Does a loose weave tear better

Almost every textile reference says the same thing: for a given yarn and a given number of threads, a twill or a satin tears stronger than a plain weave, and the reason is that fewer interlacings leave the yarns free to group at the tip of the tear and share the load between them.

The mechanism is real, the measurements are real, and the explanation is very widely repeated. This essay tries to compute it and cannot, and the failure is worth writing down, because the place where the arithmetic runs out is not where the explanation puts it.

How a woven cloth tears #

A tear is not a fabric splitting. It is a sequence of single-thread failures, one at a time, at a point that travels.

Pull the two sides of a cut apart and the load is carried by the threads crossing the line of the cut. Those threads are not straight: they run from one side of the cut to the other around the tip, and as the load rises they slide along the threads they cross, sweeping backwards toward the tip and bunching there. What forms is a roughly triangular region of gathered threads with the tip at its point — the del zone, named for its shape.

The threads in the zone are in tension together. Whichever breaks first, breaks; the zone re-forms one thread further on; the tear advances. So the force required is not the strength of one thread and not the strength of all of them. It is the strength of however many are in the zone at once, which is why the number in the zone is the quantity worth computing.

That much is agreed on by everybody and it is not in dispute here.

What decides how many gather #

A thread can only slide until it meets its neighbour. The spacing between two threads is one over the sett; the thread itself occupies one diameter of that; so the room available in each gap is

$$s = \frac{1}{\text{sett}} - d,$$

and closing $k$ gaps supplies $k s$ of movement. The opening at the tear tip has to be accommodated by that movement, so the number of threads that can gather is the opening divided by the slack.

Both quantities in that expression are properties of the yarn and of how densely it is set. Neither of them is a property of the weave. Two cloths at the same sett in the same yarn have the same slack in every gap, whatever their drafts, and the figures compute it twice and compare the results rather than saying so.

Where the geometry does speak, it says the opposite #

There is one comparison in which the weave does enter, and it goes the wrong way for the trade’s claim.

Comparing two weaves “at the same construction” is ambiguous. It can mean at the same sett, in which case the slack is identical and the geometry predicts no difference at all. Or it can mean each at the setting it is actually woven at, and cloths are not woven at arbitrary setts: a weave is set as densely as its own interlacings allow, because that is what makes it firm.

Set each weave at its own jam and the satin is the tighter cloth. In a yarn a fortieth of an inch across, a plain weave jams at twenty threads to the inch and has 0.025 inches of slack in each gap; an eight-end satin jams at thirty-two and has 0.006. Four times less room to move, so on this reading the geometry predicts that the satin gathers fewer threads and tears more easily.

So the geometric account gives two answers depending on which comparison is meant, and neither of them is the one the trade reports. It is silent when the setts match and wrong when they do not.

What is missing, and it is one word #

The step the calculation skips is the one that lets a thread slide at all.

Sliding is resisted. A warp end passing over and under the picks is gripped at every interlacing, and the grip has two parts: the normal force from the crimp — the thread is bent around its neighbours and pressing on them — and the coefficient of friction between the two surfaces. To move, a thread must overcome that at every crossing it passes.

The number of crossings is the interlacing count, and the interlacing count is exactly what the weave decides. A plain weave grips a thread at every pick; an eight-end satin grips it at one pick in eight. So the resistance to sliding falls with the interlacing count, and the group that can actually form is not the group that geometrically could form — it is however many threads the load can drag into place before something breaks.

That is the mechanism, and it is a frictional one. It has the weave in it in exactly the way the trade says. What this site cannot do is put a number to it: there is no friction in any model here, and there is no crimp force either. The trellis is a pin-jointed net with free hinges, the matrix knows nothing about yarn, and neither of them can be asked how hard a thread is to move.

One geometric effect the slack calculation does miss #

Before the negative result is accepted too comfortably, there is a second geometric quantity the weave does supply, and it belongs in the account.

The slack expression asks how far a thread can move before it touches its neighbour. It assumes the thread moves as a rigid rod, sliding sideways along its whole length. That is not what happens near a tear tip. A thread is held wherever it is interlaced and free between those points, so what it actually does is bow: the free length between two grips deflects sideways, and the length available to bow is the float.

An end in a plain weave is gripped at every pick, so no part of it is free for more than one pick’s span. An end in an eight-end satin is gripped once in eight, so a span seven times as long can deflect. For a given sideways force a beam’s deflection grows very fast with its free length, so the satin’s ends move into the zone far more readily than the plain weave’s — at the same sett, with the same slack in every gap.

That is a weave effect, it is geometric, and it points the way the trade says. It is not in the figures here for the honest reason: computing a deflection needs a bending stiffness, and bending stiffness is a mechanical property of the yarn that this site measures elsewhere and does not connect to the matrix. What can be said without a stiffness is the ratio of free lengths, which is the float, and the float is the number this whole ladder is about.

So the corrected statement is that there are two weave effects, both real, and the slack is not one of them: the grip count, which is frictional, and the free span between grips, which is geometric but needs a stiffness before it yields a number.

Why the two weave effects point the same way and cancel differently #

The essay ends with two weave effects, the grip count and the free span, and it is worth putting them side by side against the sett comparison, because the three together explain why the trade’s claim survives measurement while its stated reason does not.

The grip count and the free span are the same number read twice. A weave that interlaces rarely has few grips per millimetre and long spans between them, because the two are reciprocal: the interlacings are what divide the thread into spans. So they are not two independent mechanisms whose contributions have to be weighed against each other; they are one property of the matrix acting through two routes, and both routes favour the loose weave.

