A cloth has one budget for two directions

A woven cloth extends a few per cent along the warp without stretching a thread, and a few per cent across the weft. Written down as two numbers they look like two reserves. A shirting has 3.93 per cent of free extension one way and 8.97 the other, in the way a tank has a capacity and a battery has a charge, and it is natural to think of a cloth pulled lengthwise as having spent from the first and left the second alone.

That picture is wrong in a way that matters, and the reason is one of the oldest facts in this collection: there is one locus.

The claim #

A cloth’s two interchange budgets are the two distances from one point to the two ends of one curve, so spending one refunds the other.

Stated that way it sounds like a restatement. It has three consequences that are not.

The first is that a pre-tensioned cloth does not have less recovery than a relaxed one — it has the same total and a different distribution. The second is that the distribution is strongly asymmetric near the ends: a cloth held near its warp bound has almost no warp budget and a great deal of weft budget. The third is the one that bites in practice: a shock strain that is entirely recoverable in a relaxed cloth is partly permanent in a pre-tensioned one, and nothing about the cloth or the fibre has changed.

The argument #

The locus is one-dimensional because the closure condition is one equation. A plain weave’s two thread systems must supply the cloth’s thickness between them; the warp’s share of that thickness is the free coordinate; fix it and both spacings, both crimps and both weave angles follow. There is exactly one number to choose, so the set of states at constant thread length is a curve in the plane of the two strains and not an area.

Along that curve one dimension rises exactly as the other falls. This collection asserts it at every sampled state rather than at the ends, and the assertion has teeth: a sign error there would draw a cloth that got wider as it was pulled and would look entirely reasonable.

So define the remaining budget from wherever the cloth currently stands, rather than from where it was woven — which is what a cloth being pulled on actually has available.

At three per cent of warp strain the poplin has 0.90 per cent of warp budget left out of 3.93 — it has spent three quarters of it — and its weft budget has risen from 8.97 per cent to 17.52. The cloth has not lost recovery. It has moved it across.

The shock strain, which is where this becomes practical #

Consider two per cent of extra strain arriving suddenly: a jerk on a strap, a knee straightening, a gust in a sail.

Applied to a relaxed poplin, two per cent is entirely inside the 3.93 per cent budget. No thread has changed length, and when the load comes off the cloth is exactly the length it was. Whatever else that event did, it left nothing behind.

Applied to the same poplin already held at three per cent, the same two per cent finds 0.90 per cent of budget and 1.10 per cent of thread strain. The fibre returns a measured fraction of that, and the balance is permanent.

The comparison is worth stating in the form the arithmetic gives it. Nothing to something: the relaxed cloth keeps nothing at all, and the held cloth keeps up to 0.29 per cent of its length, permanently, from the same event. There is no ratio between the two because one of them is zero, and reporting the comparison as a multiple would have been a division by nothing dressed up as a result.

Why this is not the Poisson ratio again #

This collection already has a rung about the exchange rate between the two directions — a cloth’s Poisson ratio is not a material’s, and it is not the reciprocal of itself the other way round.

That rung is about the slope of the locus. This one is about position on it, and the two are independent pieces of information about the same curve. A cloth can be near the middle of its locus with a steep slope or near an end with a shallow one, and the questions they answer are different: the slope says what a strain does to the other dimension now, and the position says how much further the cloth can go before the answer stops being geometric.

The practical consequence of keeping them separate is that a Poisson ratio measured on a pre-tensioned specimen is a measurement of a different point on the curve, which is exactly the shape of the finding that rung reported. Here it acquires a second reason: not only is the ratio a function of position, so is the whole meaning of a subsequent test.

What was counted, and how #

The remaining budgets are read off the locus by interpolating between the two sampled states that bracket the pre-strain, rather than snapping to the nearer of them. That distinction is not fussiness: snapping left the warp budget at zero pre-strain short of the budget itself by the width of one sample — 3.86 per cent against 3.93 — which looks exactly like a small real effect and is a discretisation.

