The behavior of rubberlike materials when stretched

Abstract

Kuhn's method of computing the relationships existing between stretch, tension and double refraction in rubber-like substances has recently been extended by Wall, by Treloar and by the author. These treatments assume randomly kinked structures consisting of an (effectively) infinite number of chain-elements. In Kramers' method no assumption regarding the number of chain-elements is made. The present paper attempts to apply Kramers' theory to (1) long-chain “zwitterions” in an electric field and (2) a network of macro molecules in unidirectional extension. It is shown that the earlier theory of rubber elasticity is a result of the present one within the limit of a very large number of chain-elements per molecule, or a very small stress. It is found that the stress-strain diagram of rubber conforms closely to the requirements of the theory. A few results for molecules consisting of 1, 2 or 3 chain-elements, together with some graphs, are added with a view to their application to cellulose in the near future. The Editor of this journal kindly drew my attention to a paper of James and Guth (11), whose treatment of rubber elasticity has some features in common with the method developed in the present article. These authors have also used a potential energy of the type given in equation (14), which leads to the introduction of an exponential factor exp(λ s), (compare equation 16). In particular, they also point out the analogy with electric or magnetic dipoles in an electric or magnetic field (compare section on zwitterions). On the other hand they neglect the restriction: s > 0 in the integrations (equation 16), which accounts for the situation represented by Fig. 1. This does not affect the results obtained at high values of the stress and, in fact, at advanced degrees of stretch both the James-Guth theory and the present one approach the following result (using the symbols of the present article): v v max = Cth λ − 1 λ . A further common feature is the fact that the final stress-strain relation contains two arbitrary constants, one dependent on the total number of junction points ( G in equation 13) and one depending on the maximum extensibility ( N in, for example, equation 24). A somewhat unsatisfactory feature of James and Guth's treatment is the one-dimensional character of their considerations. On the other hand their final formula is simpler than ours and their highly interesting analysis of the network structure may eventually be helpful in an attempt to obviate the introduction of the overall force p.