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.
Highlights
The derivation of effective theories for thin structures such as plates, rods, or ribbons is a classical problem in continuum mechanics
We are interested in effective descriptions for viscoelastic ribbons, i.e., bodies with three different length scales: a length l which is much larger than the width ε which, in turn, is much larger than the thickness h
Let us start by considering a nonlinear three-dimensional model of a viscoelastic material with reference configuration at finite strains in Kelvin’s-Voigt’s rheology, i.e., a spring and a damper coupled in parallel
Summary
The derivation of effective theories for thin structures such as plates, rods, or ribbons is a classical problem in continuum mechanics. Despite the long history of the subject with contributions already by Euler, Kirchhoff, and von Karman (see [6,12] for surveys), rigorous results relating lower-dimensional theories to three-dimensional elasticity have only been obtained comparably recently. They were triggered by the use of variational methods, by Γ-convergence [13] together with quantitative rigidity estimates [24]. We are interested in effective descriptions for viscoelastic ribbons, i.e., bodies with three different length scales: a length l which is much larger than the width ε which, in turn, is much larger than the thickness h
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Nonlinear Differential Equations and Applications NoDEA
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
