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Shape Optimization, Homogenization and Optimal Control pp 1–17Cite as

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Homogenization in Magnetic-Shape-Memory Polymer Composites

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 169))

Abstract

Magnetic-shape-memory materials (e.g. specific NiMnGa alloys) react with a large change of shape to the presence of an external magnetic field. As an alternative for the difficult to manufacture single crystal of these alloys we study composite materials in which small magnetic-shape-memory particles are embedded in a polymer matrix. The macroscopic properties of the composite depend strongly on the geometry of the microstructure and on the characteristics of the particles and the polymer.

We present a variational model based on micromagnetism and elasticity, and derive via homogenization an effective macroscopic model under the assumption that the microstructure is periodic. We then study numerically the resulting cell problem, and discuss the effect of the microstructure on the macroscopic material behavior. Our results may be used to optimize the shape of the particles and the microstructure.

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Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft through Schwerpunktprogramm 1239 Änderung von Mikrostruktur und Form fester Werkstoffe durch äußere Magnetfelder and through Sonderforschungsbereich 1060 Die Mathematik der emergenten Effekte.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    Sergio Conti

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Martin Lenz & Martin Rumpf

  3. Institut für Geometrie, Technische Universität Dresden, Dresden, Germany

    Matthäus Pawelczyk

Authors
  1. Sergio Conti
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  2. Martin Lenz
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  3. Matthäus Pawelczyk
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  4. Martin Rumpf
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Corresponding author

Correspondence to Sergio Conti .

Editor information

Editors and Affiliations

  1. FB IV Abt. Mathematik, Universität Trier, Trier, Germany

    Volker Schulz

  2. Département de Mathématiques de la, Université Cheikh Anta Diop de Daka, Dakar-Fann, Senegal

    Diaraf Seck

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Conti, S., Lenz, M., Pawelczyk, M., Rumpf, M. (2018). Homogenization in Magnetic-Shape-Memory Polymer Composites. In: Schulz, V., Seck, D. (eds) Shape Optimization, Homogenization and Optimal Control . International Series of Numerical Mathematics, vol 169. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90469-6_1

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