The monotonicity is asserted in both directions along the curve, and the far end is checked: a cloth held at its own budget must have none of it left, to within a rounding error, and comes out at 0.09 per cent.

The refusal is asserted too. Asking for the budget at a pre-strain larger than the budget itself is not a small number, it is a question about a cloth that cannot be held there without stretching, and the machinery names the range instead of returning the end of it. This site has twice shipped a solver that kept the last state it could reach and handed it back as an answer, and both times the symptom was an ordinary fabric quietly declared impossible or an interval’s edge wearing a solution’s clothes.

The asymmetry, and which cloths have most of it #

The two budgets are not the same size, and how unequal they are is a property of the construction rather than a general fact about cloth.

For the poplin the ratio is 2.28 to one in the weft’s favour, and it is the most unbalanced row in this site’s table in every other respect too: thirty-two warp ends and twenty-two picks per centimetre, of two different counts. A cloth like that has most of its free extension across its width and very little along its length, which is exactly the wrong way round for a shirt whose sleeve is pulled lengthwise all day.

For the balanced cloths the asymmetry is milder but never absent, and it always runs the same way. The direction with more crimp has more to give, and in a cloth woven under warp tension the weft carries the crimp, because the warp is held straight on the beam while the picks are driven in around it. That is the standing account of why a woven cloth stretches more across than along, and it is usually given as a fact about weaving practice. Read through the locus it is a fact about position: a cloth off the loom is not standing in the middle of its own curve, it is standing near the warp end of it, and the two budgets are unequal for that reason and no other.

The consequence for a garment is not subtle. A cloth cut so that the direction of use is the warp direction has the smaller reserve in the direction it needs it, and every mechanism in this ladder then applies at once: a smaller budget means the corner in the recovery curve arrives sooner, which means an ordinary strain lands past it, which means the cloth keeps something. Cutting on the cross reverses it, and cutting on the bias avoids the question entirely by using a different mechanism, which this collection prices elsewhere.

What a designer can do with it, and what a designer cannot #

The useful half is a warning rather than an instruction.

Pre-tension in the direction where recovery is not wanted. A cloth tensioned across its width has its whole warp budget intact and rather more weft budget than it started with; a cloth tensioned lengthwise has spent the reserve it needs. An awning tensioned in one direction only, a webbing sling stitched to lie flat, a screen stretched on a frame: in each case the direction that was tensioned to make the thing work is the direction that has nothing left, and the direction nobody thought about has plenty.

What a designer cannot do is get the total up. The sum of the two distances along the curve is fixed by the two thread lengths, which is to say by the construction, and no amount of moving the cloth along its own locus adds to it. The only lever on the total is the one the neighbouring rung establishes: the cover factor, at a maximum near 0.41, and the balance, which matters more.

And the transfer is not reversible for free. Letting the pre-tension off returns the cloth to its own resting position, but “its own resting position” is a band rather than a point, because friction holds the crossings wherever they were left. So a cloth repeatedly tensioned and released does not return exactly, and the accumulated drift is in the direction it was tensioned. That is the mechanism behind a strap that has grown, and it is one this collection can only describe qualitatively here because it needs the frictional band that is deliberately switched off in every number above.

The refund is not one for one #

“Spending one budget refunds the other” is the shape of the result and it invites a reading that the arithmetic does not support: that the two budgets add to a constant, so that whatever is taken from one appears in the other. They do not, and the discrepancy is large enough to be the practical part.

Take the poplin’s own numbers. At rest it stands 3.93 per cent from the warp end and 8.97 from the weft end, which is 12.90 between them. Held at three per cent it stands 0.90 from one and 17.52 from the other, which is 18.42. The sum has grown by nearly half.

Nothing has been created. The two budgets are distances along the curve, and converting a distance along the curve into a strain uses the curve’s local slope — which is the exchange rate between the two directions and is not constant. Three points of warp strain spent bought 8.55 points of weft strain, an exchange rate near 2.8, and that rate is the slope at the part of the curve the cloth is passing through rather than a property of the cloth.

