The locus gets a force

The tensile ladder is three rungs of careful geometry with a hole through the middle of it. A cloth extends by moving its crimp computes the whole set of states a fabric can reach with no yarn changing length. Pulled both ways, only one can give works out what happens when both directions are loaded. A cloth’s Poisson ratio is not a material’s reads the exchange rate off that set.

All three describe a set of reachable states and none of them can say which state the cloth is in. That is not a gap in the writing; it is what a kinematic model is. A locus is a claim about what is possible, and possibility does not have a preferred point.

With a bending rigidity for the yarn it does, and the preferred point comes with a curve through it.

The claim #

The tension in a cloth being extended is the slope of its bending energy along its own locus, and that slope is exactly zero at one point.

The point is the state of least energy, which is where an unloaded cloth sits. So the load–extension curve of a woven fabric has a genuine zero in it, and the region around that zero — the flat start every textile testing laboratory sees and every account calls “the crimp coming out” — is flat because the energy is stationary there, not because anything is slack.

The derivative, and why it is one line #

Take a specimen of N ends and M picks. Extending it along the warp moves the picks apart; it does not change how many there are. So the specimen’s total bending energy is N·M·u, with u the energy of one modular length, and N and M are constants of the specimen.

The tension is the derivative of energy with respect to length, and the length is M·p₂:

**F = dE/dL = N·M·(du/dp₂)·(dp₂/dL) = N · du/dp₂

The picks-per-specimen cancels. What is left is the energy of one crossing differentiated with respect to the pick spacing, multiplied by the number of ends — or, per unit width, that derivative divided by the end spacing.

The derivative is taken along the locus, which is the load case a strip test actually applies: the cloth is pulled in one direction and is free to narrow in the other, so the lateral contraction is included rather than assumed away. Every state on the locus has both spacings, both crimps and both weave angles already, and the energy of one modular length is 2Bθ/D exactly, so the whole computation is a difference between neighbouring samples of a curve this site has been drawing since the tensile ladder was built.

Why the zero is a symmetry rather than a coincidence #

A stationary point of an energy is not surprising. What is worth arguing is where it is, and for a balanced cloth the answer is decided by symmetry alone.

Take a cloth whose two yarns are the same count in the same fibre and whose two setts are equal. Exchanging warp and weft is then a symmetry of the whole construction: it carries the locus to itself with its direction reversed, and it carries the energy function to itself. A symmetric function on a symmetric interval is stationary at the fixed point of the symmetry — which is the state where the two weave angles are equal.

So for such a cloth the force vanishes at the state where the crimps are equal, at every yarn stiffness, for every fibre. It is the same argument that fixes the crimp ratio, read as a statement about a slope rather than about a minimum, and it is why the two rungs share a function.

For a real cloth the symmetry is broken — every cloth in this site’s table is set with a denser warp than weft — and the zero moves off the symmetric state. It does not disappear. A continuous energy on an interval has a minimum somewhere, and the tension is zero there.

What the flat start really is #

The standard account of a woven fabric’s load–extension curve has three regions, and the first is described in every textbook in roughly these words: a low-modulus region in which the crimp is removed, followed by a knee, followed by a high-modulus region in which the yarn itself is extended.

The description is right. The explanation usually offered for the first region is not quite. It is generally put as the crimp comes out at little cost, because straightening a thread is easier than stretching it — which is true and does not explain the shape. Straightening a thread against a bending resistance costs something at every point, and there is no reason for that cost to be small at the start and large later unless something makes it so.

What makes it so is the stationary point. Near a minimum an energy is quadratic, so its slope is linear in the displacement and passes through zero — which means the force is not merely small at the start, it goes to nothing, and the region is flat for the same reason the bottom of any valley is flat.

The number for a sheeting: the force stays under one per cent of the yarn’s breaking load until the cloth has extended 2.6 per cent. Within three per cent either side of rest, the tension in an end never exceeds a twentieth of a newton against a breaking load of 3.7.

The prediction the symmetry argument makes #

An explanation that says “the toe is a stationary point” is worth more than one that says “the crimp comes out”, because the two predict different things about which cloths have a pronounced toe.

