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In YouTube I have seen various videos where someone cut with a sword into hard objects and has cut that into pieces. For example:

https://www.youtube.com/watch?v=EVCWGwvctt4

Given the mass, the velocity and the material of the cutting blade, how I can compute whether some material is cut by it completely or not? Which basic equations should I use? Are there theories that model mathematically such things?

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Real-world engineering example

To my knowledge there are no simple analytical methods of determining whether an object such as a sword can cut cleanly through any arbitrary object. In manufacturing engineering we would rather cut many small pieces of material than take one big cut (i.e. we generally use saws to cut material instead of giant guillotines). The exception would be for thin objects like sheet metal, which is cut with large punches or shears, as pictured below. Shearing operation: (a) side view of the shearing operation; (b) front view of power shears equipped with inclined upper cutting blade. Image Source: Mikell P. Groover, Fundamentals of Modern Manufacturing, 5th Ed. pp. 483

What you will notice about the shearing operation is that it requires an "ideal" shearing case. The blade has a very small amount of clearance between it and the die (light grey support) to ensure that the material fails in shear and not in bending. Furthermore, the blade is angled so that it minimizes the cutting force by not cutting the entire stock material at once. So the chances of encountering this sort of situation with a sword is unlikely. However, if you had this situation, here are the equations you would use.

The recommended clearance, $c$ is calculated using: $$c = A_ct $$ Where $A_c$ is the clearance allowance and $t$ is the stock thickness (thickness of the cut material). $A_c$ can vary from 0.045 for very soft materials (i.e. aluminum) to 0.075 for relatively hard materials (i.e. stainless steels).

The cutting force, $F$, can be calculated using: $$F = StL $$ Where $S$ is the shear strength of the material and $L$ is the length of the cut edge. This equation assumes you will cut the entire length all at once, so you can apply a factor to reduce this amount if you have angled cutting like in Figure 19.3 above.

Unfortunately it's not that simple for swords

In these types of operations we would generally apply high, constant forces at low speeds. This is very different from a sword, where you would likely be moving at higher speeds and applying time-varying forces (i.e. impact). Furthermore, the blades used in shearing operations are usually much thicker than the objects that they cut, so we can mostly ignore the stresses and deflections in the blade. This would not be the case for, say, a sword cutting through another sword, or armour. The relative hardness and fracture toughness of the materials, crack propagation, etc. would likely be relevant as well.

If you're looking to do calculations for fun anyway

If you're just doing these calculations for fun, not for serious engineering design, here's my advice. Make some assumptions (i.e. ideal shearing case) and use the cutting force equation above to check what the load on the sword would be. Assume that the force is a point load placed at the middle of the sword, apply a safety factor of 5 and check whether the sword would fail due to bending. If it doesn't fail then you could say the sword would "likely" be able to chop cleanly through the material. Then check whether or not a human could actually apply that force at that point on the sword. If not, then you are free to imagine the sword being wielded by some kind of super-strong android samurai.

...Wait a sec, you aren't building a super-strong android samurai are you? XD

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