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Buckling of Nonlinearly Elastic Plates with Microstructure

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Generalized Continua as Models for Materials

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 22))

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Abstract

In the framework of the general nonlinear plate theory we consider a buckling problem for an elastic plate with incompatible plane strains generated by continuous distributions of edge dislocations and wedge disclinations as well as other sources of residual stress (non-elastic growth or plasticity). In contrast to the Föppl-von Kármán model the plane strains are not supposed to be small. To explore buckling transition of such kind of structures, the problem is reduced to a system of nonlinear partial differential equations with respect to the transverse deflection of the plate and the embedded metrics coefficients, which naturally leads to the non-trivial plate shapes that are seen even in the absence of any external forces. In the case of very thin plate (membrane) that doesn’t resist bending we present several exact solutions for the axially-symmetric domains.

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Notes

  1. 1.

    Seung and Nelson [3] deduced this equation in the inextensional limit.

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Acknowledgments

The author thanks Prof. L.M. Zubov for fruitful discussions.

This work was supported by the Russian Foundation for Basic Research (via the grants 12-01-00038 and 12-01-91270) and the Federal target programme “Research and Pedagogical Cadre for Innovative Russia” for 2009–2013 years (state contract N P596).

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Authors and Affiliations

  1. Southern Federal University, Milchakova 8a, Rostov-on-Don, Russia, 344090

    Svyatoslav Derezin

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  1. Svyatoslav Derezin
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Correspondence to Svyatoslav Derezin .

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Editors and Affiliations

  1. , Lehrstuhl für Technische Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, Magdeburg, 39112, Germany

    Holm Altenbach

  2. , Centre des Materiaux CNRS UMR 7633, Mines ParisTech, BP 87, EVRY Cedex, 91003, France

    Samuel Forest

  3. University, Dep. of Theoretical Mechanics, St. Petersburg State Polytechnical, Politekhnicheskaya, 29, St. Petersburg, 195251, Russia

    Anton Krivtsov

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Derezin, S. (2013). Buckling of Nonlinearly Elastic Plates with Microstructure. In: Altenbach, H., Forest, S., Krivtsov, A. (eds) Generalized Continua as Models for Materials. Advanced Structured Materials, vol 22. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36394-8_4

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