Autoguru;
Considering your initial question: the piece of 'something' you want to place in between these metal rods, is it going to be constrained, or do the sides have an opportunity to 'bulge out', if you will? Do you want to adhere it to one of the rods (so there is rubber on metal friction after contact), or both sides (so there is rubber on rubber friction after contact)?
Which material works best depends on how exactly you're going to compress the material. If you need something to retain its shape well, silicones do just that. They can take huge stresses without growing cracks, and once you unload, they will recover very much to their initial shape. So, if you load it repeatedly, this would be a good option (think about strain offset / Mullins effect). "Yield" is the wrong term to use when dealing with rubbers, because a truly elastomeric material does not "yield" one bit, it recovers fully (the use of the term "yield" is how us rubber people recognize the plastic and metal people).
Silicone may not be very good, though, if it's constrained on all sides, since you're essentially compressing the material volumetrically / hydrostatically. Look for materials that have a very stiff bulk modulus, in that case (in other words, materials where the Poisson's Ratio is approaching 0.5).
Also, at a loading rate of 0.5 m/s, a piece of rubber of about 0.2 mm thick will need some serious damping qualities. I could imagine you wanting to do some storage modulus tests on some likely candidate materials. No particular materials come to mind, but if you google search for a paper called "measuring the dynamic properties of elastomers for analysis", the first hit describes an excellent dynamic test (not sure if I'm allowed to put the link in a post directly).
Of course, a simple high rate compression test of the candidate materials would get you quite a long way. I know there are some material testing labs out there for whom 0.5 m/s is quite easy, and some of the good labs can actually measure stress and strain quite accurately on the specimen.
Finally, the friction matters because the more friction is going on at the contact area (rubber-metal, or rubber-rubber), the more shearing is going on in the specimen (meaning, in turn, the more 'damage' you're doing to the material. You could / should consider lubricating the contact areas so as to minimize the shearing (but, if you use adhesives to stick the rubber to one of the rods anyway, you will have shearing there anyway).
Bottom line is, look at some of these variables and pick a couple of materials to try out; it sounds like you have the luxury to do so (without ruining your metal rods etc.). If not, you could still get the material samples tested on some or all of these criteria you find important, and judge based on those values.
hope this helps!
Ron
