A cloth loses its strength before its mass

An abrasion test reports mass loss, or it reports cycles to a hole, or it reports the appearance of a specimen against a set of photographs. Every one of those is a statement about how much material has gone. None of them is a statement about how much strength has gone, and the two are separated by a factor that nobody computes.

The claim #

When one per cent of a cloth’s solid volume has been rubbed away, between three and eight per cent of every thread’s section has gone with it — and which end of that range a cloth sits at is decided by its draft.

Two things follow, and the second is the one that reverses a piece of trade wisdom.

The two curves are not proportional and never become proportional. The mass removed is an integral over depth of the bearing area; the section removed is a property of the single depth reached. One is cumulative and the other is not, so their ratio moves as the wear proceeds.

And the weave with the most contact loses the least strength for a given mass. A plain weave at one per cent mass loss has lost 8.3 per cent of its section; a two-and-two twill 5.4; a three-and-one twill 3.8; an eight-end satin 2.6. The ordering is the ordering of the crown line, and it is the opposite of what a float count suggests.

The argument #

Rub a cloth with a hard flat abradant and it removes everything above a plane. As the plane descends, two quite different quantities change.

The mass removed is the volume between the plane and the surface, which is the integral from the top down of one minus the bearing area. It starts at nothing and grows slowly, because near the top there is almost nothing there — the crowns are thin.

The section removed is a property of the depth alone. Wherever a thread’s crown is at the plane, the cap above it is gone, and the remaining area of that thread’s section is the section less the cap. Every thread in the fabric has its crowns at the same height, so every thread is cut to the same remaining area.

And a thread breaks at its thinnest place. That result was established for the natural variation of a spun yarn, where the thinnest place is wherever the fibres happened to be fewest; here it is imposed, at a place chosen by the abradant and repeated identically along every thread in the fabric.

So the strength of a worn cloth is the strength of a thread reduced by the cap that has been cut off it, and the mass of a worn cloth is reduced by an integral that is much smaller.

What was counted, and how #

For a series of depths from nothing to the thread’s own radius, three numbers are computed and none of them is fitted.

The bearing area, from the closed form, which is where the plane is in contact.

The volume removed, by trapezoidal integration of one minus that area over the depth, converted to a fraction of the cloth’s own solid volume — which is the cover of both systems times the thickness, so the answer is comparable between fabrics.

The remaining section, from the geometry of a circular segment: a cylinder of radius r cut at depth δ loses a cap of area r²·arccos((r − δ)/r) − (r − δ)√(2rδ − δ²), and the remaining fraction is one minus that over the whole section.

The results for one sheeting in four drafts, at one per cent of mass removed: plain 8.3 per cent of section gone, two-and-two twill 5.4, three-and-one twill 3.7, eight-end satin 2.6. At five per cent of mass: 23.5, 22.4, 17.1 and 12.5.

And the depth at which half the section has gone is the same for all four, because that is a property of the thread rather than of the cloth. What differs is the mass that has to be removed to get there: 11.8 per cent for the twill against 21.3 for the satin, a factor of nearly two.

Why the ordering reverses #

The trade’s rule is that a satin wears badly, and the geometric reason usually given is that its long floats have nothing holding them down.

This collection has already separated two quantities there: a satin does not wear quickly, because its flat face spreads the rubbing over more thread than a plain weave’s crowns do, but it fails badly once a float is cut, because a severed float releases a long length of thread. Those move in opposite directions and are usually spoken of as one.

What the bearing curve adds is the size of the first effect, and it is larger than that essay could assume. The satin’s contact area at a given depth is several times the plain weave’s, so a given rubbing pressure is spread over several times the area, and reaching a given depth costs several times the mass. At one per cent of mass loss the satin has lost a third of the section the plain weave has.

The second effect is not computed here and is not affected by any of this. A cut float is still a cut float, and what happens to a fabric after its first thread parts is a question about propagation rather than about surfaces.

