applied in some order, and the order chosen was arbitrary.

Under Miner it could not matter, because a sum does not have an order. Under crack growth it can, and the mechanism is already on this page: a cycle’s contribution depends on the crack length it meets, so a spectrum that puts its large cycles first meets a shorter crack with them than one that puts them last. That effect turns out to be small. A second one is not: a single large cycle leaves a zone of yielded material at the crack tip, and the crack has to grow through its own damage before it resumes — so an overload slows what follows it. Whether the same record played backwards gives the same life, and by how much a single heavy lift can extend a detail’s life rather than shortening it, is the question after this one.