What a loop presses with

A knitted loop hangs on the loop below it, and the two press on each other. How hard has been an unanswerable question on this site for as long as there have been knitted essays on it, and the answer turns out to be about forty millinewtons for an ordinary cotton jersey — four grams-force, at one interlacing, in a fabric nobody is pulling.

The number is worth having for itself and it is worth much more as a comparison. A woven crossing in the same yarn presses with several hundred.

Where it comes from #

Nothing new is needed. The solved span between two interlacings carries a constant internal force, the two spans meeting at a crest are mirror images, their components along the fabric cancel and their components across it add. The load on the interlacing is twice the transverse component, and the transverse component is a Lagrange multiplier the solve already produced.

So the force costs nothing beyond the shape, and it is available at every configuration rather than at one.

What sets its size #

Dimensionally there is no choice: a force made out of a bending rigidity and a length can only be a rigidity over a length squared. What the solve supplies is the pure number in front, and for a relaxed jersey it is close to two.

The consequences of that form are worth more than the value. A finer yarn at the same loop length presses less, because the rigidity falls faster than the length does. A longer loop at the same count presses less, as the square. And two fabrics with the same tightness factor but different fibres differ by exactly the ratio of their rigidities over their loop lengths squared — which is a statement that can be checked and, more usefully, a statement about what Munden’s constants are silent on.

The bracket it inherits #

Every force here is linear in the yarn’s bending rigidity, and a spun yarn’s bending rigidity is a bracket a hundred and thirty wide rather than a number: the fibres may slide over one another or may not, and twist decides how far along that range a real yarn sits without anybody being able to say how far.

Forty millinewtons is the figure at the free bound, which is the lower end. It is quoted because the site’s own comparison against measured fabric rigidities found the free bound landing inside the band of real cloths for every construction in its table, and the coherent bound landing two to three orders above all of them.

So the number is a considered choice of the bottom of a bracket, and it is stated as one everywhere it is used.

Why it is an upper bound as well #

There is a second reason for caution and it points the same way, which is convenient.

The force computed is what an unset yarn presses with — a straight rod bent into a loop and springing to get out of it. A yarn that has been wet-relaxed and tumbled has taken a set: its natural shape has moved towards the loop’s own, and a yarn whose natural shape is the loop presses with nothing at all. A real relaxed fabric is somewhere between, and every force scales as one minus how far the setting has gone.

The rung that pursues this finds that the setting cannot be pinned down from the fabric’s dimensions alone. What can be said is the direction: forty millinewtons is the most a relaxed jersey’s loops press with, and the true figure is at or below it.

The comparison, and why it is honest #

The woven figure and the knitted one come from routes with nothing in common. The woven one starts from a measured thickness, notes that a round section is the thickest a thread can be for its area, and asks what force per crossing reconciles the measurement with the geometry. The knitted one starts from a loop length and a solved shape and never touches a measurement of force, thickness or anything else.

Two independent methods that disagree would be informative. Two that agree in order of magnitude would be a confirmation. What is actually available is a contrast, and its value is that neither route could have produced it: the thickness inversion cannot be run on a knit because a knit’s thickness is not set by how flat its threads are pressed, and the elastica cannot be run on a woven cloth because a woven thread has no free run.

What the ratio explains #

One number, and a list of separately-explained differences between the two fabrics falls out of it.

A knit gives up a fibre end more easily. A knit gives up its fibres more easily already argued that from the structure; the arithmetic under it is that the grip on a fibre is friction times a normal load, and the normal load is eight times smaller. A knit pills sooner, for the same reason with an extra step. A cut edge of jersey does not fray but runs, because a thread that cannot be gripped hard enough to break is a thread that always slides. A seam in a knit slips at a lower load. A knit drapes softer.

That is five trade observations with one multiplication under them, and the multiplication was not available until the normal load was.

Two terms, not one #

The last figure is the one worth reading twice, because the effect is larger than the contact force alone.

Grip per unit length of buried thread is the force per contact times the contacts per unit length. In a woven cloth a thread meets a crossing every thread spacing, which is a few tenths of a millimetre. In a knit it meets an interlacing every half loop length, which is nearly two millimetres. So the contacts are four times sparser and eight times lighter, and the product is thirty-one.

