Sandwich theory

Kragen Javier Sitaker, 2019-08-05 (17 minutes)

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.

The basic theory

Basic strength-of-materials background

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
    ---------
    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.

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
       ##
      ####
     ######  
----------------- wood
  =============
  ============= 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 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 Young’s modulus of corrugated cardboard in lateral compression is, but the things to notice about this expression are:

  1. It’s inverse-quadratic in L, so a column ten times as long can bear only one-hundredth the weight without buckling.
  2. It’s directly proportional to the modulus of elasticity, so a beam that’s made from a material that’s one-third as stiff can only bear one-third the weight without buckling.
  3. It doesn’t involve the strength of the cardboard at all; it only depends on its stiffness.
  4. Compressive and tensile strengths are proportional to the cross-sectional area, but Euler’s critical buckling load is proportional to the area moment of inertia, not the cross-sectional area.

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.

Cracks

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.

Basic sandwich-theory background

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.

Common current sandwich-panel examples

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.

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”.

Current more exotic examples

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.

Candidates for upcycling garbage into sandwich panels

Styrofoam and cardboard are the two most obvious candidate materials, but many others are possible. boPET, as in discarded chip bags and one of the layers in TetraPak drink boxes, is fairly stiff.

microcellular polyethylene

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