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Non-linear constitutive relations for magnetostrictive materials
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International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
Non-linear constitutive relations for magnetostrictive materials Yongping Wan, Daining Fang ∗ , Keh-Chih Hwang Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Abstract In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-4eld dependence, are proposed to model the magnetic-4eld-induced strain saturation of magnetostrictive materials in the region of intense magnetic 4elds. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, de4ned by the quantity of magnetic domains switched by per unit magnetic 4eld and magnetic 4eld can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coe8cient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic 4eld. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic 4eld while the HT model can, to some extent, re:ect the trend of saturation of magnetostrictive strain under a high magnetic 4eld. The model based on DDS, which is more e;ective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic 4elds. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed. ? 2002 Published by Elsevier Science Ltd. Keywords: Magnetostrictive material; Non-linear deformation; Coupling constitutive relations; Magnetostrictive strain
1. Introduction Magnetic materials have already played an important role in the development of new technologies. To investigate the mechanical laws of magnetic materials will be vital for their application and development [1,2]. Magnetic materials can show elongation ∗
Corresponding author. Tel./fax: +86-10-62772923. E-mail address: [email protected] (D. Fang).
and contraction in the magnetization direction. This is called magnetostriction that is due to the switching of a large amount of magnetic domains caused by spontaneous magnetization, below the Curie point of temperature [3]. In each domain, the lattice distorts and the magnetization direction is collinear with one of the principal axes of the spontaneous strain. Without any applied magnetic 4eld, the magnetization direction of each domain is random and there will be no macroscopic magnetization e;ect. However, the magnetization of magnetic domains will turn to the
0020-7462/03/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0 0 2 0 - 7 4 6 2 ( 0 2 ) 0 0 0 5 2 - 5
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Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
direction of the external magnetic 4eld if the material is subjected to a magnetic 4eld. The material will show elongation in the external magnetic 4eld if the direction of magnetization of magnetic domains is along the long principal axis of the spontaneous strain. This is the positive magnetostriction. That the material contracts in the direction of the external magnetic 4eld, is called as the negative magnetostriction. Generally, the positive magnetostrictive materials are employed in engineering [2,3]. For magnetostrictive materials, the strain keeps the same for either positively or negatively applied magnetic 4eld. Therefore, the vibration frequency will be twice that of the magnetic 4eld if the magnetostrictive material is only subjected to an alternating 4eld. This is the so-called double frequency e;ect. To obtain a synchronous magnetostriction under an applied alternating 4eld, a bias 4eld produced by a direct current must be applied. On the other hand, the material exhibits a sudden increase of strain at a critical magnetic 4eld when a compressive pre-stress is imposed on the material. This implies that the magnetostrictive coef4cient is more sensitive to the applied magnetic 4eld when a compressive pre-stress is imposed on the magnetic materials. Generally, magnetostrictive materials are very brittle. Their stretching yield stress is substantially smaller than their compressive yield stress. Thus, a compressive pre-stress is always applied on the magnetostrictive rod practically [4,5]. The magnetostrictive materials usually endure a coupled mechanical-magnetic 4eld when they are in application. Their constitutive relations are essentially non-linear [6,7]. This makes it di8cult to develop any theoretical models. At present, the linear or non-linear but de-coupled constitutive relations are usually employed in modeling or analyzing [8,9]. However, it is quite important to take the coupling terms into consideration if a model is designed to predict the properties of magnetostrictive materials accurately. Carman et al. [4] has proposed a non-linear coupling constitutive relation for one-dimensional problems. This model can predict some deformation characteristics of the magnetostrictive rod in a speci4c region of pre-stress. Yet, it cannot describe the saturation of magnetostrictive strain when the material is subjected to an intensive magnetic 4eld. In the present work, three phenomenological constitutive relations of magnetostrictive materials are studied. The quadratic equations of the
standard square (SS) constitutive model are derived by means of truncating the polynomial expansion of the Gibbs free energy. To produce some kind of saturation, the hyperbolic tangent (HT) constitutive equations are developed involving a HT magnetic-4eld dependence. A better physically grounded constitutive model, which is more e;ective in simulating experimental curves, is motivated in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation, and the careful analysis of the experimental results obtained by Mo;et et al. [10]. It is assumed, that the relation between domain-switching density de4ned by the quantity of magnetic domains switched by per unit magnetic 4eld and magnetic 4eld can be described by a density function with normal distribution. Furthermore, a material function is proposed to describe the dependence of the peak piezomagnetic coe8cient on the compressive pre-stress for one-dimensional cases so that the moduli in the constitutive models can be experimentally determined. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.
2. SS constitutive relation (SS model) The energy balance equation of electric–magnetic body is [8,9] d 1 u˙ i u˙ i + U dv dt v 2 = (fi u˙ i + ) dv + ti u˙ i ds; (1) v
S
where U is the internal energy density per unit mass; the electric–magnetic energy density per unit time, = −∇ · (E × H ), where E is the electric 4eld intensity vector; H the magnetic 4eld intensity vector and ∇ the gradient operator. dv and ds refer to the body element and the surface element, respectively. fi ; ti are the components of body force and surface force, respectively, and ui the component of the displacement vector. The dots in Eq. (1) represent di;erentiation with respect to time. is the mass density. Using the equations of mass conservation and
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
momentum conservation, the energy balance equation can be simpli4ed as U˙ dv = ( + ˙ij ij ) dv; (2)
compressive prestress (MPa) 1 2 3 4 5 6 7 8
3
v
where ij and ij are the components of the strain tensor and stress tensor, respectively. Considering the assumption of quasi-magnetostatics of insulators, one 4nds ˙ ∇ × E = −B:
∇ × H = 0;
Strain [10 -3 ]
v
2
6.9 15.3 23.6 32 40.4 48.7 57.1 65.4
8 7 6
5 4 3
2
1
experimental results Standard Square model
(3) 1
Then Eq. (2) renders ˙ U dv = (B˙ k Hk + ˙ij ij ) dv; v
v
0 0
(4)
where B is the magnetic induction vector. For small distortion problems, by means of the internal energy density per unit volume, there is U˙ = B˙ k Hk + ˙ij ij :
(5)
Choosing stress and magnetic 4eld as independent variables, in terms of a Legendre transform, one can get @G @G ij = ; B = (6) k @ij H @Hk in which G is the Gibbs free energy function. To obtain a constitutive relation, the Gibbs free energy function can be written in a series: G = G0 +
@G @G Nij + NHk @ij @Hk
+
1 @2 G Nij Nkl 2 @ij @kl
+
1 @2 G Nij NHk 2 @ij @Hk
+
1 @2 G NHl NHk 2 @Hl @Hk
+
@3 G 1 Nij Nkl Nmn 3! @ij @kl @mn
+
@3 G 1 Nij Nkl NHm 3! @ij @kl @Hm
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1
2
3
4
5
H [KOe]
Fig. 1. Magnetostriction curves (the calculated curves are obtained by use of the SS model).
+
@3 G 1 Nij NHk NHl 3! @ij @Hk @Hl
+
@3 G 1 NHm NHk NHl + · · · : 3! @Hm @Hk @Hl
(7)
Obviously, it is important to rationally characterize the moduli or constants in the above constitutive equations in terms of experimental results. Mo;et et al. [10,11] experimentally investigated the magnetostrictive properties of a magnetic material (Terfenol-D). The Terfenol-D rod was operated under eight di;erent bias magnetic 4elds and compressive pre-stresses, respectively. The dashed lines in Figs. 1–3 are the measured curves of strain versus the applied magnetic 4eld. Results indicate that as the compressive pre-stress becomes larger, a larger magnetic 4eld is needed to drive the rod in order to get the same stretching strain. At the same time, the tangent slope of the magnetostriction curve continues to decline, which implies that the magnetostrictive strain induced by per unit external magnetic 4eld gradually decreases as the compressive stress increases. To derive constitutive equations of the magnetostrictive materials from the series expansion of the Gibbs free energy (see (7)), one should review material properties and some experimental results. The giant magnetostrictive compound of rare earth and iron, Tb0:27 Dy0:73 Fe1:95 , possesses a crystal structure of the cubic Laves phase. The cubic phase compound
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experimental results Hyperbolic Tangent model 2
8
7 6
5
-3
Strain [10 ]
4
compressive prestress (MPa) 1 6.9 2 15.3 3 23.6 4 32 5 40.4 6 48.7 7 57.1 8 65.4
3 1
2 1
0 0
1
2
3
4
5
H [KOe]
Fig. 2. Magnetostriction curves (the calculated curves are obtained by use of the HT model).
experimental results model based on density of domain switching
-3
Strain [10 ]
1.6
3
1.2
5
6
7
8
4
2
compressive pre-stress [MPa] 1 6.9 2 15.3 3 23.6 4 32.0 5 40.4 6 48.7 7 57.1 8 65.4
0.8
1 0.4
0.0 0
1
2
3
4
5
experimental results request that the terms involving partial derivatives of odd indices should not appear in the series expansion given in (7), which means the material constant tensors of odd orders vanish. Therefore, the piezomagnetic coe8cient tensor, which is a material constant tensor of three orders and re:ects the property of piezomagnetism, vanishes for the magnetostrictive materials. Thus, the following constitutive relations are developed based on the truncated polynomial energy expansion: ij = sijkl kl + mijkl Hk Hl + rijklmn kl Hm Hn ;
(8a)
Bk = kl Hl + mklmn mn Hl + rklmnpq mn pq Hl :
(8b)
in which sijkl is the elastic compliance tensor; mijkl the 4eld magnetostrictive modulus tensor that physically denotes the magnetostrictive strain produced by per unit external magnetic 4eld. Its dimension is m2 A−2 ; rijklmn the 4eld magnetoelastic modulus tensor that physically refers to the coupling magnetostrictive strain produced by per unit external magnetic 4eld under per unit compressive pre-stress. The dimension of this modulus is m4 A−2 N−1 ; kl the permeability tensor that can be recognized as kl for isotropic materials, where ij is the Kronecker delta. 2.1. Standard square constitutive relation for one-dimensional problems
H [ KOe]
Fig. 3. Magnetostriction curves (the calculated curves are obtained by use of the DDS model).
