Preprint
Inserted: 25 sep 2025
Last Updated: 25 sep 2025
Year: 2025
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 Kármán 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: dimension reduction, rods, nonlinear elastodynamics, von Kármán
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