Sandwich theory is the theory of sandwich structures, which are a kind of anisotropic composite material consisting of two stiff “facesheets” bonded to a lightweight “core”, like a two-dimensional version of a one-dimensional I-beam; corrugated cardboard and drywall are the most common examples. I started reading about this because of an interest in cardboard furniture (see Cardboard furniture) and recycled materials.
When built with materials that aren’t full of cracks, structures stressed in tension are generally quite strong, and in a way that barely varies with size. Piano wire (aka music wire — a kind of steel used mostly for springs nowadays) has a yield stress of about 2.5 GPa, so a platform that can support my 110 kg can be supported by 0.43 mm² of piano wire, a wire 0.75 mm thick. I’d have to sit down on it very slowly to not break the wire on impact, and if you just connect the wire directly to the platform, it will be kind of tippy; but hanging by three or four such wires would be safe:
|| ||
w|| ||w
i|| me ||i
r|| o ||r
e|| /|\ ||e
s|| / \ ||s
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platform
Moreover, this holds whether the wires are connected to the ceiling 2.6 meters above my floor or to the top of a 100-meter-tall building I’m swinging at the base of. 100 meters of four 0.43 mm² piano wires would only weigh 1.4 kg, which is not a significant addition to my weight; you have to get into kilometers of height before the weight of the wire is a significant fraction of what it can support.
As Descartes observed, if we scale up the whole situation instead of just the cable length, things get a little worse: a King-Kong-sized person ten times my height, width, and breadth would weigh a thousand times what I do, about 110 tonnes, while scaling up the cables from 0.75 mm to 7.5 mm in diameter only strengthens them to be able to support 11 tonnes. You’d need to increase them to 24 mm in diameter to support the Kragen Kong. Still, though, that’s not such an extreme departure from uniform scaling.
There’s a compressive strength that works the same way; if I put a 10-cm-diameter disk of cardboard under a wooden board and start piling weight onto the board, there’s a certain amount of weight above which the cardboard will get crushed; let’s say 300 kg, which is in the ballpark but not a measured figure. If you use several different disks of the same cardboard, they’ll all crush at about the same weight, and moreover that weight doesn’t change even if you stack two or three or ten of the disks up; the same 300 kg will suffice to crush the stack of ten disks as to crush a single disk. If you use a bigger disk, the weight needed to crush it will go up proportionally.
weight
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----------------- wood
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============= disks
----------------------- floor
But if you use smaller and smaller disks (and proportionally smaller weights, including the wood), or taller and taller stacks, something weird happens. Well, not weird if you have any experience with material objects, but different. Once the stack gets taller than a certain limit — say, around ten times its width, for cardboard — taller and taller stacks can support less and less weight. So I can stand on one 10-cm cardboard disk, and I can stand on a stack of ten of them, and I can probably stand on the column of 650 of them that it would take to reach 2.6 meters, but I definitely cannot stand on a column of 100 meters of them. In fact, I probably can’t even build one. Adding more cardboard to the stack in this way reduces its compressive strength rather than leaving it unchanged.
I’m not talking about the top of the stack falling over sideways, which also gets harder to avoid when the stack is taller; the phenomenon I’m talking about happens even if you go into an elevator shaft and rest an elevator-shaft-sized wooden platform on top of the stack of smaller-than-shaft-size disks. I’m talking about buckling (pandeo in Spanish), where the middle of the stack bends outwards in some random direction and it stops being a linear stack. Buckling is governed not by the compressive strength of the column but by the rigidity of the column.
Euler worked out the basic math for this. For compressive forces below Euler’s critical value, any small bend in the column, for example due to sound waves, gets straightened back out by the column’s elasticity. But once the column is bearing a load over Euler’s critical limit, such a small bend will get amplified by the compressive stress more and more. Euler’s critical load is π²EI/(KL)², where E is the modulus of elasticity of the beam, I is the least planar area moment of inertia of the column’s cross section (πr⁴/2 for a solid circle), L is the length of the column, and K (the “column effective length factor”) varies between ½ and 2; it’s ½ in this case because the cardboard disks resting flat on the ground and the elevator-shaft platform are not free to rotate.
Sadly I don’t have the faintest idea what the modulus of elasticity of corrugated cardboard in lateral compression is, but the things to notice about this expression are:
Point #1 here means that, normally, large structures fail because of buckling before than they fail because of tensile or compressive failure, unless they’re made of materials that are full of cracks. This means that if you can improve buckling behavior somehow, you can dramatically extend the size of structures you can construct.