is a non-polar crystal, which deforms elastically without any magnetization under a stress. There do not exist such properties that are represented by the material constant tensors of odd orders. The experimental phenomenon that the magnetostrictive strain is the same under either positively or negatively applied external magnetic 4eld [1,5] indicates that the strain is the function of the magnetic 4eld with even powers. In addition, experiments have also revealed that the sti;ness depends on the external magnetic 4eld [10]. This suggests that there should exist coupling terms between stress and magnetic 4eld. Furthermore, the :ux switches to the inverse orientation when the external magnetic 4eld changes to the opposite direction. Obviously, the above material properties and
For a one-dimensional problem, the SS constitutive equations become = s + mH 2 + rH 2 ;
(9a)
B = H + mH + r2 H:
(9b)
There are three distinct regions in the curve of strain versus applied magnetic 4eld for general magnetostrictive materials (see the dashed curves in Figs. 1–3). Under a low magnetic 4eld, strain is very small. In the moderate region of the magnetic 4eld, strain responds to external 4eld quite sensitively. That is, a small increase of magnetic 4eld will produce much larger strain. Under an intensive 4eld, strain tends to saturate. To obtain larger strain response in the practical engineering application, magnetostrictive materials are usually designed to work in the moderate region. In order to simulate the strain response of magnetostrictive materials, the main attempt in this
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
work is to establish a relation between the moduli in the constitutive equations and the corresponding experimental data under a moderate external magnetic 4eld. From the viewpoint of micro-mechanism of magnetostriction, when a magnetic 4eld is applied, the magnetic domains will switch to the direction parallel to the external magnetic 4eld. At the same time, the material exhibits elongation in the direction of external magnetic 4eld. Another driving force is the external stress. When applied an external load, the magnetic domains will also switch continuously to the direction perpendicular to the external force. In the meantime, magnetocrystalline anisotropy will retard the rotation of magnetic domains. Thus, the external force must be strong enough to overcome the magnetocrystalline anisotropy e;ect. There exist di;erent critical stresses for di;erent magnetostrictive materials. From the analysis of the experimental results it can be found that the piezomagnetic coe8cient (Note that the derivative of strain with respect to magnetic 4eld, d = @ =@H | , is named the piezomagnetic coe8cient in order to be consistent with the nomenclature adopted in [10], despite that there exists no true piezomagnetism in the magnetostrictive materials discussed in this paper. The frequent use of this phrase hereafter should always be referred to this de4nition.) decreases as the compressive pre-stress increases. At the same time, the magnetic 4eld associated with the peak piezomagnetic coe8cient under a compressive pre-stress increases as the pre-stress increases. Supposing a linear dependence of the magnetic 4eld associated with the peak piezomagnetic coe8cient on the external stress, it can be veri4ed that the error between the experimental data and the value calculated by the linear function is small and the approximation is generally acceptable. Thus, a linear function can be assumed as H˜ = H˜ cr + N; (10) where H˜ cr represents the magnetic 4eld corresponding to the peak piezomagnetic coe8cient under the critical external stress. is a material constant with its dimension of A m N−1 and physically denotes the increase of the external magnetic 4eld to reach the peak piezomagnetic coe8cient due to increase of the external stress, N = − cr
(11)
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in which is the pre-stress and cr the critical stress to switch the magnetic domains and usually a material constant. Di;erentiating (9a) with respect to H , one 4nds @ = 2(m + r)H (12) d= @H and the peak piezomagnetic coe8cient renders d˜ = 2(m + r)H˜ :
(13)
˜ d˜0 , where When there is no pre-stress, H˜ = H˜ 0 and d= d˜0 and H˜ 0 are the peak piezomagnetic coe8cients and the corresponding magnetic 4eld at which the piezomagnetic coe8cient equals the largest value in the absence of pre-stress, respectively. If we assume = 0 in (13), the magneostrictive modulus can be stated as d˜0 : (14) m= 2H˜ 0 From (10), (11) and (13), it follows that the peak piezomagnetic coe8cient can be written as d˜ = d˜cr + aN + b(N)2 ;
(15)
where a and b are two coe8cients with dimensions m3 A−1 N−1 and m5 A−1 N−2 , respectively. Two coe8cients can be obtained by 4tting (15) from the experimental data. For a general case with a compressive pre-stress, replacing H˜ and d˜ in (13) by (10) and (15), the magnetoelastic modulus can be obtained as d˜0 1 d˜cr + aN + b(N)2 : (16) − r= 2(H˜ cr + N) 2H˜ 0 2.2. SS constitutive relation for three-dimensional problems There are two modulus tensors needed to be characterized in the general three-dimensional constitutive equations, i.e., the magnetostrictive modulus tensor mijkl and the magnetoelastic modulus tensor rijklmn . For isotropic materials it can be assumed that the magnetostrictive modulus tensor is an isotropic tensor that can be generally expressed as mijkl =
" #−" ( ik jl + jk il ) + ij kl ; 2 3
(17)
in which ij is the Kronecker delta; #=m1111 +2m1122 , " = m1111 − m1122 ; m1111 the strain caused by per unit
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magnetic 4eld and measured in the direction of external magnetic 4eld; m1122 the strain measured in the direction perpendicular to magnetic 4eld. De4ne magnetic Poisson ratio as q = −m1122 =m1111 . This ratio can be obtained by two experimentally measured curves, –H and ∗ –H for one-dimensional problems, where and ∗ are the longitudinal strain parallel to the direction of the magnetic 4eld and the transverse strain vertical to the direction of the magnetic 4eld, respectively. The general constitutive equations with the series expansion are ij = sijkl kl + mijkl Hk Hl + rijklmn mn Hk Hl + · · · ;
(18a)
+ rklijmn ij mn Hl + · · · :
(18b)
If stress and magnetic induction B are speci4ed as independent variables, a Legendre transform will lead to @G ∗ @G ∗ ij = ; Hk = − : (19a,b) @ij @Bk
Expanding the elastic Gibbs free energy function G ∗ into a series for isotropic materials renders ij = sijkl kl + mijkl Bk Bl + rijklmn kl Bm Bn + · · · ; Hk =
m∗ijkl =
1 mijkl ; 2
rijklmn =
(20a)
1 Bk − m∗klmn mn Bl ∗ − rklijmn ij mn Hl + · · · ;
(20b)
where m∗ijkl denotes the strain caused by per unit magnetic induction of the material and is intituled as the ∗ induction magnetostrictive modulus tensor; rijklmn the coupling magnetostriction produced by per unit magnetic induction under unit compressive stress and is named as induction magnetoelastic modulus tensor. is the permeability of the material. Replacing B in (20a) by (18b) results in ij = sijkl kl + 2 m∗ijkl Hk Hl + m∗ijpq (mpkmn + mqlmn )mn Hk Hl ∗ + 2 rijklmn mn Hk Hl + · · · :
(21)
(22a)
1 ∗ : (mijpl mpkmn + mijkp mplmn ) + 2 rijklmn (22b)
For isotropic materials, it can be assumed that the 4eld magnetoelastic modulus tensor rijklmn and the in∗ duction magnetoelastic modulus tensor rijklmn are all sixth-order isotropic tensors, and are mutually proportional tensors. By introducing a dimensionless coe8cient C, the 4eld magnetoelastic modulus tensor can be given by rijklmn =
Bk = Hk + mklmn mn Hl
B
Comparing (18a) with (21), one can get
C (mijpl mpkmn + mijkp mplmn ):
(23)
With the symmetry of the magnetoelastic tensor taken into consideration, the tensor can be written as (see the appendix) rijklmn =
1 C [(mijlp mpkmn + mijkp mplmn ) 3 + (mijmp mpnkl + mijnp mpmkl ) + (mklip mpjmn + mkljp mpimn )]:
(24)
The coe8cient C needs to be characterized in terms of the magnetoelastic modulus in the one-dimensional cases. Thus, it is seen from the above equations that three moduli measured from experiments in the one-dimensional case are needed to completely characterize the modulus tensors in the constitutive equations for the general three-dimensional cases.
