Theory of elasticity of polymer networks. The effect of local constraints on junctions

The effects of restrictions on the fluctuations of network junctions imposed by neighboring chains are treated by resort to a model in which these restrictions are represented by domains of constraint. Whereas in a phantom network devoid of such constraints the displacements of junctions from their mean positions are invariant with strain, involvements with neighbors in a real network render them dependent on the strain. The observed stress consequently is increased above predictions for the equivalent phantom network. On the plausible assumption that the dimensions of the domains are transformed linearly (affinely) by the macroscopic strain, the constraints become less restrictive in the direction of a principal extension λt with increase in λt. Hence, the relative enhancement of the stress diminishes with the strain. The characteristic departures of the observed tension (ƒ)-extension (α) curve for elongation from the expression ƒ α kT(α−α−2) of previous molecular theories are qualitatively reproduced by the present theory, which also accounts for the approximate constancy of ƒ/(α−α−2) observed in compression and the decrease of this ratio with dilution.