These last three points are crucial to sandwich theory! By stiffening columns with high-stiffness facesheets, and increasing the moment of inertia by moving those facesheets further apart by inflating the core (even at the cost of some strength), we can make sandwich-structured composites that resist buckling enormously more effectively than their component materials.
There’s also flexural stress. Sometimes large structures fail because of flexural failure — trees dropping branches or blowing over in a storm are examples. Moreover, flexural stress shares with buckling the property that you really need a great deal more material to resist the load than you would expect — a steel beam that can support my weight in flexion is generally going to need a great deal more than 0.43 mm² of cross section. I am going to mostly ignore flexural stress, except as it contributes to buckling, because ⓐ the same sandwich-panel construction that helps with buckling also helps with flexural stress, and ⓑ flexural loads for a given beam construction only fall inverse-proportional to the characteristic dimension of the structure, while buckling loads are inverse-proportional to its square, so for large enough structures buckling still dominates.
The cheapest building materials — earth, brick, other fired-clay ceramics, concrete, mortar, glass, plaster of Paris, and often even stone — are all full of cracks. As long as they’re in compression, this doesn’t matter, but once they’re in tension, the cracks form stress risers that weaken them enormously. This makes it difficult and dangerous to build large structures from them unless they’re heavily reinforced with materials with good tensile strength. Adobe is made from earth reinforced with straw, plaster of Paris is commonly reinforced with horsehair, and concrete is commonly reinforced with steel rebar. Even so, larger buildings are invariably steel-framed rather than relying on the tensile strength of even reinforced masonry.
The moment of inertia around the x-axis, Iₓ, is ∫∫y²dxdy where the area integrated over is that of the cross-section. For panels, that’s the only one we care about, because the panel is much wider than it is thick, so if we take our x-axis parallel to its surface, the y-axis moment of inertia is orders of magnitude too big to worry about.
This says that if we have some cross-sectional area of material, we can increase its moment of inertia proportionally, to an arbitrarily high level, by moving it further from the x-axis:
small moment large moment
*****
*****
--*****-- --*****--
*****
*****
The trouble with this is that once we insert empty spaces in the middle, it stops being a solid object. We need to put enough stuff in the middle to keep the parts moving together; the stuff in the middle is pretty much only stressed in shear. Also, when the column is flexing, the facesheet on the inside of the bend is in compression merely from the flexing, and that compression can cause it to buckle.
If we only have one material to work with, we can improve the situation by moving almost all the material as far as possible from the x-axis in both directions, leaving the rest of the material as fine “webs” in between to resist the shear, and we have an I-beam, or rectangular tubing, or a channel, or a sort of sandwich panel:
***** ***** ***** ***********************
* * * * * * * * * *
* * * * * * * * * *
----*---- --*---*-- --*------ --*---*---*---*---*---*--
* * * * * * * * * *
* * * * * * * * * *
***** ***** ***** ***********************
(This kind of sandwich panel, a single plastic with square channels running through it, is in common use for translucent roofing for bus stops around here; it’s called “twinwall plastic” when made from polycarbonate, and similar sandwich panels made from other plastics are called “corrugated plastic”, “corriboard”, or “coroplast”.)
More generally, you can use a different kind of material in the middle, maybe one with not much strength of any kind, just enough to resist the shear stress. Incidentally, that shear stress is also inversely proportional to the distance between the facesheets, so if your panel is thick, you can get by with a very wimpy core material. You do still need enough tensile and compressive strength provided by the cross-sectional area of the facesheets and the core to withstand the tensile and compressive stresses on the panel from its edges, but in a large structure, that is a much easier problem than preventing buckling.
XXX understand the theory
An underappreciated aspect of sandwich panels is that they are not only stronger (in flexure and buckling) but also softer than the same materials would be if solid. This can increase their impact strength.
Single-wall corrugated cardboard, as I said, is the most common kind of sandwich panel in daily life. It has a smooth “linerboard” paper on the “outside”, according to Angela Ben-Eliezer, a corrugated paper layer of “flutes”, and a second linerboard paper on the “inside” which is not smooth, all glued together, typically with sodium silicate. The linerboard layers play the role of facesheets, while the flutes play the role of the core — a highly anisotropic core, in this case.
The inner fluted layer is mostly air — typical corrugated cardboard is made from paper of about 90–130 g/m², which makes it about 150 μm thick, but has a 4-mm-thick fluted layer, so the fluted layer is about 96% air and 4% paper, and the cardboard as a whole is about 88% air and 12% paper. In fact, only part of the flutes — about half — is in the fluted layer itself. The other half is glued flat to one or the other linerboard, making it part of the facesheet.
