Asymptotic behavior of a structure made by a plate and a straight rod

TL;DR

This paper analyzes the asymptotic behavior of a combined thin plate and rod structure under nonlinear elasticity, deriving simplified models and junction conditions as the structure's size scales.

Contribution

It introduces a rigorous derivation of the limiting models for a plate-rod structure, including junction conditions, within the nonlinear elasticity framework.

Findings

Derivation of the Von-Kármán equations for the plate

Reduction to a 1D rod model at the limit

Identification of junction conditions for bending and stretching

Abstract

This paper is devoted to describe the asymptotic behavior of a structure made by a thin plate and a thin rod in the framework of nonlinear elasticity. We scale the applied forces in such a way that the level of the total elastic energy leads to the Von-K\'arm\'an's equations (or the linear model for smaller forces) in the plate and to a one dimensional rod-model at the limit. The junction conditions include in particular the continuity of the bending in the plate and the stretching in the rod at the junction.