Textile Structure
how cloth is put together
Cloth doing a job

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.
15 min read 6 figures A mechanism, not a materialExactly this manyMeasured, not claimed

Worth reading first: A knit gives up its fibres more easily · What a loop presses with · A pill is anchored, not made.

A knit gives up its fibres more easily put the case in two halves. A knit presents more yarn per unit area to be rubbed, which is a count and was computed. And it presses its threads together less hard, so a fibre is held less firmly — which is a force, and there was none to be had.

There is one now, and the second half turns out to be the larger of the two. The frictional grip on a buried fibre is thirty times weaker in a jersey than in a poplin of the same yarn.

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.
Fig. 1 The frictional grip on a thread per millimetre of the length that is buried, in a woven poplin and in a jersey of the same yarn. Each contact in the knit is lighter by eight and the contacts are four times further apart, so the grip per millimetre is thirty-one times weaker and the crossover length — at which a thread breaks rather than slides — moves with it: twelve millimetres in the cloth and four hundred and sixty in the knit.

Why thirty and not eight

Because two independent geometric facts point the same way and multiply.

Each contact is lighter. A relaxed jersey’s loops press on each other with about thirty-nine millinewtons; a relaxed poplin’s threads press with three hundred and eleven. That is a factor of eight, and both figures are relaxed ones — the knit’s from a solved shape and the cloth’s from inverting a thickness measurement.

The contacts are further apart. A woven thread meets a crossing at every thread of the other system it passes — 2.2 per millimetre in a poplin. A knitted thread meets an interlacing every half loop length, which is 0.57 per millimetre. That is a factor of 3.9.

Eight times three point nine is thirty-one. Neither factor was a surprise alone; the product is, because it only appears when both are computed in the same units for fabrics in the same state.

What grip has to do with pilling

A pill is a ball of entangled fibre held to the fabric by a few anchoring fibres, and it is a standing population rather than an event: what a fabric shows is a generation rate times a lifetime.

Grip enters both terms. It decides how readily a fibre end can be drawn out of the yarn to join the fuzz, which is generation; and it decides how firmly the anchoring fibres are held once the ball exists, which is lifetime. A fabric whose grip is thirty times weaker generates more and holds longer, and the two effects compound in the same direction.

The crossover length

The quantity that makes it concrete is the one this collection uses on the woven side: the gripped length at which a thread’s own breaking load equals the friction holding it.

Shorter than that, the thread slides out. Longer, it breaks first. For a relaxed poplin it is about twelve millimetres, which is the same range as a cut edge’s fray and as a seam allowance.

For a jersey of the same yarn the model puts it at four hundred and sixty millimetres — longer than any garment. A thread in a knit therefore never breaks by being pulled out; it always slides, at every length, however hard it is held.

Which is what a run is

That is not an aside. A run is a loop sliding out of the loop below it, over and over, and the reason it can propagate at all is that the alternative — the yarn breaking — never becomes cheaper.

A woven cloth cannot run because its crossover length is about a centimetre: pull a thread over any ordinary grip and it breaks before it can travel. A knit runs because its crossover length exceeds the fabric. One number decides which failure mode a structure has, and it is the same number that decides whether it pills.

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.
Fig. 2 What the number is computed against. The bending rigidity per unit width sets the scale of everything holding a fibre in a knitted loop, so the criterion’s number inherits the bracket that rigidity is known to within — which is stated here rather than hidden in a constant.

Both contact forces in one place

The comparison is worth seeing as a chart rather than as two sentences, because the woven figures span a range and the knit sits below all of them.

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.
Fig. 3 The normal force at one crossing of eight relaxed woven cloths, against one interlacing of a relaxed jersey. Every cloth presses harder, by between five and twenty-two times, and the two numbers come from completely different routes — a thickness inversion for the cloths and a solved shape for the knit.

The spread among the woven cloths is itself informative. An open cheesecloth presses hardest of all, because its threads are bent hardest by their own crimp relative to how few of them there are; a fine batiste presses least. So “a woven cloth” is not one number, and the knit is below the whole range rather than below an average.

