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Magnetostrictive twisting of film–substrate plates
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
Magnetostrictive twisting of film–substrate plates Victor H. Guerrero, Robert C. Wetherhold* Department of Mechanical and Aerospace Engineering, 318 Jarvis Hall, University at Buffalo, SUNY, Buffalo 14260-4400, NY, USA Received 20 December 2003
Abstract In this paper we propose a method to calculate the torsional deformations induced in film–substrate systems by magnetostrictive strains. The calculation scheme proposed assumes that the magnetostriction phenomenon can be modeled in terms of anisotropic expansional strains. These strains are incorporated into the classical plate theory to formulate an energy minimization problem, which is solved using a Ritz method. To illustrate the method we use a cantilever plate in which the film and the substrate are mechanically isotropic and a saturating magnetic field is applied in a direction parallel to the plane of the plate. Our calculations show that the deformation induced by an in-plane field generally includes twisting as well as bending deflection. The cases in which only one of these deflections is present are discussed. One of the advantages of the method proposed is that it can be used regardless of the relative thickness of the substrate with respect to that of the film. It also allows us to consider cases in which a film–substrate system deforms due to piezoelectric, thermal or hygroscopic expansional strains. r 2004 Elsevier B.V. All rights reserved. PACS: 75.80.+q; 75.70.-i Keywords: Magnetostriction; Magnetoelastic interaction; Magnetostrictive material
1. Introduction There is a growing variety of microsensors and actuators that incorporate layered structures made of a magnetostrictive thin film (single layer or multilayer) deposited on a nonmagnetic substrate. The interest in the study and the exploitation of these structures, also known as bimorphs, arise from the potential advantages that they offer to the designer who is interested in remote operation, high-energy density, short response time and manufacturability. The full realization of this *Corresponding author. Tel.: +1-716-645-2593-2241; fax: +1-716-645-3875. E-mail address: [email protected] (R.C. Wetherhold).
potential requires, among other things, further research on materials, manufacturing, modeling and design. On the materials side, significant advances have been made in recent years due to the development of materials that undergo large magnetostrictions (giant magnetostrictive materials, GMS) under the influence of relatively small fields. Unfortunately, the tools available for modeling and design of magnetostrictive layered structures are still not sufficiently developed. In this work we intend to contribute to these areas by presenting a method to calculate the torsional deformations induced in film–substrate systems. The magnetostrictively induced deformation of bimorphs has been the subject of various studies performed in the last decades. As early as 1976,
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.02.002
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Klokholm [1] investigated the deformation of a cantilevered film–substrate beam that undergoes magnetization and provided an expression to calculate its bending deflection. A few years later, several authors [2–4] challenged Klokholm’s results and presented a new expression to calculate the deflection at the free end of a magnetically saturated cantilever bimorph. In their calculations, these authors used different versions of a twodimensional model in which the thickness of the film (tf) is large enough so that the surface magnetostriction is negligible (tfZ50 nm) but relatively small compared to the thickness of the substrate (ts), such that tf/tsr0.001. Unfortunately, the limited range of validity of this model, together with several inconsistencies which may be found in the derivations, restricted its reliability and applicability. In a recent article [5], the authors have proposed a self-consistent derivation in which the magnetostrictive strains are treated simply as expansional strains. Using this method it is possible to calculate accurately not only the deflections of a cantilever bimorph beam but also its strains, stresses and energy densities, regardless of the relative magnitude of the thickness of the film. This is especially useful in cases in which the necessity of inducing larger deflections demands the use of a relatively large thickness ratio z ¼ tf =ts : The method has also been used to derive the conditions that define film-substrate cantilever beams that show maximum bending deflection, as required in the case of sensors, or that exert maximum force, as required in the case of actuators [6]. There is however a potentially very useful feature of magnetostrictive film–substrate bimorphs that has received very little attention. This is the possibility of generating not only bending but also twisting deflections by using magnetically induced strains. A literature review reveals relatively few articles that deal with magnetostrictively induced torsional deformations. Previous studies on this topic [7–9] present a limited framework for describing the phenomenon, which in turn makes very difficult to obtain qualitative and quantitative predictions. Until this point, the work has been concentrated on the use of AC magnetic fields to
induce dynamic torsional deformations in magnetically anisotropic bimorphs. These dynamic deformations have been used with relative success to infer some properties of magnetostrictive thin films whose thickness is such that z=tf/tsE1/150 [7,8]. In cases in which the values chosen for z are of the order of 0.1, the dynamic twisting deflections have been used to actuate optical scanners [9]. None of the previous studies have presented a general framework for calculating the torsional strains, stresses and deformations induced by external DC magnetic fields. The only methods proposed are those intended to calculate dynamic torsional strains. These existing methods also seem to have a number of inconsistencies. For instance, in some cases the fact that torsional and bending deformations can be induced at the same time in a bimorph is ignored. In others, the behavior of film–substrate systems in which the thickness of the film is comparable to that of the substrate are predicted by using derivations intended for use when zr0.001. The study of the behavior of bimorphs made of magnetically isotropic thin films has also been omitted, as well as the relationship between statically and dynamically induced deformations. In this article, we propose a framework for the calculation of the twisting and bending deflections induced in a film-substrate system by means of a DC magnetic field. Our intention is to provide some tools to assess, predict and optimize the response of magnetostrictive bimorphs. This should prove useful at the moment of designing devices such as those intended for micropositioning, angular motion and torque generation, compensation for thermally induced deformations, etc. We also expect that our results will make it possible to formulate self-consistent methods to calculate magnetostrictively induced dynamic deflections. Our article can be divided into three main parts. In the first part, we present the relationships between the stresses, strains and displacements observed in a magnetostrictive bimorph. In the second part, we use these relationships to formulate a Ritz method that will allow us to calculate the deformations induced in a film– substrate plate. Finally, we illustrate the use of
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
the method proposed by solving several examples for a magnetically saturated cantilever plate.
2. Stress, strain and displacements in magnetostrictive bimorphs Let us consider a bimorph made of a magnetically and mechanically isotropic magnetostrictive thin film (tfZ50 nm) deposited on a nonmagnetic substrate. The simplest geometry that can be fabricated and studied is that of a rectangular plate of length L, width W and thickness t. Such a plate is schematically shown in Fig. 1, along with a coordinate system whose axes are parallel to the edges of the plate and form an xy plane that coincides with the midsurface of the bimorph. If the film-substrate system is subjected to a magnetic field applied in a direction parallel to the xy plane (no field component in the z direction), a particular set of magnetostrictive strains will be induced in the film, and as a result the plate will deform. Among other factors, the film strains and the plate deformations will depend on the magnitude of the field applied and the initial magnetic state of the film. As the magnetic field increases, the induced strains will also increase until they reach a point that correspond, to the saturated state. In this work we will consider only moderate fields, and therefore there is no volume magnetostriction. If the magnetostrictive material is isotropically demagnetized in the initial state, the magnetically induced strains will increase gradu-
z
tf ts z2
z1
z0 L
W x
Fig. 1. Plate made of a magnetostrictive thin film deposited on a nonmagnetic substrate.
