Textile Structure
how cloth is put together
Concept

Contact force — where it appears

The force one thread presses another with where they cross, which is twice its tension times the sine of its weave angle. In a cloth that is not under tension it cannot be computed from the geometry at all, and this site reaches it by inverting a measured thickness.

Named by 42 essays across 9 fields — each of them below, with the objects they name alongside it.

The force at a crossing. One warp end of a sheeting riding over three picks, with the weave angle Peirce's geometry solves for at that construction: 36.8°. An end held at 0.50 N presses each pick it crosses with 0.599 N, which is twice the tension times the sine of the angle and has no material constant in it. What the drawing cannot show is the relaxed case: a cloth with no tension in it still holds its threads together, and what does the holding then is the yarn's own resistance to being bent, which needs an elastica this site does not have.

Every crossing is a force

A thread arrives at a crossing at an angle and leaves at its negative, so the two pulls have transverse parts that add. The force pressing one thread onto another is twice the tension times the sine of the weave angle — and for an ordinary sheeting that is more than the tension in the thread itself.

cloth · Contact Where a thread stops sliding and starts breaking. A pick of sheeting gripped over a length of cloth, drawn one crossing at a time. The resistance is 0.30 times the 0.599 N each crossing presses with, so it rises with the length held; the breaking load of 3.74 N does not. The two are equal at 7.4 mm. What the drawing cannot show is that μ is a range rather than a constant, so the mark is a band and its position is exactly inversely proportional to the friction.

A thread is held one crossing at a time

Grip a thread over a length of cloth and its resistance to being pulled out rises with that length, because it is held at every crossing it makes. Its own breaking load does not rise at all. The two curves cross, and the length at which they cross turns out to be a seam allowance, a frayed edge and a tuft's anchorage — three rules of thumb with one number under them.

cloth · Contact A thread withdrawn from a plain weave and from a satin. One pick of a sheeting being pulled from 1.60 mm of cloth, in a plain weave and in a five-end satin, at a friction coefficient of 0.30. The drawn width of each thread is the tension in it at that point, to a common scale; the faint rules are where the thread actually turns, which is at its interlacings and nowhere else. A plain weave turns 2.80 times per millimetre and a satin 1.12, so the plain weave's tension compounds 2.78-fold over this length against the satin's 1.46. What the drawing cannot show is that the wrap angle is taken from a plain-weave geometry in both panels: a satin's crimp is genuinely smaller, so its real turns are gentler than these and its grip weaker still.

A thread is gripped where it turns

The arithmetic this site has used for fraying, seam slippage and tuft anchorage counts every crossing a thread makes as a grip and adds them up. A thread lying flat on the surface of a satin presses on nothing at all, and a thread that is gripped is gripped by a friction that compounds along its length rather than adding. Both corrections were recorded as missing and both are here.

cloth · Contact Stick, slip, and the ratio between the two coefficients. The force in a thread as a cloth is agitated, with a static coefficient of 0.300 and a kinetic one of 0.225 — a ratio of 0.75, which is what fibre on fibre measures. The force climbs until it reaches the static limit, the contact breaks away, and while it is sliding it resists only at the kinetic limit. So a cloth that is being shaken can be left anywhere in the narrower band, and a cloth at rest anywhere in the wider one. What the trace cannot show is how much this buys: the band does not narrow in the ratio of the coefficients, because the restoring force stiffens away from the minimum, and the real narrowing is nearer 0.87 than 0.75.

Two coefficients, not one

Every friction on this site is a single number, and the account of why a cloth relaxes better when it is agitated depends entirely on there being two. Separating static from kinetic changes what a resting state is: a cloth at rest is held by one coefficient and a cloth being shaken by the other, and the band it can be left in narrows — by less than the ratio, because a cloth's restoring force is not linear in its extension.

cloth · Friction The step between a figure and its ground. Three figures on a plain ground, all in one sheeting's threads at its own setts. A thread presses on the thread it crosses only where it turns, and it turns at its interlacings — so the pressing a region receives per unit area is the contact force at one turn times the turns per unit area, and the second factor is a property of the matrix exactly. A five-end satin turns two fifths as often as a plain weave, is pressed two fifths as hard, flattens less, and stands 55 µm proud of it. What the rows cannot show is that both regions are given a plain weave's weave angle: a satin's crimp is genuinely smaller and its turns genuinely gentler, so the real step is larger than this, by an amount not computed here.

