We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in Kelvin’s-Voigt’s rheology to derive a viscoelastic plate model of von Kármán type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity considered in Friedrich and Kružík (SIAM J Math Anal 50:4426–4456, 2018) 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 Müller (Commun Pure Appl Math 55:1461–1506, 2002; Arch Ration Mech Anal 180:183–236, 2006), and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty (Commun Pure Appl Math 57:1627–1672, 2004), we perform a dimension-reduction from three dimensions to two dimensions and identify weak solutions of viscoelastic form of von Kármán plates.

中文翻译：

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三维粘弹性框架下von Kármán板块理论的推导

我们在 Kelvin's-Voigt 流变学的有限应变设置下对非简单粘弹性材料应用准静态非线性模型，以推导出 von Kármán 类型的粘弹性板模型。我们从 Friedrich 和 Kružík (SIAM J Math Anal 50:4426–4456, 2018) 中考虑的三维粘弹性模型的时间离散解开始，其中粘度应力张量符合时间连续框架无差异原则。结合 Friesecke、James 和 Müller 的非线性板理论推导（Commun Pure Appl Math 55:1461–1506, 2002; Arch Ration Mech Anal 180:183–236, 2006），以及度量空间中梯度流的抽象理论Sandier 和 Serfaty（Commun Pure Appl Math 57：1627-1672，2004），