On tube models of rubber elasticity: fitting performance in relation to sensitivity to the invariant I2

Abstract

This study investigates the capability of micromechanical models of rubber elasticity to predict the deformation behaviour of soft materials under various modes of deformation. The free energy of individual chains is decomposed into a freely fluctuating chain contribution and a tube contribution representing topological constraints. Full-network averaging over all chain orientations is considered, along with three-chain and eight-chain approximations. The performance of various tube formulations is analysed in relation to their implicit (or in some cases explicit) dependence on the second invariant I2. We show that micromechanical models that involve the area-stretch of the macroscale continuum lead to I2 dependence when combined with the full-network averaging scheme, whereas micromechanical models that only involve the line-stretch of the continuum show much weaker I2 sensitivity. However, I2 sensitivity can emerge from line-stretch–based micromechanical models when three-chain averaging is used. Comparisons between model predictions and experimental data confirm the direct correlation between strong I2 sensitivity and fitting performance. Overall, our study suggests that micromechanical models of rubber elasticity should involve both the line-stretch and the area-stretch to elicit I2-dependent behaviour and reproduce experimental trends.