November 9, Abstract. properties change in response to the application of a magnetic æeld. These æuids typically

Transcription

1 On the Eæective Magnetic Properties of Magnetorheological Fluids æ November 9, 998 Tammy M. Simon, F. Reitich, M.R. Jolly, K. Ito, H.T. Banks Abstract Magnetorheological èmrè æuids represent a class of smart materials whose rheological properties change in response to the application of a magnetic æeld. These æuids typically consist of small èçmè magnetizable particles dispersed in a non-magnetic carrier æuid that generally contains additives such as surfactants and anti-wear agents ëë. Due to such additives, there is an outer non-magnetic layer on the particles that keeps them from touching. The goal of this paper is to study the eæective magnetic behavior of an MR composite as a function of the interparticle distance. To this end, we present and employ a model for the eæective magnetic properties of MR æuids with periodic microstructure that is based on the theory of homogenization. Finally, we discuss an interpolating formula for the eæective permeability of MR æuids as an extension of the work of Keller ëë and Doyle ëë. Introduction Magnetorheological èmrè æuids are suspensions of micron-sized magnetizable particles èsuch as ironè in a non-magnetic carrier æuid èsuch as oil or waterè. The essential characteristic of these materials is that they can be rapidly and reversibly varied from the state of a Newtonian-like æuid to that of a stiæ semi-solid with the application of a moderate magnetic æeld. This feature, called the MR eæect, is a consequence of the fact that, in the presence of a magnetic æeld, the particles magnetize and form chain-like structures that align in the direction of the applied æeld as depicted in Figure. This columnar microstructure, in turn, dramatically increases its resistance to an applied shear strain èsee Figure è. This feature has inspired the design of new technology and various products such as semi-active dampers, brakes, clutches and numerous other robotic control systems ëë èsee also A typical MR æuid includes special additives such as surfactants and anti-wear agents that enhance its performance in MR æuid based devices ëë. These additives may form a non-magnetic layer on the inclusions that keeps them from actually touching. This minimum interparticle distance aæects the overall magnetic properties of the æuids ëë. Understanding the magnetic response of MR æuids and its dependence on microstructure is important to the design of improved æ Research supported in part by a US Department of Education Graduate Assistance in Areas of National Need ègaannè Fellowship ètmsè, the U.S. Air Force Oæce of Scientiæc Research under grants AFOSR F èhtb and KIè, AFOSR F èhtbè, AFOSR F ètmsè, AFOSR F èfrè, NSF DMS-9 èfrè. Center for Research in Scientiæc Computations, Department of Mathematics, North Carolina State University Raleigh, NC 9 School of Mathematics,University of Minnesota,Minneapolis, MN Advanced Technologies Research Group, Lord Corporation, Thomas Lord Research Center Cary, NC - 900

2 MR æuids. So here we study the eæective magnetic properties of these æuids as a function of the minimum interparticle distance. Predicting the magnetic properties of these æuids, however, is a challenging task due to the highly nonlinear and oscillatory nature of the magnetization of the constituents. As stated, the characteristic size of the particles is several orders of magnitude smaller than the characteristic size of a sample, rendering standard ænite element modeling impractical. Thus, in this paper we present a model for the eæective magnetic response of MR æuids that is based on the theory of homogenization. We obtain the constitutive equations that govern the B-H response of MR æuids with periodic microstructure in both the linear and nonlinear regimes. The corresponding numerical results exhibit good agreement with experimental data èfrom experiments carried out at Lord Corporationè. However, it becomes increasingly diæcult to obtain robust numerical calculations for very small values of the interparticle distance because the necessary mesh size becomes too small. So we also discuss an interpolating formula for the effective permeability of MR æuids for all èpositiveè values of the interparticle distance. It extends an interpolating relation due to Doyle from a simple cubic array structure to particle-chains. Doyle's work is based on that of Keller who analyzed the asymptotic behavior of the eæective property as the interparticle distance approaches zero. This paper is organized as follows. In the next section, we review the mathematical model based on homogenization theory and present analytical formulae for the eæective properties of MR æuids with periodic microstructure. In x, we present experimental data collected for three MR æuids and discuss calculating the permeability of a sample from the experimental B- H curve. Then we discuss the corresponding numerical calculations for the eæective properties of MR æuids and compare them with experimental data in x. Finally in x, we present an interpolating formula for the eæective permeability of MR æuids and investigate its applicability by comparing it with the homogenization calculations. NO MAGNETIC FIELD APPLIED MAGNETIC FIELD H APPLIED MAGNETIC FIELD H APPLIED STRAIN γ p p p p γ (a) (b) (c) H H Figure : The MR eæect: èaè the particles æow in the absence of magnetic æelds; èbè they magnetize and form columns when a magnetic æeld is applied. ècè Schematic of the microstructure under an applied shear strain. Shear Stress (kpa) Shear Strain Rate (sec- ) 000 Oersted 000 Oersted 000 Oersted 0 Oersted Figure : Shear strain rates versus shear stresses in the post-yield regime, for various values of the magnetic æeld intensity ëë.

