The steel that is stronger in a millisecond

Every strength in this collection is a number: $f_y$, $f_c$, $f_u$. They come from tests, and every one of those tests was run at a strain rate of about $10^{-4}$ per second, because that is what a universal testing machine does when it is asked to take a few minutes over a specimen. That is the same silence the stress at which nothing in particular happens already carries about how a yield point is defined: a number that looks like a property and is partly a procedure.

Nothing on a drawing records it. There is no axis in any stress–strain diagram on this site for how fast the strain arrived, and for most of the loads this subject deals with that is entirely reasonable. For a few of them it is not, and the difference between the two cases is the whole of this essay.

Which free body produced the number, and it is not a free body #

This is one of the few results on this site that is not derived from equilibrium at all. It is a statement about the mechanism of yielding, and the mechanism is dislocation motion.

A metal yields when dislocations move through its lattice, and they move by overcoming obstacles — other dislocations, solute atoms, precipitates. Overcoming an obstacle is partly thermal: at any temperature the lattice is vibrating, and a dislocation gets occasional help from a thermal fluctuation large enough to push it past. Give it more time and it is more likely to get that help, so a slow test needs less applied stress.

That is the physical content of rate sensitivity: a slow test lets thermal activation do part of the work. It follows immediately that the effect should be larger at low temperatures, where there is less thermal help to begin with, and that it should be larger for materials whose obstacles are strongly temperature-dependent — which is why body-centred-cubic metals like mild steel are markedly rate-sensitive and face-centred-cubic ones like aluminium and austenitic stainless are much less so.

The engineering description is Cowper and Symonds’ two-constant fit:

$$\frac{\sigma_d}{\sigma_s} = 1 + \left(\frac{\dot{\varepsilon}}{D}\right)^{1/q}$$

with $D = 40.4$ per second and $q = 5$ for mild steel. It has one property worth noticing before anything else: at $\dot{\varepsilon} = D$ the factor is exactly 2, so $D$ is not an arbitrary fitting constant. It is the rate at which the material is twice as strong.

Which rates are real #

The regimes decide whether any of this matters, and they are surprisingly clean.

A testing machine runs at $10^{-4}$ per second, which is the reference.

An earthquake delivers strains at $10^{-3}$ to $10^{-2}$ per second. That is one or two decades above the reference and the enhancement is a few per cent — which is why no seismic code carries a rate factor and why nobody misses it.

A vehicle impact or a dropped load gives $10^{-1}$ to $1$ per second, and the enhancement is 20 to 30 per cent.

A blast gives $10$ to $10^{3}$ per second, where mild steel is 1.5 to 2.5 times its static strength and concrete in tension is more.

So this is a correction that is either negligible or decisive, with almost nothing in between, and the reason is the fifth root. A power of $1/5$ is very flat: it takes five decades of rate to double the factor, so nothing happens for a long time and then a great deal happens at once. A correction with a different exponent would have had to be carried everywhere.

Three things that do not go up #

The enhancement is easy to describe as free strength, and three qualifications turn that into a more useful picture.

The modulus does not move. Elastic stiffness is set by the interatomic bond, not by dislocation motion, and it is rate-independent to within a per cent over the whole range above. So the elastic response of a blast-loaded structure is exactly what it always was, and only the yield point has moved — which means the elastic deflection at first yield increases, and the structure absorbs more energy elastically before it starts to be damaged.

The ultimate strength barely moves. The enhancement on $f_u$ is roughly a third of the enhancement on $f_y$, so the ratio $f_u/f_y$ closes up: from 1.56 statically to about 1.23 at blast rates. That is a real loss. A material with a small margin between yield and ultimate has less strain hardening, a shorter plateau, and less warning — the property that appears in none of the equations is being spent.

It also breaks a design argument that is used everywhere without being stated. Capacity design — making a connection stronger than the member it joins, so that the member yields first — relies on the ratio between the member’s actual yield stress and its connection’s actual capacity. Rate enhancement raises the member’s yield stress and may not raise the connection’s, particularly where the connection fails by a mechanism that is not yielding: a weld fracture, a bolt shear, a concrete cone. A hierarchy that holds at testing-machine rates can invert at impact rates.

And the fracture toughness goes down. The ductile-to-brittle transition temperature rises with rate, which is why a Charpy test — a notched bar hit with a pendulum, which is a rate test dressed as a toughness test — reads conservatively for a slowly loaded structure and correctly for an impacted one. A steel that is comfortably ductile at its service temperature under a slow load can be brittle at the same temperature under a fast one, and that combination is what the sub-grade of a steel specification exists to control.

Concrete, which is worse and matters more #

Concrete is far more rate-sensitive than steel, and its tensile strength more than its compressive.

The model code’s fit gives a compressive increase factor of about 2.3 at 100 per second and a tensile one of about 2.7, against steel’s 2.2. The mechanism is different — it is about the time available for microcracks to find their way round aggregate particles, and about the viscous resistance of water in the pores — but the consequence for design is direct: a rate-sensitive analysis can change which material governs.

That is the reason blast design of a reinforced concrete element is not simply the static design with a factor on it. Raise the concrete’s tensile strength by 2.7 and its compressive by 2.3 while the steel goes up by 2.2, and the section’s balance shifts: a member that was under-reinforced statically can be over-reinforced dynamically, and a shear failure that was comfortably avoided can become the governing mode because the shear capacity has risen less than the flexural demand.

The other direction, which nobody calls a rate effect #

There is a mirror image of all this at the slow end, and it goes by a different name for no good reason.

