One-dimensional viscoelastic von K\'{a}rm\'{a}n theories derived from nonlinear thin-walled beams

TL;DR

This paper derives a one-dimensional viscoelastic von Kármán model for thin-walled beams from a 3D Kelvin-Voigt framework, extending previous dimension reduction results by considering simultaneous shrinking of beam dimensions.

Contribution

It demonstrates that the 1D viscoelastic beam model can be obtained through simultaneous dimension reduction, complementing earlier sequential approaches.

Findings

The 1D model aligns with previous 2D and 3D derivations.

The limit process uses advanced $ ext{Γ}$-convergence techniques.

The approach applies to beams with frame-indifferent elastic and viscous stresses.

Abstract

We derive an effective one-dimensional limit from a three-dimensional Kelvin-Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via [Friedrich-Kru\v{z}\'ik '20] and [Friedrich-Machill '22] by a successive dimension reduction, first from 3D to a 2D theory for von K\'{a}rm\'{a}n plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static $\Gamma$-convergence in [Freddi-Mora-Paroni '13], on the abstract theory of metric gradient flows, and on evolutionary $\Gamma$-convergence by Sandier and Serfaty.