Asymptotic models for curved rods derived from nonlinear elasticity by Gamma-convergence

TL;DR

This paper rigorously derives one-dimensional models for thin curved beams from three-dimensional nonlinear elasticity using Gamma-convergence, analyzing different energy scalings and including elastic rings.

Contribution

It provides a unified derivation of reduced models for curved rods from nonlinear elasticity, clarifying the relation between nonlinear and linearized limits.

Findings

Limit models for various scalings match those from linearized elasticity.

The approach applies to both curved beams and elastic rings.

Different energy scalings lead to distinct limiting behaviors.

Abstract

We study the problem of the rigorous derivation of one-dimensional models for a thin curved beam starting from three-dimensional nonlinear elasticity. We describe the limiting models obtained for different scalings of the energy. In particular, we prove that the limit functional corresponding to higher scalings coincides with the one derived by dimension reduction starting from linearized elasticity. Finally we also address the case of thin elastic rings.