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ally to reach a value equal to the saturation magnetostriction constant ls in the direction in which the field is applied. In the perpendicular directions the magnetically induced strains observed at saturation will be equal to ls/2. In the typical case, the film–substrate system considered is reduced to a beam and the magnetic field is applied along its longitudinal or the transverse directions. The result is the well-known anticlastic bending deformation that has been previously studied [1–6]. In contrast, if the magnetic field is applied along a direction that forms an angle y with the x-axis, the resulting deformation of the bimorph will involve, in general, not only bending but also twisting. This is the consequence of having induced strains that are not symmetric with respect to the x-axis. In this article we will describe a method that can be used to calculate this type of deformation. Since the in-plane dimensions of a thin film– substrate plate are comparable with each other and relatively large with respect to the thickness, we can assume that the bimorph being deformed is in a state of plane stress. Under these circumstances, the magnetostrictive expansional strains induced by a magnetic field in the kth layer of the film–substrate system are given by the strain transformation [10] 8 e 9k 2 2 m > > < ex = 6 e e k ey fe g ¼ ¼ 4 n2 > : e > ; 2mn gxy
n2 m2 2mn
38 9k mn > < e1 > = 7 mn 5 e2 ; > : > ; m 2 n2 0
ð1Þ where m=cos(y) and n=sin(y). The number of values that the superscript k can take depends on the number of layers present in the plate studied. In our case the superscript k can only take the values s and f, which correspond to the substrate and the film, respectively. Even though Eq. (1) is valid for strains induced by an arbitrary magnetic field, we will only consider the case in which the film is magnetically saturated. In this situation, the vector fe1 e2 0gT represents the expansional strains measured along an x1 x2 x3 coordinate system whose x1-axis coincides with the direction of magnetic saturation. The superscript T is used to indicate the transpose of a matrix. With the
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V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
previous considerations in mind, we have that ef1=ls, ef2=ls/2, where ls represents the saturation magnetostriction value. Since the substrate is made of a non-magnetostrictive material fee gs ¼ f0g: Note that Eq. (1) could be modified to include additional expansional strains that are of piezoelectric, thermal and hygroscopic origin. For instance, if the film–substrate system is subjected to a temperature change DT, and if ak is the thermal expansion coefficient of the kth layer, we s e,f would have ee,s =ls+afDT. In x =a DT and ex k these expressions a DT represents the thermal expansional strain. The ‘‘accomodational’’ stresses set up in the kth layer to ensure compatibility between the strains of the film and the substrate are 9k 2 8 3k 0 Q11 Q12 > = < sx > 6 7 fsgk ¼ sy ¼ 4 Q12 Q22 0 5 fe ee gk ; > > ; : 0 0 Q66 txy ð2Þ where the entries of the stiffness matrix [Q] are Q11 ¼ Q22 ¼ E=ð1 n2 Þ; Q12 ¼ nQ11 and Q66 ¼ E=ð2 þ 2nÞ: E is the Young’s modulus and n is the Poisson’s ratio measured at constant applied magnetic field. The vector {e} represents the total strain of an arbitrary point located at a distance z from the midsurface of the film–substrate plate. Using the assumptions of classical thin plate theory, this strain can be expressed in terms of the midsurface strain vector {e0} and the curvature vector {k} by means of the following equation: {e}={e0}+z{k}, where 9 8 qu0 > > > > 8 0 9 > > > > qx > > e > > > > x = = < qv < 0 0 0 ; ¼ fe g ¼ ey qy > > > ; > > : 0 > > > > > gxy > > > ; : qu0 þ qv0 > qy qx 9 8 q2 w > > > > > > 9 > 8 > > qx2 > > > k > > > x = = < q2 w > < : ð3Þ fkg ¼ ky ¼ 2 > > qy > > > ; > : > > kxy > > 2 > > > > > ; : 2 q w > qxqy
e0x and e0y are the midsurface normal strains measured along the x- and y-axes and g0xy represents the in-plane engineering shear strain. u0, v0 and w represent the displacements in the x; y and z directions of an arbitrary point taken at the midsurface of the bimorph. Eq. (3) can be conveniently expressed as a matrix product as follows: 8 9 ( ) > = < u0 > 0 e ð4Þ ¼ ½q v0 ; > k ; : > w where the differential operator matrix [q] is defined as 2
q 6 qx 6 6 6 ½q ¼ 6 0 6 6 4 0
q 0 qy q q qy qx 0
0
0
0
0
0
0
0
q2 qx2
q2 qy2
2
q2 qxqy
3T 7 7 7 7 7 : 7 7 5 ð5Þ
Since we can use Eq. (2) to obtain the stresses induced in the film–substrate bimorph, we can calculate the net forces and moments per unit length developed in the plate studied by using the following definition ðN; M Þ ¼
Z
h=2
fsgð1; zÞ dz;
ð6Þ
h=2
fNg ¼ fNx Ny Nxy gT represents the normal and shear force resultants; fMg ¼ fMx My Mxy gT are the bending and twisting moment resultants. Substituting Eq. (2) into Eq. (6) and integrating over the plate thickness we obtain the integrated plate constitutive equation (
N M
)
" ¼
A
B
B
D
#(
e0 k
(
)
N MS M MS
) ;
ð7Þ
where the elements of the stiffness matrices [A], [B] and [D] are given by
n X Aij ; Bij ; Dij ¼ k¼1
Z
zk
zk1
Qkij 1; z; z2 dz;
ð8Þ
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
and the equivalent magnetostrictive forces and moments are calculated as follows:
N
MS
;M
MS
¼
n Z X k¼1
zk
½Q k fee gk ð1; zÞ dz;
347
be easily implemented to solve problems that involve a wide variety of boundary and loading conditions.
