The front that counts for half its length

The toughness that belongs to the plate took the master curve’s weakest-link term seriously. In the ductile-to-brittle transition a ferritic steel’s cleavage starts at the weakest spot the crack front passes through, so a longer front samples more spots and is weaker, and the curve scales its toughness by $(25/B)^{1/4}$ for a front $B$ millimetres long. For a crack through the thickness of a plate the front is the plate’s thickness, and that essay found how much an assessment with the standard 25 mm toughness overstates a thick plate’s strength: by 8 per cent at 6 mm and by 45 per cent at 100 mm, at the worst crack length.

It ended on the cracks inspections actually find. A fatigue crack from a weld toe is not a through crack. It starts as a thumbnail at the surface — a semi-ellipse a few millimetres deep and several times as long — and its front is a curve along which the stress intensity varies, largest at the deepest point for a long crack and at the surface for a near-semicircular one. A weakest link along such a front does not sample it evenly. This essay asks how much of the curved front counts, and what an assessment loses by counting all of it, none of it, or the plate’s thickness instead.

The stress intensity round a curved front #

The crack is 8 mm deep in a 40 mm plate under a uniform tension, and its length at the surface is varied. Newman and Raju’s solution, fitted to finite-element results and the standard description of a surface crack, gives the stress intensity at every point of the front:

$$K(\phi) = \sigma\sqrt{\frac{\pi a}{Q}}\,\big[M_1 + M_2 (a/t)^2 + M_3 (a/t)^4\big]\,g(\phi)\,f_\phi(\phi),\qquad Q = 1 + 1.464\,(a/c)^{1.65},$$

with $\phi$ the ellipse’s angle — $90^\circ$ at the deepest point, $0^\circ$ and $180^\circ$ where the front meets the surface — and $f_\phi = [(a/c)^2\cos^2\phi + \sin^2\phi]^{1/4}$ the factor that makes a long crack most intense at its deepest point.

The long crack’s front is twice as intense at its deepest point as at the surface. The semicircle’s is nearly uniform, a little higher at the surface, where the free face lets the crack open more. The 40 mm crack, five times as long as deep, sits between. A fracture assessment that reads one number off this curve reads the largest, and it is right to: the largest stress intensity is where the crack would grow if it grew by fatigue, and it is where an assessment that knows nothing of weakest links would expect cleavage to start.

Where along the front cleavage starts #

The master curve’s statement for a straight front of length $B$ at a uniform stress intensity $K$ is that it cleaves with probability

$$P = 1 - \exp\!\left[-\frac{B}{25}\left(\frac{K - K_{\min}}{K_0 - K_{\min}}\right)^4\right],$$

with $K_0$ the 25 mm specimen’s Weibull scale and $K_{\min} = 20\ \text{MPa}\sqrt{\text{m}}$. The $B/25$ is the weakest link: every millimetre of front is an independent chance, and the chances multiply. Along a front whose $K$ varies, the same statement is an integral of the same quantity:

$$P = 1 - \exp\!\left[-\frac{1}{25}\int_{\text{front}}\left(\frac{K(s) - K_{\min}}{K_0 - K_{\min}}\right)^4 ds\right].$$

The fourth power is what decides the answer. A stretch of front at nine-tenths of the largest stress intensity contributes, once $K_{\min}$ is taken off, about half as much per millimetre as the stretch at the largest; a stretch at two-thirds contributes a tenth.

For the long cracks the chance is concentrated round the deepest point and fades to almost nothing at the surface: three-quarters of the 80 mm crack’s chance of cleaving lies in the middle third of its front. For the semicircle it is spread round the whole front and peaks where the front meets the surface. This is the picture the earlier essay could not draw. A through crack’s front is all at one stress intensity, so every millimetre of it counts equally, and the thickness adjustment takes the whole front. A curved front counts only where it is intense.

How much of the front counts #

The integral can be turned back into a length: the length of straight front, at the largest stress intensity on the curved one, that would give the same chance of cleaving. That effective length is what the thickness adjustment should be fed.

The front counts for a little over half its length, and the share hardly moves with the crack’s shape or depth: between 0.54 and 0.6 for every crack drawn except the nearly circular ones, where the stress intensity is so nearly uniform round the front that almost all of it counts. The stability is not a coincidence. A long crack has a long front and a sharply peaked stress intensity; a short one has a short front and a flat one; the fourth power weights the two in ways that roughly cancel. It means a single rule — feed the adjustment about half the front — would serve most surface cracks, and that the two obvious rules, all of the front or none of it, are both wrong by a predictable amount.

