A crushed pile is not held down by its fibres

Stand on a carpet and the pile goes flat. Step off and most of it comes up, over minutes, and some of it does not — a chair leg leaves a mark that outlasts the chair, and a track across a hall is a permanent feature of the hall.

The natural explanation is that the fibres have been bent past what they can return, exactly as they are at a crease. A tuft standing five millimetres proud and pressed flat has turned through a right angle, which sounds like severe treatment.

It is not. The fibres in a crushed carpet are nowhere near their limit, and the explanation has to be somewhere else.

The claim #

The fibre strain in a crushed pile is negligible, and there is a pile height below which it stops being.

The arithmetic is one line and has nothing in it but geometry. A tuft of height h pressed flat turns through a right angle over its own length, so it lies on an arc whose radius is 2h/π. A fibre of diameter d on the outside of that arc is strained d/2R, which is πd/4h.

That is inversely proportional to the pile height, so a tall pile bends its fibres less. For a cotton fibre of 1.7 decitex — twelve microns across — a five-millimetre pile gives 0.187 per cent and a ten-millimetre pile 0.094.

The threshold is where the strain reaches the bottom of the measured range, two per cent, and it comes out at 0.469 millimetres.

What that number separates #

Half a millimetre is not an arbitrary length in this subject. It is roughly where a pile fabric stops being called a carpet and starts being called a velvet.

A carpet pile is five to fifteen millimetres — a height the fastening it is bound by has nothing to say about. Every one of those crushes with a fibre strain between a tenth and a twentieth of a per cent, which is a strain nothing in a fibre notices.

A velvet or a velveteen pile is a fraction of a millimetre — short enough that the surface reads as a solid colour rather than as a set of tufts, which is what the fabric is for, and which is why a velvet is cut apart rather than woven short. At 0.3 millimetres the strain is 3.12 per cent, inside the measured range, and cotton returns 63 per cent of it.

So the two fabrics crush by two different mechanisms, and it is a geometric threshold rather than a difference in what they are made of. A velvet’s crush mark is partly a fibre that has taken a set; a carpet’s is not.

That is consistent with the two trades’ experience in a way that is worth noting. A velvet’s pressure mark is notoriously hard to remove and is usually treated with steam, which acts on the fibre. A carpet’s is treated by raking, vacuuming or wetting, all of which act on the tufts’ arrangement rather than on the fibres in them.

The threshold is a fibre diameter, not a length #

Half a millimetre is quoted above as though it were a property of pile fabrics. It is not. Rearranging the strain expression puts the threshold at π·d ÷ 4ε, where ε is the bottom of the measured recovery range — so the boundary between the two mechanisms is proportional to the fibre’s diameter and to nothing else about the fabric.

For the twelve-micron cotton used above that is 0.47 millimetres. Change the fibre and it moves in proportion.

A coarse carpet fibre of forty-five microns puts the threshold at 1.8 millimetres. A carpet is still far clear of it, but a short-pile velour made from carpet-weight fibre is not: at a millimetre it is inside the range where fibres take a set, and it is there because of what it is made of rather than because of how short it is.

A microfibre of seven microns puts the threshold at 0.27 millimetres, and that is the interesting one, because it is below where velvets are made. A three-tenths of a millimetre pile in a seven-micron fibre strains to 1.83 per cent — just outside the measured range, where the same pile in cotton strains to 3.12 and is well inside it.

So the two fibres put the same fabric on opposite sides of the boundary.

That is a prediction rather than a description, and it is one the upholstery trade appears to have found empirically: microfibre velvets are sold on their resistance to pressure marking, and cotton velvet is the fabric with the reputation for keeping every mark it is given. The arithmetic says the difference is not the finish and not the construction. It is that a finer fibre bends round the same corner at a smaller strain, in direct proportion to its diameter.

And it inverts the usual reasoning about coarseness. A coarse fibre is chosen for bulk and for resilience under load, both of which are true of the tuft. At the fibre’s own scale it is the more vulnerable of the two, and only the pile height hides it.

What actually holds a carpet down #

If the fibres are not doing it, something is, because a crushed carpet stays crushed for a while.

Two candidates are available and this collection has machinery for one of them.

