Yield strengths and ultimate tensile strengths cover a similar-sized, but lower and narrower, range than Young’s moduli. They generally correlate, although there is substantial deviation — the plastics have immense deformations at break, the metals smaller, the ceramics smaller still.
Here’s a table of very approximate quantitative figures:
| | Young’s | yield | tensile |
| | modulus | stress | rupture |
| | (GPa) | (MPa) | (MPa) |
| diamond | 1000 | brittle | 60000 |
| tungsten carbide | 600 | brittle | 300 |
| sapphire | 400 | brittle | 400 |
| carborundum | 400 | brittle | 120-3000 (?) |
| cubic zirconia | 200 | ? | 700 |
| A36 steel | 200 | 250 | 500 |
| zircon | 200 | brittle | 300 |
| tooth enamel | 80 | brittle | 20 |
| soda-lime glass | 70 | brittle | 100 |
| 6061 aluminum | 70 | 76-370 | 130-410 |
| quartz | 80 | brittle | 40 |
| concrete | 25-50 | brittle | 4 |
| lead | 14 | creeps | 15 |
| wood (along grain) | 9-14 | brittle | 5 |
| poly(methyl methacrylate) | 3 | brittle | 70 |
| poly(ethylene terephthalate) | 3 | ? | 70 |
| gypsum plaster | 1.4 | brittle | 3 |
| high-density polyethylene | 0.8 | ? | 20 |
| styrofoam | 0.005 | brittle | 0.4 |
Critters (the technical term) use teeth, claws, and beaks to cut things, maneuvering them into position with softer tissues. A smallish tooth can have an even smaller point that digs into the material to cut, supported by a wider base rooted in a “handle” of material softer than the tooth itself, which is held in still softer material.
Machines can use the same technique, and sometimes do: lathes, fly cutters, boring bars, shapers, and D-bits all cut with a single point, often made of a cermet; it’s common to dress grinding wheels with a single-pointed diamond mounted at the end of a steel dressing tool; a woodworking plane commonly uses a steel blade held in a wooden frame; and a hand file commonly consists of case-hardened steel teeth on the surface of a piece of softer steel, held in a softer wooden handle, held in a still softer hand.
For example, you could imagine a tungsten-carbide tooth (sometimes these are called “teeth”, other times “tools”, “bits”, or “inserts”; analogous artifacts in archaeological contexts are called “microliths”) shaving a 100-micron-thick, one-millimeter-wide shaving (“chip”) as it scrapes along a steel surface. WC (the most unfortunate chemical formula ever) is several times stronger in compression, some 1500 MPa, but suppose we limit ourselves to its 300 MPa tensile strength; then the tungsten carbide will keep cutting as long as the force is less than 30 N. This is enough to get the steel to yield so that the carbide can propagate a crack under the chip. (Carbide’s higher compressive strength clearly helps a lot here.)
The carbide can be held in a softer material such as steel or even aluminum; to keep those 30 N under the 76 MPa worst-case yield stress of the aluminum, we need at least 0.4 mm² of contact area between the carbide and the aluminum. So the carbide tooth itself could be tiny, with a 100-micron-long, 1-millimeter-wide point, supported on an 800-micron-tall pyramid with an 800-micron-diameter circular base. In fact, at such a low stress, even PMMA and PET would be strong enough not to rupture, although they would certainly creep; a more conservative approach would be to use a truncated aluminum cone with, say, an 800-micron circular tip, 3 mm height, and a 3-mm-diameter circular base, supported on wood, HDPE, PMMA, PET, or many other possible materials.
It probably isn’t practical to cut most steels with something much smaller than that tooth, because the steel is too ductile; you’ll end up just forming the steel instead of cutting it. The surprising thing is that 0.2 mm³ of tungsten carbide, about π milligrams, is sufficient to enable cutting steel. 200 surface feet per minute (in the medieval units commonly used in machining in the USA) is probably achievable, which works out to 1.02 m/s in SI units, so this is a material removal rate of some 102 mm³/s of steel, about 0.8 g/s, removing the mass of the carbide tooth itself roughly every 4 milliseconds.
Assuming a 15 minute tool life, this means that this tooth can remove about a quarter of a million times its own mass in steel during its lifetime.
A single gram of tungsten carbide contains enough material to make some 300 such teeth.
It is possible to substitute an intermediate-hardness material for the base of the tooth. Steel, a harder aluminum, or yttrium-stabilized zirconia would work. (You could try zircon, but I suspect it would be too fragile.)
As economic context, here in the kitchen I have a cheap zirconia kitchen knife that’s about 100 mm × 25 mm × 1.5 mm, which is about 2800 mm³ of zirconia, enough for some 14000 such teeth; such a knife costs some US$7, about 0.05¢ (US$0.0005) per tooth. I also have a high-speed steel hacksaw blade, which is about 310 mm × 12.7 mm × 600 μm (300 mm between the mounting-hole centers), about 2400 mm³, which was even cheaper (about US$1.50 per blade), and is also suitable for cutting unhardened steel.
Traditionally, steel was cut merely with case-hardened steel, but this has its limitations. 19th-century advances in steel improved tool life considerably, but today steel is usually cut with ceramics, especially the tungsten carbide mentioned above.
Three other hard ceramics mentioned above — sapphire, carborundum, and zirconia — may be more easily produced from terrestrial materials than tungsten carbide. Tungsten atoms are outnumbered by silicon in Earth’s crust about a million to one. Sapphire is made from aluminum and oxygen; aluminum is very nearly as common as silicon, while oxygen is even more common. Carborundum is made from silicon and carbon; carbon is outnumbered by silicon only about 300 to 1, and of course diamond is entirely carbon. Zirconium is outnumbered by silicon about 3000 to 1, making it about 300 times as abundant as tungsten. Zirconia is quite brittle if not stabilized with, for example, 2–3% of yttria, but fortunately yttrium is only about one order of magnitude rarer than zirconium itself.
Zirconium has the additional merit of being relatively easy to concentrate, since it forms separate grains of zircon (jacinth, ZrSiO₄) in many igneous rocks, including most granites, which are easily separated from sand by their high density (4.6 g/cc); they are also separable by their refusal to melt at any reasonable temperature (below 2500°, though they sinter at much lower temperatures). Zircon itself, perhaps even naturally-occurring crystals, may be usable as a material for cutting metal; but zirconium is readily, if expensively, derived from it by calcining with carbon and chlorine, then reducing the resulting zirconium tetrachloride with magnesium, the same Kroll process used to reduce titanium; and zirconia is superior to zircon in almost every way.
Historically carborundum was discovered by Acheson running an electric arc through a mixture of clay and coke in an iron crucible, insulated by the granular materials themselves, in 1890; but Despretz probably made it without knowing in 1849 by joule-heating of a carbon rod embedded in silica sand, which is essentially the “Acheson process” used today; sawdust and salt are former additives now little used. (This is also the first process for making synthetic graphite, by heating the carborundum until the silicon sublimes at 4150°. XXX wouldn’t graphite sublime too? Shouldn’t that be 2830°?)
Sapphire is normally refined as an intermediate step in the production of aluminum, for example by the Bayer process: low-silica bauxite is digested in 170°–180° lye (or anhydrous 1200° sodium carbonate and coke, in the Deville process) to obtain sodium aluminate, from which crystalline aluminum hydroxide is precipitated (by cooling, by neutralization with CO₂, or simply by evaporation with seed crystals).
Zirconium carbide can be made simply by carbothermic reduction of zirconia with graphite; it is even harder than zirconia itself (?), though it has “poor oxidation resistance over 800°”.