What crimp interchange actually conserves

A woven cloth pulled along the warp gets longer and narrower, and no fibre in it stretches. The thread was longer than the cloth to begin with, because it went over and under; straighten it a little and the cloth gains that length; and the length has to be taken from somewhere, so the weft crimps more and the fabric closes in across.

That is the mechanism, and it is correct. This essay is about the arithmetic that follows it, which is usually presented as though the mechanism determined the answer. It does not. It leaves one equation short, and the equation everybody supplies to close the system is a convention with consequences.

What inextensibility actually says #

Write $l_1$ for the length of warp thread in a repeat, $L_1$ for the cloth’s length along the warp, and $c_1 = l_1/L_1 - 1$ for the warp crimp; the same with a 2 for the weft.

Extending the cloth means increasing $L_1$. The warp thread does not change length, so

$$c_1' = \frac{l_1}{L_1'} - 1,$$

and the new warp crimp is determined completely. That much is a consequence of nothing stretching, and it is exact.

Now the weft. Its thread length $l_2$ is also unchanged, so

$$(1 + c_2')\,L_2' = l_2.$$

One equation, two unknowns — the new weft crimp and the new width. Any pair satisfying that product conserves the weft’s length: the cloth could narrow a great deal and crimp a great deal, or narrow slightly and crimp slightly, and inextensibility is indifferent between them.

So “nothing stretches” fixes the extension’s effect on the warp and says nothing at all about the width. Something else has to decide, and what that something is has been left implicit in every account of interchange this site has been able to find.

The rule everybody uses #

The rule in universal use is that the crimp the warp gives up is added to the weft, in percentage points. Warp crimp falls from eight to five; weft crimp rises from four to seven; the total crimp is unchanged.

It is a natural rule and it is not a law. Nothing in the geometry says the two crimps sum to a constant, and there is no conservation principle behind it — it is bookkeeping, of the kind that closes a system so a number can come out.

The figures on this page use it, and they assert it rather than leaving it implicit. Two assertions run every time one draws: that both thread lengths are unchanged, which is the physics, and that the total crimp is unchanged, which is the rule. Separating them is the point. The first cannot fail without something being badly wrong; the second cannot fail at all, because it is what the code does — and asserting it makes the assumption visible to anybody who changes the code later.

What the geometry says instead #

Peirce’s solved geometry does supply the missing equation, and it is not the bookkeeping one.

In that model each thread runs straight between crossings and turns through a circular arc around the thread it crosses, with the closure condition that the two crimp heights fill the thickness of the cloth. Given the two spacings the whole state is determined — crimps, heights, weave angles — and the crimp is a genuinely non-linear function of the spacing.

The consequence is worth stating plainly. Constant total crimp is a first-order approximation to the geometry, exact in the limit of a small pull. It gets the direction right at any extension — the cloth always narrows — and it gets the size right near the state the cloth was measured in. At a large extension it drifts, and the drift is in the direction of over-stating how much the cloth narrows, because the real relation flattens.

This is not a defect of the rule so much as its job. A mill measuring a fabric has the two crimps and wants to know what a small tension will do, and the rule answers that question in one line. Trouble arrives only when a number produced by it is quoted as though it had come from the geometry.

How far the exchange can run #

The exchange has a hard stop and a soft one, and the hard one is easy to miss because it looks so obvious once stated.

The hard stop is the warp crimp itself. A thread cannot be straighter than straight, so the extension available from crimp interchange along the warp is exactly $c_1$ — a cloth with eight per cent warp crimp cannot be extended more than eight per cent by this mechanism, whatever is done to it. Past that, further extension has to come from somewhere else, and the only remaining sources are the fibre stretching and the fabric shearing.

The figures refuse to draw a pull that exceeds the crimp available, which is the assertion that stops the arithmetic from producing a negative crimp and drawing it as though it meant something.

