The notch that trades stress for strain

The hole that multiplies the stress by three found that a circular hole in a wide plate triples the stress at its edge, whatever its size, and the notch a crack does not feel in full found that a fatigue crack, averaging over a volume the material owns, feels less than the full factor at a sharp notch. Both were elastic arguments. The second ended on the case that is not: the elastic-plastic notch, where the root has yielded and the elastic factor no longer says what it carries. That is the state a notch in a steel member is in whenever its nominal stress is more than a third of yield — which is to say under most of the loads anyone designs for.

Three times, but only at first #

Take a notch of elastic factor $K_t = 3$ in a mild steel. Under a nominal stress $S$ its root carries $3S$ for as long as the root is elastic. The steel’s cyclic stress–strain curve — the one it settles to after a few cycles, which is the curve a fatigue calculation needs — reaches 0.2 per cent strain at 252 N/mm², and the root departs from elastic behaviour at a nominal stress of about a third of that, 84 N/mm². Above that the elastic answer is not slightly wrong; it is wrong in kind. At a nominal 200 N/mm² it would put 600 N/mm² at the root, which the steel cannot carry at any strain it can survive.

What the root does instead is decided by its surroundings. The root is a small volume of yielding steel held by a large volume of elastic steel around it, and the elastic surroundings impose the root’s deformation more than its stress. A root that yields does not shed its strain; it sheds its stress and strains more to make up. The question is how much more, and the material’s curve alone cannot answer it — it needs one more equation linking the root to the elastic field around it.

Neuber’s hyperbola #

Neuber’s answer, from the analysis of a notched bar in shear in 1961, is that the geometric mean of the stress and strain concentrations stays at the elastic factor:

$$K_\sigma K_\varepsilon = K_t^2, \qquad \text{so} \qquad \sigma\,\varepsilon = \frac{(K_t S)^2}{E}.$$

For a given nominal stress the right-hand side is a constant, so the root lies on a hyperbola in the stress–strain plane, and it also lies on the material’s curve. Where they cross is the root’s state.

At a nominal 200 N/mm² the hyperbola meets the curve at 293 N/mm² and 0.61 per cent. The elastic answer would have put the root at 600 N/mm² and 0.30 per cent: Neuber halves the stress and doubles the strain. The product, the quantity Neuber holds fixed, is the same either way — which is the content of the rule: the notch’s strain energy density, roughly, is set by the elastic field around it, and the root can arrange it as stress or as strain, but not reduce it.

There are two other rules a designer meets, and they bracket Neuber’s from below. Glinka’s equivalent strain energy density rule holds the root’s strain energy — the area under its stress–strain curve — at its elastic value, rather than the rectangle σε that Neuber holds: at 200 it gives 272 N/mm² at 0.44 per cent. The linear rule, older and cruder, holds the strain at its elastic value: 246 at 0.30 per cent. Measurements and fine finite-element analyses put the real root between Neuber and Glinka, nearer Glinka in plane strain and nearer Neuber in plane stress.

The two concentrations part company #

Below 84 N/mm² both concentrations are three. Above it they diverge, and fast. At a nominal 100 N/mm² — 40 per cent of the steel’s cyclic yield — the stress concentration has fallen to 2.20 and the strain concentration risen to 4.09. At 300 the root carries only 1.13 times the nominal stress, while its strain is eight times the nominal strain. The elastic factor stops being a stress factor at a third of yield, and survives only as the geometric mean of two factors that no longer resemble it.

Each half of that has a use. The falling stress concentration is why notched members in ductile steel are not weaker in a static test by the factor of three the elastic solution suggests: the root yields, spreads, and the member’s static strength is nearly that of the net section, as the strength the welder gives back and its successors found for soft zones. The rising strain concentration is why the same notch governs fatigue: a crack starts where the strain cycles most, and the strain is concentrated more, not less, once the root yields.

Three rules, a factor apart #

The three rules agree while the root is elastic and diverge as soon as it yields. At a nominal 250 N/mm² Neuber’s root strain is 0.87 per cent, Glinka’s 0.62 and the linear rule’s 0.37: Neuber’s is 40 per cent above Glinka’s and 2.4 times the linear rule’s. The linear rule, which assumes the root’s strain is unaffected by its yielding, is not a conservative simplification — it is the least conservative of the three, because a yielding root strains more than an elastic one, not the same.

