Textile Structure
how cloth is put together
Concept

Torsional rigidity — where it appears

How hard a thread is to twist about its own axis, which is the shear modulus times the polar second moment of its section. Like a yarn's bending rigidity it is bracketed between the fibres sliding freely and the section behaving as a solid, and the two brackets are the same factor — so the ratio between the two stiffnesses is knowable when neither of them is.

Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.

A slack twisted yarn takes a coil of one size. A 20 tex cotton at 1400 turns a metre. Its own torque is 2.060 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 0.91 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

A crepe is a yarn that will not lie still

A crepe cloth's pebbled surface is the yarn's own torque acting on a cloth that cannot resist it. The instability that makes a slack yarn snarl says how large the pebbles should be, and the answer is a millimetre — which is what a crepe looks like.

pattern · Variation A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

Why a slack yarn snarls

Let go of a twisted thread and it wraps on itself. That is not the yarn being badly behaved: it is a buckling, it has a criterion, and the criterion turns a nuisance into an instrument for measuring the one constant this collection cannot pin down.

setting · Twist How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

A yarn that has been set has no torque

Every torque on the torsion ladder assumes a yarn is elastic in twist for ever. It is not: a steamed yarn's residual torque is gone, its twist is unchanged, and the process that removes one without the other is the trade's whole answer to liveliness.

finishing · Memory How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

How much yarn has to hang

The tension a thread needs to stay straight, converted into the only unit anybody has an intuition for: the length of the yarn's own weight. One bound says two metres and the other says six hundred, and everybody who has handled thread already knows which.

setting · Twist A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

The twist a fabric gives back

A T-shirt that hung straight in the shop has its side seam round the front after three washes. The torque was there all along, the setting had hidden it, and the water gave it back — which makes spirality a finishing failure rather than a knitting one.

finishing · Memory A slack twisted yarn takes a coil of one size. A 20 tex cotton at 800 turns a metre. Its own torque is 1.177 µN·mm at the free bound, and a rod under tension T is stable while its torque stays under 2·√(B·T). Below that tension the straight state stops being a minimum and the thread wraps on itself at a radius of 1.59 mm. That radius is 2B/M, which is 2/(r·ω) once the torque is written out: it depends on the twist and on the ratio of the two stiffnesses, and on neither stiffness itself.

A snarl comes in one size

The radius a twisted thread coils to is twice its bending rigidity over its torque. Write the torque out and the bending rigidity cancels completely, leaving a number that depends on the twist and on the ratio of two stiffnesses — and on nothing else about the yarn at all.

setting · Twist Where a fold's two moments cancel, and where the trade folds. The two moments about a fold's own axis, for 2 singles of 20 tex cotton at 800 turns a metre. The falling curve is what the singles' own residual twist supplies, which the folding takes out of them; the rising one is what bending each single onto its helix costs. They cross at 161 turns a metre, a ratio of 0.201, and the closed form for that crossing is C/(B+C) — the ratio of the two stiffnesses and nothing else. The shaded band is where the trade actually folds, 0.6 to 0.75 of the singles twist. The balance point is nowhere near it, by a factor of three.

The folding rule is not a torque balance

Fold a two-fold yarn at about two thirds of its singles twist. This collection has carried that as a bracket copied from the trade and derived nowhere. It is now derivable, the derivation gives a fifth rather than two thirds, and reaching two thirds would need a fibre stiffer in torsion than in bending.

setting · Ply The fold the trade actually makes. 2 singles of 20 tex cotton at 800 turns a metre, folded at 566 — a ratio of 0.707. That is 1/√2, the ratio at which the fold's surface helix angle equals its singles' — 22.8° — because a fold of 2 singles is √2 times the diameter. The trade's own bracket for 2 folds is 0.6 to 0.75, and it contains this number.

What a balanced yarn is balanced about

A specification that says a yarn is balanced does not say which of two conditions it means, and the two have different numbers, different dependences and different consequences. One of them is what folding achieves and the other is what folding is said to achieve.

setting · Ply How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

What a high-twist yarn costs a cloth

Twist buys strength up to a point and then loses it, and everything else it does is a cost. A crepe twist is chosen knowing that, and the trade's twist limits are a balance among five quantities that this collection can now put beside one another.

setting · Twist A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it.

A thread has a second stiffness

Every mechanical number here came from one material constant: how hard a thread is to bend. A thread also resists being twisted, nothing here has ever used that, and the ratio between the two turns out to be the only stiffness number about a yarn that can be known at all.

mechanics · Torsion Twist is not torsion: a straight rod, twisted. A rod drawn with a cross painted along it, and the cross turned through 1.5 full turns from one end to the other. The rod is straight, so it has no Frenet frame at all — a straight line has no osculating plane to turn. The two discs above are the rod's ends seen down its own axis, which is where the twist is an angle rather than a foreshortened wiggle. The twist is the rotation of the painted cross about the rod's own tangent, and it is 1.5 turns whatever the centre line does. That is the quantity a torsional rigidity resists, and it is why a collection that solved a plane curve had not thereby dealt with torsion.