A figured cloth has a step in its surface

A damask is one cloth in one set of threads at one sett, and it is not flat. A thread presses on the thread it crosses only where it turns, so a region that turns less often is pressed less often, flattens less and stands thicker — and the step is a ratio of interlacing rates, read off the matrix with no yarn property in it.

pattern · Blocks A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.40 N per crossing the sections flatten to aspect ratios of 1.75 and 1.84, the cloth thins to 0.263 mm, and the warp runs flat for 0.109 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

A flattened thread is a record of a force

A yarn in cloth is not round, and this site has modelled the flattening for as long as this collection has run with the amount of it left as a number somebody chose. Give the section a stiffness and ask what the cloth prefers, and the answer is a circle — at every stiffness, for every balanced cloth in the table. Flattening does not happen by itself; it happens because something pressed.

mechanics · Compression What holds a tuft in, in newtons. The withdrawal force of a V-fastened and a W-fastened tuft over the reported range of yarn-on-yarn friction, on a duck ground with 1.00 N in each pick. Each is the friction at the wraps, with each wrap's share dragged around every wrap between it and the pulled end — a capstan series rather than a single factor. W runs from 0.79 to 6.84 N and V from 0.15 to 0.41. Both are an order of magnitude below what a carpet is specified at, which is a finding about carpets rather than about the model. What the chart cannot show is the backing, which is where the rest of a tufted carpet's anchorage comes from.

What holds a tuft in, in newtons

The pile ladder computed a tuft's anchorage as a capstan ratio and said, correctly, that a ratio was all it could offer. A ratio multiplies a tension and there was no tension anywhere on this site. There is one now — and the answer, put beside what a carpet is actually specified at, falls short by a factor of three.

fancy · Pile Which seams slip and which break. For each cloth, the grip a 10 mm seam allowance offers divided by the thread's own breaking load. Above one the fabric or the thread gives first and the seam holds until it does; below one the threads slide out and the seam opens with the cloth intact. voile, batiste, poplin, sheeting break; cheesecloth, muslin, duck, filter slip. The allowance that would save every cloth in the table is 74 mm, which is set by the openest of them alone. What the chart cannot show is the stitching itself, which has its own strength and its own way of cutting the threads it passes through.

A seam slips before it breaks

A sewn seam fails in one of two ways and the trade names them separately: the threads pull out of the cloth beside the stitching, or something breaks. Which one a given cloth does is decided by whether its seam allowance clears the length at which grip beats strength — and at the ordinary ten millimetres, the table divides four against four.

applied · Seams A crossing before and after it is pressed. One warp end of a sheeting riding over three picks, drawn twice to the same scale. Unpressed, both sections are circles and the cloth is 0.388 mm thick. At 0.42 N per crossing the sections flatten to aspect ratios of 1.79 and 1.88, the cloth thins to 0.260 mm, and the warp runs flat for 0.113 mm over each pick before it begins to curve. What the drawing cannot show is why the cloth does not do this by itself: flattening shrinks the arc radius as well as the crimp height, so it costs bending energy, and a relaxed cloth keeps its threads round.

Peirce and Kemp are one cloth at two moments

This site has run two thread sections side by side since the setting field was built — a circle and a flattened racetrack — and said honestly that they disagree and that the disagreement is the point. They are not rival descriptions of the same fabric. They are descriptions of the same fabric before and after something pressed it, and the difference between them is a pressure that can now be named.

setting · Sett How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The criterion gets a force

This site's integrity criterion decides exactly whether a draft describes one cloth, and has one standing limitation: it says a tuft bound under one pick and a tuft bound under three are both attached, and it is right, and one of those is a carpet while the other sheds. What separates them needs a normal force in a fabric that is not under tension — the number the rung that computed it recorded as unavailable, and the one a thickness gauge now supplies.

fancy · Integrity From a calender's line load to a force at one crossing. A sheeting through a nip loaded at 30 N per millimetre of bowl width, with the cloth in contact over 5.0 mm. The pressure is the first divided by the second, 6.00 N/mm², and the force at one crossing is that pressure times the area a crossing owns — the product of the two thread spacings, 0.1374 mm². So the crossing carries 0.824 N, the sections flatten to 2.42 and 2.63, and the cloth thins from 0.388 mm to 0.218 mm. What the drawing cannot show is that two cloths through the same nip are not given the same treatment: the area a crossing owns varies fivefold across this site's table, and it is a factor in the force.

