My flesh has a Young’s modulus much less than that of steel, but by holding a file or saw in my hand I can cut steel. The force on my flesh is the same as the force on the steel — except for a nearly irrelevant term of tool weight — but it is distributed over a larger contact area.
Similarly, I think we can produce plastic machinery that can cut harder materials, such as steel, by holding small “tools” or “teeth” of ceramic or hardened steel. The small tools might not themselves be of particularly ideal geometry — perhaps they are the results of crushing a zirconia knife or a silicon wafer with a hammer, for example — if the plastic machinery and control system is adapted to the particular geometry they have, if it uses sufficiently low cutting forces, and if the geometry has sufficiently sharp points (though points that are too sharp will slow down cutting).
The first part of the idea is simply that the rigidity of the connection between the tool and its holder should be at least comparable to, and ideally exceed, the rigidity of the contact between the tool and its work — each of which is a Young’s modulus multiplied by a contact area. So if, for example, the tool holder is nylon or polystyrene, with their Young’s modulus around 3 GPa, while the workpiece is a titanium alloy with a Young’s modulus around 110 GPa (steel is around 200 GPa), the contact area on the workpiece needs to be more than 40 times smaller than the contact area on the tool holder — ideally more like 400.
That’s actually the less important criterion, though; it’s actually possible to cut with a tool held by a tool holder that deflects more than the material does, although it’s going to need complicated control algorithms to get decent precision that way. The more important criterion is that the tool holder’s yield strength needs to be greater than the ultimate strength of the cut surface — the first being the yield stress multiplied by the contact area, the second being the ultimate stress multiplied by the tearing area. Titanium’s ultimate strength is 900 MPa with 6% aluminum and 4% vanadium; A36 steel’s is only 400 MPa. Meanwhile 6–6 nylon’s yield stress is around 45 MPa and polypropylene’s is around 12–43 MPa.
So, even though the strength criterion is more important than the rigidity criterion, you can meet the strength criterion easily if you can meet the rigidity criterion, because common materials vary much more in elasticity than they do in strength.
For cutting softer materials like wood, bone, or fingernails, broken glass teeth would work fine.
Suppose that the tooth is silicon carbide, with its Young’s modulus of around 450 GPa. (I don’t know what its ultimate strength is, but I don’t think it will often come into play here, since I assume everything else around it will break first under the desired low-shock conditions.) Suppose you’re pushing it into A36 steel, with its 200 GPa modulus and 400 MPa ultimate strength, and that the splitting part of the metal is comparable in size to the contact area, which is, say, about a 1 mm circle, 0.79 mm². To get the steel to cut, you need a force of 314.159 newtons, which will also have compressed the steel immediately around the cut by 0.2% and the tooth by 0.1%. If your holder is 6–6 nylon, then in order to not yield at 45 MPa, it needs a surface area of 7 mm² pressing on the root of the tooth, and that plastic will squish by 1.5%; to squish by less than 0.2% it would need to be 52 mm², an 8-millimeter-diameter circle — a rather large chunk of carborundum! The parts of the workpiece and the tooth holder (“gum”?) further away from the tooth will be resisting the same force over a larger cross-sectional area, unless the workpiece is very small, so the deformation will be less.
These rather demanding dimensions for the tooth and tooth holder can be improved by using an intermediate material between the plastic and the ceramic, such as brass, aluminum, or steel, into which the tooth will be set in order to be grasped by the plastic. In the above example, the 8-mm-diameter circle of plastic could be grasping a chunk of aluminum, brass, or steel, which in its turn grasps the tooth itself, and is perhaps brazed or soldered to it.