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Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity

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Abstract

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.

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Acknowledgements

This work was funded by the DFG Project FR 4083/1-1 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 -390685587, Mathematics Münster: Dynamics–Geometry–Structure. M.F. acknowledges support by the Alexander von Humboldt Stiftung and thanks for the warm hospitality at ÚTIA AVČR, where this project has been initiated. M.K. acknowledges support by the GAČR project 17-04301S and GAČR-FWF project 19-29646L.

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Authors and Affiliations

  1. Applied Mathematics, Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    Manuel Friedrich

  2. Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 182 00, Praha 8, Czechia

    Martin Kružík

  3. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Praha 6, Czechia

    Martin Kružík

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  1. Manuel Friedrich
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  2. Martin Kružík
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Correspondence to Martin Kružík.

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Communicated by I. Fonseca

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Friedrich, M., Kružík, M. Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity. Arch Rational Mech Anal 238, 489–540 (2020). https://doi.org/10.1007/s00205-020-01547-x

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