The crack between the two checks

The flaw that sets the strength drew the two ways a cracked plate can fail as two lines. Fracture mechanics gives a failure stress $K_c/(Y\sqrt{\pi a})$ that falls as the square root of the crack length and does not know the steel can yield; yielding gives a failure stress equal to the yield stress, which does not know there is a crack. Where the two lines cross is the transition crack length, and the essay’s main finding was that it gets shorter as the steel gets stronger. In its closing section it noted that fracture mechanics requires the plastic zone at the crack tip to be small, and that “it stops being small exactly where it matters most”.

This essay is about exactly where that is. The practical check for a cracked member takes the lower of the two lines — fracture if the crack is long, yield if it is short — and between them it has a corner. No real plate has a corner there. A crack in a steel that can yield carries a yielded strip ahead of each tip; the strip relieves the crack’s singularity and at the same time removes material from the section, and near the crossing both effects are large. What the plate actually does lies below both lines, and how far below is the whole content of a model Dugdale published in 1960 that fits in one expression.

The strip of yielded steel #

Dugdale’s idea is geometric. Ahead of each crack tip, instead of the unbounded elastic stress, there is a thin strip of steel that has yielded and holds exactly the yield stress $\sigma_Y$. The length of the strip, $\rho$, is whatever makes the stress at its far end finite: the singularity of the lengthened crack under the applied stress is cancelled by the singularity of the yield stress pulling its faces together. That one condition fixes the strip at $\rho = a[\sec(\pi\sigma/2\sigma_Y) - 1]$ for a crack of half-length $a$ in a wide plate — small at small stresses, and growing without limit as the applied stress approaches yield.

A strip that grows without limit is net-section yielding by another name. A strip that stays short is linear fracture mechanics with a correction. Between the two, the model says the crack fails when its effective stress intensity,

$$K_{\text{eff}} = Y\sigma_Y\sqrt{\pi a}\;\sqrt{\tfrac{8}{\pi^2}\ln\sec\frac{\pi\sigma}{2\sigma_Y}},$$

reaches the toughness. Solved for the stress, that is a closed form:

$$\sigma_f = \frac{2\sigma_Y}{\pi}\arccos\!\left[\exp\!\left(-\frac{\pi K_c^2}{8Y^2\sigma_Y^2 a}\right)\right].$$

For a short crack the exponential is tiny, the arccosine is $\pi/2$ and the failure stress is the yield stress. For a long one the argument is small, $\arccos(e^{-x}) \approx \sqrt{2x}$, and the expression becomes $K_c/(Y\sqrt{\pi a})$ exactly. The model contains both of the simple lines as its limits and draws a curve between them that lies under both. At the crossing, 20.1 mm for this steel, it is at 288 N/mm² against the 355 both lines claim.

The same error for every steel #

The expression depends on the crack length only through the combination $K_c^2/(Y^2\sigma_Y^2 a)$, which is $1/\pi$ times the ratio of the transition length to the crack length. So measured against its own transition length, every crack in every steel has the same strip-yield curve.

Drawn that way, the simple check’s error is a single curve. It is nothing for cracks much shorter than the transition length and nothing for cracks much longer, and it peaks at the crossing, where the lower of the two lines overstates the plate by a factor of 1.23 — for every toughness, every grade and every geometry factor. The peak is $1/[(2/\pi)\arccos(e^{-\pi^2/8})]$, a number that depends on nothing but the model. The error exceeds 5 per cent from half the transition length to four times it: a band of cracks from 10 to 85 mm in the steel of the first figure, which covers the lengths an inspection of a welded structure is typically set to find and then has to assess.

The usual correction to fracture mechanics near yield is Irwin’s: lengthen the crack by an estimate of its plastic zone, $(K/\sigma_Y)^2/2\pi$, and use the lengthened crack in the elastic formula. The dotted curve applies it, capped at yield. It removes the unsafe corner and replaces it with a slightly conservative bend, 0.97 of the strip-yield stress at the crossing — which is to say the correction works, and a check without it does not.

