The record played backwards

Miner’s rule states that damage accumulates as $\sum n_i/N_i$ and that failure arrives when the sum reaches one. The rule has been called a working hypothesis by everybody who has ever written it down, including Palmgren in 1924 and Miner in 1945, and the property it is most often criticised for is one it cannot help having: a sum has no order. Add the same terms in any sequence and the total is the same number.

Integrating the same cycles as a crack produces something that does have an order, and two separate mechanisms put it there. One is already visible in the growth law. The other is not in the growth law at all.

Two sequences, the same cycles, and two lives #

The first mechanism is arithmetic and it needs nothing beyond the growth law. $\Delta K = Y\Delta\sigma\sqrt{\pi a}$, so a block of cycles meeting a long crack does more than the same block meeting a short one. Put the large block first and it is spent on a short crack; put it last and it is spent on a long one.

That reasoning predicts that large cycles first should be the more damaging arrangement, and it is — by a factor of nearly two. The intuition that says otherwise, that a structure should be “run in” gently, is exactly backwards for a crack: gentle cycles early grow the crack slowly and leave the severe ones to act on something still small.

What the figure is really about is the Miner sum at the moment of failure. For the low-high sequence it is 0.99, so the rule is right to a per cent. For the high-low sequence it is 0.68 — the detail has failed with a third of its Miner life apparently unused. Every experiment on block loading since the 1950s reports the same asymmetry, usually as sums between 0.3 and 2.5, and this calculation produces it from the growth law alone with nothing fitted — which is the same sense in which the detail decides and the steel does not was a consequence of geometry rather than a fact about materials.

There is a reading of that worth keeping. Miner’s rule is not wrong about the cycles; it is wrong about the crack. It has no state variable — no quantity that carries the effect of the past forward — so it cannot express a mechanism whose whole content is that the structure is different afterwards.

The mechanism that is not in the growth law #

The second mechanism is not a refinement of the first. It runs the other way and it is larger.

A cycle of stress intensity $K$ leaves a zone of yielded material ahead of the crack tip of size roughly

$$r_y = \frac{1}{6\pi}\left(\frac{K}{\sigma_y}\right)^2 .$$

When the load comes off, the surrounding elastic material closes back on that zone and leaves it in compression. A crack growing into a compressive field grows more slowly, and it keeps doing so until it has grown out of the zone the overload made. Wheeler’s model writes that as a multiplier on the rate,

$$\frac{da}{dN} \to \left(\frac{r_y}{a_{\mathrm{OL}} + r_{\mathrm{OL}} - a}\right)^{p}\frac{da}{dN},$$

active while the crack’s own plastic zone is inside the one the overload left.

The arithmetic is worth doing once because the numbers are unintuitive. The zone scales as $K^2$ and so as the square of the stress range: a 160 N/mm² cycle beside a 40 N/mm² one leaves a zone sixteen times larger. The multiplier is then $(1/16)^{1.5} = 1/64$, so the cycles immediately after the overload grow the crack at one sixty-fourth of their former rate, recovering as the crack advances.

One cycle in four million adds fifty-four per cent to the life. The cycle itself grows the crack by three microns. Everything else it does is done by what it left behind.

When it happens, and what it is worth #

The benefit falls as the crack lengthens, and the reason is a ratio rather than a size. The overload’s zone and the crack’s own both scale as $K^2$, and $K$ scales as $\sqrt a$ for both — so the ratio between them stays at sixteen whatever the crack length, and the retardation’s strength does not change. What changes is how quickly the crack eats through the zone: the zone’s length grows as $a$, and the crack’s speed grows as $a^{1.5}$, so a long crack clears a long zone faster than a short crack clears a short one.

A heavy lift early in a structure’s life is worth more to it than the same lift late. That is the opposite of how an abnormal load is usually recorded, which is as an event that used something up.

The reason nobody takes credit for it #

The exponent $p$ has no derivation. It is a number chosen to make the model fit a set of tests, and different sets give different numbers: the published range for structural steel runs from about 1.3 to 3.5, and across it the same overload is worth anywhere between thirty per cent and several times the life.

So the honest position on retardation is a strange one and it is worth stating plainly. The effect is real, it is large, it is measured in every variable-amplitude test anybody has run, and it is not creditable. No code permits a designer to claim it, because a model whose one parameter spans an order of magnitude in the answer is not a model a design can be built on.

