A cloth does not mind a hole

Punch a hole in a plastic sheet and it is much weaker than the material it removed. Punch the same hole in a woven cloth and the cloth barely notices. That difference is not a fact about plastics and cotton; it is a fact about continua and thread systems, and it is exactly computable in both cases.

It is also why a garment can have buttonholes, eyelets, laces and vents in the middle of a loaded panel, which would be unthinkable in a sheet of anything.

The film: a factor that does not know how large the hole is #

A film is a continuum. Load it and the stress runs through the material as a field; put a hole in it and the field has to flow around the hole, which crowds it at the sides. For a circular hole in a wide sheet the crowding factor is three — Kirsch’s elasticity solution, from 1898, quoted here and not derived, because elasticity is not this site’s subject.

The consequence is the striking part. The factor is three whether the hole is a millimetre across or ten, because the solution for a small hole in a large sheet does not contain the hole’s size. So a film with any hole in it at all is down to about a third of its strength, and making the hole smaller does not help.

That is why a nick in the edge of a plastic bag propagates, why a film is specified with a tear-initiation test, and why nobody puts a buttonhole in one.

The cloth: exactly the threads that were removed #

A woven cloth is not a continuum. Its threads are separate bodies, and the only thing coupling a thread to its neighbour is friction where they cross — a coupling this site has computed for seams and which is very weak compared with the thread’s own stiffness along its length.

So each thread carries its own load and hands almost nothing sideways — which is also why a cloth’s own strength is a thread count times a thread’s strength. Remove some threads and the rest are undisturbed:

strength left = 1 − (hole across the load) ÷ (cloth width)

The net section, and nothing else. Three properties of that expression are worth more than the number.

It is linear in the hole, so a small hole costs almost nothing: a 1 mm hole in a 50 mm strip costs two per cent where the film loses sixty-seven.

It is independent of the sett. A hole 5 mm across severs six threads at twelve per centimetre and eighteen at thirty-six, and costs the same ten per cent of the strength either way, because what matters is the fraction of the width removed and not the number of threads in it. That is asserted rather than remarked, and it is the property that makes the expression worth writing down.

And it depends only on the hole’s projected width across the load, so the hole’s shape is irrelevant — which has a consequence a film’s arithmetic cannot produce at all.

A slit along the load costs a cloth nothing #

Cut a 40 mm slit down the warp direction of a 50 mm warp-loaded strip. It severs no threads. The strength is unchanged — exactly unchanged, not approximately — and the assertion in the code says so.

Cut the same slit across the warp and it severs 80 per cent of the threads and takes 80 per cent of the strength.

Two identical cuts, differing only in direction, with a factor of five between their consequences. A film would lose comparably in both cases, because a continuum’s stress concentration cares about the notch’s tip rather than its orientation. This is the sharpest available statement of what it means for a fabric to be a thread system rather than a material, and it is the reason a seam ripper does not destroy a garment, a warp-direction ladder in a knit does not weaken it against a warp load, and a fabric can be slit lengthwise into tapes with no loss at all.

The crossover is exactly two thirds #

The two curves meet where 1 − a/W = 1/K, which for K = 3 is at a hole two thirds of the width. Below that the cloth is stronger; above it the film is.

That is a clean number and it is worth reading carefully, because it does not say a film is better for large holes in any useful sense. At two thirds of the width removed both are at a third of their strength and both are close to useless. What the crossover really says is that the entire practical range belongs to the cloth: for holes of a few per cent of the width — buttonholes, eyelets, needle holes, punctures — the cloth is a factor of twenty or more ahead.

The factor is worth stating as its own claim: at a hole one fiftieth of the width the cloth keeps 98 per cent and the film keeps 33, so the ratio is three to one at the crossover’s own scale and twenty-nine to one at a small hole. A trade that has to put holes in things chooses the thread system.

What holds the edge of the hole, which is a different question #

None of the above says the hole is safe. It says the cloth’s strength is unchanged, and a cloth has a second failure mode a film does not: the threads at the edge of the hole are no longer held.

That is fraying, and it is the reason a buttonhole is stitched round, an eyelet is fitted with a metal ring, and a cut edge is overlocked. A woven cloth’s response to a hole is to keep its strength and lose its integrity at the boundary — which is exactly the opposite of a film’s, which keeps its integrity and loses its strength.

Why the load-sharing assumption is defensible, and where it fails #

The whole result rests on one modelling claim: that threads do not share load sideways. It is worth defending, because it is the kind of assumption that is either almost exactly true or badly wrong.

