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

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 crosssection, 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.

Nonuniform bending theory of hyperelastic beams in finite elasticity

“…This specic assumption will be veried later, showing how, for compact cross sections, the radius r(s) actually undergoes only small variations when moving away from the beam axis. 12 The three S J terms are the same because, at the beam axis, stretches and invariants are the same.…”

“…Taking these diculties into account, nonlinear analyses of beams have been carried out in the literature on the basis of approximate approaches. Audoly and Lestringant [12], using the rod theories with director approach, proposed a method for deriving equivalent one-dimensional models for slender nonlinear beams. Through a kinematic parameterization, a three-dimensional beam is modeled by means of its center-line to which local degrees of freedom, simulating the mobility of cross sections, are added.…”

“…In the papers [12], [13] and [14], the only connection between the threedimensional model and the reduced beam model is represented by the internal actions, which, however, are not evaluated according to the actual deformation of the three-dimensional solid. Moreover, no attempt is made to match the deformation and stress at any point of the three-dimensional solid with those of the proposed beam model.…”

“…This kinematic model is quite dierent from the Rivlin model, where the bending of a block was essentially formulated for a two-dimensional context, neglecting systematically the deformation of the cross sections [11]. The kinematic model also diers from that adopted in the simplied one-dimensional approaches men-tioned above [12], [13] and [14], in which the longitudinal deformation of the beam is decoupled from that of the cross sections.…”

A Rod Theory for Liquid Crystalline Elastomers

“…An extension to anelastic rods where non-elastic deformations and changes in microstructure occur was considered in [19]. A generic one-dimensional strain energy for rod models derived from large-strain finite elasticity was recently proposed in [4] where many related models were also reviewed.…”