A knee is a dome imposed a thousand times

A pair of trousers that has been worn for a season has knees in it, and the knees are there when nobody is in the trousers. The cloth has acquired a shape it did not have, in the one place it was repeatedly asked to take one, and no amount of pressing gets it entirely out.

This collection has the machinery for the deformation itself. A flat sheet of inextensible threads free to rotate where they cross cannot take a double curvature without shearing — that is the trellis, and it is the second thing this site built. A dome imposes a definite shear, the shear is available up to a locking angle at which the threads touch side by side, and past that the cloth has nowhere to go and must either wrinkle or slip.

What it has not asked is how close an ordinary knee comes to that limit. The answer is: very.

The claim #

A dome’s shear demand has no length in it, and an ordinary knee spends most of a cloth’s shear on one bend.

The first half is the surprising one and it is easy to state: the shear a sphere demands depends on the angular extent the cloth has to cover and not on the sphere’s radius. A knee of sixty millimetres and a beach ball of three hundred demand exactly the same shear, at the same fraction of their own radii — asserted across a thirtyfold range of radius and identical to six figures.

The second half follows once a number is put to the angular extent. A trouser knee wrapped to about its own radius is at 49.55 degrees of shear against a locking angle of 58.46 — a limit that is a function of the cover alone — which leaves nine degrees.

Small is not the same as gentle, and that is the whole reason a knee bags where an elbow of a loose sleeve does not.

Why the demand is scale-free #

A sphere’s departure from a plane is a statement about angles. Walk a distance s along a great circle from the pole and the circumference of the circle at that distance is 2πR sin(s/R) rather than 2πs — and the ratio of the two depends only on s/R.

A trellis laid on the sphere has to absorb exactly that deficit. Its threads cannot change length — which every draped figure on this site checks segment by segment — so the only thing available is the angle between them, and the shear it needs at each point is set by how much circumference is missing there. That is a function of s/R and of nothing else.

So doubling a dome’s radius and doubling how far the cloth reaches leaves the shear unchanged. This collection asserts it directly rather than deriving it in prose, at radii from ten millimetres to three hundred, because a scale-dependence sneaking in through a discretisation is exactly the kind of thing an assertion is for.

What a knee actually asks for #

A knee is roughly a hemisphere of sixty millimetres radius when it is bent, and the cloth over it has to reach from the patella outwards to where the leg becomes a cylinder again — of the order of one radius.

At one radius the demand is 49.55 degrees. A sheeting locks at 58.46. So an ordinary knee bend leaves nine degrees of margin, which is 15 per cent of the available shear.

That margin is what a garment lives on, and three things eat into it.

A closer cloth locks sooner. The locking angle is set by the ratio of thread diameter to spacing, so a densely set fabric has less shear available. A cloth at a cover half again as high as the sheeting’s locks well below the knee’s demand and cannot conform at all.

Reaching further costs steeply. The demand curve is convex: 0.8 radii asks 32.5 degrees, 1.0 asks 49.6, 1.2 asks 68.3. Ten per cent more reach past the equator costs nearly twenty degrees, and there is not twenty degrees to give.

And the cloth is not free at its edges. The trellis calculation assumes every crossing may rotate. A knee’s cloth is sewn into a leg, so its edges are held, and the shear it can actually deliver at the knee is less than the shear the geometry would permit.

Why repetition is not the same as magnitude #

Everything above is about one bend. A knee is not one bend.

A shear inside the locking angle is a mechanism, and this collection’s account of a mechanism is that it comes back — the trellis returns to square, nothing has stretched, no yarn is longer. That account is exactly true in the absence of friction and exactly wrong in its presence.

Every crossing that rotates is a crossing held by friction, and a crossing rotated and released does not return to its starting angle: it returns to somewhere inside a band, in the same way that a cloth extended and released returns to somewhere inside a band. The width of the band is friction over stiffness, and the residual is a fraction of each cycle.

So a knee bent once and straightened is a knee with a small residual shear in it. A knee bent a thousand times accumulates, and it accumulates in the direction the knee bends, at the place the knee is.

The permanent shape is not from the size of the deformation but from the number of them. That is why a knee bags and a single hard stretch does not produce one, and why the shape appears at the knee rather than being spread over the leg.

Why it localises so sharply #

The localisation is the part that a mechanics of a whole panel would miss, and this collection’s budget picture accounts for it directly.

