Nonlinear weakly curved rod by Gamma convergence

TL;DR

This paper derives a nonlinear model for weakly curved rods using Gamma convergence from 3D nonlinear elasticity, focusing on the case where strain energy scales with the fourth power of the thickness.

Contribution

It introduces a novel derivation of a nonlinear weakly curved rod model via Gamma convergence without presupposing material behavior, extending the understanding of rod mechanics.

Findings

Derived a nonlinear weakly curved rod model using Gamma convergence.

Connected the model to the linear shallow arch and shallow shell models.

Provided a rigorous mathematical foundation for the model's validity.

Abstract

We present a nonlinear model of weakly curved rod, namely the type of curved rod where the curvature is of the order of the diameter of the cross-section. We use the approach analogous to the one for rods and curved rods and start from the strain energy functional of three dimensional nonlinear elasticity and do not presuppose any constitutional behavior. To derive the model, by means of $\Gamma$-convergence, we need to propose how is the order of strain energy related to the thickness of the body $h$. We analyze the situation when the strain energy (divided by the order of volume) is of the order $h^4$. That is the same approach as the one when F\"oppl-von K\'arm\'an model for plates and the analogous model for rods are obtained. The obtained model is analogous to Marguerre-von K\'arm\'an for shallow shells and its linearization is the linear shallow arch model which can be found in the literature.