What a second deployment costs

Folding that gets built lists what an account of deployables still owes and ends the list with a question rather than a topic: what folding is not good for, which is anything needing to deploy repeatedly.

It is a good question because every structure in that essay deploys once. A solar array unfolds and stays; an airbag opens and is thrown away; a stent expands and remains expanded; a starshade opens and never closes. Nothing in the field cycles, and the reason turns out to be arithmetic rather than habit.

The chain, in three links #

Three facts each established elsewhere compose into the result, and none of them is new.

A crease has a radius. Nothing in a body folds on a line gives a hinge its width and the crease has a radius gives the same quantity to paper. A crease of radius $\rho$ in a sheet of thickness $t$ bends its outer fibre through a strain of about $t/2\rho$ — sharper crease, more strain.

A material takes less strain the more often it is asked. That is the ordinary shape of fatigue: a strain amplitude survivable once is not survivable a thousand times, and the usual form is $\varepsilon \approx C N^{-b}$ for a material exponent $b$ somewhere near a half. So the sharpest hinge that survives $N$ cycles has

$$\rho(N) = \frac{t}{2C}\,N^{\,b}$$

And a blunter hinge packs worse. Four materials, four optima puts the best fold count at $k^* = S/2(\pi-2)\rho$, inversely proportional to the radius, with the packing it delivers proportional to the same thing.

Compose the three and the packing a structure can reach falls as $N^{-b}$. Ten times the life costs a factor of three in smallness; a thousand times costs thirty.

What that means for the shelf #

Fourth of eight, and still not chosen for it measures every pattern on its printed shelf by how much compaction it buys per unit of crease, and finds the Miura in the middle of it. This adds a second axis to that shelf, and it is a much steeper one.

A pattern’s conversion rate varies by a factor of about eight across the shelf. The cycle count varies it by a factor of thirty between a one-shot device and one that works a thousand times, and every pattern pays the same factor, because the penalty is in the hinge rather than in the geometry. So a designer choosing between patterns is choosing within a factor of eight, and a designer choosing between one deployment and a thousand has already given up more than that choice can recover.

The map is the interesting row. It is the one application here that is cycled, it is cycled by a person rather than a mechanism, and everybody knows what happens to it: a map refolded a few hundred times tears along its creases. That is the fatigue law arriving without any arithmetic, and it is the everyday evidence that the field’s one-shot habit is not conservatism.

Reading the table #

The four rows are worth walking, because the numbers are more extreme than the phrase “a power law” suggests.

At one deployment the hinge may be as sharp as the material takes once — 0.05 on a sheet of ten — the sheet folds best at 87.6 folds and packs 43.8 times. That is the regime every structure in the deployable literature is in.

At ten the radius is three times larger, the best count is 27.7 and the packing is 13.9. Ten deployments is not a demanding requirement — it is what a piece of ground support equipment would face — and it has already cost two thirds of the compaction.

At a hundred the radius is ten times the original, the best count is 8.8 and the packing is 4.4. A pattern with eight or nine folds in it is not what anybody means by an origami deployable.

At a thousand the radius is 1.58 on a sheet of ten — nearly a sixth of the sheet per hinge — the best count is 2.8, and the packing is 1.4, which is to say the structure barely folds at all.

The interesting row is the second. The penalty is steepest at the start, because a power law with an exponent below one has its largest proportional effect over the first decade, so the difference between one deployment and ten is larger than the difference between a hundred and a thousand. A designer asking for “a few test deployments before flight” has already made the expensive request.

The exponent decides how fast, not whether #

The fatigue exponent is a material property and nobody has measured one for a folded sheet, which is a fair objection to the whole computation.

It does not matter as much as it looks. Across the range of exponents metals and polymers actually show — 0.4 to 0.6 — the packing falls by between sixteen and sixty-three times over three decades of cycle count. The conclusion is the same at every exponent and only its size moves.

What would overturn it is an exponent near zero, which is a material whose fatigue strength does not fall with cycle count at all. Such materials exist in a sense — below a fatigue limit, steel survives indefinitely — and that is the escape this arithmetic points at rather than forecloses: a hinge worked below its material’s fatigue limit cycles for ever and pays nothing. The price is that the limit is a strain, the strain is $t/2\rho$, and staying below it means a radius that does not depend on $N$ but is large.