ð9Þ
zk1
When a bimorph is not subjected to the action of any external load, as for the plates studied here, the net forces {N} and moments {M} are equal to {0}. If the bimorph is also free on all boundaries, the solution of the system of equations represented by Eq. (7) is reduced to the evaluation of the elements of the matrices [A], [B] and [D] and the vectors {N MS } and {M MS }, followed by a series of linear algebra operations. These operations will return values for the magnetostrictively induced midsurface strains {e0} and curvatures {k} of a free plate. On the other hand, if the plate is supported in a given fashion, we need to devise a method to approximate the solution of the corresponding boundary value problem. In simple cases, as when the film–substrate system is clamped at one end, the strains and curvatures calculated for the free bimorph can be used to provide estimates for the resulting deformation. In fact, a one-dimensional version of this method has been used to calculate the induced bending deflection in magnetostrictive cantilever film–substrate beams [5,6]. Even though this approach provides excellent results for the deflections of beams, it is expected to give only rough estimates for cantilever film-substrate systems in which the width/length ratio is larger than about 0.1 (plates). This is because the deformation of such a system also depends on the variation of deflection along the transverse (y) direction. This deflection represents an additional degree of freedom that is not taken into account in onedimensional approximations. In the present study, we will use the Ritz method to find the bending and twisting displacements induced in a magnetostrictive bimorph plate acted upon by a magnetic field. This method has been chosen because of its simplicity and versatility. As it will be shown below, it only requires the use of a given set of approximating functions to represent the displacements u0 ; v0 and w: Additionally, it can
3. Displacement calculation using the Ritz method To be able to find the deflections induced by a magnetic field on a magnetostrictive bimorph plate we only need to minimize the potential energy of the film–substrate system (Utot). This potential energy is given by [11] " # Z B 1 T A Utot ¼ f½q fugg f½q fugg dxdy 2 A B D ( ) Z N MS T dxdy: ð10Þ f½q fugg M MS A The next step is to express the midsurface displacements u0 ; v0 and w as follows: u0 ¼
P X
aui Ui ðx; yÞ;
i¼1
w¼
H X
v0 ¼
J X
avj Vj ðx; yÞ;
j¼1
awk Wk ðx; yÞ;
ð11Þ
k¼1
where Ui ; Vj and Wk are approximating functions that satisfy the geometric boundary conditions (also called kinematic or essential boundary conditions) imposed on the plate. The coefficients aui ; avj and awk are unknown and will be determined in such a way that they minimize Utot. Note that Eq. (10) does not include a term for the potential magnetic energy in the applied field. This is because in our case in which the film is magnetically saturated, the magnetization is parallel to the applied magnetic field; thus the contribution of such a term is constant and therefore it does not play a role during the minimization process. Substituting Eq. (11) into Eq. (10) we can express the potential energy of the bimorph as 1 Utot ¼ fagT ½K fag fagT fLgMS ; ð12Þ 2 where {a} is the unknown coefficient vector defined as {aui avj awk }T which is of dimension ðP þ J þ HÞ 1: The matrix [K] and the
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
348
magnetostrictive actuation vector {L}MS are given by ½K ¼
"
Z
f½q ½R gT
A
B
#
f½q ½R g dxdy; B D ( ) Z MS N ¼ f½q ½R gT dxdy; M MS A A
fLgMS
ð13Þ
where the approximating function matrix [R] has dimension 3 ðP þ J þ HÞ and is defined as 2 6 ½R ¼ 4
fU1 ; U2 yUP g
f0g
f0g f0g
fV1 ; V2 yVJ g f0g 3
f0g f0g
fW1 ; W2 yWH g
7 5
:
ð14Þ
3 ðPþJþHÞ
The values of the coefficients aui ; avj and awk that minimize Utot are the ones that satisfy the system of equations given by qUtot =qanm ¼ 0: Therefore, the unknown coefficient vector can be obtained as follows: fag ¼ ½K 1 fLgMS :
ð15Þ
In order to calculate the magnetostrictively induced displacements of the film–substrate plate we only need to substitute the values of the elements of the vector {a} calculated by means of Eq. (15) into Eq. (11). The last item that we need to discuss in this section is the nature of the approximating functions Ui ; Vj and Wk : Besides the fact that these functions should satisfy the geometric boundary conditions, we can also note that in this case the displacements u0 and v0 can be expressed using only a few terms. This is because u0 and v0 play a minor role in the deflection of the plate being considered. What is important is the adequate approximation of w. In the next section, we will present a particular film–substrate combination whose deformation is to be predicted by the scheme discussed.
4. Calculation examples Consider a cantilever plate made of an amorphous TbFeCo thin film deposited on a (1 0 0) Si substrate. The film and the substrate have the following properties: Ef=80 GPa, nf=0.31, ls ¼ 1020 106 ; Es ¼ 130 GPa; ns=0.28 [12]. Using data for the bimorphs fabricated by Garnier et al. [13], the thicknesses of the film and the substrate are taken to be tf ¼ 2 mm; ts ¼ 15 mm: The initial magnetization of the film is assumed to be randomly distributed in three dimensions. The length and the width of the plate are taken as L ¼ W ¼ 1 mm: Since the displacements and the slope at the fixed end of a cantilever are zero, the boundary conditions that the approximating functions must satisfy at the fixed end are u0 ¼ v0 ¼ w¼ qw=qx ¼ 0: A set of functions that satisfies these conditions is Ui ¼ x Xmi Yni ; Vj ¼ x Xmj Ynj ; ð16Þ Wk ¼ x2 Xmk Ynk ; where Xm ðxÞ and Yn ðyÞ are sets of orthogonal polynomials generated by starting with X1 ¼ Y1 ¼ 1 and using a Gram–Schmidt orthonormalization procedure over intervals [0,L] and [0,W], respectively. In the lack of additional information on how to select a weighting function to improve the performance of the method proposed, we constructed our orthogonal polynomials using a weighting function equal to 1. Using the functions proposed and the data provided, and if y=45 , the deflection profile calculated by means of the proposed method is as shown in Fig. 2. As can be seen in this figure, when the magnetostrictive thin film is saturated, the bimorph not only bends in the parallel and transverse directions but also twists. As can be inferred from Eq. (1), the relative amounts of bending and twisting change as the angle y varies between 0 and 90 . To describe these changes it is necessary to choose a set of meaningful and observable variables. The simplest variable that can be used to describe the bending of a cantilever bimorph is the displacement of the free end of the
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
0.012 − Φ (rad)
6
Φ ∆
2
0.009
-2
0.006
-6
0.003
-10
0
∆ (µm)
0.015
349
-14 0.0
0.1
0.2
0.3
0.4
0.5
θ/π Fig. 3. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial state the film is isotropically demagnetized.
plate at the point given by x ¼ L and y ¼ W =2; that is D ¼ wðL; W =2Þ: For the case of torsion, the variable selected is the twisting angle at the center of the free end of the cantilever bimorph (F), which is given by qw W L; F¼ : qy 2 Note that, in the strict sense, it is not possible to define a single twisting angle at the free end of a plate that also bends in the transverse (y) direction. For this reason we have defined F so that it is evaluated at a specific point, which in this case is the center of the rectangular cross–section of the film-substrate studied. Fig. 3 shows the change of D and F as y goes from 0 to 90 . As can be seen in this figure, as y increases the deflection D increases and changes sign while the magnitude of the twisting angle F shows a maximum at y=45 . Only when y=0 or 90 does the studied film– substrate system bend without twisting. In these cases the results obtained here agree relatively well with those presented in previous articles [5,6]. In this magnetostrictive bimorph, the torsional deformations induced cannot be separated from
0.012 − Φ (rad)
Fig. 2. Bending and twisting induced in a film–substrate plate whose film is saturated by a magnetic field applied at an angle y=45 .
