Interaction between energy and entropy in highly elastic polymers?
If I remember correctly, the relationships you have presented are related to the configurational entropy of the polymer chains between cross links of the polymer network.  In these equations, R is the spatial distance between cross links, N is the number of chain segments between cross links, and l is the length of each chain segment.  The smaller the value of R, the greater the number of configurations that the chain can exhibit (i.e., the greater the entropy).  Thus, I believe that there should be a minus sign in your equation for S, because, as R increases, the fewer the number of configurations that the chain can exhibit, and thus the lower the entropy.  The parameter you call U is, I believe the Helmholtz free energy, and is a measure of the stored elastic energy of the polymer network.  As the R gets longer, the polymer chains have been stretched more, and more elastic energy is stored in the chains.  Conceptually, the polymer chains are like springs between the cross links.