Three-dimensional constitutive equations of ferromagnetic materials with magnetoelastic coupling

Journal of Materials Processing Technology 181 (2007) 165–171

Three-dimensional constitutive equations of ferromagnetic materials with magnetoelastic coupling Determination of elastic coefficients under magnetic field Eiji Matsumoto a,∗ , Tomomi Hase a , Takaaki Suzuki b a

b

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

Abstract This paper determines the elastic coefficients in the constitutive equations of typical ferromagnetic materials. The constitutive equations are given by rate-type equations for the magnetization and the strain expressed in terms of the magnetic field and the stress. The authors have shown that the constitutive equations can describe the stress-dependent hysteretic magnetization curve, the stress magnetization effects, the stress-dependent magnetostriction curve, etc. They have determined the coefficients except the elastic coefficients as functions of the magnetization and the stress for typical ferromagnetic materials. In this paper, we attempt to obtain the elastic coefficients depending on the magnetization and the stress. To do it, applying the magnetoacoustic effect, we measure the stress–strain curves and the speeds of the longitudinal and the transverse ultrasonic waves under the magnetic field. © 2006 Elsevier B.V. All rights reserved. Keywords: Ferromagnetic material; Magnetoelastic coupling; Three-dimensional constitutive equations; Elastic coefficients; Magnetostriction

1. Introduction Magnetoelastic coupling is an interaction between the mechanical and the magnetic fields in the ferromagnetic materials. For example, the deformation caused by the magnetic field is called the magnetostriction, and the magnetization change due to the stress is called the stress-magnetization effect. Magnetoelastic coupling is also seen in the change of the magnetic or the mechanical properties of the material, respectively, by the applied stress or the magnetic field. The magnetoelastic coupling plays important roles in sensors [1], actuators [2], and the effect influences the efficiency of electromagnetic instruments and safety of electromagnetic structure under strong magnetic field [3]. Theoretical study of magnetomechanical behaviors with magnetoelastic coupling has been done by Brown [4] in the frame of continuum theory for deformable magnetic materials, which was extended to include finite deformations by Pao and Yeh [5]. Matsumoto [6] derived rate-type constitutive equations which can represent many magnetomechanical behaviors including magnetic and mechanical nonlinearity and hysteresis. Suzuki and Matsumoto [7,8] determined the constitutive param-

∗

Corresponding author. Fax: +81 75 753 5247.

0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.03.038

eters for several typical ferromagnetic materials including giant magnetostriction materials, and they examined the validity of the constitutive equations. In this paper, we propose the method to determine the elastic coefficients of the constitutive equations which have been assumed to be constant in the previous studies. In the following section, we introduce the three-dimensional constitutive equations, and analyze the magnetomechanical behavior under the magnetic field parallel to the uniaxial stress in Section 3. In Section 4, we determine all the elastic coefficients as functions of the magnetization by measuring the longitudinal and the transverse strains and the speeds of the longitudinal and the transverse ultrasonic waves polarized in the parallel and the normal directions to the magnetic field. 2. Rate-type constitutive equations for ferromagnetic materials Neglecting the effect of the exchange energy, we can assume that the mechanical and the magnetic states of a deformable magnetic material are prescribed by the current values and the past histories of the strain E and the magnetization M. In order to take into account the magnetic or the mechanical hysteresis, Matsumoto [6] and Suzuki and Matsumoto [7] proposed the rate

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type constitutive equations expressed as ± ˙ ˙ i = P(T , E, H, M)± ˙ H ij Mj + Q(T , E, H, M)ijk Ejk , ± ˙ ˙ T˙ ij = R(T , E, H, M)± ijk Mk + S(T , E, H, M)ijkl Ekl ,

(1)

where dot denotes time derivative, double indices are summed from 1 to 3, and T and H are the stress and the magnetic field, respectively. Superscripts ± of the coefficients distinguish the different functional forms corresponding to the increasing and the decreasing curves of the hysteresis. Since the constitutive equations are homogeneous with respect to the time derivatives, they exhibit rate-independent magnetoelastic behaviors. The inverse form of (1) will be more convenient to determine the constitutive parameters because the physical meanings of their coefficients are more explicit. That is, from (1) we can obtain ± ˙ ˙ i = A(T , E, H, M)± ˙ M ij Hj + B(T , E, H, M)ijk Tjk , ± ˙ ˙ ij = C(T , E, H, M)± H ˙ E ijk k + D(T , E, H, M)ijkl Tkl .

