CONTRIBUTION TO A LIQUID-LIKE THEORY OF RUBBER ELASTICITY .2. EXISTENCE OF A (LAMBDA(X)LAMBDA(Y)+LAMBDA(Y)LAMBDA(Z)+LAMBDA(Z)LAMBDA(X)) TERM

We first show that the classical gas-like theory of rubber elasticity is incorrect because the force-extension relation for an individual polymer chain is dependent on the interaction of its chain segments with segments of other chains in its neighbourhood. Two previous attempts by the author to formulate a liquid-like theory are discussed and improved. It is then argued that chain entanglements result in the value of the number of segments N in the expression f = 3k7x/N increasing with chain extension x or with increasing force f. The entropic contribution to the free energy of a rubber from this effect is then shown to be given by B(lambda(x)lambda(y) + lambda(y)lambda(z) + lambda(z)lambda(x)), where B is proportional to kT and to the number of chains per unit volume. In order to facilitate comparison with experiment, the stress-strain equations for simple extension, biaxial extension and for pure shear are displayed for three different choices of strain invariants: those of Rivlin, those of this paper, and those of the localization model of Gaylord, Douglas and McKenna. Comparisons with experiment are discussed.