ACTUATION BEHAVIOR IN PATTERNED MAGNETORHEOLOGICAL ELASTOMERS: SIMULATION, EXPERIMENT, AND MODELING

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1 Proceedings of the ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2012 September 19-21, 2012, Stone Mountain, Georgia, USA SMASIS ACTUATION BEHAVIOR IN PATTERNED MAGNETORHEOLOGICAL ELASTOMERS: SIMULATION, EXPERIMENT, AND MODELING Paris von Lockette, Rutgers University, New Brunswick, NJ 1 Rowan University, Glassboro, NJ 2 1 Visiting Scientist, Mechanical and Aerospace Engineering Department 2 Associate Professor, Mechanical Engineering Department 1 Copyright 2012 by ASME

2 ABSTRACT Magnetorheological elastomers (MREs) are an emerging class of smart materials whose mechanical behavior varies in the presence of a magnetic field. Historically MREs have been comprised of soft-magnetic iron particles in a compliant matrix such as silicone elastomer. Numerous works have experimentally cataloged the MRE effect, or increase in shear stiffness, versus the applied field. Several other researchers have derived constitutive models for the large deformation behavior of MREs. In almost all cases the arrays of embedded particles, and or the particles themselves, are assumed magnetically symmetric with respect to the external magnetic field, i.e. the bulk materials exhibit magnetic symmetry in the given experimental or analytical configuration. In this work the author presents results of dynamic shear experiments, Lagrangian dynamic analysis, and static shear simulations on MRE material systems that exhibit broken magnetic symmetry. These new materials utilize barium hexaferrite powder as the magnetically anisotropic filler combined with a compliant silicone elastomer matrix. Simulations of representative laminate structures comprised of varied arrays of magnetic particles exhibit novel actuation behaviors including reversible shearing deformation, variable magnetostriction, and most surprisingly, piezomagnetism. Results of dynamic shear experiments and analytical modeling support predicted shearing actuation responses in MREs having broken symmetry and only in those systems.. INTRODUCTION Magnetorheological elastomers (MRE) are particulate composites consisting of ferromagnetic particles in an elastomer matrix that may be cured in a magnetic field to produce some patterning of the embedded particles [1,2]. MRE behavior is driven by the interplay of magnetic forces and torques coupled to the hyperelastic response of the matrix. Magnetic forces, which are quadratic in nature and hence not reversible, will exist between particles regardless of particle shape, arrangement, and/or magnetic behavior. These forces drive magnetostrictive behaviors [3]. However, the shape of the embedded particles and their magnetization response determines whether or not magnetic torque is generated; magnetic torque behavior is linear, and hence reversible, in nature for hard-magnetic materials [4]. Magnetic symmetry, symmetry between the internal magnetization and the external applied field, must be broken at the particle level to create reversible torque-driven behaviors on a particle. It must also be broken at the aggregate level to create macroscopic torque-driven behaviors distributed throughout a sample. Past works on MREs, theoretical, computational, and experimental, have almost exclusively dealt with materials exhibiting magnetic symmetry and thus have not taken advantage of an entire class of possible behaviors that result from torque generation(see [4] for a more thorough discussion). Several models of MRE behavior exist which link dipole energy formulations of various arrangements of soft-magnetic particles with macroscopic deformation of a hyperelastic matrix, such as an Ogden or Arruda-Boyce material, to yield deformation- magnetic field-dependent elastic responses (for example [5,6]). Other researchers have formulated continuum models that link the interaction of the external field with the particle s magnetization, sometimes including interparticle affects, coupled with the mechanics of deformation to yield similar deformation- magnetic field-dependent predictive models(see for example [7,8]). In all of these cases magnetic symmetry is preserved at the particle and/or aggregate level. Consequently, these works have examined only responses available through interparticle forces, i.e. quadratic responses driven by magnetostrictive effects. Only recently have researchers begun examining the effects of broken symmetry by analyzing MRE systems comprised of ellipsoidal particles and/or particles of hardmagnetic material. A series of papers lay the homogenization framework [9] for studying such a material, however the most recent paper in the series [10], while addressing broken geometric symmetry through particle aspect ratios, employs soft-magnetic material behavior and still maintains aggregate symmetry. The particles major axes are assumed collinear with the field. The formidable problem of analytically quantifying the role broken magnetic symmetry at the micro- and macro-level plays in MRE response has yet to be thoroughly studied. Recent work by the author begins to address the role broken symmetry plays in MRE response through experimentation [4]. The work quantitatively characterizes the degree of magnetization-induced torque behavior produced in various types of hard- and softmagnetic MREs. The work highlights how aligned hardmagnetic MRE materials can be well characterized by magnetic torques while MRE comprised of soft-magnetic material are better characterized by inter-particle forces arising from demagnetizing fields. 2 Copyright 2012 by ASME

