On a molecular statistical basis for Ogden's model of rubber elasticity

Abstract

In this paper, a link is established between the statistical theory of long chain molecules and Ogden's phenomenological model of rubber elasticity. It has been shown by several authors in the past that many invariant-based phenomenological models for rubber-like materials are related to the classical statistical theories. The essential means to reach this reconciliation were methods to account for a non-affine deformation of polymer chains in the network, appropriate techniques to calculate their averaged response, and an approximation of the inverse Langevin function appearing in the non-Gaussian statistical theory. It is shown in this paper that the very same approach, if appropriately implemented, allows to express the strain-energy function of Ogden's material in terms of physical constants characterising the polymer chain and network, together with few additional parameters that account for the non-affine deformation of the polymer chains. Particularly, it is shown that Ogden's model can be represented as a non-affine non-Gaussian 3-chain model with topological constraints.