Energy functions for rubber from microscopic potentials

The finite deformation theory of rubber and related materials is based on energy functions that describe the macroscopic response of these materials under deformation. Energy functions and elastic constants are here derived from a simple microscopic (ball-and-spring) model. Exact uniaxial force-extension relationships are given for Hooke's Law and for the thermodynamic entropy-based microscopic model using the Gaussian and the inverse Langevin statistical approximations. Methods are given for finding the energy functions as expansions of tensor invariants of deformation, with exact solutions for functions that can be expressed as expansions in even powers of the extension. Comparison with experiment shows good agreement with the neo-Hookean energy function and we show how this derives directly from the simple Gaussian statistical model with a small modification. © 2007 American Institute of Physics.