10
Φ ∆
6
0.009
2
0.006
-2
0.003
-6
0
∆ (µm)
0.015
-10 0.0
0.1
0.2
0.3
0.4
0.5
θ/π Fig. 4. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial demagnetized state the magnetostrictive material has a domain structure such that the magnetization is isotropically distributed within the plane of the film.
the bending deflections. A different situation emerges when in the demagnetized state the magnetization is isotropically distributed in the plane of the film. In this case, e1=3ls/4=e2 [6] and the values taken by D and F as we change y are the ones summarized in Fig. 4. Note that in this situation it is possible to maximize the torsional deformation induced without simultaneous bending by saturating the film at y=45 . In addition to the previous cases studied, there is at least another possibility to obtain bending and twisting deflections using a magnetostrictive cantilever plate. This possibility is realized by subjecting a bimorph whose film shows a transverse uniaxial
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
anisotropy to a magnetic field. This anisotropy could be induced by post-deposition annealing of the thin film under the influence of a magnetic field applied along the y direction. Assuming that the magnetostrictive film is in a single domain state and that its magnetization is given by rotation, the expansional strain induced in this case is given by [8]: 9k 8 e 9k 8 2 > = 3 > = < ex > < cos y > eey fee gk ¼ ¼ ls cos2 y : > > ; 2 > ; : e > : sin ð2yÞ gxy
ð17Þ
Using Eq. (17) instead of Eq. (1) for the calculations performed with the proposed method, we obtain the values for D and F as shown in Fig. 5. As can be expected, in this case the maximum bending deflection is obtained when the film is saturated along the longitudinal (x) direction. The twisting angle F is zero when y=0 or 90 , and its maximum magnitude occurs when y=45 . It is important to mention that in our calculations we have assumed that in the initial state, and every time that the applied magnetic field is zero, the midsurface of the bimorph studied coincides with the xy plane. That is, we assumed that once the applied field is reduced to zero, the domain structure of the magnetostrictive material resembles the one that was found in the original demagnetized state.
0.012 −Φ (rad)
0
Φ ∆
-12.8
-4
0.009
-8
0.006
-12
0.003
-16
0
-13
∆ (µm)
0.015
-20 0.0
0.1
Besides the calculation of torsional deformations, another area that has received little attention in previous studies is the influence of the width of a bimorph on the bending deflection induced in a magnetostrictive film–substrate system. This is in part because the methods used to solve the problem in the past did not have the intrinsic capability to deal with variations introduced in the curvatures of a bimorph due to the boundary conditions to which it is subjected. Using the method proposed in this work with plates of variable width, we have obtained the results shown in Fig. 6. The expansional strain field considered in Fig. 6 was the one obtained for a system with an isotropically demagnetized film in the initial state. For comparison purposes, we also show the bending deflection predicted using expressions presented by the authors in previous articles [5,6]. In these articles it was assumed that the width to length ratio ðW =LÞ of the film–substrate system is such that the bimorph can be considered a beam with free bending along the transverse (y) direction. Note that we cannot directly compare with the results that would be predicted using the expressions proposed by Du Tre! molet de Lacheisserie and others [2–4] because the range of validity of those equations is limited to thickness ratio tf =ts r0:001: As expected, the deflection calculated by the method proposed in this work closely approximates that calculated assuming free bending in the
0.2
θ/π
0.3
0.4
0.5
Fig. 5. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial state the film exhibits a uniaxially induced anisotropy along the transverse (y) direction.
∆ (µm )
350
∆ ∆ beam
-13.2 -13.4 -13.6 0.00
0.20
0.40
0.60
0.80
1.00
W/L Fig. 6. Bending deflection (D) of a film–substrate plate calculated for different values of the thickness to length ratio ðW =LÞ: Dbeam represents the deflection estimated by assuming free bending in the transverse (y) direction.
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
transverse (y) direction only when the ratio W =L of the bimorph is smaller than about 0.1 (see Fig. 6). Similar results were obtained by Watts et al. [14] by means of a finite element approximation performed using ANSYSs and modeling the magnetostrictively induced strains as anisotropic thermal expansions. To contrast their method with the one presented here, we should mention that we do not need to calculate equivalent strains to be used as input for models conceived to predict deformations induced by thermal expansion. The proposed method allows us to calculate the deflections induced in bimorphs not only due to magnetostriction but also by additional expansional strains, such as those of piezoelectric, thermal or hygroscopic origin.
5. Summary Beams and plates made of a magnetostrictive thin film deposited on a nonmagnetic substrate represent a family of simple and effective transducers that are suitable for microsystems. These devices are used as sensors and actuators by taking advantage of the strong magnetoelastic interactions occurring in the film. In this work, we have proposed a method to calculate the bending and twisting deflections that are induced in film– substrate bimorphs as a result of the strains induced by a magnetic field. The method makes use of the classical laminate plate theory to formulate an energy minimization problem, which in this case is solved using a Ritz method. Here we have limited ourselves to the study of rectangular and layerwise isotropic cantilever plates that are deformed due to magnetostrictively induced strains. These strains were calculated only for the cases in which the film was magnetically saturated along a given direction. In the initial demagnetized state, the film magnetization was assumed to be in one of the following configurations: (1) isotropically distributed in three dimensions, (2) isotropically distributed but confined to the plane of the film, and (3) oriented according to a uniaxially induced magnetic anisotropy.
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One of the reasons why the torsional deformation of magnetostrictive bimorphs has not received more attention is the fact that previous models have considered the effects of saturating a film only at angles y=0 and 90 . When y is allowed to take intermediate values we found that, in general, an external magnetic field induces not only bending but also twisting deflections in the plates studied. However, it is possible to selectively generate only bending or only twisting deformations, depending on the direction along which a saturating magnetic field is applied and the initial state of the magnetostrictive thin film. Some of the features that distinguish the method proposed here from the ones presented by previous authors are: First, our method can be used to calculate the deformations induced in magnetostrictive bimorphs even when the film is not very thin with respect to the substrate. Second, the calculation scheme proposed is self-consistent and can be used not only on magnetostrictive filmsubstrate beams but also on plates. In the case of cantilever plates, the constrained deformation that exists in the transverse direction results in deflections that are smaller with respect to those calculated assuming free bending. Third, the method presented here deals with magnetostrictive interactions directly. This also allows us to study bimorph deformations induced not only due to magnetostriction but also by other expansional strains, including those of piezoelectric, hygroscopic or thermal nature. Finally, we should mention that the method discussed in this article can be extended to solve problems that involve bimorphs of arbitrary shapes and boundary conditions. For this purpose we only need to choose suitable functions that approximate the displacements u0 ; v0 and w for use in the Ritz formulation.
Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. CMS-00-99830. The support provided by Dr. Shih-Chi Liu, Program Manager, is gratefully acknowledged.
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References [1] E. Klokholm, IEEE Trans. Magn. 12 (1976) 819. [2] E. van de Riet, J. Appl. Phys. 76 (1994) 584. [3] M. Weber, R. Koch, K. Rieder, Phys. Rev. Lett. 73 (1994) 1166. [4] E. du Tr!emolet de Lacheisserie, J.C. Peuzin, J. Magn. Magn. Mater. 136 (1994) 189. [5] V.H. Guerrero, R.C. Wetherhold, J. Appl. Phys. 94 (2003) 6659. [6] V. H. Guerrero and R. C. Wetherhold, J. Magn. Magn. Mater. (2004) in press.
