Textile Structure
how cloth is put together
Concept

Contact pressure — where it appears

The load divided by the area actually carrying it, rather than by the area in view. On a woven cloth the ratio between the two is one over the bearing fraction — twenty to fifty at ordinary loads — and it grows without limit as the load falls, because the contact vanishes faster than the force does.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

What it costs to touch a cloth. The pressure needed to bring a stated fraction of the plan into contact, for plain, 2/2 twill, satin 8 in sheeting. Reaching two per cent of the plan takes 2.83 kPa on a plain and 0.05 on a satin 8, a factor of 55. The stiffness in this figure is fitted and is labelled as such. A yarn's resistance to being squashed out of round has no lower bound at all — a bundle of fibres free to slide is a fluid in cross-section — so no bracket exists to compute this from, and what is used is the value this collection fitted to measured fabric thickness. Every curve moves together across its published range, which is why the ratio between weaves survives and the absolute values are quoted with the fit named.

How much of a cloth is touching

Press a fabric against a flat plate with the weight of a hand and ask what fraction of it is actually in contact. The bearing curve answers, and the answer is about four per cent — of which the great majority is not the cloth's surface at all, but the hairs standing off it, which nothing in this arithmetic can see.

cloth · Contact The presser foot sinks 1.8 µm into a 2/2 twill. A thickness gauge presses a flat foot onto the cloth at 1 kPa and reads the gap. It does not read the geometric thickness. The foot sinks until the area it is touching can carry the load, and that is 2.57% of the plan at a depth of 1.8 µm — so a 2/2 twill in sheeting whose outside stands 381.6 µm apart measures 379.8 µm. How far the foot sinks is a property of the draft, because the bearing area near the top is, and a weave with plateaux stops the foot in a fraction of the distance a plain weave lets it travel. The transverse stiffness used here is fitted to measured fabric thickness rather than predicted: across its published range the reading moves between 377.0 µm and 380.6 µm.

A thickness gauge reads the draft

A presser foot does not stop at the top of a cloth. It sinks until the area it is touching can carry the load, and how far that is depends on the shape of the bearing curve near the top — which is a property of the weave. So there is a weave term inside a measurement nobody thinks of as a weave measurement, and it is worth about one per cent.

cloth · Contact Three kinds of surface, and only one of them starts open. The bearing curves of a 2/2 twill in sheeting, of a terry loop pile, and of a cut corduroy pile of 30 tex, over the first six per cent of a cloth's thickness. The woven curve opens as the square root of the depth and the loop pile's does the same, because a loop's top is a curved thread like any other. The cut pile does not open: it is a flat line at 5.2% of the plan, from a depth of nothing, because a blade severed every tuft in one pass and left every end in one plane. That is not a larger contact area, it is a different kind of contact area — one that does not vanish as the load goes to zero, which is a property no woven surface has.

A pile is the only surface with no crowns

Every other fabric touches on the tops of curved threads, so its contact vanishes as the load goes to zero and the pressure on what is touching rises without limit. A cut pile touches on flat ends left in one plane by a blade — so its contact is finite at no load at all, its pressure concentration is bounded, and it is the only fabric whose abrasion mass loss is an honest measure of its damage.

fancy · Pile At 0.1 kilopascals the plate is standing on hair. A flat foot pressed onto sheeting at 0.1 kPa, over 3 mm of cloth. It stops 25 µm above the cloth's own crowns, because that is where the hairs it is bending can carry the load: of the 10 hairs drawn, 9 reach higher than the foot and are laid over under it, and the rest are untouched. The vertical scale is set by the approach and not by the layer — 5188 pixels to the millimetre off the cloth against 192 along it — because the foot's height above the crowns is tens of micrometres and the layer it stands in is more than a millimetre, so a picture at one scale shows the second and not the first. The thickness reported is 0.432 mm against 0.382 mm for the cloth itself, so 12% of the reading is a population and not a fabric. The hair layer carries up to 0.23 kPa before the foot reaches the crowns at all. A laid-over hair is drawn as two straight segments where a real one is an elastica: the corner is a convenience and the height it turns at is the measurement. What the picture cannot show is that a bent hair leans on its neighbours, which the arithmetic behind it does not know either.

