Textile Structure
how cloth is put together
After the loom

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.
14 min read 6 figures Measured, not claimedA mechanism, not a materialStructure before fibre

Worth reading first: A dimension without a state · A knit's change of state is not its swelling · How thick a knit is.

A dimension quoted without a state is not a measurement. That is one of the firmest rules in this collection and it was arrived at the hard way: a knitted fabric’s wale and course spacings move by ten per cent between the state it comes off the machine in and the state it settles into after washing and tumbling, and two constants taken from different rows of one table put every occupancy on the ladder five per cent out.

A knitted fabric has a third dimension now. It is worth asking the same question of it, and the answer is not the same.

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.
Fig. 1 The fabric end-on, at the fully relaxed state. The tilt is the climb over the drop, and the drop carries a course spacing that changes with the state. The climb does not.

The three states

The states are what a fabric has been through, and this collection uses the standard three.

Dry-relaxed is what comes off the machine and is left alone. It is the largest, because the yarn has not been allowed to move.

Wet-relaxed is the same fabric wetted out and dried without restraint. Smaller in both directions.

Fully relaxed is wetted, tumbled and dried. Smaller again, and the state a garment ends up in after a few washes.

The wale and course spacings for a 20 tex cotton at a 3.5 mm loop, and what the model makes of each:

state wale course tilt energy a stitch force through the thickness
dry-relaxed 0.875 mm 0.700 mm 10.91° 21,761 nJ 6.80 mN
wet-relaxed 0.854 0.660 11.42° 23,300 7.41 mN
fully relaxed 0.814 0.636 11.75° 24,395 7.81 mN

Everything moves. The tilt rises eight per cent, the energy twelve, the through-thickness force fifteen.

Except one thing

The thickness is 0.334 millimetres in all three states, exactly.

Not approximately, not within a per cent: identically, because the thickness is two yarn diameters and neither the diameter nor the interlacing appears in the table above at all. Relaxation moves the spacings and the spacings are not in the thickness.

That is the finding of this rung and it is a stronger statement than it looks. A knitted fabric’s thickness does not need a state.

Why the exception exists

The two in-plane dimensions are free. Nothing fixes a jersey’s wale spacing except where friction and setting have left it, which is why this collection has to take Munden’s measurements as input and why the model can predict every force in the fabric and none of its dimensions.

The thickness is constrained. It is not a spacing the fabric can choose; it is set by the interlacing, which is two centre lines passing at a diameter because two threads of one diameter in contact are a diameter apart. The fabric could only get thicker by having its threads not touch at the crossing, and then it would not be a knitted fabric.

So the two kinds of dimension answer to different things. A free dimension relaxes because relaxation is the fabric moving toward wherever friction lets it stop. A constrained one has nowhere to move to.

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.
Fig. 2 A rib in section, where a state and a thickness are most obviously the same quantity. The bed gap sets the thickness directly and the relaxation state sets how much of the gap the yarn actually occupies — so a fabric’s state is a statement about its thickness whether or not anybody measured one.

What the states do to the forces

The forces move in the direction the earlier rungs established and by more than the dimensions do.

Bending energy rises twelve per cent from dry to fully relaxed, which is the finding that makes this ladder’s whole account of relaxation work: the more completely a knit is relaxed, the further it sits from the minimum of its own bending energy. Relaxation is not the yarn springing to a comfortable shape; it is the yarn being allowed to move to wherever friction and setting leave it, and that place holds more energy rather than less.

The through-thickness force rises with it, from 6.80 to 7.81 millinewtons — fifteen per cent. A fully relaxed jersey pushes its two faces apart harder than a dry-relaxed one, at the same thickness.

That is a small and specific prediction: a fabric gets firmer through its thickness as it relaxes, without getting thicker. It should show up as a change in the compression curve’s initial slope after washing, at an unchanged relaxed thickness.

And to the tilt

The tilt rises from 10.91° to 11.75°, which is the fabric’s course spacing shrinking while the climb stays put.

