Let us consider a bike wheel. Does anybody know a way to calculate how much does rolling resistance changes if the inner tire volume is filled with foam instead of air? Let's assume a foam hardness which equals a 2 bar inflation.
Even with a finite-element tool kit you would need a lot of physical data in order to be able to calculate the RR. In addition to data for the tyre casing and rubber you would need the visco-elastic properties of the foam. Deformation of a foam will certainly absorb more energy than deformation of an air chamber, hence result in higher RR.
There is a kind of foam tyre made of closed-cell polyurethane foam sold under the trademark name Amerityre Flatfree. You can find on the internet an investigation of this tyre by Thierry Larose Chevalier et al from the University of Ottawa. http://www.flatfreetire.ca/Ottawa%20U.pdf
The results of this study are a bit ambiguous. Based on in-shoe force measurements from cycling on training rollers the authors state that there is no significant difference in RR with a traditional clincher road tyre. However, the participant cyclists subjectively commented that they felt more resistance.