3. HT constitutive relation (HT model) The SS relations (strain is quadratic with respect to magnetic 4eld) presented in the previous section can accurately predict the experimental results only in the region of the low or moderate magnetic 4eld. However, the substantial error appears in the high-4eld region in terms of this model. The constitutive relations proposed in this section can narrow the error gap between theoretical values and experimental results in
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
the region of the high magnetic 4eld. Suppose that the Gibbs free energy function is expressed by G=
1 1 mn Hm Hn + sijkl ij kl 2 2 +
Hm Hn 1 tanh2 (k|H |)rijklmn ij kl 2k 2 |H |2
+
H m Hn 1 tanh2 (k|H |)mmnij ij k2 |H |2
(25)
in which tanh(x) is the hyperbolic tangent function; k = 1= H˜ a relaxation parameter that is adequately chosen to make the independent variable of the hyperbolic function dimensionless, where H˜ is the external magnetic 4eld corresponding to the peak piezomagnetic coe8cient as de4ned in the previous section. Substituting (25) into (6) for one-dimensional problems renders = s +
1 1 m tanh2 (kH ) + 2 r tanh2 (kH ); 2 k k
B = H + +
(26a)
2 sinh(kH ) m k cosh3 (kH )
1 2 sinh(kH ) r ; k cosh3 (kH )
(26b)
where sinh(x) is the hyperbolic sinusoidal function; cosh(x) the hyperbolic cosine; m the magnetostrictive modulus and r is the magnetoelastic modulus. By di;erentiating (26a) with respect to H , the piezomagnetic coe8cient is given as d=2
m + r tanh(kH )(1 − tanh2 (kH )) k
(27)
and the peak piezomagnetic coe8cient becomes d˜ = 2 tanh(1)(1 − tanh2 (1))(m + r)H˜ :
(28)
When there is no pre-stress, i.e. = 0, then H˜ = H˜ 0 , d˜ = d˜0 and m=
1 d˜0 : 2 tanh(1)(1 − tanh (1)) 2H˜ 0
(29)
For a general case in which a pre-stress exists, by substituting H˜ in (10) and d˜ in (15) into (28), the 4eld magnetoelastic modulus can be obtained
as r=
1 2 tanh(1)(1 − tanh2 (1)) 1 d˜cr + aN + b(N)2 d˜0 : × − 2H˜ 0 H˜ cr + N
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(30)
4. A model based on density of domain switching (DDS model) As is well known, a huge number of magnetic domains exist in the magnetic materials below the Curie point of temperature. Under an external magnetic 4eld, enormous domains switch to the direction parallel to the magnetic 4eld so that the magnetostrictive strain appears. The stronger the magnetic 4eld is, the more the domains switch and the larger the magnetostrictive strain appears. For a transducer typically made from a magnetostrictive rod in engineering, the most signi4cant parameter is the strain output induced by per unit magnetic 4eld. In order to obtain a larger strain output induced by per unit magnetic 4eld, the rod is usually designed to be under the dual action of a compressive pre-stress and a bias magnetic 4eld. As mentioned in the previous section, domain switching can also be activated by external stress. That is, the magnetic domains, surmounting the inner resistance of material, switch to the direction normal to the external stress, if the applied external stress surpasses the critical domain-switching stress that is inherent and determinate for a speci4c kind of magnetic materials. When subjected to a combined load of the external stress and the magnetic 4eld, the magnetic 4eld can be increased to a critical value that corresponds to a critical state of switching the domains. Under this circumstance, an extra small increment of magnetic 4eld will induce a large amount of domains to switch, hence, producing a signi4cant magnetostrictive strain. By introducing the domain-switching density that is de4ned as the quantity of switched domains inspired by per unit magnetic 4eld, one can easily 4nd that the domain-switching density reaches the peak value when the rod is under the magnetic 4eld associated with the maximum piezomagetic coe8cient. The giant magnetostrictive rod of the rare earth and iron compound possesses a signi4cant sensitivity of stress. When the stress surpasses the
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critical domain-switching stress, on the magnetostriction curve there usually exists an in:exion, at which the magnetostrictive strain induced by per unit magnetic 4eld reaches the peak value, and the domain-switching density attains the crest too. In the process of magnetization, the domainswitching density varies with the external magnetic 4eld. Consequently, this results in the gradual change of the tangent slope of the magnetostriction curve. For the magnetostrictive materials with a distinct sensitivity of stress, such as the giant magnetostrictive compound of rare earth and iron, the dependence of the domain-switching density on magnetic 4eld can be approximated by a density function similar to a probability density function with normal distribution in a broad range of magnetization. The domain-switching density attains the peak value when the external magnetic 4eld approaches H˜ , i.e. H = H˜ , and at this point the output strain induced by per unit magnetic 4eld also reaches the peak value. Since the domain-switching density is actually identical to the tangent slope of the magnetostriction curve macroscopically it can be described by the piezomagnetic coe8cient. Therefore, for one-dimensional problems (note that only one-dimensional magnetostrictive problem is considered in this section), the piezomagnetic coe8cient can be expressed by means of a density function related to the domain-switching density, i.e., @ (x − 1)2 ˜ d= ; (31) = d exp − @H A where d˜ is the peak piezomagnetic coe8cient, exp(x) the exponential function, x = |H |= H˜ , and A = cr =. H˜ is the magnetic 4eld associated with the peak piezomagnetic coe8cient, |H | the absolute value of external magnetic 4eld, cr the inherent critical stress of domain switching, and is the external stress. The total strain consists of the elastic strain, e , and the magnetostrictive strain, H , i.e. = e + H :
(32)
The elastic strain, e , satis4es the Hooker’s law, i.e. e = s;
(33)
where s is the elastic compliance. H is the strain induced by external magnetic 4eld. Note that the
magnetostrictive strain should be zero in the absence of magnetic 4eld. Integration of (31) with respect to H yields
' ˜ ˜ cr |H | erf −1 Hd = 2 cr H˜ − erf − ; cr H
√
(34)
in which erf (x) = exp(−x2 ) d x is the error function. Experiments [10,11] have shown that the :ux density distinctly depends on the external compressive pre-stress. In the low magnetic-4eld region, there is a linear dependence of the :ux density on magnetic 4eld. Likewise, the magnetic induction is generally assumed to be composed of two parts, i.e. the linear magnetization part, which varies linearly with the magnetic 4eld, H , and the perturbation part coupling with stress that is dependent on both magnetic 4eld and stress: B = H + B0 (H; );
(35)
where is the permeability of the material, and B0 (H; ) represents the coupling term of magnetic 4eld and stress. For one-dimensional cases, the thermodynamically based constitutive relations, (6), can be simpli4ed as @G @G = ; B= : (36a,b) @ H @H The Gibbs free energy, which is the function of magnetic 4eld and stress, G = G(H; ), is derived by using an approach similar to that presented by Hom et al. [12] for electrostrictive materials. Therefore, the combination of Eqs. (32) – (34) and (36a) results in √ ' cr 1 2 H˜ d˜ G = G0 (H ) + s + 2 2 0
|H | −1 × erf cr H − erf − d cr
(37)
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
in which G0 (H ) is the function of magnetic 4eld. Substituting (37) into (36b) renders @G0 (H ) B= d˜ exp + sign(H ) @H 0 2 |H | × − d; (38) −1 cr H˜ where sign(H ) = H=|H | signi4es the sign of external magnetic 4eld. Eqs. (35) and (38) can be used to determine the following functions as G0 (H ) =
1 H 2 ; 2
(39a)
B0 (H; ) = sign(H ) ×
0
d˜ exp − cr
2 |H | d: −1 H˜ (39b)
Note that d˜ and H˜ have been given already in (15) and (10), respectively. Hence, the constitutive equations can be obtained as √ ' ˜ = s + [H cr + ( − cr )][d˜cr + a( − cr ) 2 cr + b( − cr )2 ]
|H | −1 × erf cr H˜ cr + ( − cr ) ; (40a) − erf − cr B = H + sign(H ) [d˜cr + a( − cr ) 0
+ b( − cr )2 ] 2 |H | ×exp − d: −1 cr H˜ cr + ( − cr ) (40b) Obviously, the above constitutive equations involve several material constants, such as s; ; H˜ cr ; d˜cr ; cr and , which have de4nitely physical implications and can be determined by controlled experiments.
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5. Experimental veri&cation and discussion In this section, all three constitutive models are employed to predict the strain response. With the experimental results published in [10], the error comparison of the three models is given and an evaluation of the three models is presented. To keep the minimum energy state, the magnetic domains will switch to the direction perpendicular to the external force when a pre-stress is applied on the magnetostrictive materials. However, the direction of spontaneous magnetization in the magnetic domain is, at the same time, a;ected by the magnetocrystalline anisotropy. To let the magnetic domains switch, the external force must be strong enough to overcome the magnetocrystalline anisotropy e;ect. The response of magnetostriction appears to be different under an external magnetic 4eld when various pre-stresses are applied on the magnetostrictive material. The micro-mechanism of this phenomenon is associated with magnetic domain switching. When the pre-stress exceeds the critical domain-switching stress, an in:exion will appear on the magnetostriction curve (see the dashed line in Figs. 1–3). At this in:exion, the piezomagnetic coe8cient reaches the largest value. The reason is that a large amount of magnetic domains switch to the direction of external magnetic 4eld. From this point of view, the critical stress of the magnetostrictive material can be obtained by means of 4nding the in:exion on the magnetostriction curve under the external pre-stress. As for a Tb0:27 Dy0:73 Fe1:95 magnetostrictive rod, according to the experimental results in [10], the material constants are obtained as (see Fig. 4) cr = −15:3 MPa;
H˜ cr = 40; 000 A m−1 ;
d˜cr = 19:3 × 10−9 m A−1 ; = −2874 A m−1 MPa−1 : By means of 4tting (15) with reference to the experimental data, a = 0:3542 × 10−9 m A−1 MPa−1 ; b = 0:00288 × 10−9 m A−1 MPa−1 MPa−1 : The material constants used in the calculations are listed in Table 1. The magnetostrictive and magnetoelastic modulus of the SS model are calculated by
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means of (14) and (16), respectively. From Table 1, it can be found that the magnetostrictive modulus remaining constant for all the compressive pre-stresses. This indicates that the magnetostrictive modulus is an inherent constant of the material. Note that the calculated magnetoelastic modulus is not a constant. However, this modulus still approaches a constant as the compressive pre-stress increases. Thus, the magnetoelastic modulus can exhibit the inherent properties of the material only in a speci4c region of compressive pre-stress. Figs. 1 and 2 show the comparison of the predicted values (solid lines) and experimental results (dash lines) in terms of the SS model and the HT
~ H
ζ = _ tan α
~ H 0 = 40000A / m
σ cr = _15.3MPa
α
~ H0
0
σ
σ cr
Fig. 4. Schematic of the relation between external pre-stress and magnetic 4eld corresponding to peak piezomagnetic coe8cient.
