Particle mixtures in magnetorheological elastomers (MREs)

Transcription

1 Particle mixtures in magnetorheological elastomers (MREs) Paris R. von Lockette *a, Jennifer Kadlowec a, Jeong-Hoi Koo b a Rowan University, Glassboro, 21 Mulica Hill Rd, NJ 828; b Miami University, 51 East High Street, Oxford, Ohio 4556 ABSTRACT Keywords: Magnetorheological elastomer, nanoparticle, tunable vibration absorber, polydimethysiloxane Magnetorheological elastomers (MREs) are state-of-the-art elastomagnetic composites comprised of magnetic particles embedded in an elastomer matrix. MREs offer enormous flexibility given that elastomers are easily molded, provide good durability, exhibit hyperelastic behavior, and can be tailored to provide desired mechanical and thermal characteristics. MRE composites combine the capabilities of traditional magnetostrictive materials with the properties of elastomers, creating a novel material capable of both highly responsive sensing and controlled actuation in real-time. This work investigates the response of MRE materials comprised of varying mixtures of 4 and 1 micron iron particles. Samples are tested in compression yielding a compressive modulus and measure of the shear stiffness via Mooney plots. Samples are also tested using a tunable vibration absorber (TVA) designed specifically for this experiment. The TVA loads the samples in oscillatory shear (1 1Hz) under the influence of a magnetic field. In all samples, results show increases in the material s stiffness under the application of a magnetic field as evidenced by the frequency response function of the TVA system. Increases in stiffness of 5% at.15t were achieved with samples containing 3%-4 micron particles and 3%-4micron + 2%-1 micron particles. This yields a ratio of over 3%/T. The two-particle MRE appeared not to have reached saturation suggesting further stiffness enhancement was possible beyond the saturated single-particle 4 micron sample. However, this may be a result of the larger iron content. Results also suggest variation in the behavior of two- versus single-particle MRE behavior as evidenced by the shear modulus found in compression, but results are inconclusive. MRE materials made with nanoparticles of hard magnetic barium ferrite show stiffness increases of 7%/T which is comparable to MREs having larger iron particles. 1. INTRODUCTION Magnetorheological elastomers (or MREs), solid analogs of MR fluids (MRFs), are state-of-the-art elastomagnetic composites comprised of magnetic particles embedded in an elastomer matrix. A recently published definition of this class of materials states that MREs have the following characteristics: the particles have an asymmetric shape, preferably with a main anisotropy axis; the particles are soft ferromagnetic or small permanent magnets; and the composite has an elastic behavior, due to the matrix properties, up to a relative deformation of The study of MREs as a viable class of smart materials has gained increasing interest since they have recently exhibited large magnetorheological (or MR) effects in sample materials. 2 The MR effect, E, is measured as the relative change in a material s complex modulus when placed within a magnetic field, B, with respect to the modulus with no field present, e.g. E E = B E E B= B= vonlockette@rowan.edu, , (fax) (1) V. 2 (p.1 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