The soft stop is the weft jamming. All that crimp has to be accommodated by the weft, and a thread can only crimp so far before it is wrapped as tightly round its neighbours as the geometry permits. That limit is the same jamming condition that caps the sett, reached from the other side: as the cloth narrows the weft’s effective spacing falls, and when it reaches the jam nothing further can happen.

Which stop is reached first depends on the cloth. An openly set fabric with a lot of warp crimp will run out of crimp; a closely set one will jam first, and will jam well before the crimp is used up. Neither limit is in the bookkeeping rule, which will happily transfer crimp until the warp is straight regardless of whether the weft could have taken it.

There is a third stop that belongs in the list for completeness and is not geometric. Long before either limit, the load required to straighten the warp rises steeply, because straightening a thread against its crossings means pushing those crossings apart and the cloth resists. So the available extension and the achievable one differ, and by a large factor in a firm cloth: the crimp is there and the force needed to recover it exceeds what the fabric will see in use. That is a mechanical statement, it needs a stiffness this site does not model, and it is the reason a fabric quoted as having eight per cent crimp extension does not behave as though it had eight per cent of stretch.

Why the exchange is not symmetric #

A second thing the bookkeeping rule conceals is that pulling the other way is not the mirror image.

The two systems generally have different crimps to begin with — a warp is woven under tension and a weft is not, so warp crimp is usually the smaller of the two in a cloth that has been finished, and the larger in one straight off the loom. They also sit at different setts and are frequently different yarns.

So a cloth with eight per cent warp crimp and four per cent weft crimp can extend eight per cent one way and four per cent the other, and the widths it loses in doing so are different too, because a percentage of the width and a percentage of the length are different lengths unless the piece is square. Balance does not save this: a balanced weave is one where each system takes half the face, which says nothing about how the crimp is divided.

The practical form of the asymmetry is familiar to anybody who has laundered a shirt. Fabric relaxes toward the state with the least strain energy, and the shrinkage that follows is crimp interchange running backwards from wherever the finishing left it — which is why a cloth finished stretched lengthways shrinks lengthways and grows across, and why the two figures on a shrinkage test are rarely equal and often have opposite signs.

What the rule would have to be replaced by #

The constant-total-crimp rule closes the system with a subtraction, and it is worth saying what the honest closure is, because the answer explains why nobody uses it.

The missing equation is a force balance, not a geometric one. Inextensibility fixes both thread lengths and leaves the width free; what actually decides the width is that the cloth is in equilibrium — the weft’s tension, whatever it is, is being resisted by the warp’s crossings, and the state settles where the two agree. That needs the yarn’s bending stiffness, the contact forces at the crossings, and a solve.

So the closure is not a better geometric rule; it is a different kind of model. No amount of care with lengths supplies it, because lengths do not determine the answer at all — the same two thread lengths are consistent with a whole one-parameter family of shapes, and picking one out is a mechanics problem.

That has three consequences worth separating.

The constant-total rule is a guess at a mechanical answer, dressed as a geometric one. Its success at small extensions is not evidence that the geometry supports it; it is evidence that any smooth rule agrees with any other near the reference state, which is what first-order agreement means.

And a second guess would do as well. Holding the ratio of the two crimps constant, or holding the cloth’s area constant, or holding the total thread length in a unit of cloth constant, are all equally defensible bookkeeping rules and all agree with the real answer to first order. They disagree with each other at a large extension by amounts of the same order as the correction the geometry would supply.

Which is why the rule survives. A mill wants the answer to a small extension, every closure gives the same answer there, and the one that needs a subtraction is the cheapest. A rule that is correct in the only regime it is used in, arrived at for a reason that does not hold, is not a bad rule — it is a rule whose justification is somewhere other than where it is usually put.

The uncomfortable version of that is worth stating plainly. Nothing in this essay tests the rule, because everything here is computed under it. Testing it would need the mechanics, and the mechanics is what the rule exists to avoid.

The measurement problem behind all of it #

Every number in this essay begins with two crimps, and obtaining those is harder than the arithmetic that follows.

