I am sure that someone somewhere has done this although I don't know of it. I think that I would start off by experimenting and graphing the results. It should then be fairly easy to do a bit of curve fitting and see what comes out.

Just out of interest, some years ago myself and some colegues had the chance to drop a "super ball" down an empty lift shaft from various heights. Beyond a certain height that I do not recall, the ball did not bounce but shattered. This allowed us to work  out the maximum amount of energy that the ball could store.

With "bounce" You think of rebounce efficency, I guess?

This is the question to the reversibility of the bonce processes.

Yes processes, because there are two at least: compression

and decrompression of the air in the ball, and the deforrmation

of the rubber involved.

Any real deformation is partly irreversible, this is true for rubbers

as it is for steel or putty, the difference is just in the amount

of heat produced the bounce process instead of reacting

back purely elastic.

To know this for the rubber ball, You had to know some property

figures of the rubber (elasticity hysteresis of the rubber including

time dependence of the hysteresis, think of "Silly Putty") and

You need to know how big the deformation is, and how fast

all this happens.

The second part is the compression/decompression of the air.

Fast compression is nearly "adiabatic", this means the compression

(adiabatic means thermally perfectly isolated, the heat of compression

stays in the gas) is so fast, that the air will not heat up the ball or will

be heated by the rubber.

In that case the adiabatic gas equation applies.

But, as I wrote, only nearly adiabatic! The rubber wall will be heated

by the air somewhat, the more, the slower the bounce is. If slow enough, the heat

tranferred will flow back during expansion. This is the case for an

extremely slow "bounce". When the bounce is very fast, almost no heat

is transferred from air to the rubber, the bounce will be closer to ideality.

There is some intermediate bouce time, at which loss has a maximum.

(Same for the losses due to rubber deformation)

What do we learn: Nothing to write home about!

The higher the pressure, the shorter the bounce time and the

lesser the deformation of the rubber, the closer to 100 % will

be the efficency of the bounce. 

An exact calculation of this is very demanding, hysteresis figures

for the rubber will be not easy to get.

Georg

No, there is no simple single equation available off-the-shelf.

Yours is a complex system because the material is inhomogeneous [has vaying internal structure, unlike a solid rubber ball], the mechanics are non-linear [inelastic: energy is lost in the process], the geometry changes shape markedly during the interaction [dynamic, time-vaying].

You have two ways of building a model of the process. OK, maybe three. The first is empirical: just drop lots of balls under different conditions and take measuresments and plot the results. You will get a coefficient of restitution out of that, but not so much a single coefficient but one that changes with temperature, dot colour etc. You can then go on to empirically predict the outcomes for other situations by interpolation and extrapolation. This is a very good approach if you just want usefully accurate predictions.

The second is to use engineering analysis methods of finite element analysis and multiple mechanics stimulation. There is software to do this. It uses the basic mechanics, but solves it computatationally and puts the results out in ways that are easy to visualise what is happening. You would be surprised by the power of these methods. However the necessary software is not economically available outside of engineering, so this is not something you could do at home [unless you had a big budget].Essentially this method is numerical simulation that steps its way through small increments in time. Its limitation is that it's only as accurate as you are successful in capturing the underlying mechanics at the initial representation and set-up of the model. And while such a model does embody basic mechanics, it will not reveal to you the elegant equation describing the final solution, which your questions implies you desire. Nonetheless, this is how most engineering analysis is done, because it works sufficienty well.

The third is to prodeed along your current course of action, which is to assemble equations that describe the various parts of the problem. The independent equations then have to be inserted into each other to create an integrated mathematical model. This is necessary because the different physical phenomena of temperature, pressure, shape, etc., all interact with each other. Furthermore, the interaction is not just one instant and then its all over, but rather something that happens over some time. Therefore time will be another dynamic variable that will have to be included. It it likely that you could eventually write such a mathematical model, though you may have to make some simplifying assumptions along the way. Therein lies a limitation: This type of approach might look like it is more accurate than the other two above, but it tends to invariably involve quite serious simplifying assumptions, which degrade the accuracy. So an approach using pure physics and mathematics can result in disappointingly inaccurate results than don't stack up to reality. Another limitation is that the equations can often be impossible to solve explicitly, and in which case will require numerical solution, so the insight into the problem can be obscured at the very end anyway.

A fourth approach is to use elements of all the above. Do some simple mechanics with majorly simplifying assumptions so that a neat [but inaccurate] equation pops out at the end. In parallel make a whole lot of tests with real balls. Plot the data, and overlay the theoretical curve. Identify where there are major discrepencies, and create a fudge factor to tweak the theory curve into alignment. It doesn't sound sciency enough to call it a 'fudge factor', so just call it a 'coefficient' because it sounds much grander. A coefficient just rolls up a whole lot of uncertainty into one manageable ball. Engineers have been doing it for years, so there is plenty of proof that it works. And why stop at one? You can have lots of these coefficients, one for each little correction you need to make. A coefficient doesn't have to be a single constant either: it can be a function or a look-up chart. Understanding each of these coefficients then becomes a future research project or maybe even a whole PhD project!

Welcome to the exciting, but messy, world of generating new knowledge!