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

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.