Straight rod with different order of thickness

TL;DR

This paper analyzes the asymptotic behavior of linearly varying thickness rods in linear elasticity, deriving limit equations for bending, stretching, and torsion as the thickness varies from epsilon to epsilon squared.

Contribution

It introduces a decomposition method for displacement fields and establishes a priori estimates to derive limit models for rods with non-uniform thickness.

Findings

Derived limit equations for bending, stretching, and torsion.

Established a priori estimates for displacement fields.

Provided a framework for asymptotic analysis of variable-thickness rods.

Abstract

In this paper, we consider rods whose thickness vary linearly between $\eps$ and $\eps^2$. Our aim is to study the asymptotic behavior of these rods in the framework of the linear elasticity. We use a decomposition method of the displacement fields of the form $u=U\_e + \bar{u}$, where $U\_e$ stands for the translation-rotations of the cross-sections and $\bar{u}$ is related to their deformations. We establish a priori estimates. Passing to the limit in a fixed domain gives the problems satisfied by the bending, the stretching and the torsion limit fields which are ordinary differential equations depending on weights.