Shear locking in a composite preform

A composite part is made by laying woven reinforcement into a mould, wetting it with resin, and curing it. The reinforcement is a cloth — carbon, glass or aramid, woven plain or in a twill — and the mould is whatever shape the part is.

The problem is the one darts exist to solve, with the solution unavailable. A flat sheet cannot be laid over a doubly curved surface without deforming; a dressmaker cuts and seams; a composite engineer cannot, because a cut is a discontinuity in the fibre and a discontinuity in the fibre is where the part fails.

So the cloth has to deform, the only deformation available is shear, and the whole engineering question is whether the shear it needs is less than the shear it has.

The shear is not optional #

The starting point is the result the mechanics ladder is built on: a cloth laid over a curved surface must shear, and how much is decided by the surface rather than by the cloth.

Lay a trellis over a surface by the fishnet construction — two yarn paths chosen first, every other node then forced — and the shear at each cell is an output. A developable surface costs exactly zero: a cylinder, a cone, anything that can be unrolled. A doubly curved surface costs shear whose sign the Gaussian curvature decides: a sphere closes the cells, a saddle opens them.

None of that is negotiable and none of it involves stiffness. It is a statement about what an inextensible net can reach, and it is why the first thing a composite designer does with a difficult shape is ask whether it can be made developable.

The shear that is available #

The other half is the locking angle, and it is a property of the cloth alone.

Shearing a trellis closes the perpendicular distance between parallel threads from the pitch $p$ to $p\sin\theta$, where $\theta$ is the included angle. The threads touch when that reaches the yarn width $d$, so the cloth locks at an included angle of $\arcsin(d/p)$ — a shear of ninety degrees less that.

The number depends entirely on how openly the cloth is woven. A reinforcement at seventy per cent cover locks at about forty-six degrees of shear. One at ninety per cent locks at twenty-six. One woven solid cannot shear at all.

That is why reinforcement fabrics are woven far more openly than clothing fabrics of the same yarn. The openness is not about resin flow, though it helps that; it is about buying shear.

Putting the two together #

Now the engineering question, which is a comparison between the two numbers cell by cell.

Compute the drape over the mould. At every cell, compare the shear the curvature demands with the shear the cloth has. Where the demand exceeds the supply, the trellis has no degree of freedom left — and a cloth with no degree of freedom cannot lie on the surface. It buckles out of plane.

That is a wrinkle, and it is a geometric prediction rather than a mechanical one. The mechanics decide the shape of the wrinkle; the geometry decides whether there is one.

The prediction has a radius attached, and the radius moves with the sett. On a hemisphere of six pitches’ radius, a cloth at forty per cent cover reaches the whole of it with shear to spare. The same cloth at seventy per cent still reaches. At ninety per cent, the outer cells demand more shear than the cloth has, and the preform wrinkles at a definite distance from the pole.

Two consequences fall straight out.

A sharper mould needs a more open cloth. Halving the radius of curvature roughly doubles the shear demanded at a given distance, so the cover has to come down to compensate.

And the starting cross matters. The fishnet construction begins from two yarn paths, and everything else is forced by them. Move the starting cross and the whole shear field moves — which is why laying up a preform is a skilled operation and why the same cloth on the same mould can succeed or wrinkle depending on where it is first pinned.

What the saddle does #

The other sign of curvature is worth a section, because it behaves oppositely and the difference is easy to get backwards.

A sphere closes the cells, so the threads crowd together and the cloth locks. A saddle opens them, so the threads move apart — and threads moving apart do not jam. There is no locking angle in that direction.

What a saddle produces instead is a preform with gaps in it: regions where the fibre volume fraction has fallen because the cloth has been pulled open. That is a different failure and often a worse one, because a resin-rich region is a weak region and it does not announce itself the way a wrinkle does. A wrinkle is visible in the mould; an opened cell is visible only in a section afterwards.

So the two curvatures fail in opposite ways for the same geometric reason, and only one of them has a limit that can be computed from the cloth. The locking angle bounds the positive-curvature case and says nothing about the negative one, and a check written only against locking would pass every saddle.

Why the cloth is not simply cut #

The obvious answer to all of this is a dart, and it is worth stating clearly why composite practice resists it.

A dart in a garment removes material and closes a seam; the resulting cloth is continuous across the seam only because it is sewn. In a composite, load is carried by the fibre, and a fibre that stops carries nothing past where it stopped. A darted preform has a line across which the reinforcement is discontinuous, and that line is where the part will fail.

