In this study, three-dimensional particle-based simulations are used to model
magnetorheological fluids. The numerical model of the MRF is implemented in the framework
of the Discrete Element Method (DEM) and takes into account the coupling of the
magnetic dipoles, the hydrodynamic drag forces and steric forces between particles. To
accurately treat the magnetic interaction between particles, the magnetic field at the
particles’ position is computed and an appropriate magnetization model is implemented.
DEM simulations with different volume fractions of the MRF are carried out and the
resulting magnetization curves are put in comparison with experimental data

Bierwisch, Claas

Lagger, Hanna G.

Moseler, Michael

Peguiron, Joël

English

UPCommons. Portal del coneixement obert de la UPC

1071IV International Conference on Computational Methods for Coupled Problems in Science and EngineeringCOUPLED PROBLEMS 2011M. Papadrakakis, E. Oñate and B. Schrefler (Eds)MAGNETIZATION MODELS FOR PARTICLE-BASEDSIMULATIONS OF MAGNETORHEOLOGICAL FLUIDSHANNA G. LAGGER∗, JOËL PEGUIRON∗, CLAAS BIERWISCH∗ ANDMICHAEL MOSELER∗∗ Fraunhofer Institute for Mechanics of Materials IWMWöhlerstrasse 11, 79108 Freiburg, Germanye-mail: hanna.lagger@iwm.fraunhofer.de, www.iwm.fraunhofer.deKey words: Magnetorheological Fluid, Magnetization Model, Discrete Element Method,Numerical SimulationAbstract. In this study, three-dimensional particle-based simulations are used to modelmagnetorheological fluids. The numerical model of the MRF is implemented in the frame-work of the Discrete Element Method (DEM) and takes into account the coupling of themagnetic dipoles, the hydrodynamic drag forces and steric forces between particles. Toaccurately treat the magnetic interaction between particles, the magnetic field at theparticles’ position is computed and an appropriate magnetization model is implemented.DEM simulations with different volume fractions of the MRF are carried out and theresulting magnetization curves are put in comparison with experimental data.1 INTRODUCTIONTypical magnetorheological fluids (MRF) consist of magnetically permeable particles(e.g. iron) in carrier oil. Upon activation of an external magnetic field, the apparentviscosity of the MRF changes within a few milliseconds by orders of magnitude, inducinga change of the MRF from liquid to solid. By controlling the strength of the magneticfield, the viscosity of the MRF can be adjusted very accurately. For this reason, MRFare highly interesting for several industrial applications, such as controllable dampers orautomotive clutches.In this work we use numerical simulations to investigate the mechanisms which governthe behaviour of the MRF when subjected to a magnetic field. Inside the MRF, thecontribution of the magnetized particles to the local magnetic field cannot be neglected[1]. Therefore an accurate description of the particles’ magnetization is required. We canexperimentally assess magnetization curves of magnetorheological fluids, but not of singleparticles. Different magnetization functions to model the response of an iron particle to amagnetic field can be found in the literature [2, 3, 4], however there is no well-established11072Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael Moselermagnetization function which is suitable for every MRF. A specific magnetization func-tion is an arbitrary choice and can also be different for different materials [5]. Here wechoose to perform simulations based on different single-particle magnetization functions,compare the resulting MRF magnetization curves to experiments [4, 6] and adapt thecurve parameters.In Sec. 2 the simulation method is presented. The calculation of the magnetization ofa particle is explained in Sec. 3. Different magnetization models from the literature arediscussed. In Sec. 4 we derive an expression to evaluate the magnetization of the MRF inthe simulation. In Sec. 5 we show first simulation results and bring them in comparisonwith experimental results from the