3 Mathematical model. Periodic microstructure In this section we summarize our model for the eæective magnetic properties of MR æuids with periodic microstructure that is based on the mathematical theory of homogenization. The results presented here can be found in the literature èsee, e.g., ë, ëè. Let æ ç IR n èn= or depending on the geometry that we considerè be a sample of heterogeneous material that, in the presence ofavertical magnetic æeld, has periodic microstructure that we now describe. Let Y be the parallelogram deæned by Y = ç,c ; c ç æææææç,cn ;c n for some constants c i and assume that the permeability matrix ç of the sample is Y-periodic; that is, çèxè =çèx+c i e i è for every x IR n and for every i =;:::;n. Here the vectors e ;:::;e n are the unit direction vectors in IR n. Then, as depicted in Figure, Y is the basic periodicity cell of æ. The constituents of our MR æuids are assumed isotropic and therefore, we may write ç Periodicity Cell Y d = - µ m Figure : Schematic of the periodic microgeometry assumed here. ç =^çi where ^ç = Now we deæne ç æ :IR n!m næn by ç çp ç c in the particles, in the carrier æuid. ç x ç æ èxè = ç ; æç èè where æ is the ratio of the particle size to the characteristic size of a sample. We note that ç æ is æy -periodic in IR n. Then the magnetic æeld inside the specimen satisæes the fundamental equation of magnetostatics that, for periodic structures, can by written as ç ç x ç ç,r æ ç ræ æ =0 èè æ where æ æ is the magnetostatics scalar potential. Since the sample is in the presence of a vertical applied æeld ~H = H n e n, the boundary conditions that complement èè are æ æ = 0 at the bottom, æ æ = H n at the top èè æ = 0 on the sides of the èè where ~n is the unit normal vector to the Since ç æ is highly oscillatory, a direct treatment of the governing equations èè-èè is impractical. So, in order to ænd the macroscopic magnetic properties of MR composites, we employ the theory of homogenization.

4 . Linear homogenization For low æeld intensities, MR æuids exhibit linear behavior; that is, ç is independent of the magnetic æeld. So we study the asymptotic behavior of èè as æ! 0. We assume that çèxè M næn is Y-periodic, symmetric, uniformly positive deænite, and each ç ij is bounded. Then the theory of homogenization provides the following convergence properties as æ! 0: where æ 0 satisæes the homogenized problem ræ æ * ræ 0 weakly in L èæ; IR n è èè ç, x æ æ rææ *ç hom ræ 0 weakly in L èæ; IR n è,r æ èç hom ræ o è=0onæ: èè The èconstantè homogenized matrix ç hom is given by èç hom è ik = jy j Z Y çik èyè+ The function w k is the unique solution to the auxiliary problem ç ræèçèyèèek + rw k èyèèè = 0 in Y; w k H è èyè; nx j= ç ij ka j èè è8è where Hè èy è denotes the set of periodic functions in H èy è with zero mean. Here the convergence properties èè imply that for suæciently small æé0, èè provides the eæective magnetic permeability of the MR composite èi.e., B ~ = çhom Hè. ~ Calculating the homogenized matrix requires solving the boundary value problems è8è for w k èk =;;è and substituting w k into equation èè.. Nonlinear Homogenization Magnetic materials such as iron possess a magnetic saturation M s ; that is, a maximum achievable magnetization that, once reached, does not allow any further magnetization. Thus, for moderate to large valued æelds, the permeability is, in fact, a function of the magnetic æeld ~ H. Then we study the asymptotic behavior of,r æ èçè x æ ; rææ èræ æ è=0 è9è where the B-H map èy; çè æ æ IR n! çèy;çèç IR n is assumed to be Y-periodic in y. Furthermore, assuming the map to be Lipschitz-continuous, strictly monotone and coercive in ç, then one can prove the following convergence properties of è9è: ræ æ * ræ 0 weakly in L èæ; IR n è è0è weakly in L èæ; IR n è ç, x æ ; ræææ ræ æ *b hom èræ 0 è where æ 0 satisæes the homogenized problem,r æ èb hom èræ 0 èè=0: èè The map b hom èçè is deæned for all ç IR n by b hom èçè = jyj Z Y çèy; ç + rw ç èyèèèç + rw ç èyèèdy; where w ç is the unique solution to the auxiliary problem ç rèçèy; ç + rwç èyèèèç + rw ç èyèèè = 0 in Y w ç Hè èè èy è: èè