Double-wall and triple-wall corrugated cardboard is a less common material, used for higher-strength applications. Triple-wall cardboard has three layers of flutes and four layers of linerboard, and the interesting thing here is that the inner layer of flutes is typically substantially thicker than the outer layers. That is, it’s a sandwich panel whose facesheets are themselves sandwich panels! It’s a recursive sandwich!
Drywall is a sandwich of gypsum (plaster of Paris) between two sheets of paper. Unlike the other examples, this isn’t to help it bear buckling loads; it’s to give it tensile strength (and thus flexural strength) so you can carry it around and drive screws through it, instead of shattering the way plaster on lath would do if you treated it that way.
Corrugated sheet steel is similar to corrugated cardboard, but just the flutes, without the linerboard; it’s just a wavy sheet of steel. This is not the most efficient use of material, since the “core” consists of roughly as much steel as the “facesheets”, but it’s very cheap to make. Slightly more elaborate corrugated sheet metal, like that used for the walls of shipping containers or aluminum siding, uses sharp bends to increase the amount of material in the “facesheets” while still being inexpensive to fabricate by bending a single sheet.
Foamcore is a sandwich made of two sheets of paper with styrofoam in between, commonly used for architectural models and picture framing.
The Hexayurt is made from sandwich panels sold in the US and some other countries for house insulation; these are foamed polyisocyanurate with aluminum-flashing facesheets. Similar lightweight rigid sandwich panels made of various materials are common insulating materials; when the facesheets are some kind of structural board material like OSB or drywall, it’s called a “structural insulated panel”.
Fiberglass fabric glued onto the surface of styrofoam is a common material for small airplanes, especially model airplanes.
RV enthusiasts have taken to fabricating furniture by cutting styrofoam to shape, fitting them together, and coating the surface with some kind of stiffener, such as fiberglass window screening material stuck to the foam with latex paint.
I’m seeing sandwich panels on some of the bus lines here in Buenos Aires; they seem to have polyethylene cores and melamine facesheets.
Cement board has been a common building material for decades; it’s a sandwich with a cellulose-reinforced portland cement core and glass-fiber mesh facesheets. It’s kind of like drywall, but a lot stiffer; it’s useful as an underlayment to keep tile floors laid over wood from breaking when the wood flexes.
Common FDM 3-D printing slicers will, by default, fill the interior of the model (whatever its shape) with a honeycomb, thus making it a sort of three-dimensional quasi-sandwich-panel. This is mostly to cut down printing time, but it also helps to compensate for PLA’s abysmal impact strength.
In aerospace, sandwiches with sheet-metal facesheets and metal honeycomb cores are common.
The Grenfell Tower fire in 2017, which killed 72 people, about a quarter of those present, was caused in large part by the use of Arconic Reynobond PE sandwich panels, with aluminum facesheets around a polyethylene core (not, I think, foam), which were covering an 150-mm-thick layer of polyisocyanurate foam under a ventilation space.
Styrofoam and cardboard are the two most obvious candidate core materials, but many others are possible. boPET, as in discarded chip bags and one of the layers in TetraPak drink boxes, is fairly stiff, and so boPET and TetraPak are abundant candidates for facesheet materials; cardboard is also a good candidate.
Thin aluminum sheet, as from drink cans, is considerably stronger and stiffer in tension than either of those; it might make a good facesheet. More broadly, sheet metal is often discarded, and it makes a fantastic facesheet material, as do plywood and OSB.
Drink cans have a particularly interesting property: they are mostly coated on the inside with a thin layer of very chemically inert plastic, usually a food-safe epoxy. It might be possible to use this coating as a ready-made glue (see below about the difficulties of glues) to stick the rolled-flat shreds of shredded cans together, like wood chipboard. Some combination of heat, pressure, and previous functionalization with cold plasma (see Cold plasma oxidation) might do the trick. This should not require temperatures anywhere close to those necessary to melt the aluminum.
Expanded sheet metal — that stuff they use for security grilles and transparent nonskid steel stairs — is likely an excellent facesheet material, but it is only very stiff along one of its axes.
Fabric-backed vinyl is discarded in bulk when billboards are updated with new advertising, and it has its merits, but it is sufficiently useful that it already tends to get recycled.
Woven cloth (from discarded clothes, mattresses, etc.) is another candidate facesheet ingredient — unbroken chains of fibers running in long straight lines along the surface will contribute more stiffness than the same fibers in a more disordered mass like that of paper. Knit cloth would contribute little stiffness by itself, but “nonwoven cloth” ought to be intermediate, like paper. Glass-fiber cloth or cloth woven from Kevlar, Nomex, or Spectra would be ideal, but are hard to find, except in the form of glass-fiber window screens. Some cloths are probably not suitable: nylon and silk are elastic enough that they probably wouldn’t add enough stiffness.
