Textile Structure
how cloth is put together
Concept

Extension — where it appears

How far a fabric can be pulled without any yarn stretching, supplied by crimp interchange in a weave and by loops reconfiguring in a knit. It is finite and it is set by the crimp available, so a woven cloth's stretch runs out at a definite point and a knit's does not.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

Pull it lengthways and it narrows. The same cloth before and after a small extension along the warp. No thread has stretched: the extension came out of the warp crimp, that crimp went into the weft, and the fabric is narrower for it.

Crimp, and why cloth narrows when it is pulled

A thread in cloth is longer than the cloth it crosses. Pull the fabric one way and that extra length is taken out of one direction and put into the other, so the cloth gets narrower without a single fibre stretching.

setting · Crimp 1×1 rib — alternate wales to the back. A knitted fabric seen in section across the wales. Alternate wales pulled to the back fold the fabric like a concertina, so its relaxed width is a projection; pulling it wide unfolds the section and no yarn changes length while it happens.

Rib and interlock

A rib fabric is a plain knit folded like a concertina, and its enormous widthwise stretch is the fold opening out. Nothing in it is elastic, and the extension available is a cosine.

knits · Knit Four ways a fabric gets longer without stretching. Extension available from each mechanism, computed from the geometry that provides it. None of these numbers involves a yarn changing length; every one of them is a shape changing, and they differ by an order of magnitude.

Why a knit recovers and a woven does not

Four fabrics get longer without a single yarn stretching, and the four mechanisms are worth wildly different amounts. Three of them are recoverable and one is very nearly not, and which is which follows from where the extension came from.

knits · Knit Everywhere a sheeting can go. Every state a sheeting of 28 × 26 threads per centimetre in 25 and 25 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 4.03 per cent of extension is available along the warp, and reaching it costs 6.98 per cent of the width.

A cloth extends by moving its crimp

Everybody says the extension available along the warp is the warp's own crimp. On six of eight ordinary cloths it is not — a close sheeting has 14.61 per cent of warp crimp and reaches 4.03 per cent, because the limit lives in the weft.

mechanics · Tensile Pull it lengthways and it narrows. The same cloth before and after a small extension along the warp. No thread has stretched: the extension came out of the warp crimp, that crimp went into the weft, and the fabric is narrower for it.

What crimp interchange actually conserves

Pull a cloth lengthways and it narrows, because the crimp moves from one system to the other. Inextensibility says that much and no more — it is one equation short of an answer, and the second equation everybody uses is an assumption with a name.

setting · Crimp Everywhere a muslin can go. Every state a muslin of 24 × 22 threads per centimetre in 20 and 20 tex can reach without a yarn changing length, solved from Peirce's plain-weave geometry. The set is a curve and not a region: 6.59 per cent of extension is available along the warp, and reaching it costs 21.82 per cent of the width.

Pulled both ways, only one can give

A cloth at constant thread length has one degree of freedom, so its reachable states are a curve rather than a region. Equal extension in both directions meets that curve at exactly one point — the state the cloth is already in — so the amount available is nought.

mechanics · Tensile single float as an array. One cell per needle per course, over 2 repeats each way. K knits a new loop and casts the old one off, T tucks the yarn into the loop below without casting off, M misses the needle and floats past it. The margins carry the two conditions: a wale with no K in it never casts off, and a course with no K in it is never caught by anything. Both hold here.

The float in a knit

Five rungs of this anchor have taken the float to be a length of thread on the surface with nothing holding it down. A knit has one too, and it behaves the same way in the light and the opposite way in the hand — because a woven float lengthens its thread and a knitted one shortens its fabric.

knits · Float The loop that costs nothing to extend. A plain knitted loop at rest and extended by 35 per cent, with the arcs marked. The arcs' radius is the diameter of the yarn the loop wraps, 0.167 mm, and it is set by contact rather than by the fabric's dimensions — so extending the fabric lengthens the legs and bends nothing further. The bending energy is 0.0176 N·mm at both, and the model therefore asks no force at all for an extension a woven cloth would refuse. What the drawing cannot show is what a real knit's first few per cent do cost, which is friction and yarn flattening and is not a bending property.