Three rules against the integral #

Three rules suggest themselves to an assessor who has a surface crack and the earlier essay’s adjustment. Read the largest stress intensity on the front against the toughness of a straight front as long as the whole curved front. Read it against the toughness of a front as long as the plate is thick, which is what the adjustment does for a through crack. Or read it against the 25 mm toughness, as the standard check does with no adjustment at all.

The whole-front rule is the steady one. It always counts more front than the weakest link does, so it always gives a lower failure stress, by 3 to 8 per cent across every shape drawn. That is a safe rule with a small, nearly constant cost, and it can be applied without the integral.

The 25 mm rule is the dangerous one, because its error has a sign that depends on the crack. For a crack about as long as it is deep its front is about 25 mm long anyway, and the rule happens to be close. For a long shallow crack the front is far longer than 25 mm and nearly all of it is near the peak, so the rule leaves out most of the front that counts, and its failure stress runs 19 per cent above the weakest link’s at a crack ten times as long as deep. Long shallow cracks are the ones fatigue produces at weld toes, because adjacent initiation sites coalesce along the toe before the crack grows deep.

The plate’s-thickness rule is right only by accident. It counts 40 mm of front whatever the crack, which is about the effective length of a crack five times as long as deep in this plate, so at that shape it is close; for longer cracks it counts too little and for shorter ones too much. The next figure shows the accident from the other side.

Four failure stresses as the crack deepens #

As the crack deepens at a fixed shape, every rule’s failure stress falls, roughly as the inverse square root of its depth, and the rules keep their order. The whole-front rule stays 8 per cent below the weakest link throughout. The 25 mm rule starts below it, for shallow cracks whose fronts are shorter than 25 mm, and crosses above it once the front’s effective length passes 25 mm — at a depth of about 8 mm for this shape. The plate’s thickness falls further below the weakest link as the crack deepens, because the front grows past the plate’s thickness while the rule’s count does not.

The plate’s thickness, by accident #

Change the plate under the same crack and only one rule moves much. The weakest link and the other two rules depend on the plate only through the small $a/t$ terms in the stress intensity. The plate’s-thickness rule depends on it through the toughness, as $(25/t)^{1/4}$. So the rule that gives a through crack exactly the right front gives a surface crack the right answer only where its plate happens to be as thick as the crack’s effective front: here 26 mm. In a thinner plate it is unsafe, by 9 per cent at 15 mm; in a thicker one it is conservative, by 19 per cent at 120 mm, where it penalises the crack for a front it does not have.

That is the answer to the question the earlier essay left: the thickness adjustment should count the crack’s own front, weighted by its stress intensity, and for a surface crack that is about half its length — not the plate’s thickness, and certainly not 25 mm.

The weakest link by hand #

For the 40 mm crack at the stress the weakest link gives, 315 MPa, the largest stress intensity is at the deepest point: $0.156 \times 315 = 49.1\ \text{MPa}\sqrt{\text{m}}$. At −60 °C, 22 degrees below the reference temperature of −38 °C, the 25 mm Weibull scale is $31 + 77\,e^{0.019 \times (-22)} = 81.7\ \text{MPa}\sqrt{\text{m}}$. The front is 46 mm long, and it is worth 56 per cent of that, 25.8 mm, at $49.1\ \text{MPa}\sqrt{\text{m}}$.

Check that against the straight-front formula. A front of 25.8 mm has a Weibull scale of $20 + 61.7 \times (25/25.8)^{1/4} = 81.2\ \text{MPa}\sqrt{\text{m}}$, and its 5 per cent toughness is $20 + 61.2 \times (-\ln 0.95)^{1/4} = 20 + 61.2 \times 0.476 = 49.1\ \text{MPa}\sqrt{\text{m}}$ — the deepest point’s stress intensity, as it must be. The whole-front rule asks the same $49.1\ \text{MPa}\sqrt{\text{m}}$ of a 46 mm front, whose 5 per cent toughness is $20 + 61.7 \times (25/46)^{1/4} \times 0.476 = 45.2$, and so allows $45.2/0.156 = 290$ MPa. The 25 mm rule allows $(20 + 61.7 \times 0.476)/0.156 = 316$ MPa.

The error as a crack size #

A stress error of a fifth sounds like a margin that a safety factor would swallow. An assessment does not usually ask for a failure stress, though. It asks how deep a crack can be tolerated at the stress the structure carries, and the answer to that question is far more sensitive, because the failure stress falls only as the square root of the crack’s depth.