model, respectively. It can be seen from Fig. 1 that except for the 4rst two cases of pre-stress, the theoretical values from the SS model are in good agreement with the experimental results in the region of the low and moderate magnetic 4eld. However, large di;erences appear in the high magnetic 4eld region. From Fig. 2, however, it can be seen that the predicted values of the HT model in the high 4eld region are in better agreement with the experimental results than those of the SS model. This makes out that the HT model can, to some extent, predict the saturation of the magnetic-4eld-induced strain when the applied magnetic 4eld is intensive. Fig. 3 presents the comparison of the theoretical values of the DDS model (solid lines) and the experimental results (dashed lines). One can 4nd that the theoretical curves trace out the experimental curves in all cases of pre-stress and the predicted values are in general agreement with the experimental results. This indicates that the DDS model can re:ect the distinct dependence of magnetostrictive behavior on external stress and saturation of magnetostrictive strain under intensive magnetic 4elds. The assumption that the dependence of domain-switching density on external magnetic 4eld can be depicted by a density function with the normal distribution is generally acceptable since the DDS model captures the main characteristics of the magnetostrictive behavior of a Terfenol-D rod. In the region of high magnetic 4eld, nevertheless, comparisons of the predicted and the experimental curves (see Figs. 2 and 3) show that there still exists an underestimation of magnetostrictive strain predicted by the
Table 1 Material properties and calculated moduli Bias condition
Pre-stress (MPa)
Bias magnetic 4eld (kOe)
Piezo-magnetic coe8cient (nm A−1 )
Standard square
Hyperbolic tangent
Magnetostrictive modulus 10−12 (m2 A−2 )
Magnetoelastic modulus 10−20 (m2 A−2 Pa−1 )
m 10−12 (m2 A−2 )
r 10−20 (m2 A−2 Pa−1 )
1 2 3 4 5 6 7 8
6.9 15.3 23.6 32 40.4 48.7 57.1 65.4
0.15 0.4 0.7 1.0 1.3 1.6 1.9 2.2
25 18 16 13 12 10 9 8
0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09
−2:77 −1:0 −0:24 0.002 0.08 0.1 0.1 0.1
0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28
−8:67 −3:13 −0:75 0.006 0.25 0.31 0.31 0.31
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065 compressive stress σ =65.4MPa Hyperbolic Tangent model Standard Square model Model based on density of domain switching
200
150
error [%]
100
50
0 0 _
50
1
2
3
4
5
H [KOe]
_100
Fig. 5. Error evaluation of the three models when the pre-stress is 65:4 MPa.
DDS model as opposed to an overestimation by the HT model. As is addressed before, the strain response of magnetostrictive materials is generally divided into three regions. In the low magnetic 4eld region, the strain rises slowly as the 4eld increases. The strain increases rapidly in the moderate 4eld region. And in the high 4eld region the strain approaches saturation. On the basis of comparisons of the calculations and experimental results, an error evaluation of the three models when the pre-stress is 65:4 MPa is exhibited in Fig. 5. Obviously, the errors of the HT and the DDS models with reference to the experimental data are much smaller than that of the SS model in the high 4eld region. The HT and the DDS models can predict, to some extent, the saturation trend of the magnetic-4eld-induced strain in the high 4eld region while the SS model fails to do such a prediction. However, the fact that the SS model can describe experimental results accurately in the region of low and moderate 4elds suggests that it might be appropriate to employ the SS constitutive relation when the magnetic 4eld is not very high. From the above analysis, the validity of each model should include the following three aspects, i.e. the capability to accurately predict the experimental results in the region of small magnetic 4eld, the capability to predict the saturation trend of experimental results under intense magnetic 4eld and the capability to re:ect the notable dependence of magnetostrictive behavior on external stresses. It seems that the DDS model is superior to the other two models in terms of
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the above three aspects. Moreover, the DDS model involves only several moduli that have de4nitely physical denotations and can be determined by controlled experiments, contrasted with the SS model that are developed by means of truncating a polynomial expansion of the Gibbs free energy function. It seems to be reasonable to select the DDS model though it has a relatively more complicated formulation, if one is going to put a constitutive model into a FEA code in order to analyze the magnetomechanical response of structure in practical engineering. As is pointed out by the anonymous reviewer of this paper, there is too little freedom left to adjust the curves to 4t better with the experimental results. None of the proposed three models can predict the nonlinear deformation behavior well when the applied magnetic 4eld is extremely intensive. A more perfect constitutive relation may probably be established on the involvement of microstructures embedded in the magnetic materials. Nevertheless, the HT and the DDS models, which are related to overestimating and underestimating the experimental curves demonstrated in Figs. 2 and 3, respectively, in the region of very intensive magnetic 4eld, may provide the upper bound of estimation and the lower bound of estimation if these two models are utilized to analyze the practical structures. From this point of view, the availability of these models in this work is helpful and deserved. The strain response of magnetic materials under coupling magnetoelastic loading is complicated in essence, especially for the giant magnetostrictive compound of rare earth and iron whose magnetoelastic response is intensively sensitive to external stress. The evolvement of microstructures, such as magnetic domain switching, may be vital for uncovering basic mechanisms of the macro-response of magnetic materials. Further work is needed to develop a constitutive model that can predict the whole process of the non-linear magnetostrictive deformation well. 6. Conclusions In the present work, the phenomenological non-linear constitutive relations of magnetostrictive materials are studied and three thermodynamically based constitutive models are proposed. The standard square model (SS), which is valid for the region of
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small and moderate magnetic 4elds, is developed by means of truncating the polynomial expansion of the Gibbs free energy function. The hyperbolic tangent model (HT), proposed by including a hyperbolic tangent magnetic-4eld dependence, can predict, to some extent, the trend of saturation of magnetostrictive strain in the high 4eld region. The DDS model (based on density of domain switching), developed on the ground that magnetic domain-switching physically underlies the magnetostrictive deformation, is more e;ective in simulating the experimental curves. In order to experimentally determine the moduli in the constitutive equations, one material function is proposed to describe the relation between the compressive pre-stress and the magnetic 4eld corresponding to the peak piezomagnetic coe8cient. Besides, the SS constitutive relation for the general three-dimensional problem is presented, and an approach is introduced to characterize the modulus tensors in terms of experimental results. Further work is needed both to develop a constitutive model that can predict the whole process of the non-linear magnetostrictive deformation well and to perform more experiments to obtain experimental results of various magnetostrictive materials.
where #1 –#15 are mutually independent coe8cients. Suppose that the tensor is symmetric with respect to each of the three pairs of indices, i; j and k; l and m; n, and also symmetric with respect to alternatively arranged index pairs, the tensor can be reduced to
Acknowledgements
rijklmn =
The authors are grateful for the support by the National Natural Science Foundation of China.
+ kl ( im jn + in jm kl ) + mn ( ik jl + il jk )] + #3 [ lm ( ik ln + ik jn ) + km ( il jn + in jk ) + ln ( ik jm + im jk ) + kn ( im jl kn + il jm )]:
(A.2)
In (A.2) there are only three independent coe8cients. For the 4eld magnetoelastic tensor and 4eld magnetostrictive tensor, then rijklmn =
C (mijpl mpkmn + mijkp mplmn ):
(A.3)
Symmetrizing the 4eld magnetoelastic tensor with respect to the three alternatively arranged index pairs leads to 1 C [(mijlp mpkmn + mijkp mplmn ) 3 + (mijmp mpnkl + mijnp mpmkl ) + (mklip mpjmn + mkljp mpimn )]:
(A.4)
The 4eld magnetostrictive tensor is a four-order isotropic tensor and can be generally written as
Appendix Generally, the sixth-order isotropic tensors can be expressed as rijklmn = #1 ij kl mn + #2 ij km ln + #3 ij kn lm + #4 ik jl mn + #5 ik ml ln + #6 ik jn lm + #7 il jk mn + #8 il jm kn + #9 il jn km + #10 im jk ln + #11 im jl kn + #12 im jn kl
mijkl = # ij kl + "( ik jl + il jk )
(A.1)
(A.5)
in which # = m12 , " = 1=2(m11 − m12 ). m11 is the strain caused by per unit magnetic 4eld and measured in the direction of external magnetic 4eld; m12 the strain measured in the direction perpendicular to the magnetic 4eld. Inserting (A.5) into (A.4) renders
C 4 2#2 ij kl mn + #"[ ij ( km ln rijklmn = 3 + kn lm ) + kl ( im jn + in jm kl )
+ #13 in jk lm + #14 in jk km + #15 in jm kl ;
rijklmn = #1 ij kl mn + #2 [ ij ( km ln + kn lm )
+ mn ( ik jl + il jk )] + "2 [ lm ( ik ln
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
+ ik jn ) + km ( il jn + in jk ) + ln ( ik jm + im jk ) + kn ( im jl kn + il jm )]
;
(A.6)
where C can be characterized in terms of the magnetoelastic modulus for one-dimensional cases and should be obtained according to the experimental data. From (A.2) and (A.6), it can be known that the 4eld magnetoelastic tensor is a sixth-order isotropic tensor that is symmetric with respect to each of the three pairs of indices i; j and k; l and m; n and also symmetric with respect to three alternatively arranged index pairs. There exist three independent coe8cients that should be and can be obtained through one-dimensional experiments. References [1] A.E. Clark, Magnetostrictive rare earth–Fe2 compounds, in: E.P. Wohlfarth (Ed.), Ferromagnetic Materials, Vol. 1, North-Holland, Amsterdam, 1980, pp. 531–589. [2] R.D. Greenough, A.G. Jenner, M.P. Schulze, A.J. Wilkinson, The properties and applications of magnetostrictive rare-earth compounds, J. Magn. Magn. Mater. 101 (1991) 75–80.
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[3] H. Gracia-Miquel, D.X. Chen, M. Vazquez, Domain wall propagation in bistable amorphous wires, J. Magn. Magn. Mater. 212 (2000) 101–106. [4] G.P. Carman, M. Mitrovic, Nonlinear constitutive relations for magnetostrictive materials with applications to 1-D problems, J. Intell. Mater. Systems Struct. 6 (1996) 673–683. [5] A.E. Clark, M.L. Spano, H.T. Savage, E;ect of stress on the magnetostriction and magnetization of rare earth–Fe1:95 alloys, IEEE Trans. Magn. Magn. Mag-19 (1983) 1964–1966. [6] A.N. Abd-Alla, G.A. Maugin, Nonlinear phenomena in magnetostrictive elastic resonators, Int. J. Eng. Sci. 27 (1989) 1613–1619. [7] S.S. Antman, Synthesis of nonlinear constitutive functions, Applications to the electromagnetic control of snapping, Trans. ASME J. Appl. Mech. 66 (1999) 280–283. [8] G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam, 1988. [9] Y.H. Pao, C.S. Yeh, A linear theory for soft ferromagnetic elastic solids, Int. J. Eng. Sci. 11 (4) (1973) 415–436. [10] M.B. Mo;et, A.E. Clark, M. Wun-Fogle, J. Linberg, J.P. Teter, .A. McLaughlin, Characterization of Terfenol-D for magnetostrictive transducers, J. Acoust. Soc. Am. 89 (3) (1991) 1448–1455. [11] L. Kvarnsjo, G. Engdahl, Di;erential and incremental measurements of magneto elastic parameters of highly magnetostrictive materials, in: L. Lanotte (Ed.), Magnetoelastic E;ects and Applications, Elsevier Science, Amsterdam, 1993, pp. 63–69. [12] C.L. Hom, N. Shankar, A fully coupled constitutive model for electrostrictive ceramic materials, J. Intell. Mater. Systems Struct. 5 (1994) 795–801.