literature.2 DISCRETE ELEMENT MODELThe numerical description of the MRF is based on the Discrete Element Method(DEM), originally proposed by Cundall and Strack to compute the motion of a largenumber of particles [7]. After the computation of the forces Fi on each particle, Newton’sequation of motionmGi r̈i = Fi (1)is solved numerically for an ensemble of particles i = 1, . . . , N with masses mGi and centerof mass positions ri. We model the MRF clutch with a three-dimensional simulation boxcontaining spherical particles, terminated by solid walls in one direction and with periodicboundary conditions in the other two directions.The forces included in the model are magnetic interaction forces between the particles,elastic repulsion, and the Stokes’ drag of the fluid on the particles. The wall is modeledas a dense and flat ensemble of non-magnetic particles. Gravity and Brownian forces areneglected.2.1 Magnetic forcesThe dipole-dipole interaction energy Ei,j of two magnetic particles i and j with mag-netic moments mi and mj is [8]Ei,j =µ04π[mi · mjr3i,j− 3r5i,j(mi · ri,j) (mj · ri,j)], (2)where ri,j = ri − rj is the vector connecting the centers of particle i and j, ri,j = |ri,j| itsnorm and µ0 = 4π · 10−7 TmA the vacuum permeability.The force on dipole i caused by dipole j is given byFi,j = −∇iEi,j=3µ04π[(mi · mj) ri,j + (mj · ri,j)mi + (mi · ri,j)mjr5i,j− 5(mi · ri,j) (mj · ri,j) ri,jr7i,j].(3)21073Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerThe total magnetic force on particle i is then the sum of the contributions of all otherparticles jFi =∑jj =iFi,j. (4)The magnetic moments of the particles depend on the local magnetic field. The depen-dence is specified by a magnetization function which is described in detail in Sec. 3.2.2.2 Hydrodynamic forcesThe fluid acts on the particles via Stokes’ drag,FStokesi = 6πηRi (vfluid − vi) , (5)where vi is the velocity of particle i, Ri is the particle radius, vfluid is the velocity of thecarrier oil and η its viscosity.2.3 Contact forcesA repulsive contact force which prevents the particles from overlapping is included.The normal force of particle j on particle i is given by the Hertzian repulsionFni =(23Y1 − ν2√RiRjRi + Rjξ3/2i,j)ri,jri,j, (6)where Y is Young’s modulus, ν is Poissons’s ratio, and ξi,j = max{Ri + Rj − |ri,j|, 0} theoverlap [9].As a first approach, no tangential forces are assumed.3 TREATMENT OF THE MAGNETIC INTERACTIONSIn this section we introduce a selfconsistent algorithm for the calculation of the localmagnetic field as well as general features of the single-particle magnetization function(Sec. 3.1). Three specific magnetization models are then discussed in Sec. 3.2.3.1 Selfconsistent calculation of local magnetic induction and particle mag-netizationThe local magnetic induction at the position of particle i,Bi,ext = Bapplied +∑jj =iBi,j , (7)is given by the sum of the externally applied magnetic field Bapplied = µ0Happlied and thesum of the contributions from other particles31074Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerBi,j =µ04π[3(mj · ri,j) ri,jr5i,j− mjr3i,j]. (8)The magnetization Mi of particle i is Mi =miViwhere Vi is the volume of particle i. Themagnetic field Hi,in at the interior of particle i is given by [3]Hi,in =1µ0Bi,ext + αMi. (9)The parameter α thus describes the contribution of the particle’s own magnetization toits inner field Hi,in.For the magnetization of the particle, we consider a model of the formMi = |MS|f(b|Hi,in|)Bi,extBi,ext, (10)where MS is the saturation magnetization of the particle and f is a scaled magnetizationfunction which is linear for small |Hi,in| and tends to 1 for large |Hi,in|. The parameter bis chosen such that the slope for small fields matches the low-field susceptibility χ of thematerial.As Hin itself depends on M, Eq. (10) is an implicit equation. We solve this equationnumerically by a root-finding algorithm that combines Newton-Raphson with the bisectionmethod for a fail-safe routine [10]. We approach the solution of the coupled equations(7), (8) and (10) by the following iterative algorithm.1. Initial step: Mi = 0, Bi,ext = Bapplied2. Solve Eq. (10) with root-finding algorithm, update Mi for all i3. Update Bi,ext for all i with equations (7) and (8)4. Repeat steps 2. and 3. until the convergence criterionmaxi|Mi(n) − Mi(n − 1)| < ε (11)is reached, where Mi(n) is the magnetization of particle i in the n-th iteration.The positions of the particles are not changed during the selfconsistency loop.3.2 Different magnetization models proposed in the literatureMagnetorheological fluids are mostly suspensions of carbonyl iron powder, a highlypure iron, where magnetic hysteresis is negligible [4]. Therefore we consider only modelsof the form of Eq. (10), describing anhysteretic magnetization behaviour. In this sectionthree different choices for the magnetization function f as introduced in Eq. (10) arepresented, the role of the model parameters is discussed, and the different models arecompared.41075Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael Moseler3.2.1 Three magnetization functionsThe three models we want to compare are the following:1. A simple magnetization function for axially anisotropic materials with applied fieldalong the easy axis, found in [2]µ0M = µ0MS tanh(χMSHin) (12)2. A phenomenological model for isotropic materials, based on the modified Langevinfunction [3]µ0M = µ0MS(coth(3χMSHin) − 13χMSHin)(13)3. The Fröhlich-Kennely law for the magnetization of a particle located inside an infi-nite chain [4]µ0M = µ0MSχMSHin1 + χMSHin(14)3.2.2 How to choose the parametersSaturation magnetization MS is the saturation magnetization of the material of thespheres. This value can be obtained experimentally. Following [11], the saturation mag-netization MS of the suspension of particles and carrier oil can be related to the saturationmagnetization of the bulk magnetic solid MS,bulk by the volume fraction φ of the suspen-sion, MS = φMS,bulk.Susceptibility and α All chosen models are constructed in a way thatM ∼ χHin for small Hin, (15)where χ = χmat is the susceptibility of the material. The susceptibility χpart of a particleis given byM ∼ χpartHext for small Hext. (16)Combining equations (9), (15) and (16), we getχmat =χpart1 + αχpart. (17)Unfortunately, the single particle parameters are usually not known. In experiments, themagnetization curve of the whole MRF is measured instead [12]. Thus we have to infer51076Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael Moselerthe susceptibility of the particle from the experimentally measured magnetization curveof the MRF.Regarding the parameter α, one obtains for uniformly magnetized spheres α = −13and thus χsphere =3χmat3+χmat[8]. In other cases, α is a coupling parameter depending on thecoupling of the magnetic domains inside the particle and on the particle geometry andhas to be fitted to experiments [3, 8].3.2.3 Comparison of the magnetization modelsFigure 1 shows the different magnetization models for parameter values µ0MS = 1.709 T(taken from [4]), α = −1/3 (uniformly magnetized spheres) and a susceptibility χmat ≈ 5.7.Figure 1: Comparison of the three magnetization functions Eq. (12), Eq. (13), and Eq. (14).We see that the three models differ strongly in their behaviour for intermediate fieldstrength. The tanh function is very steep and approaches saturation early. The modifiedLangevin function deviates from the linear regime for small fields earlier and needs strongerfields to reach saturation. The same holds for the Fröhlich-Kennely function, only thatthe effects are still more pronounced.4 DEFINITION OF THE MRF MAGNETIZATIONExperimentally the magnetization of the MRF is obtained byµ0M = Bmeasured − µ0Happlied, (18)where Happlied is the externally applied magnetic field, controlled by a electromagnet, andBmeasured is the magnetic induction measured at a location close to the MRF.If we want to mimic the experimental procedure in the simulation, we need to choosea point where to measure the magnetization. We can–similarly to the experiment–applya