Why Neuber’s estimate is always the larger is a matter of shapes. Both rules start from the elastic root, where the strain energy density is half the rectangle under the line, $\sigma\varepsilon/2$. Neuber keeps the rectangle; Glinka keeps the area. On an elastic line the two statements are the same. Past yield the curve flattens, and the area under a flattening curve is more than half its rectangle — for a curve that has gone completely flat it is nearly the whole rectangle. Glinka’s root therefore reaches its target energy at a smaller strain than Neuber’s root reaches its target rectangle, and the gap widens the flatter the curve becomes. Neither rule is derived for a real notch in a real plate; each is an assumption about which quantity the elastic surroundings impose, and the truth, being neither, falls between.

Neuber’s rule is the one most design methods use, because it is the larger and because it is a closed form. That choice has a price, and the price is paid in the place the strain is used.

What the rule costs in life #

A crack starts at a notch root after a number of cycles set by the strain the root cycles through. The relation is the strain–life curve, the sum of an elastic term and a plastic term: $\varepsilon_a = (\sigma_f'/E)(2N)^b + \varepsilon_f'(2N)^c$, with constants fitted to smooth specimens cycled in strain control. Feed it the root strain amplitude from each rule and the three rules become three lives.

Under ±120 N/mm², where the root just yields on each excursion, Neuber’s root fails at 30,300 cycles and Glinka’s at 60,300 — a factor of two — and the linear rule promises 112,000. Under ±200 the factor between Neuber and Glinka is 2.2, and the linear rule is wrong by nearly seven times. In the low-cycle range the choice of notch rule decides the life to within a factor of two, which is larger than most of the material scatter a designer worries about. At small amplitudes, where the root stays elastic, the three converge and the choice does not matter; it matters exactly where the loads are large enough to be the reason for the calculation.

The notch’s own factor moves both ends of the problem. A gentler notch, $K_t = 2$, keeps its root elastic up to a nominal 126 N/mm², and under ±120 its two rules differ by a factor of 1.65 in life, 232,000 cycles against 384,000. A sharper one, $K_t = 5$, yields its root at a nominal 50 N/mm², and under the same ±120 its rules give 3,440 and 7,670 cycles, a factor of 2.23. The sharper the notch, the earlier its root yields, the more of each cycle is plastic, and the more the answer depends on which rule says how plastic. The overload’s gift scales the same way: a single excursion to a nominal 200 leaves 100 N/mm² of compression at the root of the gentle notch and 197 at the sharp one.

That is the regime of earthquakes, of crane girders under overload, of pressure vessels started up and shut down a few thousand times, and of the loads that never come near failing anything when they come near enough to yield a notch. In high-cycle fatigue under small loads, the notch is elastic and the rules collapse back into the stress concentration factor and its fatigue cousin.

One overload, a compressive root #

The root’s yielding has a consequence that outlasts the load. Load the notch to a peak and take the load away: the root unloads along an elastic line from its yielded state, and the surroundings, which were never yielded, spring back further than the root can follow. The root is left in compression.

After a single overload to a nominal 200 N/mm² the root holds 147 N/mm² of compression in a member that carries no load. The unloading is computed on the same rule with the range doubled, which is Masing’s description of a steel’s return from a peak: an unloading curve shaped like the loading curve, twice the size. This is the notch’s version of what the overstrain the release gives back built on purpose in a thick tube: a bore yielded once and released holds compression that subtracts from every later tension. A notch overloaded once has been autofrettaged by accident, and fatigue tests on notched members show the benefit as lives lengthened, sometimes several times, after a single early overload.

The benefit has a sign. An overload in compression leaves the root in tension, and a notch that sees its largest load the wrong way first has been damaged by it rather than protected.

A mean stress yields itself away #

A fatigue check needs the root’s mean stress as well as its range, because a tensile mean opens cracks and shortens lives. The elastic answer would put three times the nominal mean at the root: 450 N/mm² for a nominal mean of 150. The root cannot carry that. Its first excursion, to a nominal 250, yields it to 317 N/mm²; its return to a nominal 50 runs down the doubled curve through a range of 440 N/mm², to −122 — compression at the root while the member is still in tension — and the cycle settles about a mean of 97. Most of the mean stress yields itself away on the first cycle, and the root’s life falls only from 72,500 cycles to 60,300, where the elastic mean of 450 would have taken it to 32,700.

This is shakedown at the scale of a notch, the behaviour the structure that settles down found in a frame: the first cycle yields, a residual stress forms, and later cycles run elastically about a different mean. It is also why fatigue detail categories, fitted to welded joints full of residual stress at yield, are written in stress range alone with no mean-stress term: the mean has already gone wherever the yielding sent it.

