Archive for Rational Mechanics and Analysis

, Volume 176, Issue 2, pp 227–269 | Cite as

Mathematical Derivation of the Continuum Limit of the Magnetic Force between Two Parts of a Rigid Crystalline Material

  • Anja SchlömerkemperEmail author


The topic of this paper is a mathematically rigorous derivation of the continuum limit of the magnetic force between two parts of a rigid magnetized body. For this we start from a discrete setting of magnetic dipoles fixed to a scaled Bravais lattice, Open image in new window The limit as l→∞ corresponds to the passage to the continuum. The magnetic dipole moments are scaled in such a way that we obtain a finite total magnetic moment per unit volume. Under certain regularity assumptions on the magnetization and the boundaries we derive a force formula in the passage from the discrete setting to the continuum. Compared with a corresponding magnetic-force formula which has been previously discussed in the literature, the limiting force consists of an additional explicit local surface term, which is due to short-range effects and which reflects the lattice approximation of the underlying hypersingular integral.


Neural Network Dipole Moment Unit Volume Electromagnetism Magnetic Force 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aharoni, A.: Introduction to the Theory of Ferromagnetism. 2nd ed., International Series of Monographs in Physics 109, Oxford University Press, New York (2000)Google Scholar
  2. 2.
    Ball, J.M., James, R.D.: Fine Phase Mixtures as Minimizers of Energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)Google Scholar
  3. 3.
    Ball, J.M., James, R.D.: Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338, 389–450 (1992)Google Scholar
  4. 4.
    Bobbio, S.: Electrodynamics of Materials: Forces, Stresses, and Energies in Solids and Fluids. Academic Press, San Diego (2000)Google Scholar
  5. 5.
    Brown, W.F.: Magnetoelastic Interactions. Springer-Verlag, Berlin (1966)Google Scholar
  6. 6.
    Brown, W.F.: Micromagnetics. John Wiley and Sons, New York (1963)Google Scholar
  7. 7.
    Carbou, G., Fabrie, P.: Regular solutions for Landau-Lifschitz equation in a bounded domain. Differential Integral Equations. 14, 213–229 (2001)Google Scholar
  8. 8.
    Cauchy, A.: De la pression ou tension dans un système de points matériels. Exercices de Mathématique (1828), In: Œuvre complètes d’Augustin Cauchy, publiées sous la direction scientifique de l’Académie des sciences, Paris, IIE Séries 8, 253–277 (1890)Google Scholar
  9. 9.
    DeSimone, A., Podio-Guidugli, P.: On the Continuum Theory of Deformable Ferromagnetic Solids. Arch. Ration. Mech. Anal. 136, 201–233 (1996)Google Scholar
  10. 10.
    Döring, W.: Einführung in die theoretische Physik. Vol. 2: Das elektromagnetische Feld. Walter de Gruyter & Co., Berlin (1968)Google Scholar
  11. 11.
    Ericksen, J.L.: Electromagnetic Effects in Thermoelastic Materials. Math. Mech. Solids. 7, 165–189 (2002)Google Scholar
  12. 12.
    Eringen, A.C., Maugin, G.A.: Electrodynamics of continua. Vol. 1: Foundations and solid media. Vol. 2: Fluids and complex media. Springer-Verlag, New York (1990)Google Scholar
  13. 13.
    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. CRC Press LLC, Boca Raton (1999)Google Scholar
  14. 14.
    Fabes, E.B., Jodeit, M., Rivière, N.M.: Potential techniques for boundary value problems on C1-domains. Acta Math. 141, 165–186 (1978)Google Scholar
  15. 15.
    Federer, H.: Geometric measure theory. Reprint of the 1969 ed., Springer-Verlag, Berlin (1996)Google Scholar
  16. 16.
    Folland, G.B.: Introduction to partial differential equations. 2nd ed., Princeton University Press, Princeton New Jersey (1995)Google Scholar
  17. 17.
    Grisvard, P.: Singularities in boundary value problems. Research Notes in Applied Mathematics 22, Springer-Verlag, Berlin (1992)Google Scholar
  18. 18.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, Inc., New York (1981)Google Scholar
  19. 19.
    James, R.D.: Configurational Forces in Magnetism with Application to the Dynamics of a Small-Scale Ferromagnetic Shape Memory Cantilever. Continuum Mech. Thermodyn. 14, 55–86 (2002)Google Scholar
  20. 20.
    James, R.D., Müller, S.: Internal variables and fine-scale oscillations in micromagnetics. Continuum Mech. Thermodyn. 6, 291–336 (1994)Google Scholar
  21. 21.
    Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen. Springer-Verlag, Berlin (1998)Google Scholar
  22. 22.
    Landau, L.D., Lifschitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media. 2nd ed., Pergamon, Oxford (1984)Google Scholar
  23. 23.
    Lorentz, H.A.: The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat. 2nd ed., B.G. Teubner, Leipzig (1916)Google Scholar
  24. 24.
    Müller, S.: Variational models for microstructure and phase transitions. In: Hildebrandt, S., Struwe, M. (Eds.): Calculus of variations and geometric evolution problems. Lecture Notes in Math. 1713, Springer-Verlag, Berlin, 85–210 (1999)Google Scholar
  25. 25.
    Müller, S., Schlömerkemper, A.: Discrete-to-continuum limit of magnetic forces. C. R. Acad. Sci. Paris, Ser. I. 335, 393–398 (2002)Google Scholar
  26. 26.
    Penfield P., Haus, H.A.: Electrodynamics of Moving Media. The MIT Press, Cambridge, Mass. (1967)Google Scholar
  27. 27.
    Schlömerkemper, A.: Magnetic forces in discrete and continuous systems. Doctoral thesis, University of Leipzig (2002)
  28. 28.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton New Jersey (1970)Google Scholar
  29. 29.
    Verchota, G.: Layer Potentials and Regularity for the Dirichlet Problem for Laplace’s Equation in Lipschitz Domains. J. Funct. Anal. 59, 572–611 (1984)Google Scholar
  30. 30.
    Zagier, D.: Introduction to Modular Forms. In: Waldschmidt, M., Moussa, P., Luck, J.-M., Itzkyson, C. (Eds.): From Number Theory to Physics. Springer-Verlag, Berlin (1992)Google Scholar
  31. 31.
    Zagier, D.: Personal communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Analysis, Dynamik und ModellierungUniversität StuttgartStuttgartGermany

Personalised recommendations