Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 3, pp. 659-676, Warsaw 2010 ANALYSIS OF MAGNETIC FIELD EFFECT ON FERROMAGNETIC SPHERES EMBEDDED IN ELASTOMER PATTERN Anna Boczkowska Warsaw University of Technology, Faculty of Materials Science and Engineering, Warsaw, Poland e-mail: abocz@meil.pw.edu.pl Leszek Czechowski, Mieczysław Jaroniek Technical University of Lodz, Faculty of Mechanical Engineering, Łódź, Poland e-mail: lczechow@p.lodz.pl; mieczyslaw.jaroniek@p.lodz.pl Tadeusz Niezgoda Military University of Technology, Faculty of Mechanical Engineering, Warsaw, Poland e-mail: t.niezgoda@wat.edu.pl The results of modelling of the magnetorheological elastomers (MREs) mi- crostructure using Finite Elements Method (FEM) are shown. MREs are solids analogous to magnetorheological fluids consisting of carbonyl-iron particles and a soft elastomer matrix. Fabrication of MREs is performed in an external constantmagnetic field. Due to the presence of this field, the ferromagnetic particles tend to arrange themselves into elongated chains according to the magnetic field lines. FEM analysis of the MREs micro- structure takes into account two cases: magnetic power and elastic vectors as well as their interactions between the components of the composite. The paper describes spatial ordering of the particles based on their interactions with themagnetic field. Two phenomenawere taken into account: the force of a magnetic field and the local magnetic dipoles. Both fields have been modelled for a circular conductor with electrical current. Key words: magnetorheological elastomers, Finite ElementMethod, photo- elastic method 1. Problem formulation Magnetorheological elastomers (MREs) are solidphase analogues ofmagnetor- heological fluids. The MREs consist of ferromagnetic carbonyl-iron particles and a soft elastomer matrix. Their mechanical properties change because of 660 A. Boczkowska et al. the influence of an external magnetic field (Dorfmann and Brigadnov, 2003; Lokander and Stenberg, 2003b, Zienkiewicz and Taylor, 1994). According to the literature, the elasticity modulus of MREs can undergo a change by abo- ut 30-40%, and even 60% (Dorfmann and Brigadnov, 2003; Kankanala and Trianta fyllidis, 2004) due to the presence of the magnetic field. MREs most often consist of iron micro-dimensional particles, although in the literature we can find examples of application of various particles from carbonyl-iron and iron alloys (Lokander et al., 2004) to magnetostrictive par- ticles (Lokander and Stenberg, 2003a). Size and shape of the particles applied inMREs can be different (Farshad andBenine, 2004; de Vicente et al., 2002). A huge interest in MRE application to technology causes the necessity of their theoretical description (Connoly and McHugh, 1999). A significant obstacle to publish materials concerning MREs and their properties is the lack of description ofmagneto-elastic deformation. Some solutions on the basis of Cauchy’s elastic theory for isotropic magnetic sensitive bodies were found (Jolly et al., 1999). These theoretical results are not always agreeable with experimental re- sults. In literature, one can find a lot of mathematical models describing de- formation of MREs under the influence of a magnetic field (Dorfmann and Brigadnov, 2003; Dorfmann and Ogden, 2003; Dorfmann et al., 2004). Some of them even assume that there is various magnetization in the range of fer- romagnetic particles (Bocera and Bruno, 2001). This phenomenon does not occur for appropriately small particles of dimensions ¬ 1.5µm. In the case of greater particles, this effect has to be taken into consideration inmathematical equations, what is described in Bocera and Bruno (2001). Othermodels are focusedonanalysis of a limitednumberof particleswhich are within limits of the same radius (Yin et al., 2002). Constitutive equations for composites consisting of magnetic particles in an elastomer matrix and subjected to deformations can be described according to Eshelby’s approxi- mation. A combination of the finite element method with the ”meshfree” method, makes possible creation of a discrete model subjected to one-axial deforma- tion with or without the presence of a magnetic field (Kleiber, 1985). FEM description can be found in many works (Kleiber, 1985; Lokander ans Sten- berg, 2003a,b; Lokander et al., 2004; Turner et al., 1956; Wang et al., 2003; Woźniak, 1993, 1995; Yin et al., 2002; Zhou, 2003; Zienkiewicz and Taylor, 1994). FEM is based on the assumption that a solid might be considered as a collection of finite elements connected to each other with the use of finite nodal points. Analysis of magnetic field effect on ferromagnetic... 