3 This work begins to address the problem by investigating the effects of MRE microstructure on macroscopic response using Comsol, a commercial multi-physics finite element package, to solve the coupled magneto-elastic problem. In addition, experiments in dynamic shear response are conducted and linked to a Lagrangian dynamic analysis that highlights the need for breaking magnetic symmetries in device application situations to provide true actuation. Fundamentally, this work argues for the importance of understanding magnetic torque generation in MREs and how the manipulation of distributed magnetic torques may be used to develop novel material behaviors. Torque is manipulated by varying the microstructure through spatial variations in particle alignments and magnetization behaviors. Finally, magnetic torques are employed to generate actuation behavior that is employed in dynamic shear. (a) (b) (c) Figure 2: MRE microstructure classifications (a) uniform (b) alternating, (c) random. Blue arrows show magnetization which is characterized by (a) orientation with respect to horizontal, (b) orientation of first column with respect to the horizontal with subsequent mirrored columns, (c) orientation of individual magnetizations with respect to he horizontal. Figure 1: Schematic of MRE simulation architecture showing material domains, boundary conditions, particle orientations, and applied field orientation. SIMULATION METHODOLOGY The systems studied herein consist of a 5x5 array of ferromagnetic particles arranged in a latticework structure with defining particle orientation angle (measured with respect to the horizontal axis) embedded in an elastic matrix and surrounded by a volume of air (see Figure 1). An aligned column of five particles results in over 90% of the energy density of an infinite chain supporting this initial sizing for these investigations [6,7]. Regular arrayed structures have been assumed by previous researchers developing constitutive theories therefore, given the debate over the actual distribution of particles and magnetizations in real systems [11], a similar structure is justified here. The system studied is twodimensional as shown. The particle volume concentration is fixed at 30% (here by area) recreating the optimal mixture ratio shown repeatedly for soft-magnetic iron based materials across a range of particle size and particle size distributions [1,2,12,13]. Particle aspect ratio and orientation, however, are variable. Material properties of the composite s constituents are taken either from values in the literature or from experimental values determined through in-house magnetic and mechanical characterization(see [4] for details). The hard-magnetic inclusions are modeled as barium ferrite (BaM) with constant orthotropic remanant magnetization. For small fields a constant valued easyaxis component of the remanant magnetization of BaM and a negligible (zero) hard-axis component are warranted. BaM at a 30% volume fraction in silicone elastomer is chosen as a model constituent for comparison with experimental studies using these same material detailed later in this work. Model material parameters are given in Table 1. BaM Air Matrix [ ] [T] E [Pa] 200e9-866e3 [ ] Table 1: Material Properties Used in Simulations Multiphysics finite element simulations are developed in COMSOL, a commercial package, to solve the magnetoelastic problem, which is coupled through the Maxwell surface stress tensor, across three separate domains. The three domains are the air box surrounding the composite, the matrix material, and the set of twenty-five inclusions. Maxwell s equations are solved over all three domains while equations of elasticity are solved only over the matrix and inclusions. The formulation uses an augmented Lagrangian-Eulerian description such that field variables are calculated in the deformed configuration as defined by the current solution to the elasticity problem in an iterative fashion. Mesh points in the air box are displaced smoothly between the air box outer boundary and the matrix material s outer boundary, which conforms to the elasticity solution, using a Laplacian operator. Three separate array configurations, differentiated by the orientation of the embedded particles, are simulated each with its own set of displacement boundary conditions (see Figure 2). The uniform array (Figure 2a) has orientation 3 Copyright 2012 by ASME