certain external field H and measure the magnetic induction B at some point close tothe MRF. Figure 2 shows the value of Bmeasured − Bapplied for a setup where the MRF isbetween two discs at z = ±37.5 µm. Bmeasured is the measured magnetic induction in thesimulation (at y = 0, averaged in x-direction) as a function of the measuring position,61077Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerFigure 2: Locally measured magnetic induction (only the contribution of the MRF, without externallyapplied field). The MRF is located between the yellow bars.called z0, in z-direction. We see that the measured value depends significantly on thedistance of the measuring point from the MRF. However, we want to have a value for themagnetization of the MRF, which is independent of the measuring point. To this purpose,we analyze the contribution of the MRF to the magnetic field B at distances far from theMRF. The behaviour of the absolute value of the magnetic field for large distances fromFigure 3: Magnetic field B induced by the MRF, far from the MRF.the MRF can be described by|Bmeasured(z) − Bapplied| = cz3(19)with a constant c, as shown in Fig. 3. Thus the field of the MRF shows the same decayas a magnetic dipole field. We therefore define the magnetic moment of the MRF mMRFas the magnetic dipole moment inducing the same magnetic field B as the MRF at largerdistances from the MRF.A magnetic dipole located at the origin with dipole moment m = mez creates amagnetic fieldBdipole(z) =µ02πm|z|3 (20)71078Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael Moselerat location (0, 0, z). The value of mMRF = |mMRF| can be found by fitting the large-distance magnetic field B (19), yieldingmMRF = c2πµ0. (21)Another expression for the MRF magnetization can be obtained by considering that, seenfrom a larger distance from the MRF, the magnetic dipoles carried by the particles insidethe MRF are approximately at the same spot. With the superposition principle we getfor the field created by the MRF:BMRF(z) =∑iBi(z) =∑iµ02πmi|z|3 =µ02π1|z|3∑imi , (22)where Bi is the magnetic field created by particle i and mi is the magnetic moment ofparticle i. On the other handBMRF(z) =µ02πmMRF|z|3 . (23)Combining the equations (22) and (23) we getmMRF =∣∣∣∣∣∑imi∣∣∣∣∣. (24)Thus, we expect the two definitions (21) and (24) for mMRF to be equivalent. To test thishypothesis, we performed simulations with an externally applied field of µ0H = 0.14 T,0.42 T, and 1.4 T. The results are shown in Table 1.Table 1: Comparison of the two definitions for the magnetic moment of the MRF.µ0Happlied [T] 0.14 0.42 1.4m = c2πµ0[Am2] 2.40 × 10−8 5.94 × 10−8 6.92 × 10−8m =∑imi [Am2] 2.38 × 10−8 5.90 × 10−8 6.86 × 10−8The differences between the definitions amount to less than 1 % of the magnitude ofthe magnetic moment. Therefore the two definitions can be regarded as equivalent. Forour simulations, we take mMRF =∑i mi as the working definition. MMRF is then definedasMMRF =mMRFVMRF=N 〈mpart〉VMRF= φ 〈Mpart〉. (25)81079Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerThe MRF magnetization for monodisperse suspensions can thus be expressed as the aver-age particle magnetization 〈Mpart〉 = 〈mpart〉Vpart times the volume fraction φ =N VpartVMRF, whereVpart is the volume of one particle, VMRF the volume of the MRF, and N the number ofparticles in the MRF.Equation (25) provides a convenient definition for the magnetization of the MRF inthe simulation. The experimentally measured magnetization however still depends onthe unknown measuring point. We can get independence from the measuring point bydividing the measured magnetization by the measured high-field saturation value at thesame measuring point. This gives transformed experimental values|B(z0) − Bapplied||BS(z0) − Bapplied| =µ02πmexpMRF|z0|3µ02πmexpS,MRF|z0|3=mexpMRFmexpS,MRF(26)The same can be done for the computed magnetizationMMRFMS,MRF=mMRFmS,MRF(27)With this scaling procedure, the experimental [Eq. (26)] and computed [Eq. (27)] mag-netization values can be put in