On the network orientational affinity assumption in polymers and the micro–macro connection through the chain stretch

Resumen

We question the network affinity assumption in modeling chain orientations under polymer deformations, and the use of the stretch measure projected from the right Cauchy–Green deformation tensor (or non-affine micro-stretches derived from that measure) as a basic state variable for the micro–macro transition. These ingredients are standard, taken from the statistical theory of polymers, and used in most micromechanical polymer network and soft tissue models.

This assumption imposes a constraint in the network which results in an anisotropic distribution of the orientation of the chains and, hence, in an additional configurational entropy that should be included. This additional entropy would result in an additional stress tensor which is typically neglected. But an arguably more natural alternative, in line with the typical assumption for the chain behavior itself and with the disregard of these forces, is to consider that the network may fluctuate unconstrained to adapt to macroscopic deformations. This way, the isotropic statistical distribution of the orientation of the chains is maintained unconstrained during deformation and no additional stress is imposed. In practice, we show that this free-fluctuating network assumption is equivalent to consider the stretch projected from the stretch tensor (instead of the right Cauchy–Green deformation tensor) as the state variable for the deformation of the network chains.

Employing simple Neo-Hooke-type models and our recent data-driven macro–micro–macro approach, we show very important differences in predictions using both assumed behaviors. We demonstrate that with the free-fluctuating network assumption, we can obtain accurate predictions for all tests in polymers using just one test curve to calibrate our model. With the same macro–micro–macro approach but employing the network affinity assumption, we are capable of capturing accurately only the test used for calibration of the model, but not the overall polymer behavior. Further numerical examples are developed to give supporting evidence for the unconstrained orientation assumption.