Doing the crude experiments documented in Single-point incremental forming of aluminum foil, I was surprised to learn how thin aluminum foil is — in the neighborhood of 10 μm, though heavy-duty foil is 25 μm. I had thought it was considerably thicker because of its strength and tendency to retain creases.
So it occurred to me that maybe cut, laminated, and creased aluminum foil, or for that matter steel foil, was a potentially useful material for self-replicating machinery, along the lines of cardboard furniture. The fact that aluminum foil is so thin means that you can increase its rigidity immensely by forming it into relatively small parts, which would then have the stiffness to easily bend the still-flat foil.
Consider, for example, forming a corrugated sheet from aluminum foil similar to the corrugated iron sheets commonly used for the roofing of military bases and other slums. If you make it 10 mm thick, then you have about ¼ of the foil resisting tension on the outside of the bend with a lever arm of 5 mm, ¼ of it resisting compression on the inside in the same way, and the other ½ resisting tension and compression with an average lever arm of 2½ mm; the average lever arm is then 3¾ mm, while in the untouched sheet, the average lever arm might be 2½ μm. So the corrugated sheet is on the order of 1500 times stiffer. It’s fucking magic.
(There’s an additional work-hardening factor if you’re working with ordinary aluminum foil from the grocery stre, which is annealed, rather than whatever comes out of your aluminum-foil-making machine on the moon or whatever.)
That also works for increasing flexural strength (in the sense of the bending force on a member needed to provoke plastic deformation, not the stricter sense of the tensile stress in the fibers of the material needed to provoke plastic deformation) and resistance to buckling, but there’s no such magic for increasing tensile strength — although perhaps you could use the folded aluminum as a mold to be filled with some other material.
There’s a bunch of work in computational origami for things like unfolding satellite space shades, which unfortunately I don’t know anything about. Robert Lang is the big name in computational origami, and satellite space shades are folded with the Miura fold. My level of origami is basically that I can fold an origami crane, so I did that using this aluminum foil. It wrinkled considerably more than paper does, but it also creases better. To get the square to fold it from, I cut along the edge using my zirconia knife, a process which required only the tip of the blade, the last 30 μm or so — so presumably you could do the same thing with a 30-μm-long blade, which wouldn’t require very much sharpened zirconia or alumina or similar material.
(At some point I had dropped the knife on its tip and chipped it, so last week I bought a 750-grit diamond hone and sharpened the tip on it with some dish detergent.)
There’s a “Handbook of Compliant Mechanisms” from 2013 out of the BYU flexures research group, published by Wiley; nearly half of it is a rather poor-quality “library of compliant mechanisms”, much of which consists of things that can be cut out of a sheet but then flex into a three-dimensional shape (to which they have given the name “lamina emergent mechanisms”). The book doesn’t mention origami at all, and much of the book focuses on techniques that are difficult or impossible to apply to origami. To me it seems clear that origami is an important technique for flexure fabrication, and it turns out that Magleby and Howell, two of the editors of the “Handbook”, published a couple of papers on this, even before the Handbook came out. So I’m not sure why they didn’t include anything about this in the book.
Some traditional origami forms, like the flapping-wing bird and the jumping frog, are designed as flexures; these are called “action origami” or sometimes “kinetic origami” (a term due to Magleby and/or his coauthors in a 2011 paper). Papers typically analyze these as rigid flat panels connected by flexible hinges at the folds, a model which seems unmotivated by physical considerations — in paper typically the folds are slightly less rigid than the panels in the “hinge” direction, but in aluminum foil, they are typically slightly more rigid.
Many of the annoying features of aluminum, such as its high cost, large springback, abrasive nature, and tendency to accumulate internal stresses during heating that produce delayed distortions, are less important in this situation.
Because the aluminum is about ten times thinner than paper, but only 2.7 g/cc (according to Compressed sensing microscope), this aluminum is about a third as heavy as paper, square millimeter for square millimeter. Consequently the crane is much lighter than a paper crane of the same size would be, rather astoundingly light. Unfortunately, it isn’t very sturdy; I can plastically deform it by blowing on it. Work-hardened (unannealed) foil or a more reinforced design might help with this.
The foil is really amazingly flexible for its strength. According to the Wikipedia tensile-strength article, annealed aluminum has a yield strength in the neighborhod of 15–20 MPa, a Young’s modulus of 70 GPa, and an ultimate strength (“engineering”, I suppose, calculated according to the original material thickness) of 40–50 MPa, from which we can deduce that its plastic strain is about 0.2%–0.3%, while its yield strain is about 0.6%–0.7% (engineering, I suppose). So plastically creasing the foil involves bending it at a radius such that the inner surface is 0.2%–0.3% shorter than the centerline, and the outer surface is 0.2%–0.3% longer, so a bend radius of 1.7 to 2.5 mm.
The same article, though, points out that aluminum alloys are immensely stronger: 414 MPa yield and 483 MPa ultimate for 2014-T6 and 248 MPa ultimate for 6063-T6 (a tempered grade of a general-purpose “wrought” alloy), though their elastic modulus is about the same. The very common 6061-T6 is 275 MPa yield and 310 MPa ultimate, with 69 GPa Young’s modulus, roughly the same. These alloys are precipitation-hardened; annealed 6061 (“6061-O”) has only about 55 MPa of yield strength and only about 125 MPa of ultimate tensile strength, but elongation of 25%–30%, so it might make sense to perform the origami on the annealed material and then heat-treat the finished product to precipitation-harden it, gaining an additional factor of 5 in resistance to deformation.
How does it compare to paper? According to UHMWPE clothes could be lightweight and sturdy, the tensile strength of cellulose is in the neighborhood of 40 MPa, but when paper tears, it commonly tears because the different cellulose fibers have come apart, not because all the fibers are failing at the crack, as when you cut it with scissors; the tear tends to rotate into an orientation near parallel to the surface as it propagates in order to break even less fibers, even though this spreads the failure over a larger surface. This is also why cotton paper like that used in dollar bills is harder to tear. Unfortunately, this leaves me little wiser!
Cellulose at room temperature is normally thought of as a brittle material, one which fails without an intermediate plastic-deformation stage, and indeed when you tear paper there isn’t a noticeable stretched-paper area at the tip of the tear, nor does the edge formed by the tear crinkle in paper the way it does in aluminum foil. But this would predict that creasing paper should be impossible — like polyester napkins, the paper should just elastically return to its original form unless fibers were actually broken. Fibers are actually broken, as evidenced by the tendency of a tear to follow even a simple paper crease, but by itself that doesn’t explain the tendency of paper to elastically return to the creased orientation — it would only explain its tendency to be more flexible at the crease.
At first, I thought we couldn’t explain paper’s tendency to hold creases through some kind of non-cellulose interaction, because cellophane holds creases too, and I thought cellophane was pure cellulose; but it turns out that cellophane also contains glycerin as a plasticizer. Presumably this very plasticity is what allows it to hold creases.
This suggests that paper holds creases by altering the interactions of cellulose fibers. On the inner radius of the crease, there is presumably some thickness of the paper that is crushed into rubble, around which is wrapped a series of layers of cellulose fibers loosely stuck together, either by other materials present in the paper or by interactions among fibers that touch one another. The fibers in these layers slide past one another and dig into inner layers during the creasing process, and perhaps in the outermost layers are broken by the tension. This leaves the natural length of the layers altered, causing the paper to elastically maintain the crease. This hypothesis is Original Research™ and therefore may be entirely wrong, but it gives me a reasonable alternative to supposing that the cellulose is deforming plastically.