5 In our calculations, we use an empirical constitutive relation known as the Fríohlich-Kennelly relation given by çè Hè=+ ~ èç p,è M s èè èç p, è j ~Hj + M s to determine the permeability of the iron particles at a given æeld strength H. In comparison to the linear case, calculating the eæective constitutive law èè requires solving a continuum of the nonlinear equations èè. In x we will present numerical results for both equations èè and èè. data In this section we turn to a discussion of the experimental setup and discuss calculating the permeability of a MR æuid sample from the experimental B-H curve. Experiments were conducted èat Lord Corporationè to measure the magnetic properties of three MR æuids with particle loadings of 0è, 0è, and 0è by volume. The arrangement used to collect the data èkjs model HG-00 magnetic Hysteresisgraphè is schematically depicted in Figure : a current is passed through an electromagnet and a probe is used to measure the applied æeld ~ H where ~ H = H ^k; the magnetic induction ~B = ~Bè ~Hè can then be evaluated from the cylindrical sample with the aid of a search coil. In Figure we display the resulting intrinsic induction ~B i = ~B, ç 0 ~H as ELECTROMAGNET H SPECIMEN (MR FLUID) J 0 B(H) Figure : The experimental setup. a function of ç 0 ~H. Since hysteresis is negligible in each case, in our analysis, we only consider data in the ærst quadrant that correspond to the second part of the hysteresis loop. Intrinsic Induction [ T ] % 0% 0% µ H [ T ] 0 Figure : magnetic induction curves for three MR æuids of diæerent iron volume percents.

6 % 0% 0% Chosen value. s n r n / n. 0% 0% 0% Chosen value n Figure : Slope and residual mean square of each æt using n data points. For the linear permeability of the sample, we are interested in calculating the permeability as measured at very low æelds. Since the magnetic induction is approximately linear for small æelds, we take the permeability to be the slope of the B-H curve in this region. We calculate this slope by conducting a linear least squares æt of the data. While it is important to use enough data so that the results are accurate and robust, the æt should be conducted only over the linear regime. To this end, consider the following notation: èh k ;B k è = kth data point of the B-H curve ~H k = ç 0 H k s n = slope of the æt using the ærst n data points b n = y-intercept of the æt using n data points B~ n k = s n Hk ~ + b P n r n = n n, k= jb k, B ~ nj k èresidual mean squareè: The slope and residual mean square of each æt are shown in Figure. We selected the linear regime as the set of n points whose æt yielded the minimum residual mean square and denote this set by the interval è0;h l è. As a result, the permeabilities and H l are summarized Table. ç eæ ç 0 H l ètè 0è.0.0 0è.0.0 0è.9.08 Table. Numerical experiments In this section we present the numerical results for the linear and nonlinear eæective constitutive laws based on homogenization theory as stated in x and compare these with the experimental data that was presented in x. The speciæc geometry that we consider here consists of a n-dimensional èn= or è static conæguration of spherical iron inclusions èsuch as that in Figure è. We assume the inclusions form single chains in the direction of the applied æeld that are periodically dispersed throughout the carrier æuid. We denote by a the ratio of the vertical interparticle distance, æ, to the particle diameter. We take the èæxedè period along the direction of the æeld to be unity and the period in the other directions to be l èfor simplicity, itis assumed that all lengths are dimensionlessè. Then, the periodic microstructure is characterized by the periodicity cell Y =ë0;lë n, æë0; ë as is shown in Figure. The physical parameters used here are ç p = 000; ç c = and ç 0 M s =: T èteslaè where ç 0 =çæ0, Tm=A is the permeability ofvacuum.

