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

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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.

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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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Authors and Affiliations

  1. Mathematical Institute, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Tomáš Roubíček

  2. Institute of Thermomechanics of the Czech Academy of Sciences, Dolejškova 5, 182 00, Prague 8, Czech Republic

    Tomáš Roubíček

  3. Dipartimento di Ingegneria, Università degli Studi Roma Tre, Via Volterra 62, 00146, Rome, Italy

    Giuseppe Tomassetti

Authors
  1. Tomáš Roubíček
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  2. Giuseppe Tomassetti
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Correspondence to Tomáš Roubíček.

Additional information

This research was partly supported through the Grants Czech Science Foundation 16-03823S “Homogenization and multi-scale computational modelling of flow and nonlinear interactions in porous smart structures” and 16-34894L “Variational structures in continuum thermomechanics of solids” and by the Austrian-Czech projects FWF/MSMT ČR No. 7AMB16AT015, as well as through the institutional project RVO: 61388998 (ČR). The second author also acknowledges financial support of INdAM-GNFM through Grant “Progetto Giovani”.

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Roubíček, T., Tomassetti, G. A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis. Z. Angew. Math. Phys. 69, 55 (2018). https://doi.org/10.1007/s00033-018-0932-y

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