The pressure on the crowns, which is why it happens at all #

The concentration of damage has a companion in the concentration of load, and together they explain why a rubbing that looks trivial does so much.

A cloth carrying a nominal pressure carries it on a few per cent of its plan, so the pressure on the crowns is twenty to fifty times the nominal figure. That is where the abradant is doing its work, and the work done per unit area is what removes material.

So the crowns are being pressed hardest, worn first, and cut at the place that decides the thread’s strength — three concentrations acting on the same few per cent of the fabric. The bearing curve is the common cause of all three, which is why one geometric object answers a question about contact, a question about pressure and a question about wear.

And the three do not compound in the same direction. A weave with more contact has lower crown pressure, which slows the removal; it also has more area from which to remove, which speeds it. The net effect on the rate is not settled by anything in this collection, because the rate needs a wear law — Archard’s, or something like it — with a coefficient that has to be measured. What is settled, and needs no wear law at all, is the conversion: whatever mass has gone, the section that has gone with it is fixed by the geometry.

What this says about an abrasion test #

Three things, and all three are about interpretation rather than about method.

A mass-loss figure understates damage by a factor between three and eight, and the factor is not the same for two fabrics being compared. Two cloths that lose mass at the same rate under a Martindale are not losing strength at the same rate unless they have the same crown line.

A cycles-to-hole figure is measuring the second effect and not the first. A hole appears when threads part and start to be pulled out, which is about propagation. It correlates with the section loss rather than with the mass loss, which is why cycles-to-hole and mass-loss rankings of the same set of fabrics famously disagree.

And a residual-strength figure is the honest one, because it is measuring the thing that has actually happened. It is also the least often quoted, because it destroys two specimens instead of one.

The practical form: abrasion resistance measured as mass loss and abrasion resistance measured as residual strength are different quantities that are reported under one name, and the transformation between them is a property of the weave that can be computed before either test is run.

Which system wears, which is decided elsewhere #

There is one more concentration and it is the sharpest of the four.

A cloth’s two systems crown at different heights, and the higher one takes the whole of the rubbing until the abradant has descended past the step. On an ordinary sheeting that step is eight micrometres — which is a fifth of the depth at which a fabric has lost a serious fraction of its section. So for the first fifth of a cloth’s wearing life, one thread system is being worn and the other is untouched.

That is not a small asymmetry. A cloth whose warp crowns higher loses warp section while its weft is intact, and its strip strength in the warp direction falls while its weft strength does not move at all. Anyone testing such a fabric in one direction only would find it either much worse or entirely unaffected, depending on which direction was chosen.

And which system it is comes from the division of the crimp, which is a convention rather than a measurement. This ladder keeps arriving at the same place: the surface is computed exactly from the draft, and the one input the draft does not supply is small, unmeasured, and decides something large.

The first fifth is one system only, and a strip test can miss it entirely #

The step between the two crown levels is eight micrometres on an ordinary sheeting, and that is stated above as a fifth of the depth at which a serious fraction of the section has gone. Following it through says something sharp about what an abrasion test can report, and it is sharper than the essay’s own summary.

Below the step, the loss is not shared. The abradant is riding on one system’s crowns and has not reached the other’s at all, so the whole of the section removed comes out of one set of threads. So while that lasts:

the worn system loses section at twice the rate a shared surface would give, and the other loses none.

That is not a correction to a ratio; it is a different regime. A cloth a fifth of the way into its wearing life has one direction damaged and one direction untouched, and a residual-strength measurement made in the untouched direction reports no damage at all while the fabric has already lost a serious fraction of its strength the other way.

Three consequences, and the third is a test design.

A single-direction residual-strength test is a coin toss early on. Chosen along the worn system it reports double; chosen across it, nothing. Which is which depends on the division of the crimp — the same unmeasured convention this ladder keeps arriving at — so the choice cannot be made from a specification.

And the disagreement is largest exactly where the test is most useful. The early phase is the one a specification cares about, because it decides how long a garment looks and performs new. The late phase, where both systems are being worn and the rates converge, is the one everybody reports.

