One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

Friedrich M, Machill L (2023)

Publication Type: Journal article

Publication year: 2023

Journal

Calculus of Variations and Partial Differential Equations Springer Verlag (Germany)

Book Volume: 62

Article Number: 190

Journal Issue: 7

DOI: 10.1007/s00526-023-02525-3

Abstract

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

Authors with CRIS profile

Manuel Friedrich Professur für Angewandte Mathematik (Angewandte Analysis)

Involved external institutions

Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE)

How to cite

APA:

Friedrich, M., & Machill, L. (2023). One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams. Calculus of Variations and Partial Differential Equations, 62(7). https://dx.doi.org/10.1007/s00526-023-02525-3

MLA:

Friedrich, Manuel, and Lennart Machill. "One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams." Calculus of Variations and Partial Differential Equations 62.7 (2023).

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