The notch a crack does not feel in full

The hole that multiplies the stress by three ends by dividing the subject into three cases, and putting the whole weight of the concentration factor on one of them: under cyclic load the peak matters more than in either static case, because a fatigue crack starts where the local stress range is largest and does not care that its neighbourhood has yielded.

That is right, and it is not the whole story. A fatigue crack does not feel the elastic peak either.

Two notches with the same concentration factor and different sizes have different fatigue strengths, and the small one is much the less damaging. The elastic solution contains no length, so it cannot possibly account for that — and the account is the whole of this page.

The factor that has no size in it, and the failure that does #

Kirsch’s solution multiplies by three whether the hole is a millimetre across or a metre. That is a genuine property of the elastic field and it is exactly why $K_t$ is so useful: one number, tabulated once per shape.

What has a size in it is the gradient. The disturbance around a hole of radius $a$ decays over a distance proportional to $a$ — the same field, scaled — so a hole ten times smaller has a stress that falls away ten times more steeply. At the root of both notches the stress is $3\sigma$; a tenth of a millimetre away from the small one it is very nearly $\sigma$, and a tenth of a millimetre from the large one it is still $2.9\sigma$.

A crack does not start at a point. It starts in a volume of material containing enough grains, enough slip and enough persistent slip bands to nucleate one, and that volume has a size the material owns rather than the geometry. If the peak stress lives in a region smaller than that volume, the material initiates against something closer to the average across it.

So the quantity a fatigue crack responds to is the stress averaged over a material length, and the elastic factor is an upper bound on it that gets weaker as the notch sharpens.

Which free body produced the number #

The free body is a small block of material at the notch root, several grains across, and what crosses its faces is the elastic field the notch produced.

Two states of that block have to be distinguished. Under a blunt notch the field is nearly uniform across it: the block is in tension at $K_t\sigma$ throughout, and the crack that starts in it starts against the peak. Under a sharp notch the field falls by a factor of two across the same block, so most of the material in it is at a fraction of the peak, and the average is well below $K_t\sigma$.

Nothing about the elastic solution changes between the two. What changes is the relationship between two lengths — the notch’s root radius and the material’s own — and that ratio is the only variable in every fit ever written for this.

Peterson’s is the oldest of them:

$$q = \frac{1}{1 + a_p/r}, \qquad K_f = 1 + q\,(K_t - 1)$$

with $q$ the notch sensitivity, $r$ the root radius, and $a_p$ a material length that falls as the material gets stronger. It says exactly the argument above: when the notch is large compared with the material’s length, $q \to 1$ and the crack feels the whole factor; when it is small, $q \to 0$ and the notch disappears.

The fit is a fit. It is not derived from a mechanism, its material length is calibrated on notched fatigue tests, and Neuber’s alternative form gives numbers a few per cent different. What is not a fit is the shape of the answer, which follows from the two lengths alone.

The stronger steel keeps less of it #

Which produces the result that decides most fatigue-governed detailing, and it is the opposite of what the material selection would suggest.

The material length $a_p$ falls from 0.43 mm at 430 MPa to 0.05 mm at 1,400, because a stronger steel has a finer microstructure and a smaller plastic zone at the notch root — less material over which to average the peak away.

Put the two ends of that curve beside the plain fatigue limits they belong to and the design consequence appears. A mild steel’s plain fatigue limit is around half its tensile strength, so 215 MPa, divided by 2.40 gives 90. A 1,400 MPa steel’s plain limit is perhaps 600, divided by 2.90 gives 207. The strong steel is still better — but it started 2.8 times better and finished 2.3 times better, and every sharpening of the notch narrows the gap further.

At a 0.1 mm root the two factors are 1.38 and 2.60, and the ratio of notched strengths is 1.48 against a smooth-bar ratio of 2.8. Half the advantage bought in the material has been given back at one hole.

That is the mechanical content of a rule every fatigue code states as a bare fact: the detail decides and the steel does not. The categories in a fatigue standard carry no material strength in them at all, and this is why — not because the strength is irrelevant, but because the sensitivity rises with it at very nearly the rate the strength does.

The two ways a notch can be made harmless #

Reading the first figure as a design instruction gives two routes, and they are not equally available.

