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Bounds on the Energy of a Soft Cubic Ferromagnet with Large Magnetostriction

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Abstract

We complete the analysis initiated in Dabade et al. (J Nonlinear Sci 21:415–460, 2018) on the micromagnetics of cubic ferromagnets in which the role of magnetostriction is significant. We prove ansatz-free lower bounds for the scaling of the total micromagnetic energy including magnetostriction contribution, for a two-dimensional sample. This corresponds to the micromagnetic energy per unit length of an infinitely thick sample. A consequence of our analysis is an explanation of the multi-scale zig-zag Landau state patterns recently reported in single crystal Galfenol disks from an energetic viewpoint. Our proofs use a number of well-developed techniques in energy-driven pattern formation.

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Notes

  1. Our choice of signs here is a bit different from convention: The materials that are of interest in this paper are “negative anisotropy materials,” with \(K_a < 0\), and correspondingly, \(\varphi \) is defined by the negative of Eq. (1.13), nevertheless rendering the product \(K_a \varphi \) nonnegative.

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Acknowledgements

We would like to thank Robert V. Kohn for several useful comments on an earlier draft of this paper and an anonymous referee for catching an error in an earlier version. RV thanks Dallas Albritton for helpful conversations on Besov spaces. We thank Felix Otto for pointing out a small error in Dabade et al. (Dabade et al. 2018, Figure 2 a) of our previous paper, where the middle zig-zag lines were inverted. The correct figure is Fig. 2, making magnetization divergence free. The research of R.V was partially supported by the Center for Nonlinear Analysis at Carnegie Mellon University, by an AMS-Simons travel award, and by the National Science Foundation Grant No. DMS-1411646. The work of RDJ was supported by NSF (DMREF-1629026), and it also benefitted from the support of ONR (N00014-18-1-2766), the MURI Program (FA9550-12-1-0458, FA9550-16-1-0566), the RDF Fund of IonE, the Norwegian Centennial Chair Program and the hospitality and support of the Isaac Newton Institute (EPSRC Grant EP/R014604/1).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

    Raghavendra Venkatraman

  2. LMS, École Polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France

    Vivekanand Dabade

  3. Aerospace Engineering and Mechanics Department, University of Minnesota, Minneapolis, MN, USA

    Richard D. James

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  1. Raghavendra Venkatraman
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  2. Vivekanand Dabade
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  3. Richard D. James
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Correspondence to Raghavendra Venkatraman.

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Communicated by Irene Fonseca.

Dedicated to Peter Sternberg on the occasion of his sixtieth birthday, with respect and admiration.

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Venkatraman, R., Dabade, V. & James, R.D. Bounds on the Energy of a Soft Cubic Ferromagnet with Large Magnetostriction. J Nonlinear Sci 30, 3367–3388 (2020). https://doi.org/10.1007/s00332-020-09653-6

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