The force that holds a knit open

Everything this collection knows about the forces in a knitted fabric came out of one derivative: the rate at which a loop’s bending energy changes as its interlacings move. Move them apart along the wales and the energy falls, and the size of that fall per unit of movement is the contact force.

Once the loop is allowed out of the fabric’s plane, the interlacings can move in three directions rather than two, and the derivative has three components. Two of them were already known under other names. The third is new, it points through the fabric, and it is what this rung is about.

The number #

For a 20 tex cotton at a 3.5 mm loop, fully relaxed, with the yarn’s bending stiffness taken at the free end of its bracket:

7.81 millinewtons a stitch, pushing the fabric’s two faces apart.

That is one fifth of the whole contact force of 38.30 millinewtons, and it points at 11.75° out of the fabric — which is exactly the tilt of the loop’s own plane, as it must be, since the two are one rotation seen twice.

Why the sign is not obvious #

A reader who has followed the ladder this far has good reason to expect the opposite sign. Everything else about a relaxed knit’s bending pushes it outwards: the fabric wants to be wider and it wants to be longer, and the whole finding of the rung on the energy surface is that a relaxed jersey sits on a slope in both directions at once, held where it is by friction rather than by equilibrium.

Through the thickness the sign is the same, and it is worth seeing why it is the same rather than assuming it.

Squeezing the fabric thinner means bringing the head of one course and the feet of the next closer together through the thickness. The distance between those two points along the fabric has not changed; only their separation across it has. So the straight line between them gets shorter, and the thread — whose length is fixed — has more slack to dispose of and must bend harder to dispose of it.

A thread with more slack costs more energy. So squeezing costs energy, and the fabric pushes back.

There is a second way to see the same thing, and it is the one that makes the sign feel inevitable rather than merely correct.

The loop lies in a plane tilted twelve degrees out of the fabric. Squashing the fabric is squashing that plane — bringing its two ends together along the direction perpendicular to the fabric, which has a component along the plane’s own steepest line. So a press on the fabric is, from the thread’s point of view, a shortening of the very chord whose length decides its curvature. There is no separate through-thickness mechanism to work out. There is one mechanism, seen from a direction the flat model could not look from.

What it is as a pressure #

A force per stitch is hard to compare with anything. A pressure is not, because a pressure is what a compression measurement reports.

A stitch occupies a wale spacing by a course spacing — 0.814 by 0.636 millimetres, or 0.518 square millimetres. Dividing:

15.1 kilopascals, or 113 millimetres of mercury.

That is the pressure that would have to be applied to a relaxed jersey to take its thickness to nothing, if the yarn itself were incompressible and nothing else got in the way. Both of those conditions fail before the pressure is reached, which is the honest caveat and is dealt with below — but the order of magnitude is the useful thing, and it is a large one.

Fifteen kilopascals is well above the pressures a garment ever sees in wear. A medical compression stocking works at two to five kilopascals. A finger pressing on a fabric to judge its handle applies something in the same range. So a knitted fabric under a hand is being compressed a long way short of the force its own structure could supply, which is why the handle of a knit is a story about its surface and its hairs rather than about its loops.

It is also worth saying which pressure this is not. It is not the pressure at which a knitted fabric stops getting thinner; that is set by the yarn running out of room to be compressed, and this model has no compressible yarn in it. It is the initial slope: the pressure a fabric would need if it went on resisting at its relaxed rate all the way down. A real fabric’s resistance rises steeply long before that, so fifteen kilopascals is an underestimate of the pressure that would actually flatten a jersey, and a very large overestimate of the pressure needed to take the first few per cent off its thickness.

The thickness that comes with it #

The same geometry gives the thickness with nothing fitted.

Two centre lines pass one diameter apart. Each has a radius on either side of it. So a plain jersey is two yarn diameters thick — 0.334 millimetres for the 20 tex cotton — and that is the number the through-thickness force is holding open.

The striking part is what it does not depend on. Knit the same yarn at a 5 mm loop instead of a 3.5 mm one and the fabric is far more open, far lighter and far more extensible, and it is exactly as thick. The interlacing still passes at a diameter. Only the yarn moves the thickness.

That is a hard prediction, and how thick a knit is is where it meets what a gauge actually reads.

