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

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.