Crimp is measured by removing a thread from the cloth, straightening it under a small standard tension, and comparing its length with the length of cloth it came from. Three things about that procedure decide the answer, and none of them is a property of the fabric.

The tension. A yarn under no tension is not straight and a yarn under too much is stretched, so the standard specifies a load proportional to the yarn’s count. Change the load and the crimp changes with it, monotonically and by amounts comparable with the crimp itself in a soft yarn.

Where the thread came from. Crimp varies across the width of a piece and along its length, most sharply near the selvedges where the take-up differs. A single thread is a sample of one.

What the cloth had just been doing. Crimp interchange is exactly the point: a cloth that has been under tension has a different crimp from the same cloth relaxed, and it does not fully recover. So the measurement records the fabric’s recent history along with its construction.

The consequence for this essay is a limit on what its precision can mean. The interchange arithmetic is exact given two crimps; the two crimps carry an uncertainty of a relative tenth or worse; so a computed narrowing of 2.94 per cent is a computed narrowing of about three. Every figure here prints a decimal place because the arithmetic has one, and the input does not.

What was counted, and how #

The figures compute the exchange from the two crimps and the pull, and then check the result against the definitions rather than against themselves.

Both thread lengths are reconstructed from the after-state and compared with the before-state, and must agree to within a part in a billion. The narrowing is asserted rather than assumed, so a sign error would stop the build rather than draw a cloth getting wider. The pull is refused if it would need more crimp than the warp has. And the total crimp is asserted constant, which names the rule.

Peirce’s crimp curve is solved by bisection on the weave angle, which is legitimate because the crimp height is monotone in the angle over the range where the straight portion of the thread has not gone negative — a property of the equations rather than a hope about them. Every solution is put back through the equations it came from before it is drawn.

Where this sits among the other extensions #

Crimp is one of four ways a fabric gets longer without a fibre stretching, and it is the smallest of them by a wide margin.

That ranking is why crimp interchange is a nuisance in manufacturing rather than a design tool. Nobody chooses a fabric for its crimp extension. What the crimp does decide is how much a cloth moves in finishing, in laundering and under tension on a machine — small percentages, applied to a bolt fifty metres long, which is a great deal of cloth. Three per cent of fifty metres is a metre and a half, and it is the difference between a cutting plan that yields and one that does not.

Who worked it out #

Crimp interchange is old as an observation and mid-century as a calculation. Weavers have always known that cloth narrows under lengthways tension; the quantitative treatment came with Peirce’s 1937 geometry, which gave the first model in which the two crimps are related by something other than bookkeeping.

The constant-total-crimp rule appears to have arrived from the mill rather than from the literature, and it is stated in textbooks as a working approximation with varying degrees of caution. The most careful versions say that the sum of the crimps is approximately constant for small deformations, which is exactly right and exactly what gets dropped when the rule is repeated.

The full treatment — a cloth deforming under biaxial load with both threads inextensible and the geometry solved at each state — belongs to the fabric-mechanics literature that grew out of Peirce, and it is a substantial calculation rather than a formula. That is the usual reason a simple rule survives: the correct answer needs a solver, the approximate one needs a subtraction, and the approximate one is right in the range most questions fall in.

The lesson this site takes from it is the one it keeps taking. A mechanism quoted without its stop is the standing failure in this subject, and here there are two stops — the warp running straight and the weft jamming — and a rule in daily use that knows about neither.

Where the ladder goes next #

Below this rung is crimp itself, and beside it the two section models, which disagree about how much crimp a given cloth has in the first place.

Onward, the maximum sett is the jam this essay’s soft stop runs into, and the four extension mechanisms put crimp in its place among the others.

What the pictures here cannot show. The two rectangles in the interchange figures are cloth before and after, and nothing between them is drawn. The path from one to the other is a loading history with friction and hysteresis in it, and a fabric does not come back along the line it went out on — which is why a shrinkage measurement is quoted after a stated number of wash cycles rather than as a property of the cloth.