Derivation of a one-dimensional von K\'{a}rm\'{a}n theory for viscoelastic ribbons

TL;DR

This paper derives a one-dimensional viscoelastic von Kármán model for ribbons from a 2D plate model using dimension reduction, gradient flow theory, and $ ext{Gamma}$-convergence, including convergence of discrete approximations.

Contribution

It introduces a novel 1D viscoelastic ribbon model incorporating stretching, bending, and twisting, derived from 2D plates via rigorous mathematical analysis.

Findings

Convergence of gradient flows for the viscoelastic ribbon model

Validation of time-discrete approximation convergence

Extension of $ ext{Gamma}$-convergence results to viscoelastic case

Abstract

We consider a two-dimensional model of viscoelastic von K\'arm\'an plates in the Kelvin's-Voigt's rheology derived from a three-dimensional model at a finite-strain setting. As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty and complement the $\Gamma$-convergence analysis of elastic von K\'{a}rm\'{a}n ribbons in [Freddi et al., 2018]. Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.