2 The MR effect versus field strength has been reported for several elastomers with various volume fractions of carbonyl iron 1-1. The curves pass through linear regions before saturation. Results are also reported for the shear modulus, G. As a class of smart materials, MREs offer enormous flexibility given that elastomers themselves are easily cast, provide good durability, exhibit hyperelastic behavior, and can be tailored to provide desired stiffness, damping, environmental, and thermal characteristics. MRE composites combine the capabilities of traditional magnetostrictive materials with the flexibility of elastomers creating a novel material capable of both highly responsive sensing and carefully controlled actuation in real-time. 3,4 For example, as active elements in vibration absorbers the variable stiffness of MREs enables them to shift the damped frequency under the application of a magnetic field. Consequently, MREs have found use in a wide range of vibration attenuation and control applications in the automotive and aerospace industries which has led to the current emphasis on application-intensive research. 5-8 However, since their behavior is governed by the physics underlying both magnetic and elastomeric materials, the study of MREs must combine knowledge of both subjects in order to better understand the science governing the composite material s behavior. This work begins an examination of the behavior of MREs as the hyperelastic filled elastomers that they are. Specifically, this work begins to address the dependence of the magnetorheological response on particle size distribution and the use of nanoparticles of hard magnetic material through dynamic shear and quasistatic compression tests MREs are fabricated by combining usually asymmetric magnetic particles with an uncured elastomer compound such as natural or silicone rubber. Effective particles sizes range from nanoparticles to coarse iron filings, ~5 µm The mixture is then cured either with or without the presence of an applied magnetic field of.5-1 T. While curing in a field, particles may form lines collinear with field lines en masse. In addition, particles with a preferred magnetization axis tend to reorient with the external field H (See Figure 1). Once cured, an external magnetic field will induce a moment on the embedded particles in the matrix that is resisted by torsional constant, K. The summation of individual particle strains due to the induced torques causes macroscopic strain in the bulk. Local particle rotations of 1-2 radians can be achieved in matrix materials where the torsional elastic constant is below 1 N-m. 1 In addition, applied mechanical strains may induce a rotation with respect to the preferred axis as well. Together, these phenomena lead to the elastomagnetic constitutive equation. Consider magnetic ellipsoids in an elastomer matrix cured at an angle θ i between the z axis and the semimajor axis of the ellipsoid. For an applied strain ε z, magnetic field H z, magnetization M, vacuum permittivity µ o, and particle volume V m one finds: KAε z sinθi cosθi MVmµ H z sinθ m θ = (2) K 2 2 where A is a geometrical factor ( a b / a + b ) with a and b being the length of the principle axes of the ellipsoid. From Eq. (2), it can be seen clearly that anisotropy of the particles will enhance the MR effect. However, this effect can be further magnified by noting that the magnetization vector is not only a function of magnetic field H but also depends on particle geometry (demagnetization factor), magnetic anisotropy as well as particle size (magnetic domain walls), and particle volume, V m. A critical volume fraction of particles, often reported as roughly 26% - 36%, is needed to maximize performance while still avoiding particle-particle interference. 1,9,1,15 The maximum E reported for different combinations of matrix and filler materials varies considerably with roughly 6% at approximately.8 T, the largest reported 2,9 ; this would yield a ratio of E to field strength of 75%/T. Recent theory states that this critical volume fraction maximizes the content of magnetic material while allowing free action of the particles. 1,9 These arguments elicit a notion of filler particle packing wherein one attempts to maximize magnetic material density while retaining the elastic behavior. Figure 2 shows a 2D schematic of such packing, but also clearly highlights the availability of interstitial positions for particles with reduced dimensions which will enhance the MR effect by increasing V m in eq. (2). As a starting point, the authors assume a body centered cubic arrangement of primary particles such that the interstitial spacing and volume ratio of primary to interstitial position are set. While other studies have used finite element methods in conjunction with phenomenological models of filled rubber behavior to estimate a critical volume particle concentration of 26% 15 several studies and this work have shown effective use of higher particle concentrations. In addition, it is not clear if this phenomenological treatment accounts for bimodal particle distributions V. 2 (p.2 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

