Multiscale magneto-elastic modeling of magnetic materials including isotropic second order stress effect

Abstract

Compact and high speed electromechanical systems lead to higher and higher levels of multiaxial mechanical stress, that may strongly change the magnetic behavior of materials, making the development of highly accurate magnetic models a very important task. Among all available magnetoelastic models, multiscale approaches seem to be the most promising.The coupling effect is introduced at the single crystal scale, mainly attributed to the evolution of magnetic domain structures under magneto-mechanical loading. All these models use however a formulation of free energy where the energetic term describing magneto-elastic coupling is linearly stress dependent, that hardly allow (using artificial mechanisms) providing non-monotonic stress effect on the magnetic behavior. The proposition detailed in this paper is to consider a second order stress term in the free energy expression, allowing a linear dependance of magnetostriction tensor with stress to be defined and providing the non-monotonous stress effect. This introduction leads to a more complex description of magnetoelastic effect that needs the identification of a large number of complementary material constants. In this paper, developments are made in the frame of cubic symmetry for first order magneto-elastic term (joining the classical description) and considering an isotropic second order stress effect for the sick of simplicity. This simplification leads to only two additive physical constants to be identified. An identification procedure is proposed and applied to model the magnetoelastic behavior a non-oriented (NO) 3wt%silicon-iron electrical steel.

Multiscale Modeling of Magnetostrictive Materials

“…λ 100 and λ 111 ) considering the (usual) incompressibility condition. However a second order development in stress is theoretically possible, as recently proposed in [5], leading to the so-called morphic effect. Its practicle implementation remains however challenging since introducing this secondorder term involves the identification of the terms of a sixth-rank tensor E α (M 2 s E α is the morphic tensor) following:…”

Piezomagnetic behavior: experimental observations and multiscale modeling

“…A higher stress level progressively shifts the Villari reversal point to the lower magnetic field values. It can be noticed that the Villari reversal can be related to the change of sign of the magnetostriction vs. magnetic field behavior [10]. Figure 4 shows the magnetic response associated with the variation of stress (σ m = 100 MPa, ∆σ = 200 MPa, f mec = 0.5Hz) for H stat = (850 A/m, 1700 A/m,3400 A/m, 8500 A/m, 15000 A/m).…”

“…The model is derived from a description of reversible magneto-elastic behavior [10,11] extended recently to magnetic hysteresis [12]. This description relies on the definition of the material Gibbs free energy at the magnetic domain scale and the estimation of the domains volume fractions using a at equilibrium stochastic approach at the grain scale.…”

“…The Gibbs free energy density is written at the magnetic domain scale α. The reader can find in [10] complete and comprehensive explanation and hypotheses about the construct of this energy functional. Indeed at the domain scale, both magnetization M α and magnetostriction strain ε µ α (seen as a free deformation tensor) can be considered as homogeneous.…”

Finite element implementation of a multi-scale dynamic piezomagnetic continuum model

“…One approach to modelling is to incorporate the heterogeneity of the material within a multi-scale piezomagnetic framework. The different phases at the lower level of observation are modelled explicitly and homogenisation principles may be applied to derive effective properties [1,2,18,27,28]. This requires the identification of a suitable Representative Volume Element at the lower scale of observation, after which Fourier transforms [1,2] or Eshelby solutions [18,27,28] may be used to quantify the relevant effective properties.…”