Why the field’s habit is not a habit #

It is worth being explicit about what the arithmetic licenses, because “folding is no good for cycling” would be too strong.

What it says is that a folded structure’s compaction falls as a power of the cycle count. It says nothing about whether a folded structure can cycle — plainly it can, since a bellows does and a map does a few hundred times. The claim is about the ratio that makes folding attractive in the first place.

Folding is chosen over telescoping, inflating or hinged panels because it packs better. Take a factor of thirty off the packing and the comparison changes: a structure that must cycle a thousand times and packs at 1.4 is no longer beating a mechanism, and the reasons to fold — no sliding surfaces, one continuous sheet, no assembly — are still there but are no longer decisive.

So the field’s one-shot habit is a selection effect rather than a rule. Folding is used where it wins, it wins where compaction dominates, and compaction dominates where the structure deploys once. Panels instead of paper makes the neighbouring point about what a real mechanism costs a paper model; this makes it about what a requirement costs before any mechanism is drawn.

What a cycling structure would have to be #

Taking the arithmetic seriously gives a specification rather than a prohibition, and it is a recognisable one.

A structure that must cycle wants few folds, blunt hinges and a large sheet — because the packing at the fatigue-limited optimum is $k^*/2 = S/4(\pi-2)\rho$, and the only free variable left once $\rho$ is fixed by the cycle count is $S$. Making the sheet bigger does not improve the ratio; it improves the absolute compaction, which is what a designer actually wants.

That specification describes a bellows, and bellows are what engineering uses when something has to open and close repeatedly. A bellows has a handful of deep convolutions rather than many shallow ones, its folds have generous radii, and it packs badly by the standards of anything in the deployable literature. It is the fatigue-limited optimum of this arithmetic, arrived at independently and a long time ago.

So the answer to “what is folding not good for” is not that folding fails at cycling. It is that a cycling folded structure is a different object from a deploying one, with a different fold count, a different hinge and a different appearance, and calling both of them folding hides the factor of thirty between them.

The arithmetic also says where the boundary sits, which a prohibition would not. A structure asked for ten deployments is still recognisably a folded one — 27.7 folds, packing 13.9 — and a structure asked for a thousand is not. Somewhere between ten and a hundred cycles a folded design stops being a folded design, and that crossing is a property of the material’s exponent and the sheet’s thickness rather than of anybody’s judgement.

The crease itself, drawn #

The first link of the chain is the one a reader can see, and it is worth putting on the page because the whole result rests on it.

A crease is not a line. It is a region of the sheet bent through an angle, it has a width set by how sharply the material will bend without failing, and that width is surface the pattern does not get to use. The finer the pattern, the larger the share of the sheet that is crease rather than panel — and the fatigue law’s only job is to say how wide that region has to be.

The fatigue law does one thing to this picture: it raises the radius. Everything else follows without any further physics — a larger radius shifts the whole curve, the peak moves in, and the packing at the peak falls. Which is why the result is so insensitive to the details of the material: the material enters at one place and through one number.

The other cost, which this does not price #

There is a second reason the field deploys once and it has nothing to do with fatigue.

A deployment that happens once can be qualified once. A mechanism that must work repeatedly has to work repeatedly under conditions that change — after months folded, after thermal cycling, after whatever the first deployment did to it — and each of those is a separate case to demonstrate. From a shell to a solar array prices the twenty-five years between a pattern being published and flying, and finds the gap entirely in qualification rather than in knowledge.

None of that is in the arithmetic here, and it may well be the larger term. What the arithmetic does is show that even with qualification free, the geometry alone would keep cycling structures out of the compaction race.

The tube, which cycles and is built anyway #

There is one counterexample in the field’s own hardware and it deserves naming.

The tube that gets built observes that every folded structure leaving a laboratory is a sheet joined to itself — a boom, a stent, a bellows, an airbag, a packed antenna. Two of those do cycle: a bellows by design, and a boom in some designs that retract.

