A fabric to fill and a fabric to load

A dry reinforcement is only half a part. The other half is a liquid resin that has to reach every fibre before it gels, and it gets there by flowing through the fabric — which means the fabric is being asked for two incompatible things at once.

Stiffness wants fibre, and fibre means a close sett. Flow wants passages, and the passages are the gaps the sett leaves. Both are inequalities on one decision, they run in opposite directions, and the useful question is therefore not what is the best sett but whether any sett satisfies both — a question with a yes-or-no answer that depends on how large the part is.

Where the resin actually goes #

Before any of that can be computed, one thing has to be got right, and getting it wrong is a factor of a thousand.

A woven reinforcement has two length scales. Inside a tow, the passages are the spaces between filaments seventeen micrometres across, packed at about two thirds. Between the tows, the passages are the channels the sett leaves — hundreds of micrometres wide. Permeability goes as the square of the passage size, so the channels are not slightly better than the tows: they are three or four orders of magnitude better.

So the resin runs down the channels and the tows fill afterwards, slowly, sideways, from their own surfaces. That is not a refinement of the picture — it is the picture, and it has a consequence a fabricator will recognise: the voids in an infused part are inside the tows and not between them, because the flow front outruns the tow saturation and leaves air trapped in the middle of bundles that the advancing resin has already passed.

The ratio is not a constant of reinforcements. At four tows per centimetre the channels beat the tows by a factor of twelve thousand; at five it is seven hundred; at six and a half it is less than one. The dual-scale claim is a claim about a function of the sett with a computable crossing, not about a particular fabric — which is why the assertion in the code is on the crossing rather than on the size of the ratio. Asserting that the channels are much better would have been an assertion about the default arguments, which is this site’s commonest way for a check to go quietly bad.

Darcy’s square law is the whole difficulty #

With a permeability in hand, the fill time is one line. Resin entering under a constant pressure difference advances as the square root of time, so the time to travel a distance L is

t = μ φ L² ÷ (2 K ΔP)

with μ the viscosity, φ the porosity to be filled, K the permeability and ΔP the pressure driving it. Four of those five are properties of the materials and the process. The fifth is the size of the part, and it enters squared.

That is why a large part is a different problem rather than a bigger one. Doubling the flow length quadruples the fill time with nothing about the fabric changed, so the ceiling on the sett comes down while the floor stays exactly where it was.

The interval, and the size at which it closes #

Put the two bounds together and the answer for a given part is an interval in the sett. For a half-metre flow length at thirty minutes and a floor of 45 per cent fibre, that interval is narrow: five tows per centimetre and very little else.

Stretch the part and it closes. The critical flow length is 933 millimetres for this fabric, this resin and this pressure — bisected on the length, with the specification satisfiable at two per cent below it and unsatisfiable at two per cent above, both asserted, because a boundary whose far side nothing checks is a boundary nobody has tested.

An unsatisfiable specification is a result and not a failure. It says the answer is not a better cloth but a different process — a flow medium laid over the laminate to carry resin along the surface, an injection at pressure rather than under vacuum, a resin thinned or a cure retarded, or a prepreg that arrives with its resin already in place. Every one of those is a change to something other than the fabric, and the arithmetic is what says so.

What a fabricator is actually given, and why it is the wrong pair of numbers #

A reinforcement is sold on two numbers: an areal weight in grams per square metre, and a style number that stands for a weave. Neither is the quantity above.

The areal weight is a product of the sett and the count — n · tex, twice over, plus the crimp — so it fixes neither on its own. A 400 g/m² fabric can be five coarse tows to the centimetre or ten fine ones, and those two have the same weight, very nearly the same fibre content at the same thickness, and channels differing by a factor of four in width and sixteen in permeability. The number that decides whether a part can be filled is not on the label.

The style number stands in for the rest, and it works in the way a trade name works: it identifies a fabric somebody has already infused successfully. That is genuinely useful information and it is not a model. It transfers to a part of a different size not at all, because the size enters squared and the style number does not know the size.