That has a consequence for the friction balance, which needed the coefficient to exceed a half times the cosine of the tilt. Dry-relaxed the requirement is 0.4910; fully relaxed it is 0.4895. Both are above what any fibre in this collection’s table supplies, so the conclusion is unmoved — friction cannot hold a relaxed knit along its wales in any state.

The direction is the interesting part. Relaxing a fabric makes it very slightly easier for friction to hold. That is one of a small number of things on this ladder that push toward the observed stability rather than away from it, and it is far too small to be the answer.

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.
Fig. 3 Why the tilt matters to the balance. The contact force resolves into a part along the wales, which friction has to hold, and a part through the thickness, which it does not. A steeper tilt puts more of the force where it is not being asked to do anything.

And to everything computed from the spacings

The two free dimensions move by ten per cent between the states, and every quantity computed from them moves with them.

Stitch density rises from 1.63 to 1.93 stitches a square millimetre, so the fabric holds nearly a fifth more yarn per unit area at the end of its relaxation than at the beginning. Areal weight rises in the same proportion — which is the familiar observation that a cloth gains weight by losing size, stated for a knit.

And the fibre fraction rises with it, because the same extra yarn is being packed into an unchanged thickness. A dry-relaxed jersey is 22 per cent fibre and a fully relaxed one 27 — so a fabric gets denser as it relaxes without getting thicker, and its warmth per unit thickness falls even as its weight rises.

That last one is a small and specific prediction and it is the kind that is easy to get backwards from intuition: relaxation makes a fabric heavier, denser and very slightly less insulating per millimetre.

Which dimensions need a state, restated

The rule can now be given in a sharper form for a knitted fabric.

Wale spacing, course spacing, and everything computed from them — areal weight, stitch density, cover, extensibility, every force — need a state, and quoting one without it is quoting a number about an unknown fabric.

Thickness does not, because it is fixed by the yarn’s own diameter and the interlacing.

And the yarn’s diameter itself does, in a way this rung is not equipped to handle: a cotton yarn swells when it is wet, so a fabric measured wet has a thicker yarn in it than the same fabric measured dry. That is a fibre property rather than a structural one and it belongs with a knit’s change of state is not its swelling, which is where the two are separated.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 3 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. Single jersey never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. A two-by-two rib crosses between the beds twice a course, travelling 0.668 mm through the thickness. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 4 Where the yarn sits through the thickness, for three structures. A state moves every one of these traverses up or down, so a thickness quoted without a state is a thickness quoted for one of a family of positions the yarn could be in.

The trap this avoids

The reason to spell this out is that the opposite mistake is very easy to make and this collection has made a version of it before.

Two shape constants read from different rows of a relaxation table — one wet-relaxed, one fully relaxed — sat in this collection’s own arithmetic for several rungs and moved every occupancy by five per cent. The assertion guarding them checked that the occupancy was proportional to the tightness factor, which is true for any pair of constants, so nothing caught it.

The mirror-image mistake is available here: to decide that since two dimensions need a state, the third must too, and to go looking for a wet-relaxed thickness that does not exist. A rule that is right about most of a class is not a rule about all of it, and the way to find out which members it covers is to ask what the rule is a consequence of.

Here it is a consequence of a dimension being free rather than of it being a dimension.

What a measurement would see

None of this survives contact with a gauge unmodified, and it is worth saying how.

A measured knitted thickness is mostly hair layer at low loads, and a fabric’s hair layer changes with its state a great deal: washing and tumbling raise fibre off the yarn, and a fully relaxed fabric is hairier than a dry-relaxed one.

So a measured thickness will move between the states even though the structural one does not, and it will move upwards — the opposite direction from every other dimension. That is a prediction with a distinctive signature: the fabric gets smaller in both plan dimensions and thicker on a gauge, at an unchanged structural thickness.

If a measurement shows that, the account here is doing well. If the measured thickness falls with relaxation, something structural is moving that this model does not have.

What it says about specifying a fabric

A trade specification for a knitted fabric usually gives its areal weight, its stitch density and sometimes its thickness, with the state named for the first two and often not for the third.

