Abstract

We show the existence of periodic solutions to a model describing a rate-independent hysteresis response in bulk ferromagnets. The magnetic microstructure is treated in terms of Young measures and the whole formulation is based on energy functionals for the Gibbs and dissipative energies. Our proof of the periodicity of solutions is mainly based on uniqueness of a solution to time incremental problems and on the Tychonoff fixed point theorem.

Author information

Contact details are reproduced from the original publication and may be historical.

Martin Kružík

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

kruzik@utia.cas.cz

Keywords

  • Ferromagnetism
  • fixed point theorem
  • hysteresis
  • periodic solution

Mathematics Subject Classification

35Q60, 49S05, 78A30, 78M30

References

  1. V. Barbu, T. Precupanu: Convexity and Optimization in Banach Spaces, 2nd ed., D. Reidel, Dordrecht (1986).Google Scholar
  2. A. Bergqvist: Magnetic vector hysteresis model with dry friction-like pinning, Physica B 233 (1997) 342–347.https://doi.org/10.1016/S0921-4526(97)00319-0Google Scholar
  3. W. F. Brown, Jr.: Magnetostatic Principles in Ferromagnetism, Springer, New York (1966).Google Scholar
  4. C. Carstensen, A. Prohl: Numerical analysis of relaxed micromagnetics by penalised finite elements, Num. Math. 90 (2001) 65–99.https://doi.org/10.1007/s002110100268Google Scholar
  5. S. Chikazumi: Physics of Magnetism, Wiley, New York (1964).Google Scholar
  6. G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  7. K. Deimling: Nonlinear Functional Analysis, Springer, Berlin (1985).https://doi.org/10.1007/978-3-662-00547-7Google Scholar
  8. A. DeSimone: Energy minimizers for large ferromagnetic bodies, Arch. Rat. Mech. Anal. 125 (1993) 99–143.https://doi.org/10.1007/BF00376811Google Scholar
  9. R. Engelking: General Topology, Heldermann Verlag, Lemgo (1989).Google Scholar
  10. A. Hubert, R. Schäffer: Magnetic Domains, Springer, Berlin (1998).Google Scholar
  11. R. D. James, D. Kinderlehrer: Frustration in ferromagnetic materials, Continuum Mech. Thermodyn. 2 (1990) 215–239.https://doi.org/10.1007/BF01129598Google Scholar
  12. D. Jiles: A self consistent generalized model for the calculation of minor loop excursions in the theory of hysteresis, IEEE Trans. Mag. 28 (1992) 2603.https://doi.org/10.1109/20.179570Google Scholar
  13. M. Kružík: Periodicity properties of solutions to a hysteresis model in micromagnetics, in: Proceedings of ENUMATH 2003 (Prague, August 2003), M. Feistauer et al. (eds.), Springer, Berlin (2004) 605–614.https://doi.org/10.1007/978-3-642-18775-9_58Google Scholar
  14. L. D. Landau, E. M. Lifshitz: On the theory of the dispersion of magnetic permeability of ferromagnetic bodies, Physik Z. Sowjetunion 8 (1935) 153–169.Google Scholar
  15. L. D. Landau, E. M. Lifshitz: Course of Theoretical Physics 8, Pergamon Press, Oxford (1960).Google Scholar
  16. A. Mielke: Energetic formulation of multiplicative elastoplasticity using dissipation distances, Continuum Mech. Thermodyn. 15 (2003) 351–382.https://doi.org/10.1007/s00161-003-0120-xGoogle Scholar
  17. A. Mielke, T. Roubíček: A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Modeling Simul. 1 (2003) 571–597.https://doi.org/10.1137/S1540345903422860Google Scholar
  18. A. Mielke, F. Theil: Mathematical model for rate-independent phase transformations, in: Models of Cont. Mechanics in Analysis and Engineering, H.-D. Alber et al (eds.), Shaker, Aachen (1999) 117–129.Google Scholar
  19. A. Mielke, F. Theil: On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl. 11 (2004) 151–189.https://doi.org/10.1007/s00030-003-1052-7Google Scholar
  20. A. Mielke, F. Theil, V. Levitas: A variational formulation of rate-independent phase transformations using extremum principle, Arch. Rat. Mech. Anal. 162 (2002) 137–177.https://doi.org/10.1007/s002050200194Google Scholar
  21. P. Pedregal: Relaxation in ferromagnetism: The rigid case, J. Nonlin. Sci. 4 (1994) 105–125.https://doi.org/10.1007/BF02430629Google Scholar
  22. P. Pedregal: Parametrized Measures and Variational Principles, Birkhäuser, Basel (1997).https://doi.org/10.1007/978-3-0348-8886-8Google Scholar
  23. X. Ren, L. Truskinovsky: Finite scale microstructures in nonlocal elasticity, J. Elasticity 59 (2000) 319–355.https://doi.org/10.1007/978-94-010-0728-3_18Google Scholar
  24. R. Rogers, L. Truskinovsky: Discretization and hysteresis, Physica B 233 (1997) 370–375.https://doi.org/10.1016/S0921-4526(97)00323-2Google Scholar
  25. I. V. Roshchin, J. Yu, A. D. Kent, G. W. Stupian, M. S. Leung: Magnetic properties of Fe microstructures with focused ion beam-fabricated nano-constrictions, IEEE Trans. Mag. 37 (2001) 2101–2103.https://doi.org/10.1109/20.951066Google Scholar
  26. T. Roubíček: Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin (1997).https://doi.org/10.1515/9783110811919Google Scholar
  27. T. Roubíček, M. Kružík: Microstructure evolution model in micromagnetics, Z. Angew. Math. Phys. 55 (2004) 159–182.https://doi.org/10.1007/s00033-003-0110-7Google Scholar
  28. T. Roubíček, M. Kružík: Mesoscopic model for ferromagnets with isotropic hardening, Z. Angew. Math. Phys. 56 (2005) 107–135.https://doi.org/10.1007/s00033-003-2108-6Google Scholar
  29. A. Visintin: Modified Landau-Lifshitz equation for ferromagnetism, Physica B 233 (1997) 365–369.https://doi.org/10.1016/S0921-4526(97)00322-0Google Scholar
  30. A. Visintin: On some models of ferromagnetism, in: Free Boundary Problems I (Chiba, 1999), N. Kenmochi (ed.), Gakuto Int. Series in Math. Sci. Appl. 13 (2000) 411–428.Google Scholar
  31. A. Visintin: A Weiss-type model of ferromagnetism, Physica B 275 (2000) 87–91.https://doi.org/10.1016/S0921-4526(99)00712-7Google Scholar
  32. J. Warga: Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).https://doi.org/10.1016/B978-0-12-735150-6.50011-7Google Scholar
  33. L. C. Young: Generalized curves and existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. Lett. Varsovie, Classe III 30 (1937) 212–234.Google Scholar

Suggested citation

M. Kružík. “Periodic Solutions to a Hysteresis Model in Micromagnetics.” Journal of Convex Analysis 13 (2006), No. 1, 81–99. https://doi.org/10.68381/jca13006

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press.