The tufts lean on one another. A pile fabric at any usable density has its tufts touching, and a tuft pressed over is a tuft resting on its neighbours. Getting up again means sliding past them, which needs the tuft’s own bending to exceed the friction where it touches — the same competition between a restoring bending moment and a frictional barrier that decides where a woven cloth comes to rest, and where a knitted loop settles.

And the tuft’s anchorage may have moved. This collection computes what holds a tuft in, and it is a capstan: the pile yarn wraps round the ground picks that bind it, and the friction along that wrap is what resists a pull. A tuft that has been repeatedly bent over is a tuft whose wrap angle has been cycled, and a cycled frictional grip is not the grip it started with.

The first of these is almost certainly the larger, and the reason to say “almost certainly” rather than to compute it is that the geometry of a leaning tuft resting on its neighbours is not something this collection has built.

Why the strain falls with height rather than rising #

The result is counter-intuitive and the reason is worth spelling out, because the intuition is about the wrong quantity.

A tall tuft looks as though it is being bent more, because it moves further: the top of a ten-millimetre tuft travels ten millimetres and the top of a one-millimetre tuft travels one. That is a displacement, and displacement is not what strains a fibre.

What strains it is curvature, and a tuft turning through a fixed angle over a longer length is a gentler curve. The angle is fixed at ninety degrees whatever the height, because that is what lying flat means, so a taller tuft distributes the same total turn over more length and curves less.

The same reasoning appears in this collection’s account of a crease from the other side: a fold’s severity is set by its radius, not by how far the two faces have moved, and the tightest fold is the one where the yarn’s own diameter sets the radius. A tuft has no such floor, because nothing on the inside of the bend is touching anything.

The two marks a pile keeps, and how to tell them apart #

An account that separates the mechanisms should predict something a reader can check, and this one predicts that a carpet and a velvet keep different kinds of mark.

A carpet’s mark should respond to disturbance and not to heat. If what holds it down is tufts leaning on one another against friction, then anything that lowers the friction or supplies a disturbance lets them up: raking, brushing, vacuuming, damp. Time alone should do a little, because the tufts’ own bending is working against the barrier continuously, and the recovery should be gradual and never quite complete.

A velvet’s mark should respond to steam and not to brushing. If part of it is a fibre that has taken a set, then the fix is whatever lets the fibre release — moisture and heat for the natural fibres, heat alone for the melt-spun ones — and brushing the surface rearranges the tufts without touching the fibres in them.

Both of those are what the two trades actually do, and neither was arrived at by computing anything. What the arithmetic adds is that the two remedies are not two versions of one practice: they act on two different mechanisms that happen to be separated by a pile height of about half a millimetre.

There is a third prediction that follows and is more useful. A pile made short for appearance and expected to recover like a carpet will disappoint, because it is on the wrong side of the threshold. A half-millimetre pile in an upholstery that is sat on has fibres inside their measured recovery range and will keep a proportion of every crush permanently, and no amount of the treatment appropriate to a carpet will help.

What was counted, and how #

The strain at each height is computed from the arc radius and the fibre diameter, and the machinery asserts the relation rather than the values: a tuft is inside the measured range exactly when it is shorter than the threshold the closed form gives.

That assertion is written that way because of a mistake this collection has made three times. An earlier version asserted that the list of heights straddled the threshold, which is a property of the default arguments rather than of the finding, and it failed the first time a figure asked about a single height. The rule of the house is to assert the relation or the regime and never the value the defaults happen to produce.

The monotonicity is asserted too: a taller pile must bend its fibres less, which is the whole content of the inverse proportionality.

The one number this rung would most like #

The account ends at a mechanism it cannot compute, and it is worth naming exactly what would close it, because the gap is narrow.

What is needed is the moment a leaning tuft applies to its neighbours, against the frictional moment resisting it. Both halves are within reach of machinery this collection already has. The restoring moment is a bending stiffness times a curvature, and the bending stiffness of a pile yarn is the same bracket every other yarn on this site has. The frictional resistance is a normal force at the contacts times a coefficient times a lever arm, and the normal force is what the neighbouring tufts’ own bending supplies.

What is missing is the geometry: how many neighbours a leaning tuft touches, where along its length, and at what angle. That depends on the tuft density and the pile height together, and it is a packing problem rather than a mechanics one.

