This is the final version of the article. Available from EDP Sciences via the DOI in this record.JEMS 2012 – Joint European Magnetic SymposiaBy application of a magnetic field on an amorphous metallic strip, the orientation of magnetization of Weiss domains can be changed. When the strip changes its length, this effect is called magnetostriction. We simulate this effect using a finite element method. In particular we calculate the change of the mechanical resonance frequency of a magnetic platelet as a function of the applied field. This gives a quantitative model of the influence of the applied magnetic field on the effective Young's Modulus of the material. © 2013 Owned by the authors, published by EDP Sciences

Bance, S

Exl, L

Fischbacher, J

Gusenbauer, M

Hrkac, G

Oezelt, H

Praetorius, D

Reichel, F

Schrefl, T

Suess, D

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Mechanical Oscillations of Magnetic Strips under the Influence of ExternalFieldF. Reichel1,a, T. Schrefl1, D. Suess2, G. Hrkac3, D. Praetorius3, M. Gusenbauer1, S. Bance1, H. Oezelt1, J. Fischbacher1,and L. Exl11. University of Applied Science, Industrial Simulation, Matthias Corvinus-Strasse 15, 3100 St.Poelten, Austria2. Vienna University of Technology, Institute of Solid State Physics (Inst. E 138), Wiedner Hauptstrasse 8-10, 1040 Wien,Austria3. Vienna University of Technology, Institute for Analysis and Scientific Computing (Inst. E 101), Wiedner Hauptstrasse8-10, 1040 Wien, AustriaAbstract. By application of a magnetic field on an amorphous metallic strip, the orientation of magnetizationof Weiss domains can be changed. When the strip changes its length, this effect is called magnetostriction.We simulate this effect using a finite element method. In particular we calculate the change of the mechanicalresonance frequency of a magnetic platelet as a function of the applied field. This gives a quantitative model ofthe influence of the applied magnetic field on the effective Young’s Modulus of the material.1 IntroductionIn-plane Weiss domains occur in soft ferromagnetic rib-bons as a consequence of the minimization of the Gibbsfree energy. The formation of magnetic domains reduce themagneto-static energy at the expense of the wall energy. Inmagnetic platelets exceeding the single domain limit mag-netic domains are always present. These domains can bereoriented by applying an external magnetic field. Owingto the magnetoelastic coupling energy a change in mag-netization changes the length of the ribbon. Conversely, achange of length of the ribbon changes the orientation ofthe magnetization in the magnetic domains. Mechanical vi-brations lead to a periodic change of the magnetization ofthe ribbon. The oscillation frequency shifts with an appliedmagnetic DC field. We compute this frequency shift for thesecond harmonic of the longitudinal oscillation.The magnetostrictive effect is especially effective if themagnetic field is applied perpendicular to the easy axis asshown in Fig.1.2 EquationsIn an one-dimensional model the oscillations of a magneticstrip are given by the following equationρ∂2u∂t2=∂σ∂x(1)Here u is the displacement, σ is the stress, ρ is the densityof the material. t and x are the time and space coordinates.The stress is given by Hook’s law as:σ = E$e (2)a e-mail: franz.reichel@fhstp.ac.atFig. 1. Orientation of magnetization of Weiss domains on ametallic strip. The magnetization angle φ is defined as the an-gle between local magnetization vector J and the magnetizationfield H. A transverse induced anisotropy creates the perpendicu-lar domain pattern.where is E is the Young’s Modulus and $e is the elasticcontribution of the strain, which causes a change in stress.Similar to the thermal strain the magnetic strain, $m doesnot change the stress. The total strain is the sum of themagnetic and elastic contribution.$ = $m + $e (3)According to [3] in an isotropic material the magnetic strainis:$m =3λs2(cos2θ − 13)(4)Here θ is the angle of the magnetization with respect to thex axis (long axis of the magnetic strip). λs is the magne-tostriction constant.The total strain $ is given as the partial derivative of thedisplacement u with respect to x.EPJ Web of Conferences  40, 13004 (2013) DOI: 10.1051/epjconf/20134013004 © Owned by the authors, published by EDP Sciences, 2013 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20134013004EPJ Web of Conferences$ =∂u∂x(5)Using the above equations (3)-(5) we can rewrite Hook’slaw as:σ = E($ − $m) = E(∂u∂x− 3λs2(cos2θ − 13))(6)Replacing σ in equation (1) with (6) givesρ∂2u∂t2=∂∂xE(∂u∂x− 3λs2(cos2θ − 13))(7)This is the one-dimensional wave equation for the longitu-dinal mechanical oscillation of a magnetostrictive strip.The right-hand side of equation (7) contains the magneto-mechanical coupling that links the magnetization angle θwith the mechanical displacement. For the simulations it isconvenient to non-dimensionalize the variables in time andspace.We definex′ =xL, (8)where L is the length of the ribbon andt′ =√Eρ∗ 1Lt (9)For this transformation we assume that the Young’s Mod-ulus is uniform within the sample. Then the equation ofmotion reads:∂2u∂t′=∂2u∂x′2− ∂∂x′(3λsL2(cos2θ − 13))(10)This coupling is nonlinear as the magnetization angle de-pends on the external field H and the local stress σ(x) viathe following set of equations:cosθ =HHAσ(11)HAσ =2Ku − 3λsσJs(12)HAσ is the local