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Homogenization in micro-magneto-mechanics

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Abstract

Ferromagnetic materials are characterized by a heterogeneous micro-structure that can be altered by external magnetic and mechanical stimuli. The understanding and the description of the micro-structure evolution is of particular importance for the design and the analysis of smart materials with magneto-mechanical coupling. The macroscopic response of the material results from complex magneto-mechanical interactions occurring on smaller length scales, which are driven by magnetization reorientation and associated magnetic domain wall motions. The aim of this work is to directly base the description of the macroscopic magneto-mechanical material behavior on the micro-magnetic domain evolution. This will be realized by the incorporation of a ferromagnetic phase-field formulation into a macroscopic Boltzmann continuum by the use of computational homogenization. The transition conditions between the two scales are obtained via rigorous exploitation of rate-type and incremental variational principles, which incorporate an extended version of the classical Hill–Mandel macro-homogeneity condition covering the phase field on the micro-scale. An efficient two-scale computational scenario is developed based on an operator splitting scheme that includes a predictor for the magnetization on the micro-scale. Two- and three-dimensional numerical simulations demonstrate the performance of the method. They investigate micro-magnetic domain evolution driven by macroscopic fields as well as the associated overall hysteretic response of ferromagnetic solids.

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Notes

  1. There exist materials that show even higher deformations under applied magnetic field, given by ferromagnetic shape memory alloys (these show strains in the order of a few percent). The governing effects taking place in shape memory alloys are however different from those in pure ferromagnets and will not be considered in the present contribution. For more information see, for example, Kiefer et al. [30].

  2. The magnetostrictive strain for a cubic response in (57) can be explicitly written as,

    $$\begin{aligned} {\varvec{\varepsilon }}_0=\frac{3}{2} \begin{pmatrix} E_{100}(m_1^2-1/3) &{} E_{111}m_1m_2 &{} E_{111}m_1m_3 \\ E_{111}m_2m_1 &{} E_{100}(m_2^2-1/3) &{} E_{111}m_2m_3 \\ E_{111}m_3m_1 &{} E_{111}m_3m_2 &{} E_{100}(m_3^2-1/3) \end{pmatrix} _{CF} \end{aligned}$$

    The material modulus \({\varvec{\mathcal {K}}}\) can be filled with the entries \({\mathcal {K}}_{1111}{=} E_{100}, {\mathcal {K}}_{2222}{=} E_{100}, {\mathcal {K}}_{3333}{=} E_{100}, {\mathcal {K}}_{1212}{=} {\mathcal {K}}_{1221}{=}E_{111}/2, {\mathcal {K}}_{2112}{=} {\mathcal {K}}_{2121}{=}E_{111}/2, {\mathcal {K}}_{1313}= {\mathcal {K}}_{1331}=E_{111}/2, {\mathcal {K}}_{3113}= {\mathcal {K}}_{3131}=E_{111}/2, {\mathcal {K}}_{2323}= {\mathcal {K}}_{2332}=E_{111}/2, {\mathcal {K}}_{3223}= {\mathcal {K}}_{3232}=E_{111}/2\). The fourth order cubic mechanical modulus \({\mathbb {C}}^{cub}\) in (58) is given as

    $$\begin{aligned} {\mathbb {C}}^{cub}= \begin{pmatrix} {\mathbb {C}}_{11} &{} {\mathbb {C}}_{12} &{} {\mathbb {C}}_{12} &{} &{} &{} \\ {\mathbb {C}}_{12} &{} {\mathbb {C}}_{11} &{} {\mathbb {C}}_{12} &{} &{} &{} \\ {\mathbb {C}}_{12} &{} {\mathbb {C}}_{12} &{} {\mathbb {C}}_{11} &{} &{} &{}\\ &{} &{} &{}{\mathbb {C}}_{44} &{} &{}\\ &{} &{} &{} &{}{\mathbb {C}}_{44} &{} \\ &{} &{} &{} &{} &{}{\mathbb {C}}_{44} \end{pmatrix}. \end{aligned}$$

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Acknowledgments

The authors thank the German Research Foundation (DFG) for financial support of the projects under the grants Mi 295/71-1 and KE 1849/2-2 within the research unit FOR 1509. M.-A.K. thanks the “Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg” as well as the DFG in the framework of the Cluster of Excellence in “Simulation Technology” (EXC 310/2) at the University of Stuttgart.

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  1. Institute of Applied Mechanics (CE), Chair I, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    A. Sridhar, M.-A. Keip & C. Miehe

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Sridhar, A., Keip, MA. & Miehe, C. Homogenization in micro-magneto-mechanics. Comput Mech 58, 151–169 (2016). https://doi.org/10.1007/s00466-016-1286-y

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