Asymptotically exact strain-gradient models for nonlinear slender elastic structures: a systematic derivation method

TL;DR

This paper introduces a systematic, nonlinear method for deriving one-dimensional strain-gradient models for slender elastic structures, capturing large deformations and providing asymptotically exact results without prior assumptions.

Contribution

It presents a novel, Ansatz-free derivation approach for nonlinear strain-gradient models that are asymptotically exact for slender structures with large deformations.

Findings

Derivation of models for hyper-elastic cylinders, membranes, and elastic blocks.

Models account for large deformations and strain gradients.

Method is asymptotically exact when strain varies on larger scales.

Abstract

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.