Abstract In this contribution, we propose to cope with the problem of soft magnetic materials heterogeneity and non-uniformity in terms of domains structure. This non-uniformity expresses itself with space variations of domains and walls geometry and characteristic properties from the bulk towards the surface. We investigate the possibility of describing and predicting these changes from a mesoscopic point of view. We begin with an introduction of typical subdivisions and define a tensor state variable [Λ2] to represent the diversity of magnetic structures with domains and walls. We then explain the material structuring thanks to an energy balance between the mesoscopic magnetic exchange, magneto-crystalline anisotropy, self-magnetostriction anisotropy, stress induced anisotropy and the dipolar demagnetizing energy. We write every contribution as a function of [V2] = [Λ2]−1. After minimizing the total energy, we derive a formulation compatible with classical numerical methods. [Λ2] is deduced thanks to a partial differential equation and surface boundary conditions. When a time varying field is applied to the material, damping effects occur either in the mass or at the surface. Eddy currents induced within domains lead to consider a volume dissipation energy. The surface magnetic field is also dampened by both the static hysteresis mainly due to defects and the dynamic hysteresis which stems from eddy currents around magnetic walls, added to the an-hysteretic field. The surface magnetic field, magnetic structure, and thus the polarization being known on the external surface, time variations of the volume magnetic structure can be calculated within the mass. Using the static or dynamic magnetic field coupling at the surface, the magnetic polarization can be rebuilt in the mass to calculate the apparent magnetic permeability. Finally, finding the geometry and frequency dependent vector magnetic behavior and iron losses becomes possible. The tensor magnetic phase theory is able to account for the sensitivity of the magnetic structure to the geometry, the macroscopic anisotropy partly influenced by the metallography, the residual or induced stress, some surface effects such as the texture, the rugosity or even any scribing patterns, at a mesoscopic scale. Two test cases for GO and NGO electrical steels are presented. Sensitivity analysis on the test case with GO steel are discussed. Results are then compared to static and dynamic measurements of GO SiFe sheet samples. This paper contributes to the investigations carried out on the geometry dependent magnetic behavior of soft magnetic materials.

中文翻译：

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软磁材料细观体积结构的张量磁相理论 – 准静态和动态矢量极化、表观磁导率和损耗 – 低感应水平下 GO 钢的实验鉴定

摘要 在此贡献中，我们建议解决软磁材料在畴结构方面的异质性和非均匀性问题。这种不均匀性表现在域和壁几何形状的空间变化以及从体块到表面的特征特性。我们研究了从细观角度描述和预测这些变化的可能性。我们首先介绍典型的细分并定义张量状态变量 [Λ2] 来表示具有畴和壁的磁性结构的多样性。然后，由于介观磁交换、磁晶各向异性、自磁致伸缩各向异性、应力诱导各向异性和偶极退磁能之间的能量平衡，我们解释了材料结构。我们将每个贡献写为 [V2] = [Λ2]−1 的函数。在最小化总能量后，我们推导出与经典数值方法兼容的公式。[Λ2] 是由偏微分方程和表面边界条件推导出来的。当对材料施加随时间变化的场时，会在质量或表面发生阻尼效应。域内感应的涡流导致考虑体积耗散能量。表面磁场也被主要由缺陷引起的静态磁滞和由磁壁周围的涡流引起的动态磁滞抑制，添加到非磁滞场。表面磁场、磁结构以及外表面的极化是已知的，体积磁结构的时间变化可以在质量内计算。使用表面的静态或动态磁场耦合，可以在质量中重建磁极化以计算表观磁导率。最后，找到几何形状和频率相关的矢量磁行为和铁损成为可能。张量磁相理论能够解释磁结构对几何形状的敏感性、部分受金相影响的宏观各向异性、残余应力或诱导应力、一些表面效应，如纹理、粗糙度甚至任何划线图案，在细观尺度上。介绍了 GO 和 NGO 电工钢的两个测试案例。讨论了对 GO 钢测试用例的敏感性分析。然后将结果与 GO SiFe 片材样品的静态和动态测量结果进行比较。