That practice turns out to be defensible for the wrong reason. It is defensible because the structural thickness genuinely does not need a state; it is undermined because the measured thickness does, through the hair layer and through the load the gauge applies.

The honest specification is a thickness with its load and its state, and the reason to give both is that the two influences push opposite ways.

Where a course of yarn sits through the thickness. The through-thickness position of one course of yarn, half period by half period, for 3 structures at a bed gap of 3 yarn diameters — 0.501 mm — on a 20 tex yarn whose diameter is 0.167 mm. Single jersey never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. A one-by-one rib crosses between the beds 4 times a course, travelling 0.501 mm through the thickness. A tubular fabric never leaves the bed it started on: it oscillates by 0.167 mm and comes straight back, because a loop's feet were drawn through the head below and are on the far side of it, and that is the whole of its third dimension. The horizontal is a count of half periods rather than a length, because the repeats are not the same width and the comparison is not about their widths.
Fig. 5 The same picture for the three structures whose thicknesses are computed most differently. A jersey’s is two loop planes, a rib’s is a bed gap and a tube’s is undefined — and all three move with the state, which is the general statement this rung is making.

What this does not settle

Wet swelling. A cotton yarn’s diameter rises by several per cent when it is wet, which moves the thickness directly. This rung holds the diameter fixed, which is right for comparing dry fabrics in different states and wrong for comparing a wet fabric with a dry one.

Whether the three states are the right three. They are the standard ones and this collection uses them because the measurements exist for them. A fabric goes on relaxing for as long as it is allowed to, and “fully relaxed” is a procedure rather than an asymptote.

And what setting does. Every force here scales with how completely the yarn has taken the loop as its natural shape, and the states are not that: they are dimensional states, and the set fraction is a separate parameter this collection cannot measure from the dimensions.

The exception has a shape worth generalising

A dimension that does not need a state is a dimension that is constrained by contact rather than left by friction, and once that is said the rule generalises past knitting.

A woven cloth’s thickness is also partly constrained: two thread diameters plus a crimp height, where the diameters are contact and the crimp height is not. So a woven cloth’s thickness needs a state and a knitted one does not, and the difference is that a woven cloth’s crimp relaxes and a knitted fabric’s interlacing has nothing to relax.

That prediction is checkable against this collection’s own woven work, which finds that relaxation is the crimp coming back — a cloth’s thickness rises as it relaxes because its crimp does. A knitted fabric’s does not, because there is no crimp height in it.

So the rule is about which term in a thickness relaxes, and the two fabrics differ in whether they have such a term at all.

Which measurement would show it

The experiment is a swatch, a gauge and a wash, and it is worth setting out because the signature is distinctive.

Measure a jersey dry-relaxed, at a stated load. Wash and tumble it, dry it flat, measure again at the same load. The fabric will have shrunk by about ten per cent in both plan dimensions, gained about a fifth in areal weight, and — the model says — kept exactly the same structural thickness.

What the gauge will read is the structural thickness plus a hair layer, and washing and tumbling raise fibre off a yarn. So the prediction is: smaller in plan, heavier per unit area, and slightly thicker on the gauge, with the extra thickness being hair.

If the measured thickness falls with relaxation, something structural is moving that this account does not have. That is a clean falsification and it costs an afternoon.

What is genuinely new here

One number that does not move, and one that does.

A knitted fabric’s structural thickness is the same in every relaxation state, exactly, because the interlacing that sets it does not relax.

And its through-thickness force rises fifteen per cent from dry to fully relaxed, so a washed fabric is firmer under a finger at an unchanged thickness — which is the kind of small, specific, checkable prediction that a model earns its keep by making.

What the pictures cannot show

The end-on drawing is at one state, and the difference between the states would be invisible in it: an eight per cent change in an eleven-degree angle is a tenth of a degree, which is less than the width of the line.

That is the honest reason there is no three-state version of the picture. The differences here are in the numbers and not in the shapes, and a figure showing three indistinguishable lines would imply the opposite of what the table says.

