Asymptotic derivation of high-order rod models from non-linear 3D elasticity

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