The coupling between two parallel threads in a woven cloth is friction at the crossings with the transverse system. To transfer load from a broken thread to its neighbour, that friction must act over the crossings between them — and the capstan arithmetic says how much it can carry: a great deal, over enough crossings, but only after a slip that is large compared with the elastic strains involved. So the transfer happens slowly and at large deformation, not at the moment of loading.

Two consequences follow, and they are the model’s boundary. At first loading the assumption is good — each thread carries what it was given, and the net section is the strength. At large extension it fails: the threads bunch, the load does redistribute, and a torn cloth pulls into a group of threads at the tear’s tip. That regime is the subject of the next rung, and the capstan is what governs it there too.

The cloth’s own concentration factor is not one #

The film’s stress concentration factor is three and the cloth’s, on the no-sharing assumption, is one — the load simply reduces and nothing is crowded anywhere. That is the cleanest way to state the whole finding, and it is the limiting case rather than the answer.

The sharing is not zero; it is slow. A severed thread picks its load back up over the crossover length — about twelve millimetres in a relaxed woven cloth — so within that distance either side of the hole the surviving threads are carrying part of what the severed ones lost.

Count them. A hole of width a severs a·n threads at a sett n, and the threads sharing their load are those within a crossover length ℓ on each side, which is 2ℓn. So the overload on a neighbour is

K = 1 + a ÷ 2ℓ.

A woven cloth’s stress concentration factor is one plus the hole’s width over twice the crossover length, and the sett cancels out of it exactly as it cancels out of the net section.

For a five-millimetre hole that is 1.21. For a twenty-millimetre hole, 1.83. Not one, and not three.

Which gives a length scale for the whole distinction #

The expression reaches the film’s factor of three when a = 4ℓ, which is about five centimetres.

So a hole smaller than a centimetre is a thread-system hole and a hole larger than five is a continuum hole, and the crossover length is the scale that separates them. Between the two the cloth is neither: it concentrates stress, less than a film and more than not at all.

That is a much more useful boundary than the strength crossover at two thirds of the width, because it does not depend on the specimen’s width at all. The width decides how much strength a hole costs; the crossover length decides whether the cloth is behaving like a thread system while it costs it.

And it says which of the essay’s two claims is the robust one. The net section is exact regardless — the section through the hole carries what its surviving threads carry, whatever is happening either side. The absence of a concentration is not, and it decays with the hole’s size on the scale of a centimetre.

Which explains the two trades’ different holes #

The scale accounts for the essay’s own pairing of a buttonhole with a bolt hole, and it explains why only one of them needs a calculation.

A buttonhole is fifteen millimetres of slit along the warp — no threads severed at all in the load direction — and even a round eyelet is three or four millimetres across. At a/2ℓ that is a concentration of 1.15 or less, which is inside anything a garment’s safety margin covers. A garment maker is always in the thread-system regime, and the trade’s practice of ignoring the strength consequence entirely is correct.

A bolt hole in a laminate is six to twelve millimetres and the reinforcement is resin-impregnated, which raises the coupling enormously and shortens the crossover length towards nothing. A cured laminate is a continuum, its factor is the film’s, and the joint has to be designed as one — which is exactly why the width-to-diameter ratios the essay quotes are what they are.

So the same fabric is a thread system dry and a continuum cured, and the quantity that moved is the crossover length. That is the sharpest form of what a matrix does to a reinforcement: it does not make the fibres stronger, it makes them share.

The same arithmetic decides a bolted joint in a composite #

The net section is not only a garment result, and its most consequential use is at the other end of this field.

A bolt through a composite laminate removes fibre, and the joint can fail in two ways: through the net section, where the remaining fibre breaks, or in bearing, where the material crushes against the bolt. Which one happens is decided by the ratio of the hole’s diameter to the strip’s width, and the arithmetic on the first is exactly the expression above — one minus the hole over the width.

That is why a laminate joint is designed with a width-to-diameter ratio of five or six and an edge distance of three diameters: the numbers are chosen to keep the joint out of net-section failure and in bearing, because bearing fails gradually and the net section fails suddenly. A woven reinforcement makes it worse than a unidirectional stack in one respect and better in another — the crimp is irrelevant here, but the fabric’s ability to redistribute load around a hole is greater than a unidirectional ply’s, for exactly the coupling reason discussed above.

So one expression covers a buttonhole in a shirt and a bolt through a wing rib. The difference is that the shirt’s hole is two per cent of the width and the wing rib’s is twenty, and the second is close enough to the crossover for a designer to need the calculation.

Why a nick in the selvedge is the exception everybody knows about #

There is one place where a small cut in a woven cloth really does behave like a cut in a film, and the reason is instructive.