A calender spends the compression for good

Calendering was described on this site as moving a cloth from one thread-section model to another, which was right and had no number in it because the amount of the move was a free parameter. It is a pressure now — and the same nip setting turns out to give two cloths quite different treatments, because the force at a crossing is the pressure times the area a crossing owns.

finishing · Lustre finish The pressure a wetting generates in a close cloth. The pressure a sheeting's yarn is compacted at, against how far its fibres have swollen. Below 9.29% the cloth accommodates the swelling as a shape change and the pressure is nothing; above it there is no state at constant thread length, so the yarn must be compacted back to a diameter the geometry can hold and van Wyk's cube law prices it. At cotton's 20% it is 7.80 MPa. The other route out — stretching the threads until they are long enough to wrap the swollen partner — needs 9.1% of strain against a breaking strain of 6.6% computed from the site's own tenacity and modulus, so the thread would break first and there is one route rather than two. What the curve cannot show is its own uncertainty: van Wyk's constant runs from 0.003 to 0.011, so the height of this curve is known to a factor of nearly four and its shape is not.

A wetting supplies the force the criterion needs

This collection's criterion decides whether a cloth is one cloth, exactly, and cannot see friction — so pricing what it misses needed a contact force, and the only one available had a measured fabric thickness inside it. A wetted close cloth generates one from geometry alone, and it lands within seven per cent of the measured route's answer.

fancy · Integrity The size of the capstan correction. The ratio of the capstan crossover to the crossover a sum of independent contacts gives, for every cloth in the table at a friction coefficient of 0.30. It is exactly ln(1 + z)/z, where z is the thread's breaking load times the wrap angle, over the contact force — a quantity with no friction coefficient in it at all. That is why the earlier result that μ·L* is exactly constant survives this correction to twelve figures: μ was only ever in the factor outside the logarithm. The correction is largest for the duck, whose coarse strong yarn makes z large, and smallest for the batiste. What the rows cannot show is that a real cut edge frays at a friction nobody measured on that particular cloth.

How far a cut edge frays

A seam allowance, a fray width and a tuft's bound length are the same number wearing three hats, and the earlier estimate of it was four times too long. Correcting it moves the whole table across the boundary an ordinary allowance sits on — from four cloths holding and four slipping, to all eight holding — and turns a specification argument into a different one.

applied · Seams How much too thick a round section is, and what reconciles it. For each cloth in this site's table: the thickness a circular Peirce section predicts, the thickness a cloth of that construction measures, and the force per crossing that makes the flattened model reproduce the measurement. The over-prediction runs from 36% to 81%. The reconciling forces span a factor of 4.6 across a table whose counts span a factor of six, and every one of them is of the order of the contact force the cloth's own warp tension supplies — which is what makes this a model rather than eight fitted parameters. What the rows cannot show is that the thicknesses are trade figures for cloths of these constructions rather than measurements of these particular fabrics, so what is being read is an ordering.

The relaxed cloth's contact force

How hard two threads press on each other in a cloth that is not being pulled is the number this site's own integrity criterion has needed since its first essays, and the route to it was an elastica nobody had. A thickness gauge supplies it instead — because a cloth's thickness is a record of how flat its threads are, and how flat they are is a record of how hard they are pressed.

mechanics · Compression The crossover length in four weaves. The gripped length at which a pick of a sheeting breaks rather than slides, in four weaves at a friction coefficient of 0.30. It is a millimetre or two for a plain weave and 8.0 mm for an eight-end satin, which is what a cut edge of each does. The interlacing rate appears in the answer only as a factor outside the logarithm, so the ratios between the four are exactly the ratios of their interlacing rates — in this model and in the sum of independent contacts alike. The rung below expected the float correction to change that ordering; it does not. It changes the size, by a factor of 3.7. What the rows cannot show is that all four use one cloth's crimp, so a satin's genuinely gentler turns are not in them.

A float presses on nothing

The rung below expected the float correction to change how a satin's hold compares with a plain weave's, by something like the ratio of their interlacing rates. It does not change the comparison at all — the interlacing rate leaves the answer outside the logarithm and divides straight out of any ratio. What it changes is the absolute answer, by a factor of four, for every weave alike.

weaves · Float The two halves of a beat-up force. The force the reed must apply per metre, for a sheeting, against the number of picks that are still sliding against the warp. The elastic half — the warp tension times the crimp's elasticity with respect to the pick spacing — is 573 N/m and does not depend on the zone at all; that cancellation is exact and is the result the rung below established. The frictional half is 1133 N/m per sliding pick and is nothing but zone. Against a reported 400–1500 N/m, that leaves room for at most 0.82 picks sliding — so the fell region a weaver can see, ten to fifty picks deep, is not the same quantity as the picks that are still moving. What the rows cannot show is that this is a static friction throughout, and a beat-up is a blow.

