The same steel, brittle in January

Nearly every material property in this collection is a number with a unit. A yield stress, a modulus, a density, a coefficient of expansion: measure it once and use it.

Toughness is not like that. It is a curve, and the axis it runs along is temperature. A structural steel absorbs two hundred joules in a Charpy test at room temperature and fifteen at minus twenty, and in between it does not get gradually weaker. It changes the mechanism by which it fails — from a ductile tearing that consumes energy through the whole section to a cleavage that runs along crystallographic planes and consumes almost none.

Which free body produced the number #

The material at the tip of a crack, and the question is whether it can yield.

A crack tip is a stress concentration of unbounded severity in an elastic material — the hole that multiplies the stress by three is the blunt version, and a sharp crack has no finite factor at all. What stops the stress being infinite in a real metal is that the material yields, and the plastic zone that forms blunts the tip and consumes energy as the crack advances.

Two things can suppress that yielding.

Low temperature. Cleavage is a stress-controlled mechanism whose critical stress barely changes with temperature; yielding is a thermally activated one whose flow stress rises sharply as the temperature falls. Below the temperature at which the two cross, the material reaches its cleavage stress before it reaches its yield stress, and the crack runs.

Constraint. Near the tip of a crack in a thick plate the material cannot contract laterally — the surrounding material holds it — so it is in a triaxial tensile state. Yielding depends on shear, and the shear stress in a triaxial tension is small relative to the direct stresses, so the material can be at three times its yield stress in every direction and still not be yielding. Plane strain suppresses plasticity, and a thick plate is in plane strain where a thin one is not.

That is why the same steel behaves differently as a 10 mm plate and a 60 mm one, and why it is the opposite of every strength rule: a thicker section is more brittle.

One free parameter #

The piece that turns this from anecdote into prediction is Wallin’s master curve, and it is a remarkable result.

The observation is that the transition of every ferritic steel has the same shape:

$$K_{Jc,med} = 30 + 70\exp\left[0.019(T - T_0)\right] \ \text{MPa}\sqrt{\text{m}}$$

with a single free parameter $T_0$, the temperature at which the median toughness is 100 MPa√m. Fix $T_0$ from a small number of tests at one temperature and the whole curve for that steel is determined.

A material property that is a temperature rather than a strength is an unusual object, and it has a practical consequence worth having: it collapses a family of curves into a single axis shift. Two steels with $T_0$ of −60 and −20 °C are the same material, forty degrees apart, and everything true of one is true of the other with the axis moved.

That is what makes the design rules possible. A code’s table of limiting thicknesses against service temperature is the master curve inverted: for each combination of steel sub-grade, service temperature and stress level, find the thickness at which the critical flaw falls to the size a fabrication might contain and stop there.

The flaw, which is what all of it is about #

Toughness is only ever spent on a defect, and the arithmetic linking them is one expression:

$$a_c = \frac{1}{\pi}\left(\frac{K_c}{Y\sigma}\right)^2$$

so the critical crack length goes as the square of the toughness. A factor of two in toughness is a factor of four in the flaw that can be tolerated, which is why the transition curve’s steepness matters so much more than its height.

Put numbers on it. A plate at $T_0 = -60$ °C carrying 200 N/mm²: at +20 °C the toughness is 347 MPa√m and the critical flaw is 764 mm — larger than anything that could hide in a member. At −20 °C the toughness is 177 and the flaw is 199 mm — still large, but now a real crack that could grow there by fatigue. At −60 the toughness is 100 and the flaw is 64 mm, which is a weld defect that an inspection might miss.

Forty degrees of weather turns an inspectable flaw into an uninspectable one, with no change to the steel, the stress or the section.

Rate, which shifts it another forty degrees #

The third variable is how fast the load is applied, and it moves the curve by about as much as a substantial change of thickness does.

Yielding is time-dependent: dislocations need time to move, so a higher strain rate raises the flow stress, which is the steel that is stronger in a millisecond seen as an increase in strength. At a crack tip that same increase is a loss, because raising the flow stress brings the cleavage stress within reach.

The shift is roughly 40 °C between a static test and an impact test. Which means the Charpy test — a swinging hammer, a strain rate of hundreds per second — measures the dynamic transition, and using it for a statically loaded structure is conservative by that forty degrees. It also means a structure that is impact-loaded, blast-loaded, or subject to a sudden member loss, is operating on the dynamic curve and should be assessed on it.