A light touch never reaches the crowns

This collection found that a plain weave touches at points and every other cloth touches along lines, and that the difference is an exponent rather than a factor. It is a real result about a real surface, and at the pressures a fabric is actually touched at, nothing ever reaches that surface.

cloth · Contact The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.

What a thickness gauge reads on a knit

The structure says two yarn diameters and the gauge says more, and the gap is not an error in either. A gauge lands on the highest crowns, through a canopy of protruding fibre, under a load that has already begun to compress both — and it does that on a surface that is nothing but crowns.

cloth · Contact The contact force turns as the climb grows. The two components of the contact force against the climb, for a 20 tex cotton jersey at a 3.5 mm loop. The force along the wales is what friction has to hold and the force through the thickness is what holds the fabric open, and the second is bought at the expense of the first. At a jersey's own climb of one diameter they are 37.50 mN and 7.81 mN; at four diameters, which is a rib on an open gap, they are 22.39 mN and 18.61 mN. The friction balance is the ratio: friction has the whole force to work with and only the along-the-wales part to hold, so the coefficient a relaxed knit would need falls from a half to 0.490.

Every fabric's thread lies in a plane

A woven thread's crimp wave lies in a plane at right angles to the cloth. A knitted loop lies in a plane twelve degrees off it. Both halves of this collection turn out to be one picture with one angle in it, and the angle decides how much of a fabric's contact force acts through its thickness.

cloth · Contact Two exponents in a compression curve. How far a flat plate sinks into plain, 2/2 twill, satin 8 in sheeting, against the pressure it is applying, on logarithmic axes where a power law is a straight line. The measured slopes over the light end of the range are 0.50 for the plain, 0.67 for the 2/2 twill, 0.67 for the satin 8 — against two thirds predicted for any weave carrying a float and one half for a weave carrying none. The prediction is one line of algebra: pressure is a stiffness times a strain times a bearing fraction, the bearing fraction is a square root of depth for a plateau and linear in it for a point, so the pressure is the three-halves power in the first case and the square in the second. Nothing is fitted to produce it; the slopes are measured afterwards and compared. At the heavy end every curve bends, because the crowns have merged and the cloth has stopped being a surface and started being a solid — which is a different regime with a different law, and it belongs to the compaction of a fibre mass rather than to the geometry of an interlacement.

A cloth compresses along its own bearing curve

A fabric's pressure–thickness curve is always fitted with an empirical power law and the exponent is reported without explanation. It is not empirical. At light loads it is two thirds for any weave carrying a float and one half for a weave carrying none, and the two numbers come out of one line of algebra with nothing fitted in it.

mechanics · Compression A rib is quietest at a gap of two diameters. The through-thickness force of a one-by-one rib against the bed gap it is knitted at, in units of the yarn's own diameter. It does not rise from the bottom, and the dip is geometry rather than noise: a crossing is shared between the two half periods either side of its sinker loop, and each of those also carries the interlacing's own diameter — one climbing with it and one against. At a gap of two diameters the second half period climbs nothing at all, and the fabric is at its quietest through its own thickness. Above that both halves climb the same way and everything rises together, which is the regime a real rib is knitted in: 14.8 mN at 5 diameters against 7.0 mN at two.

A rib is quietest at two diameters

Open the beds of a rib and everything about it should get stronger. It does not. The through-thickness force falls to a minimum at a bed gap of exactly two yarn diameters and rises on both sides of it, because a crossing's climb and an interlacing's own climb cancel there.

knits · Two-bed A thickness is a property of the pressure it was measured at. What a gauge reports for sheeting against the pressure it presses with. At 1 kPa it reads 0.382 mm, which is the cloth; at 0.02 kPa it reads 0.555 mm, which is the cloth plus 87 µm of hair on each face. The difference is 31% of the reading and it is not a compressibility: nothing in the fabric has been squashed, the foot has simply stopped in a different place. That is why every thickness standard specifies its pressure to two figures, and why comparing a thickness from one standard with a thickness from another is comparing two different measurements of two different objects. The dashed line is the cloth's own geometric thickness, which no reading below the crossover ever reaches.

