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Gradient Polyconvexity in Evolutionary Models of Shape-Memory Alloys

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Abstract

We show the existence of an energetic solution to a model of shape-memory alloys in which the elastic energy is described by means of a gradient polyconvex functional. This allows us to show the existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Admissible deformations do not necessarily have integrable second derivatives. Under suitable assumptions, our model allows for solutions which are orientation preserving and globally injective everywhere in the domain representing the specimen.

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Acknowledgements

We are indebted to the two referees for many helpful comments, in particular, for an overview of the history of non-simple materials. The research of MK was partly supported by the GAČR Grants 17-04301S and 18-03834S. PP moreover gratefully acknowledges the financial support by GAUK Project No. 670218, by Charles University Research Program No. UNCE/SCI/023, and by GAČR-FWF Project 16-34894L. This work was partially supported also by the Project PPP 57212737 with funds from the BMBF.

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Authors and Affiliations

  1. The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou věží 4, 182 08, Praha 8, Czech Republic

    Martin Kružík

  2. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Praha 6, Czech Republic

    Martin Kružík

  3. Faculty of Mathematics & Physics, Charles University, Sokolovská 83, 186 00, Praha 8, Czech Republic

    Petr Pelech

  4. Institute of Mathematics, University of Würzburg, Emil-Fischer-Straße 40, 97074, Würzburg, Germany

    Anja Schlömerkemper

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  1. Martin Kružík
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  2. Petr Pelech
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  3. Anja Schlömerkemper
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Correspondence to Martin Kružík.

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Kružík, M., Pelech, P. & Schlömerkemper, A. Gradient Polyconvexity in Evolutionary Models of Shape-Memory Alloys. J Optim Theory Appl 184, 5–20 (2020). https://doi.org/10.1007/s10957-019-01489-9

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  • DOI: https://doi.org/10.1007/s10957-019-01489-9

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