The crimp is the price of being cloth

A fabric woven from glass or carbon is bought for one number. Not for its handle, its drape or its cover, but for the stiffness it delivers along the direction the load will run — and every other property of it is either a means to that or a nuisance on the way.

That makes it the one place on this site where the crimp is unambiguously a defect. Everywhere else it is the mechanism behind something useful: it is why a woven cloth extends a little in its own directions with no yarn stretching, why it narrows when it is pulled, and why it comes back to a smaller size after a wash. In a laminate it is only a loss.

The account usually given of that loss is that the crimp lowers the fibre volume fraction — that a wavy tow wastes room a straight one would not. It is worth computing, because it is wrong, and the direction it is wrong in points at what the real cost is.

What a fibre volume fraction is made of #

Vf is the fraction of a laminate that is fibre rather than resin, and it is normally measured: burn the resin away, weigh what is left, divide by the density. It can also be computed before the cloth is woven, from four things this site already has.

The tows per centimetre are a decision. The count of each tow, in tex, is on the package. The fraction of a tow that is actually fibre rather than air is its packing factor, which is where the same arithmetic already lives for a spun yarn. And the thickness the cloth ends up at is geometry. Put them together and

Vf = (fibre volume per unit area) ÷ (cloth thickness)

with the numerator a sum over the two systems of n · A · (1 + c): tows per unit width, times the solid area of one, times the crimp, because a crimped tow puts more length of fibre into a given area of cloth than a straight one does.

The crimp is not an input #

The interesting part is that the crimp does not have to be supplied. It follows from the tows.

At a crossing the two tows stack, so one of them sits above the mid-plane by half the other’s thickness and below it by the same amount when the roles reverse. The amplitude of a tow’s path is half the thickness of the tow it crosses — not its own — and the cloth is exactly as thick as the two of them together, because each one’s surface is its own half-thickness beyond its own extreme centreline.

The wavelength is the other half, and the wavelength is the weave. A plain weave sends the tow up and down every two picks. A 2/2 twill takes four. A five-end satin takes five. Same rise, longer cycle, gentler path — so the crimp is decided by the float, which is the quantity this site has been computing since its first essay and which turns out to be the composite designer’s stiffness knob.

For a 400 tex glass tow at five ends per centimetre, delivered eight times as wide as it is thick, the arithmetic gives a tow 1.39 mm wide and 0.174 mm thick, a cloth 0.347 mm thick, 402 grams per square metre, and a warp crimp of 0.46 per cent in a plain weave. Every one of those is a consequence of the four inputs rather than a measurement.

Where the usual account has the sign wrong #

Now the claim to test. If the crimp cost fibre content, a woven fabric would hold less than the same tows laid flat.

The comparison is not hypothetical. A non-crimp fabric is two unidirectional plies at right angles held together by a light stitch, and it is sold precisely because a woven laminate is less stiff than it might be. Its thickness is the two tow thicknesses, exactly as the woven fabric’s is, and its crimp is zero.

At the same thickness, therefore, the woven fabric holds more fibre — by 0.46 per cent, which is the crimp, because a wavy tow is a longer tow. The number is small, and its sign is the point: there is nothing to find here in the direction the account points.

What does move the fibre content, and by a great deal, is how flat the tow is delivered. That is the second figure above, and it is a factor of a third. It has nothing to do with the interlacing.

What the crimp actually costs #

The cost is in the direction nobody looks for it, because it is not a matter of volume at all.

A tow in a woven cloth does not run straight, so along its own length it is locally off-axis — by nothing at the crowns and by its steepest angle on the flanks. An off-axis ply is much less stiff than an on-axis one, and the average has to be taken in series: each short segment of the tow carries the same load, so the compliances add and the stiffnesses do not.

That is Ishikawa and Chou’s crimp model, from 1982, and naming it matters because it is a model. It treats the undulating tow as a stack of infinitesimal off-axis plies, ignores the resin between them, ignores the transverse tow entirely, and is one-dimensional. It gets the size and the sign of the effect and it is not a laminate analysis.

Half a per cent of crimp costs six per cent of the stiffness. That is an exchange rate of twelve to one, and it is the whole reason a woven reinforcement is not simply the obvious choice.