Tear a piece of cloth by starting a nick at the edge and pulling: it goes, and it goes easily. That is not a contradiction of anything above — it is a different loading. The net-section arithmetic is about a cloth in uniform tension, where the threads at the hole’s edge are unloaded and the rest carry on. Tearing loads the cloth so that the whole force arrives at the tip of the cut, on one or two threads at a time, and one thread at a time is a very small strength.

So the two statements sit together: a hole costs a cloth only the threads it removes, and a cut propagates catastrophically if the load is applied so as to concentrate on its tip. The first is a strength; the second is a mode. Which is why fabric specifications carry both a tensile strength and a tear strength, and why the numbers are not comparable — and why the next rung is about the second.

What was counted, and how #

The arithmetic is two expressions and the interesting work is in the assertions.

Sett-independence is checked by computing the same hole in a cloth at twelve and at thirty-six threads per centimetre and requiring the answers to be identical — not close, identical — while the count of severed threads differs by a factor of three. That pair is the claim.

Shape-independence is checked by cutting a slit along the load and requiring the strength to be exactly one and the severed count exactly zero.

The film’s independence of hole size is checked across the whole ladder of holes: its curve must not move at all, which would catch a stress-concentration factor accidentally made a function of the hole.

And the crossover is arithmetic — 1 − 1/K — rather than a search, with the search’s only job being to confirm which row first falls below it.

What the picture cannot show #

The hero figure draws two curves and cannot draw the reason they differ.

What separates a cloth from a film is load sharing — whether a thread hands force to its neighbour — and that is invisible in any drawing of either material. Both would look like a rectangle with a hole in it. The curves are the consequence; the mechanism is a modelling assumption, defended above and drawn nowhere.

Nor can the figure show the film. Its curve is a horizontal line at one third, which is the whole of what Kirsch’s solution says at this scale, and a horizontal line is a poor picture of a stress field crowding round a hole. That field belongs to elasticity and this site does not draw it — the line is a quoted number given an axis, and the caption says so.

What the figures do carry, and prose would carry worse, is the pair: the same hole in the same width at two setts, with identical strengths and different severed counts. That is the assertion made visible, and it is the one thing about this arithmetic a reader might otherwise doubt.

Where the model stops #

Kirsch’s factor of three is for a circular hole in an infinite isotropic plate. A real film is finite, often anisotropic and often ductile, so the factor is three only as an idealisation — a slit is worse, a large hole in a narrow strip is different, and a ductile film yields at the hole’s edge and redistributes. The figure is quoted as the textbook idealisation it is.

A cloth’s threads are not perfectly independent. They are held at every crossing, they are usually finished or coated, and a densely set cloth behaves measurably more like a continuum than an open one. The net section is the low-coupling limit.

Nothing here is a tear. A hole under static load and a cut that propagates are different problems: the second involves what happens at a crack tip, where the threads have bunched and the load is anything but uniform. That is the next rung’s subject and it needs the friction this arithmetic deliberately neglects.

And nothing here is the hole’s edge. Fraying, ravelling and the loss of the threads around a cut are real failures that leave the net-section strength untouched and the garment unusable, which is why the trade’s answer to a hole is always a finishing operation rather than a strength calculation.

There is a converse worth recording, because it is the reason this arithmetic matters to a garment maker rather than only to an engineer. A cloth’s indifference to holes is what makes it fastenable at all. Buttonholes, eyelets, laces, hooks, studs, belt loops and vents are all interruptions in a loaded panel, and every one of them is affordable because the loss is the fraction of the width removed. A garment made of film has to be fastened at its edges — a bag with a zip, a sleeve with a seam — and a garment made of cloth can be fastened anywhere. That is a design freedom with a one-line justification, and it belongs beside ravelling as the other half of what a cut edge means.

Who found it, and when #

Net-section thinking is as old as riveted plate design and is the first thing any structural course does with a hole. Kirsch’s solution is 1898. Neither is textile work.

What appears to be missing from the textile literature is the contrast drawn as a pair. Fabric mechanics measures tear strength, tongue tear, trapezoid tear and puncture resistance; continuum mechanics computes stress concentrations; and the statement that a woven cloth has no stress concentration factor at all — it has a thread count sits between the two and belongs to neither. It is not a deep result. It is one line of arithmetic and a modelling assumption, and it explains a whole trade’s practice about buttonholes.

Where the ladder goes next #

A hole is static. A cut propagates, and what stops it is either a reinforcing thread heavy enough to survive the tip or a cloth loose enough for its threads to bunch and share the load. An earlier essay here reached that question and had to leave it: the geometry supplies a grip count and a slack and neither is a force. The next rung closes it with the friction the fancy weaves brought.