The alternatives are all attempts to avoid the cut.

More open cloth, buying shear, at the cost of a lower fibre volume fraction and a weaker laminate.

Multiple smaller pieces, overlapped rather than butted, which restores continuity at the cost of thickness and weight.

Braided preforms, where the braid angle itself changes as the braid is pulled over a mandrel of varying diameter, so the reinforcement adapts without shearing a fixed weave.

And weaving the shape, which is the expensive answer: a three-dimensional woven preform made to the part’s geometry, so that no draping is needed at all.

What shearing does to the part #

A wrinkle is the failure everyone looks for. Shear short of a wrinkle is not a failure and it is not free either, and the consequences are worth listing because they are what makes the shear field itself a design output rather than an intermediate.

The fibre direction moves. Shear rotates the two thread systems relative to one another, so a cloth that was at zero and ninety degrees becomes zero and sixty. Since a laminate’s stiffness is highly directional, the finished part’s properties vary from point to point in a pattern the drape decides. A designer who assumed a uniform zero-ninety layup has designed a different part from the one that comes out of the mould.

The fibre volume fraction rises. Closing the cells crowds the tows, so a sheared region has more fibre per unit area than an unsheared one. That is usually welcome for strength and unwelcome for resin flow: the sheared regions are the hardest to wet out, and dry spots in a part cluster where the shear was highest.

The thickness changes. More fibre per unit area in the same nominal ply means a locally thicker laminate, which shows up as a dimensional error on the finished surface and is the reason difficult mouldings need more machining than easy ones.

All three of those are computed from the same shear field the wrinkle prediction uses. That is the useful consequence of doing the geometry properly: one construction, several outputs, and no need to measure the finished part to find out what was going to happen to it.

The two variables an engineer has #

Everything above reduces to a negotiation between two numbers, and it is worth being explicit about which of them is under whose control.

The demand is the mould’s. It follows from the Gaussian curvature and the distance the cloth has to travel, and a designer changes it only by changing the part — which is usually why the part exists and therefore not available.

The supply is the cloth’s. It follows from the cover factor, and a designer changes it by specifying a more openly woven reinforcement or a different weave. A satin reinforcement shears more readily than a plain one at the same sett, because its long floats let the tows rotate with less resistance and because fewer interlacings allow a more open setting in the first place.

Where the two cannot be reconciled the answer is the starting cross, which is free and is the least-discussed variable of the three. Moving where the cloth is first fixed moves the whole shear field, and on a mould with one difficult region and one easy one it is usually possible to spend the available shear where it is needed. That is precisely the operation a skilled laminator performs by hand and by eye, and the fishnet construction is the arithmetic version of it.

How much shear a hemisphere demands, in a number #

The comparison above was run cloth by cloth and mould by mould. The demand side of it can be closed in one line, and the line turns out to say something a designer would not guess.

The trellis is inextensible, so every yarn segment stays one pitch long. What the curvature does is change the distance round a circle drawn on the surface: on a sphere of radius R, a circle at geodesic distance s from the pole has circumference 2πR sin(s/R) rather than 2πs. The cloth has to fit a hoop of unshortened threads into a shorter hoop, so it takes up the difference

ε = 1 − sin(s/R) ÷ (s/R)

by closing its cells. And closing a rhombus of side p shortens one diagonal by √2 sin(θ/2), so a contraction of ε forces an included angle of 2 arcsin((1 − ε)/√2) — which for small ε is a shear of twice the contraction, in radians.

Two things about that are worth pausing on. The pitch has cancelled: a finer cloth at the same cover demands exactly the same shear, because both the segment length and the hoop scale together. And R appears only through s/R, so the demand over a spherical cap depends on how much of the sphere is being covered and not on how big it is.

The last column is the locking condition read backwards. A cloth locks at an included angle of arcsin(d/p), which is arcsin of the cover, so the shear it has available is arccos of the cover — and the cover that just survives a demand γ is cos γ.

A full hemisphere therefore needs a reinforcement woven below about eighty per cent cover, whatever its size and whatever its pitch. That is the result, and it is scale-free in a way the rest of the essay’s variables are not: a wrinkling hemisphere cannot be rescued by making the part larger, by using a finer tow, or by using more plies. Only openness helps.