Neither factor is a surprise on its own. The product is, and it is the kind of thing that only shows up when both are computed in the same units by the same machinery.

The crossover length #

The quantity that makes it concrete is the one this collection already uses on the woven side: the gripped length at which a thread breaks rather than slides.

For a relaxed poplin it is about twelve millimetres, which is the range a cut edge frays over, the range a seam allowance is cut to, and the range a tuft is anchored across. For a jersey of the same yarn the model puts it at four hundred and sixty millimetres — longer than any garment.

A crossover length longer than the fabric means the thread never breaks by being pulled out. It always slides. That is not a curiosity: it is the definition of a run, and the rung that prices one starts from exactly this.

What the figures cannot show #

A force. Every drawing on this ladder shows a curvature or a shape, and the force is a slope of one of those along the thread.

Worse, the drawings cannot show the direction of the contact load either. A smooth contact pushes along the line joining the two centre lines, which at a knitted interlacing is roughly across the fabric — out of the page in a plan view, which is the one direction a plan view has no room for. Where a figure here needs it, the number is printed rather than drawn.

A knit’s contacts, counted #

The count matters as much as the force, and it is worth doing explicitly because the two fabrics count differently.

Two interlacings per stitch on the yarn’s own path, each shared with one neighbouring course. A woven thread meets a crossing at every pick it passes — twenty-two to the centimetre in a poplin, against five and seven tenths per centimetre of yarn in a jersey.

So the density of contacts along the thread differs by about four, and it differs in the same direction as the force. Two independent geometric facts about the two fabrics point the same way, which is why the grip differs by thirty-one rather than by eight.

The energy the fabric is holding #

The same solve returns the loop’s stored bending energy, and it is worth converting into a quantity a reader can weigh.

An unset 20 tex cotton jersey holds about twenty-five microjoules of bending energy per stitch, and there are nearly two million stitches in a square metre. That is fifty joules a square metre — a large number, and about a thousand times the energy it takes to bend the same fabric double.

It is large because the yarn’s curvature in a loop is sixty times the curvature of a fabric folded over, and energy goes as the square. What it means practically is that a knit is a fabric with a great deal of elastic energy stored in it at rest, if its yarn is unset — which is exactly the premise the next two rungs put under a microscope, and exactly the premise that turns out to be wrong.

The count cancels, and what is left is the fibre #

The three levers listed above are the yarn’s rigidity, the loop length and the setting. Two of those are not independent, and eliminating between them collapses the expression in a way worth writing down.

At the free bound a spun yarn’s rigidity is the sum of its fibres’, so it is the fibre count times one fibre’s rigidity — and the fibre count is the yarn’s tex over the fibre’s. The tightness factor is the square root of the tex over the loop length, so the loop length is the square root of the tex divided by the tightness factor.

Put both into a force that goes as rigidity over loop length squared, and the tex appears once on top and once on the bottom:

contact force = (a pure number) × (one fibre’s bending rigidity ÷ the fibre’s tex) × (tightness factor)².

The yarn count cancels exactly. A jersey knitted from a 10 tex cotton and one knitted from a 40 tex cotton, both at the tightness factor their knitters would choose, press at their interlacings with the same force — because the coarser yarn’s greater stiffness is exactly undone by the longer loop it is knitted at.

Three things follow, and none of them is obvious from the force as first written.

The fibre enters only as a rigidity per unit linear density, which is its modulus divided by its density, times a shape factor for its section. That is a material constant in the strict sense — it is the same for a 1.3 decitex polyester and a 6 decitex one — so the fibre’s fineness drops out along with the yarn’s count. What is left is the polymer and the cross-section.

The tightness factor is the whole of the construction, squared. Everything a knitter decides — gauge, loop length, count — reaches the contact force only through that one combination, which is the same collapse Munden found in the dimensions and is the reason the two results belong to one anchor.

And it explains the range the section above measured. Across the tightness factors a cotton jersey can be knitted at, the force ran from about fifteen millinewtons to thirty-five — a factor of 2.3, against a factor of about 1.5 in the tightness factor itself. The square of 1.5 is 2.3, which is the arithmetic arriving at the measured spread from a different direction and is the check the cancellation deserves.

