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Model of a magnetizable elastic material
Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
Model of a magnetizable elastic material V.A. Naletova *, V.A. Turkov, Y.M. Shkel, D.J. Klingenberg Department of Mechanics and Mathematics, Moscow University, Vorobievy gory, 119899, Moscow, Russia Institute of Mechanics, Moscow University, Michurinskii Pr. 1, 119899, Moscow, Russia Department of Chemical Engineering and Rheology Research Center, University of Wisconsin, 53706 Madison, WI, USA Received 30 October 1998; received in revised form 19 February 1999
Abstract A model of a paramagnetic isotropic elastic material with small Young's modulus is proposed, wherein the stress tensor contains additional terms involving products of strain and "eld components. The new &cross' terms produce an e!ective Young's modulus which depends on the magnetic "eld, whereas the classical model does not predict an in#uence of a magnetic "eld on deformation. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.80.#q; 83.20.Bg Keywords: Magnetizable elastic composite material; Stresses in magnetizable materials; Magnetic "eld; Deformation; Young's modulus
1. Introduction A model describing "eld induced stresses in paramagnetic elastic isotropic materials may be proposed as the analogy of a previous model of elastic isotropic dielectrics in an electric "eld [1]. The main assumption of the model [1] is that a tensor of dielectric permeability depends linearly on a strain tensor u . Using the same assumption GH about magnetic permeability we have calculated the free energy of paramagnetic elastic isotropic materials in an external magnetic "eld. Our expression for the free energy has additional &cross' terms
* Corresponding author. Fax: #7-095-9392090. E-mail address: [email protected] (V.A. Naletova)
(terms containing products of the magnetic "eld and the strain tensor components). Analogical terms (terms containing products of the electric "eld and the strain tensor components) are neglected in the model [1]. Same &cross' terms occur in an expression for the free energy of elastic anisotropic ferromagnetic materials [2]. However, for rigid ferromagnetic materials (small deformation and large Young's modulus) additional &cross' terms are small. As it is proven hereinafter the new model (with additional &cross' terms) describes equilibriumly magnetizable (paramagnetic) media with small Young's modulus when deformations are large. For example, the model may be valid for isotropic magnetizable composite materials on the basis of nonrigid elastomers or rubbers. It should be noted that this model may be valid for other paramagnetic homogeneous materials with small Young's modulus.
0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 3 6 6 - 2
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
Here we consider the e!ect of the additional &cross' terms on the stress tensor. The contribution of the &cross' terms to the stress tensor of an elastic material in a magnetic "eld can be described by e!ective values of the Young's modulus that depend on the magnetic "eld. Here we do not describe the well-known *E-e!ect for anisotropic ferromagnetic materials. The model predictions for two examples of deformation of material in an applied magnetic "eld are compared with the previous model (without &cross' terms) [1] in which there is no in#uence of the applied "eld on the elastic response.
2. Stress tensor of magnetizable elastic materials in a magnetic 5eld Let us consider a equilibriumly magnetizable (paramagnetic) elastic isotropic material with a magnetic permeability tensor k that depends on GH the strain tensor components u , GH k (u )"k (kg #a u #a uIg ), GH IJ GH GH I GH
(1)
where k ("4p;10\ NA\) is the permeability of free space, g is the metric tensor, k is the perGH meability of the nondeformed material, and a and a are scalar functions of thermodynamic para meters. The parameters a and a can be estimated for isotropic magnetizable composite media through the analogy with heterogeneous dielectric materials [3] a "!(k!1), a "!(k!1) ( k# ).
B ) dH, B "k HH. G GH
F (H"0)"0.5j(uI)#gu uGH, (4) I GH where j and g are the LameH coe$cients. Substituting Eq. (1) into Eq. (3), the thermodynamic potential FI becomes H H HGHH FI "F !k k !k a uG !k a u . G 2 GH 2 2 (5) The stress tensor in a magnetizable elastic medium in the presence of a magnetic "eld may be obtained by analogy with the development in Ref. [1],
*FI p "FI g # #H B . (6) GH GH *u H G H GH 2 Upon substituting the expression for the free energy, Eq. (5), into Eq. (6), the stress tensor p beGH comes p "juIg #2gu I GH GH GH k#a 2k!a Hg H H !k #k G H GH 2 2
#k a u H HI!u HI G IJ
HIHJ g 2 GH
H #k a uJ H H ! g . J G H 2 GH
(7)
The products of the strain tensor and magnetic "eld components (&cross' terms in Eq. (7)) were not accounted for in the model developed in Ref. [1].