661 TheFEMconcept introduced byTurner et al. (1956) and Zienkiewicz and Taylor (1994) assumesfinite elementswhich areboundedbyanumberof nodal points. FEM enables analysis of objects with complex geometry and can be applied for a material model in a combined internal structure (Wang et al., 2003; Zienkiewicz and Taylor, 1994) provided that a proper model can be created and its division into finite elements can be carried out. Many works, related to some extent with the subject of this paper were devoted to analysis of the magnetic field using the finite element method as well. Only few of them are described here in short.Matsui andOkuda (1998), for instance, proposed a scheme for calculating the magnetic field in a sphe- rical shell, based on the Earth’s outer core. According to their results, they suggested that the accuracy of the dipole field depends on the radius of the simulation domain. In Tomczyk and Koteras (2008), the authors presented results of an experimental and numerical study of the influence of the air gap between the coils in a transformer. They obtained a good agreement between both investigations. Broeh et al. (1996) showed results of the finite-element method in combination with the Biot-Savart law concerning the magnetic field distribution generated by a dipolar source within a homogeneous volume conductor of an arbitrary shape. Simulations were performed to evaluate the numerical accuracy for a homogeneous spherical volume conductor. In this paper, the effect of the magnetic field on the elastomer pattern containing steel spheres is shown. The theoretical description of the influence of magnetic field lines on ferromagnetic particles and the determination of forces is rather complicated. Thewhole analytical calculations concerning the knowledge of attraction of the spheres are usually simplifiedby the assumption ofoneconcentrated forceacting ingravity centre of thebody.Suchanapproach allows one to obtain only approximated results, however, the numericalmodel makespossible thedeterminationof forces in eachpointof thediscretebody. In this manner, the resultant forces can be found, which allows one to compute stresses and displacements in the elastomer pattern. The fundamental aim of the work was to determine the limit value of the magnetic field in order to destruct elastomer continuity. Two tests (numerical and empirical) were performed. In the numerical model, the source of the magnetic field is a solenoid through which an electrical current passes. Numerical computations for the spatial model which consists of the solenoid generating themagnetic field and of the elastomer patternwith two spheres have been conducted.The elastomer pattern has nomagnetic property and the source of appearing stresses in it is caused bydisplacements of the spheres due to themagnetic field.Theproblem 662 A. Boczkowska et al. is solved by applying the finite elementmethod inANSYS11.0 code. For given current values, results have been presented in forms ofmagnetic field intensity, magnetic induction, magnetic forces, stresses and displacements. The applied numerical method is particularly presented in the third Section. In the experimental model, real elastomer patterns in which two spheres with diameter 12.7mmhad been set were prepared. The experimental appro- ach allowed one to observe stress distribution in the material. 