A coated cloth fails at its holes

A tarpaulin, an airbag, a truck curtain, an inflatable boat and a breathable jacket are all the same object: a woven fabric with a film on it. The fabric carries the load and the film keeps things in or out, and the two are usually specified separately — the cloth by its weight and sett, the coating by its grams per square metre.

That separation misses the one place they meet. A film spread over a fabric is supported everywhere it lies on a thread and unsupported over every hole, so under pressure the only part of it doing anything is the part spanning a gap. Which makes the coating’s job a property of the hole, and this site has already established the one fact that makes that a sharp statement rather than a vague one.

A woven cloth’s holes are all the same size. Not approximately: exactly, because a repeat is a repeat and every hole in it projects to the same rectangle — though the passage behind that projection is not the same in every weave. So the film over a woven substrate has one span to bridge, and the fabric has a burst pressure rather than a burst-pressure distribution.

The bound, and what cancels out of it #

The mechanics is the thin-membrane one and it is stated rather than derived. A clamped circular film of radius a under pressure bulges into a spherical cap; the membrane tension per unit width is half of p·R for a cap of radius R, and R cannot be smaller than a. So a film of thickness t and strength σ holds at most

p = 2·t·σ / a

It is a bound and not a prediction. Adhesion, the film’s own bending, the shape of the thread it curves over and every viscoelastic property of a real coating are outside it, and each of them makes a real fabric hold less. What the bound is good for is the ratio, and every result below is one.

Two cloths under the same coating differ in burst pressure by exactly the inverse ratio of their openings. The thickness cancels, the strength cancels, the cap geometry cancels, the polymer cancels. So:

• a cloth at 24 threads per centimetre holds 3.47 times what one at 12 holds, under the identical coat;
• across the whole weavable range of a 40 tex polyester, from 8 threads per centimetre to 26, the bound rises by a factor of 7.33;
• and the coating weight is the same at every point on that curve.

A specification quoting grams of coating per square metre has said nothing about which end of that range a fabric is at.

Where the distribution comes in #

A nonwoven has no repeat. A web of fibres laid down at random has a pore-size distribution, and this site’s own nonwoven work treats it exactly that way — a threshold rather than an exact criterion, because there is no periodic object to compute on.

A membrane over a distribution of spans fails at the largest one. So the comparison between a coated woven and a coated nonwoven is not between two mean pore sizes; it is between one exact size and the tail of a distribution.

Take a nonwoven whose mean pore is finer than the woven cloth’s opening — 202 micrometres against 252 — with a three-and-a-half-fold spread, which is ordinary for a needled or spunbonded web. On its mean pore the same PVC film would hold 476 bar against the woven cloth’s 381: the nonwoven wins. On its largest pore it holds 136. The woven cloth is better by a factor of 2.80, and the quantity that decided it is not the pore size but the spread.

That is the practical form of a fact this site has stated from the other direction. A woven filter is specified by one number and a nonwoven by a curve, because one has a pore size and the other has a distribution. Coating turns the same difference into a strength: the exactness of a woven cloth’s geometry is worth something structural, and not only something to a filtration engineer.

Which of the two actually gives way #

The bound above is what the film will hold. The cloth under it has its own strength, and which of the two gives first is the question a burst test is actually answering.

A burst test presses a diaphragm through a clamped circle of fabric, so the cloth takes the shape of a spherical cap and carries a tension of half of p·R per unit width — the same arithmetic as the film over one hole, at a scale ten thousand times larger. The cloth’s tension at break is the threads per unit width times what one thread carries, and for 40 tex polyester at 0.6 N/tex and twenty threads per centimetre that is 48 newtons per millimetre, giving 64 bar on a standard diaphragm.

Compare that with the film’s bound and both curves fall as the cloth opens — the film because its span grows, the cloth because it has fewer threads — but they fall at different rates. So they cross, and which side of the crossing a fabric is on decides what a burst test measures.

