# Derivation of von K\'arm\'an plate theory in the framework of three-dimensional viscoelasticity

## TL;DR

This paper derives a viscoelastic von Kármán plate model from three-dimensional nonlinear viscoelasticity using a rigorous dimension reduction approach, combining nonlinear plate theory and gradient flow methods.

## Contribution

It introduces a novel derivation of viscoelastic von Kármán plates from 3D models within a finite-strain, Kelvin-Voigt framework, integrating gradient flow theory.

## Key findings

- Derived a 2D viscoelastic plate model from 3D viscoelasticity.
- Established weak solutions for the viscoelastic von Kármán plate equations.
- Unified nonlinear plate theory with gradient flow approach.

## Abstract

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von K\'arm\'an type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and M\"{u}ller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von K\'arm\'an plates.