Then coefficient A is the differential susceptibility tensor under fixed stress and B the slopes of the stress–magnetization curves under fixed magnetic field. Coefficient C is the slopes of the magnetostriction curves under fixed stress and D the elastic compliances (the inverse of the elastic coefficients) under fixed magnetic field. When the magnetomechanical process is reversed, M and E are governed by the constitutive equations with the other set of coefficients. In this case, we implicitly assume that the initial values of M and E for the new process are given by the final values for the previous process such that M and E change continuously. 3. One-dimensional magnetomechanical process Let us consider that the magnetic field and the uniaxial stress are applied in the same direction along X1 axis to an isotropic ferromagnetic material. The magnetic and the mechanical changes in the material are sufficiently slow so that the process can be regarded as quasistatic. From the symmetry, the components of the magnetization and the strain tensor take the form M = (M, 0, 0),

E22 = E33 ,

Eαβ = 0(α = β).

Fig. 1. Magnetization curve of low carbon steel under uniaxial stress.

(2)

(3)

that the final magnetomechanical state does not depend on the path while the magnetic or the mechanical process lies in one side of the hysteresis. That is, the coefficients A4 can be determined by A2 . They have obtained coefficients A1 , A2 , A3 , A4 and A7 as functions of the stress and the magnetization for typical ferromagnetic materials. On the other hand, coefficients A5 , A6 , A8 and A9 were assumed to be constant from difficulty of measurement. Let us first apply the uniaxial stress and then the magnetic field with the stress fixed. For the magnetization process under constant stress, from (4) we have the relations  ˙  M dM ±  A1 = = , (5)  ˙ dH H T =const A± 2

 ˙ 11  E dE11  = = ,  ˙ dH H T =const

 ˙ 22  E dE22  = = .  ˙ dH H T =const (6)

Here the right hand sides of (5) and (6) mean the derivatives of M–H, E11 –H and E22 –H curves with respect to H for fixed T. Hence, coefficients A1 is the magnetic differential susceptibility and A2 and A3 the slopes of the longitudinal and the transverse magnetostriction curves for fixed T. Fig. 1 shows the magnetization curves of a typical structural material, low carbon steel SM490A, at several stress levels. We see that in coefficient A1 , the dependence on M or H is much larger than

Substituting (3) into the constitutive Eq. (2), we have the one-dimensional version of the constitutive equations, i.e., the relations among the longitudinal components H, M, T of the magnetic field, the magnetization and the stress tensor, and the longitudinal and the transverse strains E11 and E22 : ⎧ ⎫ ⎡ ⎤⎧ ⎫ ˙ ˙ A1 (H, M, T )± A4 (H, M, T )± A7 (H, M, T )± ⎪ ⎪ ⎨ M ⎪ ⎬ ⎨H ⎪ ⎬ ⎢ ⎥ ˙ 11 = ⎣ A2 (H, M, T )± A5 (H, M, T )± A8 (H, M, T )± ⎦ T˙ . E ⎪ ⎪ ⎩˙ ⎪ ⎭ ⎩ ⎪ ⎭ 0 E22 A3 (H, M, T )± A6 (H, M, T )± A9 (H, M, T )± The authors [7] have shown that coefficient A1 represents the hysteretic magnetization curve, and A2 and A3 the longitudinal and the transverse magnetostriction curves, respectively. They also proved that there exists a symmetry in the coefficients matrix in (4), if we adopt the local equilibrium condition such

A± 3

(4)

the stress dependence from comparison of nonlinearity of the curve with its shift by the stress. The authors showed that this term with A1 involves the Jiles-Atherton and the Preisach models extended to the stress dependence as special cases, and that the

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obtained theoretical magnetization curves are in good agreement with the experimetanl results for typical ferromagnetic materials. The authors also proposed the model of the higher order stress-dependent magnetostriction curves based on the above formulation. When we apply a uniaxial stress under a fixed magnetic field, similarly to the above discussion, we can derive  ˙  M dM ±  A4 = = , (7)  ˙ dT T H=const

terms and the magnetic dependence of the coefficients in the constitutive equations. In the case of variation of the uniaxial stress at fixed magnetic field, the stress and the strain is governed by (8) where the magnetoelastic coupling terms have no effect. Thus, for ultrasonic waves or variation of uniaxial stress, the stress–stain relation is given by a similar form to the pure mechanical one, while the elastic coefficients are determined by the bios magnetomechanical state. In conclusion, considered variation of the stress σ and the strain ε is governed by the generalized Hooke’s law with uniaxial anisotropy

 ˙ 11  E dE11  = , dT T˙ H=const  ˙ 22  E dE22  A± = = 6 dT T˙ H=const

σij = S(M, T )± ijkl εkl

A± 5 =

(8)