The half of the beat-up that is all zone

The elastic half of the beat-up force is exact and the length of the beat-up zone cancels out of it, which is this site's own result and disagrees with every practical account of weaving. The frictional half is nothing but zone — and requiring the total to match the force a loom is actually built to apply puts the number of picks still sliding at about one.

setting · Beat-up A sheeting's crossing, dry and wetted. One crossing of a sheeting in section at three swellings: dry, at the swelling where its geometry has its last state, and fully wetted at 20%. The closure condition is that the two systems' crimp heights add to the cloth's thickness, and each can supply at most the height it reaches when its straight portion has just vanished. Swelling raises the demand in proportion and the supply more slowly, so the margin closes and then goes negative — at 9.29% for this cloth against cotton's 20%. The bottom panel is drawn as far as the threads reach and no further, because there is no state to draw. What the drawing cannot show is what happens instead, which is that the yarn is compacted.

The swelling a cloth cannot take

Three of the eight cloths in this collection's table have no wet state at all. A thread of fixed length cannot wrap a partner that has grown by a fifth, so the closure condition fails and the geometry has nothing to offer. What happens instead costs megapascals, and the alternative route is not merely dearer but unavailable — the thread would break first.

mechanics · Compression What a tensioned sheeting has left of its load. A sheeting pulled to a strain, clamped at that length and left. Its length does not change, and its load does: crossings rearrange locally until the load has fallen to what friction alone can hold, which is 0.0756 N per end and is the same number whatever the cloth was pulled to. So the fraction retained is that floor over the load applied, and it falls — a cloth tensioned to 4.94 per cent keeps 29 per cent of what it was given. Below the resting band's own edge nothing is lost at all, because the cloth was never outside what friction could hold. What the plot cannot show is time: nothing here says how long the rearrangement takes, only where it stops.

A tensioned cloth loses its load

Clamp a fabric at a fixed length and its tension falls overnight. Nothing crept and nothing flowed: the crossings rearranged locally until the load had dropped to what friction alone can hold, and that level is the same number whatever the cloth was pulled to — so the harder it was tensioned, the smaller the share it keeps.

mechanics · Set Whether a cloth's hairs can reach one another. n_A λ² for each construction in this site's table — the hairs per square millimetre times the square of their own length, which is the pure number that asks whether a hair can touch its neighbour. It is a count times an area, so it has to be a pure number. Every one of them is under one, which means no ordinary woven cotton cloth has a hair layer at all: it has isolated whiskers on a bare surface. The dashed line is the threshold. The spread across the whole table is only 1.9-fold, because the density goes as the sett times the root of the count and those move in opposite directions as a cloth is made finer — so construction is almost powerless here, and everything that crosses this threshold does so by finishing rather than by weaving.

A knit gives up its fibres more easily

Knitwear pills and shirting does not, and the fibres are often the same fibres. The difference is a count of yarn per unit area and a pressure between threads, and both of them push a knit over a threshold that a woven cloth of the same yarn cannot reach.

knits · Knit Everything a plain knit can be, at one loop length. Bending energy over the two spacings a plain knit has to choose, for a 20 tex cotton yarn at a 3.5 mm loop, as a multiple of the energy the fully relaxed fabric holds. Darker is more. The solid edge is where the straight line between two interlacings reaches the yarn between them — the geometry's own limit, with nothing elastic in it — and there is no state beyond it at any force. Munden's three relaxation states are marked, and the thing to see is that they are not in a hollow: they lie along a slope, in order, with the most completely relaxed of them the highest. An unset yarn would slide down and to the right until it met the edge. Real fabrics sit where they were left.

