Jigsaw blades break a lot. In a sense that’s because the stroke of the saw is greater than the elastic limit of the sawblade material. But this is entirely avoidable.
If the stroke of the saw is too short, it won’t cut, because all of the motion will be taken up by the elastic deformation of the sawblade and the workpiece. This is how saws for removing plaster casts avoid cutting skin: their stroke length is shorter than the skin’s elastic limit. But the saw blade is typically much, much longer than the distance the chips from the workpiece have to move to detach from the workpiece; typical numbers might be 100 mm and 100 μm. The elastic limit of a hard steel might be ½% permitting about a 250-μm stroke without risk of breaking the blade, which is plenty to cut the workpiece; if this is not long enough, the jigsaw can be built bigger.
It’s also necessary for the stroke length to be larger than the tooth size, or each tooth will cut a separate hole, rather than joining the holes together into a slot. A higher movement frequency can be used with smaller teeth and the same total material removal rate; moreover thinner teeth and the elimination of the breakage risk sometimes permit using thinner blades and thus lower total power; but sometimes this undesirably reduces the achievable kerf curvature.
A typical electric jigsaw blade might move 1 m/s at 50 Hz. The speed of sound in steel is about 4 km/s, so a 100-mm-long stretched-tight steel jigsaw blade will move more or less as a rigid body at frequencies below about 40 kHz. Moving 250 μm twice per cycle at 40 kHz would be 20 m/s, so such an ultrasonic jigsaw could probably cut at a higher speed than a regular jigsaw without risk of breaking the blade, at least if there’s some way to clear the chips. If the blade is 100 μm square, like one of my beard hairs or these hair-fine copper wires I’ve been trying to solder with, and has an extra 50% of non-tensile-load-bearing mass of teeth on one side, it weighs 120 μg/mm and so has a total mass of some 12 mg. Accelerating it by 40 m/s in half of a 40 kHz would require 3.2 Mm/s/s, or 330 thousand gees of acceleration, which works out to almost 40 newtons with this mass, thus a stress of 40 MPa at the pulling end, about 4% of the strength of steel.
This kind of sounds like an ultrasonic cheesecutter that can cut through brass, mild steel, glass, bone, fingernails, hard plastics, fired clay, concrete, and maybe wood and granite, but not actual cheese as such, or your skin, or turkey.
Other possibilities to alleviate these compromises include using a blade with omnidirectional teeth (for example a single helical tooth, like a buttress-thread screw), which has no minimum kerf curvature radius and can also be rotated between cutting strokes; mounting hard teeth (whether high-speed steel or something like tungsten carbide) on a softer blade that can stretch further; and force feedback through electronics that stop pulling on the blade when overload is detected; or coupling the saw frame to the jigsaw blade through a lightweight spring that limits the force over the saw’s normal stroke. But I’m kind of excited about this ultrasonic cheesecutter thing.
To actually make it work you probably need synchronized but mechanically weakly coupled pullers at the two ends of the sawblade, like a two-man sawing team, rather than hoping the saw frame will move as a sufficiently solid body. By controlling the amount of slack with some sort of feedback, they ought to be able to keep the tension on the blade relatively constant. Strain gauges in a lightweight saw frame occur to me as one possibility.
The mechanical power going into the wire is about 20 m/s · 40 N = 800 W, but almost all of that is being transmitted from one sawblade puller to the other over the wire, then returned 25 μs later; only a small amount of it goes into the workpiece being cut. You’d still probably have to water-cool it.
An interesting feature of this device is that, because it runs at 40 kHz, its cutting action should be uncannily almost silent.
A vibrating engraver or scraper that works at scales, frequencies, and powers like this, rather than the usual 50 Hz or so, would also be very interesting. It could push its hardened tip into the workpiece as per normal.