Load a material very slowly — over years rather than seconds — and it is weaker, not stronger. Concrete under sustained compression fails at about 85 per cent of its short-term strength, a fact so well established that the coefficient sits in every design code without anybody calling it a rate effect. Timber is worse: its strength under a permanent load is about 60 per cent of its five-minute value, and the modification factor for load duration is one of the largest numbers in a timber code.

The mechanism at that end is not thermal activation past an obstacle; it is time-dependent damage — microcracking that propagates slowly under a sustained stress. But the shape is continuous with the fast end, and a single curve of strength against loading time runs from the years-long sustained-load value through the testing-machine value to the blast value without a break in it.

Seen that way, the testing machine’s rate is not a natural reference point at all. It is a convention, chosen because a few minutes is a convenient length for a test, and it happens to sit in the flat middle of a curve that falls to the left and rises to the right. Everything in this collection is calibrated against an arbitrary point on a slope, and the reason that works is that most structural loads sit near the same point.

What it is worth to a structure #

The place the enhancement earns its keep is in an energy balance rather than in a strength check.

A blast delivers an impulse — a pressure over a very short time — and for a structure whose natural period is long compared with the pulse, the peak pressure is irrelevant and only the impulse matters. The structure then absorbs that impulse’s kinetic energy by deforming plastically, and the deflection it reaches is set by

$$\text{kinetic energy in} = \text{resistance} \times \text{deflection}$$

so the deflection goes inversely with the resistance. Raise the plastic moment by the rate factor and the collapse load rises exactly in step, and the deflection under a given impulse falls by the same ratio. That is the energy argument a dropped weight makes, with the resistance rather than the stiffness as the variable.

For mild steel at blast rates that is a factor of two on the resistance and a halving of the damage, from a material property nobody paid for. Blast design without rate enhancement over-predicts the deflection by a factor of two, which is the difference between a structure that is repairable and one that is written off — and it is the reason blast codes carry the factor while every other code ignores it. The same energy balance decides how much ductility an earthquake asks for, at a rate where the factor is worth nothing.

How the numbers were got, which is the hard part #

Measuring a stress–strain curve at a thousand strains per second is not a matter of running a testing machine faster, and the instrumentation problem is worth a paragraph because it explains why the data are as sparse as they are.

At those rates the specimen deforms in a millisecond or two. A conventional load cell cannot report a force in that time — it has a natural frequency of its own and rings — so the force reading at the moment of interest is the load cell’s dynamics rather than the specimen’s. The specimen itself has the same problem: a stress wave takes a finite time to travel its length, so for the first several transits the two ends of it are carrying different stresses and there is no single “the stress in the specimen” to report.

The instrument that solved it is the split Hopkinson pressure bar, and it works by not measuring the specimen at all. A short specimen is sandwiched between two long elastic bars; a striker hits the first bar and sends a compressive pulse along it; part of the pulse reflects off the specimen and part transmits into the second bar. Strain gauges on the bars, far from the specimen and in material that stays elastic, record the incident, reflected and transmitted pulses, and the specimen’s stress and strain history is inferred from those three by one-dimensional wave theory.

That inference has assumptions in it — the specimen has to be short enough to be in equilibrium after a few wave transits, and the bars have to stay elastic and one-dimensional — and the technique works over a band of about $10^2$ to $10^4$ per second. Below it, servo-hydraulic machines reach perhaps $10^2$. Between the testing machine’s $10^{-4}$ and the Hopkinson bar’s $10^2$ there is a stretch of six decades that is genuinely hard to measure, and it is exactly where the vehicle-impact regime sits.

So the two-constant fit is spanning a gap in the evidence rather than summarising a dense dataset, and its behaviour in the middle of the range is an interpolation. That is worth knowing about a number being used to divide a deflection by two.

Where the model stops #

Cowper and Symonds’ fit does not return one at the static rate. At $10^{-4}$ per second it says 1.076, so the constants carry a seven per cent offset at the very rate the static strength was measured at. That is not a small error in a two-parameter fit and it is a fair measure of how much of this model is a curve rather than a mechanism. Dividing through by the value at the reference rate fixes it and is what should be done; not everybody does.

The fit has no temperature in it. Since the mechanism is thermal activation, the rate sensitivity and the temperature sensitivity are two readings of one thing, and a proper description — Johnson and Cook’s, or a thermally activated flow model — carries both. Cowper and Symonds at room temperature is an isotherm through a two-dimensional surface.

And the strain rate is not uniform. Everything above quotes a rate as though a structure had one. A member responding to a blast has a rate that varies through its section, along its length and through time — highest at the extreme fibre near a hinge at the instant the hinge forms, and near zero elsewhere. Design practice picks a representative rate from the expected response time, which is circular: the response depends on the strength and the strength depends on the response. The usual resolution is one iteration.

The generalisation #

The habit worth carrying is a question to ask of any material number.

Every property in this collection was measured under conditions, and the conditions are not on the drawing. A yield stress has a rate in it. A concrete strength has an age and a curing regime in it. A fatigue category has a stress ratio and a weld geometry in it. A fire rating has a heating curve in it that no real fire follows. A creep coefficient has a humidity in it.

Most of the time the design conditions are close enough to the test conditions that the difference is inside the factors, and asking the question wastes a minute. Occasionally they are five decades apart, and then the number on the drawing is describing a different material from the one in the structure.

The way to tell which case is which is not to look harder at the number. It is to ask what the test was, and whether the structure is doing the same thing.