The consequence runs both ways and only one direction is useful.

Tensioning the warp is an efficient way to buy weft budget. Every point of warp strain given up returns nearly three points across the width, near this part of the poplin’s curve.

Tensioning the weft is an inefficient way to buy warp budget. The same exchange rate, inverted, means 8.55 points of weft strain buy 3.03 points along the warp — a rate near 0.35. So the cloth’s two directions are not interchangeable even as a transfer: the direction with the larger budget is the expensive one to spend from.

That asymmetry is the same fact as the asymmetry of the budgets themselves, seen as a rate rather than as a total, and it points the same way. A cloth off the loom stands near its warp end, where the curve is steep, so small movements of the warp move the weft a great deal. Move it towards the weft end and the curve flattens, so the same movements buy less.

Two practical readings, and both correct the advice the section above gives.

Pre-tensioning across the width to preserve the warp’s recovery is a poor trade. It costs a great deal of weft budget for very little warp budget, and near the weft end it costs the cloth its remaining width freedom entirely. The transfer that works is the other one, which is the direction nobody wants.

And the total is not a quantity to design against. The sum of the two strains is not conserved, and it is not a capacity in any useful sense — a cloth held near one end of its locus has a larger strain sum than a relaxed one and is nearer to failing than it was. Reading the sum as a reserve, the way the two budgets invite, gets the direction of the risk exactly backwards.

Where the model stops #

A real cloth is not held uniformly. Every number here is for a specimen at one strain throughout. A garment is pre-tensioned in patches — tight over a shoulder, slack under an arm — so the budget left is a field rather than a number, and the places that keep something are the places that were already tight. That is the correct account of why wear localises, and it is an account this rung cannot make quantitative.

The refund is not free in energy. Moving along the locus costs work, and the two directions do not cost the same; a cloth held at three per cent of warp strain has more weft budget and needs more force to reach the weft end than it would have from rest. Nothing here prices that, and it is the reason the practical advice — pre-tension in the direction where recovery is not wanted — is less useful than it sounds.

And friction is off. The whole of this rung treats the locus as frictionless, so a cloth released from any point returns exactly to where it started. It does not: it rests somewhere in a band, and the band is wide enough to matter. Switching it on shifts every number here by a fraction of a per cent and changes no ordering.

The generalisation #

When two apparently independent reserves are two ends of one degree of freedom, spending one is not spending, it is moving. The distinction is invisible in the totals and decisive in the consequences.

It recurs wherever a constraint reduces a two-dimensional space to a curve. A pretensioned cable net has one length budget shared between two directions; a gearbox synchroniser has one axial travel shared between two engagements; a suspension at full droop has spent its rebound and gained its bump. In each the naive account — two capacities, drawn down separately — gives right answers at the middle of the range and wrong ones near the ends, and near the ends is where things fail.

The narrower lesson is about what a specification should say. A fabric’s free extension is not a property of the fabric alone but of the fabric and the state it is held in, so a number quoted without the pre-tension is a number quoted at zero, which is a state few working fabrics are in.

Who found it, and when #

Crimp interchange and its one-dimensionality are Peirce’s, from 1937, and the fact that pulling a cloth one way narrows it the other is older than any of the arithmetic. Fabric mechanics has always understood the biaxial state as a surface rather than a point.

What appears not to be standard is the budget framing: reading the two ends of the locus as what is left rather than as where the cloth can go, and noticing that the arithmetic of a pre-tension is a transfer rather than a consumption. The reason is probably that the question only arises once a cloth’s recovery has been split into a geometric part and a fibre part, which is a recent enough move in this collection that the consequence had not been chased.

Where the ladder goes next #

The next rung asks what happens when a cloth is held at a fixed length rather than a fixed strain, and finds that its load falls to a frictional floor that does not depend on how hard it was pulled.

Sideways, the fibre half of the split has a ladder of its own: what a fibre returns is a measurement and nothing predicts it. And a knee is the same argument in shear rather than extension, where the budget is a locking angle and a dome spends most of it at once.