  We determine stability boundaries for the wrinkling of highly
uni-directionally stretched, finely thin, rectangular elastic sheets. For a
given fine thickness and length, a stability boundary here is a curve in the
parameter plane, aspect ratio vs. the macroscopic strain; the values on one
side of the boundary are associated with a flat, unwrinkled state, while
wrinkled configurations correspond to all values on the other. In our recent
work we demonstrated the importance of finite elasticity in the membrane part
of such a model in order to capture the correct phenomena. Here we present and
compare results for four distinct models:(i) the popular F\"oppl-von K\'arm\'an
plate model (FvK), (ii) a correction of the latter, used in our earlier work,
in which the approximate 2D F\"oppl strain tensor is replaced by the exact
Green strain tensor, (iii) and (iv): effective 2D finite-elasticity membrane
models based on 3D incompressible neo-Hookean and Mooney-Rivlin materials,
respectively. In particular, (iii) and (iv) are superior models for elastomers.
The 2D nonlinear, hyperelastic models (ii)-(iv) all incorporate the same
quadratic bending energy used in FvK. Our results illuminate serious
shortcomings of the latter in this problem, while also pointing to inaccuracies
of model (ii), in spite of yielding the correct qualitative phenomena in our
earlier work. In each of these, the shortcoming is a due to a deficiency of the
membrane part of the model.
