  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.