What this does not reach

A fabric under tension. All three states are unloaded. A fabric on a frame or on a body is at dimensions of its own and none of the constants apply.

A partially relaxed fabric. The three states are procedures rather than points on a continuum, and a fabric halfway through its relaxation is not described by any of them.

And the thing every quantity here scales with. How completely the yarn has taken the loop as its natural shape is a separate parameter from the dimensional state, it multiplies every force on the page, and this collection has established that it cannot be measured from the dimensions because it cancels out of every balance the dimensions can be put into.

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.
Fig. 6 The dimension that has no state. Two centre lines a diameter apart because two threads in contact are a diameter apart, with a radius outside each: relaxation moves the fabric’s plan and has nothing here to move.

The rule in its final form

For a knitted fabric:

A dimension that friction leaves needs a state. The wale spacing, the course spacing, and everything computed from them — stitch density, areal weight, cover, extensibility, every force.

A dimension that contact fixes does not. The thickness, and only the thickness.

And a dimension of the yarn rather than of the fabric needs a state of its own. The yarn’s diameter changes when it is wet, which is a fibre property and belongs with swelling rather than with relaxation.

Three categories where the collection previously had one rule, and the middle one has exactly one member.

Which rungs this depends on

That the thickness is two diameters, from the interlacing, with nothing in it that a relaxation could move.

That the states are what this collection says they are — three dimensional states with measured constants, established at a dimension without a state and used unchanged here.

And that relaxation raises a loop’s bending energy rather than lowering it, which is the direction the measurements run and is the reason the forces on this page rise as the fabric settles.

The number that would be worth measuring

One measurement would test the whole of this rung and it is a thickness at three states rather than one.

The prediction is specific and slightly counter-intuitive: the structural thickness is identical in all three, and the measured thickness rises. Rises, while every other dimension of the fabric falls — because washing and tumbling raise fibre off the yarn and the hair layer is most of a light gauge’s reading.

A fabric that measured thinner after relaxation would say something structural is moving that this account does not carry, and the obvious candidate would be the crossings closing up as the fabric’s plan tightens around them.

That is a swatch, a wash, a tumble and three readings on a gauge, and this collection has not made it.

Where the ladder goes next

The constants that supply the two dimensions that do need a state say nothing at all about the third, which is the constants say nothing about thickness.

And a thickness that is fixed while everything around it relaxes is a thickness that will be measured wrongly, which is what a thickness gauge reads on a knit.

Where this goes next

How much a cloth can give back. Relaxation shrinkage against how much of the warp crimp the loom took out, with the area shrinkage above it and the ceiling drawn across. The ceiling is the crimp itself, read as c/(1+c): it is all the length there is to give, and no tension reaches past it. A dimension without a state Every essay in this field turned on the same omission, and collecting them produces a result none of them had on its own — the quantities a cloth's geometry determines divide cleanly into those a finishing works can move and those it cannot, and the division is not the one the vocabulary suggests. Munden's states against the fibre swelling. What a plain knit does between its relaxation states, beside the swelling of the fibre it is made of. Going from dry-relaxed to wet-relaxed a jersey loses 5.66% along its courses and 2.44% across its wales, and going on to fully relaxed it loses 9.1% and 7.0%. The fibre swells 20%. Two things rule the swelling out as the cause: it is several times the whole change, and the change is markedly anisotropic while a swelling enters both of a loop's dimensions through the same loop length. What the bars cannot show is what does cause it, which is friction — water lets the loops move to where the yarn's own bending had been trying to put them. A knit's change of state is not its swelling A jersey is smaller wet-relaxed than dry-relaxed, by 5.7 per cent along its courses and 2.4 across its wales. Water is obviously involved, so the swelling is the obvious cause. Two things rule it out, and both are properties of the constants rather than measurements of a fabric. 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.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

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.

Cloth thicknessContact forceDimensional stabilityLoopMunden constantsPermanent setRelaxationSpecificationTightness factor