Dimension reduction for time-dependent von K\'arm\'an rods
Analysis of PDEs
2025-10-24 v1
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.
Keywords
Cite
@article{arxiv.2510.20623,
title = {Dimension reduction for time-dependent von K\'arm\'an rods},
author = {Federico Cianci and Bernd Schmidt},
journal= {arXiv preprint arXiv:2510.20623},
year = {2025}
}