3. Deformation of materials in a magnetic 5eld
H
The free energy F (H"0) is represented by Hooke's law,
(2)
The thermodynamic potential of the material in a magnetic "eld H is FI ";!¹S!H ) B, where ; and S are the internal energy and the entropy of the medium and "eld per unit volume, ¹ is the temperature, and B is the magnetic induction. According to Gibbs' identity, we can represent the free energy as FI "F (H"0)!
571
(3)
We will consider deformation of an elastic body which is a parallelepiped in a Cartesian coordinate system with axes directed along the edges of the parallelepiped when a uniform magnetic "eld H is applied (H is a magnetic "eld without the body). A uniform surface force with density F"$Fe is X applied to the xy faces of the parallelepiped (e is the G base vector along the i-axis). The other surfaces of the parallelepiped are stress-free.
572
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
We assume that the magnetic "eld H"H is a uniform one everywhere (H M, where M is a vector of magnetization of the material, noninductive approximation). The stress tensor components satisfy the equations of statics, because we search for uniform deformations and H is a uniform one. The boundary conditions on the parallelepiped surfaces R , R , and R are WX XV VW +p ,"RWX"0, +p ,"RXV"0, +p ,"RVW"F. (8) Here +A,"A !A , with indices m and a denotK ? ing values of A on the material and air sides of the surfaces, respectively. Our objective is to determine the deformation of the material in the uniform magnetic "eld caused by the external surface force density F. Below we consider the two di!erent cases, H "H e and H "H e . V X H parallel to the x-axis: Taking into account expression (7), we write down Eq. (8) in the form H juJ#2gu #k (a uJ#a u ) "A, J J 2
1!k H#k H , 1#k H g(3j#2g) E" . j#g E"E W
H juJ#2gu !k (a uJ#a u ) "F#B, J J 2 H A(H )"k (1!k#a #a ) , 2
(13)
Assuming j g, the expression for u follows from Eq. (11): 1#2/3G , u "F/E, E"E 1#1/2G
(9)
H juJ#2gu !k (a uJ#a u ) "B, J J 2
Taking into account Eq. (9), the expression for the strain u is obtained: (F#2B)(1#k H) j#g u " g(3j#2g) (1!k H#k H) A(1#2k H) B#A # ! , (1!k H#k H) 2g k a k a , k " , k " 2(3j#2g) 4g
H juJ#2gu !k (a uJ#a u ) "B, J J 2
k a (2j#g) k " . 2g(3j#2g)
(12)
E "3g, G"k a H/2g"2k H. W Thus if the extending force is su$ciently large and j g, the e!ective value of the Young's modulus in the magnetic "eld E depends essentially on the "eld if "G" is not small. The classical model does not predict any in#uence of the magnetic "eld on deformation for this case. H parallel to the z-axis: In this case, the bound ary conditions (8) yield
H juJ#2gu !k (a uJ#a u ) "B, J J 2
H B(H )"k (k!1#a ) . 2
In the classical model [1] k , k , k equal zero. Assuming F A and F B, we obtain the following expression for u : j#g F(1#k H) . (11) u " g(3j#2g) (1!k H#k H) Thus if the extending force is su$ciently large, then the in#uence of the magnetic "eld can be described by the introduction of the e!ective value E of the Young's modulus in the magnetic "eld as follows:
(10)
H juJ#2gu #k (a uJ#a u ) "F#A, J J 2 From these equations we can calculate u as fol lows:
!(2j#g) (F#A)(1#2k H) u " g(3j#2g) 1!k H#k H 2B(1#k H) F#B#A # # . 1!k H#k H g
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
Consider the case of large forces: F A and F B. Then the deformation can be determined by the formula !(2j#g) F(1#2k H) F # . u " g(3j#2g) 1!k H#k H g When the extending force is su$ciently large and j g, the in#uence of the magnetic "eld can be described by the introduction of the e!ective value E of the Young's modulus in the magnetic "eld is as follows:
E"E 1#G . The examples of deformation of the material considered above illustrate that for large forces the classical model predicts no in#uence of the magnetic "eld on deformation while the new model in this case allows us to introduce the e!ective value of the Young's modulus, which depends on the "eld. Obviously, the &cross' terms represented in the formula for the stress tensor (7) should be taken into
573
account when the deformation of materials is not small and the LameH coe$cient g is small enough: "G"""k a H/2g"&1. So this model can describe the properties of isotropic composite materials that are created on the basis of nonrigid polymers or rubbers.
Acknowledgements This work was funded in part by the Russian Foundation for Basic Research No. 97-01-00570.
References [1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York, 1984. [2] G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, Elsevier, New York, 1988. [3] Y.M. Shkel, D.J. Klingenberg, J. Appl. Phys. 83 (1) (1998) 415.
