Methodical Fitting for Mathematical Models of Rubber-like Materials

TL;DR

This paper introduces a three-stage method for modeling the non-linear response of rubber-like materials, providing accurate fits to experimental data and insights into the material's deformation stages.

Contribution

The paper presents a novel fitting approach that decomposes rubber response into three stages, with strain-energy functions aligned with elasticity theory and statistical mechanics.

Findings

Accurately fits experimental data across all deformation stages.

Reveals robustness of non-linear elasticity theory.

Provides interpretable models for large extension behavior.

Abstract

A great variety of models can describe the non-linear response of rubber to uni-axial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the "upturn" of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage, that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak non-linear elasticity theory; and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of non-linear elasticity is much more robust that seemed at first sight.