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A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis

  • Tomáš Roubíček
  • Giuseppe Tomassetti
Article
  • 42 Downloads

Abstract

A theory of elastic magnets is formulated under possible diffusion and heat flow governed by Fick’s and Fourier’s laws in the deformed (Eulerian) configuration, respectively. The concepts of nonlocal nonsimple materials and viscous Cahn–Hilliard equations are used. The formulation of the problem uses Lagrangian (reference) configuration while the transport processes are pulled back. Except the static problem, the demagnetizing energy is ignored and only local non-self-penetration is considered. The analysis as far as existence of weak solutions of the (thermo) dynamical problem is performed by a careful regularization and approximation by a Galerkin method, suggesting also a numerical strategy. Either ignoring or combining particular aspects, the model has numerous applications as ferro-to-paramagnetic transformation in elastic ferromagnets, diffusion of solvents in polymers possibly accompanied by magnetic effects (magnetic gels), or metal-hydride phase transformation in some intermetallics under diffusion of hydrogen accompanied possibly by magnetic effects (and in particular ferro-to-antiferromagnetic phase transformation), all in the full thermodynamical context under large strains.

Keywords

Elastic magnets Large strains Viscous Cahn–Hilliard equation Heat transfer Weak solutions Existence Galerkin method 

Mathematics Subject Classification

35K55 35Q60 35Q74 65M60 74A15 74A30 74F15 74F10 76S99 78A30 80A17 80A20 

Notes

Acknowledgements

The authors are thankful to Miroslav Šilhavý for fruitful discussions about modelling aspects. Also, many conceptual and other comments of three anonymous referees have been very useful for improving the presentation.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteCharles UniversityPrague 8Czech Republic
  2. 2.Institute of Thermomechanics of the Czech Academy of SciencesPrague 8Czech Republic
  3. 3.Dipartimento di IngegneriaUniversità degli Studi Roma TreRomeItaly

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