7 α = a r 0 r l Figure : Cross section of the periodicity cell. Effective Permeability Expermental a=.008 a=.0 a=.0 a= Volume Fraction Figure 8: Linear eæective permeability computed for the two-dimensional model geometry. First, we present the numerical results based on our two-dimensional model geometry èn = è for the linear case. The elliptic equation è8è was solved to compute èç hom è èwhere èç hom è represents the eæective permeability in the direction of the applied æeldè using a standard ænite element method based on piecewise-linear elements. In Figure 8, the experimental data are shown along with the results of our homogenization calculations for values of the ratio a = :008;:0;:0;:0. We remark that the numerical calculations for a = :0 agree well with the experimental values for all three æuids. In Figures 9-, we depict the eæective B-H curves èfrom linear regime through saturationè based on equations èè-èè assuming a = :00;:0. We constructed the eæective B-H curves by interpolating the discretely calculated values of b hom èçè for ç = H ~ =è0;h è, ç 0 H è0; 0:8è. For each æuid, the estimates for a = :0 provides a good æt not only in the linear region, but through saturation as well. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a= µ H [ T ] Figure 9. Eæective magnetization for the 0è sample èa = :00;:0è compared to experimental data.

8 Next we discuss the numerical results for the three-dimensional model geometry èn = è and we only consider the linear case. The linear elliptic equations è8è were solved using a commercial ænite-element software package ë9ë. The results for èç hom è are shown in Figure for a = :000;:00, :00, :00. We ænd the agreement to be reasonable, given the idealized nature of the assumed microstructure. Indeed, some degree of polydispersity was certainly present within our samples and, of course, the true columnar structures were more complex than simple linear chains. In fact, this complexity is accentuated at larger volume fractions. This may explain the slight increase in the deviation as the particle concentration is increased. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a= µ 0 H [ T ] Figure 0: Eæective magnetization for the 0è sample èa = :00;:0è compared to experimental data. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a= µ H [ T ] Figure : Eæective magnetization for the 0è sample èa = :00;:0è compared to experimental data. Effective permeability 8 a=.000 a=.00 a=.00 a= Volume Fraction Figure. Linear eæective permeability computed for the three-dimensional model geometry. 8

9 Other approximations for ç eæ In this section we discuss an interpolating formula for ç eæ of MR composites for n=. We seek an interpolating relation which provides a good approximation for all values of a. To this end, we consider linear composites that contain spherical inænitely permeable inclusions èfor iron inclusions, ç p = 000 so this is a reasonable case to considerè. It becomes increasingly diæcult to obtain accurate numerical estimates as a! 0. However, J.B. Keller showed that, when the particles are uniformly dispersed èon a cubic latticeè, the asymptotic behavior of ç eæ is logarithmic as a! 0 ëë. Based on this work, Doyle suggested the interpolating formula ç eæ = ç c ëa + B logèç c, çèë èè where ç is the volume fraction of the inclusions and ç c = ç= is the maximum volume fraction attained at a =0ëë. To ænd the constants A and B, Doyle then considered the following conditions on ç eæ at ç =0: ç eæ = ç c èè dç eæ dç = ç c : èè Equation èè was obtained using the well-known asymptotic estimate attributed to Maxwell ë0ë given by ç çp ç +ç c +çèç p,ç c è ç eæ ç ç c : è8è ç p +ç c,çèç p,ç c è Maxwell's estimate is valid for small volume fractions and becomes exact as ç! 0. From èè-èè, one can determine A and B and èè can be written as çç : è9è ç eæ = ç c ç, ç log ç, ç ç Equation è9è, known as Doyle's estimate, is now compared with our homogenization estimates. As is seen in Figure, the agreement is good throughout the entire range of volume fractions. 8 Homogenization Interpolation Effective Permeability Volume Fraction Figure : Homogenization estimates and equation è9è for the eæective permeability. Doyle's estimate is for cubic structures whereas the microstructure for MR æuids is columnar. In fact, cubic geometry is a special case of the model geometry depicted in Figure with l =. So it is our intent to seek an extension of Doyle's estimate for l ç. Our preliminary investigations suggest that the e -component ofç eæ is logarithmic as a! 0 or, equivalently, asç!ç c èthe 9