The relaxed knit is not at a minimum

Differentiate a loop's bending energy along the fabric instead of across it and the answer should be zero, because a relaxed fabric is one nothing is pulling. It is not zero. It is tens of newtons a metre, downhill in both directions at once — and the three relaxation states everybody measures run the wrong way up the slope.

knits · Elastica A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

A rib climbs a gap

A jersey's yarn crosses one diameter between interlacings because that is what a crossing of two threads is. A rib's crosses the whole distance between the two beds. Nothing else in the model changes, and that one length is the whole mechanical difference between the fabrics.

knits · Two-bed 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 What a knit gives when it is pulled. Force against course-wise extension for a 20 tex cotton jersey at a 3.5 mm loop, computed from the loop's own bending with the relaxed shape as the yarn's natural one. The loop length is the same at every point on the curve: nothing here is the yarn stretching. It reaches 92% extension at 2.96 N per metre and then stiffens by a factor of 85 over the rest of the range, as the straight line between two interlacings runs up against the yarn between them. This is the number the collection's second phase recorded as a lower bound it could not compute, and it is still a lower bound in one respect: friction at the contacts is not in it, so a real fabric is stiffer than this and does not come back along the same curve.

A loop is set and not sprung

The way out of a model that predicts a jersey should spread is to stop treating its yarn as a straight rod bent into a loop. A yarn that has been wetted, heated and dried has taken the loop as its own natural shape — and once the natural shape is the loop, every force downstream becomes computable with the relaxed fabric as the origin.

finishing · Elastica 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 What a raising machine can catch in a 2/2 twill. The draft with every weft float long enough for a raising wire to lift marked on it. The teeth need an unsupported length of thread on the surface, so a cloth in which every thread is bound at every crossing offers them nothing at all — which is why a napped fabric is always a twill or a satin and never a plain weave.

What holds a nap in a knit

A fibre buried in a woven cloth breaks rather than slides once about thirty-four millimetres of it is held, and a cotton staple buries about fourteen. In a knit the same figure is over a metre — seventy-five times the burial available — so nothing is ever close, and a raised knit sheds for the whole of its life.

finishing · Nap What holds a thread in, as two factors. The two quantities whose product is the grip on a buried thread, for a woven poplin and a jersey of the same yarn. Each contact in the knit is lighter by 8.0 times, and the contacts are further apart by 5.6 — one per half loop length against one per thread spacing. They multiply rather than competing, so the grip per millimetre of buried thread is 31 times weaker in the knit, and the crossover length at which a thread breaks rather than slides moves with it: 11.6 mm in the cloth and 461 in the knit. The two factors are drawn separately because their product is two orders of magnitude and a bar chart of it would put the knit's bar below the width of a line.

The knitted pilling criterion gets its number

Knitwear pills and shirting does not, and the standing explanation here has been that a knit presents more exposed yarn under less pressure between its threads. The second half had no number. It has one now, and it is bigger than the argument needed: the grip on a buried fibre is thirty times weaker in a knit than in a woven cloth of the same yarn.

applied · Pilling 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.

How thick a knit is

Two centre lines pass at a diameter and each has a radius on either side, so a plain jersey is two yarn diameters thick with nothing fitted. It does not depend on the gauge, it lands inside the band of this collection's woven cloths, and it is lower than any gauge will read.

knits · Knit A knitted loop is a plane curve in a plane that is not the fabric's. A relaxed 20 tex jersey at a 3.5 mm loop, seen from the end of a course. Each course is a straight line because it is one: a half period leaves and arrives along the course direction, so the plane it bends in contains that direction, and a plane curve seen along a line in its own plane projects to a segment. The upper panel expands the thickness 3× so the arrangement can be seen and no angle may be measured off it; the lower panel is the same fabric at one scale, where the tilt is what it really is. The angle is 11.75°: a climb of one yarn diameter, 0.167 mm, against a drop of a course spacing and a diameter, 0.803 mm. Successive courses overlap by exactly one diameter, which is the interlacing, and the fabric is therefore 0.334 mm thick — two yarn diameters, with nothing fitted.

A state is a thickness too

This collection's rule is that a fabric dimension quoted without its relaxation state is not a measurement. A knitted fabric has three dimensions and only two of them obey the rule: its thickness is the same in every state, because the interlacing that sets it does not relax.

finishing · State Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.00, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not.

