Journal of Materials Processing Technology 161 (2005) 141–145

Comparison of Jiles–Atherton and Preisach models extended to stress dependence in magnetoelastic behaviors of a ferromagnetic material Takaaki Suzukia,∗ , Eiji Matsumotob,1 a b

Department of Mechanical Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan Department of Energy Conversion Science, Kyoto University, Yoshida-Honmachi, Sakyo-Ku, Kyoto 606-8501, Japan

Abstract The Jiles–Atherton and the Preisach models are extended and employed in the rate-type constitutive equations of magnetoelastic materials. These models have individual advantages in applicable materials, suitability of numeric calculations, easiness of parameter identification, etc. From the comparison of the induced stress magnetization effects, it is proved that the constitutive equations maintain the original magnetization characteristics and can unifiedly describe the various magnetoelastic behaviors. In particular, it is confirmed that the constitutive equations with the extended Preisach model can express the stress magnetization curve of nickel more accurately than the one with the extended Jiles–Atherton model. © 2004 Elsevier B.V. All rights reserved. Keywords: Ferromagnetic material; Magnetoelastic coupling; Constitutive equation; Jiles–Atherton model; Preisach model

1. Introduction Recently, there is increasing evidence that structural and electromagnetic materials are subjected to strong magnetic fields and/or mechanical forces. For efficiency and safety of electromagnetic instruments and structures, it becomes important to analyze the magneto-mechanical behaviors of ferromagnetic materials, cf. [1]. Most conventional magnetization models with magnetoelastic couplings do not have generality to describe complicated magnetoelastic phenomena. Because these models do not include a mechanical constitutive equation with magnetoelastic couplings, they cannot describe the mechanical behavior under the magnetic field, e.g. the stress–strain curve under the magnetic field and the magnetostriction curve under the stress and the magnetic field. Suzuki and Matsumoto [2] proposed the magnetic and the mechanical constitutive equations applicable to more complicated phenomena or more wide class of materials. The constitutive equations can involve the conventional typical ∗

Corresponding author. Tel.: +81 75 753 3565; fax: +81 75 753 3565. E-mail addresses: [email protected] (T. Suzuki), [email protected] (E. Matsumoto). 1 Tel.: +81 75 753 5247; fax: +81 75 753 5247. 0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.016

magnetization models extended to the stress dependence. In this paper, as a magnetic constitutive equation, we extend the Jiles–Atherton model (JA model) and the moving Preisach model (MP model). Assuming that there exists a local equilibrium magneto-mechanical state, it is shown that the stress magnetization effect can be derived from the stress dependencies of the magnetization and the magnetostriction curves [2]. For two models, the stress magnetization effects are analyzed by the magnetic and the mechanical constitutive equations, and the numerical results are compared with the experimental result on nickel.

2. Magnetic and mechanical constitutive equations based on continuum theory According to Suzuki and Matsumoto [2], one-dimensional magnetic, mechanical and magnetoelastic behaviors of ferromagnetic materials under parallel uniaxial stress and magnetic field are governed by


˙ M ˙ E



 =

A1 (H, M, T, ξ)± A2 (H, M, T, ξ)±

A3 (H, M, T, ξ)± A4 (H, M, T, ξ)±




˙ H T˙

 (1)

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where M is the magnetization, E the longitudinal strain, H the magnetic field and T the uniaxial stress. The dot denotes the time derivative, superscripts ± distinguish the different functional forms, which give rise to the magnetic or the mechanical hysteresis. Variable ξ = {H1 , H2 , . . . , Hn } denotes the turning points of the magnetic field, i.e. the ordered values at which the direction of the magnetic field is reversed. Consider that the magnetic field is applied under the constant stress. Substituting T˙ = 0 into the constitutive Eq. (1), we obtain  ˙  M ∂M  A1 (H, M, T, ξ)± = = (2) ˙ T =const ∂H H  ˙ E ∂E ±  A2 (H, M, T, ξ) = (3) =  ˙ ∂H H T =const Assuming that H and T uniquely determine M and E in the present magnetization process, we can identify the coefficient A1 , the differential susceptibility defined by the slope of the magnetization curve under the constant stress, and the coefficient A2 , the slope of the magnetostriction curve with respect to the magnetic field. As a specified expression for the magnetostriction curve, we employ a polynomial series of the magnetization, cf. [2]. By considering the magnetic field as a function of the magnetization in the present process under the fixed stress, we have A± 2 =

