Pulled both ways, only one can give

Every account of crimp interchange describes a cloth pulled one way. That is the honest case to describe, because it is the one a testing machine sets up and the one a bolt of cloth on a roll is in.

It is not the case most fabric in service is in. An air-supported roof, a fire hose, a pressurised sleeve, a filter under differential pressure, a sail, a balloon — every one of them is loaded in both directions at once, and the question of what a cloth does when both directions are pulled has an answer this site can now compute rather than guess at.

The answer is unusually clean. It is: nothing.

One degree of freedom, and what it forbids #

The rung below holds both thread lengths fixed in Peirce’s geometry and finds that the closure condition — the two crimp heights fill the thickness of the cloth — is then one equation in the two weave angles. One equation, two unknowns, so the reachable states are a curve.

That is a stronger statement than it looks. A curve in the plane of the cloth’s two dimensions means the cloth has exactly one degree of freedom: name the length and the width follows, with nothing left to choose. It cannot get longer and wider. It cannot get longer and stay the same width. It cannot get shorter and narrower. Every one of those would be a move off the curve, and off the curve a thread has changed length.

And the curve falls. Along its whole length an increase in one dimension is a decrease in the other, which is the interchange said in the geometry rather than in percentage points.

The reachable set, counted rather than believed #

“One equation, two unknowns, therefore a curve” is algebra, and algebra of that shape is right almost always and wrong occasionally — a degenerate equation can carve out a region, and a badly behaved one can produce several branches. So the dimension is measured.

The closure condition is rewritten as a function of the two cloth dimensions rather than of the two angles. Given a length, the warp’s thread length names its weave angle; given a width, the weft’s does the same; and $h_1 + h_2 - D$ is then a single number that is zero exactly on the reachable set. For a grid of lengths, that number is scanned across a range of widths and its sign changes are counted.

Every reachable length admits exactly one width. On the muslin, seventeen of the twenty-one lengths scanned are reachable at all, and every one of the seventeen has a single root, with no gaps in the middle of the branch; on the warp-dense poplin it is fifteen of twenty-one, and again every one of them has exactly one. A region would give a band of widths at each length; a set with two branches would give two roots somewhere. Neither happens.

That is the assertion this essay rests on, and it can fail. It is written as a check that refuses a scan with two roots at some length or a hole in the middle of its range, rather than as a sentence saying the set is one-dimensional.

The equal-biaxial line meets it once #

An equal biaxial extension is a state in which both dimensions have grown in the same proportion. Every such state lies on the ray through the origin at unit slope, so the question is how many times that ray meets the reachable curve.

The answer is exactly once, and it can be shown rather than counted. Moving out along the ray raises both spacings; raising a spacing lowers that system’s weave angle; lowering a weave angle lowers its crimp height. So the residual $h_1 + h_2 - D$ falls strictly and monotonically along the ray, from positive on the compressed side to negative on the extended side, and a strictly monotone function has one zero.

The numbers say so too. At a scale factor of 0.96 the residual is 6.51 hundredths of a millimetre positive; at 1.04 it is 9.27 hundredths negative; and at 1.00 it is smaller than a ten-million-millionth of a millimetre, which is zero as far as double-precision arithmetic can tell. Forty-one samples along the ray fall strictly, every one lower than the last.

So a cloth pulled equally in both directions cannot move. Not a little, not slowly, not less than a woven fabric would if it were slacker: the reachable set contains exactly one equally-biaxial state and the cloth is standing on it.

What was counted, and how #

The eight nominal plain weaves of the rung below, from an open scrim to a close sheeting, with diameters from the counts by conservation of volume at a packing factor of 0.6.

For each, the reference state is solved from Peirce’s equations and checked by running them forwards on the answer. The two thread lengths are then held and the curve sampled at 241 states, with both thread lengths reconstructed at every one of them and compared with the originals. The worst departure over the whole table is one part in ten thousand million million.

Three separate claims are asserted rather than described. That the two dimensions move in opposite senses at every consecutive pair of samples, so a sign error would stop the build rather than draw a cloth widening as it lengthens. That the scan finds one width per length. And that the residual along the equal-biaxial ray falls at every step, so the single crossing is a measurement of the function rather than a reading of the picture.