International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
Non-linear constitutive relations for magnetostrictive materials Yongping Wan, Daining Fang ∗ , Keh-Chih Hwang Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Abstract In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-4eld dependence, are proposed to model the magnetic-4eld-induced strain saturation of magnetostrictive materials in the region of intense magnetic 4elds. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, de4ned by the quantity of magnetic domains switched by per unit magnetic 4eld and magnetic 4eld can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coe8cient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic 4eld. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic 4eld while the HT model can, to some extent, re:ect the trend of saturation of magnetostrictive strain under a high magnetic 4eld. The model based on DDS, which is more e;ective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic 4elds. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed. ? 2002 Published by Elsevier Science Ltd. Keywords: Magnetostrictive material; Non-linear deformation; Coupling constitutive relations; Magnetostrictive strain
1. Introduction Magnetic materials have already played an important role in the development of new technologies. To investigate the mechanical laws of magnetic materials will be vital for their application and development [1,2]. Magnetic materials can show elongation ∗
Corresponding author. Tel./fax: +86-10-62772923. E-mail address: [email protected] (D. Fang).
and contraction in the magnetization direction. This is called magnetostriction that is due to the switching of a large amount of magnetic domains caused by spontaneous magnetization, below the Curie point of temperature [3]. In each domain, the lattice distorts and the magnetization direction is collinear with one of the principal axes of the spontaneous strain. Without any applied magnetic 4eld, the magnetization direction of each domain is random and there will be no macroscopic magnetization e;ect. However, the magnetization of magnetic domains will turn to the
0020-7462/03/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII: S 0 0 2 0 - 7 4 6 2 ( 0 2 ) 0 0 0 5 2 - 5
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Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
direction of the external magnetic 4eld if the material is subjected to a magnetic 4eld. The material will show elongation in the external magnetic 4eld if the direction of magnetization of magnetic domains is along the long principal axis of the spontaneous strain. This is the positive magnetostriction. That the material contracts in the direction of the external magnetic 4eld, is called as the negative magnetostriction. Generally, the positive magnetostrictive materials are employed in engineering [2,3]. For magnetostrictive materials, the strain keeps the same for either positively or negatively applied magnetic 4eld. Therefore, the vibration frequency will be twice that of the magnetic 4eld if the magnetostrictive material is only subjected to an alternating 4eld. This is the so-called double frequency e;ect. To obtain a synchronous magnetostriction under an applied alternating 4eld, a bias 4eld produced by a direct current must be applied. On the other hand, the material exhibits a sudden increase of strain at a critical magnetic 4eld when a compressive pre-stress is imposed on the material. This implies that the magnetostrictive coef4cient is more sensitive to the applied magnetic 4eld when a compressive pre-stress is imposed on the magnetic materials. Generally, magnetostrictive materials are very brittle. Their stretching yield stress is substantially smaller than their compressive yield stress. Thus, a compressive pre-stress is always applied on the magnetostrictive rod practically [4,5]. The magnetostrictive materials usually endure a coupled mechanical-magnetic 4eld when they are in application. Their constitutive relations are essentially non-linear [6,7]. This makes it di8cult to develop any theoretical models. At present, the linear or non-linear but de-coupled constitutive relations are usually employed in modeling or analyzing [8,9]. However, it is quite important to take the coupling terms into consideration if a model is designed to predict the properties of magnetostrictive materials accurately. Carman et al. [4] has proposed a non-linear coupling constitutive relation for one-dimensional problems. This model can predict some deformation characteristics of the magnetostrictive rod in a speci4c region of pre-stress. Yet, it cannot describe the saturation of magnetostrictive strain when the material is subjected to an intensive magnetic 4eld. In the present work, three phenomenological constitutive relations of magnetostrictive materials are studied. The quadratic equations of the
standard square (SS) constitutive model are derived by means of truncating the polynomial expansion of the Gibbs free energy. To produce some kind of saturation, the hyperbolic tangent (HT) constitutive equations are developed involving a HT magnetic-4eld dependence. A better physically grounded constitutive model, which is more e;ective in simulating experimental curves, is motivated in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation, and the careful analysis of the experimental results obtained by Mo;et et al. [10]. It is assumed, that the relation between domain-switching density de4ned by the quantity of magnetic domains switched by per unit magnetic 4eld and magnetic 4eld can be described by a density function with normal distribution. Furthermore, a material function is proposed to describe the dependence of the peak piezomagnetic coe8cient on the compressive pre-stress for one-dimensional cases so that the moduli in the constitutive models can be experimentally determined. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.
2. SS constitutive relation (SS model) The energy balance equation of electric–magnetic body is [8,9] d 1 u˙ i u˙ i + U dv dt v 2 = (fi u˙ i + ) dv + ti u˙ i ds; (1) v
S
where U is the internal energy density per unit mass; the electric–magnetic energy density per unit time, = −∇ · (E × H ), where E is the electric 4eld intensity vector; H the magnetic 4eld intensity vector and ∇ the gradient operator. dv and ds refer to the body element and the surface element, respectively. fi ; ti are the components of body force and surface force, respectively, and ui the component of the displacement vector. The dots in Eq. (1) represent di;erentiation with respect to time. is the mass density. Using the equations of mass conservation and
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
momentum conservation, the energy balance equation can be simpli4ed as U˙ dv = ( + ˙ij ij ) dv; (2)
compressive prestress (MPa) 1 2 3 4 5 6 7 8
3
v
where ij and ij are the components of the strain tensor and stress tensor, respectively. Considering the assumption of quasi-magnetostatics of insulators, one 4nds ˙ ∇ × E = −B:
∇ × H = 0;
Strain [10 -3 ]
v
2
6.9 15.3 23.6 32 40.4 48.7 57.1 65.4
8 7 6
5 4 3
2
1
experimental results Standard Square model
(3) 1
Then Eq. (2) renders ˙ U dv = (B˙ k Hk + ˙ij ij ) dv; v
v
0 0
(4)
where B is the magnetic induction vector. For small distortion problems, by means of the internal energy density per unit volume, there is U˙ = B˙ k Hk + ˙ij ij :
(5)
Choosing stress and magnetic 4eld as independent variables, in terms of a Legendre transform, one can get @G @G ij = ; B = (6) k @ij H @Hk in which G is the Gibbs free energy function. To obtain a constitutive relation, the Gibbs free energy function can be written in a series: G = G0 +
@G @G Nij + NHk @ij @Hk
+
1 @2 G Nij Nkl 2 @ij @kl
+
1 @2 G Nij NHk 2 @ij @Hk
+
1 @2 G NHl NHk 2 @Hl @Hk
+
@3 G 1 Nij Nkl Nmn 3! @ij @kl @mn
+
@3 G 1 Nij Nkl NHm 3! @ij @kl @Hm
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1
2
3
4
5
H [KOe]
Fig. 1. Magnetostriction curves (the calculated curves are obtained by use of the SS model).
+
@3 G 1 Nij NHk NHl 3! @ij @Hk @Hl
+
@3 G 1 NHm NHk NHl + · · · : 3! @Hm @Hk @Hl
(7)
Obviously, it is important to rationally characterize the moduli or constants in the above constitutive equations in terms of experimental results. Mo;et et al. [10,11] experimentally investigated the magnetostrictive properties of a magnetic material (Terfenol-D). The Terfenol-D rod was operated under eight di;erent bias magnetic 4elds and compressive pre-stresses, respectively. The dashed lines in Figs. 1–3 are the measured curves of strain versus the applied magnetic 4eld. Results indicate that as the compressive pre-stress becomes larger, a larger magnetic 4eld is needed to drive the rod in order to get the same stretching strain. At the same time, the tangent slope of the magnetostriction curve continues to decline, which implies that the magnetostrictive strain induced by per unit external magnetic 4eld gradually decreases as the compressive stress increases. To derive constitutive equations of the magnetostrictive materials from the series expansion of the Gibbs free energy (see (7)), one should review material properties and some experimental results. The giant magnetostrictive compound of rare earth and iron, Tb0:27 Dy0:73 Fe1:95 , possesses a crystal structure of the cubic Laves phase. The cubic phase compound
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experimental results Hyperbolic Tangent model 2
8
7 6
5
-3
Strain [10 ]
4
compressive prestress (MPa) 1 6.9 2 15.3 3 23.6 4 32 5 40.4 6 48.7 7 57.1 8 65.4
3 1
2 1
0 0
1
2
3
4
5
H [KOe]
Fig. 2. Magnetostriction curves (the calculated curves are obtained by use of the HT model).
