general electromagnetic theory for the continuum

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literature review In the late 90s, stress calculations in electric motors have appeared as a result of noise and vibrations concerns (Reyne et al., 1987; Reyne, Sabonnadière, and Imhoff, 1988). At the time already, concerns regarding the intri-cate coupling between magnetic and mechanical aspects in ferromagnetic materials were raised. These materials with high magnetic permeability constitute the bulk of electric motors. They are used to enhance and channel the magnetic flux within the machine to improve performance. They have intrinsic strongly coupled magnetic and mechanical behavior (Cullity and Graham, 2011; Fonteyn, 2010; Fonteyn et al., 2010b; Fonteyn et al., 2010a). First, they develop important magnetization, which triggers body forces of magnetic origin in addition to the Lorentz force due to cur-rents (e.g. see Reyne et al., 1987). They foremost expand when applied a magnetic field, a phenomenon called magnetostriction (Joule, 1847; Lee, 1955; Du Trémolet De Lacheisserie, 1993). More recently, interest in computing stresses in electric mo-tors also arose for questions of motor performance, as the magnetostriction effect is accompanied by it reverse phenomena, referred to as inverse magnetostriction or Villari effect: stresses in ferromagnetic materials stretch and reorient the magnetic domains within the material. This modifies the material’s permeability to the mag-netic field and hence its magnetization. In turn the machine’s performance that relies on magnetization may be affected. This led to studies of the effect of stresses due to centrifugal forces on magnetization (Rekik, Hubert, and Daniel, 2014), or stresses due to manufacturing and assembling processes (for instance Daikoku et al., 2005; Bernard and Daniel, 2015 for shrink fitting, or Takezawa et al., 2006 for punching of electrical steel sheets). Consequently, important research is now led on the characterization of electrical steel sheets used in electric motors (e.g. see Aydin et al., 2017). For the design of future electric motors with optimized performance, modeling tools are required that can properly account for these fine material char-acterization and their coupled magneto-mechanical behavior. This is the ground to the current work.
As pointed out by Reyne et al., 1987, the first difficulty in the formulation of magneto-mechanical problems was the evaluation of the distribution of local elec-tromagnetic body forces, for which various expressions were suggested but none achieving general agreement. Since then, continuum mechanics has advanced this problem calling to consistent thermodynamic formulations of continuum electro-magnetism1, specifying material behaviors through the definition of a continuum’s specific free energy from which derives combined thermo-magneto-mechanical stresses giving the interface and body forces of electromagnetic origin. The multi-plicity of the different formulations, direct as well as variational, is however still to-day a source of confusion. Different (albeit equivalent) expressions for the Maxwell stress and electromagnetic body forces can be obtained and are thus responsible for the difficulty in the correct modeling of stresses in electric motors. For further discussion on this issue, the interested reader is referred to the article by Kankanala and Triantafyllidis, 2004 and book by Hutter, van de Ven, and Ursescu, 2006.
Regarding the direct approach of continuum mechanics, which uses conserva-tion laws to derive the equations of the problem, particularly helpful is the work of Kovetz, 2000 where the body forces due to the electromagnetic field are accounted for through a generalized linear momentum and a generalized Cauchy stress. Their precise form is not postulated but further given as deriving from a specific free en-ergy by application of fundamental principles of mechanics and thermodynamics. The postulate is on the form of the flux of electromagnetic-energy, namely Poynt-ing’s vector. The theory developed in this book has been used in many domains (Kankanala and Triantafyllidis, 2004; Thomas and Triantafyllidis, 2009; Dorfmann and Ogden, 2003; Dorfmann and Ogden, 2004; Dorfmann and Ogden, 2005) and Steigmann, 2009 makes the case of its equivalence with other formulations such as those reviewed in Hutter, van de Ven, and Ursescu, 2006. It has been taken to the modeling of electric motors in the recent work by Fonteyn, 2010; Fonteyn et al., 2010b; Fonteyn et al., 2010a. However several approximations are used (e.g. a small strain approximation involving non frame-indifferent invariants and the angular momentum balance principle is not imposed) and may be revisited.
It should be noted that in the following approach, we adopt a macroscopic view of the continua and derive macroscopic constitutive relations. Other recent ap-proaches address the characterization of the magneto-mechanical couplings through multiscale approaches (Daniel, 2018; Daniel, Bernard, and Hubert, 2020). Because these approaches include more physics, their advantage lies in the constitutive laws being described with fewer parameters (Aydin et al., 2017). These methods are very well suited to the understanding of the underlying physics behind magneto-mechanical couplings, and as such offer great insight for the optimization of mag-netic behavior. They however have the drawback of much greater computational costs when applied to the modeling of electric motor problems (e.g. see Daniel, Bernard, and Hubert, 2020). Because of this last point, and because recent, they where not included in the scope of the present work.
As a final note, other phenomena of importance in electric motors are: magnetic hysteresis that creates losses, and in particular the influence of stress on magnetic hysteresis is studied (e.g. see Daniel, Rekik, and Hubert, 2014; Bernard and Daniel, 2015; Rasilo et al., 2016); anisotropy, with grain oriented electrical steel sheets now used to improved machines performance (Lopez et al., 2009; Cassoret et al., 2014; Sugawara and Akatsu, 2013). Moreover, strong currents influence temperature due to ohmic effects, temperature may influence magnetization and electrical conduc-tivity but also stresses through thermal expansion, and so on. Although they are beyond the scope of the present work, the theory exposed is aimed general enough to be able to account for these effect. outline To set the stage, and in an effort, for consistency, to start from the most general problem before applying restrictions suitable for electric motor problems in a later Chapter 2, the general formulation for coupled electro-magneto-thermo-mechanical boundary value problems are presented in this chapter. The derivation is mostly taken from Kovetz, 2000, and presented here for self-sufficiency and clarity of the work exposed in this thesis.
The method adopted is the current configuration, direct approach of contin-uum mechanics.
The governing equations and interface conditions are derived from conservation principles of electromagnetism, mechanics and thermodynamics. As in Kovetz, 2000, the conservation principles are written in current configuration (or Eulerian coordinates). The equations in reference configuration (or Lagrangian coordinates) are then derived from the current configuration’s equations using kinematic rela-tions.
In Section 1.1, we introduce a few formal conventions as preliminaries. In Section 1.2 the governing equations of the problem are derived based on conservation principles of electromagnetism, mechanics and thermodynamics. In Section 1.3, these governing equations are taken to the reference configuration. In Section 1.4, the constitutive relations are derived based on the restrictions imposed by the second principle of thermodynamics and the angular momentum balance. They are finally taken to the reference configuration in Section 1.5