2. Material properties The values of mechanical properties of the elastomer are found by performing a one-directional tensile test. In numericalmodelling of themagnetic field, the effect of elastic deformation of spheres and elastomer is assumed (Table 1). Because of good cohesion of the spheres and elastomer material condition of deformation continuity at the border of two bodies has been assumed. Table 1.Assumedmaterial properties Ferritical steel Elastomer Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio E [MPa] ν [–] E [MPa] ν [–] 200000 0.3 0.12 0.45 3. Numerical model (ANSYS 11.0 code) In the numericalmodel, a homogeneousmagnetic field has been assumed.This was achieved by incorpolation of a solenoid, which generates a homogeneous magnetic field as a result of an electric current flowing through a wire. The assumption that inside a real solenoid the magnetic field is predominantly homogeneous is to some degree a simplification, although in numerical com- putations it is acceptable. The current passing through a wire of a solenoid produces the magnetic field according to the relation H = nI l (3.1) where n is the number of coils (in theFEMmodel amounts 1), I [A] – current passing through one coil, l [m] – height of the solenoid. Analysis of magnetic field effect on ferromagnetic... 663 The value of magnetic induction B is expressed by the equation as B=µH (3.2) where µ=µairµ0 is the absolute magnetic permeability, µ0 =4π10 −7H/m – magnetic permeability of vacuum, µair = 1 – relative magnetic permeability of the air. In Fig.2, the elements type source are shown as the solenoid producing the magnetic field.Dimensions of the elastomer pattern and spheres are presented in Fig.1. Fig. 1. Dimensions of the elastomer pattern and spheres, [mm] In Fig.2, the solenoid for modelling of the magnetic field is shown. Fig. 2. (a) Dimensions of the solenoid and spheres (solenoid: length, width, height, thickness: 130mm×34mm×50mm×10mm), (b) partially numerical model for computations in Ansys 11.0 code 664 A. Boczkowska et al. The elastomer pattern of dimensions 50mm × 50mm × 13mm with two steel spheres has been set inside the solenoid and the whole space around it is surrounded by the air (dimension of the air space amounts 100mm× 100mm× 200mm). The relative magnetic permeability for steel spheres is assumed µsteel =10 4, and for the elastomer pattern µelastomer =1. Numerical calculations have been carried out in two stages, namely: firstly, to find magnetic forces appearing only on spheres (it is assumed that the elastomer pattern has no magnetic property) and secondly, to determine the stress anddisplacement distribution for these forces. Innumerical calculations, Biot-Savart’s formula was used for finding themagnetic forces. The three-dimensional numerical model has been divided into finite ele- ments (for the spheres and elastomer pattern 13958 and 13608 elements, re- spectively). For investigation of the magnetic field, an element type solid98 with six degrees of freedom at each node was applied. In order to determine stresses and displacements, an element type solid92 was used. For the elasto- merpattern, the followingboundaryconditionhasbeenassumed:displacement on two external surfaces is equal to zero in all directions (Fig.3). Fig. 3. Division into finite elements and assumed boundary conditions 4. Physical model of MRE (photoelastic model) A physical model of a MRE is made of polyurethane polymer (PU) formed by synthesis of two polyols VORALUX R© HF 505 and 14922 and Isocyanate HB 6013 compound, delivered by Dow Chemical Company. There is a com- posite model performed in an analogous way as the real MRE created at Warsaw University of Technology. Inside the elastomer pattern of dimensions 50mm× 50mm and thickness 10mm, two low alloy steel spheres of diame- Analysis of magnetic field effect on ferromagnetic... 665 ters 12.7mm were placed. The distance between the gravity center of steel elements is 15.875mm. ElastomerPUisoptically activewhatmeans thatunder the influenceof the stress state appearing in it and in a polarized light it indicates birefringence. This property enables one to analyse the stress and strain state caused by the magnetic field using the photoelastic method. 