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

TL;DR

This paper introduces a comprehensive method for deriving high-order one-dimensional rod models from complex three-dimensional non-linear elasticity, accommodating various strains, rotations, and cross-sectional properties with asymptotic accuracy.

Contribution

It presents a novel, broadly applicable approach for deriving asymptotically exact high-order rod models from non-linear 3D elasticity, including finite strains and arbitrary cross-sections.

Findings

Method handles finite strains, rotations, and complex cross-sections.

Derived models are asymptotically exact to higher order.

Validated against existing linear and weakly non-linear solutions.

Abstract

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.