The ceiling the bore sets

The overstrain the release gives back worked through autofrettage. A thick tube is pressurised past yield once, so that when the pressure is released the yielded bore is squeezed by the elastic outer wall and left in compression; the working pressure then has to overcome that compression before it can do harm. The essay found the method’s limit. The release itself must stay elastic, so the bore can be taken from yield in tension to yield in compression and no further, and the most autofrettage buys is twice the first-yield pressure — less, if the steel yields early in reverse, a Bauschinger factor β taking it to $(1 + \beta)$ times.

Its last section named the older way to put a bore in compression. Shrink an outer tube on to an inner one: heated, the outer tube expands enough to slide over; cooled, it grips the inner, in hoop tension itself and squeezing the inner into compression. Nothing yields. Built so that both layers reach yield together at the working pressure, a compound tube has no Bauschinger effect to lose. The essay asked whether a shrink fit beats an overstrain for a given wall, and whether the two combined — an autofrettaged inner tube with a shrunk-on jacket — could carry a tube past the factor of two that limits either alone.

Layers that yield together #

A. V. Gadolin’s design for a compound gun barrel is the shrink fit at its best. Split the wall into $n$ layers, each with the same ratio of outer to inner radius, $k^{1/n}$ for a wall of overall ratio $k$, and choose the interference at each fit so that at the working pressure the bore of every layer reaches yield at the same moment. By Tresca’s criterion a layer’s bore yields when the pressure difference across it is $(\sigma_y/2)(1 - k^{-2/n})$, and with every layer carrying that much the tube carries

$$p_n = n\,\frac{\sigma_y}{2}\left(1 - k^{-2/n}\right).$$

For one layer that is the plain tube’s first-yield pressure, $p_y = (\sigma_y/2)(1 - 1/k^2)$. For two it is $\sigma_y(1 - 1/k)$. As the layers become many it tends to $\sigma_y \ln k$, the pressure that would make the whole wall plastic.

Take the earlier essay’s tube: a wall 2.5 times its bore in a steel yielding at 900 N/mm². A plain tube first yields at 378 N/mm². Two layers, split at the geometric mean of the radii, carry 540 — 1.43 times as much. At the working pressure both bores are at yield, the liner’s and the jacket’s, which is the point of the design. At rest the liner’s bore is held at 0.43 of yield in compression, and the jacket’s bore, where it grips the liner, at 0.43 in tension.

The same wall autofrettaged to its best carries 756, twice the plain tube, its bore resting at full yield in compression. At this wall ratio two layers are far behind.

More layers, and a ceiling #

Against the wall ratio, the layers gain slowly and steadily: two layers give 1.20 times the first-yield pressure on a wall 1.5 times its bore, 1.43 at 2.5, 1.60 at 4. Four layers give 1.32, 1.75 and 2.00. Autofrettage rises faster and then stops dead at 2, or at 1.7 for a steel whose reverse yield has been cut by its first yielding.

Every curve stops at 2. That is the result the rest of this essay is about, because it is not a property of either method.

Gadolin’s formula, followed without limit, would carry the 2.5 to 1 tube to $\sigma_y \ln 2.5$, 2.18 times its first-yield pressure. The layers do not get there. Three give 1.63, four 1.75, six 1.88, eight 1.95, and from there the gain is capped at exactly 2, however many layers are added. The shrink fit, which never yields its steel, is held to the same ceiling as the overstrain that yields it on purpose.

The range is the wall’s #

The reason is in what a prestress can and cannot change. Once the layers are assembled — or once the overstrained tube has sprung back — the tube answers its working pressure as one elastic wall. Every point in it moves by the same Lamé solution for the whole wall, whatever was locked into it beforehand. At the bore, the stress difference swings by

$$\Delta(\sigma_\theta - \sigma_r) = \frac{2p\,k^2}{k^2 - 1}$$

between rest and work. That range is fixed by the pressure and the wall. No prestress can change it; a prestress can only choose where it sits. A residual stress, in the sense of the stress that was there before the load, is a field in equilibrium with nothing outside the body, and adding it to an elastic response shifts every stress by a constant at each point without touching how much that stress changes.

The top of the range may reach yield in tension. The bottom may reach yield in compression. So the range can be at most twice the yield stress, which is what $p \le \sigma_y(1 - 1/k^2) = 2p_y$ says, and that is the ceiling of 2. The range belongs to the wall; the floor it may hang down to belongs to the steel.