4.1. Optical and mechanical properties of the elastomer Photoelastic measurements enabled examinations of deformations and stresses in the whole structure of the elastomer. On the basis of photoela- stic measurements, we can directly define the difference of stresses or main strains σ1−σ2 = kσm ε1−ε2 = 1+ν E (σ1−σ2)= 1+ν E kσm (4.1) where: kσ =(σ1−σ2)/m, fε = kσ(1+ν)/E aremodel constants corresponding to the difference of stresses andmain deformations. The values of model constants corresponding respectively to the difference of main stresses kσ or main strains fε on the basis of the compression test of a cuboid of dimensions 50mm×50mm×10mm located in polariscope space have been determined. The results of the compression test are as in Table 2. Table 2.Results of the compression test Stress σ=P/(gb) [MPa] Order of isochro- matic m Photoelastic Force Thickness Width constant P [N] g [mm] b [mm] k=σ/m [MPa/is.ord.]∗ 9 10 50 0.018 1 0.018 26.5 10 50 0.053 3 0.018 ∗ is.ord. – isochromatics order The material characteristic and Young modulus obtained experimentally were assumed in the investigations. Using the material characteristic and Young modulus found on the basis of compression tests performed in polariscope, the optical properties (model constants) of elastomer have been then determined. As it is well known, the lines for which the differences between stresses andmain strains have a constant value and the same color are called the iso- chromatics. Basing on isochromaticsmeasurements, the stress distributionhas 666 A. Boczkowska et al. Fig. 4. Strain-stress curves for the pure elastomer pattern in the case of (a) tensile test, (b) tensile-compression test been defined. Applying the stress-strain relation, in an elastic body deviator strain components are proportional to deviator stress components, as written σ1−σsr ε1−εsr = σ2−σsr ε2−εsr (4.2) The result of photoelastic measurements can also be applied to strain and stress analysis concerning elastic-plastic materials. In an elastic-plastic body, the proportion of deviator strain components and deviator stress components is described by the relation σ1−σsr ε1−εsr = σ2−σsr ε2−εsr =2G′ G′ = τ0 γ0 ( γ γ0 )N−1 (4.3) where N is the hardening exponent, in the elastic-plastic body for γ ­ γ0, τ = τ0(γ/γ0) N, τ0 and γ0 are the yield stress and yield strain, and G ′ is secant shear modulus. For a simple tensile test, when σ1 = σ, σ2 = σ3 = 0, the parameters ε1 = εint, ε2 and ε3 can be determined on the basis of the shape change law ε2 = ε3 =−(εint−3εsr)/2, where σint = σ =Aε P and εint = (σint/A) 1/p or σsr =σ/3. Themain strain difference amounts ε1−ε2 = 3 2 εint−3εsr (4.4) Themodel constant should be replaced with the function presenting themain strain difference in terms of isochromatics loops ε1−ε2 = fε(m)m (4.5) Analysis of magnetic field effect on ferromagnetic... 667 For the considered materials, the substituted Poisson’s ratio νs−pl value ranges from 0.45 to 0.49. In the case of a disc loaded by focused force, well known formulas are used. The model constant of the elastomer for the assumed characteristic amo- unts kσ =0.0183MPa/(is.ord). Fig. 5. Isochromatics distribution of the sample loaded by a focused force for great deformations shown on bright and dark photos For the material with a nonlinear characteristic σ=σ0 ( ε ε0 )p Es = σ0 (ε0)p εp−1 σθ−σr = 2P πg (cosnθ)p r n2 = 1−2µ µ2 r= 2P πg (cosθ)p σθ −σr (4.6) Themodel constant of the elastomer for the nonlinear characteristic is kσ(m)= σ0 ε p 0 εp m [ 1− 2 3 (1−2ν0) ( ε ε0 )p−1] σ=σ0 ( ε ε0 )p (4.7) where: σ0 =0.0183MPa, ε0 =0.1375, ν0 =0.45, E0 =0.12MPa. 5. Results 5.1. Numerical investigation Delivering a suitablemagnetic field value that have an effect on the spheres was performed by a current flow passing through a coil wire. The numerical 668 A. Boczkowska et al. Fig. 6. Isochromatics distribution found on the basis of nonlinear characteristic of the sample loaded by focused force for great deformations Fig. 7. Elastomermodel used in the investigations: (a) dimensions, (b) model photo without load in polariscope in sodium light model differs from the real one since there is one-coil solenoid, whereas in the numerical study this fact does not has a significant influence. Numerical computations have been carried out for a various current flow values to determine the magnetic field influence on stresses in the elastomer pattern. The