SR Time Dilation in Rigid Structure Clocks Special relativity says that all clocks will show same time dilation, irrespective of clock mechanism. But Time period, T, of a clock is a formula that must continue to hold even if time dilates. Let us look at a tuning fork clock. When T in other frame changes due to relative motion of clock, the value of physical quantities in other frame must "somehow" then work out to give this T.
Formula for frequency of a tuning fork clock is:

where
l is the length of the prongs.
E is the Young's modulus.
I is the second moment of area of the cross-section.
ρ is the density of the material.
A is the cross-sectional area of the prongs.
We have l, A and I which are affected by Length contraction and for these we can do an exact calculation. E, Elasticity and ρ, density remain. For E - no nice formula. ρ, density = mass/volume, we could have used relativistic mass, and for relativistic volume we could use the cuboid prong of the tuning fork being contracted. But is that correct for ρ under special relativity?
E and ρ remain. So by putting in all the rest, we have a nice formula connecting E and ρ for relativistic speeds. Would this be correct methodology that yields a new formula, and if not why not? In classical physics there is no such suggested connection between E and ρ.
In a rigid structure clock such as a tuning fork clock, do we not have a paradox of sorts?
 A: My advice is to set aside slogans like "moving clocks run slow" or (worse still) "clocks in different frames of reference tell different times". They help only if you know how to interpret them. Time dilation is this:
The time interval between two events, measured in a frame of reference in which the events occur in different places (requiring two synchronised clocks to measure the interval) is greater than the time interval between the same two events in the frame of reference in which the events occur at a single place (and therefore only one clock is needed).
It's not clocks in general in one frame that we're comparing with clocks in another frame. Nor are we comparing one clock in one frame with one clock in another frame. And as the paragraph in bold makes clear (I hope) the clocks' spatial positions within their frames in order for them to be present at both events is what matters; space and time interact. Any attempt to compare clock mechanisms in different frames misses the point. Sorry if this was not the sort of answer you wanted.
Maybe we can look at it this way. The clock whizzes past us in the laboratory. Two successive ticks are registered by sensors in the lab at the places where the ticks occur, and the (dilated) time interval between the ticks is calculated from subtraction of readings of recording clocks at these places. We find also, that the simultaneous positions (in the lab frame) of the front and back of the clock are separated by a shorter distance than in the clock's rest frame. But the length contraction we observe in the lab frame doesn't affect the clock's ability to keep time! The clock hasn't been 'affected', it's just that in the lab we're measuring the time interval between events (ticks) in different places, just as we have to measure the length of a moving object differently from the way we would measure a stationary object, because we have to be so careful about simultaneity!
