A tear stops where the grip is

An earlier essay here reached this question and recorded a negative result, which is the most useful kind of thing to leave for a later essay:

The trade’s account — fewer interlacings let the yarns group — is standard. The slack in each gap is 1/sett − d, which contains no weave at all… What the weave supplies is the grip count and the free span between grips, and both need a friction or a stiffness this site does not have.

The compound-cloths field then brought a friction, for a completely different purpose: the capstan, applied to a pile tuft’s anchorage, at a stated coefficient. With it in hand the tearing question can be asked properly, and the answer has the shape the trade’s account implies and the geometry alone could not supply.

Two ways for a thread at a tear’s tip to fail #

A tear running across a cloth loads the threads at its tip. Each of them can do one of two things.

It can slide: pull out of the weave, against the friction of the crossings holding it, and travel until it joins its neighbours in a bundle. That is what the trade means by the yarns grouping, and a bundle of threads sharing a load is much harder to break than one thread alone — so a cloth whose threads slide tears at a high force and tears untidily, with a ragged edge.

Or it can break: fail in tension where it is, one thread at a time, with the tear advancing thread by thread. That is a clean tear at a low force, because the load is only ever on one or two threads.

Which happens is a comparison between two forces, and the capstan supplies one of them.

Why the sett enters a friction calculation at all #

The capstan knows nothing about spacing: it multiplies tension by e^(μθ) per wrap and does not care how far apart the wraps are. The sett enters through the length over which the tip’s load is shared.

The zone at a tear’s tip in which the threads are engaged is a few millimetres of cloth — a length, not a thread count — and a closer sett fits more crossings into it. So the grip available at the tip rises with the sett twice over: through the crossing count, and through the weave angle, which steepens as the cloth crowds.

Both are computed. The zone’s length is a stated parameter — three millimetres by default — because this site has no model for it, and every number below moves if it is chosen differently. What does not move is the existence of a crossover, and where it sits at a stated μ.

For a plain weave in a 0.15 mm yarn at μ = 0.3, the crossover is at thirty-six threads per centimetre: at thirty-two the grip is 31 and the thread slides; at thirty-six it is 137 and the thread breaks.

What the friction buys, and what it does not #

It is worth being precise about what has been closed and what has not.

Closed: the mode. Whether the threads at the tip slide or break is now a computation, the crossover is a sett, and the crossover moves with the friction in the direction it should — more friction makes a cloth break at a looser sett, which is asserted rather than observed. So the trade’s account has a mechanism, a variable, and a boundary.

Not closed: the force. A tear strength in newtons needs the number of threads engaged at the tip, the load they share, and the statistics of which fails first. The capstan gives a ratio, and a ratio is not a force. Nothing in this essay is a tear strength and nothing should be read as one.

That is the same division the fancy weaves made and the finishing field repeated: compute the missing quantity separately, print it beside the topological answer, and do not blend them. A single number combining a criterion’s verdict with a friction’s ratio would have no model behind it.

A ripstop grid, and a bound with no free parameter #

The applied answer to tearing is not a sett. It is a grid: a heavier end and pick every few millimetres, in an otherwise light cloth — the pattern visible in a parachute, a spinnaker, a lightweight jacket.

Its bound is exact and has nothing in it. A tear cannot pass a reinforcing thread it has not broken, so the longest tear a ripstop can suffer is one grid spacing — whatever the load, the yarn, the friction or the weave. That is the kind of statement this site collects: no fitted constant, no measured coefficient, just the observation that a tear advances thread by thread and one of the threads is heavier.

The cost is exact too. A reinforcing end every s millimetres, in both directions, at Δtex heavier than the ground, adds 2Δtex/s grams per square metre. For a 22 tex ground reinforced with 78 tex ends every 8 mm, that is 14 g/m² on a 132 g/m² ground: a ten per cent surcharge for a tear that cannot exceed eight millimetres.

The cheapest grid is the one at the limit, and that is a closed form #

Both constraints are monotone in the spacing and they run opposite ways, so the optimum is not a compromise: it is a boundary.

The tear limit is an upper bound on the spacing. The weight budget is a lower one. So the lightest grid that meets a stated tear limit has its spacing equal to the limit, and the specification is satisfiable exactly when

2 (ripTex − baseTex) ÷ maxTear ≤ budget

which is an inequality with no search in it. The search is run anyway and asserted to agree, because a closed form that nothing checks is a closed form with a factor of two in it — and here the factor of two is real: it is the two systems, and forgetting it would halve every answer.

The arrest itself is not parameter-free, and the essay says so #

The bound above says a tear cannot pass a reinforcement it has not broken. Whether it breaks it is a different question and this arithmetic does not pretend to answer it.