What the arithmetic above is for is precisely that transfer. Given a sett, a count, an aspect ratio and a weave, it produces a fibre content, a channel width and a fill time for a stated flow length, and the whole of the design work is then reading an interval off an axis. That is the same move thread count needs and never gets: a number that is a product of two decisions cannot substitute for either of them, and quoting it as though it could is how a trade ends up with a figure everybody uses and nobody can act on.

The same inequality, three trades apart #

The shape of this argument is not about resin, and recognising the shape is worth more than the numbers in it.

A filter cloth must hold a soil back and still pass water. The first wants a small hole, the second a large one, both are inequalities on the sett, and for a fine enough soil the interval is empty — at which point the answer stops being a woven fabric and becomes a nonwoven, exactly as the answer here stops being a fabric and becomes a process.

A seam must grip the cloth’s own threads harder than the load pulls them and must not perforate the cloth doing it. More stitches buys the first and spends the second, so there is an optimum rather than a maximum, and it sits where two computed curves cross.

A ripstop must arrest a tear within a stated length and must not spend more weight than the garment can carry. Both bounds are exact, they run opposite ways in the grid spacing, and the cheapest grid that meets the tear limit is the one whose spacing is the tear limit.

Four trades, four pairs of inequalities, one geometry underneath — the clear gap between two threads, which is the spacing less the diameter, and which this site has been computing since it first asked how close threads can be set. The gap is the resin’s channel, the filter’s pore, the needle’s room and the tear’s slack. That is not an analogy: it is the same length, measured for four purposes.

What the pictures in this field cannot show, and it is worth being explicit about, is the inequality itself. A figure can draw a fabric at one sett and print what it does; it cannot draw an interval being empty. So every trade-off figure in this field is a picture of a function with limits across it rather than a picture of a fabric, and the drawing rule they share exists because the object being drawn is a specification rather than a cloth.

What was counted, and how #

The permeability of each scale is Kozeny–Carman: K = d²ε³/(16k(1 − ε)²), with ε the porosity and k a fitted constant. Two of those are computed from the fabric and one is not, and the one that is not is stated at every use. Reported Kozeny constants run from below one for flow along a fibre bed to about five for a random one, and the permeability goes as its reciprocal — so every number here is a number at a stated k, and the argument uses only the scalings, which do not contain it at all.

The porosities are the fabric’s own. Inside a tow the porosity is one minus the packing factor. Between the tows it is one minus how much of the cloth’s volume is tow, which is the quantity the fibre volume arithmetic already computes and asserts to be at most one. The channel’s width is the spacing less the tow’s width, which is the same clear gap the filter cloth essays measure with the same expression.

Three monotonicities are asserted while the rows are built, because the shape of the answer depends on them: a closer sett holds more fibre, a closer sett takes longer to fill, and the ratio between the two permeabilities falls. Any one of them coming out backwards would leave the interval in the wrong place while every number still looked reasonable.

Where the model stops #

Capillary pressure is missing. Resin wets a fibre and is drawn into a tow by surface tension, which is a second driving force acting at the scale where Darcy’s law is being applied to the wrong medium. It is what actually saturates the tows, and it is the reason the void content of a real part is not the void content this arithmetic implies.

The fabric moves while the resin flows. Under vacuum the plies compact, and they compact more where the resin has not yet arrived, so the permeability and the fibre content are both functions of position and time. The calculation here is for a fabric held at one thickness.

Race-tracking is not modelled at all. A gap at the edge of a mould, or between two plies cut short, is a channel orders of magnitude more permeable than either scale here, and it will carry the resin around the part rather than through it. Every practitioner’s first infusion failure is this one, and no geometric model of a fabric can see it.

And the flow is one-dimensional. A real infusion is a two-dimensional front, often anisotropic, with the warp and weft directions differing by a factor of two or more — which the arithmetic here could carry and does not, because the square law on the length is what decides the answer and a factor of two in the direction moves it very little.

The pressure is not a free variable, which is why the length matters so much #

One term in the fill time deserves separating out, because it is the one a fabricator cannot buy their way past.