comparison.5 RESULTSHere we present DEM-simulation results at different volume fractions with sphericalparticles of 1 µm in diameter, comparable to the experiments reported in [4]. We used thetanh model with values from Sec. 3.2.3. In the simulations we see that the magnetizationcurve of the MRF deviates significantly from the single sphere curve (Fig. 4). Figure 5Figure 4: Scaled magnetization curve of the MRF (φ = 35 %) compared with the single sphere curve.shows the unscaled MRF magnetization curves simulated at different volume fractions φ.In Ref. [4], the relative differential permeability curves of MRF are shown instead of the91080Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerFigure 5: Magnetization curves of the MRF for different volume fractions φ.magnetization. From simulations, we get the relative differential permeability µr,dif of theMRF asµr,dif(B) = 1 + µ0dMMRFdB(28)The simulation results are shown in Fig. 6. As stated in Sec. 3.2.2, we can get experimentalFigure 6: Relative differential permeability µr,dif for different volume fractions φ of the MRF from DEM-simulations.values for the low-field susceptibility χMRF of the MRF, whereas the susceptibility of thematerial χmat is unknown. The susceptibility χMRF is related to the relative differentialpermeability byχMRF = µr,dif(0) − 1. (29)The computed values are shown in Fig. 7. The linear dependence of χMRF on volumefraction is also observed in experiments [4]. The susceptibility there is much higher though.By increasing the susceptibility of the material χmat we also get higher susceptibilities forthe MRF (see figure 8). However, the exact dependence still has to be investigated.101081Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerFigure 7: Low-field susceptibility of the MRF from DEM-simulations vs. volume fraction φ.Figure 8: Low-field susceptibility of the MRF vs. material susceptibility χmat for different volumefractions φ.6 CONCLUSIONSA numerical model of a magnetorheological fluid based on the Discrete Element Methodwith emphasis on the magnetization model was presented. The magnetic interactionswere treated using a selfconsistent algorithm. For the description of the anhystereticmagnetization curve of the particles, three magnetization models from the literature werecompared. An expression for the magnetization of the MRF in the simulation was derived,thus enabling the comparison of experimental and computed magnetization curves.From simulations we see that the magnetization curve of the MRF deviates significantlyfrom the single sphere curve. The low-field susceptibility of the MRF depends linearlyon volume fraction. The material’s susceptibility, which is needed as a parameter for themagnetization model, can be obtained by comparing the resulting MRF susceptibility toexperiments.111082Hanna G. Lagger, Joël Peguiron, Claas Bierwisch and Michael MoselerREFERENCES[1] Furst E M and Gast A P, Phys. Rev. E 61, 6732 (2000).[2] Raghunathan A, Melikhov Y, Snyder J E, and Jiles D C, Appl. Phys. Lett. 95, 172510(2009).[3] Jiles D C and Atherton D L, J. Magn. Magn. Mater. 61, 48 (1986).[4] de Vicente J, Bossis G, Lacis S, and Guyot M, J. Magn. Magn. Mater. 251, 100(2001).[5] Jiles D C, Thoelke J B, and Devine M K, IEEE Trans. Magn. 28, 27 (1992).[6] Ehrlich J, Böse H, Private communication.[7] Cundall P A and Strack O D L, Geotechnique 29, 47 (1979)[8] Jackson J D, Classical Electrodynamics. Wiley (1998).[9] Hertz H, J. reine angew. Math. 92, 156 (1881).[10] Press W H, Teukolsky S A, Vetterling W T, and Flannery B P, NUMERICALRECIPES in Fortran 77. Cambridge Univ. Press (1992).[11] Rosensweig R E, Ferrohydrodynamics. Dover Publications (1997).[12] Jolly M R, Carlson J D, and Muñoz B C, Smart Mater. Struct. 5, 607 (1996).12

Magnetization models for particle-based simulations of magnetorheological fluids

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Magnetization models for particle-based simulations of magnetorheological fluids

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