Standing elbows are impressive when they land, but I do agree in the sense that elbows on the ground shouldn't be allowed. Sure, it's a legitimate form of martial arts, but so are several other techniques that are illegal in MMA.
When you break it down, elbows on the ground are basically dudes taking almost all the force of a punch and directing it into a much smaller area.
To get a little more scientific about why they cause cuts, the mathematical formula for pressure is P=F/A
Pressure = Force / Area
There's no distinct ratio for the size of the point of an elbow to a fist, but let's say it's 1:5 for example, that is, that the point of an elbow is five times smaller than a fist.
Taking that back to the equation, that would mean that if a fighter maintained the force of a punch while throwing an elbow, the pressure would be five times larger with the elbow.
It takes under 500 pounds per square inch of pressure, on average, to break skin. The average boxer can create around 1000 pounds per square inch of pressure in their punches, so if an MMA fighter hits as hard as a boxer, they could potentially break the skin with elbows 100% of the time if they connected properly.
All that being said, it's much harder to connect with elbows on the feet than on the ground, so it should be pretty okay to use them standing up. In all the fights I've seen where elbows aren't allowed on the ground, they're all more technical in terms of grappling, and they're all usually more entertaining than those where elbows are allowed (and utilized) on the ground. So basically, elbows shouldn't be allowed on the ground, at least not to the head/face of an opponent.
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