Textile Structure
how cloth is put together
Concept

Elastica — where it appears

The curve of least bending energy joining two fixed points at a fixed length, which is the shape a free thread takes between two crossings. It is the only thread path with a derivative, so it is the only one from which a contact force can be recovered — a path assembled from arcs and straights has an energy and no forces.

Named by 39 essays across 7 fields — each of them below, with the objects they name alongside it.

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 A knitted loop, solved rather than drawn. 3 courses by 3 wales of a 20 tex cotton jersey at a 3.5 mm loop, tightness factor 12.8, with one stitch picked out. The centre line is the curve that minimises the yarn's own bending between one interlacing and the next, and the yarn is drawn at its own width of 167 µm so that the crowding is the fabric's rather than the drawing's. It is rounder than the horseshoe a knitting diagram draws, and deliberately so: a diagram draws the topology and an elastica draws the mechanics, and a rod with a fixed length between two fixed points does not hug a rectangle. Half the yarn between two interlacings is spare — the straight line between them is 51% of the yarn available — which is what lets a loop be solved as a free elastica at all. The tightest bend anywhere on it is 1.00 times one over the yarn diameter, the curvature of a yarn wrapped hard round another of the same size. Nothing arranged that: the only things imposed are the loop length and the two spacings.

A loop is nine tenths free run

Half the yarn in a knitted stitch is slack — the straight line between two interlacings is a little over half the thread available to span it. That is two orders of magnitude more room than a woven thread has, and it is why a knitted loop is a shape that can be solved rather than a shape that has to be constructed.

knits · Elastica 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 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.

What a knit gives when it is pulled

How far a knit stretches by rearranging its loops is usually given as a bound rather than a number, because saying more needs a loop with bending stiffness in it. Solved from the loop's own bending, the answer is a curve: soft for a hundred per cent, then stiffening by a factor of eighty as the yarn between two interlacings runs out of ways to be anywhere but straight.

knits · Knit A jersey gets taller before it gets shorter. How much a 20 tex cotton jersey shortens along its wales as it is pulled along its courses, with the course spacing at every extension chosen to minimise the loop's energy rather than assumed. Over the first 81% it is negative — the fabric gets 2.0% taller as it is pulled wider — and only then does it start to contract, reaching 88% at the geometric limit. A material with a negative Poisson ratio is a curiosity; a knit has one over part of its range for a reason with no material in it at all, which is that widening a wale at a fixed loop length first lets the loop's tightest bends open and only later starts taking height away from it.

A jersey gets taller before it gets shorter

Pull a knit along its courses and the first thing it does is grow along its wales — by two per cent, over the first eighty per cent of extension, before it turns round and contracts. The transverse response changes sign, and there is no material in the explanation at all.

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.

How far a knit could go if its yarn were the limit

The yarn in a stitch allows three hundred and twenty per cent course-wise extension before the straight line between two interlacings reaches the thread spanning it. A jersey jams at about a hundred. The factor of three is the finding: what stops a knit stretching is not the loop running out of yarn.

knits · Knit Why a cuff is ribbed. The force a knit pulls back with, over the range a cuff is used across. It rises the whole way — 0.79 N per metre at 23% to 3.38 at 104% — and it is small throughout, which is the combination a cuff needs and almost nothing else supplies. A rib gets its extension by geometry, folding alternate wales to opposite faces so that its relaxed width is about half its opened one, and it gets its recovery from the loop reconfiguring. Neither is the yarn stretching, which is why a cuff made of a fibre with no elastic recovery at all still works.

A rib pulls back on a force the loop supplies

A cuff has to give a great deal at almost no load and come back reliably, and no ordinary material does both. A rib gets its extension from folding, which is geometry, and its return from the loop reconfiguring, which is now a computable force — under four newtons a metre over the whole range a cuff works across.

knits · Two-bed 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 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 A loop is bent about as hard as its yarn allows. The tightest curvature anywhere on a relaxed loop, against the knitter's own tightness factor, in units of one over the yarn diameter — which is the curvature of a yarn wrapped hard round another of the same size, and the tightest bend any fabric asks for. Across the whole range a knitter can reach it stays between 0.73 and 1.27, crossing one at a tightness factor of about thirteen — which is where the trade's own usable band begins. Nothing arranged that. The only things imposed are the loop length, the yarn diameter and the two measured spacings, and the curvature is whatever the minimisation returns.

Two knits with one tightness factor are one knit

The loop model has exactly one dimensionless group in it — the yarn's diameter over the loop length — so two fabrics that share it have the same loop, to fifteen figures, whatever they are made of. That group is the knitter's own tightness factor, and it explains why an index quoted as empirical works as well as it does.

finishing · Knit geometry 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 Two courses of a knitted tube, and the number between them. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, wrapped onto a tube 12 wales round, drawn with the course below it. Each is a closed curve, which is what a linking number needs. The Gauss integral over the pair returns 0.0000 — zero, to four places. In a knitted fabric the answer is 12: every needle loop of one course is drawn through the loop below it. The model puts the interlacing at a point where two centre lines pass one yarn diameter apart, and two curves that pass beside one another are not linked however close they come.

A jersey's course has no writhe

The mechanism everybody quotes for why a hard-twisted jersey leans is that the fabric relieves the yarn's twist by writhing. This collection's own solved course has a writhe of minus six parts in a million, and it is not small — it is exactly zero, by a symmetry, and the symmetry is a statement about what the model left out.

knits · Topology 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 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 seam must give what the knit gives

A knitted seam fails because it is too short, not because it is too weak: the thread in it is nearly two thousand times stronger than the load it carries. What decides whether it survives is one line of geometry — the extension a seam can reach is twice the fabric's thickness times the stitches per unit length.

applied · Seams What links what, in the two ways of making cloth. The linking number between two adjacent courses, for a knitted tube of 12 wales, for the same tube as this collection's model draws it, and for a woven cloth's two thread systems. The fabric's is 12 — one for every needle loop drawn through the loop below. The model's is -0.0000, because it places the interlacing at a point where two centre lines pass a diameter apart and two curves passing beside one another are not linked. The woven cloth's is -0.0000 and always will be, at any crimp and for every weave. That last row is not a defect of any model: a woven cloth really is unlinked, and it is the reason it frays where a knitted fabric runs.