What it is for is the other direction. It explains why constant-amplitude tests are conservative for real spectra, why Miner sums measured in service scatter above one as often as below, and why a detail that has survived an unexpected overload is in a better state than a calculation would suggest. It is a reason to be unsurprised rather than a reason to be generous, and it belongs with the rest of what a damage-tolerant design has to hold in mind and cannot put in a calculation.

The same zone, under a different name #

The plastic zone in this essay is the same object that appears in three other places here, and naming the connection is worth more than the model is.

It is the zone that makes a notch less damaging than its elastic stress concentration says: the notch the crack does not feel in full is a notch whose peak stress has been relieved by yielding over a volume of the same kind. It is the mechanism behind shakedown, where a structure that yields once on its first overload and is elastic ever afterwards is a structure whose residual stresses have rearranged to make the subsequent cycles survivable. And it is what a proof load is for: applying a load larger than any the structure will see, once, in order to leave it in a better state.

Retardation is shakedown at the scale of a crack tip, and the three ideas are the same idea at three sizes. Each of them says that a structure which has yielded somewhere is not thereby damaged — it may be improved — and that the improvement is a residual compression left behind by the excursion.

Which is also why the objection to all three is the same. The benefit is real and it is not creditable, because it depends on the overload actually having happened at the size assumed, and a design cannot assume an event it does not control.

Which of the two mechanisms wins #

Both effects have the same sign for the same arrangement and they are not the same size.

Sequence alone — the $\sqrt a$ effect — made the difference between 12.4 and 6.4 million cycles in the first figure, and that is a large number because the two blocks differ by a factor of 2.25 in range and by a factor of 27 in count. On a spectrum whose bands are closer together it is much smaller: running the five-band traffic spectrum of the essay that integrated the crack with its bands in ascending and descending order changes the life by under a per cent, because the bands interleave a few thousand times a year and no arrangement of them meets a meaningfully different crack.

Retardation does not care about that. It is triggered by a single cycle and it lasts for as long as the crack takes to clear a zone, which can be millions of cycles.

So the practical ordering is: for ordinary traffic, order is a per-cent effect and Miner’s blindness to it is not the problem. For a structure that sees occasional extreme events — a crane that lifts one heavy load a year, a bridge with abnormal-load permits, an offshore structure with storms — the order is a tens-of-per-cent effect and it runs in the structure’s favour.

And for a block-loaded test rig it is the dominant effect, which is where the whole literature on Miner sums between 0.3 and 2.5 comes from. Block loading is the one loading history that maximises the disagreement, and it is the one almost every test has used.

The whole of it, once, by hand #

A detail cycling at 40 N/mm² with a crack at 3 mm, $Y = 1.12$, $\sigma_y = 355$ N/mm².

Its own stress intensity is $\Delta K = 1.12 \times 40 \times \sqrt{3\pi} = 137$ N/mm^1.5, and its plastic zone is

$$r_y = \frac{1}{6\pi}\left(\frac{137}{355}\right)^2 = 0.0079\ \text{mm}.$$

One cycle at 160 N/mm² gives $K = 549$ and a zone of $r_{\mathrm{OL}} = 0.127$ mm — sixteen times larger, as the square of the ratio of ranges requires. The overload’s plastic front sits at $a + r_{\mathrm{OL}} = 3.127$ mm.

Immediately afterwards the retardation factor is

$$\left(\frac{0.0079}{3.127 - 3.000}\right)^{1.5} = \left(\frac{1}{16}\right)^{1.5} = \frac{1}{64},$$

so the ordinary traffic grows the crack at one sixty-fourth of its former rate. As the crack advances the denominator shrinks and the factor rises; it reaches one when $a = 3.119$ mm, at which point the retardation is spent.

That is 0.119 mm of crack growth to be done at a rate that starts at one sixty-fourth. At the unretarded rate of $5.21\times10^{-13}\times137^3 = 1.34\times10^{-6}$ mm a cycle those 0.119 mm would take 89,000 cycles; held at one sixty-fourth throughout they would take 5.7 million. The factor recovers as the crack advances, so neither number is the answer — the integration gives 2.3 million extra cycles, against a remaining life of 3.3 million without the overload.

The zone is small and the time to cross it is not. That is the whole shape of the effect: a tenth of a millimetre of crack, at a fraction of the usual speed, costing more cycles than the rest of the detail’s life. It is also why the answer is so sensitive to the exponent — the exponent decides the fraction, and the fraction multiplies a number of cycles that is already large.