3 Several other studies have also examined the effect of particle size on the maximum MR effect. The largest E values have been seen in materials with particles roughly 1-1 microns. 11 Though particle size distributions are not normally reported, it can be assumed that the materials have been made with particles of a nominally unimodal size distribution. This work begins to examine the differing magnetorheological stress-stretch response between MRE materials with unimodal versus bimodal particle distributions to loading in uniaxial compression. Compression testing yields the compressive modulus, E, as well as the shear modulus, G, through the use of the Mooney equation. Dynamic shearing tests are performed on MRE samples with bimodal particle distributions to compare the bimodal material's response in shear to the unimodal material s response under the same conditions. 2. RESEARCH METHODS The materials used in this work are all cast from using Dow Corning HS II RTV Silicone Elastomer Compound as the matrix material. The compound is preheated in an oven to 6 C before mixing to decrease viscosity. Once heated the desired amount of iron particles are added and mixed thoroughly. The catalyst is added in a 15:1 compound:catalyst ratio in the final step. Specimens were cast in cylindrical molds, mm height by 19.5 mm diameter for compression tests and rectangular prism molds 19.5 mm length by 12.7 mm width and 12.7 mm height for the dynamic shearing specimens. To investigate the role of particle distribution samples were made with the compositions in Table 1. Table 1: MRE sample compositions Volume Fraction[%] Average Particle Size Cured in Field Sample ID Primary Secondary Primary Secondary B 28 4 µ NA No C µ 1 µ No D 3 4 µ NA No E µ 1 µ No F µ 1 µ No I 26 4 nm NA Yes Samples of type B and C used 4 and 1 micron particles to achieve total iron concentration of 28% by volume, which approaches the critical volume concentration. Sample B and C moved from nominally unimodal to nominally bimodal particle distributions in order to test the effects of particle distributions on the MRE effect for a fixed total volume concentration. Additionally, Samples D and F went from 3% to 32% iron with the addition of the secondary particles to determine the effects of adding smaller iron particles to increase the total volume concentration. As a starting point, the ratio of 4 to 1 micron particles was set at 1:12 by number to follow the volume ratio of primary to interstitial spacing in a body centered cubic unit cell structure containing perfect spheres of 4 micron diameter. Nanoparticles of barrium ferrite (~4 nm), a hard magnetic material, were used as the sole filler particles in a set of MRE samples. Cylindrical and rectangular barium ferrite samples were cured in a uniform magnetic field of 2T with the field oriented at 45 degrees to either the axis of the cylinder or the short axis of the rectangle. This process poles the barium ferrite along this orientation. A schematic of the experimental setup for quasistatic compression tests is shown in Figure 3. Tests were performed using a MTS servo-hydraulic frame and a permanent magnet 2" in diameter to produce a uniform magnetic field through the specimen of.18t. The upper fixed platen, 16 in length, was machined from aluminum to avoid direct magnetic attraction between the magnet and the load cell. A quasistatic strain rate of.1% s -1 was used to minimize back electromotive force in the aluminum cylinder; the applied force due to back EMF was not measurable on the load cell used. The specimens were compressed to λ = L/L o =.7 (or ε=-.35). Dynamic tests were conducted using a shearing tunable vibration absorber (TVA) apparatus designed for these V. 2 (p.3 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