(Aluminum or steel window screens would work too.)
Of course, these materials by themselves need some kind of glue to turn them into facesheets. (See below about glues.)
Discarded circuit boards are extremely stiff: they are made of glass-fiber cloth impregnated with a plastic such as phenolic resin. They are relatively abundant in the waste stream, and would be ideal.
Since the core is ideally very thick, it can be made from a quite weak material, including plastic foams, especially microcellular foams. So foaming discarded thermoplastics, for example with nitrogen, is a potentially very interesting way to upcycle them.
Most organic matter can be reduced to some kind of charcoal by heating at high enough temperature without oxygen. Charcoal is very rigid — more so the purer it is — and lightweight, because it’s basically a foam. It might make a very reasonable core material. It’s usually full of cracks, so unless you crush it into a powder, its tensile strength is very low. One possible approach to that problem is to form the organic matter into a fine open-cell foam before carbonizing it, for example by baking it into bread, and then heating and cooling it very slowly. That might yield a carbon foam with reasonable tensile strength but I wouldn’t bet on it.
Carbon black — charcoal formed as smoke rather than in solid form — is routinely used to add tensile strength to rubber, for example in car tires and shoe soles. It, too, can be produced from almost any organic matter.
Facesheets are only strong when in tension; when in compression they rely on the core material to support them against buckling. If there’s a way to pretension them somehow, this could increase the strength of the whole structure. One way would be to simply pull on the facesheets during the lamination process, so they are stretched while the bond is being formed. Alternatively, some plastics might enter into tension when heated, the way PET drink bottles and shrinkwrap do. Another alternative: in cloth aircraft construction, after construction, the cloth surface is impregnated with a substance called “dope”, which shrinks as it cures, setting up tension in the cloth. Is there a “dope” that could be used to pretension sandwich panel facesheets in this way?
Alternatively, if the core material is a closed-cell foam, you might be able to pretension the facesheets this the other way around, by setting up compression in the foam. A couple of ways to do this: you could assemble the sandwich panel in a chamber under high pressure, thus partly collapsing the cells of the foam; or you could arrange for air to diffuse through the foam into the cells after assembly, for example by baking the foam in a vacuum before assembly, long enough for a substantial fraction of the air to diffuse out.
Typically the blowing agents diffuse out of the foam after it is produced, and this is often considered problematic when it happens faster than air diffuses in — it means that the foam contracts shortly after fabrication, then later expands again after absorbing air — but it produces a critical period during which sandwich-panel fabrication would pretension the facesheets. So, perhaps the same phenomenon could be provoked intentionally, either by foaming the plastic with the usual blowing agents or by provoking a very volatile gas like hydrogen, helium, or methane to diffuse into the foam first. (A chamber full of methane is a lot easier to produce than a vacuum chamber, if perhaps more dangerous.)
Because the primary purpose of the facesheet is increasing rigidity, not strength, in order to resist buckling, it might make sense to use materials with high rigidity and low strength; glass, plaster, and cement come to mind.
Generally, filled plastic systems can get a lot of rigidity from the filler, so using a high-rigidity filler like rocks, sand, clay, cullet, grog, eggshells, carbon, or machining swarf might make sense. (The usefulness of such fillers isn’t strictly limited to “plastics” as such; they are of course a key part of concrete, mortar, and fired-clay ceramics, in which cases they mostly contribute strength rather than rigidity.) Such fillers work better with large aspect ratios, so that the individual filler particles can more easily be much closer together than their large dimension, so that when the material as a whole is stressed in tension or compression, the binder between them is stressed in shear over a much larger cross-sectional area than that of each interacting filler particle. This way, most of the tensile deformation of the bulk material comes from stretching filler particles rather than stretching or shear-deforming the less-rigid binder.
A hexagonal, herringbone-brick, or jigsaw-puzzle pattern of facesheet tiles, with a little space between them, could provide rigidity without causing the facesheet to break when its elastic limit is exceeded.
These approaches aren’t viable if ⓐ the core material is so weak that you’re depending on the facesheet to provide tensile strength as well; or ⓑ the sandwich panel is sufficiently stressed in flexion, as opposed to buckling, to depend on the facesheet’s strength as well as its rigidity.
Masonite, medium-density fiberboard, wood, paper, PMMA, CDs, wet wipes, fabric-softener sheets, and many other things are also plausible facesheet materials.