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governing equations in current configuration

As a foreword to the introduction of the electromagnetic equations, we mention here that the governing equations of electromagnetism derive from 3 fondamen-tal principles: the charge conservation principle, the electromagnetic field princi-ple, and the aether frame principle. The charge conservation principle results in Maxwell-Gauss and Maxwell-Ampère’s laws, which state the behavior of two elec-tric charge and current potentials: the electric displacement field d and h-field h. The electromagnetic field principle expresses the behavior of the electromagnetic field (e, b). The aether frame principle links the charge and current potentials (d, h) to the electromagnetic field (e, b) via two material response fields: the polarization p and magnetization m. The interested reader is referred to Kovetz, 2000 for a more detailed account on electromagnetism.

Charge conservation principle

As we mentionned above, the conservation of electric charge is usually split into two equations, commonly referred to as Maxwell-Gauss’ and Maxwell-Ampère’s laws.
maxwell-gauss’s law states that the volumetric electric charges q are sources of electric displacement d. Applied to an arbitrary moving control volume v(t), where @v(t) is the surface boundary of the control volume v(t) and n is the out-ward pointing unit normal to the surface.
From standard arguments involving the arbitrariness of the control volume and Gauss’s divergence theorem, one obtains the pointwise form of Gauss’s law with its associated interface conditions2,