4 experiments, see Figures 4a and 4b. The TVA isolates the primary mass (the upper mount, induction coil, and accelerometer) from the direct actuation of the hydraulic actuator while allowing the primary mass and absorber mass (the upper accelerometer ) to respond to the input together as a floating, base-excited two degree of freedom system, see Figure 5. The TVA includes integrated induction coils (3 turns/coil) capable of producing.1 T total at 1 amps across the gap where the PDMS silicone elastomer MRE samples are suspended in shear. In this system, as the primary mass vibrates, driven by the MTS servo-hydraulic frame, a portion of that vibration is transferred to the absorber mass (herein the upper accelerometer) which operates at a natural frequency, ω a. The natural frequency of the absorber is governed by its mass and the stiffness of the two MRE samples which suspend it; those samples vibrate in a shearing mode. By applying a magnetic field through the MRE samples, their stiffness is altered which shifts the natural frequency of the absorber. The actuator was driven over a frequency range of 1-1 Hz with uniform white noise. Data was collected using a Bruel and Kaer PULSE data acquisition system with accelerometers placed on the middle crossbeam (the primary mass) and between the two MRE samples (the absorber mass). 3. EXPERIMENTAL RESULTS Figure 6 shows a plot of nominal stress versus engineering strain in for a virgin sample of the barium ferrite MRE material, I-type, in five consecutive uniaxial compression tests. The plot shows slight weakening at high strain with successive testing as per the well known Mullins effect in filled elastomers 12. Similar results were observed for all material compositions to varying degrees. All samples were cyclied repeatedly before data collection in compression and in dynamic shear. Compression tests yielded stress versus strain curves which were nearly identical with and without the application of a magnetic field an all cases. This behavior is in line with published results from analytical modeling that shows only slight changes in the compressive response under an applied axial magnetic field 13,14.To gain a better understanding of the material response to compressive loading, those data are better represented by plots of the tangent modulus versus strain. The tangent modulus is the moving average of the slope of the true stress versus true strain curve. Figures 7 and 8 show the tangent modulus versus true strain for various samples with and without a magnetic field applied along the axis of compression. The plots (and data for other samples, not shown) highlight a reduction in stiffness at high stretch under the application of the magnetic field for some samples, and not others. The effect is most pronounced in F-type materials which have 3%-4 micron + 2%-1 micron particles. Figures 11 and 12 show Mooney plots for B- and C-type samples. In the Mooney plot where f * is the nominal stress, the left hand side of (3) is plotted versus 1/λ yielding an equivalent graph of the Mooney constants c 1 and c 2. f = c c + 2 λ λ 2 1 (3) 2 λ ( ) In tension the plot yields c 1 along the y-intercept and c 2 is the slope of the line. In shear, the graph is representative of the shear stiffness of the material owing to the proportional relationship between the constant C 1 and the shear modulus G. 12 Both Figures 11 and 12 show strain softening initially before strain hardening at large strains with and without the magnet. In both cases, however, the material shows an increased stiffness with the magnet in shear. The increase in stiffness is much larger for the bimodal samples of type C. Dynamic shearing tests using a tunable vibration absorber were conducted to compare the response in dynamic shear. Figure 13 shows a frequency response plot of the TVA absorber referenced to the input from the primary mass for an F- type bimodal sample. The graph shows a clear shift in the natural frequency of the TVA from just below 1 Hz to roughly 124 Hz which indicates an increase in the shearing stiffness of the material. A similar plot for the C-type bimodal sample is shown in Figure 14. However, whereas in Figure 13 the amplitude of the frequency response function remains constant as the natural frequency is shifted higher, the amplitude decreases in the C-type bimodal specimen of Figure 14. A trend in this reduction in amplitude could not be discerned between unimodal and bimodal samples. Figure 15 shows a frequency response function plot for an I-type 26% barium ferrite nanoparticle MRE sample. The material is shown to stiffen under a magnetic field as well. Samples of type D tested also showed a magnetic-fielddependent increase in natural frequency (not shown) V. 2 (p.4 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

5 The increase in stiffness of the sample can be calculated by treating the absorber as a vibrating spring-mass system with k ω = (4) m where ω is the natural frequency of the absorber-mre system as measured, k is the shear stiffness of the MRE sample, and m is the mass of the absorber-mre system. Since the absorber-mre geometry is constant (negligible magnetostriction is assumed) as well as the mass, changes in the natural frequency must stem from the material s shear modulus which is proportional to its stiffness, k. This allows calculation of the MRE effect in shear as 2 ωb ω G = ω 2 B= 2 B=. (5) Figure 16 shows a graph of the MRE effect in shear versus field strength for C-, D-, F-, and I-type samples. The graphs near saturation in both the D- and F- type materials at the same value for the MRE effect, approximately 5% however the bimodal sample shows a slightly more linear response which may indicate further MRE effect is possible. The critical volume concentration for these samples may exist at a higher volume percent of iron. For the I-type sample the shows little MRE effect. The results across all mixtures highlight the use of a compliant matrix material such as silicone elastomer in that the ratio of the MRE effect to field strength applied reduces to over 3%/T in the D and F samples, the best performing, as compared to 75%/T as reported in the literature. The nanoparticulate barium ferrite-filled MRE shows a ratio 7%/T which is also comparable to the best previously reported results. 4. CONCLUSIONS This work has tested MRE materials fabricated using mixtures of two particles sizes. Uniaxial compression tests have shown slight reductions in uniaxial stiffness at high strains for some unimodal and bimodal samples, but not for others. The bimodal versus unimodal trends are inconclusive in this regard. Some evidence that MRE materials with bimodal distribution of particles may exhibit different characteristic behaviors than MRE materials with unimodal particle distributions has been shown in the MRE Effect versus field strength curves of the bimodal F sample over the unimodal D sample (Figure 16) though this may reflect the effect of the increased iron content. Barrium ferrite was used in an MRE composite to assess the response of a nanoparticulate filled MRE. Compression testing of barium ferrite nanoparticulate-filled MREs has shown a slight Mullin s effect in the initial stress-strain response. Similar results where seen for other MRE compositions (not shown). The effect disappears after roughly ten cycles. The ratio of the MRE effect to field strength in the barium ferrite sample was roughly 7%, which is comparable to the literature for MREs with soft magnetic materials of 1-15 micron sized filler particles. The positive response of this material highlights the need to investigate further compounds using nanoparticles in possibly trimodal distributions with microparticles. The most promising materials tested here, D and F, having 3% of 4 micron and 3%-4 micron plus 2%-1 micron particles, respectively, showed an MRE effect of roughly 35% at.15t. The ratio of the MRE effect to field strength reduces to a controllable stiffness increase of over 3%/T (which would be saturation limited). This is well above the 75%/T reported elsewhere and highlights the utility of using soft matrix material such as silicone rubber. This increase would allow broader control of stiffness with less power consumption. The F-type bimodal sample also appeared to be farther from saturation suggesting higher MRE effects are possible using a mixture of particle sizes and/or increased iron content. These MRE materials having high MRE effect vs. magnetic field responses make them ideal materials for controls applications, especially where varying stiffness elements are needed, such as adaptive TVAs. With their relatively low stiffness, these materials would be best suited applications with low load levels and/or small vibration amplitudes. By V. 2 (p.5 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