[7] J.C. Peuzin, K. Mackay, J. Appl. Phys. 81 (1997) 1833. [8] Zs. Sarkozi, K. Mackay, J.C. Peuzin, J. Appl. Phys. 88 (2000) 5827. [9] T. Bourouina, et al., J. MEMS 11 (2002) 355. [10] R.C. Wetherhold, V.H. Guerrero, J. Magn. Magn. Mater. 262 (2003) 218. [11] H. Kioua, S. Mirza, Smart Mater. Struct. 9 (2000) 476. [12] N.H. Duc, K. Mackay, J. Betz, D. Givord, J. Appl. Phys. 79 (1996) 973. [13] A. Garnier, et al., Sensors and Actuators A 84 (2000) 156. [14] R. Watts, M.R.J. Gibbs, W.J. Karl, H. Szymczak, Appl. Phys. Lett. 70 (1997) 2607.
Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
Magnetostrictive twisting of film–substrate plates Victor H. Guerrero, Robert C. Wetherhold* Department of Mechanical and Aerospace Engineering, 318 Jarvis Hall, University at Buffalo, SUNY, Buffalo 14260-4400, NY, USA Received 20 December 2003
Abstract In this paper we propose a method to calculate the torsional deformations induced in film–substrate systems by magnetostrictive strains. The calculation scheme proposed assumes that the magnetostriction phenomenon can be modeled in terms of anisotropic expansional strains. These strains are incorporated into the classical plate theory to formulate an energy minimization problem, which is solved using a Ritz method. To illustrate the method we use a cantilever plate in which the film and the substrate are mechanically isotropic and a saturating magnetic field is applied in a direction parallel to the plane of the plate. Our calculations show that the deformation induced by an in-plane field generally includes twisting as well as bending deflection. The cases in which only one of these deflections is present are discussed. One of the advantages of the method proposed is that it can be used regardless of the relative thickness of the substrate with respect to that of the film. It also allows us to consider cases in which a film–substrate system deforms due to piezoelectric, thermal or hygroscopic expansional strains. r 2004 Elsevier B.V. All rights reserved. PACS: 75.80.+q; 75.70.-i Keywords: Magnetostriction; Magnetoelastic interaction; Magnetostrictive material
1. Introduction There is a growing variety of microsensors and actuators that incorporate layered structures made of a magnetostrictive thin film (single layer or multilayer) deposited on a nonmagnetic substrate. The interest in the study and the exploitation of these structures, also known as bimorphs, arise from the potential advantages that they offer to the designer who is interested in remote operation, high-energy density, short response time and manufacturability. The full realization of this *Corresponding author. Tel.: +1-716-645-2593-2241; fax: +1-716-645-3875. E-mail address: [email protected] (R.C. Wetherhold).
potential requires, among other things, further research on materials, manufacturing, modeling and design. On the materials side, significant advances have been made in recent years due to the development of materials that undergo large magnetostrictions (giant magnetostrictive materials, GMS) under the influence of relatively small fields. Unfortunately, the tools available for modeling and design of magnetostrictive layered structures are still not sufficiently developed. In this work we intend to contribute to these areas by presenting a method to calculate the torsional deformations induced in film–substrate systems. The magnetostrictively induced deformation of bimorphs has been the subject of various studies performed in the last decades. As early as 1976,
0304-8853/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.02.002
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Klokholm [1] investigated the deformation of a cantilevered film–substrate beam that undergoes magnetization and provided an expression to calculate its bending deflection. A few years later, several authors [2–4] challenged Klokholm’s results and presented a new expression to calculate the deflection at the free end of a magnetically saturated cantilever bimorph. In their calculations, these authors used different versions of a twodimensional model in which the thickness of the film (tf) is large enough so that the surface magnetostriction is negligible (tfZ50 nm) but relatively small compared to the thickness of the substrate (ts), such that tf/tsr0.001. Unfortunately, the limited range of validity of this model, together with several inconsistencies which may be found in the derivations, restricted its reliability and applicability. In a recent article [5], the authors have proposed a self-consistent derivation in which the magnetostrictive strains are treated simply as expansional strains. Using this method it is possible to calculate accurately not only the deflections of a cantilever bimorph beam but also its strains, stresses and energy densities, regardless of the relative magnitude of the thickness of the film. This is especially useful in cases in which the necessity of inducing larger deflections demands the use of a relatively large thickness ratio z ¼ tf =ts : The method has also been used to derive the conditions that define film-substrate cantilever beams that show maximum bending deflection, as required in the case of sensors, or that exert maximum force, as required in the case of actuators [6]. There is however a potentially very useful feature of magnetostrictive film–substrate bimorphs that has received very little attention. This is the possibility of generating not only bending but also twisting deflections by using magnetically induced strains. A literature review reveals relatively few articles that deal with magnetostrictively induced torsional deformations. Previous studies on this topic [7–9] present a limited framework for describing the phenomenon, which in turn makes very difficult to obtain qualitative and quantitative predictions. Until this point, the work has been concentrated on the use of AC magnetic fields to
induce dynamic torsional deformations in magnetically anisotropic bimorphs. These dynamic deformations have been used with relative success to infer some properties of magnetostrictive thin films whose thickness is such that z=tf/tsE1/150 [7,8]. In cases in which the values chosen for z are of the order of 0.1, the dynamic twisting deflections have been used to actuate optical scanners [9]. None of the previous studies have presented a general framework for calculating the torsional strains, stresses and deformations induced by external DC magnetic fields. The only methods proposed are those intended to calculate dynamic torsional strains. These existing methods also seem to have a number of inconsistencies. For instance, in some cases the fact that torsional and bending deformations can be induced at the same time in a bimorph is ignored. In others, the behavior of film–substrate systems in which the thickness of the film is comparable to that of the substrate are predicted by using derivations intended for use when zr0.001. The study of the behavior of bimorphs made of magnetically isotropic thin films has also been omitted, as well as the relationship between statically and dynamically induced deformations. In this article, we propose a framework for the calculation of the twisting and bending deflections induced in a film-substrate system by means of a DC magnetic field. Our intention is to provide some tools to assess, predict and optimize the response of magnetostrictive bimorphs. This should prove useful at the moment of designing devices such as those intended for micropositioning, angular motion and torque generation, compensation for thermally induced deformations, etc. We also expect that our results will make it possible to formulate self-consistent methods to calculate magnetostrictively induced dynamic deflections. Our article can be divided into three main parts. In the first part, we present the relationships between the stresses, strains and displacements observed in a magnetostrictive bimorph. In the second part, we use these relationships to formulate a Ritz method that will allow us to calculate the deformations induced in a film– substrate plate. Finally, we illustrate the use of
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
the method proposed by solving several examples for a magnetically saturated cantilever plate.