experimental results model based on density of domain switching
-3
Strain [10 ]
1.6
3
1.2
5
6
7
8
4
2
compressive pre-stress [MPa] 1 6.9 2 15.3 3 23.6 4 32.0 5 40.4 6 48.7 7 57.1 8 65.4
0.8
1 0.4
0.0 0
1
2
3
4
5
experimental results request that the terms involving partial derivatives of odd indices should not appear in the series expansion given in (7), which means the material constant tensors of odd orders vanish. Therefore, the piezomagnetic coe8cient tensor, which is a material constant tensor of three orders and re:ects the property of piezomagnetism, vanishes for the magnetostrictive materials. Thus, the following constitutive relations are developed based on the truncated polynomial energy expansion: ij = sijkl kl + mijkl Hk Hl + rijklmn kl Hm Hn ;
(8a)
Bk = kl Hl + mklmn mn Hl + rklmnpq mn pq Hl :
(8b)
in which sijkl is the elastic compliance tensor; mijkl the 4eld magnetostrictive modulus tensor that physically denotes the magnetostrictive strain produced by per unit external magnetic 4eld. Its dimension is m2 A−2 ; rijklmn the 4eld magnetoelastic modulus tensor that physically refers to the coupling magnetostrictive strain produced by per unit external magnetic 4eld under per unit compressive pre-stress. The dimension of this modulus is m4 A−2 N−1 ; kl the permeability tensor that can be recognized as kl for isotropic materials, where ij is the Kronecker delta. 2.1. Standard square constitutive relation for one-dimensional problems
H [ KOe]
Fig. 3. Magnetostriction curves (the calculated curves are obtained by use of the DDS model).
is a non-polar crystal, which deforms elastically without any magnetization under a stress. There do not exist such properties that are represented by the material constant tensors of odd orders. The experimental phenomenon that the magnetostrictive strain is the same under either positively or negatively applied external magnetic 4eld [1,5] indicates that the strain is the function of the magnetic 4eld with even powers. In addition, experiments have also revealed that the sti;ness depends on the external magnetic 4eld [10]. This suggests that there should exist coupling terms between stress and magnetic 4eld. Furthermore, the :ux switches to the inverse orientation when the external magnetic 4eld changes to the opposite direction. Obviously, the above material properties and
For a one-dimensional problem, the SS constitutive equations become = s + mH 2 + rH 2 ;
(9a)
B = H + mH + r2 H:
(9b)
There are three distinct regions in the curve of strain versus applied magnetic 4eld for general magnetostrictive materials (see the dashed curves in Figs. 1–3). Under a low magnetic 4eld, strain is very small. In the moderate region of the magnetic 4eld, strain responds to external 4eld quite sensitively. That is, a small increase of magnetic 4eld will produce much larger strain. Under an intensive 4eld, strain tends to saturate. To obtain larger strain response in the practical engineering application, magnetostrictive materials are usually designed to work in the moderate region. In order to simulate the strain response of magnetostrictive materials, the main attempt in this
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
work is to establish a relation between the moduli in the constitutive equations and the corresponding experimental data under a moderate external magnetic 4eld. From the viewpoint of micro-mechanism of magnetostriction, when a magnetic 4eld is applied, the magnetic domains will switch to the direction parallel to the external magnetic 4eld. At the same time, the material exhibits elongation in the direction of external magnetic 4eld. Another driving force is the external stress. When applied an external load, the magnetic domains will also switch continuously to the direction perpendicular to the external force. In the meantime, magnetocrystalline anisotropy will retard the rotation of magnetic domains. Thus, the external force must be strong enough to overcome the magnetocrystalline anisotropy e;ect. There exist di;erent critical stresses for di;erent magnetostrictive materials. From the analysis of the experimental results it can be found that the piezomagnetic coe8cient (Note that the derivative of strain with respect to magnetic 4eld, d = @ =@H | , is named the piezomagnetic coe8cient in order to be consistent with the nomenclature adopted in [10], despite that there exists no true piezomagnetism in the magnetostrictive materials discussed in this paper. The frequent use of this phrase hereafter should always be referred to this de4nition.) decreases as the compressive pre-stress increases. At the same time, the magnetic 4eld associated with the peak piezomagnetic coe8cient under a compressive pre-stress increases as the pre-stress increases. Supposing a linear dependence of the magnetic 4eld associated with the peak piezomagnetic coe8cient on the external stress, it can be veri4ed that the error between the experimental data and the value calculated by the linear function is small and the approximation is generally acceptable. Thus, a linear function can be assumed as H˜ = H˜ cr + N; (10) where H˜ cr represents the magnetic 4eld corresponding to the peak piezomagnetic coe8cient under the critical external stress. is a material constant with its dimension of A m N−1 and physically denotes the increase of the external magnetic 4eld to reach the peak piezomagnetic coe8cient due to increase of the external stress, N = − cr
(11)
1057
in which is the pre-stress and cr the critical stress to switch the magnetic domains and usually a material constant. Di;erentiating (9a) with respect to H , one 4nds @ = 2(m + r)H (12) d= @H and the peak piezomagnetic coe8cient renders d˜ = 2(m + r)H˜ :
(13)
˜ d˜0 , where When there is no pre-stress, H˜ = H˜ 0 and d= d˜0 and H˜ 0 are the peak piezomagnetic coe8cients and the corresponding magnetic 4eld at which the piezomagnetic coe8cient equals the largest value in the absence of pre-stress, respectively. If we assume = 0 in (13), the magneostrictive modulus can be stated as d˜0 : (14) m= 2H˜ 0 From (10), (11) and (13), it follows that the peak piezomagnetic coe8cient can be written as d˜ = d˜cr + aN + b(N)2 ;
(15)
where a and b are two coe8cients with dimensions m3 A−1 N−1 and m5 A−1 N−2 , respectively. Two coe8cients can be obtained by 4tting (15) from the experimental data. For a general case with a compressive pre-stress, replacing H˜ and d˜ in (13) by (10) and (15), the magnetoelastic modulus can be obtained as d˜0 1 d˜cr + aN + b(N)2 : (16) − r= 2(H˜ cr + N) 2H˜ 0 2.2. SS constitutive relation for three-dimensional problems There are two modulus tensors needed to be characterized in the general three-dimensional constitutive equations, i.e., the magnetostrictive modulus tensor mijkl and the magnetoelastic modulus tensor rijklmn . For isotropic materials it can be assumed that the magnetostrictive modulus tensor is an isotropic tensor that can be generally expressed as mijkl =
" #−" ( ik jl + jk il ) + ij kl ; 2 3
(17)
in which ij is the Kronecker delta; #=m1111 +2m1122 , " = m1111 − m1122 ; m1111 the strain caused by per unit
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magnetic 4eld and measured in the direction of external magnetic 4eld; m1122 the strain measured in the direction perpendicular to magnetic 4eld. De4ne magnetic Poisson ratio as q = −m1122 =m1111 . This ratio can be obtained by two experimentally measured curves, –H and ∗ –H for one-dimensional problems, where and ∗ are the longitudinal strain parallel to the direction of the magnetic 4eld and the transverse strain vertical to the direction of the magnetic 4eld, respectively. The general constitutive equations with the series expansion are ij = sijkl kl + mijkl Hk Hl + rijklmn mn Hk Hl + · · · ;
(18a)
+ rklijmn ij mn Hl + · · · :
(18b)
If stress and magnetic induction B are speci4ed as independent variables, a Legendre transform will lead to @G ∗ @G ∗ ij = ; Hk = − : (19a,b) @ij @Bk
Expanding the elastic Gibbs free energy function G ∗ into a series for isotropic materials renders ij = sijkl kl + mijkl Bk Bl + rijklmn kl Bm Bn + · · · ; Hk =
m∗ijkl =
1 mijkl ; 2
rijklmn =
(20a)
1 Bk − m∗klmn mn Bl ∗ − rklijmn ij mn Hl + · · · ;
(20b)
where m∗ijkl denotes the strain caused by per unit magnetic induction of the material and is intituled as the ∗ induction magnetostrictive modulus tensor; rijklmn the coupling magnetostriction produced by per unit magnetic induction under unit compressive stress and is named as induction magnetoelastic modulus tensor. is the permeability of the material. Replacing B in (20a) by (18b) results in ij = sijkl kl + 2 m∗ijkl Hk Hl + m∗ijpq (mpkmn + mqlmn )mn Hk Hl ∗ + 2 rijklmn mn Hk Hl + · · · :
(21)
(22a)
1 ∗ : (mijpl mpkmn + mijkp mplmn ) + 2 rijklmn (22b)
For isotropic materials, it can be assumed that the 4eld magnetoelastic modulus tensor rijklmn and the in∗ duction magnetoelastic modulus tensor rijklmn are all sixth-order isotropic tensors, and are mutually proportional tensors. By introducing a dimensionless coe8cient C, the 4eld magnetoelastic modulus tensor can be given by rijklmn =
Bk = Hk + mklmn mn Hl
B
Comparing (18a) with (21), one can get
C (mijpl mpkmn + mijkp mplmn ):
(23)
With the symmetry of the magnetoelastic tensor taken into consideration, the tensor can be written as (see the appendix) rijklmn =
1 C [(mijlp mpkmn + mijkp mplmn ) 3 + (mijmp mpnkl + mijnp mpmkl ) + (mklip mpjmn + mkljp mpimn )]:
(24)
The coe8cient C needs to be characterized in terms of the magnetoelastic modulus in the one-dimensional cases. Thus, it is seen from the above equations that three moduli measured from experiments in the one-dimensional case are needed to completely characterize the modulus tensors in the constitutive equations for the general three-dimensional cases.