A compression curve is two laws in series

A published fabric compression exponent is a fitted number with no derivation attached, and this site has already said why: the range it is fitted over straddles two regimes. One of them turns out not to be in the cloth at all.

mechanics · Compression What a knitted band presses a limb with. Pressure against extension for a 20 tex cotton band at a 3.5 mm loop, wrapped round a 30 mm radius — a wrist. The pressure is the fabric's own tension per unit width divided by that radius, and the tension is the loop's bending with the relaxed shape as the yarn's natural one, so nothing here is fitted. Over the range a cuff is actually used across it runs from a twentieth of a millimetre of mercury to 0.85. The shaded bands are what a compression garment is specified at, and the curve does not reach the lowest of them until 277 per cent — which is not a cuff, it is a fabric stretched almost to the point where its yarn runs straight.

What a cuff presses with

A rib cuff holds a sleeve on a wrist, so it must be pressing. Divide its own recovery force by the radius it is wrapped round and the pressure comes out at eight tenths of a millimetre of mercury — a fiftieth of the lightest medical compression, and two orders below what the same fabric resists being squashed with.

applied · Pressure The force at an interlacing, which no longer lies in the fabric. A crossing of a relaxed jersey in section on the left, with the thickness expanded 3×, and on the right the contact force drawn at the angle the solve gives it — which is a true angle, unlike anything in the section beside it. The head of one course and the feet of the next lie one yarn diameter apart through the fabric, 0.167 mm, so the interlacing is not a point in a plane and the force at it is not in one either. It comes out at 38.30 mN a stitch, turned 11.75° out of the fabric: 37.50 mN along the wales, which is what friction has to hold, and 7.81 mN through the thickness, which is what holds the two faces apart. Divided by the area a stitch occupies the second is 15.1 kPa, or 113 mmHg — a pressure, and the quantity a compression measurement reports.

The force that holds a knit open

Resolving a knitted loop's contact force out of the fabric's plane leaves a fifth of it pointing through the thickness. That fifth is 7.8 millinewtons a stitch, fifteen kilopascals over the area a stitch occupies, and it is the whole reason a jersey has a thickness rather than a plan.

mechanics · Elastica A tube of an elastic band knitted 157 mm round, up one leg. A tube knitted 157 mm round from an elastic band, pulled up an illustrative leg and read at four stations. At each the tube is stretched by the leg's circumference over its own, pulls back with the band's tension at that stretch, and presses with that tension over the leg's radius: ankle 22 cm, stretched 40%, 20.0 mmHg; lower calf 29 cm, stretched 85%, 32.1 mmHg; calf 36 cm, stretched 129%, 39.4 mmHg; below the knee 34 cm, stretched 116%, 37.6 mmHg. The calf is pressed 1.97 times as hard as the ankle. What the bars cannot show is the leg's own give, which a firm tube flattens and which changes the radius the law divides by.

A tube of one size presses the calf harder than the ankle

A band presses a limb with its tension over the limb's radius, so it is easy to conclude that a band grips hardest where the limb is thinnest. That is true of a band held at one tension, and no knitted tube is. A tube knitted to one size is stretched further wherever the leg is thicker, and its tension rises faster than the radius does: an elastic tube pressing an ankle at twenty millimetres of mercury presses the calf at thirty-nine, and a cotton jersey tube presses its calf three and a half times as hard as its ankle. A stocking graduated the other way has to pull hardest where it presses less.

applied · Pressure What leaving the plane costs, which is less than nothing. Bending energy a stitch against how far the yarn climbs between one interlacing and the next, for a 20 tex cotton jersey at a 3.5 mm loop. The curve falls. A climb is not an extra bend added to a curve that was otherwise unchanged; it is part of the straight line the thread has to span, so a longer climb leaves less slack to be spent on curvature. At a jersey's own climb of one diameter the energy is 24395 nJ against 25060 for the planar model — 2.7% lower, where an earlier estimate on this site put the ride at three per cent of the loop's bending and had the sign the other way round. At a rib's climb of three or four diameters the fall is 33%.

What a knit gives up when it is pressed

The woven half of this collection has had a compression curve for several rungs — a thickness that falls under load, a bearing area that grows, a pressure at every point. The knitted half had a plan and no depth. It has a relaxed thickness and an initial slope now, and the two fabrics turn out to resist for different reasons.

mechanics · Compression

Named alongside it

The objects these essays reach for when they reach for this one.

Cloth thicknessBearing curveContact forceLoopReal contact areaCompression energyElasticaSpecificationCrown heightCrown lineTwo-bedBuckling load

All concepts