calculations were initiated at the current I = 5 · 103A and is further gradual increase until the spheres got in touch with each other. On the basis of theoretical relation (Eq. (3.2)), one can determine the desired magnetic induction inside the solenoid, assuming the conditions as in pure air. Hence, the current I =5 ·103A flowing through this solenoid corresponds to magnetic induction 0.125T. Most of the obtained results has been shown in the middle plane (passing through the gravity centre of the spheres) in order to observe the part of the elastomer pattern between the spheres. In Fig.8, a scalar magnetic field Analysis of magnetic field effect on ferromagnetic... 669 Fig. 8. Scalar magnetic field distribution for the whole numerical model [Vs/m3] Fig. 9. Resultant magnetic field intensity Hsum [A/m] for the plane passing through the gravity centre of the spheres distribution for thewholenumericalmodel is shown. InFigs.9a,b, the resultant magnetic field is presented. Between the spheres, the greatest intensity of magnetic field amounting to 300000A/m was noticed. On the other hand, in Figs.10a,b, the magnetic induction distribution, where the limited value reaches 0.44T, is displayed. The essential part of numerical calculations was to outline the magnetic forces.TheresultantdisplacementsandvonMises stressdistributionhavebeen calculated for the determined magnetic forces, what is presented in Figs.11 and 12. The maximum resultant al summary displacement for any discrete element atmagnetic induction 0.125Twas equal to about 0.11mm.By these parameters, von Mises stress in the elastomer came to 0.011MPa. Repeating the above described calculation course for higher magnetic in- ductionvalues, itwasdecided in the farther part ofwork to showonlydisplace- ments and stresses in the elastomer pattern.The limiting current value flowing through the solenoid was assumedwhen the spheres contacted each other. On this basis, the diagram (Fig.13) which shows the relation between themagne- 670 A. Boczkowska et al. Fig. 10. Resultant magnetic induction Bsum [T] for the plane passing through the gravity centre of the spheres Fig. 11. Resultant displacement usum [m] Fig. 12. VonMises stress distribution σred [Pa] tic inductionand themaximumresultant displacement ormaximumvonMises stress in the elastomer has been created. It can be stated that, theoretically, the contact of spheres occurs at the magnetic induction 0.44T. Analysis of magnetic field effect on ferromagnetic... 671 Fig. 13. Maximum displacements and vonMises stresses depending onmagnetic induction for the spheres of diameter 12.7mm 5.2. Experimental results In Fig.14, the vertical displacement distribution obtained by FEM in AN- SYS 11.0 code as well as the result determined experimentally (Fig.14b) on the surface of the real elastomer pattern are illustrated. The displacement di- stribution found in the empiricalmodel, enables determination of stresses and deformation only on the external surface of the elastomer. However, the map in Fig.14a presents displacements in the middle plane of the elastomer. The photoelastic study makes it possible to evaluate the average stresses in the plane of the model. Fig. 14. Vertical displacement distribution [m] 672 A. Boczkowska et al. Fig. 15. Isochromatics between the spheres for magnetic induction 0.3T: (a) σ1−σ2 =0.064MPa (numerical model) and (b) σ1−σ2 = kσm=0.063MPa (experimental model) for kσ =0.0183MPa/is.ord., m=3.5 Fig. 16. Isochromatics between the spheres for magnetic induction 0.35T: (a) σ1−σ2 =0.088MPa (numerical model) and (b) σ1−σ2 = kσm=0.091MPa (experimental model) for kσ =0.0183MPa/is.ord., m=5 6. Final conclusions In the paper, numerical investigation results as well as experimental results of the effect of magnetic field on steel spheres embedded in an elastomer have been presented. In the numerical computation, anANSYS 11.0 codewas used Analysis of magnetic field effect on ferromagnetic... 