The electromagnetic field principle

A second principle of electromagnetism is based on the primitive concept of an electromagnetic field (e, b) behaving according to two laws often referred to as Maxwell-Faraday’s law and the no magnetic monopole law.
the no-magnetic monopole law – also referred to as Maxwell-Thomson’s equation – states the conservation of the flux of the magnetic field b. In other terms, there are no local monopolar sources of magnetic field contained within an arbitrary moving control volume v(t),
where as introduced before @v(t) is the closed surface boundary of v(t) and n is the outward unit normal to the surface.
Using the standard arguments introduced before, the pointwise form of Maxwell-Thomson’s equation then writes r b = 0 , n b = 0 . (1.18) law states that the circulation of the electromotive in-maxwell-faraday’Js K. tensity e e + x b5 on a closed contour @s(t) balances the time variations of the flux of the magnetic field b through the enclosed surface s(t), Z@s(t) e sdl = – d
Here again, n is the outward pointing normal to the surface. s is the tangent to the contour @s(t) oriented according to the right hand rule.
From arguments similar to those used with Maxwell-Gauss’ law, Maxwell-Fara-day’s induction law transforms in pointwise form into r e = -b , n JeK = 0 ,
where () is the flux derivative introduced in (1.4).
From the expression of the electromotive intensity e and the flux derivative ex-pression (1.4), the pointwise form of Maxwell-Faraday’s law can also be expressed in the more usual form, As a final important comment: applying the divergence operator to the second form of Maxwell-Faraday’s law (1.21), one gets @t@ (r b) = 0. As a consequence, r b is a constant in time, function of the space variable x only. The only element that the no-monopole law adds is the initial condition r b = 0 everywhere. As a result, the no-monopole law is not to be viewed as an additional governing equation but rather as an initial (or gauge) condition. For that reason, Kovetz, 2000 (§8.) mentions that Maxwell-Faraday’s law and the no-monopole law should be viewed as a single principle of conservation of magnetic flux that includes a governing equation and a gauge.

Table of contents :

A brief descriptions of a typical electric motor
Motivation and scope
$i a magneto-thermo-mechanical theory based on the direct
approach of continuum mechanics
1 general electromagnetic theory for the continuum
1.1 Conventions
1.2 Governing equations in current configuration
1.2.1 Charge conservation principle
1.2.2 The electromagnetic field principle
1.2.3 Aether frame principle
1.2.4 Conservation of mass
1.2.5 Conservation of linear momentum
1.2.6 Conservation of angular momentum
1.2.7 First principle of thermodynamic
1.2.8 Second principle of thermodynamic
1.3 Governing equations in the reference configuration
1.3.1 Electromagnetic conservation laws
1.3.2 Principles of mechanics
1.3.3 Principles of thermodynamics
1.4 Constitutive relations and dissipation
1.4.1 Completeness of the system of equations and constitutive relations
1.4.2 Restrictions from the second principle of thermodynamics
1.4.3 New form of the governing equations accounting for the constitutive relations
1.4.4 Restrictions from the angular momentum balance
1.5 Constitutive relations in reference configuration
2 direct formulation for electric motor problems
2.1 Eddy current approximation
2.1.1 Applicability
2.1.2 Application to the problem
2.1.3 In reference configuration
2.2 Constitutive behavior
2.2.1 Assumptions on the materials considered
2.2.2 Resulting body forces of electromagnetic origin
2.2.3 Resulting traction at interfaces
3 analytical motor boundary value problems
3.1 Rotor boundary value problem
3.1.1 Problem description
3.1.2 Dimensionless boundary value problem
3.1.3 Results and discussion
3.1.4 Conclusion for the rotor problem
3.2 Stator boundary value problem
3.2.1 Problem description
3.2.2 Dimensionless boundary value problem
3.2.3 Results and discussion
3.2.4 A comparison with other models
3.2.5 Conclusion for the stator problem
4 conclusion of part i 
ii a variational formulation for finite element analysis
5 variational formulation for electric motor problems
5.1 A formulation for general electro-magneto-mechanical problems
5.1.1 A Lagrangian density for purely electromagnetic problems
5.1.2 A general coupled electromagnetic-mechanical Lagrangian
5.1.3 Application of Hamilton’s principle of variations
5.2 A formulation in the eddy current approximation
5.2.1 Lagrangian of the problem
5.2.2 Application of Hamilton’s principle of variations
5.2.3 The particular 2D, Quasi-static case
6 numerical implementation 
6.1 Matrix form of the problem
6.1.1 Lagrangian of the problem
6.1.2 Element and vector of unknowns
6.1.3 Force vector and stiffness matrix
6.2 Choice of a specific free energy
6.3 Other implementation details
6.3.1 Abaqus variables used
6.3.2 Treating infinite magnetic field values in 0
6.3.3 Treating air and copper domains
6.3.4 Boundary conditions
6.3.5 Load definitions
7 simulation of electric motor problems 
7.1 FEM simulation of the idealized stator problem
7.1.1 Problem description
7.1.2 Results at small magnetic field values
7.1.3 Results at large magnetic field values
7.2 FEM simulation of a refined stator geometry
7.2.1 Problem setup
7.2.2 Results at large magnetic field values
8 conclusion of part ii


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