Thus, coefficients A4 , A5 and A6 are the slopes of the stress magnetization curve and stress–longitudinal strain and stress–transverse strain curves for fixed H. By applying the local equilibrium condition of magnetomechanical processes [7], we have the symmetry of the coefficients matrix of (4) in the form A2 (H, M, T )± A4 (H, M, T ) = , µ0 ±

(9)

where µ0 is the magnetic permeability of the vacuum. We see that from the above symmetry the functional form of coefficient A4 can be determined by A2 , in other word, the stress–magnetization curve can be determined by the longitudinal magnetostriction curve. Coefficients A7 , A8 and A9 play no role in the present process, but we note that by exchanging the roles of X1 and X2 axes, derived are the symmetry between (A3 , A7 ), (A5 , A9 ) and (A6 , A8 ). However, in this case argument H, M, T in the coefficients should be also replaced with the X2 components, so that the summetry cannot be used to determine A7 , A8 and A9 from A3 , A6 and A5 . In the next section we shall determine all the elastic coefficients including these coefficients under the one-dimensional magnetoelastic proces (3) by measuring the strains and the ultrasonic wave speeds under the magnetic field and the uniaxial stress. 4. Elastic coefficients under magnetic field Let us consider that a small dynamic motion or a static deformation is superposed on the bios state discussed in the previous section. It is easy to see that the mechanical property of the material becomes anisotropic, i.e., the uniaxial anisotropy along the applied magnetic field and the uniaxial stress through the magnetoelastic coupling and the mechanical nonlinearity. In general, a mechanical wave or a deformation follows the magnetization change from the magnetoelastic coupling corresponding to the terms with R or C in the constitutive Eq. (1) or (2), respectively. In the case of ultrasonic waves, however, the variation of the mechanical field is very small and fast, so that movement of the magnetic domain walls does not occur, which implies that for the ultrasonic wave we can neglect the magnetoelastic

or

σij = S(H, T )± ijkl εkl ,

(10)

where the independent elastic coefficients are five as expressed in the form ⎞ ⎡ ⎤ ⎛ S1111 S1122 S1122 σ11 0 0 0 ⎜σ ⎟ ⎢S 0 0 0 ⎥ ⎥ ⎜ 22 ⎟ ⎢ 1122 S2222 S2233 ⎟ ⎢ ⎥ ⎜ ⎜ σ33 ⎟ ⎢ S1122 S2233 S2222 0 0 0 ⎥ ⎟=⎢ ⎥ ⎜ ⎜σ ⎟ ⎢ 0 0 0 S2323 0 0 ⎥ ⎥ ⎜ 23 ⎟ ⎢ ⎟ ⎢ ⎥ ⎜ ⎝ σ31 ⎠ ⎣ 0 0 0 0 S1212 0 ⎦ σ12

0 ⎛

0 ⎞

0

ε11 ⎜ ε ⎟ ⎜ 22 ⎟ ⎜ ⎟ ⎜ ε33 ⎟ ⎜ ⎟. ×⎜ ⎟ ⎜ 2ε23 ⎟ ⎜ ⎟ ⎝ 2ε31 ⎠

0

0

S1212

(11)

2ε12 Here arguments M or H and T are omitted in the coefficients and S2323 is determined by S2222 and S2233 as S2323 = 21 (S2222 − S2233 ).

(12)

4.1. Uniaxial tension in X1 direction We have already assumed the local equilibrium condition, so that the total strain for the prescribed stress and the magnetic field does not depend on the path while the process lies on the same part of the hysteresis. Then by exchanging the order of applying the magnetic field and the uniaxial stress, we have   Eij (H, T ) = Eij (0, T ) + EijM (H, T ) T =const  M  = Eij (H, 0) + Eij (H, T ) H=const , (13) where EM is the magnetostriction tensor, i.e., the change of the strain tensor due to the magnetic field H. From the symmetry similar to the total strain as shown in (3), the nonvanishing magM , EM = EM . netostrictions are three normal components E11 22 33 Fig. 2 shows the longitudinal and the transverse magnetostricM and EM at each stress level for SM490A. We see that tions E11 22 the material elongates in the direction of the applied magnetic field, and shrinks in the lateral direction. The absolute values of the magnetostrictions increase with the magnetization up to around 1.5T but they start to decrease for the magnetization

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stress–strain relation and the strain–displacement relation: ρ

∂σij ∂ 2 ui = , ∂t 2 ∂Xj

(17a)

σij = Sijkl εkl ,   1 ∂ui ∂uj , εij = + 2 ∂Xj ∂Xi

(17b) (17c)

where the elastic coefficients take the form in (11). A plane harmonic wave propagating in X2 direction is expressed as ¯ i(kX2 −ωt) , u = ue

(18)

where k is the wave number, ω the frequency and u¯ is the amplitude indicating the polarized direction. In the case of the longitudinal wave, substituting the solution (18) with the amplitude u¯ = (0, u¯ 2 , 0)

(19)

into (17c) and then (17b), we have the non-vanishing stress component σ22 = S2222 ε22 .