The weave chooses the stiffness and not the fibre content #

Put the two calculations side by side over four weaves at one sett, in one yarn, and the pair of answers is the finding.

Across a plain weave, a 2/2 twill, a five-end satin and an eight-end satin, the fibre content varies by 0.43 per cent and the delivered stiffness by 6.3 per cent — a factor of fifteen between the two spreads. A designer choosing a weave is choosing stiffness, and is not choosing fibre content in any meaningful sense.

And the mechanism is one this site has drawn many times for other reasons. A float is a run where a thread stays on one face, so a long float is a long wavelength, a gentle path and a small angle. Why satin shines and why a satin reinforcement is stiffer are the same geometrical fact seen from two directions — an uninterrupted length of thread — which is exactly the kind of reach that made the float worth an anchor of its own.

The dependence on the fibre is worth a sentence, because it is the reason the trade’s habits differ between the two materials. The knock-down is a ratio between the axial modulus and what the off-axis directions supply, so a fibre whose axial stiffness dwarfs its transverse stiffness suffers more from being turned. Carbon is fifteen to one where glass is four to one — and carbon reinforcements are correspondingly more often specified as satins, as non-crimp fabrics, or as unidirectional tape.

The exchange rate, and why anyone weaves a reinforcement at all #

Twelve to one is a bad exchange rate, and it invites the question of why a woven reinforcement exists. The answer is that the arithmetic above prices only one of the two things a weave supplies, and the other is what the crimp buys.

An interlacement holds the fabric together. That is this site’s founding observation and it is not decorative: a stack of unidirectional plies is not a fabric at all until something binds it, and until the resin has cured the binding is the only thing keeping the tows where a designer put them. A non-crimp fabric needs a stitch through it for exactly the reason a draft needs its threads to interlace — and the stitch is itself a third thread system, punched through the plies, which the arithmetic above does not charge for either.

So a fabricator laying a curved mould has three things to weigh and only one of them is in the knock-down. A woven fabric drapes: it shears, it stays where it is put, it can be cut and handled, and it will not fan apart on a compound curve. The stiffness it gives up is six per cent in a plain weave and under one per cent in an eight-end satin, and the handling it gives up if the tows are not bound at all is total.

This is why the useful form of the result is not “the crimp costs stiffness” but the shape of the trade. Every increment of float length buys stiffness back and gives handling away, because the same long float that gentles the tow’s path is the one that lets it slide: an eight-end satin reinforcement is limp, distorts when it is cut, and is specified with edge tapes for that reason. The two ends of the ladder are a plain weave that is easy to handle and costs six per cent, and a satin that costs almost nothing and has to be handled with care. Nothing in between is a compromise between mistakes; it is a position on a computed curve.

And the curve has an end. At zero crimp the fabric is not a fabric, which is the non-crimp product with its stitch — so the limit of the sequence is a construction that has left the field this site is about. The arithmetic points at its own boundary, which is the most that a geometric model of a mechanical trade can be asked to do.

Both knobs are squares, and only one of them is the weave #

The four weaves span six per cent of stiffness and the tow’s aspect ratio moves the fibre content by a third. Those look like two unrelated levers because they were computed in different sections, and putting the scaling of each in one place says which is worth reaching for.

Take the tow’s path as a sinusoid. Its amplitude is half the crossing tow’s thickness and its wavelength is the float cycle times the tow pitch, and for a shallow path the arc excess — which is the crimp — is π² times the square of the amplitude over the wavelength. So

crimp ∝ (tow thickness ÷ float cycle)²,

and the knock-down, over this range, is very nearly proportional to the crimp.

Two consequences, and both are squares.

Lengthening the float cycle divides the crimp by its square. Plain weave takes two picks, a 2/2 twill four, a five-end satin five and an eight-end satin eight, so the crimps stand in the ratio 1 : 1/4 : 1/6.25 : 1/16. Against the plain weave’s 6.0 per cent knock-down that predicts 1.5, 0.96 and 0.38 — which is the six-per-cent spread the census measured, arriving from a formula rather than from four separate computations. A designer can therefore quote the knock-down for a weave nobody has run: take the plain weave’s and divide by the square of the cycle ratio.