Three failures that met their specifications #

The history is unusually clear-cut here, and worth the paragraph because the pattern repeats.

The Liberty ships, 1942–46. Around 1,500 of some 4,700 all-welded cargo ships suffered brittle fractures; a dozen or so broke completely in half, several while lying at anchor in cold water. The steel met the specification, which contained a tensile strength, an elongation and a chemistry, and no toughness requirement whatever. The combination that did it was a rimmed steel with a high transition temperature, welded construction with no crack arrestors, cold water, and square hatch corners — a stress concentration at a weld in a plate that was brittle at the service temperature.

The Hasselt bridge, Belgium, 1938. A welded Vierendeel truss failed in March, at −20 °C, having been in service a year. It was the first of about a dozen European welded bridges to do the same in the following two decades.

King’s Bridge, Melbourne, 1962. Fifteen months old, a span dropped after a girder cracked through. The cause was brittle fracture from a fatigue crack that had started at a welded cover plate — the toughness was low, the temperature was low, and the detail had produced the crack.

The pattern in all three: welding, which makes a structure continuous so a crack can run and leaves residual stresses at yield level to drive it; a detail that produced a flaw; a low temperature; and a specification that measured the wrong property. The detail decides and the steel does not is the same conclusion reached from fatigue, and the two failures are usually the same failure — a fatigue crack that grows until it is critical, and then goes brittle.

The Charpy test, which measures the wrong thing well #

It is worth being clear about what a Charpy test is, because the whole of design practice rests on it and it is not a fracture mechanics test at all.

A 10 mm square bar with a 2 mm V-notch is struck by a pendulum and the energy absorbed in breaking it is recorded. There is no crack — a machined notch is blunt by comparison — the specimen is small, the loading is dynamic, and the result is an energy rather than a stress intensity. Nothing about it can be substituted into $a_c = (K_c/Y\sigma)^2/\pi$.

What it is, is a reliable and cheap indicator of where the transition sits, and it correlates well enough with $T_0$ across a wide range of steels that a correlation is used in practice. The 27 J temperature is roughly $T_0 + 18$ °C; there are several such correlations and they disagree by ten or fifteen degrees.

So the property that decides is $K_{Jc}$, the property that is specified is the Charpy temperature, and between them is a correlation. That is a familiar arrangement in this subject — a compressive cube test standing in for a tensile strength, a penetration test standing in for a friction angle — and it works for the same reason: the surrogate is cheap, repeatable, and correlated well enough that the scatter it adds is smaller than the scatter already present.

The failure mode of such an arrangement is also familiar. A surrogate holds only over the population it was calibrated on, and a new steel, a new process route or a new thickness range can move outside it silently. Thermomechanically rolled steels and quenched-and-tempered steels sit differently on the Charpy-to-$T_0$ correlation than the normalised steels it was fitted to.

Where the temperature comes from #

One practical point deserves separating out, because it is regularly got wrong in the direction that matters.

The temperature to use is not the air temperature. It is the metal temperature, and for an exposed steel structure on a clear night the metal radiates to a cold sky and sits several degrees below the air. Codes therefore define a reference temperature that is the minimum air temperature with a radiation allowance, a stress-and-detail adjustment and — for a dynamically loaded structure — a strain-rate shift, all applied as temperature shifts to be subtracted before entering the table.

That is an unusual way to do arithmetic and it is worth appreciating. Every effect in this essay has been converted into a shift along one axis, because the master curve’s single free parameter makes it possible: thickness, rate, stress level, residual stress and cold work all become degrees Celsius, added up, and compared with one number.

It is the same move as an effective length in a column — the ends decide the length that matters — where a family of restraint conditions is collapsed onto a single axis so that one curve can serve all of them. Where the analogy is useful is in knowing what to distrust: a shift-based rule is only as good as the assumption that the effects are independent and additive, and near the ends of the range they are neither.

What a specification does about it now #

The response was to add a property to the specification, and the way it was added is worth knowing because it is not a stress.

Steel sub-grades — JR, J0, J2, K2 in the European system — are defined by the temperature at which a Charpy specimen absorbs 27 joules: +20, 0, −20 and −20 at 40 J respectively. That is a transition temperature by another name, and buying a sub-grade is buying a shift of the master curve.

