论文标题
源自非线性薄壁梁的一维粘弹性vonKármán理论
One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams
论文作者
论文摘要
我们从三维开尔文 - voigt模型中得出有效的一维极限，用于粘弹性薄壁梁，其中弹性和粘性应力张量符合框架的指示原理。方程式的限制系统包括在弹性和粘性应力下伸展，弯曲和扭曲。它与已经通过[Friedrich-Kružík'20]和[Friedrich-Machill '22]的模型相吻合，该模型是连续降低的降低，首先是从3D到vonKármán板的2D理论，然后从2D到1D理论。在本文中，我们通过表明限制也可以通过同时发送梁的高度和宽度来获得上限来补充先前的分析。我们的论点依赖于[Freddi-Mora-Paroni '13]中的静态$γ$ convergence，基于公制梯度流的抽象理论，以及桑德尔和奴隶制的进化$γ$ -Convergence。
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-Kružík '20] and [Friedrich-Machill '22] by a successive dimension reduction, first from 3D to a 2D theory for von Kármán 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 $Γ$-convergence in [Freddi-Mora-Paroni '13], on the abstract theory of metric gradient flows, and on evolutionary $Γ$-convergence by Sandier and Serfaty.