3. HT constitutive relation (HT model) The SS relations (strain is quadratic with respect to magnetic 4eld) presented in the previous section can accurately predict the experimental results only in the region of the low or moderate magnetic 4eld. However, the substantial error appears in the high-4eld region in terms of this model. The constitutive relations proposed in this section can narrow the error gap between theoretical values and experimental results in
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
the region of the high magnetic 4eld. Suppose that the Gibbs free energy function is expressed by G=
1 1 mn Hm Hn + sijkl ij kl 2 2 +
Hm Hn 1 tanh2 (k|H |)rijklmn ij kl 2k 2 |H |2
+
H m Hn 1 tanh2 (k|H |)mmnij ij k2 |H |2
(25)
in which tanh(x) is the hyperbolic tangent function; k = 1= H˜ a relaxation parameter that is adequately chosen to make the independent variable of the hyperbolic function dimensionless, where H˜ is the external magnetic 4eld corresponding to the peak piezomagnetic coe8cient as de4ned in the previous section. Substituting (25) into (6) for one-dimensional problems renders = s +
1 1 m tanh2 (kH ) + 2 r tanh2 (kH ); 2 k k
B = H + +
(26a)
2 sinh(kH ) m k cosh3 (kH )
1 2 sinh(kH ) r ; k cosh3 (kH )
(26b)
where sinh(x) is the hyperbolic sinusoidal function; cosh(x) the hyperbolic cosine; m the magnetostrictive modulus and r is the magnetoelastic modulus. By di;erentiating (26a) with respect to H , the piezomagnetic coe8cient is given as d=2
m + r tanh(kH )(1 − tanh2 (kH )) k
(27)
and the peak piezomagnetic coe8cient becomes d˜ = 2 tanh(1)(1 − tanh2 (1))(m + r)H˜ :
(28)
When there is no pre-stress, i.e. = 0, then H˜ = H˜ 0 , d˜ = d˜0 and m=
1 d˜0 : 2 tanh(1)(1 − tanh (1)) 2H˜ 0
(29)
For a general case in which a pre-stress exists, by substituting H˜ in (10) and d˜ in (15) into (28), the 4eld magnetoelastic modulus can be obtained
as r=
1 2 tanh(1)(1 − tanh2 (1)) 1 d˜cr + aN + b(N)2 d˜0 : × − 2H˜ 0 H˜ cr + N
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(30)
4. A model based on density of domain switching (DDS model) As is well known, a huge number of magnetic domains exist in the magnetic materials below the Curie point of temperature. Under an external magnetic 4eld, enormous domains switch to the direction parallel to the magnetic 4eld so that the magnetostrictive strain appears. The stronger the magnetic 4eld is, the more the domains switch and the larger the magnetostrictive strain appears. For a transducer typically made from a magnetostrictive rod in engineering, the most signi4cant parameter is the strain output induced by per unit magnetic 4eld. In order to obtain a larger strain output induced by per unit magnetic 4eld, the rod is usually designed to be under the dual action of a compressive pre-stress and a bias magnetic 4eld. As mentioned in the previous section, domain switching can also be activated by external stress. That is, the magnetic domains, surmounting the inner resistance of material, switch to the direction normal to the external stress, if the applied external stress surpasses the critical domain-switching stress that is inherent and determinate for a speci4c kind of magnetic materials. When subjected to a combined load of the external stress and the magnetic 4eld, the magnetic 4eld can be increased to a critical value that corresponds to a critical state of switching the domains. Under this circumstance, an extra small increment of magnetic 4eld will induce a large amount of domains to switch, hence, producing a signi4cant magnetostrictive strain. By introducing the domain-switching density that is de4ned as the quantity of switched domains inspired by per unit magnetic 4eld, one can easily 4nd that the domain-switching density reaches the peak value when the rod is under the magnetic 4eld associated with the maximum piezomagetic coe8cient. The giant magnetostrictive rod of the rare earth and iron compound possesses a signi4cant sensitivity of stress. When the stress surpasses the
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critical domain-switching stress, on the magnetostriction curve there usually exists an in:exion, at which the magnetostrictive strain induced by per unit magnetic 4eld reaches the peak value, and the domain-switching density attains the crest too. In the process of magnetization, the domainswitching density varies with the external magnetic 4eld. Consequently, this results in the gradual change of the tangent slope of the magnetostriction curve. For the magnetostrictive materials with a distinct sensitivity of stress, such as the giant magnetostrictive compound of rare earth and iron, the dependence of the domain-switching density on magnetic 4eld can be approximated by a density function similar to a probability density function with normal distribution in a broad range of magnetization. The domain-switching density attains the peak value when the external magnetic 4eld approaches H˜ , i.e. H = H˜ , and at this point the output strain induced by per unit magnetic 4eld also reaches the peak value. Since the domain-switching density is actually identical to the tangent slope of the magnetostriction curve macroscopically it can be described by the piezomagnetic coe8cient. Therefore, for one-dimensional problems (note that only one-dimensional magnetostrictive problem is considered in this section), the piezomagnetic coe8cient can be expressed by means of a density function related to the domain-switching density, i.e., @ (x − 1)2 ˜ d= ; (31) = d exp − @H A where d˜ is the peak piezomagnetic coe8cient, exp(x) the exponential function, x = |H |= H˜ , and A = cr =. H˜ is the magnetic 4eld associated with the peak piezomagnetic coe8cient, |H | the absolute value of external magnetic 4eld, cr the inherent critical stress of domain switching, and is the external stress. The total strain consists of the elastic strain, e , and the magnetostrictive strain, H , i.e. = e + H :
(32)
The elastic strain, e , satis4es the Hooker’s law, i.e. e = s;
(33)
where s is the elastic compliance. H is the strain induced by external magnetic 4eld. Note that the
magnetostrictive strain should be zero in the absence of magnetic 4eld. Integration of (31) with respect to H yields
' ˜ ˜ cr |H | erf −1 Hd = 2 cr H˜ − erf − ; cr H
√
(34)
in which erf (x) = exp(−x2 ) d x is the error function. Experiments [10,11] have shown that the :ux density distinctly depends on the external compressive pre-stress. In the low magnetic-4eld region, there is a linear dependence of the :ux density on magnetic 4eld. Likewise, the magnetic induction is generally assumed to be composed of two parts, i.e. the linear magnetization part, which varies linearly with the magnetic 4eld, H , and the perturbation part coupling with stress that is dependent on both magnetic 4eld and stress: B = H + B0 (H; );
(35)
where is the permeability of the material, and B0 (H; ) represents the coupling term of magnetic 4eld and stress. For one-dimensional cases, the thermodynamically based constitutive relations, (6), can be simpli4ed as @G @G = ; B= : (36a,b) @ H @H The Gibbs free energy, which is the function of magnetic 4eld and stress, G = G(H; ), is derived by using an approach similar to that presented by Hom et al. [12] for electrostrictive materials. Therefore, the combination of Eqs. (32) – (34) and (36a) results in √ ' cr 1 2 H˜ d˜ G = G0 (H ) + s + 2 2 0
|H | −1 × erf cr H − erf − d cr
(37)
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
in which G0 (H ) is the function of magnetic 4eld. Substituting (37) into (36b) renders @G0 (H ) B= d˜ exp + sign(H ) @H 0 2 |H | × − d; (38) −1 cr H˜ where sign(H ) = H=|H | signi4es the sign of external magnetic 4eld. Eqs. (35) and (38) can be used to determine the following functions as G0 (H ) =
1 H 2 ; 2
(39a)
B0 (H; ) = sign(H ) ×
0
d˜ exp − cr
2 |H | d: −1 H˜ (39b)
Note that d˜ and H˜ have been given already in (15) and (10), respectively. Hence, the constitutive equations can be obtained as √ ' ˜ = s + [H cr + ( − cr )][d˜cr + a( − cr ) 2 cr + b( − cr )2 ]
|H | −1 × erf cr H˜ cr + ( − cr ) ; (40a) − erf − cr B = H + sign(H ) [d˜cr + a( − cr ) 0
+ b( − cr )2 ] 2 |H | ×exp − d: −1 cr H˜ cr + ( − cr ) (40b) Obviously, the above constitutive equations involve several material constants, such as s; ; H˜ cr ; d˜cr ; cr and , which have de4nitely physical implications and can be determined by controlled experiments.
1061
5. Experimental veri&cation and discussion In this section, all three constitutive models are employed to predict the strain response. With the experimental results published in [10], the error comparison of the three models is given and an evaluation of the three models is presented. To keep the minimum energy state, the magnetic domains will switch to the direction perpendicular to the external force when a pre-stress is applied on the magnetostrictive materials. However, the direction of spontaneous magnetization in the magnetic domain is, at the same time, a;ected by the magnetocrystalline anisotropy. To let the magnetic domains switch, the external force must be strong enough to overcome the magnetocrystalline anisotropy e;ect. The response of magnetostriction appears to be different under an external magnetic 4eld when various pre-stresses are applied on the magnetostrictive material. The micro-mechanism of this phenomenon is associated with magnetic domain switching. When the pre-stress exceeds the critical domain-switching stress, an in:exion will appear on the magnetostriction curve (see the dashed line in Figs. 1–3). At this in:exion, the piezomagnetic coe8cient reaches the largest value. The reason is that a large amount of magnetic domains switch to the direction of external magnetic 4eld. From this point of view, the critical stress of the magnetostrictive material can be obtained by means of 4nding the in:exion on the magnetostriction curve under the external pre-stress. As for a Tb0:27 Dy0:73 Fe1:95 magnetostrictive rod, according to the experimental results in [10], the material constants are obtained as (see Fig. 4) cr = −15:3 MPa;
H˜ cr = 40; 000 A m−1 ;
d˜cr = 19:3 × 10−9 m A−1 ; = −2874 A m−1 MPa−1 : By means of 4tting (15) with reference to the experimental data, a = 0:3542 × 10−9 m A−1 MPa−1 ; b = 0:00288 × 10−9 m A−1 MPa−1 MPa−1 : The material constants used in the calculations are listed in Table 1. The magnetostrictive and magnetoelastic modulus of the SS model are calculated by
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means of (14) and (16), respectively. From Table 1, it can be found that the magnetostrictive modulus remaining constant for all the compressive pre-stresses. This indicates that the magnetostrictive modulus is an inherent constant of the material. Note that the calculated magnetoelastic modulus is not a constant. However, this modulus still approaches a constant as the compressive pre-stress increases. Thus, the magnetoelastic modulus can exhibit the inherent properties of the material only in a speci4c region of compressive pre-stress. Figs. 1 and 2 show the comparison of the predicted values (solid lines) and experimental results (dash lines) in terms of the SS model and the HT
~ H
ζ = _ tan α
~ H 0 = 40000A / m
σ cr = _15.3MPa
α
~ H0
0
σ
σ cr
Fig. 4. Schematic of the relation between external pre-stress and magnetic 4eld corresponding to peak piezomagnetic coe8cient.