anisotropy field which is composed fromthe magnetocrystalline anisotropy energy and the magne-toelastic coupling energy. Equations (11) and (12) followfrom the minimization of the magnetic energy:E(θ) =(Ku − 32λσ)cos2θ − HJscosθ (13)Here the first term is the net anisotropy energy which iscomposed of the magnetocrystalline anisotropy energy withthe magnetocrystalline anisotropy constant Ku with the mag-netoelastic coupling energy due to a longitudinal stress σ.The second term is the Zeeman energy of the magne-tization in the external field. Here we assume that the do-main structure is fixed and the application of a field justchanges the orientation of the magnetization in the do-mains (see Fig.1 and ref [3]).For the numerical treatment of the problem we applythe method of lines: We apply the Galerkin method, in or-der to discretize equation (7) in space. The shape and basisfunctions are piecewise linear polynomials. This results inN second order differential equations in time, where N isthe number of nodes in the finite element mesh.Mij∂2ui j∂t′2+ Ai ju j = fi(cosθ), i = 1, ...,N (14)Mij is the finite element Mass Matrix [4], Ai j is the finiteelement Stiffness matrix [4] and the load vector fi is givenas:fi =∫∂Ni∂x3λs2(cos2θ − 13)dx, (15)where Ni is the finite element basis function of node i.We transform these equations into corresponding set of2N ordinary differential equations. We solve the system ofequations with the Adams Bashforth method as providedby the software library “SUNDIALS“ [5].∂ui∂u= vi, i = 1, ...,N (16)Mij∂v j∂t2+ Ai ju j = fi(cosθ), i = 1, ...,N (17)The right-hand side of equation (17) contains the magne-tization angle θ which depends on the applied field andstress σ. For solving the set of ordinary differential equa-tions with sundials [5] we have to provide the right-handside of equation (15). For evaluating fi we apply a fixedpoint iteration to calculate cosθ.Table 1. Fixed point scheme for computing cosθfor(n = 0,1,2,...)compute(∂u∂x)ncompute σn from equation (6)compute cosθn from equations (11) and (12)3 ResultsIn the simulations we compute u = u(x, t) by integrat-ing equation (7) numerically, whereby we augmented theright-hand side with a damping term −λdu/dx. From theanalysis of ui = u(xi, t) we can compute the frequency ofthe oscillations, here i is the node index of the finite ele-ment mesh.As the initial state of the simulation we use the sec-ond harmonic of a longitudinal standing wave. The initialdisplacement u(x, t = 0) is shown in Fig. 2. Starting fromthis initial condition we integrate the equation of motionin time. Figure (3) shows the displacement, ui, of selectednodes of the finite element mesh.13004-p.2Joint European Magnetic Symposia 2012       Displacement functionx¹x u¹x¹xunstressedinitial conditionFig. 2. The upper part of the picture shows an unstressed metallicstrip with equidistant measuring points xi. The lower part showsthis metallic strip with the displacement function superimposedfor the initial conditions of the simulations (t = 0).Fig. 3. Displacement u of selected nodes as function of time t.The first point u0 and the last point un are fixed and thereforezero at every timestep. Same for u4 because it is the root of thedisplacement function.From the Fourier transform of ui(t) we compute the os-cillation frequency (see Fig. 4.a). This procedure was ap-plied for different external fields. Fig. 4.b shows the changein oscillation frequency with the applied field. The fre-quency decreases with increasing external field. This ef-fect is known as the ”∆E effect” [3]. This effect is used inmagnetic tapes for article surveillance [6].4 ConclusionA finite element method for the simulation of the mechani-cal vibrations of magnetostrictive platelets was developed.With this method we computed the ∆E effect for simpletransverse domain structures. This is the first step for a dy-namic elastic-micromagnetic solver that will compute themechanical response of magnetostrictive samples by appli-cation of an excitation field. In the future we will replaceFig. 4. a) Fourier transform ui(t) for i = 7 of the displacement ofa specific node.b) Change of oscillation frequency with respect to the exter-nal field. f0 is the frequency with zero external field. HA is theanisotropy field. The axes are normalized by f0 and HA.equation (11) with a micromagnetic solver for the compu-tation of the magnetization distribution.References1. Lu´bomı´r Ban˘as, On dynamical Micromagnetism withMagnetostriction (University of Gent, PhD Thesis June2005) 1422. Craig A. Grimes, Sensors and Actuators 84, (2000)205-2123. J.D.Livingston , Phys. Stat. Sol. (a) 70, (1982) 5914. Bernhard Bergmair, Applied Physics Letters 101,042412 (2012)5. Thomas Huber, Applied Physics Letters 101, 042402(2012)6. Giselher Herzer, Journal of Magnetism and MagneticMaterials 254-255, (2003) 598-6027. Gennadiy Nikishkov, Programming Finite Elements inJava (Springer Verlag, 2010) 4028. A.C. Hindmarsh, P.N. Brown, K.E. Grant, S.L. Lee, R.Serban, D.E. Shumaker, C.S. Woodward, Sundials: Suiteof nonlinear and differential/algebraic equation solvers(ACM Transactions on mathematical software, 31 (3),2005) 363-39613004-p.3

Mechanical oscillations of magnetic strips under the influence of external field

Mechanical oscillations of magnetic strips under the influence of external field

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