A knit is soft because it bends

Ask the same energy question of a woven cloth and a knitted one and the answers are not different by a factor — they are different in kind. A woven cloth's bending energy changes the moment it is extended. A knitted loop's does not change at all, exactly, over the whole of its extension, because its arcs are held to a radius by contact rather than by the fabric's dimensions.

knits · Knit What a shed costs, in newtons. The tension the shed puts into one end at each shaft of a 24-shaft harness, for a 25 tex cotton yarn. The strain is set by the loom's geometry alone; the tension is that strain times the yarn's modulus. The front shaft holds 0.52 N and the back 1.95 N, which is 14 and 52 per cent of the yarn's breaking load. What the chart cannot show is the rest of the warp tension, which the let-off adds on top of all of these and which no geometry decides.

What the shed costs, in newtons

The shed's strain has been computed here and could not be priced: a strain is a length over a length and says nothing about how hard a thread is being pulled. A modulus turns it into a tension — and the back shaft of a twenty-four-shaft harness turns out to hold its ends at half their breaking load, all day, from the geometry alone.

mechanics · Yarn stiffness The load–extension curve, computed from a stiffness. The tension in one end of a sheeting against how far the cloth has been extended, computed as the slope of its bending energy along its own constant-thread-length locus. The curve passes through zero at the state of least energy, which is where an unloaded cloth sits, and rises either side of it. What the curve cannot show is what happens after the crimp runs out: past the end of the locus the load is carried by stretching yarn rather than by straightening it, and that is a modulus three orders of magnitude higher and a different figure.

The locus gets a force

This site has drawn the set of states a cloth can reach without stretching any yarn, and has never been able to say which of them it is in or what it would cost to move. Both questions are one derivative of a bending energy — and the answer explains the flat start every fabric's load–extension curve has, which is not slack yarn but a symmetry.

mechanics · Tensile Where the strain goes, at a 25° twist. Extend a twisted yarn and the fibres in it are not all strained alike. If each stays at its own radius — the affine model — a fibre at helix angle θ takes cos²θ of the yarn's strain, so the fibre on the axis takes all of it and the fibre at the surface takes 82.1% of it. The core reaches its breaking extension first and the yarn cannot realise its own fibres. If instead every fibre migrates between the core and the surface, every fibre has the same mean strain and they all break together. The two are exactly computable — 0.8214 against 0.9509 of what the fibres could give — and the ratio between them, 2/(cos α(1 + cos α)) = 1.158, is what migration is worth. The shading is that arithmetic and the two panels use the same scale.

A straight fibre cannot share the load

Extend a twisted yarn and its fibres are not all strained alike: the one on the axis takes the whole of it and the one at the surface takes cos²α. So they do not break together, and what a wandering fibre is worth comes out as one expression with nothing fitted in it.

mechanics · Migration 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 The extension ceiling, with the yarn given a thickness. As a jersey is pulled along its courses the wale spacing grows and the course spacing has to fall, because the yarn between two interlacings is a fixed length. The upper curve is how small the course spacing may be before the yarn runs out; the lower is how small it may be before two adjacent courses occupy the same space. The geometric ceiling is 322% and the contact one 299% — 7% lower. That is the result and it is a negative one: a measured jersey extends by about a hundred per cent, so contact between courses is not what puts the computed ceiling three times beyond a real one. The candidate this ladder was written to test is ruled out.

Contact is not why a jersey stops

The model says a jersey can be pulled to three hundred and twenty per cent along its courses. Real ones stop at about a hundred. The recorded diagnosis was that nothing stops adjacent courses passing through one another — and giving the yarn a thickness closes seven per cent of a gap of two thirds.

knits · Elastica

Named alongside it

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

CrimpCrimp interchangeJammingInextensiblePeirce's geometryBending rigidityLoop lengthConstant length locusElasticaLoopSettSpecification

All concepts