6 using bimodal materials, however, it may be possible to both strengthen the materials and enhance the MRE effect. It should be noted that in addition the shift in stiffness that can be achieved upon the application of a magnetic field, compressive testing shows clearly that a combination of magnetic field and precompression could yield an even wider range of controllable behavior. This control can be utilized in both shearing and compressive modes. REFERENCES 1. L. Lanotte, G. Ausiano, C. Hison V. Iannotti, C. Luponio, and C. Luponio, Jr., JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS 6:2, (24). 2. Zhou, G Y. SMART MATERIALS & STRUCTURES 12:1, (23) 3. Zhou, G Y. Wang, Q. Proceedings of SPIE - The International Society for Optical Engineering Smart Structures and Materials 25 - Damping and Isolation 576, (25). 4. Zhou, G Y. Wang, Q. Proceedings of SPIE - The International Society for Optical Engineering Smart Structures and Materials 25 - Smart Structures and Integrated Systems 5764, (25). 5. T. Shiga, A. Okada, and T. Kurauchi, J. APPL. POLYM. SCI. 58, (1995). 6. M.E. Nichols, J.M Ginder, J. L. Tardiff, and L.D. Elie, The Dynamic mechanical behaviour of magnetorheological elastomers. in 156th ACS Rubber Division Meeting Orlando, Florida. 7. J. M. Ginder, M.E. Nichols, L.D. Elie, and J.L. Tardiff Proceedings of SPIE The International Society for Optical Engineering Smart Structures and Materials: Smart Materials Technologies 3675, (1999). 8. J.M. Ginder, W.F. Schlotter, and M.E. Nichols. Proceedings of SPIE -- The International Society for Optical and Engineering Smart Structures and Materials: Damping and Isolation 4331, (21). 9. M. Lokander and B. Stenberg, POLYMER TESTING 22, (23). 1. M. Lokander and B. Stenberg, POLYMER TESTING 22, (23). 11. D. Szabó and M. Zrínyi, Int. J. MOD. PHYS. B (22). 12. L. R. G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, Oxford, U.K. (1975). 13. L. Borcea and O. Bruno, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 49, (21). 14. S.V. Kankanala and N. Triantafyllidis, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 52, (24). 15. L.C. Davis, JOURNAL OF APPLIED PHYSICS 85:6, M θ m Particle Matrix θ i z y H Figure 1: Alignment and torque action due to external field acting on particle V. 2 (p.6 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

