Derivation of the nonlinear bending-torsion model for a junction of elastic rods

TL;DR

This paper derives a one-dimensional nonlinear bending-torsion model for junctions of elastic rods from three-dimensional elasticity using $ extGamma$-convergence, establishing transmission conditions at the junction.

Contribution

It introduces a rigorous derivation of the nonlinear bending-torsion model for rod junctions from 3D elasticity via $ extGamma$-convergence, including transmission conditions.

Findings

The model includes continuity of displacement and rotation at the junction.

Balance of forces and couples is established at the junction.

The derivation is rigorous from 3D nonlinear elasticity to 1D models.

Abstract

In this paper we derive the one-dimensional bending-torsion equilibrium model modeling the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin rod-like bodies. By taking the limit as the thickness of the 3D rods goes to zero, and by using ideas from the theory of $\Gamma$-convergence, we obtain that the resulting model consists of the union of the usual one-dimensional nonlinear bending-torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections and balance of contact forces and contact couples.