Across four ordinary coatings, two never cross and two do:

A tarpaulin bursts as cloth everywhere ordinary tarpaulins are made. A breathable laminate on an open cloth bursts at its holes. Both are ordinary products and nothing in either specification says which it is.

What was counted, and how #

The opening comes from the same computation the filter-cloth work uses: four threads bound a hole of side p − d, the open area is one minus the cover, and the two are computed by different routes and required to agree to twelve decimal places.

The bound is then arithmetic in consistent units — film thickness in millimetres, strength in newtons per square millimetre, half-span in millimetres — and the ratio result is checked rather than asserted: the ratio of two burst pressures is computed and compared against the inverse ratio of the two openings, and they must agree to nine decimal places. If they did not, the bound would depend on something besides the span and the whole essay would be about something else.

The sett sweep stops where the cloth jams, because past that point there is no hole for the film to bridge and the bound has nothing to say. The machinery refuses a jammed cloth rather than dividing by nothing.

Two refusals guard the nonwoven comparison. A distribution with no spread is a woven cloth, and the comparison is empty; a nonwoven that is coarser on average as well would lose for the ordinary reason and the finding would be nothing. Both are fed to the machinery and both are refused.

And the crossings table asserts both halves: some coating must be the weak link somewhere, and some coating must never be, because a table in which every row said the same thing would have no finding in it.

What the substrate is chosen for instead #

Nobody chooses a woven substrate because its pores are uniform, and the reasons they do choose one are worth listing, because the finding above sits underneath all of them without being any of them.

A woven base is chosen for strength in a direction: the threads are straight enough to carry a load along their own axes, which no random web can match at the same weight. It is chosen for dimensional stability, because two thread systems at right angles resist distortion in a way a web does not. And it is chosen for tear behaviour, because the threads can gather — though as the next rung records, a coating removes exactly that.

The uniform pore is a by-product of the same periodicity that gives the first two, and it arrives free. That is a common shape: a property chosen for one reason brings another that nobody specified, and the unspecified one turns out to matter somewhere the first was never aimed at. Here the somewhere is a burst test, and the number it produces is a factor rather than a percentage.

The crossing happens once, and past it the film only gets safer #

The two curves fall at different rates, which is enough to make them cross. It is worth doing the arithmetic that says they cross exactly once, because the consequence is a design rule rather than a caution.

The film’s bound is inversely proportional to the half-span, and the clear span is the thread pitch less a diameter — one over the sett, less d. So the film’s bound goes as the sett divided by (1 − d·sett). The cloth’s own burst goes as the sett, because it is a count of threads across a width times what each carries.

Divide one by the other and the sett cancels: the film’s advantage over the cloth is one over (1 − d·sett), and nothing else. That expression rises monotonically with the sett, from a fixed value at an open sett towards infinity as the cloth approaches its jam, where the span goes to nothing and the film has no hole left to bridge.

Three things follow at once. There is at most one crossing, because a monotone ratio passes any level once. Above it the film is never the weak link again, however far the cloth is set. And a jammed cloth always bursts as cloth, whatever film is on it, which is the limiting case the table’s two never-crossing rows are approaching from a distance.

So the rule for a designer is directional and does not need the crossing to be located. A fabric that bursts at its film can always be cured by setting the cloth closer, and the cure is guaranteed to work eventually rather than merely likely to help. A fabric that bursts as cloth cannot be pushed back the other way by any coating decision, because no coating changes the ratio at all.

What spread the nonwoven would need #

The nonwoven comparison turns on a tail rather than a mean, and the tail’s critical value is one division.

The web loses when its largest pore exceeds the woven cloth’s opening — that is the whole of it, since a membrane fails at the worst span it meets and the woven cloth has only one. So the web wins only while its mean times its spread stays under the woven opening, which for a mean of 202 micrometres against an opening of 252 means a spread below 1.25 to one.

