A new physics-motivated constitutive model of hyperelastic polymer networks

This study is motivated by a conceptual inconsistency in the physical interpretation of eight-chain hyperelastic theory, which arises from the combined effect of two distinct issues: the use of the marginal projection distribution p z ( | r z | ) as a surrogate for the full probability density of end-to-end distance p r ̄ ( r ̄ ) , and the subsequent reliance on a root mean square (RMS) approximation step in the micro–macro averaging of chain stretch. We first revisit this probabilistic mismatch by reformulating the probability density function of freely-jointed chains (FJCs) in terms of the squared end-to-end vector r 2 , thereby restoring consistency on chain-level statistics. Building on this formulation, the micro–macro mapping averaging of chain conformational free energy is constructed directly in terms of r 2 , leading to a one-step mean-field approximation that avoids RMS averaging. The modified probability transformation is examined by Monte Carlo sampling at the microscopic level. To account for interchain interactions, q -mean statistical description of micro tube confinement was incorporated, leading to the appearance of the general invariant I q = λ 1 q + λ 2 q + λ 3 q . The resulting continuum constitutive model is assessed against multiaxial experimental data for several polymer networks, including vulcanized natural rubber, Entec Enflex S4035A thermoplastic elastomer, Tetra-PEG, and isoprene rubber vulcanizate. Comparisons with three existing hyperelastic strain energy formulations, the extended eight-chain, extended tube models, and the four-parameter ”comprehensive” model, demonstrate comparable phenomenological accuracy of the current model while providing a clearer and more consistent micro–macro physical interpretation of model parameters. A parametric study further illustrates how the dimensionless parameters n and q govern the shape of the macroscopic stress–strain responses. The present formulation provides a consistent theoretical basis within the scope of hyperelasticity and admits potential extensions toward more complex irreversible phenomena.