The cloth over the knee has spent most of its shear on the first bend and has nine degrees left. The cloth two hundred millimetres up the thigh has spent almost none. When the leg bends again, the deformation goes where there is room for it — which is not the knee, so the knee’s cloth is loaded rather than deformed, and a loaded crossing is a crossing that slips.

The same logic runs through the extension budget from the other side: a cloth held near the end of its budget hands any further demand straight to its threads, while a cloth in the middle of its range absorbs the same demand for nothing. A knee is the shear version of a pre-tensioned cloth, and it is pre-tensioned by its own shape.

Where the accumulation stops #

The repetition argument is left as a direction — a residual per cycle, accumulating — and it has an endpoint, which is the more useful half.

A crossing slips while the shear it is being asked for exceeds what friction can hold, and stops when the difference falls inside the frictional band. So the permanent set does not creep towards the applied shear indefinitely; it stops a band’s width short of it:

permanent shear = applied shear − band width.

With a knee’s demand at 49.55 degrees and a band of a few degrees, the cloth settles at something like forty-five degrees of permanent shear, and it gets there in a number of cycles of order the demand divided by the band — a few tens of bends, not thousands.

That is why a new pair of trousers acquires knees within a day or two of being worn and then stops acquiring them. The accumulation is fast and it saturates, and both halves are consequences of the band rather than of anything wearing out.

Which is why pressing cannot get it out #

The same expression accounts for the essay’s opening observation, which is otherwise an appeal to experience.

Pressing can move the cloth by at most the band’s width, because that is the whole of what is not frictionally held. Press a knee flat and the cloth returns to within a band of where it was — which is forty-five degrees of shear, not nought.

So a pressed knee is a knee with a few degrees taken out of a forty-five degree set. The improvement is real, small, and exactly the size of the quantity that made the set permanent in the first place. Nothing about a hotter iron or a heavier press changes it, because the limit is not how hard the cloth is pushed but how far it can go before friction holds it again.

The only treatment that would reach further is one that lowers the friction while the cloth is being pushed — which is steam, which is why steam works where dry pressing does not, and why it works partially rather than completely.

And what is left decides whether the knee bags or fails #

The margin after saturation is the number that separates two garments.

A sheeting locks at 58.46 degrees and the knee has taken 45, so fourteen degrees remain. An ordinary bend asks for the five it is short by, comfortably inside — so a bagged knee goes on bending for the life of the garment and merely looks bad.

Kneeling asks for more. Going from a bend to a full kneel takes the cloth further round the joint, and the demand curve is convex: 1.2 radii asks 68 degrees against one radius’s 50. That is past the locking angle and past what the fourteen degrees can supply, so the cloth has no configuration left and must either wrinkle out of plane or slip at the yarn level.

Slipping at the yarn level is damage rather than deformation. So the prediction is that a garment worn by somebody who kneels does not bag at the knee — it goes at the knee, and it goes suddenly rather than gradually, at whatever bend first passes the lock.

That is the difference between an office trouser and a workwear one, and it is why workwear knees are reinforced, panelled or made of an openly set cloth with more shear to spend. The two failures are not the same failure at two severities; they are on opposite sides of a locking angle, and which side a garment lives on is decided by how far round the joint its cloth is asked to reach.

What a maker can do about it #

Three levers exist and they are of very different strengths.

Change the grain. A cloth’s shear is available in the same amount whichever way it is loaded, but a panel is not free at its edges, and a knee whose bend runs along the bias has the trellis’s own mechanism working with it rather than against the seams. Cutting a knee on the cross is expensive in cloth and is the reason a trouser cut on the bias hangs and moves as it does.

Open the cloth. The locking angle rises as the threads get further apart relative to their diameter, so a more openly set fabric has more shear to spend and a larger margin at the knee. That runs directly against every other requirement — cover, weight, abrasion, opacity — which is why it is rarely the lever chosen.

Or take the shape out of the geometry. A dart, a seam or a shaped panel builds the double curvature into the cut so that the cloth does not have to find it, and the shear demand drops to whatever is left over. This collection prices that elsewhere and it is the strongest of the three by a wide margin: a knee that has been shaped into the pattern asks the cloth for very little, and a knee that has not asks it for 85 per cent of everything it has.

The reason the third lever is not always used is that a shaped knee is a knee shaped for one angle of bend, and a leg spends its time at many. A garment that fits perfectly bent is baggy straight, which is the trade-off every tailored trouser is a compromise on and which no arithmetic here resolves.