Solving it would give the quantity a carpet specification actually wants, which is not a fibre property and not a tuft anchorage but the pile density at which a fabric recovers from being stood on. Too sparse and the tufts have nothing to lean on and fall over; too dense and they cannot get past one another. There is presumably an optimum, in the same way that there is an optimum cover for a woven cloth’s interchange, and this collection can see the shape of the question without being able to answer it.

Where the model stops #

The tuft is treated as bending in a circular arc, and it does not. A real tuft pressed flat is loaded at its tip and restrained at its root, so its curvature is largest at the root and smallest at the tip — an elastica rather than a circle. The strain at the root is therefore larger than the figure here and the strain at the tip smaller, with the same average. For a carpet, where the answer is an order of magnitude below the threshold, that redistribution does not change the conclusion. For a velvet, where the answer is inside the measured range, it might: the root could be at twice the average strain, and the threshold height would move up accordingly.

The fibre is treated as bending about its own middle, which is the free bound of this collection’s bending bracket. That is the right end for a fold and it is the right end here for the same reason, but a pile yarn is usually more highly twisted than a weaving yarn and a twisted yarn is further up the bracket.

Nothing here computes the recovery. For a carpet the strain is off the bottom of the measured range and the honest answer is a refusal. For a velvet it is inside, and the figure quoted is an immediate recovery, which excludes the delayed part — the part steaming a velvet is exploiting.

And the leaning is described and not modelled. The whole of the mechanism this rung concludes with is qualitative here, and it needs a geometry of tufts in contact that this collection does not have.

Where the pile height comes from, and what it costs #

The threshold makes pile height a design variable with a boundary in it, which is worth putting beside what else pile height decides.

A taller pile is more yarn per square metre and therefore heavier and dearer, at the same tuft density. It is also less stable laterally — a tall tuft leans further before it meets its neighbours — and it wears differently, because abrasion acts at the tip and a tall pile has more length to lose before the ground shows.

Against all of that, this rung adds one item on the other side: a taller pile strains its fibres less when it is crushed, in inverse proportion. Going from one millimetre to five divides the fibre strain by five and takes it from just inside the measured range to well outside it.

So the usual reasons for a tall pile — depth, warmth, the appearance of quality — turn out to have a mechanical companion that nobody quotes. And the usual reason for a short pile, which is that the surface reads as a solid rather than as tufts, is bought at the cost of a fibre strain that has entered the range where fibres take a set.

That is a genuine trade and it has a computable crossing point. What this collection cannot say is where on either side of it a particular fabric should sit, because the appearance requirement is not a quantity in any of these ladders.

The generalisation #

Bending severity is a curvature and not a displacement, so making a member longer makes its fibres safer at the same imposed angle. The intuition runs the other way because a longer member visibly moves further, and the two quantities are inversely related.

That is a recurring source of design error wherever something must be repeatedly deflected: a flexure hinge, a cable at an entry, a wire in a moving loom, a leaf spring. In each case the fix for a fatigue problem is to lengthen the bending region rather than to make it stiffer, and in each case the instinct is the opposite.

The second lesson is about where a threshold in a subject’s vocabulary comes from. A distinction the trade draws in words — carpet against velvet — sometimes turns out to sit on a computable boundary, and finding the boundary explains why the two are treated differently without anybody having decided that they should be.

Who found it, and when #

Pile crush and recovery are a standard concern of the carpet trade, and the standard tests are all fabric tests: apply a static load or a rolling one, measure the thickness lost and the thickness recovered. The mechanisms usually named are fibre bending, fibre-to-fibre friction and the tuft’s anchorage, in that order.

This rung’s contribution is to remove the first of those from the list for anything of carpet height, on a geometric argument with two measured inputs. The fibre diameter and the recovery table are somebody else’s; the arc radius and the threshold are arithmetic.

The threshold’s coincidence with the carpet-and-velvet boundary is an observation rather than a claim about how either fabric came to be made at the height it is made at.

Where the ladder goes next #

The mechanism this rung is left with is the same one that decides where a woven cloth rests and how much load a tensioned one keeps, which is a frictional barrier against a restoring bending moment.

Sideways, the fold arithmetic in the opposite regime — where the radius is set by the yarn itself and the strain is a hundred times larger — is what a crease is, and what holds a tuft against a pull rather than against a bend is a capstan with a measured coefficient in it.