A thread between two crossings is an elastica

Every crossing force Peirce's thread path can give comes from a tension, and a cloth on a table has none. What presses its threads together there is their own unwillingness to be bent — and recovering that needs a shape nobody had, because a path assembled from an arc and a straight line has a bending moment that jumps.

mechanics · Elastica Where a loop's bending actually is. Half a stitch — from the crown of a needle loop's head to the bottom of the next sinker loop — with the curvature at every station drawn as a spine standing off the curve, in units of one over the yarn diameter. The centre line is drawn as a line rather than at the yarn's own width here, because the subject of this figure is the curvature and a yarn drawn at true width covers its own spines. The peak is 1.18, at 28% of the way along, and the curvature varies smoothly from one end to the other with no jump anywhere. That smoothness is the whole point: Peirce's construction joins an arc of constant curvature to a straight line of none, so its bending moment steps at the join, and a step in moment is a point force no thread can carry. An elastica has no steps in it, which is why its forces exist at all and his do not.

A force is what an energy does when a crossing moves

The force a thread presses its neighbour with is the rate its bending energy changes as the crossing is displaced. Solved as a constrained minimisation, that number arrives with the shape rather than after it — and the same force, recovered a second time from the curve's own equilibrium, agrees to a tenth of a per cent.

mechanics · Elastica How much of a thread is spent going round the one it crosses. The share of a warp end's length that lies inside the wrap — the arc of radius half the combined diameter, which is as close as two centre lines can get — for every cloth in this collection's table, with a jersey at the foot for comparison. It runs from 7% on an open scrim to 54% on a sheeting, and what is left over is a straight run with no shape to solve. A knitted loop's figure is zero: its peak curvature never reaches the wrap's, so it touches at points and is free in between. That is the whole reason the same solver refuses a shirting and converges on a jersey, and it is a statement about the two fabrics rather than about the arithmetic.

A woven thread has no room to bend

Set an elastica solver on an ordinary shirting and it refuses the problem. The refusal is the finding: a woven thread's whole crimp is spent going round the thread it crosses, between a fourteenth and a half of its length lies inside that wrap, and what is left has no slack to take a shape with.

setting · Elastica What a run has to overcome. The friction holding one loop in the loop below it, against how tightly the fabric is knitted. It is the coefficient of friction times the contact force times the two interlacings a stitch makes, and it runs from 14.7 to 35.1 millinewtons across the knittable range — a factor of 2.4 for a factor of under two in tightness factor, because the contact force and the tightness move together. That is the arithmetic behind a rule every knitter has: a slack fabric runs and a tight one does not, and tightening it is the only lever that works. The figures are upper bounds, since a set yarn presses less.

A run is a race between two energies

A dropped stitch travels when a loop can be pulled out of the loop below it, and there are two candidate drivers: the energy the loop releases by unravelling, and the load the garment is under. One of them turns out to be negligible, and knowing which changes what a knitter can do about it.

applied · Damage How hard a relaxed fabric presses on itself. The normal force at one crossing of a relaxed cloth, against the force at one interlacing of a relaxed jersey. The woven figures were recovered by inverting a thickness measurement through a compression energy; the knitted one comes from a solved shape and no measurement at all, so the two are genuinely independent rather than two readings of one number. Every cloth in the table presses harder than the knit — by between 5 and 22 times — and the knitted figure is an upper bound besides. One ratio is behind a list of differences usually explained separately: which fabric gives up a fibre end, which pills, which frays, which lets a seam slip.

What a loop presses with

A knitted loop hangs on the loop below it and presses on it with a force nobody has been able to state. Solved from the loop's own bending it comes to about forty millinewtons a stitch — an order of magnitude under a woven crossing's, by two independent routes that have nothing in common.

mechanics · Elastica Two numbers that stay at zero however large the fabric gets. The linking number of two adjacent courses, and the writhe of one course per wale, for tubes of 6 to 20 wales. Both sit at zero and stay there: the largest departure anywhere on the plot is 1.5e+1, which is the sampling. A quantity that should grow with the fabric and does not is the cleanest kind of null result: the model has the geometry of knitting and none of its topology, and making the fabric bigger does not make the topology appear.

Five symptoms of one omission

Five things this collection recorded as unexplained, found in four different ladders over three years of work. They are the same defect seen from five directions, and the defect is one sentence written for good reasons with no visible cost at the time.

knits · Topology A rib crosses a gap where a jersey crosses a diameter. A one-by-one rib in section across 5 wales, drawn at a bed gap of 3 yarn diameters — 0.501 mm — because the relaxed gap of a rib is a measurement this collection does not hold and every figure of one says what it was drawn at. Alternate wales sit on opposite beds, so every sinker loop between them travels the whole gap. In a jersey the same yarn climbs one diameter, 0.167 mm. That single difference takes the contact force from 38.30 mN at 11.7° out of the fabric to 72.51 mN at 16.2°, and the through-thickness part from 7.81 mN to 20.17 mN.

