  We derive a non-linear one-dimensional (1d) strain gradient model predicting
the necking of soft elastic cylinders driven by surface tension, starting from
3d finite-strain elasticity. It is asymptotically correct: the microscopic
displacement is identified by an energy method. The 1d model can predict the
bifurcations occurring in the solutions of the 3d elasticity problem when the
surface tension is increased, leading to a localization phenomenon akin to
phase separation. Comparisons with finite-element simulations reveal that the
1d model resolves the interface separating two phases accurately, including
well into the localized regime, and that it has a vastly larger domain of
validity than 1d model proposed so far.