And the sett effect runs the other way, but only under one reading. A satin set at its own jam is the tighter cloth and has less room to gather, which is the geometric point the essay makes. It is also, at that sett, still gripping its threads at a fifth of the crossings a plain weave does — so a satin at its jam has less room and less resistance, and the two do not obviously cancel.

That is what makes the trade’s observation hard to compute rather than merely uncomputed. Two of the three effects favour the loose weave and one opposes it, and only the opposing one has a number. An account that has the geometric term and not the frictional term is an account of the one effect that points the wrong way, which is why running the arithmetic on its own produces the opposite of what is measured.

The useful form of that is a prediction rather than a lament. The room-based term is exactly computable and the two weave-based terms are not, so any experiment that holds the sett fixed removes the term that can be computed and leaves the two that cannot. Comparing two weaves at one sett is therefore the cleanest possible test of the frictional account — the geometry contributes nothing, and whatever difference appears is the grip and the span. That is a measurement anybody with a tear tester can make and it is not the comparison the trade’s tables report, which are at each cloth’s own construction.

What was counted, and what was not #

The figures on this page compute the slack and the group size and assert three things while they draw.

That the cloth is set openly enough to be woven at all — a cover of one or more is refused, because a cloth whose threads do not fit is not a cloth and a tear calculation on it is arithmetic about nothing.

That the slack is the same in two named weaves at one sett. This is the finding, and computing it twice and comparing is the only way it stays a measurement rather than an assertion in prose.

And that crowding the cloth reduces the slack, which is the direction the whole argument depends on.

What is not computed is the force. No figure here reports a tear strength in newtons, because nothing here could produce one. The unit on every number is inches or threads.

The claim in its defensible form #

Stripped of what cannot be supported, what remains is still worth having and is sharper than the version usually given.

The room a thread has to move is decided by sett and diameter alone. Anybody comparing two weaves at one sett and attributing a difference in tear to “more room to group” is attributing it to a quantity that did not change.

The resistance to moving is decided by the interlacing count. That is where the weave enters, it is the number the matrix supplies, and it is frictional rather than geometric.

So the two halves of the usual explanation belong to different quantities. “Fewer interlacings let the yarns group” is true, and it is true because fewer interlacings means less grip and a longer free span — not because it means more room. The room was the same, and it is the room that the phrase most naturally suggests.

The distinction is not pedantic. It predicts that a finish which lowers yarn-to-yarn friction should raise tear strength without changing the weave, and that a resin finish which welds the crossings should destroy it — and both of those are well known to be the case, dramatically so in the second instance. A room-based explanation predicts neither, since a finish changes no spacing.

The rest of the tear picture, briefly #

Three other things are known to matter and none of them is in the model either. They are listed because an essay that has just spent several sections on what its own machinery cannot reach should be honest about the size of the gap.

Yarn strength and extensibility. A group of threads shares load only if they can all reach their breaking extension together. A very extensible yarn allows more of the group to be loaded before the first one fails, which raises the tear far more than any grouping argument does. This is why nylon tears so well.

The direction of the tear. Tearing warpwise breaks picks and tearing weftwise breaks ends, and the two systems are usually different yarns at different setts, so a fabric has two tear strengths and they are frequently not close. The same asymmetry runs through balance and through the twill angle, and it is the standing reason a single number about a cloth should be treated with suspicion.

The test itself. The tongue tear, the trapezoid tear and the wing tear give different numbers for the same cloth, because they impose different openings at the tip, and the opening is one of the two terms in the group-size expression. A tear strength without a method named beside it is not a measurement.

Who worked it out #

The del zone and the load-sharing account of tearing came out of mid-century textile mechanics, and the standard reference is a 1959 paper in the Journal of the Textile Institute by H. M. Taylor, which set out the geometry of the zone and derived tear strength from the number of threads in it and the strength of each. Earlier work at the Textile Research Institute and the Fabric Research Laboratories in the 1940s had established the phenomenon and the test methods; Taylor’s contribution was to make it a calculation.

What that literature also established, and stated more carefully than the summaries do, is that the parameters governing the zone are the yarn’s breaking load and extension, the friction between the yarns, and the ease with which they can be displaced. Friction is in the list. It is usually the first thing dropped when the account is compressed into a sentence, because it is the term with no closed form, and what is left — “fewer interlacings, more grouping” — reads as though it were geometric.

The reason this essay ends where it does is the same reason the friction term is hard: yarn-on-yarn friction depends on the fibre, the twist, the finish, the pressure and the history, and none of those is a property of a draft. This site is built on the claim that a great deal about cloth is decidable from a small integer matrix. Tear is where that claim runs out, and saying so is part of the method rather than an admission against it.

Where the ladder goes next #

The rung after this is the float limit, which is a constraint that can be computed exactly and is therefore the opposite kind of essay to this one.

Sideways, firmness is the same interlacing count read as a static property, and the maximum sett is the number that decided which comparison was even available here. Crimp supplies the normal force at every grip, which is the missing half of the friction term, and the two section models disagree about the thread thickness that sets it.

What the pictures here cannot show. Every figure on this page draws slack and none of them draws force. The gathered group is shown at the size the geometry permits, not at the size a real tear produces, and the difference between those two is the frictional term the essay is about. A picture cannot show a quantity its generator does not compute, and drawing one that looked as though it had would be the exact failure this site exists to avoid.