Under vacuum the pressure difference driving the resin is at most one atmosphere, and it is less than that in practice — the vacuum is imperfect, the resin’s own head helps or hinders, and the pressure at the flow front falls as the compaction ahead of it changes. So ΔP is not a knob with a wide range on it: it is a bounded quantity a little under 10⁵ pascals, and the arithmetic above is written at that bound.

Everything else in the expression is either fixed by the material or fixed by the geometry. The viscosity is the resin’s, chosen from a shortlist, and lowering it usually means a faster cure — which lowers the time available in the same movement, so the two changes very nearly cancel. The porosity is one minus the fibre content, so it is the decision already being made. Which leaves the length, squared, as the only term with orders of magnitude in it.

That is the sense in which part size is the governing variable of the whole process. It is not that large parts are harder; it is that the one term with a square on it is the one nobody can change, so the specification’s ceiling falls quadratically while every other lever is worth a factor of two at best. Injection at pressure raises ΔP by a factor of ten or more, and that is why it exists — a process change to the single term that is bounded under vacuum.

Which lever moves the critical length, and by how much #

The critical flow length is 933 millimetres for one fabric, one resin and one pressure, and the useful question is what moves it. Every term in the fill time is a candidate and they are not comparable, because most of them enter under a square root and one does not.

At the critical length the two bounds meet at a single sett, so setting the fill time equal to the gel time and solving gives

L_crit ∝ √( K · t_gel · ΔP ÷ (μ φ) ),

with K evaluated at whatever sett the fibre floor demands. So every process lever is a square root and the fibre floor is not, because the floor moves the sett and the sett moves the permeability by a power.

Doubling the pot life buys forty per cent of part size. That is a real gain and it is far less than a doubling, which is the arithmetic behind a familiar disappointment: a resin chosen for a long open time does not let a fabricator make a part twice as large, and the reason is the square root rather than anything about the chemistry.

And relaxing the fibre content is worth more than everything else combined. Five points of fibre content moves the admissible sett down, and the permeability of the channels goes as the square of their width — which is the spacing less the tow, so it opens faster than the sett closes. The critical length doubles or better for a change a specification writer would call marginal.

That inverts the way the trade-off is usually presented. The fibre content is treated as the requirement and the process as the thing to be tuned, which is right for a small part and exactly backwards for a large one: on a part near its critical length, five points of fibre volume is the largest lever available and every process change is a square root.

Three consequences follow.

A stiffness specification should carry the part size. A floor of forty-five per cent is a different demand on a half-metre part and on a two-metre one, and quoting it without the size is quoting one half of an inequality.

And the two ways out are not equivalent. Injecting at pressure raises ΔP by ten and buys a factor of three; laying a flow medium over the laminate changes the geometry entirely, cutting the through-thickness flow length to the laminate’s own thickness and moving the whole problem out of this arithmetic. The second is not a bigger version of the first, and it is the one large parts actually use.

And the ranking is computable before anything is bought. Every term above comes from a datasheet or a drawing, so a fabricator facing an empty interval can see in a moment which of the four levers is worth pulling — which is more than the style number on a roll of reinforcement will ever supply.

Who found it, and when #

Darcy’s law is from 1856 and was about water through sand. Kozeny’s relation is from 1927 and Carman’s revision of it from 1937 — the same year as Peirce’s plain-weave geometry, which is a coincidence and a nice one, since the two are the halves this essay puts together.

The dual-scale reading of a woven reinforcement is much later, from the resin-transfer-moulding literature of the 1990s, and it arrived because parts had voids that the single-scale models said they should not have. That is the ordinary way a length scale gets discovered: something the model cannot account for turns up in a part somebody has cut open.

Where the ladder goes next #

Both of the last two rungs have leaned on a fact taken as given: that a reinforcement’s tow is flat rather than round, and that its width rather than its diameter is what has to fit between its neighbours. That is a change to the geometry every model on this site rests on, and it moves the jammed sett, the crimp, the cover and the thickness at once. The next rung takes the racetrack section this site has carried since its earliest essays to its flat limit and reads off what changes.