2. Stress, strain and displacements in magnetostrictive bimorphs Let us consider a bimorph made of a magnetically and mechanically isotropic magnetostrictive thin film (tfZ50 nm) deposited on a nonmagnetic substrate. The simplest geometry that can be fabricated and studied is that of a rectangular plate of length L, width W and thickness t. Such a plate is schematically shown in Fig. 1, along with a coordinate system whose axes are parallel to the edges of the plate and form an xy plane that coincides with the midsurface of the bimorph. If the film-substrate system is subjected to a magnetic field applied in a direction parallel to the xy plane (no field component in the z direction), a particular set of magnetostrictive strains will be induced in the film, and as a result the plate will deform. Among other factors, the film strains and the plate deformations will depend on the magnitude of the field applied and the initial magnetic state of the film. As the magnetic field increases, the induced strains will also increase until they reach a point that correspond, to the saturated state. In this work we will consider only moderate fields, and therefore there is no volume magnetostriction. If the magnetostrictive material is isotropically demagnetized in the initial state, the magnetically induced strains will increase gradu-
z
tf ts z2
z1
z0 L
W x
Fig. 1. Plate made of a magnetostrictive thin film deposited on a nonmagnetic substrate.
345
ally to reach a value equal to the saturation magnetostriction constant ls in the direction in which the field is applied. In the perpendicular directions the magnetically induced strains observed at saturation will be equal to ls/2. In the typical case, the film–substrate system considered is reduced to a beam and the magnetic field is applied along its longitudinal or the transverse directions. The result is the well-known anticlastic bending deformation that has been previously studied [1–6]. In contrast, if the magnetic field is applied along a direction that forms an angle y with the x-axis, the resulting deformation of the bimorph will involve, in general, not only bending but also twisting. This is the consequence of having induced strains that are not symmetric with respect to the x-axis. In this article we will describe a method that can be used to calculate this type of deformation. Since the in-plane dimensions of a thin film– substrate plate are comparable with each other and relatively large with respect to the thickness, we can assume that the bimorph being deformed is in a state of plane stress. Under these circumstances, the magnetostrictive expansional strains induced by a magnetic field in the kth layer of the film–substrate system are given by the strain transformation [10] 8 e 9k 2 2 m > > < ex = 6 e e k ey fe g ¼ ¼ 4 n2 > : e > ; 2mn gxy
n2 m2 2mn
38 9k mn > < e1 > = 7 mn 5 e2 ; > : > ; m 2 n2 0
ð1Þ where m=cos(y) and n=sin(y). The number of values that the superscript k can take depends on the number of layers present in the plate studied. In our case the superscript k can only take the values s and f, which correspond to the substrate and the film, respectively. Even though Eq. (1) is valid for strains induced by an arbitrary magnetic field, we will only consider the case in which the film is magnetically saturated. In this situation, the vector fe1 e2 0gT represents the expansional strains measured along an x1 x2 x3 coordinate system whose x1-axis coincides with the direction of magnetic saturation. The superscript T is used to indicate the transpose of a matrix. With the
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previous considerations in mind, we have that ef1=ls, ef2=ls/2, where ls represents the saturation magnetostriction value. Since the substrate is made of a non-magnetostrictive material fee gs ¼ f0g: Note that Eq. (1) could be modified to include additional expansional strains that are of piezoelectric, thermal and hygroscopic origin. For instance, if the film–substrate system is subjected to a temperature change DT, and if ak is the thermal expansion coefficient of the kth layer, we s e,f would have ee,s =ls+afDT. In x =a DT and ex k these expressions a DT represents the thermal expansional strain. The ‘‘accomodational’’ stresses set up in the kth layer to ensure compatibility between the strains of the film and the substrate are 9k 2 8 3k 0 Q11 Q12 > = < sx > 6 7 fsgk ¼ sy ¼ 4 Q12 Q22 0 5 fe ee gk ; > > ; : 0 0 Q66 txy ð2Þ where the entries of the stiffness matrix [Q] are Q11 ¼ Q22 ¼ E=ð1 n2 Þ; Q12 ¼ nQ11 and Q66 ¼ E=ð2 þ 2nÞ: E is the Young’s modulus and n is the Poisson’s ratio measured at constant applied magnetic field. The vector {e} represents the total strain of an arbitrary point located at a distance z from the midsurface of the film–substrate plate. Using the assumptions of classical thin plate theory, this strain can be expressed in terms of the midsurface strain vector {e0} and the curvature vector {k} by means of the following equation: {e}={e0}+z{k}, where 9 8 qu0 > > > > 8 0 9 > > > > qx > > e > > > > x = = < qv < 0 0 0 ; ¼ fe g ¼ ey qy > > > ; > > : 0 > > > > > gxy > > > ; : qu0 þ qv0 > qy qx 9 8 q2 w > > > > > > 9 > 8 > > qx2 > > > k > > > x = = < q2 w > < : ð3Þ fkg ¼ ky ¼ 2 > > qy > > > ; > : > > kxy > > 2 > > > > > ; : 2 q w > qxqy
e0x and e0y are the midsurface normal strains measured along the x- and y-axes and g0xy represents the in-plane engineering shear strain. u0, v0 and w represent the displacements in the x; y and z directions of an arbitrary point taken at the midsurface of the bimorph. Eq. (3) can be conveniently expressed as a matrix product as follows: 8 9 ( ) > = < u0 > 0 e ð4Þ ¼ ½q v0 ; > k ; : > w where the differential operator matrix [q] is defined as 2
q 6 qx 6 6 6 ½q ¼ 6 0 6 6 4 0
q 0 qy q q qy qx 0
0
0
0
0
0
0
0
q2 qx2
q2 qy2
2
q2 qxqy
3T 7 7 7 7 7 : 7 7 5 ð5Þ
Since we can use Eq. (2) to obtain the stresses induced in the film–substrate bimorph, we can calculate the net forces and moments per unit length developed in the plate studied by using the following definition ðN; M Þ ¼
Z
h=2
fsgð1; zÞ dz;
ð6Þ
h=2
fNg ¼ fNx Ny Nxy gT represents the normal and shear force resultants; fMg ¼ fMx My Mxy gT are the bending and twisting moment resultants. Substituting Eq. (2) into Eq. (6) and integrating over the plate thickness we obtain the integrated plate constitutive equation (
N M
)
" ¼
A
B
B
D
#(
e0 k
(
)
N MS M MS
) ;
ð7Þ
where the elements of the stiffness matrices [A], [B] and [D] are given by
n X Aij ; Bij ; Dij ¼ k¼1
Z
zk
zk1
Qkij 1; z; z2 dz;
ð8Þ
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
and the equivalent magnetostrictive forces and moments are calculated as follows:
N
MS
;M
MS
¼
n Z X k¼1
zk
½Q k fee gk ð1; zÞ dz;
347
be easily implemented to solve problems that involve a wide variety of boundary and loading conditions.