It is the same statement the load it can carry once made about a continuous beam under loads that come and go. Melan’s theorem says a structure shakes down if some residual stress keeps every state of loading inside yield, and a residual stress can move a range but never shorten it: any section whose elastic range under the varying load exceeds twice its yield fails by alternating plasticity, whatever is done to it. A thick tube under a pressure that comes and goes is that theorem at a single point — the bore — and every prestressing method is a choice of residual stress. Autofrettage and shrink-fitting are two ways of finding the best residual field. Neither can find a range.

Why the Bauschinger effect costs the overstrain and not the fit #

The floor is where the two methods differ. A shrink-fitted liner has never yielded. Its steel will yield in compression at the same stress it yields in tension, so its bore may hang all the way down to $-\sigma_y$ at rest. An autofrettaged bore has already yielded once, forward, in the overstrain, and yielding one way, and then the other is the experience that lowers a steel’s reverse yield — the Bauschinger effect. If reverse yield comes at $\beta\sigma_y$, the autofrettaged bore may hang down only to $-\beta\sigma_y$, and its range, and its pressure, are capped at $(1 + \beta)p_y$.

So the comparison depends on the steel. For a steel with no Bauschinger effect, $\beta = 1$, autofrettage reaches the ceiling of 2 at any wall over 2.22 times its bore, and no practical number of layers catches it. For a steel whose reverse yield falls to 0.7, autofrettage is capped at 1.70, four layers beat it, and eight come within a few per cent of the ceiling it can no longer reach. A gun steel or a high-pressure vessel steel, quenched and tempered to a high yield strength, often shows a pronounced Bauschinger effect; that is the case in which the old method keeps its place.

The combination is capped like its liner #

The earlier essay’s last question was whether combining them — autofrettaging an inner tube and then shrinking a jacket over it — could carry the tube past the factor of two. The range argument answers it without a calculation of the combination at all.

The assembled tube still answers its working pressure as one elastic wall, so the bore still swings by the whole wall’s range. The bore is the liner’s, and the liner has been autofrettaged, so its floor is $-\beta\sigma_y$. The combination is capped at $(1 + \beta)p_y$ — exactly the liner’s own cap. The jacket can supply some of the bore’s compression that the overstrain would otherwise have had to supply, which lets the overstrain be gentler and leaves less of the wall yielded; it cannot lower the floor. A combination can never do better than the most-yielded layer at the bore allows.

The only way past 2 is to change the rule the range is measured against. A liner of a stronger steel than its jacket, its yield stress higher than the jacket’s, can carry a longer range at the bore, and the jacket, seeing a smaller range further out, can be of ordinary steel. That is a different design, and it is not a prestress at all: it puts the strong material where the range is largest.

A quarter of a millimetre #

The practical difficulty with a shrink fit is in how small its essential number is. For the two-layer tube on a 100 mm bore the interface is at 79 mm radius, and the jacket must be made 0.20 mm smaller on its radius than the liner it is shrunk over — 2.6 parts in a thousand. Heating the jacket by about 214 °C opens that much, and on cooling it grips the liner with 116 N/mm² of contact pressure.

The design depends on that interference being right. Too little, and the liner’s bore is not compressed enough to reach yield at the same pressure as the jacket’s; the liner yields first, at less than the design pressure. Too much, and the liner’s bore at rest is pushed past its own reverse yield on assembly, before the tube has seen any pressure. The tolerance on the interference has to be a small fraction of the interference itself, which on this tube is a few hundredths of a millimetre on a 158 mm bore — and every additional layer is another fit, each with its own tolerance, its own heating and its own chance of being wrong. That is why compound tubes of many layers were built for guns, where the cost of the barrel was nothing against the cost of its failing, and why the many-layer limit is approached in practice by winding wire or tape under tension rather than by shrinking on dozens of tubes.

Autofrettage needs no fit at all. A single forging, pressurised once, finds its own residual field, because the yielded zone sets its own boundary. What it gives up is the floor: it cannot squeeze the bore without first yielding it.

At the same pressure, a different place for the range #

The ceiling compares the two methods at the most each can carry. A designer more often has a pressure to carry and asks which tube does it better, and there the range argument says something sharper about fatigue.