The stationary-point account says the toe is deep and symmetric for a balanced cloth and lopsided for an unbalanced one. A cloth set far more densely in one direction than the other has its minimum well away from the symmetric state, and the curve either side of it is not the same shape — extending it in the direction that has more crimp to give is cheap for a long way, and extending it the other way runs into the end of the locus quickly.

That is a testable statement about a poplin against a sheeting and it does not require any measurement of stiffness, because the position of the minimum is decided by the geometry alone for a cloth of two equal counts. The site’s own table puts a sheeting’s minimum along its locus at a crimp ratio of 1.22 and a poplin’s at 12.97 — states these cloths relax to rather than states they are quoted at, which is a distinction that matters elsewhere and not here, since a load–extension curve is a walk along the locus. A poplin’s curve in the warp direction should therefore have a much longer, flatter toe than a sheeting’s, followed by a much harder knee.

What was counted, and how #

The chain is three steps and each has its own check.

The reference state is Peirce’s solution at the cloth’s quoted construction, fed back through the equations it was solved from; the worst residual is at machine precision.

The locus is the site’s existing construction, sampled at 481 states, with both thread lengths reconstructed at every sampled point rather than at the ends. That check is the discipline the whole ladder rests on, and it is at every point because a path that conserves length at its endpoints and drifts in the middle is exactly the failure a two-state figure cannot show.

The energy is 2(B₁θ₁ + B₂θ₂)/D, which is closed-form: Peirce’s path bends only in its arcs, the arcs have constant curvature 2/D, and their length is Dθ. Nothing is integrated numerically.

Only the last step — the derivative — is numerical, and it is a central difference between neighbouring locus samples. That is worth being explicit about, because a reader could reasonably assume the whole thing was a numerical exercise, and the quantity being differentiated is exact.

One consequence of that split is worth drawing out, because it decides what the curve is good for. The extension axis is geometry: it comes from thread lengths and spacings and is as firm as Peirce’s solution. The force axis is a stiffness, and a stiffness on this site is known to a factor of three at best. So the curve should be read as a shape with a firm horizontal scale and a soft vertical one — which is exactly the opposite of how a measured load–extension curve should be read, since a testing machine knows its load cell far better than it knows what its jaws are doing to the specimen.

The rigidity enters as a scale on the whole curve and not as a shape. Doubling B doubles every force on this page and moves nothing, which is why the position of the toe and the extension at which it ends are firm while the newtons are known only as well as the stiffness bracket is.

The toe has a modulus, and it is a fourth power of the yarn #

A zero is the first thing a derivative gives; the second is the slope through it. Near the minimum the energy is quadratic, so the tension rises linearly and the constant of proportionality is the cloth’s initial modulus — the number a testing laboratory reports as the low-modulus region’s gradient, and the one every account of the toe describes qualitatively and none of them derives.

It is the second derivative of the same closed-form energy, so it costs nothing extra. Per unit width it is the curvature of u with respect to the pick spacing, divided by the end spacing, and for the sheeting on this page it comes out at about 0.9 newtons per centimetre per unit strain — three orders of magnitude below the same cloth’s modulus once the yarn is carrying the load, which is the ratio the three-region picture is a drawing of.

The useful part is what the number depends on. The energy carries the bending rigidity as a linear factor, so the modulus does too, and a bending rigidity goes as the fourth power of the yarn’s diameter. Everything else in the expression is geometry — spacings, angles, the shape of the locus — and is fixed once the construction is quoted.

So the toe splits cleanly into two halves that scale differently:

Its width is geometric. How far the cloth extends before the crimp runs out is decided by thread lengths and spacings, and a finer yarn at the same sett and the same crimp reaches the end of its locus at the same extension. The 2.6 per cent does not move.

Its steepness is a fourth power. Halving the yarn’s diameter at constant construction divides the initial modulus by sixteen.

That is a strong prediction and an unusually clean one, because the two halves are separately checkable on the same specimen and the confounded quantity — the stiffness bracket, known here to a factor of three — enters only one of them. Two cloths of the same construction in two counts should have toes of the same length and wildly different heights, and the ratio of the heights should be the ratio of the diameters to the fourth.