The comparison with a woven cloth reads differently in this light. A woven crossing’s contact force is set by the warp tension, which is a thing done to the cloth on a machine; a knitted interlacing’s is set by the fibre and the tightness factor, which are decisions about the fabric. That is why the woven figure varies across the eight cloths in the table and the knitted one barely moves, and it is a cleaner statement of the difference than the ratio of the two numbers.

What is being pressed #

A caution about the object. The load is between two loops of different courses, at the place where one hangs on the other. It is not a pressure spread over the fabric, and it is not the force pressing a knit against a table.

Nor is it symmetric in an obvious way: a loop has two such interlacings on its own path — its head, where the course above hangs on it, and its sinker, where it hangs on the course below — and both carry the same magnitude only because every course in a plain jersey is the same. In a rib, an interlock or anything with two beds, that is no longer true and the count has to be redone.

What moves it #

Three things and no others, which is a short enough list to be useful to somebody choosing a construction.

The yarn’s bending rigidity, linearly — so a coarser or stiffer yarn presses harder, and the fibre matters through its modulus rather than through anything else. The loop length, as the inverse square — so a tighter fabric presses very much harder, which is the lever a knitter actually has. And the relaxation state, through the setting, which cannot be computed here and which pushes the answer down.

Nothing about the machine gauge appears, and nothing about the fabric’s width or the take-down tension appears. Both drop out with the loop length, which is the same collapse Munden found in the dimensions turning up again in the forces.

The tightest fabric a knitter reaches #

Working the lever the other way puts a number on the range. Across the tightness factors a knitter of a 20 tex cotton can reach, the contact force runs from about fifteen millinewtons at the slack end to thirty-five at the tight end — a factor of two and a half.

That is a smaller range than the fourfold change in the inverse square would suggest, because the loop length also moves the spacings and the two partly cancel. It is still the largest single lever available, and it is the reason a tight knit resists a run, sheds less and pills less, all at once.

Against a hand #

Forty millinewtons is four grams-force, and it is worth putting beside something a reader can feel.

A single human hair weighs about a microgram and a sewing needle about a tenth of a gram. Four grams-force is roughly the weight of a small coin, applied at a contact a sixth of a millimetre across, and there are two million such contacts in a square metre of jersey. Summed as a pressure it is a few kilopascals — comparable with a firm pinch, and concentrated at points.

That is worth keeping in view when the friction arithmetic arrives, because it is easy to read “forty millinewtons” as negligible. It is small compared with a woven cloth and it is not small compared with the things it has to hold.

The number as a function #

Since the force costs a millisecond, it can be plotted rather than quoted, and the plot says more than the value.

A quantity plotted against a lever a maker actually has is a different object from the same quantity quoted at one construction. The second is a fact about a fabric; the first is an instruction about how to change one.

And the lever here is the one a knitter has least intuition about, which is why the plot is worth more than the number. Loop length and yarn count are set at the machine and their effects are felt daily; relaxation state is set by what happens to the fabric after it leaves, in a wet finishing room somebody else runs, and it moves this quantity by more than either of them. A fabric that presses on itself harder is a fabric that resists being pushed out of shape, so the same construction can come back from finishing measurably firmer than it left — and nothing on the knitting specification records the difference.

The cost of computing it is part of the point. The force is the derivative of a solved energy with respect to one of the loop’s own dimensions, and the solve is fast enough that the derivative can be taken numerically at every point of a sweep rather than once at a nominal construction. That is what turns a quantity into a curve, and a curve is what can be read for a slope — which is the thing a maker is actually choosing between.

Who found what #

The idea that a knitted fabric’s mechanics is a bending problem is old and is due to the Leeds and Manchester school of the 1950s and 1960s; minimum-energy loop models were being solved by hand before there were computers to solve them on. What has been missing is not the concept.

What has been missing is a contact force that anybody could put in a sentence beside a woven one. Both figures here existed as ideas and neither existed as a number in the same units, which is why the comparison had never been drawn.

Where the ladder goes next #

The force is now available, and the first place to spend it is not a consequence at all. It is a check.

Differentiate the same energy along the fabric rather than across it and a second force appears — the force the loops exert on their own spacings. In a relaxed fabric that ought to be zero, because a relaxed fabric is one nothing is pulling. It is not zero, and it is not small, and what that means is the most useful thing on this ladder.