The fibre moves it more than the count does, which is not obvious either. A 20 tex wool and a 20 tex polyester have almost the same diameter as the cotton — 0.180 and 0.175 millimetres against 0.167 — but their forces are 12.1 and 12.9 millinewtons against the cotton’s 7.8, because the force is linear in a bending stiffness and the stiffness carries the fibre’s own modulus and packing. A polyester jersey and a cotton jersey of the same count and the same gauge stand their thickness up with two thirds more force in the polyester.

How hard it varies #

The force is a bending stiffness over a length squared, so it is far more sensitive to the gauge than the thickness is. Over the range of loop lengths a single machine can knit:

A factor of five in the force and a factor of twenty in the pressure, across a range of tightness the same yarn on the same machine can be knitted at. The pressure moves faster than the force because the stitches get further apart as well as softer, so there are fewer of them per unit area to share the load.

The tightness factor is the group that collapses it: two fabrics matched on it have the same loop shape to fifteen figures, and their forces differ only by the stiffness over the loop length squared.

What it does to a friction condition #

There is one published result on this ladder that the third dimension moves, and it moves it by two per cent in a direction worth stating.

The condition for friction to hold a relaxed knit along its wales came out with everything cancelling: friction supplies its coefficient times the contact force at each of two interlacings, the force to be held is the contact force, and so the condition is that the coefficient exceed a half. No rigidity, no loop length, no count, no fibre.

That derivation used a force that was entirely in the fabric’s plane. It is not. Friction has the whole contact force to work against — that is what presses the two threads together at the crossing — while what has to be held is only the part along the wales. The two differ by a cosine, so:

the condition is that the coefficient exceed cos(11.75°) ÷ 2, which is 0.4895.

A two per cent relaxation of a condition that fails by a factor of two is not a rescue. Yarn-on-yarn friction runs from 0.2 to 0.5 for every fibre in this collection’s table, and a condition of 0.4895 is as unmet as one of 0.5. The setting of the yarn is still a requirement rather than an explanation, and a loop is set and not sprung still carries the argument.

What has changed is that the condition is no longer a pure number. It has an angle in it, and the angle depends on the fabric.

Which makes the condition testable in a way it was not #

That is worth more than the two per cent, because a condition with no parameters in it cannot be checked against anything.

The old condition said a half, for every knitted fabric ever made, and there is no experiment that distinguishes that from a slightly different constant. The new one says the required coefficient falls as the tilt rises — so a tighter fabric, or a coarser yarn at the same tightness, needs a slightly lower coefficient to hold itself along its wales.

Across the range in the table above the requirement runs from 0.4830 at a 2.6 mm loop to 0.4941 at 5 mm. That is a two per cent spread, and it is a prediction about which fabrics sit closest to holding themselves. It is not much of a lever, but it is a lever where there was none.

What the force is not #

Three things it is not, because each is an easy misreading.

It is not the fabric’s compression resistance. The pressure quoted above is the initial slope of a resistance, computed at the relaxed thickness with the yarn treated as an incompressible line of a fixed diameter. A real fabric under a real load flattens its yarn, and the flattening of a thread is a record of a force that this model does not carry at all.

It is not a measurement. It is a bending stiffness over a length squared, at the free end of a bracket a hundred and thirty wide. The ratio of two such forces carries no bracket; the number in kilopascals carries all of it.

And it is not what makes a knit feel thick. A fabric’s handle under a hand is decided in the first few per cent of its compression, which is the regime where the hair layer is being crushed and the loops have not begun to move. This force is what waits underneath.

The comparison with a woven cloth #

A woven cloth’s crossings press one another too, and this collection computes that force by a completely different route — inverting a measured thickness against a compression model rather than differentiating a solved shape.

Those forces run from 185 millinewtons for an open cheesecloth to 851 for a duck, against a knitted 38.3. Every woven cloth presses harder, by between five and twenty-two times.

But the comparison that matters here is the one through the thickness, and a woven cloth’s is of a different kind. A woven crossing’s normal force is entirely through the cloth: the two threads are perpendicular and their contact normal is the cloth’s own normal. So a woven cloth’s whole crossing force holds it open, where a knitted one contributes a fifth.

That is a structural statement rather than a numerical one, and it says something about which fabric a compression measurement is measuring. Pressing a woven cloth works against the crossings. Pressing a knitted one works against a fifth of them, and against friction and hairs for everything else.