∂E ∂M ∂E ∂E ± = = A ∂H ∂M ∂H ∂M 1

(4)

which implies that the coefficient A2 is expressed by the product of the differential susceptibility A1 and the slope of the magnetostriction curve expressed in the magnetization. On the other hand, consider that the stress is applied un˙ = 0 into the der the constant magnetic field. Substituting H constitutive Eq. (1), we obtain,  ˙  M ∂M  A3 (H, M, T, ξ)± = = (5) ∂T T˙ H=const  ˙ E ∂E ± A4 (H, M, T, ξ) =  (6) = ˙ ∂T T H=const Similarly to the above equations, we can identify the coefficient A3 , the slope of the stress–magnetization curve, and the coefficient A4 , the slope of the strain–stress curves, respectively, under the constant magnetic field. Coefficient A4 exhibits the elastic compliance for uniaxial stress. Consider the magnetic or the mechanical processes starting from the same state corresponding to either of the upper or the lower magnetization curves in the major loop. In the above discussions, we have implicitly assumed that the magnetization and the strain are determined only by the final values of the stress and the magnetic field. Then, there exists the symmetry of the coefficients [2], A3 (H, M, T, ξ)± =

A2 (H, M, T, ξ)± µ0

(7)

where µ0 is the magnetic permeability of the vacuum. In the following discussion, we extend the JA and the MP models to include the stress dependence as specified forms of coefficient A1 . 2.1. Stress-dependent Jiles–Atherton model (EJA model) Jiles and Atherton [3] proposed the differential equation, which exhibits the magnetization curve at free stress. The expression describes the hysteresis using the concept of the impedances to domain wall motion caused by pinning effects. Since the stress dependence of the magnetization process is mainly caused by variation of the magnetic domain structure due to the stress, we may assume that the material parameters in the JA model are stress dependent [4]. Such an extended JA model is characterized by means of the specified form of coefficient A1 : A1 (H, M, T, 0)± =

dM Man − Mirr = (1 − c) dH ±k − α(Man − M) +c

dMan dH

(8)

where c is the ratio of the initial differential susceptibilities of the normal and the anhysteretic magnetization curves, α a mean field parameter representing interdomain coupling and k a parameter characterizing the pinning energy. The order of complex signs is preserved, and the complex signs bring hysteresis in the magnetization curve. In Eq. (8), Man is the anhysteretic magnetization expressed in terms of Langevin function L(x) in the above theory,    H + αM a Man = Ms coth + (9) a H + αM Here Ms is the saturation magnetization, a a parameter with the dimension of the magnetic field characterizing the shape of the anhysteretic magnetization curve, and parameters a, α and Ms depend on T. Since the JA model has the local memory property, the past turning points ξ do not explicitly appear in coefficient A1 . Similarly to the original model, the EJA model is also convenient for identification of parameters and numerical analysis. 2.2. Stress-dependent moving Preisach model (EMP model) As another specified form of coefficient A1 , we divide the differential susceptibility into the irreversible and the reversible parts. For the irreversible susceptibility, we extend the moving Preisach model, which has been used to describe the hysteretic magnetization curve at free stress [5]. According to Vecchio [6], the change in the magnetic domain structure is expressed by the integral of distribution function p(a,b), i.e. the existence probability of elementary domains with respect to the switching fields a, b. Consider that the magnetic field changes from H1 to H2 . The magnetization

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change is expressed by following forms for H1 < H2 or H1 > H2 , respectively,

M(H1 , H2 ) =

M(H1 , H2 ) =



H2 H1

da

H1 H2

da

a H1 a H2

dbp (a, b), db p(a, b).

(10)

As a specified form of the distribution function, several expressions were proposed [7–9]. In this paper, we extend the distribution function, which Basso et al. [9] applied to soft ferromagnetic materials, i.e. the product of Lorentz functions, p(α, β, T ) =

c [1 + (α/H0 − d) ][1 + (−β/H0 − d)2 ] 2

(11)

Because the domain structure varies with the stress, we have assumed that the material parameters H0 , c and d depend on the stress. Substituting Eq. (11) into Eq. (10) and differentiating the result with respect to the magnetic field, we obtain the irreversible differential susceptibility, Lirr (H, M, T, ξ)± =

c (H/H0 ) ∓ 2d(H/H0 ) + 1 + d 2    H −1 · tan d± H0   H(ξ)tp −tan−1 d ± H0 2

Fig. 2. Numerical stress magnetization curve using EJA model.

