The nonlinear bending-torsion theory for curved rods as Gamma-limit of three-dimensional elasticity

TL;DR

This paper rigorously derives one-dimensional nonlinear bending-torsion models for curved elastic rods from three-dimensional elasticity using Gamma-convergence, considering different energy scalings as the beam diameter shrinks.

Contribution

It provides a rigorous variational derivation of nonlinear bending-torsion theories for curved rods as Gamma-limits of 3D elasticity models.

Findings

Derivation of nonlinear string and rod models from 3D elasticity

Identification of different scalings leading to distinct models

Validation of the models through variational convergence

Abstract

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending-torsion theory for rods are deduced.