Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons [Herleitung einer eindimensonalen von Kármán-Theorie für viskoelastische Bänder]

Friedrich, Manuel; Machill, Lennart

Abstract

We consider a two-dimensional model of viscoelastic von Kármán plates in the Kelvin's-Voigt's rheology derived from a three-dimensional model at a finite-strain setting in Friedrich and Kružík (Arch Ration Mech Anal 238: 489--540, 2020). 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 (Commun Pure Appl Math 57:1627--1672, 2004) and complement the Gamma-convergence analysis of elastic von Kármán ribbons in Freddi et al. (Meccanica 53:659--670, 2018). Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.