That matters for the claim being made, because a comparison against a single cloth could be a comparison against an unrepresentative one.

Where the contacts are

The spacing argument is easier to believe from the drawing than from the arithmetic.

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.
Fig. 4 Why a knit gives its fibres up more readily than a weave. Almost none of a loop’s length is inside a wrap, so there is far less of the fibre held and far less friction holding it — and the criterion’s number is smaller in a knit for exactly that reason.

That is the factor of four, visible. What the picture cannot show is the factor of eight, because a force has no extent — so the drawing carries half of the argument and the other half has to be read.

Both halves are geometric rather than material, which is why the conclusion is about structures rather than about fibres and why the same yarn behaves so differently in the two.

The lever a knitter has

Tightness, and it moves both terms again.

A tighter fabric has a shorter loop, so its contacts are closer together, and it presses harder, so each is heavier. Across the range a knitter of a 20 tex cotton can reach, the grip per unit length changes by about a factor of three — which is a large lever on a quantity that decides pilling, shedding and run resistance together.

That is why the trade’s advice on pilling is to knit tighter, and why the advice works. It has been a rule of thumb; it is an arithmetic with two multiplied factors in it.

What this does not settle

The fibre. A pill survives on its strongest anchor, which is an order statistic over a small sample, and that argument is about the fibre’s own strength rather than about the fabric’s grip.

The two are independent and both are needed. Grip decides how much fuzz is generated and how firmly a ball is held; fibre strength decides whether the anchors break before the ball is abraded away. A strong fibre in a loosely-gripping structure is the worst case, which is exactly what a lambswool–nylon knitwear blend is.

Why a woven cloth of the same yarn does not

Putting the two accounts together gives the whole answer to the oldest question in this area, and it has three parts rather than one.

A knit has more yarn per unit area at the surface to be rubbed. Its hair population clears the entangling threshold that a woven cloth of the same yarn does not. And its grip on the fibres is thirty times weaker.

Three factors, all in the same direction, none of them about the fibre. That is why the same yarn woven and knitted gives a shirting that does not pill and a jumper that does.

The three rates, with the middle one now priced

A pill is a standing population and the count on a fabric is a generation rate times a lifetime, with abrasion removing pills at a third rate. Grip enters two of the three and it is worth being explicit about which.

Generation needs a fibre end drawn out of the yarn far enough to reach its neighbours, and the resistance is the grip along the buried length. Sixty times weaker means a given rubbing action liberates far more fibre in a knit.

Lifetime is the anchoring fibres holding, and each anchor is held by the same grip. So a knit’s pills are also easier to pull off, which pushes the population the other way — and that is the term the fibre’s own strength competes with.

Removal by abrasion is about the ball rather than the anchors and is untouched by any of this.

So the net effect on the standing population is a race between a larger generation rate and a shorter lifetime, and which wins depends on the fibre. That is the arithmetic behind the trade’s most confusing observation: a weak fibre knitted loosely fuzzes and never pills, and a strong fibre in the same structure pills badly.

What the picture cannot show

A buried fibre. Every figure here is a yarn or a fabric, and the object being gripped is a single fibre inside a yarn, held by its neighbours as well as by the contacts between yarns.

That second grip — fibre on fibre inside the yarn — is not computed anywhere on this site, and it is not small. What is computed is the contribution from the inter-yarn contacts, which is the part that differs between a knit and a weave; the intra-yarn part is the same in both and cancels from the comparison. So the ratio is sound and the absolute grip is an underestimate in both fabrics.

The bracket, and why the ratio survives it

Both grips are friction times a contact force, and the knitted contact force is a bending rigidity over a length squared with a bracket a hundred and thirty wide on it. The woven one comes from a thickness measurement and has its own uncertainty.

So neither absolute figure is worth much. The ratio is, provided the two are not systematically biased in opposite directions — and the knitted figure is an upper bound, because a set yarn presses less than an unset one, so the true ratio is larger than thirty rather than smaller.