Model of a magnetizable elastic material V.A. Naletova *, V.A. Turkov, Y.M. Shkel, D.J. Klingenberg Department of Mechanics and Mathematics, Moscow University, Vorobievy gory, 119899, Moscow, Russia Institute of Mechanics, Moscow University, Michurinskii Pr. 1, 119899, Moscow, Russia Department of Chemical Engineering and Rheology Research Center, University of Wisconsin, 53706 Madison, WI, USA Received 30 October 1998; received in revised form 19 February 1999
Abstract A model of a paramagnetic isotropic elastic material with small Young's modulus is proposed, wherein the stress tensor contains additional terms involving products of strain and "eld components. The new &cross' terms produce an e!ective Young's modulus which depends on the magnetic "eld, whereas the classical model does not predict an in#uence of a magnetic "eld on deformation. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.80.#q; 83.20.Bg Keywords: Magnetizable elastic composite material; Stresses in magnetizable materials; Magnetic "eld; Deformation; Young's modulus
1. Introduction A model describing "eld induced stresses in paramagnetic elastic isotropic materials may be proposed as the analogy of a previous model of elastic isotropic dielectrics in an electric "eld [1]. The main assumption of the model [1] is that a tensor of dielectric permeability depends linearly on a strain tensor u . Using the same assumption GH about magnetic permeability we have calculated the free energy of paramagnetic elastic isotropic materials in an external magnetic "eld. Our expression for the free energy has additional &cross' terms
* Corresponding author. Fax: #7-095-9392090. E-mail address: [email protected] (V.A. Naletova)
(terms containing products of the magnetic "eld and the strain tensor components). Analogical terms (terms containing products of the electric "eld and the strain tensor components) are neglected in the model [1]. Same &cross' terms occur in an expression for the free energy of elastic anisotropic ferromagnetic materials [2]. However, for rigid ferromagnetic materials (small deformation and large Young's modulus) additional &cross' terms are small. As it is proven hereinafter the new model (with additional &cross' terms) describes equilibriumly magnetizable (paramagnetic) media with small Young's modulus when deformations are large. For example, the model may be valid for isotropic magnetizable composite materials on the basis of nonrigid elastomers or rubbers. It should be noted that this model may be valid for other paramagnetic homogeneous materials with small Young's modulus.
0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 3 6 6 - 2
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
Here we consider the e!ect of the additional &cross' terms on the stress tensor. The contribution of the &cross' terms to the stress tensor of an elastic material in a magnetic "eld can be described by e!ective values of the Young's modulus that depend on the magnetic "eld. Here we do not describe the well-known *E-e!ect for anisotropic ferromagnetic materials. The model predictions for two examples of deformation of material in an applied magnetic "eld are compared with the previous model (without &cross' terms) [1] in which there is no in#uence of the applied "eld on the elastic response.
2. Stress tensor of magnetizable elastic materials in a magnetic 5eld Let us consider a equilibriumly magnetizable (paramagnetic) elastic isotropic material with a magnetic permeability tensor k that depends on GH the strain tensor components u , GH k (u )"k (kg #a u #a uIg ), GH IJ GH GH I GH
(1)
where k ("4p;10\ NA\) is the permeability of free space, g is the metric tensor, k is the perGH meability of the nondeformed material, and a and a are scalar functions of thermodynamic para meters. The parameters a and a can be estimated for isotropic magnetizable composite media through the analogy with heterogeneous dielectric materials [3] a "!(k!1), a "!(k!1) ( k# ).
B ) dH, B "k HH. G GH
F (H"0)"0.5j(uI)#gu uGH, (4) I GH where j and g are the LameH coe$cients. Substituting Eq. (1) into Eq. (3), the thermodynamic potential FI becomes H H HGHH FI "F !k k !k a uG !k a u . G 2 GH 2 2 (5) The stress tensor in a magnetizable elastic medium in the presence of a magnetic "eld may be obtained by analogy with the development in Ref. [1],
*FI p "FI g # #H B . (6) GH GH *u H G H GH 2 Upon substituting the expression for the free energy, Eq. (5), into Eq. (6), the stress tensor p beGH comes p "juIg #2gu I GH GH GH k#a 2k!a Hg H H !k #k G H GH 2 2
#k a u H HI!u HI G IJ
HIHJ g 2 GH
H #k a uJ H H ! g . J G H 2 GH
(7)
The products of the strain tensor and magnetic "eld components (&cross' terms in Eq. (7)) were not accounted for in the model developed in Ref. [1].
