  We propose a method for deriving equivalent one-dimensional models for
slender non-linear structures. The approach is designed to be broadly
applicable, and can handle in principle finite strains, finite rotations,
arbitrary cross-sections shapes, inhomogeneous elastic properties across the
cross-section, arbitrary elastic constitutive laws (possibly with low symmetry)
and arbitrary distributions of pre-strain, including finite pre-strain. It is
based on a kinematic parameterization of the actual configuration that makes
use of a center-line, a frame of directors, and local degrees of freedom
capturing the detailed shape of cross-sections. A relaxation method is applied
that holds the framed center-line fixed while relaxing the local degrees of
freedom; it is asymptotically valid when the macroscopic strain and the
properties of the rod vary slowly in the longitudinal direction. The outcome is
a one-dimensional strain energy depending on the apparent stretching, bending
and twisting strain of the framed center-line; the dependence on the strain
gradients is also captured, yielding an equivalent rod model that is
asymptotically exact to higher order. The method is presented in a fully
non-linear setting and it is verified against linear and weakly non-linear
solutions available from the literature.