That is the useful direction for the argument being made. A conclusion that survives its own error bars in the direction the error runs is worth more than one that needs them to be small.

What was known before

That knits pill more than wovens, universally, for at least as long as knitwear has been industrial. That the reason involves both the surface yarn available and the pressure between threads, from the fabric-mechanics literature. That tightening a knit helps, from the trade.

What has not been available is the second factor as a number, so the argument has been made qualitatively and the two contributions have never been ranked. They can be now, and the ranking is not what the qualitative account suggests: the pressure term is larger than the exposure term, and it is larger because it arrives twice.

What would test it

Withdrawal, directly. Grip a single yarn at a stated buried length in each fabric and pull it out on a tensile tester, which is a standard test for woven cloth and is awkward but possible in a knit.

The prediction is a factor of thirty in the force per unit length, and it is a strong prediction because both fabrics can be made from one bobbin. If it comes out at eight, the contact force is right and the contact spacing argument is wrong. If it comes out at four, both are.

A prediction about finishing

Anything that changes the friction coefficient changes both fabrics equally, and anything that changes the contact force changes only the knit.

So a softener applied to a jersey and a poplin of the same yarn should worsen both fabrics’ pilling in the same proportion. But a setting treatment — a heat-set, a mercerisation, a resin finish — changes the loop’s natural shape and therefore its contact force, and should affect the knit alone.

That asymmetry is testable and, as far as this collection can tell, has not been looked for. The trade treats anti-pill finishes as chemistry acting on the fibre surface, and the arithmetic here says part of the effect should be mechanical and structural.

The same number, in a raised fabric

The arithmetic transfers directly to a fabric that has had fibre deliberately pulled out of it, which is what raising is.

Relaxation moves a knit away from its own energy minimum. The bending energy an unset yarn would hold in each of Munden's three relaxation states, at a constant 3.5 mm loop. A fabric taken from dry relaxation to wet relaxation to full relaxation gets smaller in both directions, and the loop therefore holds more bending energy at every further stage — a rise of 12 per cent from the first to the last. The bars run from zero, so twelve per cent is a small difference on them and the rule marks the first state's value to make the ordering readable; the percentages beside each bar are the quantity the claim is about. If the yarn springing were what set a knit's dimensions the ordering would be the other way round, and it is strict in every published set.
Fig. 5 And where in the relaxation sequence the number is quoted. The energy a loop holds rises through the three states, so the force holding a fibre rises with them — a pilling criterion quoted without a state is quoted to within the spread of this figure.

A raised knit — a fleece, a brushed jersey — is a fabric where a large amount of fibre has been drawn out on purpose, and the grip on what remains is the same thirty-times-weaker figure. That is why a fleece sheds for its whole life while a raised woven cloth stops after the first few washes, and it is why the pilling grade of a brushed knit is the hardest specification in ordinary apparel to meet.

What a standard test is measuring

The pilling tests in use are abrasion tests with a grading scale, and the arithmetic here says what they are sensitive to and what they are not.

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.
Fig. 6 The friction holding one loop in the loop below it, against tightness. Every quantity in a pilling argument scales with this curve, because every one of them is a grip — and tightness is the only construction lever that moves it.

A test that rubs a specimen for a fixed number of cycles and counts the pills is measuring a standing population at one point in time, which is a rate times a lifetime rather than either. Two fabrics with the same grade can therefore have entirely different generation rates, and the one that generates faster will look worse after further wear.

Reading a grade as a prediction of service life is the mistake this makes available, and the fix is to grade at two cycle counts rather than one. That is not what the standards ask for, and it is the change this arithmetic argues for most directly.

The number for other constructions

The comparison above is a jersey against a poplin, and both ends of it move with construction rather than being fixed properties of “knitted” and “woven”.

On the woven side the contact force spans a factor of five across this collection’s eight cloths, so a jersey against a cheesecloth is a factor of twenty-two on the force alone and a jersey against a batiste is five. A loosely-set open weave grips its own threads far harder than intuition suggests, because its threads are bent hardest by their own crimp.

