Some Refinements to Rubber Elasticity

The statistical theory of rubber elasticity has undergone significant and continuous refinement, resulting in a series of correction terms. These are sometimes omitted and sometimes included in scientific and engineering research, as the need for them arises. In this section we briefly consider some of these. 

The Gaussian statistics leading to equation (9.34) are valid only for relatively small strains—that is, under conditions where the contour length of the chain is much more than its end-to-end distance. In the region of high strains, where the ratio of the two parameters approaches j to j, this limit is exceeded. 

Kuhn and Grtin (65) derived a distribution function based on the inverse Langevin function. The Langevin function itself can be written 

The stress of an elastomer obeying inverse Langevin statistics can be written (42,63) 

At intermediate values of a (and hence of n l), equation (9.70) predicts a sharp upturn in the stress at a s greater than 4, as observed in experiments. Because of the complexity of equation (9.70), the Gaussian-based (9.34) is preferred where possible. 

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Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

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