Why the fabric does not simply open further #

If the force pushes the two faces apart, the obvious question is why the fabric is not thicker than two diameters. Nothing here is pulling it shut.

The answer is that the thickness is not free to grow. It is not a spacing the fabric can choose, like its wale spacing or its course spacing; it is fixed by the interlacing, which is two centre lines passing at a diameter because two threads of one diameter in contact are a diameter apart. The fabric could only get thicker by having its threads not touch at the crossing, and then they would not be interlaced.

So the through-thickness force is a force against a constraint rather than a force driving a dimension. That is the difference between it and the two in-plane forces, which push against nothing and are held only by friction — and it is why the thickness is the one dimension of a relaxed knit that this collection can predict outright, while the other two have to be measured and quoted.

What holds a knit open when the yarn is set #

The whole of this ladder carries a bracket on the setting of the yarn: an unset yarn is a straight rod bent into a loop and pressing to get out of it, a fully set one has the loop as its natural shape and presses with nothing, and a real fabric is somewhere between.

The through-thickness force scales with that bracket exactly as every other force does, by one minus the set fraction. So a well-set fabric holds itself open with less than 7.8 millinewtons a stitch, and a fabric set to the point where it holds no bending energy at all holds itself open with nothing.

That is not the absurdity it sounds like. A fully set fabric does not collapse, because collapsing means the yarn deforming further from its natural shape — which costs energy again, from the other side. Setting does not remove the resistance; it moves the state the resistance is measured from. What the number above is is the resistance of an unset fabric measured from the relaxed state, which is the largest it can be.

What a second bed does to it #

Everything here is one bed, where the climb is a yarn diameter because that is what an interlacing is. Put the loops on two beds and the climb becomes the gap between them — three to five diameters rather than one.

The through-thickness force does not simply scale, and the way it fails to is the subject of a rib is quietest at two diameters. Broadly, though, it grows: at a gap of five diameters the same 20 tex cotton pushes its faces apart with 14.8 millinewtons a stitch against a jersey’s 7.8, and the fabric it is holding open is three times as thick.

A rib is springier through its thickness than a jersey by more than the ratio of their thicknesses, which is a statement anybody who has squeezed a cuff will recognise.

What this makes computable that was not #

Two things downstream of this ladder need a through-thickness force and could not have one before.

A knit’s compression curve. The collection’s woven side has a bearing curve — how much of a cloth is in contact at a given depth, and what pressure that takes — built on Peirce’s geometry and a measured thickness. The knitted side had nothing, because a flat loop has no depth to bear at. It has a first point now: a relaxed thickness of two diameters and a slope of fifteen kilopascals per unit strain, and what a knit gives up when it is pressed is where the two sides are put beside one another.

A garment’s grip. A cuff, a waistband and a sock top all work by pressing on a body, and the pressure they exert is a fabric tension divided by a radius. That is a different force from this one — a hoop tension rather than a through-thickness push — and the two turn out to differ by two orders of magnitude, which is the finding of what a cuff presses with.

What is genuinely new here #

Two numbers and one correction.

A knitted fabric’s through-thickness force, 7.8 millinewtons a stitch and 15 kilopascals, which nothing in this collection had before and which no flat model could have had.

Its thickness, predicted, at two yarn diameters, independent of gauge and depending on the yarn alone.

And the friction condition acquires an angle. It was a half exactly, for every knitted fabric there is. It is now a half times a cosine, the cosine belongs to the fabric, and a condition with a parameter in it is a condition that can be argued with.

What the pictures cannot show #

The section drawing has its thickness expanded three times against its width, because a fabric a third of a millimetre thick and four fifths of a millimetre to the wale is otherwise a line. So the angle in the drawing is not the angle in the fabric, and the figure says so in its own note.

The force vector beside it is drawn at the true angle, on its own axes, precisely so that there is one true angle in the picture. A reader who measures the section’s geometry with a protractor will get thirty-five degrees. The fabric’s is twelve.

Where the ladder goes next #

A fabric with a thickness can be asked a question that a fabric without one cannot: whether it curls. Curl is a bending moment, a bending moment is a force at an offset, and an offset needs a thickness to be measured across.

The answer the model gives is not the one the question expects, and how little asymmetry a curl needs is what to do about it.