4 (a) (b) (c) Figure 3: Representative finite element results of (a) random and (b) uniform microstructures showing magnetizations (blue arrows), Maxwell traction on three particles (red arrows) and magnetic flux density (background color, red=high, blue-low); (c) shear strain response vs. orientation angle for three field strengths. The random orientation shows negligible response, <10% of maximum 15mT strain response. angle fixed for all particles. The alternating (Figure 2b) array has orientations fixed by columns which are mirrored. The random array (Figure 2c) has particle orientations that are randomized. In all configurations, however, particle centers adhere to the lattice structure with a fixed distance on centers by row and column. The uniform array is used to study the shear actuation behavior of hard-magnetic MREs with symmetry broken by particle aspect ratio in conjunction with orientation. The alternating array is used to investigate how the roles of interparticle forces and external-field induced torques vary with broken symmetry in microstructure. The random array highlights the effect of complete isotropy on MRE response. The displacement and applied field boundary conditions vary by array configuration and are depicted in Figure 1. In all three configurations external fields of are applied in the direction across all three domains using a magnetic vector potential on the vertical faces while the top and bottom faces of the air box are perfect magnetic conductors; the magnetic boundary conditions are the same for all three configurations. In addition, for all array configurations the outer boundary of the air box has fixed displacement. Inner matrix-air boundary conditions on displacement vary to accommodate varying deformation responses being studied (See Figure 1). For the uniform and random arrays, the bottom of the matrix-air boundary is fixed to allow free shearing deformation while all other faces of the matrix-air boundary are free. For the alternating arrays, roller boundary conditions are used on the left and bottom faces of the matrix-air boundary to recreate quarter symmetry in displacement. The system of equations governing the problem account for linear elasticity, e.g. small deformation and linear material behavior, as well as hard-magnetic behavior also in a linear regime. A linear magnetization solution is acceptable due to the small magnetic fields studied, 30% of the coercive field. The resulting small deformations expected warrant linear elasticity theory given appropriate initial moduli for the hyperelastic material. Solving the problem requires solution of Maxwell s equations for the magneto-static case with zero applied electric field over all three domains. The governing equations for the reduced magneto-static case can be given as where is the applied magnetic field, is the applied surface current density, and is the magnetic flux density. Employing the potential formulation where is a magnetic vector potential defines as the solution variable. The general constitutive equation for the magnetic response of the air medium and the soft-magnetic particulate domains is given by yielding the governing equation for the air medium and soft-magnetic particle domains in terms of the solution variable, (1) where, is the relative permeability of the material and 410 is the permeability of free space. The general constitutive equation of a hard magnetic material can be expressed by therefore, the governing equation for the hard-magnetic domains is given by 4 Copyright 2012 by ASME

5 (a) (b) Figure 4: a) Finite element model of periodic MRE structure showing flux density streamlines and magnetization arrows b) magnetostriction (strain) response vs. orientation for 45mT c) magnetostriction (strain) vs. for In (b) the magnetostriction is shown to change sign with field for angles near 20 degrees. In (c) the response is shown to transition from quadratic (concave down) near 0 degrees to nearly linear at 15 degrees back to quadratic (concave up) at 45 degrees. (2) where is the initial modulus and Possion s ratio. The small strain tensor can be represented as (c) which in terms of the solution variable yields (3) The elastic and magnetic problems are directly coupled through the Maxwell surface stress, given as (4) where is the outward normal to the particle surface and is a traction boundary condition applied to each particlematrix boundary. The elasticity problem is defined by the conservation of linear momentum in the static case without body forces, or where is the Cauchy stress. For linear material behavior generalized Hooke s Law is employed where is the stress tensor, and is the stiffness matrix defined in 2D plane stress by 1 0 (5) (6) where, the displacement field, is the solution variable sought. The resulting governing equation for the elasticity problem in terms is given as, (7) The system of equations is solved in an iterative fashion with updates to the displacement field affecting particle orientations and thereby the magnetic vector potential solution which in turn necessitates recalculation of the Maxwell surface stress boundary tractions on the particles. Simulations require roughly 2M degrees of freedom for a converged mesh solution with respect to magnetostriction or shear strain. To illustrate the effects of microstructure, Figure 3a shows deformed mesh results (not to scale) of simulations of hard-magnetic particles with aspect ratio 1 embedded in the silicone elastome matrix subjected to a 0.03 external field where follows the +-axis for varying orientation angle configurations. In Figure 3a the particles have random magnetization orientation while in Figure 3b the particles have uniform magnetization 5 Copyright 2012 by ASME