On the other hand, as the reversible susceptibility, we also extend the expression proposed by Basso et al. [9] as follows,   −|H| Lrev (H, T ) = µi exp (13) Hrev where the parameters µi and Hrev are assumed to depend on the stress. The total differential susceptibility A1 is determined by the sum of the irreversible and reversible susceptibilities (Eq. (12) and Eq. (13)): A1 (H, M, T, ξ)± =

(12)

where the function H(ξ)tp denotes the influential turning point, the specified magnetic field within ξ which affects the present magnetization state.

∂M ∂Mrev (H, T ) = ∂H ∂H ∂Mirr (H, M, T, ξ) + ∂H

= Lrev (H, T ) + Lirr (H, M, T, ξ)±

(14)

This model as well as the original MP model, has non-local memory property and can describe more diverse magnetic behaviors with higher accuracy.

Fig. 1. Stress magnetization curves under bias magnetic field 3.0 kA/m.

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Fig. 3. Reversible and irreversible magnetization changes due to bias magnetic field.

3. Stress magnetization change of nickel In this section, as a typical phenomenon of the magnetoelastic couplings, we analyze the stress magnetization effect, the magnetization change by the cyclic stress under the bias magnetic field. Fig. 1(a) shows the experimental stress magnetization change under bias magnetic field 3.0 kA/m. For the stress given as a function of time, we numerically solve the constitutive Eq. (1) for the magnetization using the EJA or the ˙ = 0. For the EJA model, the numerical EMP models with H stress magnetization change from the upper magnetization curve is shown in Fig. 2. We find that there is considerable disagreement with the experimental result (Fig. 1(a)). The numerical magnetization completely converges to the neutral magnetization due to the cyclic stress with the small amplitude 10 MPa. The experimental magnetization, however, does not seem to reach the neutral magnetization by the small stress amplitude, and its curve closes after the irreversible magnetization change. A similar closure behavior can be seen in M–H loops below the saturation magnetization, i.e. in the minor and the asymmetric loops. The restriction for such closure behaviors has not been taken into account in the original JA model. Therefore, we impose the closure restriction in the EJA model similarly to the case of the ferromagnetic materials with negative magnetostriction. That is, we introduce the concept of effective volume and impose the negative slope in M–T curve, cf. [2]. Then, we obtain Fig. 1(b) for the modified EJA model with the closure restriction, and Fig. 1(c) for the EMP model, respectively. It is seen that the physical inconsistencies and shortages of the EJA model do not appear in the EMP model in the stress magnetization change. From the figures, the constitutive equations using the modified EJA and the EMP models can describe the closure loop after the irreversible magnetization change of nickel. In particular, the EMP model can exhibit the stress magnetization curve more accurately than the EJA model.

For analysis of the experimental and the numerical results, we define indices of the stress magnetization change as in Fig. 3(a). From Fig. 1(b) and (c), the reversible and the irreversible magnetizations are calculated as in Fig. 3(b) and (c). Here, it can be seen that numerical results of two models are in good agreement with the experimental result. For comparison of the experimental and the numerical magnetization changes, we calculate the sum of the absolute values of the differences between the experimental and the numerical results. Then, it is seen that the disagreement of the EMP model is from 1/2 to 1/20 of that of the EJA model, except for the reversible magnetization from the upper magnetization curve. In a result, the constitutive equations with the EMP model can represent the stress magnetization curve of nickel more accurately than those with the EJA model. The equivalence of the JA and the MP models is demonstrated by referring to fundamental energy relations [10]. The difference in two models lies in the specification of the material functions, i.e., the EJA model has specified the material functions, and the EMP model reserves the arbitrariness of the function. From those characteristics, the EJA model has advantages in easiness of identification of the parameters and numerical calculations. On the other hand, the EMP model can represent wider class of materials from the arbitrariness of the material function. The original rate-type constitutive equations can involve both models as well as other models, including the extension to magnetoelastic couplings, so that they will be applicable to more complicated phenomena and more wide class of materials.

Acknowledgements This research was supported by the Grant-in-Aid for Research Fellows of the Japan Society for the Promotion of Science.

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