Blunt it, and the factor falls. This is the familiar one and it works on $K_t$ rather than on $q$ — a generous radius has a lower elastic factor to start with. It is what a fillet is for.

Or sharpen it past the point where the material can feel it, which sounds absurd and is what a small drilled hole at the tip of a crack does. Below about a tenth of a millimetre in a mild steel, $q$ falls fast enough that the increase in $K_t$ is outrun, and the notch stops mattering. That is why a scratch is not a crack, and why the boundary between them is a material property rather than a geometric one.

The second route is the one that connects this page to the subject that replaced it. As the root radius goes to zero, $K_t$ goes to infinity and $K_f$ goes to 1: the elastic factor and the fatigue factor diverge completely at exactly the geometry where the notch becomes a crack. A concentration factor cannot describe a crack, which is the observation fracture mechanics was invented to answer, and notch sensitivity is the same observation approached from the blunt side.

What it does to a member rather than a specimen #

A structural detail is not a machined notch, and the translation is worth making explicit because it is where the numbers on this page stop being the ones used.

A fatigue standard does not give a detail a $K_f$. It gives it a category — a stress range at two million cycles — measured on specimens of that detail, so the notch sensitivity, the residual stresses, the weld geometry and the initial defects are all inside the number and none of them is separable afterwards. A category is an experiment rather than a calculation, which is why a fatigue check is a lookup and a summation rather than a stress analysis.

What the argument here is for, then, is not computing a category. It is knowing which way a detail moves when it changes:

A machined radius helps twice — it lowers $K_t$ and raises $r$ — which is why grinding a weld toe is worth a category or two rather than a few per cent.

A small defect at a sharp feature may be worth nothing at all, because the feature is already below the material’s length and the crack is initiating against an average that the defect does not change.

And a high-strength steel in a welded detail buys nothing whatever, since the weld’s own geometry sets the category and the sensitivity of the parent metal is at the wrong end of the curve. That is the strongest form of the rule, and it is the reason the categories are written without a grade.

Where it decides something in a structure #

Three places on a real member, in decreasing order of how often the argument is needed.

A bolt hole in a fatigue-loaded plate. The elastic factor is three; the root radius is the hole’s own, ten millimetres or so, and in a structural steel that is fifty times the material length — $q$ is 0.96 and the crack feels the whole factor. A large hole gets no relief at all, which is why the categories for a plain hole and for the elastic solution agree so much better than the categories for a weld toe do. It is also why the static check on the same hole — the net section, torn diagonally — and the fatigue check on it disagree so completely about what the hole costs.

A weld toe. The root radius is a fraction of a millimetre and wildly variable — 0.1 to 1 mm along one weld — so the same toe is at $q = 0.19$ in places and 0.70 in others, on a detail whose category is a single number. The toe also carries the tensile residual stress the shop put there, which is a mean stress rather than a range and is the other half of why a welded detail is worse than its geometry. That variability is one of the reasons a fatigue category has the scatter it has, and it is why grinding a toe is worth so much: it moves the radius by an order of magnitude in the steepest part of the curve.

And a change of section. A shoulder, a cope, a curtailed flange: the radius is whatever the fabricator’s cutter had, the elastic factor is between 1.5 and 3, and the relief is large. Specifying a radius on a drawing at that location is one of the few fatigue interventions that costs nothing and moves the answer by a factor.

What the categories already contain #

A category is what this whole argument looks like after it has been measured rather than computed, and the two are worth holding side by side because each explains a limitation of the other.

The calculation on this page gives a factor and no life. It says how much of a geometrical peak a crack will feel and stops, because turning that into cycles needs a plain-material fatigue curve, a mean-stress correction and a statistical treatment of scatter — three things that vary more between laboratories than the notch factor varies between steels.

The category gives a life and no factor. It is the outcome of testing that detail, so it contains the notch sensitivity, the weld’s residual stress, the initial defect population and the eccentricity of a real joint, all compounded — and none of them can be recovered from it. Change any one of them and the category no longer applies, which is why a fatigue standard is a catalogue of pictures rather than a formula.

The relationship between them is that the calculation says which way the category moves. A detail with a larger root radius sits higher; the same detail in a stronger steel sits in the same place; a detail whose feature is smaller than the material’s own length may sit two categories higher than its elastic factor suggests. None of that is available from the catalogue, and all of it is what a designer needs when a detail is not in the catalogue at all.