10 details are to be reported in future workè. Following the procedure described above, we obtain the following extension for l ç çç ç eæ = ç c ç, ç c log ç, ççc è0è where ç c = ç=l is the maximum volume fraction. Of course, è0è coincides with Doyle's estimate for l =. The results for the extreme case ç c ç :8è èl = 8è are listed in Table. Also shown is the relative diæerence of the interpolation relation from the homogenization estimates. The methods are in reasonable agreement throughout the entire range of possible volume fractions. çèèè Homogenization Interpolation Relative Diæerence : æ 0, : æ 0, : æ 0, :8 æ 0, :9 æ 0, :8 æ 0, : æ 0, : æ 0, Table. In Figures - we compare the interpolating relation with estimates obtained from homogenization theory èsee Figure è for a = :00;:00;:00;:000. data are also shown. The interpolating formula is based on iniænitely permeable particles, and therefore, yields larger permeabilities than the homogenization model. In comparison to experimental data, the interpolating formula for a = :000 exhibits good agreement with experimental data for all three æuids. Interpolation Homogenization a=.00 Interpolation Homogenization a=.00 Effective Permeability Effective Permeability % volume fraction % volume fraction Figure. Homogenization estimates and equation è0è for the eæective permeability compared with experimental data for a = :00;:00. 0

11 Interpolation Homogenization a=.00 8 Interpolation Homogenization a=.000 Effective Permeability Effective Permeability % volume fraction % volume fraction Figure. Homogenization estimates and equation è0è for the eæective permeability compared with experimental data for a = :00;:000. Conclusions We presented experimental data that we collected for three MR æuids: 0è, 0è and 0è by volume. We discussed calculating the linear permeabilities using least squares and found them to be approximately in the range from,. In order to predict the experimental data, we proposed a model for the eæective magnetic properties of MR æuids that is based on the theory of homogenization. Analytical formulas for linear and nonlinear media with periodic microstructure were discussed. For the numerical results, we considered microstructures of monodispersed spherical particles arranged periodically in single particle chains. Interestingly, even with this simpliæed model, the homogenization results predicted the experimental data with acceptable accuracy. However, the method is not restricted to these speciæc microstructures or particle shapes. We also derived an interpolating formula for the eæective permeability of composites that contain inænitely permeable inclusions arranged in particle-chains. This relation showed good agreement with experimental data. Hence, it may provide a simple yet reasonably accurate method for computing the eæective permeability of MR æuids. References ëë P. P. Phulçe and J. M. Ginder, The Materials Science of Field-Responsive Fluids, MRS Bulletin è998è, 9í. ëë T. M. Simon, F. Reitich, M. R. Jolly, K. Ito, H. T. Banks, Estimation of the Eæective Permeability in Magnetorheological Fluids, CRSC Technical Report CRSC-TR98-, N.C. State Univ., October, 998; J. Intell. Material Systems and Structures, submitted. ëë J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. è9è, 99í99. ëë W. T. Doyle, The Clausius-Mossotti problem for cubic arrays of spheres, J. Appl. Phys. 9 è98è, 9í9. ëë J. D. Carlson, D. M. Catanzarite and K. A. St Clair, Commercial magnetorheological æuid devices, Int. J. Mod. Phys. B 0 è99è, 8í8. ëë M. R. Jolly, J. D. Carlson and B. C. Mu~noz, A model for the behavior magnetorheological materials, J. Smart Mater. & Struct. è99è, 0í.

12 ëë A. Defranceschi, An Introduction to Homogenization and G-convergence, Lecture Notes, School on Homogenization, ICTP, Trieste èsept. -8, 99è. ë8ë R. M. Bozorth, Ferromagnetism, van Nostrand, Toronto è9è. ë9ë Maxwell D Field Simulator, Ansoft Corporation, Pittsburgh, PA. ë0ë J. C. Maxwell, A Treatise on Electricity and Magnetism, rd Ed., Dover Publications, New York è89è.