Twist is not torsion

A curve has a torsion and a material has a twist, they share a word, and only one of them is what a torsional rigidity resists. Getting them the wrong way round would have made this collection conclude that a knitted loop carries no twist, on the strength of a theorem that says nothing of the kind.

mechanics · Torsion A fibre's two stiffnesses, as the ratio that survives the bracket. C over B for every fibre in the table, at 20 tex, which is 2G/E and nothing else. The mark beyond each bar is the range the shear modulus is reported over. Glass is the control and is not a measurement: it is a drawn isotropic solid, so its ratio must be 1/(1+ν) and at ν = 0.2 that is 0.833, which is what the table says. Every fibre with molecules drawn out along its axis sits below the isotropic value, and the ordering is the subject rather than an accident — a fibre is stiff along its axis because its molecules are drawn out along it, and the same orientation that raises E leaves G to whatever holds one chain to the next. Aramid, the most oriented thing here, is twenty times softer in torsion relative to its bending than the glass beside it.

The one fibre whose answer is known

A table of measured constants is worth what its worst row is worth, and nobody can tell which row that is. This one has a member whose answer was known before anybody measured it, and the ordering of the other nine turns out to say something about how fibres are made.

mechanics · Torsion Linked: a knitted interlacing: one loop drawn through the next. Two closed curves and the Gauss linking integral taken over them, which returns -1.0002 at 200 segments a curve. A knitted interlacing: one loop drawn through the next. A linking number is an integer, so a value coming back at a few thousandths of one is the discretisation reporting itself rather than a fabric that is slightly linked. The two arrangements are the two ways of making cloth: a knitted fabric's courses link and a woven cloth's threads do not, at any crimp and for ever.

What a closed thread cannot choose

A thread whose ends are held has a quantity it cannot change without breaking: the total number of times its material winds about its own axis, plus the number of times that axis winds about itself. The two can trade, and everything a twisted yarn does when it is let go is that trade happening.

mechanics · Topology Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one.

Where a torsion model stops

A second stiffness was added because an earlier ladder named its absence as the first thing to disbelieve. It settled four things, refuted one trade explanation, and left the question it was built for exactly where it found it.

mechanics · Torsion What flattening a section does to the ratio of the two stiffnesses. C/B is 2G/E for a circular section, because a circle's polar second moment is exactly twice its flexural one. A flattened section has two different flexural moments — easy about the long axis, hard about the short one — and the polar moment is still their sum, which is the perpendicular axis theorem and holds for any section whatever. So a flattened thread has three constants rather than two, and the multiplier on C/B depends on which way it is being bent. At the 0.78 a knitted fabric's own geometry demands, the easy direction multiplies the ratio by 1.322 and the hard one by 0.804. A knitted loop bends in the easy direction, so the collection's headline ratio is a lower bound for a yarn in cloth.

The section that changes both stiffnesses

A thread's two rigidities are in the ratio 2G/E, and that is a fact about a circular section: a circle's polar second moment is exactly twice its flexural one. A yarn in cloth is not circular, so a yarn in cloth has three constants rather than two — and the ratio a whole ladder rests on is a lower bound.

mechanics · Contact Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one.

A fabric reads its own bracket four ways

A yarn's stiffness is unknown to a factor of three hundred, and no laboratory measurement has closed it. Four unrelated everyday observations — a snarl, a knot, a flattened yarn and a cloth's own thickness — all say the same thing about which end of it a yarn sits at.

mechanics · Yarn stiffness Two stiffnesses, one bracket, and a ratio that does not move. The bending and torsional rigidities of a cotton yarn against its count, each drawn at both ends of its own bracket, on a logarithmic scale. The four curves are two parallel pairs: the torsional bracket is the bending bracket exactly — the fibre count over the square of the packing factor, 327 at 20 tex — so the vertical gap between the two bounds is the same for both deformations. The consequence is the useful part: the ratio C/B is 0.250 at the top of the bracket and at the bottom and everywhere between, because the polar second moment of a circle is exactly twice its flexural one.

Where this collection's thread model now stands

A thread has two stiffnesses and a thickness, and this collection's model has had one stiffness and no thickness. Both were added in one phase, neither reached the question it was built for, and the accounting is worth more than either.

mechanics · Elastica

Named alongside it

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

TwistStiffness ratioBending rigidityTwist factorSpecificationWritheBucklingBalanceElasticaHelix angleMeasurementAnisotropy

All concepts