The history, which is a fit that outlived its explanation #

Notch sensitivity was measured before it was explained, and the measurement is older than most of the theory around it.

August Thum’s laboratory at Darmstadt through the 1920s and 1930s produced the systematic notched-fatigue data the whole subject rests on: the same notch at several radii, in several steels, with the finding that the fatigue strength did not fall by $K_t$ and that the discrepancy grew as the material got stronger. Neuber gave the first averaging argument in 1937 — a stress averaged over a small block, with the block’s size a material constant — and Peterson gave the fitted form used here in the 1950s.

What is striking about the sequence is that the mechanism arrived last and is still contested. The averaging block, the critical distance, the plastic zone at the notch root and the weakest-link argument all reproduce the same curve, and they disagree about what to do at very sharp notches, in the short-life regime and under multiaxial loading — which is where the differences between the theories are exactly the size of the effect being predicted.

So this is a place where a well-established design rule rests on a fit whose explanation is not settled. That is not unusual in fatigue and it is worth saying rather than glossing: the reason the curve is trusted is that it was measured on thousands of specimens over eighty years, and every mechanism proposed for it has been fitted to those same specimens afterwards.

Where the model stops #

Peterson’s length is fitted to notched rotating-bending tests, mostly in the 1950s, mostly on steels between 400 and 1,400 MPa. Outside that range — an aluminium alloy, a casting, a weld — the form survives and the coefficient does not.

It is a fatigue limit argument. $q$ is calibrated at the endurance limit, where a crack must nucleate to do anything. At short lives the crack grows through the notch field rather than nucleating in it, and the governing quantity is the stress intensity of a crack in a gradient rather than an averaged stress.

The averaging volume is a fiction with a length in it. The theories of critical distances — point, line and area methods — make the fiction explicit and get better agreement, and all of them still need one calibrated length. Nothing here derives the length from the microstructure.

And a real notch has residual stress in it. A drilled hole, a punched hole and a reamed hole differ by more than their radii: the punched one has a cold-worked, tensile-residual rim, and its fatigue strength is well below anything on this page. This is why holes in fatigue-loaded members are specified by their process and not by their geometry, and why an oversize hole is a different structural object from a drilled one in more ways than its diameter.

What the pictures cannot show #

The curves plot a factor, and a factor is a ratio of two stresses. What decides whether a detail cracks is a stress range over a life, so nothing here says whether the member is safe — it says only how much of the geometry the crack will feel.

They also cannot show the sequence. The relief in the first figure is the initiation factor, and a crack that has started no longer cares about the notch that started it: it grows under the field ahead of its own tip, which within a fraction of a millimetre is the nominal field with a crack in it. So the notch decides the first ten per cent of the life and rather little of the rest — which is the reverse of the situation in a large flaw that sets the strength from the beginning.

The assumption the figure rests on #

That the material has one length, and that it is a property of its strength.

Both halves are approximations of the same shape as the size effect in concrete, and they fail in the same way. A material with two microstructural scales — a duplex steel, a composite, a weld with a coarse-grained heat-affected zone beside a fine-grained parent — has two lengths, and the notch that is large compared with one is small compared with the other. The average that a crack initiates against is then taken over whichever region is weakest rather than over a fixed volume, and the fitted length becomes a property of the pair rather than of either.

Which is worth stating as the general form: wherever a failure is governed by an average over a material length, a specimen and a structure are not comparable unless their notches are. That is true of fatigue here, of fracture in concrete, of shear bands in soil, and of the rotation capacity of a plastic hinge — the same argument each time, with a different length in it.

The ladder from here #

Later rungs on this anchor: the theories of critical distances set out properly, where the averaging is explicit and the same length predicts both notch fatigue and short-crack behaviour. The elastic-plastic notch, where Neuber’s rule relates the local stress and strain past yield and is what a strain-life calculation runs on. Residual stress at a notch, which is a mean-stress problem rather than a range one and which processes rather than geometry decide. The notch under multiaxial loading, where a hole in a biaxial field has a different factor and a different gradient. And the statistical version of the whole argument, in which a larger notch is more dangerous because it samples more material rather than because it averages less — the weakest-link account, which reaches the same conclusion by a different route and separates from this one at very sharp notches.