A new physics-motivated constitutive model of hyperelastic polymer networks

This study is motivated by a question on the necessity of root mean square approximation, which is widely used in existing physics-based constitutive models of hyperelastic polymer networks. In this work, we developed a new constitutive model that reduces the classical two-step approximation to a simple one-step mean field approximation in the micro-macro mapping of the chain stretch free energy density function. Correspondingly, we modified the classical free energy function of the inverse Langevin chain on the microscopic level and validated this modification via a Monte Carlo simulation. A semi-empirical non-affine tube constraint is introduced as the correction term to improve the model prediction under multi-axial loading scenarios, which has long been a challenge in the study of general constitutive models for polymer networks. The current model is validated with various materials, including vulcanized natural rubber, Entec Enflex S4035 TPE, Tetra-PEG, and isoprene rubber vulcanizate, under multiple biaxial loading profiles. The results are compared with two successful physics-based models (extended eight-chain model and extended tube model) with the general biaxial loading data of isoprene rubber vulcanizate. The comparison shows that the current model has a compatible phenomenological accuracy without sacrificing the physical interpretations of parameters or changing the mathematical form of the constitutive relation at the asymptotic limit of the fitted parameter. Further, we underscore that a physics-based model containing chain stretch and interchain tube constraint shall include at least four parameters, of which two control the magnitudes and the other two represent the changing tendency of microscopic effects against macroscopic deformation. At the end, we demonstrate how two dimensionless parameters n and q influence the shape of the macroscopic stress-strain curve via a parametric study.