A statistically based strain energy function for polymer chains in rubber elasticity

A statistically based strain energy is proposed for rubber-like materials at large stretches. It is based on the micro-mechanically vectorial modeling of a single polymer chain, and its entropy is used in order to account for the entropic elasticity of rubbery macromolecules. We propose a framework for derivation of a microscopic free energy function based on a multidimensional form of a generic normal (Gauss) probability distribution function (pdf). Homogenization of the microscopic free energy by means of statistical tools renders a macroscopic free energy. The random variables of the general formulation are specified as bond angle differences, representing bending and torsion, respectively, for each bead of the single chain. A further step is a formulation of both quantities in terms of the applied stretch, which eventually renders the macroscopic strain energy as a hyperelastic energy function. Additionally, we propose a methodology to satisfy a normalization condition for the related integral of the pdf over the constraint statistic domain. A numerical example illustrates the capability of the proposed energy function to simulate the S-shape behavior of the well-known experimental data for vulcanized rubbers by Treloar (Trans Faraday Soc 40:59–70, 1944).