The concrete that makes the same bridge #

The third curve belongs to a different material and a different essay. The bigger one is the weaker one found that concrete’s strength falls with the size of the member, and that Bažant’s law for it, $\sigma_N = Bf_t/\sqrt{1 + D/D_0}$, runs from a plateau at small sizes to a fracture-mechanics decline at large ones, with a transition at a size set by a material length. Written in the variable of this essay — the crack length over its transition length, which is the same thing as the member’s size over its transition size — it is $1/\sqrt{1 + a/a_t}$, and it has exactly the two asymptotes Dugdale’s curve has.

It bridges them lower down. At the crossing it is at $1/\sqrt{2} = 0.71$ of both lines, and the simple check against it would be 41 per cent too high. The difference is the material in the strip. Dugdale’s strip holds the full yield stress however far it opens, which is what a ductile steel does; the fracture process zone ahead of a crack in concrete softens as it opens, carrying less stress the more it has cracked, so it contributes less and the bridge sags further. The size effect and the cracked-plate check are one problem: a strength that holds at small scale, a fracture that governs at large scale, and a zone of damaged material at the tip whose length is comparable to the crack exactly where the two meet.

A plastic zone longer than the crack #

Fracture mechanics is an elastic theory with a small region of plasticity at the crack tip that it agrees to ignore. The model gives the size of that region at the moment of failure, and it shows why the ignoring cannot be done where it matters.

At the transition length the yielded strip at failure is 49 mm long — 2.4 times the crack it sits ahead of. Irwin’s estimate, the one usually quoted when the validity of fracture mechanics is being judged, is 13 mm, which beside a 20 mm crack looks like a correction rather than a regime change. It is a quarter of the strip-yield length because it is computed from the elastic stress intensity at failure and ignores that the yielded strip itself redistributes load onto more material. For every crack shorter than 36 mm the strip at failure is longer than the crack.

That settles the question the earlier essay left: the plastic zone is not small beside the crack anywhere in the band where the simple check is wrong, and a fracture calculation there is outside its own hypothesis. It is also the reason the elastic-plastic parameters exist, and why a toughness, like a ductility, turns out to depend on what it was measured over. The crack-tip opening displacement is, in Dugdale’s model, simply how far the faces of the strip have separated at the original crack tip — $\delta = (8\sigma_Y a/\pi E)\ln\sec(\pi\sigma/2\sigma_Y)$ — and a toughness measured as a critical opening is the same criterion as the one above, stated in a way that survives when the zone is large.

Both checks passed, and the plate fails #

The strip-yield curve can be redrawn with the applied stress over the yield stress on one axis, $L_r$, and the crack’s elastic stress intensity over the toughness on the other, $K_r$. In those coordinates the two simple checks together accept everything inside a unit square, and the strip-yield model accepts everything under the curve $K_r = L_r[(8/\pi^2)\ln\sec(\pi L_r/2)]^{-1/2}$. That is the failure assessment diagram, the form in which structural integrity assessments of cracked steel structures are made; this model’s curve was the one the diagram was first drawn with, and the curves in use today are fitted close to it.

The square and the curve agree at two points only: the top left, where the stress is negligible and the crack is at its toughness, and the bottom right, where the crack is negligible and the stress is at yield. Everywhere else the curve cuts the corner, and at nine tenths of yield it is at 0.73 where the square is at one. Put a row of cracks in the grade 355 plate at a working stress of 302 N/mm² and the 20 and 25 mm cracks land in the cut corner: each has a stress intensity below the toughness and a stress below yield, each passes both of the checks a designer would make, and each fails.

The diagram also shows what the corner means physically. A point near the top of the square is a crack failing mostly by fracture with a little help from yield; a point near the right is a section yielding with a little help from the crack. The corner is where each mechanism is helping the other a lot, and the two simple checks, each of which assumes the other mechanism is absent, have nothing to say about help.