A needled or spunbonded web runs at about three and a half to one. It is not a near miss; the requirement is nearly three times tighter than what the process delivers, and closing that gap is a manufacturing problem of a different order from making the mean pore twenty per cent finer.

That reframes the comparison usefully. Making a nonwoven finer on average is the obvious improvement and it is nearly worthless here, because it moves the mean and the whole distribution together and the tail moves with it. What would help is narrowing the distribution at a constant mean — and a process that laid fibres down at random is the one process guaranteed not to.

That is a general shape rather than a fact about webs: where a series element fails at the worst instance it meets, the mean of a distribution is not the quantity to improve, and improving it can be perfectly ineffective while looking like progress on the specification sheet.

Where the model stops #

The bound is optimistic and is not a prediction. The film is treated as spanning a circular hole where the hole is square, and the circular reading is the generous one; adhesion, penetration into the yarn, the film’s own bending stiffness and its time-dependent behaviour are all absent, and every one of them reduces what a real fabric holds. Measured burst pressures for such fabrics are well below these numbers.

The nonwoven’s spread is an argument, and it is the only quantity here that is. Three and a half to one from mean to largest is ordinary for a needled or spunbonded web; the finding is the shape — that a finer mean can lose to a wider tail — and the crossing point moves with the spread.

The fibre’s tenacity is a published property, quoted rather than derived. So is the diaphragm radius. Both enter the cloth’s own burst and neither is computed here.

And the cloth’s burst is read low on purpose. A real burst is biaxial and both thread systems share the tension; treating it as carried in one direction understates the cloth, which makes the crossings conservative in the direction that matters — a coating that this arithmetic says is the weak link is at least as likely to be one in reality.

The generalisation #

The shape here is a series element whose strength is set by the worst instance it contains, and the whole content is that a periodic structure has only one instance.

That is worth stating carefully because it inverts a usual instinct. Regularity is normally an aesthetic property or a manufacturing convenience. Here it is a strength: a fabric with one hole size gives a membrane one problem, and a fabric with a distribution gives it the tail of a distribution. The mean is not what matters and is what gets specified.

The second half generalises differently and is the more useful diagnostic. Two elements in series both weaken as a parameter moves, at different rates, so they cross — and on either side of the crossing the other element’s properties are irrelevant. Improving the yarn on the film-limited side buys nothing; improving the coating on the cloth-limited side buys nothing. Which side a product is on is not usually written down, and the two sides are not distinguishable from a specification sheet that lists both components’ properties separately.

Who found it, and when #

Coated fabrics are an old industry and the practice is sound. Burst tests are standard, the pore-size difference between wovens and nonwovens is well known to filtration engineers, and coating weight is specified because it is what a coater can control.

The membrane-over-a-hole bound is textbook thin-shell mechanics and is not anybody’s discovery. What this site adds is putting it next to a fact it already had — that a woven cloth’s holes are exactly one size — and noticing that the exactness is the thing that makes the bound sharp. The nonwoven comparison follows immediately and goes the way nobody would guess from the mean pore sizes.

The crossing is a small piece of arithmetic that appears not to be done, and the reason is probably that the two curves belong to two suppliers. A coater knows the film’s properties and a weaver knows the cloth’s, and the crossing is a property of the pair.

Where the ladder goes next #

The next rung stops asking what a coating adds and asks what it takes away, which turns out to be most of this site. A coating bonds the crossings, and the bias is a mechanism because the crossings are free to rotate; a tear runs because threads gather; a cloth takes a hole without minding because the neighbours pick the load up; and wicking happens in channels a film closes. Every one of those changes character, two of them reverse outright, and coated turns out to be a state — the same move a dimension without a state made on a different axis.

Sideways, the hole this whole essay is about is the hole between four threads; the specification interval that same hole produces in a filter is a filter cloth’s two jobs; and the substrate with no repeat in it is a nonwoven, where the exact criterion this site is built on has nothing to work on and is replaced by a threshold.