Project topics and materials
Get Complete Project Material File(s) Now! »
Magneto-mechanical modeling
In this chapter, the magneto-mechanical model considered to take into account the effect of punching process on magnetic properties of ferromagnetic materials is discussed. First the original magneto-plastic model proposed by Sablik and its theoretical background is presented. Secondly, the proposed modifications to the model are detailed, it concerns the introduction of a new modeling method of the anhysteretic and magnetostrictive behaviors, and the introduction of an equivalent stress to take into account the multiaxial stress effect. Finally, the identification methodology has been presented in detail.
Magneto-plastic model – Sablik’s approach
Sablik model
As explained in the previous chapter, section I.4.2, to take into account the plastic deformation effect in the magnetoelastic Jiles-Atherton model, Sablik rewrites the parameters (equation I.46) and (equation I.49) as a function of the dislocation density.
Since the parameter is related to the pinning sites density, it is proportional to the coercivity and hence has the same dependence. Based on the experimental works [77-80] which show that coercivity is proportional to the square root of the dislocation density, Sablik in [70][81][82] proposes the expression of the wall pinning parameter given by (II.1).
Regarding the scaling parameter , this latter is proportional to the domain density in the demagnetized state, which is determined by the pinning site density, which is in turn proportional to the pinning parameter . Thus, Sablik proposes the expression (II.2) of the scaling parameter where it exhibits the same dependence as on the dislocation density .
with G1, G2, G3 and G4 are constants related to the grain size d, 0 and 0 are respectively the wall pinning and scaling parameters prior to plastic deformation.
where 0 is the initial dislocation density prior to the plastic deformation, is the specimen shear modulus as given by (II.4), is the appropriate Burgers vector magnitude for the specimen’s dislocations, is the Poisson ratio and is the Young modulus.
The plastic deformation which is represented by the hardening stress is directly related to the dislocation density as expressed in (II.3). The hardening stress as illustrated in Fig.II.1 is given by = − σy where σy represents the yield stress and, the applied stress modeled by the Hollomon law such as ( ) = , where is the hardening coefficient and the exponent.
Stress (MPa)
Strain (%)
Fig.II.1 Illustration of the hardening stress
In the following we will only consider the anhysteretic behavior. Thus, the magneto-mechanical model is expressed by equation (I.46) which gives the anhysteretic magnetization as a function of the scaling parameter (II.2), the effective field (I.56), and the magneto-mechanical field (I.57) which is related to the magnetostriction function λ. Its expression will be discussed in more details in this section.
Anhysteretic functions
An accurate representation of the magneto-mechanical behavior also relies on the ability of the magnetic model, in our case the anhysteretic function, to reproduce the observed experimental behavior. In that context, most of the magnetic models exploit the Langevin function for description of the anhysteretic magnetic behavior. Although it exhibits interesting advantages, limited number of parameter and physical behavior, the Langevin function can lead to accuracy problems, especially when employed for hysteresis modeling [83][84]. In this section two anhysteretic functions based on the Brillouin function, which were developed to model the anhysteretic behavior, are presented and adapted for the magneto-mechanical modeling.
The Brillouin function was originally derived for paramagnet materials, where the magnetic moments are assumed oriented in random direction without interaction in the absence of an external magnetic field. However, like the Langevin function used in the Sablik model, it can be used for ferromagnet by considering the effective field. In [85], the author proposes, based on the Brillouin model, an anhysteretic function that is a linear combination of two Langevin functions. Its expression is given in (II.6) where, on the one hand, 1 and 2 are parameters related to the saturation magnetization such as = 1 + 2 and, on the other hand, 1 and 2 are the scaling parameters.
Magnetostriction model
In this study, the magnetostriction refers to the magnetostrictive deformation along the magnetization direction. As reported in chapter I, it has been observed experimentally that is an even function of the magnetization, which means that the deformation due to the magnetization is independent of its direction. Furthermore, the magnetostriction has an asymmetrical and nonlinear dependency on the tensile and compressive stress [87]. Since the scientific community began to be interested in magnetostrictive behavior under mechanical loading, different models have been proposed.
Through their several works on the magnetoelastic behavior modeling, Jiles and Sablik used different magnetostriction functions. In [88], a magnetostriction model, which depends on the magnetization and the elastic stress, has been proposed. The function given in (II.8) is expressed in term of the saturation magnetostriction that depends on the elastic stress and the square of magnetization 2.
Another more elaborated magnetostriction function has been proposed in [89] [90]. It is derived from magnetoelastic considerations by considering the mechanical equilibrium state of the material. The magnetostriction is function of the mechanical parameters and that are the Poisson ratio and Young modulus, respectively. refers to the magnetic energy and is a constant.
