Investigating a Tensor Formulation to describe the Magnetic Structure of Soft Magnetic Materials

O. Maloberti [1], M. Nesser [2], E. Salloum [2], J. Fortin [2], S. Panier [2], P. Dassonvalle [1], C. Pineau [3], M. Caruso [4], J-P. Birat [3], I. Tolleneer [4],
[1] ESIEE Amiens, France
[2] UPJV-LTI, France
[3] IRT-M2P, France
[4] CRM Group, France
发布日期 2018

Most of existing material representations [1], even including the dynamic hysteresis and iron losses [2], ignore local microscopic non-uniformities. These concern the magnetic structure (domains and walls [3]) and are mainly due to surface effects, anisotropy and exchange [4]. One previous contribution [7] proposed to cope with the problem of soft magnetic materials heterogeneity in terms of domains structure at a static equilibrium. This non-uniformity expresses itself with space variations of domains geometry and properties from the bulk towards the surface. It involves a material parameter kappa, which represents a new material mesoscopic property that can be related to a ratio between the anisotropy and the exchange energy density of walls, taking the grain size and boundaries into account. In the second part of previous contribution [8], a way is proposed to couple static and dynamic relationships between the magnetic field and the magnetic polarisation to domains and walls structuring. This theory is supposed to provide a deterministic method that predicts the geometry dependent vector behaviour, including static and dynamic hysteresis and iron losses, of every soft materials. We investigate further this possibility to describe a magnetic structure from a mesoscopic point of view. Magnetic objects have got typical and characteristic physical properties that we propose to describe thanks to one tensor variable [L2], statistically gathering main topological information. We thus introduce the typical subdivisions by defining a tensor state variable [V2]=[L2]-1 with 6 unknowns. The material structuring can be explained thanks to energy tendencies, for a given value of the tensor components at the surface of the sample. The aim of this paper is to derive and implement a weak formulation compatible with classical numerical methods and especially the Finite Element Method used in COMSOL. This model provides a way to either identify or analyse and quantify the impact of a surface magnetic structure modification induced by laser onto the volume magnetic structure and consequently onto the global magnetic behaviour. In this work we propose to: - Introduce the theory, the physics and the formulation o Tensor state variable o Energy tendencies and materials properties o Symmetries, Constraints and borders conditions o Governing equations and weak formulation - Develop and Implement the formulation for 1D test case with NO and GO materials o Degrees of freedom and dependencies o 1D Test case (symmetries, constraints, limit conditions, EDP) o FEM and analytical solutions, comparisons, figures and discussions - Develop and Implement the formulation for 2D test case with NO and GO materials o Degrees of freedom and dependencies o 2D Test case (symmetries, constraints, limit conditions, EDP) o FEM and analytical solutions, comparisons, figures and discussions References [1] M-A. Raulet, B. Ducharne, J-P. Masson, and G. Bayada, “The magnetic field diffusion equation including dynamic hysteresis”, IEEE Transactions on Magnetics, 2004, vol. 42, n°2, pp. 872–875. [2] O. Maloberti, V. Mazauric, G. Meunier, A. Kedous-Lebouc, O. Geoffroy, “An Energy-Based Formulation for Dynamic Hysteresis and Extra-Losses”, IEEE Transactions on Magnetics, 2006, vol. 42, n°4, pp. 895-898. [3] A. Hubert, R. Schafer, Magnetic Domains, Springer Verlag, 2000. [4] H.A.M. Van Den Berg, A.H.J.V.D. Brandt, “Self-Consistent Domain Theory in Soft Ferromagnetic Media II: Basic domains structures in thin film objects”, Journal of Applied Physics, 1986, vol. 60, n°3, pp. 1105-1113. [5] H.A.M. Van Den Berg, A.H.J.V.D. Brandt, “Self-Consistent Domain Theory in Soft Ferromagnetic Media III: Composite domains structures in thin film objects”, Journal of Applied Physics, 1987, vol. 62, n°5, pp. 1952-1959. [6] A. De Simone, “Energy Minimizers for Large Ferromagnetic Bodies”, Arch. Rat. Mech. Anal., 1993, vol. 125, pp. 99-143. [7] O. Maloberti, G. Meunier, A. Kedous-Lebouc, V. Mazauric, “How to Formulate Soft Materials Heterogeneity? 1. Quasi-Static Equilibrium and Structuring”, submitted to J.M.M.M., conference SMM’18 in Cardiff 2007. [8] O. Maloberti, A. Kedous-Lebouc, G. Meunier, V. Mazauric, “How to Formulate Soft Materials Heterogeneity? 2. Hysteresis, Dynamic Motions and Diffusion”, submitted to J.M.M.M., conference SMM’18 in Cardiff 2007.

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