It is worth checking against the three cases the essay already ran, because they bracket the threshold and were computed a different way. Forty per cent cover reaches; seventy per cent reaches; ninety per cent wrinkles. The bound above puts the boundary at 0.80, which sits between the highest passing case and the failing one — exactly where a correct bound should sit and where a wrong one would have no reason to land.

One honesty about what the line is. The hoop argument gives the shear averaged round a circle, and the fishnet’s shear is not uniform round one: it is zero along the two yarn paths through the pole and largest on the diagonals between them. So 36.5° is a lower bound on the peak demand, and 0.804 is an upper bound on the cover that will do. The real limit is somewhat more open than the table says, and the table is the right side of the answer to be wrong on — it forbids a cloth that would fail and permits none that would.

The transferable form of the whole calculation is shorter than the calculation:

the shear a preform needs is twice the circumference deficit of the surface, and the shear it has is the arccosine of its cover.

Both sides are one number, neither needs the drape to be constructed, and the comparison can be made on the back of a drawing before anybody has cut a piece of cloth.

What the model does not have #

The list is the same as the mechanics ladder’s standing list and it applies with more force here, because a preform is a load-bearing part rather than a garment.

No friction. The model says the cloth can reach a shape. It says nothing about the force needed to get it there, and in practice the force is what limits an automated lay-up. A cloth with high inter-tow friction wrinkles well before its locking angle simply because nobody can push it that far.

No bending stiffness. A wrinkle is a buckling event, and buckling needs a stiffness. The geometry says where the cloth runs out of shear; it does not say whether the result is a single large fold or many small ones, and that distinction matters for the finished part.

No slippage between tows. The trellis has pin joints, so tows rotate and do not slide. Real reinforcement does both, and inter-tow slip is a second deformation mechanism that lets a cloth reach shapes the trellis says are unreachable.

And the fishnet construction is one lay-up among many. It gives a valid drape, not the drape a particular operator will produce. Two people starting from different cross positions get different shear fields, both of them geometrically correct, and the model has no way of preferring one.

Where the construction came from #

The fishnet construction is older than composites and was invented for clothing.

Mack and Taylor published it in 1956 as a way of working out how a woven cloth lies on a curved surface — a garment problem, addressed at a textile research institute. The construction is exactly the one used here: fix two yarn paths, then determine every remaining node as the point one pitch from two of its neighbours. Nothing is optimised, which is what makes it reproducible: two people starting from the same cross get the same answer.

Composites adopted it wholesale in the 1980s and 1990s, because it turned out to be the right model for a quite different reason. In clothing the trellis assumption is an approximation — real cloth has friction, bending stiffness and some yarn extension. In a dry carbon or glass reinforcement it is very nearly exact: the tows really are inextensible, they really do rotate at the crossings, and the fabric really has no other deformation available. A model that was a useful idealisation for a skirt is close to the literal truth for a preform.

That is an unusual direction of travel. Most engineering models get less accurate as they leave the domain they were built for, and this one got more so.

The extensions since have mostly been about the parts the construction refuses to supply: adding friction, allowing inter-tow slip, and treating the lay-up as an energy minimisation rather than a geometric construction. All of them are more accurate and none of them is reproducible in the same way, because an optimisation has a starting point and a tolerance and a geometric construction has neither.

The measurement that ought to exist #

A last observation, because it points at a gap.

Everything in this essay compares a demanded shear against an available one, and only the first of the two is computed here from first principles. The available shear is the locking angle, which comes from a cover factor — and a cover factor for a real reinforcement is not easy to get, because a tow is not a thread with a diameter but a flat ribbon of filaments whose width changes as it is sheared.

That is the genuinely unsatisfactory part of the model. As a tow rotates it also spreads, so the width in the locking calculation is a function of the shear it is being asked about. Treating it as constant, which is what the calculation here does, over-estimates the locking angle at large shears and therefore under-estimates the wrinkling.

The direction of the error is at least known, which is worth something: the prediction is optimistic, so a preform this model says will just barely reach is one that in practice will not. Naming the sign of a model’s error is a poor substitute for not having one, and it is better than leaving it unstated.

Where the ladder goes next #

This is the top of the bias ladder as it stands. Below it are the locking angle, which supplies the limit, and the bias itself, which supplies the mechanism.

The companion in the neighbouring ladder is why clothes need darts, which is the same impossibility met by a trade that is allowed to cut, and bending stiffness and the drape coefficient, which is what happens when a cloth drapes under its own weight rather than being pushed onto a form.