A run cannot cross a bed

A dropped stitch unroves because its neighbour above can pull it out along a path that costs almost nothing. In a rib the neighbour above is on the other bed, and the path goes through the gap — so a run in a two-bed fabric has to pay for a climb before it can take a single loop.

applied · Damage How hard a fabric is to bend, per unit width. Bending rigidity in micronewton metres per unit width, computed the same way for eight woven cloths and for a jersey in each of its two directions: the yarn's own rigidity, times the length of yarn per unit area, times the fourth power of the cosine of the angle each element makes with the bending direction. The knit lands inside the band of the woven cloths rather than below it, which is worth knowing because a knit is usually called the softer fabric. It is not softer to bend; it is softer to stretch, by three decades. What it does have is direction: 2.17 to one between its two axes, where a balanced plain weave is near one, and the soft axis is the one a jersey rolls about at its top and bottom edges.

What friction has to hold in a relaxed knit

A knit's bending energy slopes away from the fabric everybody measures, so something is holding it there. Along its courses friction holds comfortably. Along its wales the driving force is exactly the contact force, so the whole balance collapses to one condition — the friction coefficient must exceed a half — and no yarn in this collection reaches it.

mechanics · Friction A bouclé loop at 80 per cent overfeed, pressed to four heights. One loop of a bouclé at 80 per cent overfeed on 3 millimetre binder spacing, drawn to scale at 100, 80, 50, 20 per cent of its free height, with the force it pushes back with beneath each. The pressed loop is the same elastica the free one is, with one condition added: at its apex the tangent is along the core again, so a pressed loop is two half-loops each with its rise prescribed. At 100 per cent it stands 1.95 millimetres and pushes with -0.00 millinewtons; At 80 per cent it stands 1.56 millimetres and pushes with 12.19 millinewtons; At 50 per cent it stands 0.97 millimetres and pushes with 10.02 millinewtons; At 20 per cent it stands 0.39 millimetres and pushes with 4.90 millinewtons. What the drawing cannot show is the stiffness bracket: the force is the lower bound, with the fibres free to slide.

A loop has a maximum force in it

A bouclé loop pressed by its neighbour is the same elastica the free one is, with one condition added — at its apex the tangent lies along the core again, so a pressed loop is two half-loops with their rise prescribed and needs no contact solve at all. Solved, the eighty-per-cent loop pushes back with nothing at its free height, twelve millinewtons at four fifths of it, and two and a half at a tenth. A curve with a maximum in it is a softening spring, and the promise this account made — a cloth's position read from its beat-up — cannot be kept, because the whole bracket sits at one pressure.

fancy · Fancy yarn 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 leaving the plane costs

Every number the flat loop model produced, beside the same number with the climb in it. Four of the five fall, none moves by four per cent, and the estimate that priced the third dimension beforehand had the sign the wrong way round for a reason worth naming.

mechanics · Elastica 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 How little eccentricity a curl needs. The radius a jersey would curl to, against how far the fabric's neutral surface fails to bisect its loops. The model's own loop is bisected exactly — its crest sits half a diameter behind the mid-surface and its trough half a diameter in front — so it carries no curling moment at all, and the curve here is what any departure from that would buy. The moment is proportional to the eccentricity, so one solve settles the whole curve. A radius of three millimetres, which is about what a fine jersey rolls to, needs 5.6% of a yarn diameter — 9.4 micrometres on a yarn 167 micrometres across. That is why curl is easy to see and hard to model: the whole of it lives inside a rounding error on the geometry. Only the course-wise edges are drawn, because bending about the other axis turns the thread's end tangents out of the fabric and a free thread given turned ends buckles clean out of it — 3.2 times the fabric's own thickness, for 37% off its energy. That configuration is not available to a thread with neighbours, so the second moment is refused rather than computed.

How little asymmetry a curl needs

A model with a thickness can finally be asked why stockinette rolls. It answers that it does not — its curling moment is exactly zero, by a symmetry — and the useful part is what that costs to break: five per cent of a yarn diameter buys the whole of the curl anybody has ever seen.

mechanics · Elastica 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.

FrictionElasticaCloth thicknessSpecificationLoopCapstanTightness factorBending energyCrimpBending rigidityCompression energyCrossover length

All concepts