The velocity threshold is a proxy for energy. It takes energy to cause a material to deform. The amount of energy required to deform a given material can be determined by integrating the area underneath its stress strain curve. Once this value is known if you can divide by the mass of the projectile and take the square root of that value to determine the minimum velocity required for expansion.
This doesn’t tell the whole story as the bullet imparts energy into the object it is striking. This imparted energy is referred to as “work” in physics and can be defined as the change in kinetic energy (projectile mass x velocity squared). More “work” roughly translates into a bigger wound channel. Shooting over a chronograph is a lot easier than whatever contraption you’d need to measure the energy of a bullet, so that’s the number you see published.
FWIW, the US military uses ft-lbs to gauge whether or not a projectile will incapacitate a soldier, not velocity. I believe they’ve determined that 350 ft-lbs will incapacitate a soldier 50% of the time.