Where it shows in service, and where it is being relied on without anybody saying so #

Three places where the sequence effect is already load-bearing in practice, in the sense that removing it would change what is observed.

Test rigs disagree with bridges, and this is why. A variable-amplitude test that reproduces measured traffic gives Miner sums near one. A block-loading test on the same detail gives anything from 0.3 to 2.5. The traffic is the realistic history and the blocks are the convenient one, and the whole scatter in the literature on Miner’s rule is an artefact of how the tests were sequenced rather than a property of the rule.

Proof testing a structure is not neutral. A load test applied to an existing bridge is an overload at every detail in it simultaneously, and it leaves every one of them retarded. The test is done to demonstrate capacity; it also, as a side effect nobody claims, extends the fatigue life of whatever it did not break.

And an ageing structure’s own history matters. A crane that lifted its rated maximum in its first year is in a different state from one that has never exceeded half of it, and the difference is not recorded anywhere. This is the one place where a structure’s log book is a fatigue input — and it is a log book kept for other reasons, in which the entry that matters is the one that looked like a near miss.

The fourth place is the one to be careful of. A designer who knows about retardation and applies a spectrum with its overloads stripped out — because the overloads were rare, or because the load model does not contain them — has removed the retardation and kept the damage, and has produced a calculation more conservative than either the full spectrum or the truncated one. That is the usual reason a fatigue check comes out worse than the structure’s service record, and it is a modelling error rather than a safety margin.

Which free body produced the number #

The free body is the same through-thickness crack in a wide plate, $\Delta K = Y\Delta\sigma\sqrt{\pi a}$ with $Y = 1.12$. The integration runs an explicit sequence of blocks once each, in the order given, and accumulates a Miner sum alongside so that what a sum would have said at the moment of failure can be read off rather than assumed.

The two sequences in the first figure share the same critical crack length, because it follows from the largest range present and both sequences contain the same ranges. The overload figures hold the critical length fixed at 44 mm across every run, so that what is being compared is the sequence and not the structure — an overload raises the peak stress and would otherwise shorten the critical crack, which is a real effect and a different one.

What the picture cannot show #

Crack closure, which is the better explanation. Wheeler’s model attributes retardation to a plastic zone ahead of the crack. The mechanism most now accepted is Elber’s: the crack’s wake is left with a plastically stretched layer that holds the faces together for part of the cycle, so the effective $\Delta K$ is smaller than the applied one. The two predict similar retardation and different things about what happens after a compressive underload, which Wheeler’s model cannot represent at all.

Underloads. A compressive overload does the reverse — it flattens the wake and accelerates subsequent growth. A spectrum containing both is not the sum of two effects.

Multiple overloads. The model here keeps one plastic front, the largest so far. Two overloads close together interact, and the second may be entirely inside the first’s zone and do nothing.

Anything about the load’s rate or duration. Every cycle here is a number. A crack in a corrosive environment has a rate that depends on how long the load is held as well as how large it is, and no sequence effect in this essay survives that.

The assumption this rests on #

That a plastic zone is the state variable. Everything in this essay follows from the claim that a crack carries one number forward from its history — where the largest plastic front so far is — and that everything else about the past is forgotten.

That is a strong claim and it is the reason the model is tractable. It is also why the model cannot represent the wake, the closure level, the residual stress redistribution or the order of two overloads, and why every one of those is a known deficiency rather than a subtlety. A better model carries more state; the limit of that programme is a finite-element analysis of a growing crack with cyclic plasticity, which is done, costs days, and is not a design method.

What survives the objection is the direction of the effect, which is the part a designer needs. Overloads retard, underloads accelerate, high-low is worse than low-high, and a Miner sum is exact for one arrangement of cycles and optimistic for another. None of that depends on the exponent.

Still open: the same detail in a thicker plate #

Every crack here and in the essay before it has grown in a plate whose thickness never appeared. The geometry factor was 1.12, the crack was through-thickness, and the only lengths in the calculation were the crack and the critical crack.

Real details are not thickness-independent, and the fatigue codes say so in a form that looks arbitrary: the same detail in a thicker plate sits in a lower category, by a factor of $(25/t)^{0.2}$, with no more explanation than that thicker is worse. That is a strange rule — it makes a property of a detail depend on a dimension at right angles to the crack — and there is a crack-growth reason for it that the S-N form conceals. Whether it is about the stress gradient under a weld toe, about the crack’s shape, or about the probability that a large plate contains a large flaw, is the question after this one.