At 200 MPa, a crack ten times as long as it is deep in the same 40 mm plate reaches a 5 per cent chance of cleaving at a depth of 8.9 mm by the weakest link along its front. The 25 mm toughness tolerates it to 11.5 mm — 28 per cent deeper. The plate’s thickness tolerates it to 10.5 mm. The whole front’s length stops it at 8.1 mm. For a crack five times as long as deep the four answers are 10.9, 13.0, 11.8 and 9.8 mm.

Those depths are what an inspection interval is built on. A crack designed to be found in time is one whose growth from the smallest size an inspection can detect to the critical size takes longer than the interval between inspections, and the critical size is the end of that race. Overstate it by a quarter and the race is longer on paper than in the plate by the time a fatigue crack takes to grow the last quarter of its depth — which, because growth accelerates with depth, is a much larger share of its remaining life than a quarter. An error of a fifth in stress is an error of more than a quarter in critical depth, and a larger one still in the time to reach it.

The scatter makes the same point from the other side. The master curve already describes the toughness of a given front as a spread, and the strength no specimen had showed why a design value is a fractile of a spread rather than an average. The 5 per cent level used throughout is that fractile. What the curved front changes is not the shape of the spread but how much front it is the spread of, and getting that wrong moves the whole distribution rather than its tail.

What the rules already know #

Assessment procedures for cracked structures already treat a surface crack’s front as more than a point: they assess it at the deepest point and at the surface, and some allow the thickness adjustment to use the length of crack front rather than the plate’s thickness. What the integral adds is a statement of how much of that length should be used, and it turns out to be a stable fraction. It also explains why the issue was invisible for a long time: for the cracks a 25 mm specimen most resembles — short and roughly circular — the 25 mm toughness is close, and it is only for the long thumbnail cracks of a weld toe that the front the specimen measured and the front in the structure part company.

The earlier essays on the crack that sets the strength and the crack between the two checks treated every crack as a straight through crack, which made its front a single number. A fatigue crack grows from a thumbnail towards that shape — the loops a crack grows on and the rule that points sideways both noted that real cracks start semi-elliptical and grow through the thickness first — and as it grows its front lengthens and its effective share changes, so the toughness an assessment should use changes during the crack’s life as well as its stress intensity.

Which free body the stress intensity comes from #

The stress intensity is not a force, but it comes from one. Cut the plate along the crack’s plane: on the uncracked ligament the tension that the crack faces no longer carry has to pass round the crack, concentrated at its front, and the stress intensity is the amplitude of that concentration. Round a curved front the free body is different at every point — the deepest point is held by the ligament beneath it, the surface points by the material either side — which is why the factor $g(\phi)$ raises the stress intensity near the surface and $f_\phi$ lowers it for a long crack. The integral then asks, of every point on the front, how likely it is that the weakest grain within it meets a stress large enough to cleave.

Where an elastic front in uniform tension stops #

The stress intensity is elastic and the crack is in a uniform tension. A weld toe crack sits in a stress field that falls with depth and carries a residual stress from welding; both change $K$ round the front, the residual stress most near the surface. Newman and Raju’s fit is for tension and a wide plate, accurate to a few per cent for $a/c$ from 0.2 to 1 and less so outside it.

The weakest link is the master curve’s. The integral assumes every millimetre of front is an independent chance with the same Weibull scale, which is the assumption the master curve was built on for straight fronts. A curved front also has a varying constraint — the surface points are less constrained than the deepest point — and lower constraint raises the local toughness, which would shift the chance further from the surface than the figures show.

Cleavage is the only mechanism. Above the transition, tearing replaces cleavage and the weakest-link argument no longer applies; well below it, the toughness is at its floor and the front’s length matters less. The figures are drawn 22 degrees below the reference temperature, in the transition, where the argument is strongest.

The crack does not grow. A real assessment allows some stable tearing before cleavage, which moves the front and with it the stress intensity round it.

Still open: the crack that is many cracks #

A weld toe does not grow one thumbnail. It grows several, a few millimetres apart along the toe, each with its own front, and they coalesce as they deepen into one long shallow crack. Before they coalesce the toe carries several short fronts whose combined length may exceed the coalesced crack’s, at stress intensities each lower than the long crack’s peak. Whether a toe with a row of small cracks is closer to cleaving than the single crack they will become — because it has more front, though each piece is less intense — and whether an inspection that finds the row and assesses only its largest member is then counting the crack it can see rather than the front that cleaves, is the question the coalescing toe puts to a weakest link that has so far been given one front at a time.