  We propose a general method for deriving one-dimensional models for nonlinear
structures. It captures the contribution to the strain energy arising not only
from the macroscopic elastic strain as in classical structural models, but also
from the strain gradient. As an illustration, we derive one-dimensional
strain-gradient models for a hyper-elastic cylinder that necks, an axisymmetric
membrane that produces bulges, and a two-dimensional block of elastic material
subject to bending and stretching. The method offers three key advantages.
First, it is nonlinear and accounts for large deformations of the
cross-section, which makes it well suited for the analysis of localization in
slender structures. Second, it does not require any a priori assumption on the
form of the elastic solution in the cross-section, i.e., it is Ansatz-free.
Thirdly, it produces one-dimensional models that are asymptotically exact when
the macroscopic strain varies on a much larger length scale than the
cross-section diameter.