Fig. 2. Magnetostriction curve under uniaxial stress.

higher than 1.5T, which is called the Villary effect. The longitudinal and the transverse magnetostriction curves shifts with the applied stress, and it seems that each curve takes the maximum absolute value around the tensile stress 30 MPa. From (13), the strain change due to the stress at fixed H can be expressed as    Eij (H, T )H=const = Eij (0, T ) + EijM (H, T ) T =const

− EijM (H, 0).

(14)

From the discussion in the previous section, small change in the stress and the strain is governed by the generalized Hooke’s law (10), so that substituting the uniaxial stress into (11), we have dε11 S2222 + S2233 =− 2 , dT 2S1122 − S1111 (S2222 + S2233 )

(15)

dε22 S1122 = 2 . dT 2S1122 − S1111 (S2222 + S2233 )

(16)

(20)

Further substituting (20) into the equation of motion (17a), we obtain the speed of the longitudinal wave as  ω S2222 vL = = . (21) k ρ The uniaxial anisotropy is a special case of orthotropic anisotropy. Then it is known that only two polarized directions are allowed for the transverse wave propagating along one of the anisotropy axes. That is, the polarized directions coincide with the other two axes of the orthotropic anisotropy, i.e., X1 and X3 axes in our case. For the transverse wave polarized in X1 direction, the amplitude and the non-vanishing stress component are given by u¯ = (u¯ 1 , 0, 0),

σ12 = 2S1212 ε12

(22)

and the wave speed is calculated from the governing equations and the above conditions in the form  S1212 ω vT 1 = = . (23) k ρ

From (15) and (16), we may obtain the slopes of the stress–strain curves at each magnetic field by taking into account the upper and the lower sides of the hysteresis curves of the magnetization.

In the case of the transverse wave polarized in X3 direction, by a similar process to the above, the wave speed is given by   ω S2323 S2222 − S2233 = . (24) vT 3 = = k ρ 2ρ

4.2. Ultrasonic waves propagating in uniaxial anisotropic state

If we remember that the uniaxial anisotropy has been induced by the magnetization M and the uniaxial stress T, the speeds of the longitudinal and the transverse waves may be functions of M and T, or equivalently, of H and T:

Let us consider the ultrasonic wave propagating along X2 direction in the uniaxial symmetric material with X1 axis. The governing equations consist of the equation of motion, the

± VL,T = VL,T (M, T )

or

± VL,T = VL,T (H, T ).

(25)

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In general, dependence of the wave speed on the stress is called the acoustoelastic effect which comes from the mechanical nonlinearity, i.e., from the dependence of the elastic coefficients on T. Furthermore, as well as the elastic coefficients, the above wave speed may have hysteresis for increasing and decreasing magnetization as indicated by superscript ±. 4.3. Determination of elastic coefficients under uniaxial stress and magnetization We perform uniaxial tension test under the magnetic field and measure the ultrasonic wave speeds as discussed in Sections 4.1 and 4.2 for SM490A. Then the elastic coefficients can be determined in the following order. S2222 = ρv2L ,

S1212 = ρv2T 1 ,

S2323 = ρv2T 3 ,

Fig. 4. Speed of transverse wave polarized in parallel direction to magnetic field.

S2233 = S2222 − 2S2323 , dε22 dε11 S1122 = −(S2222 + S2233 ) / , dT dT   dε22 dε11 S1111 = 1 − 2S1122 . / dT dT

(26)

We apply the uniaxial tension up to 100 MPa, and at each stress level, we apply sufficiently slow cyclic magnetic field in the same direction to the uniaxial stress. After the magnetization process becomes steady cycle, at each magnetic field on the magnetization curve, we measure the longitudinal and the transverse strains, the magnetostrictions and the speeds of three ultrasonic waves. Figs. 3–5 show the obtained wave speeds drawn by curves for each fixed stress. We see that the wave speeds changes according to the magnetization and each curve has hysteresis and shifts with the applied stress. In fact, the speeds of all the waves increase according as the magnetization becomes large and the speed–magnetization curve shifts lower for the transverse wave polarized in X1 direction as the stress increases. Conversely, the speed–magnetization curves for the longitudinal wave and the transverse wave polarized in X3 direction move higher as