What was counted, and how #

The shear demand is computed on a trellis laid over a sphere, with every node’s position solved and every segment’s length checked — the drape machinery asserts inextensibility at every node rather than at the boundary, which is the discipline that keeps a drape figure from being a drawing.

The trellis is drawn at a pitch that makes it reach a stated fraction of the dome’s radius, and the locking angle it is compared against is computed at the cloth’s own thread spacing rather than at the net’s. That distinction is not cosmetic: computing the lock at the net’s pitch made a knee’s margin depend on how many nodes the figure was drawn with, which is exactly the sort of number that looks like a result.

Two assertions carry it. Reaching further round a dome must demand more shear, monotonically. And the same reach at five different radii must demand the same shear, to a tolerance of a millionth — which is the scale-freedom, and is the claim this rung is built on.

Where the model stops #

The trellis has no bending stiffness and no friction. It says what is geometrically possible and never what a fabric will do, which is a caution this collection has attached to the trellis since it was built. Every number here is a permission.

The accumulation is described and not computed. How much of each cycle’s shear is retained needs the frictional band in shear, and this collection has the band in extension only. Building it would need the normal force at a crossing under a shear rather than under a tension, which is a different contact problem.

The knee is a sphere and it is not. A real knee is closer to a pair of cylinders with a saddle between them, and a saddle has the opposite sign of curvature — it demands shear of the other sense. What that does to the accumulation is unexplored here and is probably not small.

And the garment is not modelled. A trouser leg is a tube of cloth sewn from flat panels with a grain direction, and the direction the knee’s shear is demanded in relative to the warp decides how much is available. A knee on the bias has the whole of the trellis’s range to work with; a knee on the straight grain has considerably less.

Why an elbow is easier than a knee #

The scale-freedom makes a prediction that is worth testing against ordinary experience, because it says something unobvious about which garments bag.

If the demand depends only on how far round a joint the cloth reaches in that joint’s own radii, then the joints that bag are the ones the cloth is obliged to follow closely, not the ones that are sharply curved. A knee inside a fitted trouser leg is followed closely: the cloth has nowhere to stand off, so it must reach right round. An elbow inside a loose sleeve is not: the sleeve stands away, the cloth reaches perhaps half a radius, and half a radius asks for 18 degrees rather than 50.

That is the observed pattern. Fitted garments bag at their joints and loose ones do not, and the usual explanation — that a fitted garment is under more tension — is at best half of it. The other half is geometric and larger: a loose garment is not asked for the shear at all.

It also explains the one exception everybody knows. A tight sleeve bags at the elbow exactly as a trouser bags at the knee, and a loose trouser does not bag at all. What decides it is the fit and not the joint.

The generalisation #

A geometric demand that depends only on a ratio is a demand that does not get easier with size, and the instinct that a small feature is a gentle one is wrong wherever the governing quantity is dimensionless.

The pattern recurs in every covering problem: sheet metal over a die, a label on a curved bottle, a plate on a hull, a map on a globe. In each the difficulty is set by how far round the curvature the material must reach relative to the radius of that curvature, and a small tight feature is exactly as hard as a large gentle one that subtends the same angle.

The second lesson is about repetition. A deformation that is reversible in principle accumulates in practice at the rate friction leaves a residual, so a thousand small cycles are not equivalent to one large one — they are equivalent to a thousand small permanent sets. The design consequence is that the place to look for a permanent shape is not where the largest deformation happened but where the most repeated one did.

Who found it, and when #

The trellis model of woven fabric is from the middle of the twentieth century and the locking angle is standard in composites draping, where the same calculation decides whether a reinforcement can be laid into a mould without wrinkling. Draping a sphere and reporting the shear as a function of distance from the pole is routine in that literature.

Bagging at a knee or an elbow is a standard garment defect with a standard test and a substantial literature, most of which treats it as a viscoelastic recovery problem in the fabric.

What this collection adds is the scale-freedom stated as such, the comparison of a knee’s demand with a specific cloth’s locking angle, and the observation that the accumulation is frictional rather than viscoelastic — which is consistent with everything else in this field and needs no material property that this site does not already carry.

Where the ladder goes next #

The frictional residual this rung ends on is the same quantity that decides where a washed cloth rests and what a tensioned one keeps, and it is computed in extension in what a cloth held at a fixed length has left of its load.

Sideways, the shear a curved surface demands is priced across surfaces in what a hemisphere costs, and the alternative to spending it — cutting the shape into the cloth rather than persuading the cloth into the shape — is what a dart is for.