A point cannot link

A knitted fabric of n wales has a linking number of n between every pair of adjacent courses. This collection's model of the same fabric has zero, and it has zero because the interlacing was declared to be a point where two centre lines pass a diameter apart — which is a near miss, and a near miss is not a knot.

knits · Topology 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 1.14, 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.

What a loop model still cannot say

A planar rod with a natural curvature and point contacts gets a knit's forces, its modulus and its extension. It does not get torsion, it does not get the third dimension the interlacing actually needs, and it does not stop adjacent courses passing through one another — which is why its extension ceiling sits three times beyond any jersey.

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 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.

The modulus a knit has instead of one

A fabric's stiffness in extension is usually its yarn's modulus with the geometry taken out. A knit's is not: dimensionally it can only be a bending rigidity over a length cubed, and the same yarn laid straight and parallel is thirty thousand times stiffer than the fabric made of it.

mechanics · Knit Two courses at the yarn's own width, and the place they overlap. The solved course of a 20 tex cotton jersey at a 3.5 mm loop, drawn in plan with the course below it, each strand at the yarn's own diameter of 0.167 mm. Where the two overlap, the fabric is occupying the same space twice. The interlacing is where the model placed them one diameter apart. The closest they come is 0.130 mm — 0.780 of a diameter — and it is not at the interlacing. A round yarn cannot occupy this arrangement; a yarn flattened to 78% of its round diameter can, and flattened is what a yarn in a fabric measurably is.

The fabric that does not fit

Every solve in this collection minimises an energy over a centre line, and a centre line has no thickness. Nobody had checked whether the fabric that comes out of it can be built. It cannot: two adjacent courses of the relaxed jersey approach to four fifths of a yarn diameter, so the yarn passes through itself, at rest, everywhere.

knits · Contact 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.

A knit bends more easily along its courses

A jersey is not a soft fabric to bend. Computed the same way as this collection's woven cloths it lands inside their band, at the limp end — and the useful number is not the magnitude but the direction: two point two to one between its two axes, with the soft one being the axis it rolls about.

mechanics · Knit Bouclé loops on 3 mm binder spacing at overfeeds of 30%, 80%, 150%, 300%, 500%. Loops of an overfed effect thread drawn to one scale, each between two binder points on a straight core, as the curve of least bending energy with its ends along the core. Overfeed 30%: 1.30 times its base in thread, 1.10 mm tall, a bell, against a semicircle of 0.79 mm; Overfeed 80%: 1.80 times its base in thread, 1.95 mm tall, a bell, against a semicircle of 2.10 mm; Overfeed 150%: 2.50 times its base in thread, 2.92 mm tall, overhanging, against a semicircle of 3.94 mm; Overfeed 300%: 4.00 times its base in thread, 4.83 mm tall, overhanging, against a semicircle of 7.88 mm; Overfeed 500%: 6.00 times its base in thread, 7.25 mm tall, overhanging, against a semicircle of 13.14 mm. The dashed arcs are the semicircles, drawn where they fit. What the drawing cannot show is the binder's own path and thickness, which the loop's feet are taken to sit exactly on.

A bouclé loop is an elastica, not a semicircle

A bouclé's loop was taken to be a semicircle, and the semicircle cannot be right: it meets the core at a right angle, and at eighty per cent overfeed it is wider than the gap it stands in. Solved as what it is — a length of thread leaving the core along the core at two binder points — the loop is taller than the semicircle below an overfeed of two thirds and shorter above it, overhangs at 119 per cent and closes its neck at 559. Nothing of the fibre is in its shape, and the bouclé's jamming bracket at eighty per cent is 11.7 wide rather than 12.6.

fancy · Fancy yarn 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 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 loop is a plane curve in another plane

A knitted loop was solved as a flat curve because two curves in one plane cannot pass through one another and a loop must. Letting it out of the plane turns out to change nothing about its shape: the loop is still planar, and its plane is the fabric's turned through twelve degrees.

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 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 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 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 The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

What holds a crest apart

Two half periods meet at every crest of every course and, in this collection's model, run within a fiftieth of a yarn diameter of one another for more than a millimetre. In a fabric what holds them apart is the loop of the next course drawn between them — which is the loop this model does not have.

knits · Contact 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 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 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 The crest, and the loop that ought to be holding it open. One course of the solved fabric in plan, with the two half periods that meet at a crest marked. They approach to 0.003 mm — 0.018 of a yarn diameter — and run within that of one another for more than a millimetre of arc. The ring drawn between them is the needle loop of the next course, which is what holds them apart in a fabric and what this model does not have: the interlacing was declared a point, and a point holds nothing open. The same omission is what makes the course's writhe zero and its linking number zero, so three of this collection's findings are one defect seen three ways.

What a contact model would have to do

This ladder has measured a fabric that does not fit and priced nothing. The repair is a different class of problem from the one this collection solves, it costs fifteen per cent of the yarn in a stitch, and it buys back four results — which is an unusually good return for a piece of modelling.

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.

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.

Contact forceLoopLoop lengthBending rigiditySpecificationBending energyCloth thicknessJammingInterlacingStitch densityTightness factorCurvature

All concepts