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Dimension reduction for time-dependent von Kármán rods

Federico Cianci, Bernd Schmidt

TL;DR

The article analyzes time-dependent nonlinear elastodynamics for a slender rod in the von Kármán regime with energy scaling $h^6$, establishing convergence from the 3D problem to a 1D dynamic von Kármán rod model as $h\to0$. Using a Γ-convergence-inspired dimension reduction, it constructs near-rigid rotations $R^h$, derives convergences for the scaled displacements and stress, and identifies limiting moments $E^0$, $E^2$, $E^3$ and the stress $E$. The limiting system comprises two wave-type equations for $(v_2,v_3)$ coupled to static relations for $(u,w)$, and it accounts for possible energy dissipation due to high-frequency torsional vibrations, yielding extra terms unless a no-blow-up condition holds. Overall, the work provides a rigorous derivation of a dynamically consistent, reduced 1D model from 3D nonlinear elastodynamics for slender rods, with implications for long-time behavior and efficient simulations.

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.

Dimension reduction for time-dependent von Kármán rods

TL;DR

The article analyzes time-dependent nonlinear elastodynamics for a slender rod in the von Kármán regime with energy scaling , establishing convergence from the 3D problem to a 1D dynamic von Kármán rod model as . Using a Γ-convergence-inspired dimension reduction, it constructs near-rigid rotations , derives convergences for the scaled displacements and stress, and identifies limiting moments , , and the stress . The limiting system comprises two wave-type equations for coupled to static relations for , and it accounts for possible energy dissipation due to high-frequency torsional vibrations, yielding extra terms unless a no-blow-up condition holds. Overall, the work provides a rigorous derivation of a dynamically consistent, reduced 1D model from 3D nonlinear elastodynamics for slender rods, with implications for long-time behavior and efficient simulations.

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.
Paper Structure (5 sections, 7 theorems, 160 equations)

This paper contains 5 sections, 7 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Approximation and Compactness Results
  4. Proof of the main theorem
  5. Auxiliary results

Key Result

Theorem 2.3

Assume eq:centroids-Wlineargrowth and conditionsonF-eqenergiadatiinizialiyh. Assume that there exists $h_0>0$ such that for every $h \in (0,h_0)$ there exists a weak solution $y^h$ of defyhspazifunz-condizinizyhforte that satisfies the energy inequality eqenergiayh. Let $u^h$, $v^h_2$, $v^h_3$, $w^h For $k=2,\,3$, there exist functions $v^P_k \in H^1(0,L)$ and $v^S_k \in L^2(0,L)$, such that More

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 5 more