model, respectively. It can be seen from Fig. 1 that except for the 4rst two cases of pre-stress, the theoretical values from the SS model are in good agreement with the experimental results in the region of the low and moderate magnetic 4eld. However, large di;erences appear in the high magnetic 4eld region. From Fig. 2, however, it can be seen that the predicted values of the HT model in the high 4eld region are in better agreement with the experimental results than those of the SS model. This makes out that the HT model can, to some extent, predict the saturation of the magnetic-4eld-induced strain when the applied magnetic 4eld is intensive. Fig. 3 presents the comparison of the theoretical values of the DDS model (solid lines) and the experimental results (dashed lines). One can 4nd that the theoretical curves trace out the experimental curves in all cases of pre-stress and the predicted values are in general agreement with the experimental results. This indicates that the DDS model can re:ect the distinct dependence of magnetostrictive behavior on external stress and saturation of magnetostrictive strain under intensive magnetic 4elds. The assumption that the dependence of domain-switching density on external magnetic 4eld can be depicted by a density function with the normal distribution is generally acceptable since the DDS model captures the main characteristics of the magnetostrictive behavior of a Terfenol-D rod. In the region of high magnetic 4eld, nevertheless, comparisons of the predicted and the experimental curves (see Figs. 2 and 3) show that there still exists an underestimation of magnetostrictive strain predicted by the
Table 1 Material properties and calculated moduli Bias condition
Pre-stress (MPa)
Bias magnetic 4eld (kOe)
Piezo-magnetic coe8cient (nm A−1 )
Standard square
Hyperbolic tangent
Magnetostrictive modulus 10−12 (m2 A−2 )
Magnetoelastic modulus 10−20 (m2 A−2 Pa−1 )
m 10−12 (m2 A−2 )
r 10−20 (m2 A−2 Pa−1 )
1 2 3 4 5 6 7 8
6.9 15.3 23.6 32 40.4 48.7 57.1 65.4
0.15 0.4 0.7 1.0 1.3 1.6 1.9 2.2
25 18 16 13 12 10 9 8
0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09
−2:77 −1:0 −0:24 0.002 0.08 0.1 0.1 0.1
0.28 0.28 0.28 0.28 0.28 0.28 0.28 0.28
−8:67 −3:13 −0:75 0.006 0.25 0.31 0.31 0.31
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065 compressive stress σ =65.4MPa Hyperbolic Tangent model Standard Square model Model based on density of domain switching
200
150
error [%]
100
50
0 0 _
50
1
2
3
4
5
H [KOe]
_100
Fig. 5. Error evaluation of the three models when the pre-stress is 65:4 MPa.
DDS model as opposed to an overestimation by the HT model. As is addressed before, the strain response of magnetostrictive materials is generally divided into three regions. In the low magnetic 4eld region, the strain rises slowly as the 4eld increases. The strain increases rapidly in the moderate 4eld region. And in the high 4eld region the strain approaches saturation. On the basis of comparisons of the calculations and experimental results, an error evaluation of the three models when the pre-stress is 65:4 MPa is exhibited in Fig. 5. Obviously, the errors of the HT and the DDS models with reference to the experimental data are much smaller than that of the SS model in the high 4eld region. The HT and the DDS models can predict, to some extent, the saturation trend of the magnetic-4eld-induced strain in the high 4eld region while the SS model fails to do such a prediction. However, the fact that the SS model can describe experimental results accurately in the region of low and moderate 4elds suggests that it might be appropriate to employ the SS constitutive relation when the magnetic 4eld is not very high. From the above analysis, the validity of each model should include the following three aspects, i.e. the capability to accurately predict the experimental results in the region of small magnetic 4eld, the capability to predict the saturation trend of experimental results under intense magnetic 4eld and the capability to re:ect the notable dependence of magnetostrictive behavior on external stresses. It seems that the DDS model is superior to the other two models in terms of
1063
the above three aspects. Moreover, the DDS model involves only several moduli that have de4nitely physical denotations and can be determined by controlled experiments, contrasted with the SS model that are developed by means of truncating a polynomial expansion of the Gibbs free energy function. It seems to be reasonable to select the DDS model though it has a relatively more complicated formulation, if one is going to put a constitutive model into a FEA code in order to analyze the magnetomechanical response of structure in practical engineering. As is pointed out by the anonymous reviewer of this paper, there is too little freedom left to adjust the curves to 4t better with the experimental results. None of the proposed three models can predict the nonlinear deformation behavior well when the applied magnetic 4eld is extremely intensive. A more perfect constitutive relation may probably be established on the involvement of microstructures embedded in the magnetic materials. Nevertheless, the HT and the DDS models, which are related to overestimating and underestimating the experimental curves demonstrated in Figs. 2 and 3, respectively, in the region of very intensive magnetic 4eld, may provide the upper bound of estimation and the lower bound of estimation if these two models are utilized to analyze the practical structures. From this point of view, the availability of these models in this work is helpful and deserved. The strain response of magnetic materials under coupling magnetoelastic loading is complicated in essence, especially for the giant magnetostrictive compound of rare earth and iron whose magnetoelastic response is intensively sensitive to external stress. The evolvement of microstructures, such as magnetic domain switching, may be vital for uncovering basic mechanisms of the macro-response of magnetic materials. Further work is needed to develop a constitutive model that can predict the whole process of the non-linear magnetostrictive deformation well. 6. Conclusions In the present work, the phenomenological non-linear constitutive relations of magnetostrictive materials are studied and three thermodynamically based constitutive models are proposed. The standard square model (SS), which is valid for the region of
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small and moderate magnetic 4elds, is developed by means of truncating the polynomial expansion of the Gibbs free energy function. The hyperbolic tangent model (HT), proposed by including a hyperbolic tangent magnetic-4eld dependence, can predict, to some extent, the trend of saturation of magnetostrictive strain in the high 4eld region. The DDS model (based on density of domain switching), developed on the ground that magnetic domain-switching physically underlies the magnetostrictive deformation, is more e;ective in simulating the experimental curves. In order to experimentally determine the moduli in the constitutive equations, one material function is proposed to describe the relation between the compressive pre-stress and the magnetic 4eld corresponding to the peak piezomagnetic coe8cient. Besides, the SS constitutive relation for the general three-dimensional problem is presented, and an approach is introduced to characterize the modulus tensors in terms of experimental results. Further work is needed both to develop a constitutive model that can predict the whole process of the non-linear magnetostrictive deformation well and to perform more experiments to obtain experimental results of various magnetostrictive materials.
where #1 –#15 are mutually independent coe8cients. Suppose that the tensor is symmetric with respect to each of the three pairs of indices, i; j and k; l and m; n, and also symmetric with respect to alternatively arranged index pairs, the tensor can be reduced to
Acknowledgements
rijklmn =
The authors are grateful for the support by the National Natural Science Foundation of China.
+ kl ( im jn + in jm kl ) + mn ( ik jl + il jk )] + #3 [ lm ( ik ln + ik jn ) + km ( il jn + in jk ) + ln ( ik jm + im jk ) + kn ( im jl kn + il jm )]:
(A.2)
In (A.2) there are only three independent coe8cients. For the 4eld magnetoelastic tensor and 4eld magnetostrictive tensor, then rijklmn =
C (mijpl mpkmn + mijkp mplmn ):
(A.3)
Symmetrizing the 4eld magnetoelastic tensor with respect to the three alternatively arranged index pairs leads to 1 C [(mijlp mpkmn + mijkp mplmn ) 3 + (mijmp mpnkl + mijnp mpmkl ) + (mklip mpjmn + mkljp mpimn )]:
(A.4)
The 4eld magnetostrictive tensor is a four-order isotropic tensor and can be generally written as
Appendix Generally, the sixth-order isotropic tensors can be expressed as rijklmn = #1 ij kl mn + #2 ij km ln + #3 ij kn lm + #4 ik jl mn + #5 ik ml ln + #6 ik jn lm + #7 il jk mn + #8 il jm kn + #9 il jn km + #10 im jk ln + #11 im jl kn + #12 im jn kl
mijkl = # ij kl + "( ik jl + il jk )
(A.1)
(A.5)
in which # = m12 , " = 1=2(m11 − m12 ). m11 is the strain caused by per unit magnetic 4eld and measured in the direction of external magnetic 4eld; m12 the strain measured in the direction perpendicular to the magnetic 4eld. Inserting (A.5) into (A.4) renders
C 4 2#2 ij kl mn + #"[ ij ( km ln rijklmn = 3 + kn lm ) + kl ( im jn + in jm kl )
+ #13 in jk lm + #14 in jk km + #15 in jm kl ;
rijklmn = #1 ij kl mn + #2 [ ij ( km ln + kn lm )
+ mn ( ik jl + il jk )] + "2 [ lm ( ik ln
Y. Wan et al. / International Journal of Non-Linear Mechanics 38 (2003) 1053 – 1065
+ ik jn ) + km ( il jn + in jk ) + ln ( ik jm + im jk ) + kn ( im jl kn + il jm )]
;
(A.6)
where C can be characterized in terms of the magnetoelastic modulus for one-dimensional cases and should be obtained according to the experimental data. From (A.2) and (A.6), it can be known that the 4eld magnetoelastic tensor is a sixth-order isotropic tensor that is symmetric with respect to each of the three pairs of indices i; j and k; l and m; n and also symmetric with respect to three alternatively arranged index pairs. There exist three independent coe8cients that should be and can be obtained through one-dimensional experiments. References [1] A.E. Clark, Magnetostrictive rare earth–Fe2 compounds, in: E.P. Wohlfarth (Ed.), Ferromagnetic Materials, Vol. 1, North-Holland, Amsterdam, 1980, pp. 531–589. [2] R.D. Greenough, A.G. Jenner, M.P. Schulze, A.J. Wilkinson, The properties and applications of magnetostrictive rare-earth compounds, J. Magn. Magn. Mater. 101 (1991) 75–80.
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