The exchange rate, at several points #

Since the cloth cannot go anywhere except along the curve, the useful number is the curve’s slope: how much width is given up per unit of length gained. It is an exchange rate, and it is not one number.

For the muslin, reading the slope at a series of states from three per cent short of the measured one to three per cent long:

0.687, 0.797, 0.933, 1.106, 1.333, 1.636, 2.074.

For the poplin over the same span: 0.906, 1.076, 1.295, 1.583, 1.978, 2.568, 3.589. For the voile, which is the openest cloth that reaches the range: 0.545, 0.678, 0.857, 1.098, 1.459, 2.053, 3.237.

Two things are worth reading off those rows. The rate is near one at the measured state on every cloth in the table, which means an ordinary fabric gives up about as much width as it gains in length — a fact worth knowing on its own, since nothing about the geometry requires it. And the rate triples across a six per cent span on every cloth, which is a much stronger dependence than anybody quoting a single figure for a fabric’s contraction would expect.

An exchange rate of about one at the measured state turns up on every cloth in the table, and it is worth pausing on because it looks like a conservation law and is not one.

Nothing in the geometry sets it. The rate at a state is decided by how the two threads’ spacings respond to a change in the share of the thickness each crimp height takes, and that depends on both weave angles, both thread lengths and the sum of the two diameters. There is no reason for the four to combine into a number near one, and on the poplin they do not — it sits at 1.58, half again as steep as its neighbours in the table, because its two systems are set very differently.

What the near-one values do reflect is that most of the table is balanced or nearly so: equal counts, similar setts, and a crimp ratio taken as one because Peirce’s geometry cannot supply it. A cloth woven with the warp held hard and the weft beaten lightly divides the thickness unevenly, and the rate moves with it. So the number is a property of the construction and of the loom that made it, and reading it as a property of woven cloth in general is the mistake this whole ladder keeps guarding against.

The area is stationary where the cloth was woven #

The exchange rates cluster near one at every cloth’s measured state, and that is remarked above as a fact worth knowing rather than a conservation law. It is worth one step further, because an exchange rate of exactly one has a meaning and the cloths are sitting almost on it.

A cloth’s area goes as the product of its two dimensions, so differentiating along the locus gives

dA/dε₁ = 1 − (exchange rate),

at the reference state. The area is stationary exactly where the rate is one, and it falls on the side where the rate exceeds one.

Read the rows against that. The muslin’s rate at rest is 1.106, so its area is falling at about a tenth of a per cent per per-cent of extension; the poplin’s is 1.583 and it is falling at six tenths. Both are small, and both are negative — so every cloth in the table loses area when it is extended and gains area when it is shortened, and the maximum-area state sits a little on the shortening side of where the loom left it.

Three things follow.

A cloth’s areal weight is robust to handling in a way its length and width are not. The dimensions move to first order and the area moves to second, so a fabric pulled or slackened a per cent in handling changes its grams per square metre by a hundredth of that. That is why a weight measurement on a piece under mild tension is a good measurement and a width measurement on the same piece is not — a fact every mill knows operationally and which has a one-line reason.

And the cover is nearly as robust, being an inverse area. So the two quantities a cloth is most often specified by are the two least sensitive to the state it happens to be in, which is a fortunate arrangement and not a designed one.

The near-coincidence has a cause rather than being luck. The exchange rate is one where the two systems’ responses balance, and a cloth comes off the loom with its crimps divided in a way that is close to that balance — the warp under tension, the weft beaten to a similar geometry. The poplin, which is the most unbalanced construction in the table, is correspondingly the furthest from the stationary point at 1.58, and its area falls six times faster than a muslin’s. The rate at rest is a measure of how unbalanced a cloth’s construction is, readable off one number.

The caution is the same one the rates carry throughout. All of them are computed at a crimp ratio Peirce’s geometry does not supply, so a cloth whose thickness divides unevenly sits somewhere else on its own curve, and the stationary point moves with it.

Why an inflated cylinder cannot use any of this #

The practical consequence is a real thing in the trade and it explains an otherwise puzzling piece of design practice.