Take a working pressure of 540 N/mm², what the two-layer tube is designed for. Under it the bore’s stress difference swings by the whole wall’s range, 1.43 times yield, in either tube. In the two-layer tube the range runs from −0.43 at rest to yield at work. In the autofrettaged tube, overstrained to its best, the bore rests at −1.00 and the same range carries it only to +0.43 at work. The range is identical; where it sits is not. The load that never came near failing anything found that a fatigue crack grows with the range of stress, and the essay before this one found that it grows far more slowly when the range is spent mostly in compression, where the crack is held shut. At the same pressure, the autofrettaged bore spends most of every cycle closed; the two-layer bore spends most of it open.

That is a second reason an overstrain wins for a pressure that cycles, and it is independent of the ceiling. A shrink fit could be built with more interference to push its range down, but only by putting its bore nearer reverse yield at rest — and with two layers it cannot both do that and keep the jacket’s bore inside yield at work. The overstrain finds the deepest floor for itself; the fit has to be designed to it, layer by layer.

A fit that is never unloaded #

There is a cost the elastic calculation hides. The jacket of a compound tube is in hoop tension permanently, from the day it is shrunk on, and its contact pressure on the liner — 116 N/mm² in the tube drawn — is held by nothing but the jacket’s elastic stretch. A stress held by a fixed stretch is exactly the state the strain that was imposed described: if the material can creep, the stress leaks away while the stretch stays, and the fit loosens.

For a forged steel tube at room temperature that leak is negligible. For a gun barrel heated by every round, or a pressure vessel in hot service, it is not, and the shrink fit’s prestress is lost at the rate the jacket relaxes at its working temperature. The stress that leaks away worked out how fast a stress held at a fixed strain falls in a creeping material; here the strain is the interference, and what falls is the compression at the bore that the whole design depends on. An autofrettaged tube’s residual field relaxes too, but its source is a plastic strain that stays where it was put; a fit’s source is an elastic strain that is the very thing relaxation removes.

So the two methods also differ in how they age. A shrink fit begins with no yielded steel and can lose its prestress to heat; an overstrain begins with yielded steel and keeps its prestress as long as the yielded zone is not yielded back. Which is the better trade depends on whether the vessel’s enemy is the number of cycles or the temperature.

What the calculation takes for granted #

The layers share one steel. Gadolin’s equal-yield design, and the range cap of 2, are both for layers of one yield strength. With different steels the cap is set by the bore’s own steel and the design changes; the rule that a prestress cannot shorten a range does not.

Tresca’s criterion, plane stress, no end load. A closed vessel has an axial stress from its end caps, and von Mises’s criterion gives slightly different numbers. The structure of the argument — a range fixed by the wall, a floor fixed by the steel — is the same with either.

There is no temperature gradient. A vessel heated from inside, or a barrel after a round, has a bore hotter than its outside, and the thermal stress that follows puts the bore into compression while it is hot and back towards tension as it cools. That is a second range at the bore, added to the pressure’s, and it uses up the same budget of twice the yield stress; a tube designed to the ceiling for pressure alone has none left for heat.

The fits do not slip. The assembled tube is taken to respond as one wall, which needs the contact pressure at each fit to stay positive at work. In the equal-yield design it does; a fit loosened by a too-small interference would let the layers respond separately and the range at the bore would be the liner’s alone, larger.

The Bauschinger effect is one number. A real steel’s reverse yield depends on how far it was strained forward and on how it hardens, and what is left when the load comes off is the more careful picture of unloading. The figures take the simplest description, a reverse yield at $\beta\sigma_y$.

Still open: the tube that is wound #

The many-layer limit is reached in practice by winding: a liner wrapped in high-strength wire or tape under tension, each turn a layer shrunk on by its own winding tension. Every turn has its own prestress, set by how hard it was pulled, and a wire’s yield strength is several times a forged liner’s, so the jacket is far stronger than the liner it compresses. Whether a wire-wound tube’s ceiling is the liner’s range — still 2, or $1 + \beta$ if the liner is autofrettaged first — or whether the wire’s strength lets the design put the bore’s floor lower than the liner alone could hold it, and how the winding tension should change from the first turn to the last, is the question the ceiling puts to the tube built of a hundred layers instead of four.