At a tear’s tip the load is shared by a few threads — how many is a mechanical question — so a reinforcing end must be stronger than the ground by at least that factor to survive what arrives. The default here is three threads sharing, so a reinforcement needs three times the ground’s tex: 66 tex against a 22 tex ground, and the 78 tex in the figures clears it.

That number, tipThreads, is the missing model, and it is stated as an argument with the arrest reported as an inequality at a stated value rather than as a fact. A ripstop with too light a grid is a fabric whose reinforcement tears along with everything else — a real failure, and one this arithmetic can only flag conditionally.

Two answers to a tear, and they are opposites #

The field’s two remedies for tearing pull in opposite directions, and holding them side by side is the clearest thing this rung can offer a designer.

Make the cloth loose. Its threads slide, they group at the tip, a bundle carries the load, and the tear force is high. That is the mechanism the essay that asked it examined and could not put a number on, and it is why a loosely woven cotton is genuinely hard to tear. The cost is everything a loose cloth costs: it is open, it distorts, its seams slip, and it is not a fabric anybody makes a jacket from.

Make the cloth tight and interrupt it. Its threads break rather than sliding, so the tear force is low — and then a grid stops the tear after one spacing whatever the force. That is the ripstop, and it accepts a low tear resistance in exchange for a bounded tear length.

Those are different specifications, not a better and a worse answer. A sail wants the second, because a tear that runs the length of a panel is a lost sail whatever force it took to start. A workwear cotton wants the first, because a garment that does not tear at all is preferable to one that tears eight millimetres. And the arithmetic says which regime a fabric is in — the crossover sett — so a designer can tell whether the loose-cloth mechanism is even available in the fabric they have.

The pairing also explains a piece of practice that looks contradictory. Parachute fabrics are woven tightly and reinforced with a grid, which by the first mechanism is the worst of both worlds; the reason is that a parachute’s requirement is a bounded tear at a low areal weight, and the tight weave is bought for air permeability rather than for tearing. Three specifications, one fabric, and the tear requirement is met by the grid because the sett has been spent elsewhere.

What was counted, and how #

Three computations, three checks.

The mode column asserts exactly one sign change across the sett range — a count of crossings from sliding to breaking, which would catch a grip computed the wrong way round — and asserts that the crossover moves the right way with the friction, checked at 0.6 and 1.4 times the stated μ.

The ripstop’s weight is areal-density arithmetic: threads per millimetre times tex is grams per square metre directly, which is the identity the fibre-volume work established. The tear bound is a statement rather than a computation and the code carries it as the returned maxTearMm.

The band asserts that the closed form and the search agree about satisfiability, and that the lightest satisfying grid sits at the tear limit to within one ladder step. Both would fail if the factor of two for the second system were dropped, which is the mistake this arithmetic is most likely to make.

And a jam is reported rather than thrown: a sett past what the yarn allows appears in the rows as unweavable instead of taking the figure down, because the top of a sett range is a fact about the yarn and belongs in the picture.

The two ripstop conditions, combined #

The essay gives the weight condition as a closed form and the arrest condition as a separate inequality at a stated tip-thread count. Putting them together removes the reinforcement’s count from the specification entirely and leaves an inequality in the two numbers a designer actually chooses.

The arrest condition says the reinforcement must be at least t times the ground’s count, where t is the number of threads sharing the load at the tip. Substituting that minimum into the weight condition, the ground count cancels and the condition becomes

maxTear ≥ 2 (t − 1) × baseTex ÷ budget.

At the essay’s own figures — a 22 tex ground, three threads at the tip, a fifteen gram budget — that is 5.9 millimetres. So an eight-millimetre limit is satisfiable and a five-millimetre one is not, which is exactly what the two figures report, arrived at without a search.

The shortest tear a ripstop can guarantee is fixed by the ground’s count and the weight budget, and the reinforcement’s own count drops out because it is determined by the arrest condition rather than chosen.

That inverts one lever in a way worth noticing. A lighter ground shortens the achievable tear in proportion, because a lighter ground needs a lighter reinforcement to survive the tip. So the way to bound a tear more tightly, at a fixed weight budget, is to make the fabric it is bounding weaker — which is not what anybody’s intuition suggests and is exactly what a parachute fabric is.

The surcharge has no yarn in it #

Running the same substitution through the weight fraction rather than the weight gives a result cleaner still.

The grid’s weight is 2(t − 1)·baseTex per grid spacing, and the ground’s is 2·sett·baseTex per unit length. Dividing, the ground’s count cancels again and the surcharge is

(t − 1) ÷ (the number of ground threads between reinforcements).