The design rule then combines four things: the sub-grade, the lowest service temperature, the stress level, and the detail category, and returns a limiting thickness. It is a table rather than a calculation, and behind it is exactly the fracture arithmetic above with an assumed flaw size — usually taken as what a proportionate inspection would find.

Two things about that are worth noticing. The assumed flaw is the load-bearing assumption and it is invisible in the table. And the rule is expressed as a thickness because thickness is the variable a designer chooses, not because thickness is the mechanism — the mechanism is constraint, and a thin plate with a highly constrained detail can be worse than a thick one without.

Where the model stops #

The master curve is for ferritic steels. Austenitic stainless steels have a face-centred cubic lattice with far more available slip systems, and they do not have a transition at all — they stay ductile to cryogenic temperatures, which is why liquefied gas tanks are made of them or of 9% nickel steel. Aluminium likewise has no transition.

The scatter is large and is part of the model. Cleavage is a weakest-link process — it initiates at whichever particle in the sampled volume is worst placed — so toughness measurements scatter enormously, and the master curve is a median with a defined statistical distribution around it. A single test does not fix $T_0$; a handful does.

The weld metal and the heat-affected zone were treated as the parent. They are not: the zone beside a weld has been through a thermal cycle that changes its grain structure, and the strength the welder gives back is the aluminium version of a change that in steel is a toughness change rather than a strength one. The lowest toughness in a welded joint is usually in the coarse-grained heat-affected zone, a millimetre or two from the fusion line.

Residual stresses were absent. A weld leaves residual stresses at yield level, which add to the applied stress in the $K$ calculation without appearing in any load case — the stress that was there before the load is that field, and near a weld it can double the driving force on a crack.

Ductility and toughness were treated as the same thing. They are related and not identical: elongation at failure is measured on a smooth specimen with no constraint, and a steel can have twenty per cent elongation and very poor toughness. The property that appears in none of the equations is the ductility argument, and this essay’s property is the one that survives when a crack removes the freedom to yield.

And the flaw was assumed to be there. The whole apparatus computes what size of defect is tolerable; it says nothing about whether one exists. That is an inspection question, and the honest link between them is that the design assumes a flaw of the size the inspection regime would reliably find — so a specification that tightens the steel and loosens the inspection has changed both sides of the inequality. That is the same coupling the smallest of six failures meets in a timber joint, where the governing mode depends on a dimension nobody controls tightly.

Where else the transition shows up #

Three consequences follow that are not usually filed under brittle fracture and should be.

Cold-formed sections. Bending a plate cold work-hardens it and raises its transition temperature at the corners, sometimes by tens of degrees. A cold-formed member is therefore more brittle where it is most stressed, and the codes’ rules about minimum bend radii are a toughness rule wearing a forming rule’s clothes.

Punched holes. Punching leaves a work-hardened and micro-cracked edge, which is why holes in members that will be fatigue-loaded or that work at low temperature have to be reamed or drilled. It is the same mechanism at a smaller scale.

Hot-dip galvanizing. Immersing a fabricated member in molten zinc at 450 °C relieves some residual stress and can also cause liquid metal embrittlement in a highly stressed detail. The failures are rare and are always at a location where a residual stress and a stress concentration coincide.

Each of those is a fabrication decision with a fracture consequence, which is why the design office cannot own this failure mode alone.

The generalisation #

The idea to carry away is that a property that is a curve cannot be specified as a number, and specifying it as a number is how a whole generation of structures was got wrong.

Strength is a number, near enough, over the range of temperature and rate that structures see. Toughness is not: it varies by an order of magnitude over forty degrees, by a factor of three with thickness, and by another forty degrees with rate. Reducing it to a single value — “the toughness of S355” — is a category error, and the reason it took a war and a dozen bridges to notice is that the number is perfectly adequate on a warm day in a thin plate.

The second idea is about which side of a design a property lives on. Strength is compared against a load, and a load is something a designer computes. Toughness is compared against a flaw, and a flaw is something a fabricator leaves and an inspector finds. That moves the whole question out of the calculation and into the specification, the procedure and the inspection regime — which is why brittle fracture is one of the few structural failure modes where the design office cannot fix the problem alone.