6 Magnet pole face Acrylic vibrating mass, MRE sample with in-plane poling Magnet pole face Figure 5: Schematic of single lap shear experiment showing magnet pole faces, poled MRE sample, and acrylic vibrating mass. orientation, both denoted by groupings of (blue) arrows. In Figures 3a and 3b the three particles along the northwest-southeast diagonal also have the Maxwell surface stress vector depicted along the particle boundaries by (red) arrows. Figure 3c shows the resulting shear strain versus magnetization orientation for three field strengths. The strain is zero when the array of particles is symmetric with respect to the field ( 90 ) and maximized, in magnitude, when the array is orthogonal to the field ( 180,0, 180 ). In the orthogonal microstructures the strain response is also linear with respect to applied field and consequently not driven by traditional quadratic magnetostrictive forces. The strain direction is also seen to reverse as the orientation traverses from 180 to 0 and back to -180 degrees which is expected as it gives the right hand rule for magnetic torque actuation. Finally, the strains are reversible with the sign of external field (not shown). Figure 4 shows simulation results of an alternating array of hard-magnetic particles with aspect ratio 2 and subject to a varying -axis magnetic field. Figure 4a shows a representative deformed mesh (not to scale) for 45 with remanent magnetizations depicted by groupings of (blue) arrows within the particles and magnetic flux density as streamlines. Figure 4b shows the normalized -displacement, the magnetostriction, in the sample versus first column orientation angle for two field strengths, (intermediate values omitted for clarity). Within a range 5 20 the magnetostricion is shown to change signs with field strength. This is critically important because in classical definitions magnetostriction is quadratic and hence not reversible. Figure 4c highlights the microstructure in which the response is reversible by recasting the data as normalized -displacement vs. applied field strength ( for six discrete values of orientation angle 0,7.5,15,30, 37.5,45. The simulation results clearly show transition from typical quadratic magnetostrictive strain behavior to a linear and reversible response at 15. The results suggest that microstructures described by the 15 alternating array may show piezomagnetic behavior, e.g. reversible axial strain vs. field response, which is uncommon in traditional magneto-active materials. DYNAMIC SHEAR ACTUATION EXPERIMENTS AND MODELING The ability of MREs with broken magnetic symmetry to exhibit novel actuation mechanisms coupled with the observed importance of the magnetic symmetry in the particles and the aggregate suggests a reexamination the use of MREs as shear-stiffening elements in typical vibration absorbers. Specifically an examination of the effect of broken symmetry in a typical shear setup provides insight that may be useful in modeling the response of H-MREs in application contexts. Consider an acrylic oscillating mass,, attached in single shear to a hard-magnetic MRE material bushing with thickness, as shown in Figure 5. The material is poled during curing such that the easy- (shown with block arrows) and hard-magnetic (not shown) axes, and respectively are aligned as shown. The material is also subjected to an external time varying field aligned vertically as shown. A Lagrangian,, description of the system,, where is the kinetic energy of the mass and is the potential energy (elastic plus magnetic ), will allow direct determination of the equations of motion based on sample bulk magnetizations and. The kinetic energy in small deformation is given by where is the sample height and is the shear strain; the elastic energy by where A is the cross sectional area and is the purely elastic shear modulus; and the magnetic potential energy by 1/2 where is the demagnetization field, 0,0; 0,1 is the demagnetizing tensor for the macroscopic sample. The equation of motion is then given from the well known form where in this instance the generalized displacement,, follows and the generalized velocity, follows. Next consider the components of the magnetization of a bulk sample,, poled in plane such that 6 Copyright 2012 by ASME

7 which can be approximated by 0 for small fields. Under a shear strain the magnetization will rotate following where cos, sin ; sin, cos. Given these assumptions the equation of motion can be shown to be (8) γ γ This equation has the form of standard spring-massdamper configurations used to model traditional MRE field-dependent shearing behavior but includes an additional forcing term on the right hand side representing a magnetic actuation which Is not present in soft-magnetic MRE models. In eq. (8) is the damping coefficient, is proportional to the elastic spring stiffness and is proportional to a constant valued magnetic stiffness. The magnetic forcing term highlights the expected ability of this material in this configuration to produce active shearing behavior that is reversible in and is independent of deformation. This behavior stands in contrast to traditional soft-magnetic MRE materials in which the elastic stiffness term (and only that term) shows quadratic field dependence, and therefore is not reversible, and which is linear in displacement and therefore provides no response at zero deformation. An experimental setup was developed to test for the ability of H-MREs poled in plane to produce shearing actuation. The test configuration follows the arrangement show in Figure 5 using an MRE bushing, in single lap shear attached to one pole face of a C-shaped electromagnet; the other side of the bushing was affixed to a 13g acrylic mass. The material was fabricated using nominally 40 BaM powder at 30% v/v in Dow Corning HS II silicone elastomer