673 basing on the finite elements method. The obtained results have been shown in form ofmaps of physical magnitudes (magnetic field intensity, magnetic in- duction, displacements and stresses). In the experimental study, the knowledge about elasto-optics was applied. The source of real magnetic field generation was a two-pole electromagnetmade by SEIKOcompany. After conducting the examinations, the following remarks can be stated: • Distribution of themagnetic field intensity in pure air created by the so- lenoid inwhich there is a steel element, depends on themagnetic perme- ability and shape of the ferromagnetic body (see H and B distribution in Fig.9 and Fig.10). • Complete contact of spheres under the influence of the magnetic field occurs at 0.44Tand themaximumeffective stress according to theMises formula was equal to 0.15MPa (Fig.14). • In the numerical model, it can be assumed that in the space inside the solenoid filled with air, the magnetic field intensity is homogeneous and can be determined according to Eq. (3.1). • Comparison between formulation of the numerical model (assumed bo- undary conditions, see Fig.3) and the boundary condition in the real model is of no importance. The support for the realmodel has been also verified numerically. • On the basis of the elastomer pattern model, the photo-elasticity con- stant was determined. Next, isochromatic distributions corresponding to the magnetic induction at 0T-0.35T were set. A good enough ac- cordance of the results obtained numerically and experimentally can be stated. In this case, the fact that both studies have a comparative cha- racter should be noticed. However, the suggested investigation method can be applied to numerical computations for a real MRE filled with carbonyl-iron particles of dimensions from several to over a dozen µm. References 1. 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TomczykB.,KoterasD., 2008, Influence of air gapbetween coils on thema- gnetic field and the transformerwith amorphousmodular core,Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Poliechniki Wrocław- skiej, 62, 92-97,Wrocław 18. TurnerM.J., CloughR.W.,Martin H.C., Topp L.J., 1956, Stifness and deflection analysis of complex structures, J. Aero. Sci., 23, 805 19. de Vicente J., Bossis G., Lacis S., Guyot M., 2002, Permability measu- rements in cobalt ferrite and carbonyl iron powders and suspensions, Journal of Magnetism and Magnetic Materials, 251, 100-108 20. Wang D., Chen J.-S., Sun L., 2003, Homogenization of magnetostrictive particle-filled elastomersusing an interface-enriched reproducing kernel particle method, Finite Elements in Analysis and Design, 39, 765-782 21. Woźniak C., 1993, Nonlinear macro-elastodynamics of microperiodic compo- sites,Bull. Ac. Pol. Sci.: Tech. Sci., 41, 315-321 22. Woźniak C., 1995,Microdynamics: continuum.Modelling the simple compo- site materials, J. Theor. Appl. Meeh., 33, 267-289 23. Yin H.M., Sun L.Z., Chen J.S., 2002, Micromechanics-based hyperelastic constitutive modeling of magnetostrictive particle-filled elastomers,Mechanics of Materials, 34, 505-516 24. Zhou G.Y., 2003, Shear properties of magnetorheological elastomer, Smart Materials and Structures, 12, 139-146 25. ZienkiewiczO.C., TaylorR.L., 1994,The Finite ElementMethod, 4thEd., McGraw-Hill, London Badanie wpływu pola magnetycznego na ferromagnetyczne kule w osnowie elastomerowej Streszczenie W pracy przedstawiono wyniki modelowania mikrostruktury magnetoreologicz- nych elastomerów (MRE) z wykorzystaniemmetody elementów skończonych (MES). MRE są odpowiednikami w stanie stałym cieczy magnetoreologicznych, złożonymi z cząstek żelazakarbonylkowego imiękkiej osnowyelastomerowej.WytwarzanieMRE prowadzone jest w stałym polu magnetycznym. Pod wpływem pola cząstki ferroma- gnetyczne układają się w łańcuchywzdłuż linii polamagnetycznego, tworząc struktu- rę kolumnową.W analizie MES mikrostruktury MRE rozpatrzono siły magnetyczne 676 A. Boczkowska et al. i sprężystości oraz ich wzajemne oddziaływania pomiędzy składnikami kompozytu. Przedstawiono przestrzenne rozmieszczenie cząstek w oparciu o ich oddziaływania w polu magnetycznym.W rozważaniach wzięto pod uwagę rozkład pola magnetycz- nego i momenty dipolowe cząstek. W obu przypadkach pole było zamodelowane dla kołowego przewodnika prądu elektrycznego. Manuscript received November 8, 2009; accepted for print January 12, 2010