A physically-based hydrostatic strain energy model for rubber-like materials inspired by Flory-Orwoll-Vrij equation of state theory

Abstract

In a compressible hyperelastic model for rubber-like materials, the strain energy function is generally composed of the hydrostatic part corresponding to changes in interchain interactions and the compressible elastic part attributable to the network structure. It is well known that the hydrostatic part dominates the compressibility of the material. In this study, inspired by the Flory-Orwoll-Vrij (FOV) equation of state (EOS) theory for pure polymer liquids (note: a cell-like EOS theory), we assume that (i) the compressibility of rubber-like materials corresponds to changes in free volume of polymer chain segments; (ii) the hydrostatic strain energy of a rubber-like solid is attributable to changes in interchain interaction energy and chain segments’ external motion degrees of freedom (the latter solely depends on interchain forces). With a focus on the reversible isothermal deformation process, we construct a physically-based hydrostatic strain energy function based on the Helmholtz free energy formulation in FOV EOS theory. With a view towards applications, we provide a specific compressible hyperelastic model by incorporating the new hydrostatic strain energy function and compressible eight-chain model, where the latter is utilized as the elastic part of the strain energy function. Our model is capable of predicting various volume data of rubber-like materials from the literature, such as the nonlinear pressure-volume response at finite volume changes in hydrostatic compression (HC), the volume change-stretch and stress-stretch data in uniaxial tension (UT), and the stress-volume data in constrained uniaxial compression (CUC). Given the severely limited volume change data for finite stretches in UT and other modes of deformation, we simulate the volume change-volume modulus-stretch responses in UT, equibiaxial tension (ET), pure shear (PS), and uniaxial compression (UC) over their respective theoretical range of stretch and successfully predict some available (qualitative) experimental observations. Together with the simulations of the volume change-stretch responses in UT, ET, PS, and UC based on the Ogden’s and Bischoff et al.’s compressible models, we summarize the characteristics of these responses and analyze the possible deformation mechanisms. These simulations can provide some guidance for future corresponding experiments. Finally, we analyze the deformation mechanisms of HC, CUC, UT, ET, PS, and UC by simulating the responses for total strain energy and its components. This study provides a new physical picture and corresponding theoretical model for the hydrostatic strain energy function of rubber-like materials and finally proposes the general research strategy for constructing new hydrostatic strain energy functions based on the EOS theories for pure polymer liquids.