I just tried friction-cutting some PVC pipe with cotton string. A thicker braided string worked very reliably, but required a lot of work to melt the large amount of plastic, also leaving a rougher cut with bigger burrs. A thin twisted cotton string worked with a lot less effort, leaving a cleaner cut, but also broke several times.
I was thinking that you could probably do a kind of low-energy friction bandsaw/jigsaw for thermoplastics, as a kind of low-budget substitute for laser cutting and waterjet cutting, at least those that soften enough at a low enough temperature; cellulose string works up to about 200°, while steel wire should work well at higher temperatures. Nylon reputedly also works for PVC, since its softening temperature is so much higher than PVC’s; cellulose reportedly works for nylon itself, while other sources claim cotton won’t cut PVC. I think UHMWPE cord will not work at all, though I haven’t tried it.
Ideally you’d have much, much less mass attached to the string than my hands and arms, so that when it encounters extra resistance it just stops instead of breaking the string, and then maybe backs off a bit to extricate the string from the melt before trying again.
In theory the energy for making such a cut is strictly proportional to the volume of the cut, so as the cut becomes thinner, the energy to make it decreases proportional to the width. However, the force available decreases proportional to the cross-sectional area of the cord, which (if it stays round rather than becoming a strip like an actual bandsaw or even jigsaw) is proportional to the square of the width; so applying the same amount of power per unit material requires increasing the speed in inverse proportion to the thread thickness.
I was thinking that for straight cuts, slots, and non-through cuts, you might be able to use a paper or card disc in a sort of angle-grinder configuration — clamped somewhat close to the edge between two thicker discs to provide it with a sufficiently stiff forward force, to keep the paper from buckling too much. In this case, at a high enough speed (for a small enough disc) the centrifugal force of the paper itself would provide most of the pressure against the material.
Plumbers say PVC is rated for up to 154° F continuous service, which works out to be almost 68° in modern units and softens at 250° F (121°), but doesn’t melt until 360° F (182°); Wikipedia claims the Tg is 82° and the melting point can be as high as 260°. Probably somewhere in the middle is when it gets soft enough to be easily moved by the string.
The variability in PVC’s heat-stability is partly because it needs heat stabilizers to work at all; otherwise it starts to release HCl at only 70°, according to Wikipedia.
According to the plumbers’ discussion thread, its specific heat is 0.25 “Cal/°C/gm”, which I think means a quarter of that of water, working out to 1.05 J/K/g in SI units, pretty close to the 0.9 kJ/kg/K in Wikipedia. I think we’re talking about a ΔT of around 150 K, so around 150 J/g; due to PVC’s density of about 1.4 g/cc, this is roughly 210 J/mℓ. The cotton string I was using might have been 0.8 mm wide, the pipe was about 1.5 mm thick and about 20 mm across, and it took me less than a minute to cut through it at about 1 m/s of string motion at close to its breaking tension, which is about 30 N — about 30 W and about 30 seconds, so about 900 J. The annulus of π((10 mm)² - (8.5 mm)²) ≈ 90 mm² had a volume of about 70 mm³ and thus weighed about 70 mg or 0.07 g. So the heat needed to soften it adequately should be about 15 J, about 60× smaller than what was being applied. (More experienced people report shorter times to cut even thicker pipe.)
Presumably the extra heat was lost to inconsistent pressure, conduction into the plastic with its ≈0.2 W/m/K conductivity (in part because the contact area was large — cutting worked much faster once I had gotten through the first wall), heating of the air, and heating of the string. So probably using much higher power would have been more efficient; perhaps instead of 1.7% efficient we could reach 5% or 10%.
To be concrete, maybe 10 m/s would have been a better speed for the 800-μm string, cutting through the pipe in more like 3 seconds, or maybe 1 second if the hoped-for efficiency advantages materialize; if we were using an 80-μm cellulose string, we might prefer to run it at 100 m/s and only about 2–3 N; this would theoretically melt the same amount of material per second, but in an 80-μm-wide path, so it would use 0.3 seconds or 0.1 seconds to cut through the same pipe. This works out to about “500–1500 inches per minute” in archaic units, which is a quite fast cutting speed for panel-cutting machines like laser cutters, CNC plasma torches, CNC oxy-acetylene torches, and waterjet cutters.
How much efficiency improvement could we expect from cutting faster? A W is a J/s, so we have 0.2 J/s/m/K of conductivity. Presumably if we put the heat in in one-tenth as many seconds, we’ll have the heat spread out into the same set of nested cylinders it would have had if we had been cutting at the same speed in a material with one-tenth the thermal conductivity, 0.02 W/m/K. I don't know how to estimate how much this improves your efficiency.
Is it plausible to run an 80-μm cellulose thread at 100 m/s (220 mph in archaic units) and 2–3 N (7–11 ounces)? You could maybe use a 10-meter long loop of it at 10 Hz; it would be 50 mm³ and thus about 50 mg. 3 N could accelerate or decelerate it at 60000 m/s/s, so it could start or stop in 1.7 milliseconds at the edge of breaking. That sounds fast, but it’s 85 mm of travel at that speed, which is how much stretch would need to happen.
So that might be too much, but a slightly less ambitious goal is probably feasible, though.
Doing this in Styrofoam or other foams might be even more interesting, since there’s so much less mass, its thermal conductivity is so much lower, and the quality of a melted cut through Styrofoam is so much better than that of a cold cut, due to sealing off the cells.
You might think polyethylene would be much easier to cut this way, given its lower melting point, but in practice it’s more difficult; I think this is because it’s softer and has a higher specific heat (2.3 J/g/K, 2.5 times higher) and higher thermal conductivity (0.3–0.5 W/m/K rather than 0.2), but also because it lubricates much better, so much higher normal forces are needed to get the same frictional force. Presumably stepping up the power as outlined above would be sufficient to solve the problem.