and cured in a 2 magnetic field to produce magnetic alignment. A sinusoidal magnetic field was provided by an amplifier driven electromagnet. The field was varied from Hz while a laser Doppler velocometer was used to measure vibrations of the mass. Baseline measurements of the mass s vibration taken with no field and the magnet s vibration taken with an oscillating field but no sample present show negligible background vibration in the system compared to subsequently measured oscillations of the magnetic-bushing-mass system. Figure 6 shows the measured peak velocity of the mass versus frequency of the applied magnetic field. The figure clearly shows the ability of the system to actuate the mass in a shearing mode. It must be noted that the characteristic shape of the curve stems from the inherent dynamics of the system described by equation (8) in conjunction with the impendence characteristics of the Figure 6: Measured peak velocity of vibrating mass vs. frequency of applied magnetic field. The magnetamplifier system is tuned to provide 150 mt DC; the field decays with frequency. amplifier-electromagnet system; the applied field, nominally 150 DC, diminishes with frequency CONCLUSIONS This work has presented results of simulations of the effect of varying microstructure on the actuation behavior of MRE materials. Specifically, the role of broken symmetry in magnetization has been explored. Shear simulations of uniformly ordered particles show that hardmagnetic particles having an easy-axis misaligned with the external field will produce actuation driven by particle level torques; this is in addition to interparticle forces which are also present in soft magnetic materials. Furthermore, magnetic isotropy (symmetry), either from alignment with the external field or from completely random configurations precludes this actuation. Additional simulations of alternating microstructures show the ability to transition from magnetic torque dominated to magnetic force dominated behavior by highlighting the transition from linear to quadratic magnetostritive behavior. Finally, a Lagrangian dynamic analysis of a single lap shear oscillator based on coupled elastic and magnetic energies predicts the occurrence of shearing actuation for materials with the easy-axis lying in plane, perpendicular to the applied field. This geometry would produce mislignment (broken symmetry) between the external field and the internal magnetization of the sample thereby producing a distributed torque throughout similar to the results of simulated misaligned microstructures. 7 Copyright 2012 by ASME

8 Single lap dynamic shear experiments conducted to test this hypothesis were successful in showing macroscopic shear actuation resulting from an alternating magnetic field. These results strongly suggest that distributed torque generation via careful patterning of hard magnetic MRE microstructures provides a means of producing reversible shear actuation beyond the well known stiffness increase of traditional soft-magnetic MREs and possibly a reversible magnetostriction or piezomagnetism. ACKNOWLEDGEMENTS The author would like to acknowledge the support of NSF CMMI and Rutgers University, New Brunswick, NJ where the experimental work was conducted during the author s tenure as a visiting scientist. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation REFERENCES 1.J. M. Ginder, M.E. Nichols, L.D. Elie, and J.L. Tardiff, Magnetorheological Elastomers: Properties and Applications, SPIE 3675, 131 (1999). 2.L. Lanotte, G. Ausiano, C. Hison V. Iannotti, C. Luponio, and C. Luponio, Jr., State of the art and development trends of novel nanostructured elastomagnetic composites, J. Optoelectronics Adv. Materi., 6, 523 (2004). 3.J.M. Ginder, S.M. Clark, W.F. Schlotter, and M.E. Nichols, Magnetostrictive Phenomena in Magnetorheological Elastomers, Int. J. Mod. Phys. B 16, 2412 (2002). 4. P.R. von Lockette, Lofland, S., Mineroff, J., Roche, J., Babock, M. (2011) Investigating new symmetry classes in magneto-rheological elastomers, Smart Materials and Structures, 20, Xianzhou Zhang, Wen Peng, Suili Weijia, and Weihua Li, Analysis and fabrication of patterned magnetorheological elastomers, Smart Materials and Structures, 17, (2008). 6.Y. Shen, M. F. Golnaraghi, and G. R. Heppler, Experimental research and modeling of magnetorheological elastomers, J. Intell. Mater. Syst. Struct. 15, 27 (2004). 7. Dorfmann, A., Ogden, R.W. (2004). Nonlinear magnetoelastic deformations. Q. J. Mech. Appl.Math.. 57, S.V. Kankanala and N. Triantafyllidis, On finitely strained magnetorheological elastomers, J. Mech. Phys. Solids 52, 2869 (2004). 9. E. Galipeau and P. Ponte Castañeda. The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites. International Journal of Solids and Structures 49 (2012a): P. Ponte Castañeda and E. Galipeau. Homogenization-based constitutive models for magnetorheological elastomers at finite strain. Journal of the Mechanics and Physics of Solids 59 (2011): Boczkowska A, Awietjan S F, Wejrzanowski T, Kurzydłowski K J 2009 Image analysis of the microstructure of magnetorheological elastomers J. Mater. Sci. 44, Lokander M and Stenberg B 2003a Performance of isotropic magnetorheological rubber materials Polymer Testing 22, Lokander M and Stenberg B 2003b Improving the magnetorheological effect in isotropic magnetorheological rubber materials Polymer Testing 22, Copyright 2012 by ASME