Accurate constitutive models for rubber-like materials based on statistical mechanics

Although the statistical mechanical basis for the elasticity of rubber-like materials is well-understood, early constitutive models for these materials were unable to describe satisfactorily experimental data from multiple deformation modes of these materials. The various models that improve on these results achieve more accurate fits by increasing the number of fitting parameters calibrated using experimental data. We instead propose new constitutive models for rubber-like materials by revisiting the relationship between the imposed macroscopic deformation and the microscopic polymer chain stretch. Using the Irving-Kirkwood-Noll procedure to connect continuum fields and statistical mechanics, we construct a Cauchy stress tensor from forces along polymer chains. By imposing a consistency relation on this Cauchy stress, we obtain a new micro-macro chain stretch relation involving the exponential function. We also propose a hybrid chain stretch by combining our new chain stretch with the well-known affine stretch. Comparison of model predictions to experimental data in the literature shows that our two micro-macro chain stretch relations result in new two-parameter constitutive models that outperform previously proposed two-parameter models, achieving good fits to data from multiple deformation modes. Therefore, our physics-based approach to constitutive modeling has improved accuracy without increasing the number of fitting parameters.