Why the corner has done so little harm #

A 23 per cent error on the unsafe side at the most ordinary crack lengths in the most ordinary steel ought to have a record of failures behind it, and it largely does not. Three things have kept it hidden, and each is worth knowing because each can be removed without anyone noticing.

The first is the partial factors. A member checked at its design resistance works at perhaps two thirds of its yield stress under characteristic load, and a point at two thirds of yield on the assessment diagram is well inside the curve for any crack below the transition length: at $L_r = 0.67$ the curve is at 0.89, and the lost corner is mostly beyond the loads a structure sees. The error is real and the margin has been paying for it. A structure assessed rather than designed — its loads known more precisely, its factors reduced accordingly — has had that margin taken away on purpose, and the corner is then inside the loads.

The second is the yield stress itself. The value in a calculation is a specified minimum, and the stress it names is usually exceeded in the actual plate by 10 to 20 per cent; the strip carries the actual flow stress, not the specified one, and a real plate’s strip-yield curve sits above the one drawn. That too is a margin that disappears in exactly the material where it is most needed — the heat-affected zone beside a weld, where the flaws are, and where the joint is the part that runs out of ductility and its toughness is the lowest in the assembly.

The third is that real flaws are mostly not sharp cracks at full length when they start. A notch or a hole multiplies the stress without the singularity, and even the fatigue that eventually starts a crack there feels the notch only in part, through an average over a volume the material owns. By the time a flaw is a sharp crack at the transition length it has usually been growing by fatigue, and the question has become when it will be found rather than whether this one load will break it.

What a stronger grade still buys #

The flaw that sets the strength found that a stronger steel has a shorter transition length, so a flaw harmless in a weaker grade decides a stronger one. On the two simple lines that is a sudden loss: the stronger grade keeps its whole advantage up to its own transition and then loses all of it, landing on the same fracture line as the weaker grade. The strip-yield curves make the loss gradual and, for the stronger grades, early.

Grade 690 is 2.51 times as strong as grade 275. With a 2 mm crack it carries 2.45 times as much, which is nearly its whole advantage. With a 10 mm crack — nearly twice grade 690’s own transition length of 5.3 mm — it carries 1.66 times as much, which keeps less than half of the advantage. With a 30 mm crack, still shorter than grade 275’s own transition of 33.5 mm, it carries 1.22 times as much: most of the extra strength has gone before the weaker steel has even reached the point where the crack begins to matter to it. A crack of 30 mm is not a large one. It is a fatigue crack a few years into its growth, or a lack-of-fusion defect a routine ultrasonic inspection is set to find, and at that length the premium paid for the stronger steel buys about a seventh of its nominal value.

That is the strength-toughness trade stated as a number rather than as a warning. The stronger grade needs more toughness in proportion to its strength to keep its advantage across the same range of cracks, and steels are not usually supplied that way: toughness is specified by a subgrade against temperature and thickness, the same subgrade whatever the strength.

Toughness buys the last few per cent dearly #

Hold the crack and the grade, and ask instead what toughness the plate needs.

On the simple lines the plate reaches its yield stress at a toughness of 100 $\text{MPa}\sqrt{\text{m}}$ and nothing more is needed. On the strip-yield curve it is at 0.81 of yield there. Ninety per cent of yield takes 122 $\text{MPa}\sqrt{\text{m}}$; ninety-five, 143; ninety-nine, 183 — nearly twice the toughness the simple check asks for. Toughness buys strength steeply below the crossing and slowly above it, because above it the strip is already long and each increment of toughness only lengthens a strip that is carrying the yield stress anyway.

The same curve at lower toughness shows what the cold does to it.