7 d 2 d 1 Figure 2: 2D schematic of available interstitial positions. Load cell Aluminum Platen MRE sample δ Magnet Lower platen Figure 3: Schematic of compression test setup showing permanent magnet in load train V. 2 (p.7 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

8 Please verify that (1) all pages are present, (2) all figures are acceptable, (3) all fonts and special characters are correct, and (4) all text and figures fit within the B Figure 4a: Tunable vibration absorber experiment schematic. MRE Absorber Coils Absorber mass accelerometer Primary mass accelerometer MTS (driving) accelerometer Hydraulic Actuator Figure 4b: Tunable vibration absorber experiment showing MTS, three accelerometers, coils, absorber mass and the MRE samples in shear V. 2 (p.8 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

9 Absorber mass, m a x a k a, MRE x Primary mass, m k, counter spring X(t), forcing displacement Figure 5: Schematic of two degree of freedom system highlighting variable stiffness MRE, k a, and absorber mass Test 1 Test 2 Test 3 Test 4 Test5 Stress [kpa] -3-2 Test # Strain Figure 6: Stress versus strain response of virgin I-type material in uniaxial compression for five consecutive tests V. 2 (p.9 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

10 14 Tangent Modulus [kpa] B = T B =.18T Strain Figure 7: Tangent modulus versus strain for C-type sample, 28% total, 26.4%-4 micron and 1.6%-1 micron particles with (triangles) and without (circles) a magnetic field. Tangent Modulus [kpa] B = T B =.18T Strain Figure 8: Tangent modulus versus strain plot for F-type sample, 3%4 micron with 2%-1 micron particles with (triangles) and without (circles) a magnetic field showing strain softening at high strain with the magnetic field V. 2 (p.1 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

11 14 13 B = T B =.18T f*/(λ-1/λ 2 ) [Kpa] Figure 9: Mooney plot of B-type unimodal sample, 28%-4 micron, in compression with (triangles) and without (circles) magnetic field where f* is the nominal stress. The graph shows only slightly increased shear stiffness in compression for λ -1 >1.3. λ 1 13 f*/(λ-1/λ 2 ) [psi] B = T B =.18T λ -1 Figure 1: Mooney plot of C-type bimodal sample, 26.4%-4 micron and 1.6%-1 micron particles, in compression with (triangles) and without (circles) magnetic field where f* is the nominal stress. The graph shows increased shear stiffness in compression for λ -1 > V. 2 (p.11 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

12 FRF [db/1. (m/s 2 / m/s 2 )] B = T B =.5T B=.1 B =.15T Frequency [Hz] Figure 11: Frequency response function (FRF) plot for F-type samples, 3%-4 + 2%-1 micron particles, in TVA testing. Vertical lines show locations of resonance peaks. Field strength increases as curves shift from left to right. 2 FRF [db/1.(m/s 2 /ms 2 )] T.1 T.143 T.286 T.357 T.429 T.5 T.1 T.137 T Field Strength Frequency (Hz) Figure 12: Frequency response function (FRF) plot for C-type samples, 26.4% %-1 micron particles, in TVA testing. Field strength increases as curves shift from left to right. The FRF decays as resonant frequency increases V. 2 (p.12 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM

13 15 FRF [db/1. (m/s 2 /m/s 2 )] T.14 T.36 T.5 T.1 T.14 T Field Strength Frequency (Hz) Figure 13: Frequency response function (FRF) plot for I-type samples, 26%-4 nm barium ferrite particles, in TVA testing. Field strength increases as curves shift from left to right. Shearing MRE Effect 6% 5% 4% 3% 2% (32%) 3%-4 + 2%-1 micron (3%) 3%-4 micron (28%) 26.4% %-1 micron (26%) Barium Ferrite - 48 Hz (26%) Barrium Ferrite - 96Hz 1% % Field Strength (T) Figure 14: MRE effect in shear (percentage change in modulus) versus field strength for multiple samples found from TVA testing. Barrium ferrite response from a secondary resonance peak at 96 Hz (not shown) is plotted with dotted lines and crosses V. 2 (p.13 of 13) / Color: No / Format: Letter / Date: 2/3/26 1:58:26 AM