ð9Þ
zk1
When a bimorph is not subjected to the action of any external load, as for the plates studied here, the net forces {N} and moments {M} are equal to {0}. If the bimorph is also free on all boundaries, the solution of the system of equations represented by Eq. (7) is reduced to the evaluation of the elements of the matrices [A], [B] and [D] and the vectors {N MS } and {M MS }, followed by a series of linear algebra operations. These operations will return values for the magnetostrictively induced midsurface strains {e0} and curvatures {k} of a free plate. On the other hand, if the plate is supported in a given fashion, we need to devise a method to approximate the solution of the corresponding boundary value problem. In simple cases, as when the film–substrate system is clamped at one end, the strains and curvatures calculated for the free bimorph can be used to provide estimates for the resulting deformation. In fact, a one-dimensional version of this method has been used to calculate the induced bending deflection in magnetostrictive cantilever film–substrate beams [5,6]. Even though this approach provides excellent results for the deflections of beams, it is expected to give only rough estimates for cantilever film-substrate systems in which the width/length ratio is larger than about 0.1 (plates). This is because the deformation of such a system also depends on the variation of deflection along the transverse (y) direction. This deflection represents an additional degree of freedom that is not taken into account in onedimensional approximations. In the present study, we will use the Ritz method to find the bending and twisting displacements induced in a magnetostrictive bimorph plate acted upon by a magnetic field. This method has been chosen because of its simplicity and versatility. As it will be shown below, it only requires the use of a given set of approximating functions to represent the displacements u0 ; v0 and w: Additionally, it can
3. Displacement calculation using the Ritz method To be able to find the deflections induced by a magnetic field on a magnetostrictive bimorph plate we only need to minimize the potential energy of the film–substrate system (Utot). This potential energy is given by [11] " # Z B 1 T A Utot ¼ f½q fugg f½q fugg dxdy 2 A B D ( ) Z N MS T dxdy: ð10Þ f½q fugg M MS A The next step is to express the midsurface displacements u0 ; v0 and w as follows: u0 ¼
P X
aui Ui ðx; yÞ;
i¼1
w¼
H X
v0 ¼
J X
avj Vj ðx; yÞ;
j¼1
awk Wk ðx; yÞ;
ð11Þ
k¼1
where Ui ; Vj and Wk are approximating functions that satisfy the geometric boundary conditions (also called kinematic or essential boundary conditions) imposed on the plate. The coefficients aui ; avj and awk are unknown and will be determined in such a way that they minimize Utot. Note that Eq. (10) does not include a term for the potential magnetic energy in the applied field. This is because in our case in which the film is magnetically saturated, the magnetization is parallel to the applied magnetic field; thus the contribution of such a term is constant and therefore it does not play a role during the minimization process. Substituting Eq. (11) into Eq. (10) we can express the potential energy of the bimorph as 1 Utot ¼ fagT ½K fag fagT fLgMS ; ð12Þ 2 where {a} is the unknown coefficient vector defined as {aui avj awk }T which is of dimension ðP þ J þ HÞ 1: The matrix [K] and the
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
348
magnetostrictive actuation vector {L}MS are given by ½K ¼
"
Z
f½q ½R gT
A
B
#
f½q ½R g dxdy; B D ( ) Z MS N ¼ f½q ½R gT dxdy; M MS A A
fLgMS
ð13Þ
where the approximating function matrix [R] has dimension 3 ðP þ J þ HÞ and is defined as 2 6 ½R ¼ 4
fU1 ; U2 yUP g
f0g
f0g f0g
fV1 ; V2 yVJ g f0g 3
f0g f0g
fW1 ; W2 yWH g
7 5
:
ð14Þ
3 ðPþJþHÞ
The values of the coefficients aui ; avj and awk that minimize Utot are the ones that satisfy the system of equations given by qUtot =qanm ¼ 0: Therefore, the unknown coefficient vector can be obtained as follows: fag ¼ ½K 1 fLgMS :
ð15Þ
In order to calculate the magnetostrictively induced displacements of the film–substrate plate we only need to substitute the values of the elements of the vector {a} calculated by means of Eq. (15) into Eq. (11). The last item that we need to discuss in this section is the nature of the approximating functions Ui ; Vj and Wk : Besides the fact that these functions should satisfy the geometric boundary conditions, we can also note that in this case the displacements u0 and v0 can be expressed using only a few terms. This is because u0 and v0 play a minor role in the deflection of the plate being considered. What is important is the adequate approximation of w. In the next section, we will present a particular film–substrate combination whose deformation is to be predicted by the scheme discussed.
4. Calculation examples Consider a cantilever plate made of an amorphous TbFeCo thin film deposited on a (1 0 0) Si substrate. The film and the substrate have the following properties: Ef=80 GPa, nf=0.31, ls ¼ 1020 106 ; Es ¼ 130 GPa; ns=0.28 [12]. Using data for the bimorphs fabricated by Garnier et al. [13], the thicknesses of the film and the substrate are taken to be tf ¼ 2 mm; ts ¼ 15 mm: The initial magnetization of the film is assumed to be randomly distributed in three dimensions. The length and the width of the plate are taken as L ¼ W ¼ 1 mm: Since the displacements and the slope at the fixed end of a cantilever are zero, the boundary conditions that the approximating functions must satisfy at the fixed end are u0 ¼ v0 ¼ w¼ qw=qx ¼ 0: A set of functions that satisfies these conditions is Ui ¼ x Xmi Yni ; Vj ¼ x Xmj Ynj ; ð16Þ Wk ¼ x2 Xmk Ynk ; where Xm ðxÞ and Yn ðyÞ are sets of orthogonal polynomials generated by starting with X1 ¼ Y1 ¼ 1 and using a Gram–Schmidt orthonormalization procedure over intervals [0,L] and [0,W], respectively. In the lack of additional information on how to select a weighting function to improve the performance of the method proposed, we constructed our orthogonal polynomials using a weighting function equal to 1. Using the functions proposed and the data provided, and if y=45 , the deflection profile calculated by means of the proposed method is as shown in Fig. 2. As can be seen in this figure, when the magnetostrictive thin film is saturated, the bimorph not only bends in the parallel and transverse directions but also twists. As can be inferred from Eq. (1), the relative amounts of bending and twisting change as the angle y varies between 0 and 90 . To describe these changes it is necessary to choose a set of meaningful and observable variables. The simplest variable that can be used to describe the bending of a cantilever bimorph is the displacement of the free end of the
ARTICLE IN PRESS V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
0.012 − Φ (rad)
6
Φ ∆
2
0.009
-2
0.006
-6
0.003
-10
0
∆ (µm)
0.015
349
-14 0.0
0.1
0.2
0.3
0.4
0.5
θ/π Fig. 3. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial state the film is isotropically demagnetized.
plate at the point given by x ¼ L and y ¼ W =2; that is D ¼ wðL; W =2Þ: For the case of torsion, the variable selected is the twisting angle at the center of the free end of the cantilever bimorph (F), which is given by qw W L; F¼ : qy 2 Note that, in the strict sense, it is not possible to define a single twisting angle at the free end of a plate that also bends in the transverse (y) direction. For this reason we have defined F so that it is evaluated at a specific point, which in this case is the center of the rectangular cross–section of the film-substrate studied. Fig. 3 shows the change of D and F as y goes from 0 to 90 . As can be seen in this figure, as y increases the deflection D increases and changes sign while the magnitude of the twisting angle F shows a maximum at y=45 . Only when y=0 or 90 does the studied film– substrate system bend without twisting. In these cases the results obtained here agree relatively well with those presented in previous articles [5,6]. In this magnetostrictive bimorph, the torsional deformations induced cannot be separated from
0.012 − Φ (rad)
Fig. 2. Bending and twisting induced in a film–substrate plate whose film is saturated by a magnetic field applied at an angle y=45 .