Fig. 3. Speed of longitudinal wave under magnetic field.

the stress increases. In particular, variation of the longitudinal wave speed due to the magnetization becomes small according as the stress increases, and a similar tendency is found in the speed of the transverse wave polarized in X3 direction. As for the slopes of the stress–strain curves at each magnetic field, we may utilize the magnetostriction curves at each stress level discussed in Section 4.1. In order to obtain the slopes of the stress–strain curves, we used the third order polynomial approximation of the stress–strain curves at each prescribed magnetic field on the hysteretic magnetization curve. From the above data, we can calculate all the elastic coefficients by (26), where we have used ρ = 7.8 × 103 kg/m3 . Figs. 6–10 show the obtained elastic coefficients. We see that each elastic coefficient change with the magnetization and a strong anisotropy is induced in the direction of the uniaxial stress and the applied magnetic field. In general, the longitudinal and the transverse elastic coefficients, S2222 , S1111 and S1212 , S2323 become large for increasing magnetization, but S1122 varies in more complicated manner with the magnetization. Comparing the shear moduli S1212 and S2323 , we see that S1212 is higher than S2323 , which may come from the difference between the experimental conditions like temperature or from the texture

Fig. 5. Speed of transverse wave polarized in normal direction to magnetic field.

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Fig. 6. Elastic coefficient S2222 under magnetic field. Fig. 9. Elastic coefficient S1122 under magnetic field.

Fig. 7. Elastic coefficient S1212 under magnetic field. Fig. 10. Elastic coefficient S1111 under magnetic field.

anisotropy induced by material processing. If the material is isotropic and the experimental conditions are the same, two moduli should be the same at the nonmagnetized and free stress state, M = 0 and T = 0. All the elastic coefficients have hysteresis

for increasing and decreasing magnetization. We should note that the uniaxial tension test gives larger error compared with the ultrasonic speed measurement which is carried by the sing around method. This fact and (26) imply that elastic coefficients S1111 and S1122 have larger fluctuation for the magnetization change. 5. Conclusion

Fig. 8. Elastic coefficient S2323 under magnetic field.

The rate-type constitutive equations have been proposed as a theoretical model of deformable ferromagnetic materials which have magnetoelastic coupling, the mechanical and magnetic nonlinearity, the magnetic hysteresis, etc. In this paper, presented is the method to determine all the independent elastic coefficients by means of uniaxial tension test and measurement of ultrasonic wave speed under the stress and the magnetic field. It is shown that the obtained wave speeds change in a hysteretic manner with the magnetization, and each curve shift with the applied stress. The calculated elastic coefficients have also similar tendency, and it is confirmed that the uniaxial anisotropy

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is induced from the magnetoelastic coupling and the mechanical nonlinearity. The magnetomechanical constitutive equations taking into account the change of the elastic coefficients under the magnetic field may be useful for precise design of the electromagnetic instruments and the structure utilizing strong magnetic field. References [1] A. Zhukov, A.F. Cobeno, J. Gonzalez, J.M. Blanco, P. Aragoneses, L. Dominguez, Magnetoelastic sensor of liquid level based on magnetoelastic properties of co-rich microwires, Sens. Actuators, A Phys. 81 (1–3) (2000) 129–133. [2] H. Eda, Y. Yamamoto, Giant magnetostriction materials, Int. J. Jpn. Soc. Eng. 31 (2) (1997) 83–86.

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[3] K. Horiguchi, Y. Shindo, Bending tests and magneto-elastic analysis of ferritic stainless steel plate in a magnetic field, Trans. JSME A 64 (1998) 1296–1301. [4] W.F. Brown Jr., Magnetoelastic Interactions, Springer-Verlag, 1966. [5] Y.-H. Pao, C.-S. Yeh, A linear theory for soft ferromagnetic elastic solids, Int. J. Eng. Sci. 11 (1973) 415–436. [6] E. Matsumoto, Stress-magnetization effects based on rate-type constitutive equations of deformable ferromagnetic materials, Rom. J. Tech. Sci. Ser. E & E 43 (3) (1998) 391–396. [7] T. Suzuki, E. Matsumoto, Magnetoelastic behavior of ferromagnetic materials using stress dependent Preisach model based on continuum theory, Int. J. Appl. Electromag. Mech. 19 (2004) 485–489. [8] T. Suzuki, E. Matsumoto, Accurate model of ferromagnetic materials for optimal design of giant magnetostrictive actuator, JSAEM Stud. Appl. Electromag. Mech. 15 (2005) 151–158.