It also says which cloths have a toe worth noticing at all. A cloth of very fine yarn has a toe so shallow that a testing machine’s own compliance and the specimen’s own weight are comparable with it, which is why the low-modulus region of a lightweight cloth is hard to measure and why the reported initial modulus of such fabrics scatters far more than the yarn properties do. The scatter is not carelessness; it is a fourth power sitting under the noise floor.

And it sharpens what the knee is. The knee is where the two moduli cross over, so the finer the yarn the sharper the knee, since the toe falls as the fourth power of the diameter while the yarn-stretching modulus falls only as its square. A coarse canvas has a rounded transition and a fine voile has almost a corner in it — the same shape argued about above, seen as a ratio between two regions rather than as a property of either.

Where the model stops #

The curve stops where the locus stops. A locus ends when one system runs out of crimp to give or the other jams, and past that point the cloth is being extended by stretching yarn — a modulus something like a thousand times higher. The knee in a real curve is that transition and this model reaches it and cannot cross it.

There is no compression term. Threads flatten where they cross, and flattening stores energy. The site’s racetrack section is its model of a flattened thread and it has no stiffness in it, so the compression contribution is unavailable rather than neglected. In a closely set cloth it is not small.

There is no friction here. The curve above is the reversible part. A real cloth’s load–extension curve does not retrace itself on unloading, and the hysteresis is friction at the crossings — which is the subject of the relaxation rung and which turns the zero into a band a few per cent wide.

And the specimen has no edges. A strip test’s specimen is cut, so its outermost ends are not restrained sideways the way the middle is, and the narrowing that this model allows freely is resisted at the jaws. Both are real and both are outside a model whose whole content is one repeat.

The generalisation #

The statement that survives leaving textiles is about what a kinematic model can and cannot be asked.

A model that computes a set of reachable configurations is answering the question “what is possible”, and no amount of refinement will make it answer “what happens”. The second question needs a cost on the set, and the cost is always of a different kind from the geometry — an energy, a potential, a price. This is the same move that turns a feasible region into an optimum, a space of states into a trajectory, and a set of admissible mechanisms into the one a structure actually takes.

The second half is the more practical one. Where the cost function inherits a symmetry from the geometry, its stationary point is at the symmetric configuration and the cost’s own coefficients drop out. So a symmetric case can be solved exactly without knowing the material at all, and the material re-enters only as a scale on the answer’s steepness. Here that separates two very different qualities of result on one curve: where the zero is, which is geometry, and how fast the curve rises from it, which is a stiffness known to a factor of three.

Who found it, and when #

The three-region description of a fabric’s load–extension curve is old and belongs to the fabric-testing tradition rather than to any one paper. Peirce’s 1937 geometry gives the shapes; the mechanics of extending them belongs to the work of the 1960s and 1970s, and Olofsson’s energy formulation of 1964 is the direct ancestor of the argument here — a cloth as an elastic structure whose state is found by minimising, rather than a shape found by construction.

That literature was after the whole curve, including the parts this model cannot reach, and it fitted parameters to get there. What is done here is smaller and cleaner: the reversible bending part only, computed from a stiffness with nothing fitted, on a locus that already existed for another purpose.

The observation that the toe is a stationary point rather than a slack region is this site’s way of putting it. It is not a new mechanical fact — anybody who wrote down the energy would see the zero — but it is a different explanation from the one the textbooks give, and it makes a prediction the textbook version does not: that a balanced cloth’s toe is symmetric and an unbalanced cloth’s is not.

Where the ladder goes next #

Directly above this rung is the missing compression term, which would let the same energy argument predict a cloth’s thickness rather than take it from the closure condition.

Sideways, the same energy read as a depth rather than a slope gives the band a relaxed cloth comes to rest in, and read at its minimum it gives the crimp ratio Peirce’s geometry could not supply. The same derivative taken at the fell rather than in a testing machine is the beat-up force.

And the same function applied to a knitted loop gives nothing at all, which is not a failure but the sharpest result in this group: a knit’s arcs are held to their radius by contact, so extending the fabric bends nothing further and the whole extension is free.