In his book, Relativity: The Special and General Theory, Einstein claimed that the clocks located on a rotating disc run slower solely due to their tangential velocities as appear in the Lorentz factor, no matter how much acceleration they take. He then replaces potential per unit mass with velocity square $(r^2\omega^2)$:

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by $\phi$ , i.e. the work, considered negatively, which must be performed on the unit of mass against the centrifugal force in order to transport it from the position of the clock on the rotating disc to the centre of the disc, then we have

$$\phi=\frac{\omega^2r^2}{2}$$

However, I cannot really understand why the centrifugal acceleration does not affect clocks at all. Assume we have two concentric rings one with a large radius and the other with a very small one. If the rings rotate at the same tangential velocity, according to Einstein, the clocks run slower at the same rate as measured by an inertial observer at rest with respect to the plate's center. However, according to the centrifugal acceleration formula:

$$a=\frac{v^2}{r}\space,$$

the clock on the ring with a smaller radius experiences much more acceleration than one located on that with a larger radius. How can it be possible that such a large centrifugal force/acceleration, which can easily mash the nearer clock to the center of rotation (if the radius is small enough), is ineffective in altering time rates? (Forget about the viewpoint of the rotating observers.)

Remember that if the radius approaches zero, the centrifugal acceleration tends to infinity, yet the tangential velocity can remain unchanged. It is really hard for me to understand why an infinite acceleration/force cannot affect clock rates!