On the knitted side the loop length spans a factor of three in grip across the knittable band. So the ratio between a particular knit and a particular cloth runs from about five to about eighty, and quoting thirty as the number would be quoting a midpoint as a law.

What is invariant is the direction and the reason: every knit grips less than every woven cloth of the same yarn, because both of the two factors go the same way in every construction either trade makes.

What it would take to make a knit behave like a cloth

The ratio is thirty and the crossover length is four hundred and sixty millimetres, and it is worth asking what would have to change for a knit to fall on the woven side of the line — because the answer is nearly reachable and it names an existing product.

The crossover length is inversely proportional to the grip, so bringing a jersey’s from 460 millimetres to a hundred — inside a garment, which is the condition for a thread to break rather than run — needs the grip to rise by a factor of 4.6.

Two levers are available and neither is chemistry acting on a fibre.

Tightness is worth about three. Across the band a knitter of a 20 tex cotton can reach, the grip per unit length changes by a factor of three, because the contacts move closer together and press harder at once. That is the largest structural lever there is and it is the one the trade already uses.

Friction is worth about one and a half. A finish that raises the coefficient from the bottom of the cotton-on-cotton range to the top multiplies the grip in proportion, and both terms in the grip carry it linearly.

Three times one and a half is 4.5, which is just short of the 4.6 the arithmetic asks for. So:

A jersey knitted at the tight end of its band and finished to raise its friction is, to within the accuracy of any of this, at the point where its threads break rather than slide.

Three things follow and the second is the interesting one.

Such a fabric should stop running. A run is a thread sliding out of interlock after interlock, and it is available only while sliding is cheaper than breaking. A knit at its crossover has no run mechanism, which is a categorical change rather than an improvement in degree.

And such a fabric exists. A tightly knitted, resin-finished jersey — the fabric of a technical base layer or a firm interlock lining — is exactly that construction, and it is sold on its dimensional stability and its resistance to laddering. The arithmetic says those two properties are one property, and that the second is not a coating preventing a ladder but a grip that has crossed a threshold.

The cost is the whole of what a knit is for. A fabric at that grip has lost the sliding that gives it its extension, its recovery and its drape — because what stops a knit extending is the same friction, and multiplying it by four and a half multiplies the resistance to extension by the same factor. A knit that will not run is a knit that will not give, and the two are the same number.

That is a sharper statement of the trade-off than “a tight knit pills less”. It says there is a threshold rather than a slope, that the threshold is reachable at the extreme of two ordinary levers, and that crossing it converts a knit into something that behaves like a woven cloth in every respect the grip decides.

Where the ladder goes next

To the same arithmetic asked of a surface rather than of a fibre. A raised nap in a knit is held by the same contacts, and what holds it is a question this collection recorded as unanswered for want of exactly this number.

Where this goes next

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. The strong fibre is the one that pills. Standing pills per unit area by fibre, relative to wool, at one and the same fuzz supply — every row is the same cloth raised the same amount, so the only thing varying is how long a pill survives once it exists. A pill is not made, it is kept: rubbing generates it and rubbing breaks the anchor fibres that hold it, and an anchor survives in proportion to how much force it takes to break. So polyester carries 18 times wool's standing population from the same generation rate, and the ordering here is exactly the ordering of tenacity and nothing else. Wool sheds its pills because wool anchors break. No two real fabrics have the same fuzz supply, which is why a wool knit still pills more than a cotton shirting in practice — the comparison drawn here isolates the anchor and says nothing about the generation, and reading it as a ranking of fabrics would be wrong. A pill is anchored, not made Every account of pilling starts with how a ball of fibre forms and stops there, which explains why fabrics pill and not why some of them stay pilled. A pill is not a thing that happens; it is a standing population, and the number on a fabric at any moment is a generation rate times a lifetime. 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.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Contact forceCrossover lengthFrictionLoop lengthPillingPull-outReal contact areaSpecificationTightness factor