3. Deformation of materials in a magnetic 5eld
H
The free energy F (H"0) is represented by Hooke's law,
(2)
The thermodynamic potential of the material in a magnetic "eld H is FI ";!¹S!H ) B, where ; and S are the internal energy and the entropy of the medium and "eld per unit volume, ¹ is the temperature, and B is the magnetic induction. According to Gibbs' identity, we can represent the free energy as FI "F (H"0)!
571
(3)
We will consider deformation of an elastic body which is a parallelepiped in a Cartesian coordinate system with axes directed along the edges of the parallelepiped when a uniform magnetic "eld H is applied (H is a magnetic "eld without the body). A uniform surface force with density F"$Fe is X applied to the xy faces of the parallelepiped (e is the G base vector along the i-axis). The other surfaces of the parallelepiped are stress-free.
572
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
We assume that the magnetic "eld H"H is a uniform one everywhere (H M, where M is a vector of magnetization of the material, noninductive approximation). The stress tensor components satisfy the equations of statics, because we search for uniform deformations and H is a uniform one. The boundary conditions on the parallelepiped surfaces R , R , and R are WX XV VW +p ,"RWX"0, +p ,"RXV"0, +p ,"RVW"F. (8) Here +A,"A !A , with indices m and a denotK ? ing values of A on the material and air sides of the surfaces, respectively. Our objective is to determine the deformation of the material in the uniform magnetic "eld caused by the external surface force density F. Below we consider the two di!erent cases, H "H e and H "H e . V X H parallel to the x-axis: Taking into account expression (7), we write down Eq. (8) in the form H juJ#2gu #k (a uJ#a u ) "A, J J 2
1!k H#k H , 1#k H g(3j#2g) E" . j#g E"E W
H juJ#2gu !k (a uJ#a u ) "F#B, J J 2 H A(H )"k (1!k#a #a ) , 2
(13)
Assuming j g, the expression for u follows from Eq. (11): 1#2/3G , u "F/E, E"E 1#1/2G
(9)
H juJ#2gu !k (a uJ#a u ) "B, J J 2
Taking into account Eq. (9), the expression for the strain u is obtained: (F#2B)(1#k H) j#g u " g(3j#2g) (1!k H#k H) A(1#2k H) B#A # ! , (1!k H#k H) 2g k a k a , k " , k " 2(3j#2g) 4g
H juJ#2gu !k (a uJ#a u ) "B, J J 2
k a (2j#g) k " . 2g(3j#2g)
(12)
E "3g, G"k a H/2g"2k H. W Thus if the extending force is su$ciently large and j g, the e!ective value of the Young's modulus in the magnetic "eld E depends essentially on the "eld if "G" is not small. The classical model does not predict any in#uence of the magnetic "eld on deformation for this case. H parallel to the z-axis: In this case, the bound ary conditions (8) yield
H juJ#2gu !k (a uJ#a u ) "B, J J 2
H B(H )"k (k!1#a ) . 2
In the classical model [1] k , k , k equal zero. Assuming F A and F B, we obtain the following expression for u : j#g F(1#k H) . (11) u " g(3j#2g) (1!k H#k H) Thus if the extending force is su$ciently large, then the in#uence of the magnetic "eld can be described by the introduction of the e!ective value E of the Young's modulus in the magnetic "eld as follows:
(10)
H juJ#2gu #k (a uJ#a u ) "F#A, J J 2 From these equations we can calculate u as fol lows:
!(2j#g) (F#A)(1#2k H) u " g(3j#2g) 1!k H#k H 2B(1#k H) F#B#A # # . 1!k H#k H g
V.A. Naletova et al. / Journal of Magnetism and Magnetic Materials 202 (1999) 570}573
Consider the case of large forces: F A and F B. Then the deformation can be determined by the formula !(2j#g) F(1#2k H) F # . u " g(3j#2g) 1!k H#k H g When the extending force is su$ciently large and j g, the in#uence of the magnetic "eld can be described by the introduction of the e!ective value E of the Young's modulus in the magnetic "eld is as follows:
E"E 1#G . The examples of deformation of the material considered above illustrate that for large forces the classical model predicts no in#uence of the magnetic "eld on deformation while the new model in this case allows us to introduce the e!ective value of the Young's modulus, which depends on the "eld. Obviously, the &cross' terms represented in the formula for the stress tensor (7) should be taken into
573
account when the deformation of materials is not small and the LameH coe$cient g is small enough: "G"""k a H/2g"&1. So this model can describe the properties of isotropic composite materials that are created on the basis of nonrigid polymers or rubbers.
Acknowledgements This work was funded in part by the Russian Foundation for Basic Research No. 97-01-00570.
References [1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York, 1984. [2] G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, Elsevier, New York, 1988. [3] Y.M. Shkel, D.J. Klingenberg, J. Appl. Phys. 83 (1) (1998) 415.