Compared to the function proposed in (II.8), this model is able to reproduce the asymmetrical magnetostrictive behavior with respect to the compressive and the tensile stresses. However, its identification is not straightforward.
Recently, based on experimental observations, a mathematical model of the magnetostriction has been proposed by Deepak et al [91]. The proposed model consists in the product of two distinct functions as given in (II.11).
it considers a 2 degree polynomial dependency on the magnetization. The second function scales the magnetostriction depending on the stress, using the hyperbolic tangent which is controlled by the parameters 1, 2, , 0.
The authors showed that the function presents a good agreement with magnetostriction measurements under different elastic stresses. The nonlinear dependency on the compressive and tensile stresses, which was the weakness of the previous models, is well respected. Within the context of this work, this function will be integrated in the original Sablik model.
Equivalent stress approach
The main limitation of Sablik model is that the effect of stress on the magnetic behavior is restricted to uniaxial stress. Yet, stress is multiaxial in most of industrial applications. To overcome this limitation, an equivalent stress is the adaptative solution in the case of the scalar modeling approach used in the Sablik model. In this section, a brief review of the different definitions of “equivalent stress” proposed in the literature will be introduced, then the equivalent stress proposed by Hubert and Daniel [92] will be presented in more details.
Table of contents :
Introduction
1. Background and aims of the thesis
2. Scientific contribution
3. Outline of the thesis
Chapter I: Literature review
I.1 Magnetism and ferromagnetic materials
I.1.1 Magnetic quantities definitions
I.1.2 Ferromagnetic materials
I.1.2.1 Magnetic domain theory
I.1.2.2 Magnetization mechanism
I.2 Microstructure and mechanical properties
I.2.1 Crystalline structure and defects
I.2.2 Strain stress curve
I.2.2.1 Tensile test – conventional curve
I.2.2.2 Rational characteristic
I.2.3 Indentation measurement
I.3 Impact of manufacturing processes
I.3.1 Cutting process effects
I.3.1.1 Effects of plastique deformation on magnetic properties
I.3.1.2 Effect of plastic deformation on dislocation density
I.3.1.3 Plastic deformation at the cutting edge
I.3.2 Magneto-elastic coupling
I.3.2.1 Magnetostriction process
I.3.2.2 Effect of elastic stress on magnetostriction
I.3.2.3 Effect of elastic stress on magnetization
I.3.2.4 Measurement devices under elastic loading
I.4 Magneto-mechanical modeling
I.4.1 Multiscale model
I.4.2 Macroscopic models
I.4.3 Magneto-mechanical formulas
I.5 Conclusion
Chapter II: Magneto-mechanical modeling
II.1 Magneto-plastic model – Sablik’s approach
II.1.1 Sablik model
II.1.2 Anhysteretic functions
II.1.3 Magnetostriction model
II.1.4 Equivalent stress approach
II.1.4.1 Daniel and Hubert (D-H) equivalent stress
II.1.4.2 Validation of the D-H equivalent stress
II.1.5 Summary
II.2 Model identification
II.2.1 Initial identification
II.2.2 Elastic identification
II.2.2.1 Elastic identification under uniaxial stress
II.2.2.2 Elastic identification under biaxial stress
II.2.3 Plastic identification
II.3 Discussions and conclusion
Chapter III: Finite element implementation of the magneto-mechanical model
III.1 Numerical tools
III.1.1 Maxwell equations for magnetostatics and behavior laws
III.1.2 Boundary conditions
III.1.3 Vector potential formulation
III.1.4 Finite element method
III.1.4.1 Finite element formulation
III.1.4.2 Discretization
III.1.4.3 Interpolation functions
III.1.4.4 Discretization of the magnetostatic formulation
III.1.4.5 Numerical integration – Gauss method
III.2 Inverse of Sablik model
III.3 Determination of the plastic strain distribution
III.3.1 Punching process simulation
III.3.1.1 Principle of the industrial punching process
III.3.1.2 Material characteristics
III.3.2 Parametric study of the punching of a steel sheet
III.4 Finite element implementation
III.4.1 Exploitation of the punching process simulation result
III.4.2 Academic test – 2D problem
III.5 Conclusion
Chapter IV: Applications
IV.1 Application of the punching effect simulation
IV.1.1 Academic application – Single phase inductance
IV.1.2 Industrial application – Synchronous machine
IV.1.3 Synthesis
IV.2 Comparison of full plastic strain and degradation profile
IV.2.1 Finite element analysis – Steel sheet
IV.2.2 Finite element analysis – Tooth of an electrical machine
IV.3 Analysis of the degradation profile choice
IV.4 Conclusion
Conclusion and perspectives
Annex A 2D magnetostatic formulation
References .