10
Φ ∆
6
0.009
2
0.006
-2
0.003
-6
0
∆ (µm)
0.015
-10 0.0
0.1
0.2
0.3
0.4
0.5
θ/π Fig. 4. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial demagnetized state the magnetostrictive material has a domain structure such that the magnetization is isotropically distributed within the plane of the film.
the bending deflections. A different situation emerges when in the demagnetized state the magnetization is isotropically distributed in the plane of the film. In this case, e1=3ls/4=e2 [6] and the values taken by D and F as we change y are the ones summarized in Fig. 4. Note that in this situation it is possible to maximize the torsional deformation induced without simultaneous bending by saturating the film at y=45 . In addition to the previous cases studied, there is at least another possibility to obtain bending and twisting deflections using a magnetostrictive cantilever plate. This possibility is realized by subjecting a bimorph whose film shows a transverse uniaxial
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anisotropy to a magnetic field. This anisotropy could be induced by post-deposition annealing of the thin film under the influence of a magnetic field applied along the y direction. Assuming that the magnetostrictive film is in a single domain state and that its magnetization is given by rotation, the expansional strain induced in this case is given by [8]: 9k 8 e 9k 8 2 > = 3 > = < ex > < cos y > eey fee gk ¼ ¼ ls cos2 y : > > ; 2 > ; : e > : sin ð2yÞ gxy
ð17Þ
Using Eq. (17) instead of Eq. (1) for the calculations performed with the proposed method, we obtain the values for D and F as shown in Fig. 5. As can be expected, in this case the maximum bending deflection is obtained when the film is saturated along the longitudinal (x) direction. The twisting angle F is zero when y=0 or 90 , and its maximum magnitude occurs when y=45 . It is important to mention that in our calculations we have assumed that in the initial state, and every time that the applied magnetic field is zero, the midsurface of the bimorph studied coincides with the xy plane. That is, we assumed that once the applied field is reduced to zero, the domain structure of the magnetostrictive material resembles the one that was found in the original demagnetized state.
0.012 −Φ (rad)
0
Φ ∆
-12.8
-4
0.009
-8
0.006
-12
0.003
-16
0
-13
∆ (µm)
0.015
-20 0.0
0.1
Besides the calculation of torsional deformations, another area that has received little attention in previous studies is the influence of the width of a bimorph on the bending deflection induced in a magnetostrictive film–substrate system. This is in part because the methods used to solve the problem in the past did not have the intrinsic capability to deal with variations introduced in the curvatures of a bimorph due to the boundary conditions to which it is subjected. Using the method proposed in this work with plates of variable width, we have obtained the results shown in Fig. 6. The expansional strain field considered in Fig. 6 was the one obtained for a system with an isotropically demagnetized film in the initial state. For comparison purposes, we also show the bending deflection predicted using expressions presented by the authors in previous articles [5,6]. In these articles it was assumed that the width to length ratio ðW =LÞ of the film–substrate system is such that the bimorph can be considered a beam with free bending along the transverse (y) direction. Note that we cannot directly compare with the results that would be predicted using the expressions proposed by Du Tre! molet de Lacheisserie and others [2–4] because the range of validity of those equations is limited to thickness ratio tf =ts r0:001: As expected, the deflection calculated by the method proposed in this work closely approximates that calculated assuming free bending in the
0.2
θ/π
0.3
0.4
0.5
Fig. 5. Bending deflection (D) and twisting angle (F) of a magnetostrictive bimorph calculated for y A [0,90 ]. In the initial state the film exhibits a uniaxially induced anisotropy along the transverse (y) direction.
∆ (µm )
350
∆ ∆ beam
-13.2 -13.4 -13.6 0.00
0.20
0.40
0.60
0.80
1.00
W/L Fig. 6. Bending deflection (D) of a film–substrate plate calculated for different values of the thickness to length ratio ðW =LÞ: Dbeam represents the deflection estimated by assuming free bending in the transverse (y) direction.
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transverse (y) direction only when the ratio W =L of the bimorph is smaller than about 0.1 (see Fig. 6). Similar results were obtained by Watts et al. [14] by means of a finite element approximation performed using ANSYSs and modeling the magnetostrictively induced strains as anisotropic thermal expansions. To contrast their method with the one presented here, we should mention that we do not need to calculate equivalent strains to be used as input for models conceived to predict deformations induced by thermal expansion. The proposed method allows us to calculate the deflections induced in bimorphs not only due to magnetostriction but also by additional expansional strains, such as those of piezoelectric, thermal or hygroscopic origin.
5. Summary Beams and plates made of a magnetostrictive thin film deposited on a nonmagnetic substrate represent a family of simple and effective transducers that are suitable for microsystems. These devices are used as sensors and actuators by taking advantage of the strong magnetoelastic interactions occurring in the film. In this work, we have proposed a method to calculate the bending and twisting deflections that are induced in film– substrate bimorphs as a result of the strains induced by a magnetic field. The method makes use of the classical laminate plate theory to formulate an energy minimization problem, which in this case is solved using a Ritz method. Here we have limited ourselves to the study of rectangular and layerwise isotropic cantilever plates that are deformed due to magnetostrictively induced strains. These strains were calculated only for the cases in which the film was magnetically saturated along a given direction. In the initial demagnetized state, the film magnetization was assumed to be in one of the following configurations: (1) isotropically distributed in three dimensions, (2) isotropically distributed but confined to the plane of the film, and (3) oriented according to a uniaxially induced magnetic anisotropy.
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One of the reasons why the torsional deformation of magnetostrictive bimorphs has not received more attention is the fact that previous models have considered the effects of saturating a film only at angles y=0 and 90 . When y is allowed to take intermediate values we found that, in general, an external magnetic field induces not only bending but also twisting deflections in the plates studied. However, it is possible to selectively generate only bending or only twisting deformations, depending on the direction along which a saturating magnetic field is applied and the initial state of the magnetostrictive thin film. Some of the features that distinguish the method proposed here from the ones presented by previous authors are: First, our method can be used to calculate the deformations induced in magnetostrictive bimorphs even when the film is not very thin with respect to the substrate. Second, the calculation scheme proposed is self-consistent and can be used not only on magnetostrictive filmsubstrate beams but also on plates. In the case of cantilever plates, the constrained deformation that exists in the transverse direction results in deflections that are smaller with respect to those calculated assuming free bending. Third, the method presented here deals with magnetostrictive interactions directly. This also allows us to study bimorph deformations induced not only due to magnetostriction but also by other expansional strains, including those of piezoelectric, hygroscopic or thermal nature. Finally, we should mention that the method discussed in this article can be extended to solve problems that involve bimorphs of arbitrary shapes and boundary conditions. For this purpose we only need to choose suitable functions that approximate the displacements u0 ; v0 and w for use in the Ritz formulation.
Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. CMS-00-99830. The support provided by Dr. Shih-Chi Liu, Program Manager, is gratefully acknowledged.
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V.H. Guerrero, R.C. Wetherhold / Journal of Magnetism and Magnetic Materials 279 (2004) 343–352
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