Both are exactly the specification this arithmetic produces. A bellows has few, deep, generously radiused convolutions; a retractable boom rolls rather than folds, which is a crease of continuously varying position and so no crease at all in the fatigue sense — the material is bent everywhere and permanently creased nowhere.

Rolling is the escape the arithmetic does not see, and it is worth stating because it is the one arrangement that cycles without paying. A rolled sheet has no hinge with a fixed location, so nothing accumulates cycles at a point, and the strain is set by the roll radius rather than by a crease radius. Everything in this essay is about patterns with creases in fixed places, which is most of the subject and not all of it.

The one number that would settle it #

Everything above turns on a quantity nobody has measured, and naming it is more useful than hedging.

The fatigue exponent of a creased sheet is unknown. The values used here — 0.4 to 0.6 — are what metals and polymers show in bulk fatigue tests, and a crease is not a bulk specimen: it is a region worked past yield on its first fold, so its second cycle starts from a material that has already changed. Whether that makes the exponent larger or smaller is not obvious, and either would change the numbers without changing the shape.

Measuring it would not be hard. Fold a coupon to a stated radius, cycle it, count the folds to failure, repeat at several radii, and fit. That is an afternoon’s work for somebody with a sheet and a jig, and it would replace three figures of assumption with one measured line. It is the cheapest missing number in this subject, and this essay’s whole quantitative content depends on it.

What the model assumes #

Fatigue follows a single power law. Real materials have a low-cycle regime, a high-cycle regime and sometimes a limit, and one exponent across four decades is a simplification that the exponent sweep is meant to bound rather than repair.

The crease’s strain is $t/2\rho$. That is pure bending of a sheet of thickness $t$ around a radius $\rho$, with no allowance for the through-thickness compression a fold actually involves.

The packing optimum is the corrugation’s. $k^* = S/2(\pi-2)\rho$ comes from a corrugation whose hinges each consume $(\pi-2)\rho$ of surface, which is a specific geometry and not every pattern’s.

And the hinge is the only thing that fatigues. A panel, a joint, a coating and an actuator all have lives of their own, and every one of them would shorten the structure’s rather than lengthen it.

The cycles are all full cycles. A structure part-deployed and re-stowed strains its hinges less than one taken flat and folded shut, and a fatigue law counts amplitude as well as number. Treating every deployment as a full one is the conservative reading and it is the one a specification usually means.

How the numbers were checked #

The packing is required to fall at every step of the cycle table, which follows from the composition of three monotone relations and would fail if any of them had been implemented with the wrong sign.

The fall is checked against the power law, not merely observed: the ratio between the first and last rows must equal the cycle range raised to the exponent, to within two per cent.

Every exponent drawn must give a fall of more than five times over the range, which is the claim that the conclusion does not depend on the exponent.

And the constant is set from a stated radius — a hinge asked to work once may be as sharp as 0.05 on a sheet of ten, which is the radius used throughout these essays — so the table’s first row is calibrated against the rest of the collection rather than chosen.

Still open: what the crease count costs in confidence #

The cycle count is one way a deployment’s requirements eat its compaction and there is a second, entirely independent of fatigue, that nobody has priced.

A deployment that needs every hinge to work is the hinge reliability raised to the crease count. A pattern folded more finely has more hinges, so the very thing that buys compaction spends the probability that the structure opens at all — and at a plausible hinge reliability the spending is not small. Sixty hinges at 0.999 each is a system at 94 per cent; three hundred is 74.

That turns the packing optimum into a different optimum. The quantity a one-shot mission wants is not the compaction a pattern reaches but the compaction times the chance of getting it, and the fold count maximising the second is lower than the one maximising the first by an amount the arithmetic can state.

Sideways from here, the same arithmetic prices the testing. A flight article deploys once and cannot be tested, so confidence has to come from testing hinges, and the number of consecutive successes needed to demonstrate a system reliability grows with the hinge count. How many tests a finely folded deployable actually needs is a number nobody appears to have written down, and it is one line of arithmetic away.

The habit worth carrying is about requirements that look like details. Ask what a requirement does to the free variable rather than to the design. “It must work a thousand times” sounds like a durability specification and is really a factor of thirty on the compaction, applied before any pattern is chosen — which makes it the first decision rather than a constraint on the last one.