Halve the toughness, as a drop of a few tens of degrees does to an ordinary structural steel in its transition range, and the transition length falls by four, to 5.0 mm. The error at the crossing is still 23 per cent — it always is — but the crossing has moved to cracks that are routinely present in welded joints, and the 20 mm crack of the first figure now sits well into the fracture-governed range, failing at 169 N/mm² with the fracture line a modest 5 per cent above it.

The arithmetic at the crossing #

For grade 355, a toughness of 100 $\text{MPa}\sqrt{\text{m}}$, which is 3,160 $\text{N/mm}^{1.5}$, and an edge-crack factor of 1.12, the transition length is $(K_c/Y\sigma_Y)^2/\pi = (3{,}160/397.6)^2/\pi = 7.95^2/\pi = 20.1$ mm. At that crack the exponent in the strip-yield formula is $\pi K_c^2/(8Y^2\sigma_Y^2 a) = \pi \times 63.2/(8 \times 20.1) = \pi^2/8 = 1.234$, and it is $\pi^2/8$ at every transition, which is why the error there is universal. Then $e^{-1.234} = 0.291$, $\arccos 0.291 = 1.275$ radians, and $\sigma_f = (2 \times 355/\pi) \times 1.275 = 288$ N/mm².

The strip at failure is $a[\sec(\pi\sigma_f/2\sigma_Y) - 1] = 20.1 \times (1/\cos 1.275 - 1) = 20.1 \times (3.43 - 1) = 49$ mm. And the exponent that makes the plate reach 95 per cent of yield is $-\ln\cos(0.95\pi/2) = 2.54$, so the toughness it needs is $\sigma_Y Y\sqrt{8 \times 2.54\,a/\pi} = 355 \times 1.12 \times \sqrt{8 \times 2.54 \times 20/\pi} = 4{,}530$ $\text{N/mm}^{1.5}$, which is 143 $\text{MPa}\sqrt{\text{m}}$.

A thin strip, a wide plate and an elastic crack #

The model’s free body is a wide plate containing a crack whose faces are free, extended at each tip by a strip whose faces are closed by the yield stress, under a remote tension. Superposing the plate with the applied stress and the plate with the strip’s closing stress, and requiring the two stress intensities to cancel at the strip’s far end, gives the strip’s length; the effective stress intensity is the one that, in an elastic plate, would produce the crack opening the strip-yield plate actually has.

The strip is thin and holds exactly the yield stress. That is a plane-stress idealisation — a thin sheet in which the yielded zone really is a narrow band — and it is least good in thick plate, where constraint raises the stress in the zone above yield. The geometry factor is carried through unchanged, which is exact for the wide-plate centre crack Dugdale solved and an approximation for the edge crack used here. The steel does not harden; a hardening steel’s strip would carry more than the yield stress as it opened, and the flow stress is often taken midway between yield and ultimate for that reason. And the plate is wide: in a narrow member the net section yields before the strip grows long, and the square’s right-hand side moves in by the fraction of the width the crack removes.

What the pictures cannot show #

That the crack is still growing. Every curve here is for a crack of fixed length meeting a rising load once, and in a structure the length is the variable: fatigue lengthens it, and the plastic zone an overload leaves slows it for a while afterwards. The strip-yield curve says which length becomes critical at a given stress; it says nothing about when the crack will reach it. Nor can the figures show the scatter in the toughness itself, which in the transition range of a structural steel is as large as the factor of two between the two runs drawn here, from one test piece to the next.

Still open: the toughness that belongs to the plate’s thickness #

Every result above uses one toughness for each steel. But toughness depends on thickness — a thin plate yields through its thickness at the crack tip and is tougher than a thick one of the same steel, which is constrained — and the strip-yield model assumes the thin case while the toughness tests that supply $K_c$ are made on thick specimens to be conservative. Which toughness belongs in the model for a given plate, and whether the strip-yield curve for a 12 mm web and the one for a 60 mm flange of the same steel differ by more than the simple check’s 23 per cent, is the question that decides whether this correction is the whole of the story for real sections.