Delivering the tow flatter divides the crimp by the square of the thickness. A tow at an aspect ratio of eight is half as thick as the same tow at four, so it has a quarter of the crimp — the plain weave’s 0.46 per cent against 1.84, and a knock-down of six per cent against something near twenty.

That is the finding worth carrying out of this rung, because it changes what a specification should say. The tow’s aspect ratio moves the fibre content and the stiffness at once and in the same direction, and the weave moves only the stiffness. A flatter tow packs more fibre into a given thickness and runs a gentler path through it; a longer float buys the second and, as the census shows, essentially nothing of the first.

So the two levers are not equals. Buy the flattest tow the supplier will spread, and then choose the weave for handling — which is close to what the industry does, and the arithmetic says the order is right rather than merely conventional. A spread-tow fabric at an aspect ratio of twenty or more is the limit of that argument, and its selling point is usually given as thin plies; the crimp square says it is also the cheapest stiffness in the catalogue.

The caution is the same one the shallow-path approximation carries everywhere. Both squares hold while the path is gentle, which is while the amplitude is a few per cent of the wavelength — true for every construction on this page and false for a coarse tow in a plain weave at a close sett, where the crimp is large enough that the arc excess is no longer quadratic and the knock-down has to be integrated rather than scaled.

What was counted, and how #

Three computations, each with a check that is not a formality.

The fibre volume fraction is computed twice — once from the tow areas and the setts, once from the areal weight and the fibre density — and the two are asserted equal to a part in 10⁹. They share only the definition of tex, so a factor of ten in a unit conversion shows up as a disagreement rather than as a plausible number. The same routine asserts that Vf never passes the packing factor of the tow it is made of, and refuses outright a sett that puts more tow into the cell than the cell holds.

The crimp comes from inverting an arc-length integral: the slope amplitude of the sinusoid is bisected until its arc per wavelength is 1 + c, and the result is fed back through the integral to confirm it recovers the crimp it was solved from. That inversion is shared with the knock-down calculation rather than written twice, so a crimp means the same shape in both places.

The knock-down is a quadrature of the off-axis compliance along one wavelength. Three things about it are asserted rather than assumed: that a straight tow loses exactly nothing, that more crimp is never stiffer, and that the answer is the same at wavelengths of one, three and seventeen — the last being the claim that lets a knock-down be quoted without a sett.

Where the model stops, and it stops in four places #

The thickness is an un-nested bound. Real cloth settles: between crossings each system drops into the other’s gaps, so a measured fabric is thinner than the sum of two tow thicknesses and its Vf is correspondingly higher. The arithmetic here takes the bound, and the figures take a stated nesting fraction where a comparison needs one, because a bound that is stated is worth more than a fitted number that is not.

The tow is treated as a rectangle of constant width. Real tows spread at the crossings and neck between them, and the width that matters for the jam is not quite the width that matters for the cover.

The knock-down is one-dimensional and ignores the transverse tow, the resin, and the fact that a laminate has several plies whose crimps do not line up. A real laminate analysis stacks the plies and solves the whole; this stacks one tow against itself.

And nothing here says anything about what the laminate does out of plane. Interlaminar strength, impact tolerance and delamination resistance are the properties a woven reinforcement is actually chosen for over a stack of unidirectional plies, and the reason the exchange rate above is a trade rather than a mistake. They need a third dimension in the model, which is the next rung but one.

Who found it, and when #

Peirce’s plain-weave geometry is from 1937 and it is the reason the crimp here is solved rather than assumed. The crimp model for woven composites is Ishikawa and Chou’s, 1982, and it arrived with the mosaic and bridging models beside it — three idealisations of the same fabric, offered as bounds rather than as a single answer, which is a habit this site recognises.

The non-crimp fabric is the industry’s own answer to the arithmetic above, and its existence is the strongest evidence that the arithmetic is about the right quantity. A product sold on the promise of removing an effect is a product whose maker has measured the effect.

Where the ladder goes next #

A fabric that holds fibre must also let resin in, and the two requirements pull opposite ways: the channels the resin travels down are the gaps between the tows, and closing them is exactly what raising the fibre content means. The next rung computes both bounds and finds the interval between them — which is sometimes empty, and an empty interval is the most useful answer this machinery gives.