A pressurised cylinder loads its cloth in both directions at once, and not equally — the hoop direction takes twice the load of the axial one, which is why such a cloth is deliberately unbalanced. A tensioned membrane is biaxial by construction. A filter cloth under differential pressure is loaded biaxially over its whole area.

In every one of those, the cloth is being asked to move in a direction the mechanism does not have. Crimp interchange is available only for a move along the curve, and a biaxial load with any equal component in it is asking for a move off it. What the fabric does instead is whatever a fabric does when the kinematics run out: the yarns take the load, the crossings deform, the fibres extend a fraction of a per cent, and the compliance the cloth showed on a uniaxial test is simply absent.

That is why an inflatable’s dimensional behaviour is so different from the same cloth’s behaviour on a tensile tester, and why a membrane’s cutting pattern has to be computed from a biaxial test rather than from two uniaxial ones. The two uniaxial numbers describe travel along a curve, and the structure is being pushed at right angles to it.

The asymmetry nobody quotes #

One more thing the curve says, and it is visible in every one of these figures.

The reachable set is far longer on the shortening side than on the lengthening one. The muslin extends 6.59 per cent and contracts 26.87. The sheeting extends 4.03 and contracts 10.19. A batiste extends 6.56 and contracts 29.23. The travel available in the two directions differs by a factor of four or more, on every cloth in the table.

That is because a cloth being shortened along the warp lets its warp crimp grow, and a thread wrapping further round its neighbours has much more travel in it than a thread flattening out — while the weft, which is straightening, is never the thing that stops. The consequence is that a cloth has a great deal of room to come back and very little to go on, which is exactly the shape of relaxation shrinkage and exactly why finishing spends its effort putting cloth into the compressed part of the range rather than taking it out.

Where the model stops #

The curve is a set of reachable states and not a path. Nothing here says how a cloth gets from one point to another or what it takes to move it. A load appears in this essay only as the thing that decides which point a cloth sits at, and it appears as no number at all — which is deliberate and which is the site’s own recorded shortfall rather than an oversight.

So “cannot move” means cannot move kinematically. A real cloth under equal biaxial load does move, by a fraction of a per cent, because the yarns are not literally inextensible and the crossings flatten. What this essay says is that the mechanism contributes nothing to that motion, which is a statement about which explanation applies and not a prediction of zero.

The section is Peirce’s circle, and the racetrack would move every number here. The curve would still be a curve, the ray would still cross it once, and the exchange rates would all be different.

And the analysis is a plain weave. A twill’s threads run over several crossings before turning, so the arc-and-straight construction changes, and a cloth with three thread systems — a leno, a pile, a double cloth — has more degrees of freedom than one and is not described by any of this.

Who found it, and when #

The one-degree-of-freedom statement is implicit in Peirce and has been since 1937, in the sense that anyone holding his thread lengths fixed would arrive at it in an afternoon. It does not appear to be stated anywhere as a fact about cloth.

What the fabric-mechanics literature that grew out of Peirce does have is the biaxial deformation problem, treated properly and at length, with the load in it — Kawabata’s biaxial work in the 1970s is the standing reference, and it measures rather than assumes because a real crossing does deform. That work reports what a cloth does under equal biaxial load, and reports it as a stiffness far higher than the uniaxial one. The kinematic reason for the difference is the sentence above and it is not usually the sentence given.

The trade’s own version is characteristically brief. A maker of inflatable structures will say that woven cloth “has no give under pressure”, which is a true statement about a mechanism that is not available and is invariably heard as a statement about a stiff fabric.

Where the ladder goes next #

The slope of this curve is the useful number and it already has a name. Everywhere else in mechanics, the ratio of a transverse contraction to an axial extension is Poisson’s ratio — and a cloth’s version of it breaks every rule the name carries, including one that ought to be impossible to break.

Below this rung is the extension available and where it stops, and beside it the same kinematic argument made about shear, where the degree of freedom is a different one and worth ten times as much.

What the pictures here cannot show. Every figure on this page draws a cloth as two thread systems and nothing else. A real fabric under biaxial load is also being held at its edges, and how a load is introduced into cloth is a subject with its own failures — seam slippage among them — which no drawing of an interior repeat can contain.