Dimension reduction for time-dependent von K\'arm\'an rods

TL;DR

This paper investigates how solutions of 3D nonlinear elastodynamics for thin rods converge to a 1D von Kármán model as the cross section shrinks, including effects of high-frequency torsional vibrations and energy dissipation.

Contribution

It provides a rigorous derivation of the dimensionally reduced von Kármán equations from 3D elastodynamics, accounting for high-frequency vibrations and energy dissipation effects.

Findings

Solutions converge to a 1D von Kármán model as cross section shrinks.

High-frequency torsional vibrations lead to additional terms in the limit.

Energy dissipation occurs in the limit under certain conditions.

Abstract

This paper aims to study the convergence of solutions in three-dimensional nonlinear elastodynamics for a thin rod as its cross section shrinks to zero for displacements that are comparable to the small radius of the rod. Assuming the existence of solutions and proper control of the torsional velocity, we show how these converge to the solutions of an effective dimensionally reduced model which is a version of the the time dependent von K\'arm\'an equations for a one-dimensional rod. In the presence of high-frequency torsional vibrations, energy can dissipate in the limit and we obtain additional contributions in the limiting equations.