Where a structural design meets it #

A structural engineer rarely computes a root strain, and the reason is not that notches yield less in bridges than in machines. It is that the fatigue rules for steel structures are written in nominal stress range, with each detail — a weld toe, a bolt hole, a cope — given a category fitted to tests of that detail. The tests already contain the notch, its yielding, its residual stresses and its mean-stress relaxation, which is why the detail decides and the steel does not. The local strain approach, Neuber’s rule and the strain–life curve, is the mechanical engineer’s tool, for components whose shapes are too varied to have categories.

The two meet in two places. The first is low-cycle loading — seismic, or a plant with a few thousand start-ups — where the nominal stress ranges are near or past yield and the detail categories, fitted to high-cycle tests, are extrapolated beyond their data; there the root strain is the only honest measure, and the choice of rule matters by the factor of two the figure shows. The second is assessment: a structure that has seen an overload, a fire or an impact has had its notch roots yielded and its residual stresses rewritten, and only a local calculation can say whether that helped or harmed.

The root by hand #

At a nominal 200 N/mm², the elastic root would carry 600 N/mm², and Neuber’s constant is $600^2/203{,}000 = 1.773$ N/mm². The cyclic curve $\varepsilon = \sigma/E + (\sigma/772)^{1/0.18}$ at 293 N/mm² gives $0.00144 + (0.380)^{5.56} = 0.00144 + 0.00460 = 0.00604$, and $293 \times 0.00604 = 1.77$: the crossing. The stress concentration is $293/200 = 1.47$, the strain concentration $0.00604 \times 203{,}000/200 = 6.13$, and their product is 9.0 — three squared.

Glinka’s rule asks instead that the area under the curve equal the elastic $600^2/(2 \times 203{,}000) = 0.887$ N/mm². At 272 N/mm² the elastic part of the area is $272^2/(2 \times 203{,}000) = 0.182$ and the plastic part $272/1.18 \times (272/772)^{5.56} = 0.70$, together 0.88: the crossing again, a little lower down the curve, because the area under a curve that bends over is less than the rectangle Neuber counts.

Unloading from there, the range is a nominal 200 again, and on the doubled curve — Masing’s, the same shape twice the size, which is what yielding one way and then the other found a steel’s return from a peak to be — Neuber gives a range of 440 N/mm²; the root ends at $293 - 440 = -147$ N/mm².

What the rules leave out #

The notch is in plane stress. Neuber’s rule was derived for a shear-loaded notch and holds best where the root is free to contract across its thickness. A notch in a thick plate is constrained, the yielding is suppressed, and the root’s strain is nearer Glinka’s or below; the bracket is still the right picture, but the truth slides along it with the thickness.

The cyclic curve is the stabilised one, plotted in engineering stress, which at these strains is within a fraction of a per cent of the true stress the curve was always following. A real steel hardens or softens over its first cycles, and the root’s first excursion follows the monotonic curve, which for a mild steel has a yield plateau the cyclic curve lacks. The residual stress after one overload is therefore somewhat different from the figure’s, and the mean stress takes a few cycles to settle rather than one.

The material constants are representative. The cyclic and strain–life constants are of the kind tabulated for a hot-rolled mild steel, and a particular steel, a weld metal or a heat-affected zone has its own. The rules’ differences survive a change of constants; the lives do not.

And the notch is a notch, not a crack. A sharp crack has no root radius and no elastic factor, and the question becomes one of fracture, not of local strain.

The yielded zone and the strain gradient #

They cannot show the yielded zone itself, which spreads from the root into the plate as the load grows and whose size decides how good Neuber’s assumption of an elastic surrounding is. They cannot show the strain gradient at the root, which the earlier essay found decides how much of the elastic factor a crack feels, and which yielding flattens. And they cannot show the sequence of a real load history, in which the order of overloads and small cycles decides the residual stresses that every later cycle runs about.

Still open: the history that decides the mean #

Every cycle here is one of a kind, run after one first loading. A real load history mixes a few large excursions with many small ones, and each large excursion resets the root’s residual stress — to compression after a tensile peak, to tension after a compressive one. The small cycles that follow run about whatever mean the last large one left. Whether a notch’s life under a realistic spectrum depends more on the order of its large cycles than on their number, and whether counting cycles by their ranges alone, as the standard methods do, misses the mean stress that the sequence of peaks writes into the root, is the question the single overload leaves for a structure whose largest loads arrive in an order nobody chooses.