So the test should be run in both directions and reported as a pair. The difference between the two residual strengths is a direct measurement of the step — which is the quantity this ladder computes and which nothing else measures — and it goes to zero as the wear passes into the shared phase. A pair of residual strengths against cycles is a curve that starts apart and converges, and where it converges is the step in micrometres divided by the wear rate.

That is a genuinely new use for a test everybody already runs, it costs a second specimen, and it returns a surface quantity from a strength measurement. The prediction is specific enough to fail: two directions, diverging at first and converging later, with the crossing at a depth the surface arithmetic names in advance.

Where the model stops #

The abradant is a rigid plane and no real one is. A Martindale rubs a fabric against a woollen abradant cloth, which has its own bearing curve, so the contact is between two surfaces rather than one and the plane assumption overstates how flat the cut is.

The material is removed rather than displaced. Real abrasion of a fibre assembly begins by fraying and lifting fibres before it severs any, so the first phase of a wear test produces fuzz and loses almost no mass and no section at all. Nothing here has a fuzz stage in it.

The section loss is geometric, and a cut fibre is not a cut solid. Removing a cap from a yarn severs the fibres crossing that cap; the remainder can still carry load, and by the site’s own grip arithmetic a severed fibre picks its load back up within a critical length. So the strength lost is less than the section lost, by an amount that depends on the fibre length and the twist and that nothing here computes.

And the hairs go first. Everything above assumes the abradant meets the yarn surface. On a spun-yarn cloth it meets a hair layer, and the first several micrometres of any wear test is the removal of that layer — which loses mass, loses no section at all, and is why a fabric’s mass-loss curve has a fast initial phase that every standard tells the operator to discard.

What a specification could ask for instead #

The result suggests a number that no standard reports and that would take no extra testing to produce.

Every abrasion standard already measures mass loss. Every fabric already has a computable crown line. The ratio between mass loss and section loss follows from the second, so a conversion factor could be printed beside the mass-loss figure, turning a measurement everybody makes into an estimate of the quantity everybody wants.

For the four drafts here the factor at one per cent mass loss is 8.3, 5.4, 3.8 and 2.6. Those are large numbers and they differ by a factor of three between two ordinary shirting weaves, which is more than enough to reverse a ranking.

The obstacle is not the arithmetic, it is the surface. Computing the crown line needs the draft, and a great deal of fabric is bought and specified without the draft being stated at all — a “twill” or a “sateen” on a delivery note is a family rather than a matrix. That is a recurring theme in this collection: the quantity that decides a property is one the trade does not record, and the properties are then measured empirically and found to scatter.

The generalisation #

Where damage is concentrated and loss is spread, a loss measurement is not a damage measurement, and the ratio between them is a geometric property.

The transferable statement is about any system in which a failure is set by a minimum and a measurement is set by a total. A worn gear tooth, a corroded pipe, a scratched fibre — in each case the mass gone is an integral and the strength gone is a property of the deepest point, and the two can be made to differ by any factor by changing the shape of the surface.

The corollary is a design rule: to make something wear well, spread the contact. Not because a spread contact removes less material — it removes more, because it is touching more — but because it removes it from a shallower depth, and the depth is what the strength answers to.

Who found it, and when #

The distinction between mass loss and strength loss in abrasion is old and empirical; textile testing standards have carried both for decades and have noted their disagreement without explaining it. The weakest-link account of a yarn’s strength is Peirce’s, from 1926.

What is new here is the arithmetic joining them: that the ratio is the section loss at a depth divided by the integral of the bearing area to that depth, that both are computable from the draft, and that the ratio is therefore a design quantity rather than an empirical nuisance.

Where the ladder goes next #

To the surface that has no crowns at all: a pile is the only surface with no crowns, which is what a carpet is for and why its wear behaves like nothing else here.

Sideways, the same concentration argument explains where a garment fails: a seam stands proud and wears first, because a seam is a step in the bearing curve and takes the whole of the rubbing on a few per cent of the area.