Particle board is made from sawdust pressed into panels under pressure with a glue, typically melamine resin or phenolic resin. This approach, possibly with different glues (see below about glues), might be feasible for fabricating facesheet panels from a variety of waste-stream fiber sources: sawdust, wood shavings, coffee grounds, yerba mate, grass clippings, hair clippings, machining swarf, dry leaves, chipped prunings, shredded paper, shredded cardboard, dehydrated sliced vegetables, fiberglass from demolition (at the risk of asbestos), shredded cloth, bagasse, straw, pencil shavings, peanut shells, sunflower-seed shells, wicker, or chicken feathers, for example. As mentioned above, adding some high-rigidity fillers like sand, clay, cullet, grog, or eggshells may improve the rigidity of the resulting material, just as it would a filled plastic system.
(Needless to say, such composite materials could be used for things other than sandwich panels, too, or even panels.)
How can you do the lamination, so the facesheets can’t slide along the surface of the core?
You might be able to use just standard modern adhesives: epoxy, melamine, and phenolic resins. But these are entirely impractical to obtain from the waste stream, and there are cheaper alternatives.
Some materials can be bonded just with heat, and in some other cases you can use small amounts of thermoplastics to bond together layers of non-thermoplastic materials, like the EVA in a hot-glue gun. And, as mentioned above, latex paint works well enough under some circumstances, but it’s difficult and somewhat dangerous to find discarded. Porous facesheet materials like cloth and window screen allow air access to the binder, which is essential for the curing of many binders.
In some cases the strength requirement for the adhesive may be low enough that even paraffin wax (or polyethylene or PET) could be used as a hot-melt adhesive, perhaps with a filler such as clay to stiffen it and add strength. Wet clay also makes an excellent weak adhesive which hardens and contracts on drying.
The five traditional adhesives are slaked lime, wheat paste, mucilage, hide glue, and birch tar, the first four of which can be made from garbage at varying levels of personal risk, or from very inexpensive non-garbage materials. They might work. Wheat paste and mucilage don’t stick very well to nonpolar materials like common plastics, and slaked lime is caustic. Birch tar is probably not practical in the modern context. Most of these need some degree of care to keep them from rotting, blue vitriol being a traditional pH-neutral antifungal.
Blood is also sticky and dries hard, due to its albumin content, and there’s a lot of animal blood in the waste stream, but for whatever reason it isn’t widely used as an adhesive, perhaps due to excessive clotting.
“Pressure-sensitive” or “constant-tack” adhesives like those used in duct tape, packing tape, and postit notes are not really suitable for structural use, because, being viscous liquids, they will flow continuously if under strain. They can resist compression, but in shear or, especially, tension, they will gradually release their bond. In structural applications this is potentially disastrous: weeks, months, or years after initial construction of a structure, its sandwich panels could delaminate and fail, apparently spontaneously.
Asphalt or pitch, as used in roofing membranes, is on the edge here. It’s a viscous liquid, but at room temperature its flow rates are low enough that it might be acceptable, if the panels won’t be exposed to heat.
Thin-enough porous facesheets could be simply painted with boiled linseed oil, but oxygen needs to be able to diffuse to the facesheet-core interface to be able to polymerize the oil there. More generally, it might be possible to polymerize such “drying oils” with oxidizing agents like sodium percarbonate or ozone. (Linoleum flooring is polymerized linseed oil with a filler painted onto a coarse cloth backing.)
Some kinds of core or facesheet materials could be welded with a small amount of solvent, such as acetone, MEK, or ethyl acetate. Doing this on styrofoam is tricky because the solvent tends to eat the cells it touches; perhaps a little vapor, rather than drops of liquid, would do the trick.
Such solvents can also be used to make glues from other waste plastics, such as PVC pipe or polystyrene foam, peanuts, or food containers.
A final, more exotic, glue is sodium silicate, which can be made in bulk very inexpensively, simply by boiling quartz in lye; this has the benefit of hardening on timescales measured in seconds when gassed with CO₂. As I mentioned above, this is commonly used to make corrugated cardboard; it also waterproofs concrete. (Potassium silicate is nearly equivalent.)
Since the objective of gluing the lamination interface is resisting shear, not tension, the facesheets could also be connected to the core by non-adhesive means, such as nails, tacks, staples, or sewing — a thread or wire passed repeatedly through holes in the sandwich panel to keep the facesheets from sliding relative to the core. These approaches are more practical with fairly rigid facesheets, since in between the mechanical attachment points, the facesheet on a side in compression must resist buckling on its own without any real support from the core.