At three threads at the tip, a thirty-per-centimetre ground and an eight-millimetre grid — twenty-four ground threads per cell — that is 2/24, or 8.3 per cent. The essay’s ten and a half is the same cloth with a reinforcement heavier than the arrest condition requires.

So a ripstop’s minimum weight penalty is a pure number: two thread-equivalents added for every twenty-four, with no tex, no fibre and no cloth weight in it anywhere.

Which makes the trade a hyperbola #

Multiplying the two expressions together gives the design curve in one line:

maxTear × surcharge = (t − 1) ÷ sett.

For a thirty-per-centimetre ground at three threads sharing, the constant is 0.67 millimetres. So one per cent of added weight buys a sixty-seven millimetre tear bound, ten per cent buys 6.7, and thirty per cent buys 2.2 — a hyperbola, with no optimum on it and no diminishing returns either, since the product is exactly constant.

That is a much more useful statement than a feasibility check. A designer picks a point on the curve rather than solving an inequality, and the constant that positions the curve is the ground’s sett divided by the tip-thread count — which says that a more finely set ground is a cheaper ground to protect, in exact proportion, because a grid spacing of a given length contains more of its threads.

The uncertainty sits where the essay puts it. t is the unmodelled number, it enters everything here linearly, and every figure above should be read as at three threads sharing. What does not depend on it is the shape: the bound and the surcharge trade reciprocally, exactly, at every construction.

What the picture cannot show #

The grip curve rises through the sett and crosses a line, and the crossing is the essay’s result. Two things are not in it.

The first is the tear itself. Nothing in these figures draws a propagating cut: the arithmetic is a comparison of two forces at a tip whose state is assumed rather than solved, and a picture of a tear would be an illustration rather than a computation. The slack figure borrowed from that essay is the nearest available, and its own caption says that the quantity it draws contains no weave at all.

The second is the bundle. The whole of the loose-cloth mechanism is threads grouping at the tip and sharing a load, and how many group is exactly what this site cannot compute. So the sliding side of the crossover is a mode with no magnitude attached, and the figure marks it as sliding rather than showing what sliding achieves.

That is the shape of this rung’s honesty. The mode is computed, the boundary is computed, the ripstop’s bound and cost are exact, and the force is missing — recorded here rather than supplied by a plausible number.

Where the model stops #

The tip zone is a stated length with no model behind it. Everything about where the crossover sits depends on it, and it deserves the same treatment as μ: quoted, never assumed. A real tear’s tip zone depends on the fabric’s shear stiffness and on how far the threads have already bunched, which is a mechanical problem this site does not solve.

Grouping is not modelled, only permitted. The essay says the threads slide and bundle; it does not compute how many end up in the bundle or what the bundle carries. Those are the quantities a tear strength would need.

The reinforcement’s own crimp is ignored. A heavier end in a lighter ground does not lie the way its neighbours do: it crimps less, stands higher, and locally changes the cloth’s thickness and cover. A ripstop is a slightly unbalanced cloth by construction, and none of that is here.

And the grid is not free of the weave. The reinforcing ends have to be threaded, and a grid at a spacing that is not a multiple of the repeat produces a drafting problem and sometimes a visible pattern. The weight arithmetic is exact and the weaving is a separate cost.

Who found it, and when #

Tear strength has been a measured property of fabrics for as long as fabrics have been specified, with the tongue and trapezoid tests standardised in the mid twentieth century, and the observation that loosely woven cloths tear at higher forces is older than that. Ripstop as a construction is a mid-century development and its bound — a tear stops at the next reinforcement — is not a discovery so much as the reason the fabric exists.

What this rung adds is the thing that essay could not have: a mechanism for the trade’s account, with a variable in it and a boundary that can be computed. And what it records honestly is the half that is still missing. This site asked itself the question long before it could answer it, and the answer arrived because a different field went looking for a friction coefficient for a carpet tuft.

That is worth stating as a general observation about this collection. The machinery that closes a recorded shortfall is very often built for something else entirely — the capstan for pile anchorage, the racetrack for a spun yarn, the criterion for a plain weave — and the reason the shortfalls get written down is that nobody can tell in advance which field will supply the tool.

Where the ladder goes next #

This is the last rung of the applied field, and the field’s own thesis is the one the whole of it has been demonstrating: an applied requirement is an inequality, the interval it leaves can be empty, and an empty interval is a result. The nine fields the plan committed to now exist. What comes after is breadth — every field broadened towards its full set of anchors — and the arguments left unfinished are recorded in this site’s own plan rather than in an essay.