Characterization and modeling of hard magnetic particle–based magnetorheological elastomers

Abstract

Hard magnetic particle–based magnetorheological elastomers are novel magnetoactive materials in which, unlike the soft particle–based magnetorheological elastomers, the particles provide magnetic poles inside the elastomeric medium. Therefore, the stiffness of the hard magnetic particle–based magnetorheological elastomers can be increased or decreased by applying the magnetic field in the same or opposite direction as the magnetic poles, respectively. In the present work, the viscoelastic properties of hard magnetic particle–based magnetorheological elastomers operating in shear mode have been experimentally characterized. For this purpose, hard magnetic particle–based magnetorheological elastomers with 15% volume fraction of NdFeB magnetic particles have been fabricated and then tested under oscillatory shear motion advanced rotational magneto-rheometer to investigate their viscoelastic behavior under varying excitation frequency and magnetic flux density. The influence of the shear strain amplitude and driving frequency is examined under various levels of applied magnetic field ranging from −0.2 to 1.0 T. Finally, a field-dependent phenomenological model has been proposed to predict the variation of storage and loss moduli of hard magnetic particle–based magnetorheological elastomers under varying excitation frequency and applied magnetic flux density. The results show that the proposed model can accurately predict the viscoelastic behavior of hard magnetic particle–based magnetorheological elastomers under various working conditions.

Keywords

magnetorheological elastomers hard magnetic particles phenomenological model storage and loss moduli viscoelastic properties

1. Introduction

Magnetorheological elastomers (MREs) are an emerging class of smart materials that their properties can be changed in a reversible manner under the application of an external magnetic field (Dargahi et al., 2019). The viscoelastic properties of MREs, namely, the storage and loss moduli, can be controlled in a rapid, continuous, and reversible manner by applying an external magnetic field (Ginder et al., 1999; Woods et al., 2007). MREs are used in various applications such as adaptive tuned vibration absorber (Qian et al., 2017), smart material–based isolators (Li et al., 2013), vehicle seat suspension (Du et al., 2011), microcantilevers (Lee et al., 2014), microvibrations (Ying et al., 2013), sensors (Wang et al., 2009), touch-screen displays (Chen et al., 2019), magnetometers (Du and Chen 2012), prosthetic devices (Jonsdottir et al., 2015), noise barrier systems (Farshad and Le Roux 2004), sandwich beams (Zhou and Wang, 2005), negative changing stiffness isolators (Sun et al., 2015), bumper of a vehicle (Bogdanov et al., 2009), and particularly in shifting the fundamental frequency of controlled devices.

MREs consist of an elastomeric matrix such as silicon rubber impregnated with micron-sized iron particles and can be classified according to several parameters like particle types, matrix, structure, and distribution of particles. In terms of distribution of particles, MREs divided into isotropic and anisotropic (Asadi Khanouki et al., 2019; Lu et al., 2012). Curing the mixture of the elastomer and iron particles in the absence and presence of the applied magnetic field results in isotropic and anisotropic MREs, respectively. In terms of particle types, particles are divided into soft and hard magnetic particles. Soft magnetic particles, such as carbonyl iron (CI) powders, have been widely used in magnetorheological (MR) materials, because of their high saturation magnetization and low remanence (Zhao et al., 2017). Recently, particles of high-coercivity ferromagnetic materials, or hard magnetic materials, such as neodymium–iron–boron (NdFeB) have been embedded in soft matrices. The high remnant characteristics of hard magnetic materials allow them to retain high residual magnetic flux density even in the absence of magnetic fields once they are magnetically saturated (Wen et al., 2017). One of the main problem with soft MREs is that their storage modulus can be only increased by applying external magnetic field irrespective of the applied magnetic field direction which limits their potential applications in adaptive MRE–based devices such as MRE-based vibration absorbers and isolators in which the reduction in the stiffness (or natural frequency) may be required to enhance their control authority. In order to alleviate this issue, an initial magnetic field may be applied either by electromagnet or permanent magnet (Yang et al., 2014) so that the stiffness could be decreased from the initial value by decreasing the applied magnetic field. However, this approach is not effective due to the added cost, weight, and power consumption associated with generation of the initial magnetic field (Wen et al., 2017). Thus, hard magnetic particle–based MREs (H-MREs) capable of providing negative MR effect and reducing their stiffness seem to be a viable technology to considerably enhance the bandwidth of the MRE-based adaptive devices.

The actuation capability and viscoelastic properties of H-MREs have been recently investigated. Koo et al. (2012) concluded that H-MREs are capable of being used as bending-type actuators. They also measured the block force and tip displacement of the samples to characterize actuation properties of H-MREs. Borin et al. (2013) experimentally determined the tensile modulus of H-MREs and considered the influence of the remanence magnetization on the tensile modulus. The viscoelastic properties of H-MREs also investigated by Stepanov et al. (2012, 2014), and it was shown that the modulus could be changed in two ways (to increase and decrease) by applying the magnetic field in opposite directions. Moreover, they concluded that the MR effect of such material is lower than that in magnetically soft composites. Stepanov et al. (2017a) also investigated the mechanism of the re-magnetization of the magnetoactive elastomers with hard magnetic fillers. They explained that this mechanism depends on the re-magnetizing of the magnetic filler and mechanical rotation of particles inside the polymer matrix. Moreover, they investigate the potential of the hybrid magnetic elastomers as active and passive dampers (Stepanov et al., 2017b). Kramarenko et al. (2015) synthesized MREs based on a silicone matrix with magnetically hard NdFeB particles, and they concluded that the response of MRE samples depends on the mutual orientation of the external magnetic field and the internal sample magnetization. Moreover, it was noted that the loss factor increases abruptly when the magnetic field is turned on in the opposite direction to that of sample magnetization due to the particle rotation within the polymer matrix. The loss factor then decreased with time. Moreover, examining the dynamic properties of both isotropic and anisotropic MREs (soft and hard) showed that the anisotropic MREs are stiffer (Anderson et al., 2015). The influence of soft and stiff elastomer matrices on the magnetization hysteresis loops of NdFeB-based MREs was compared by Kalina et al. (2017) in order to investigate the influence of the matrix material on the rotation of hard magnetic particle. Moreover, they presented microscale model to analyze the macroscopic behavior of H-MREs as a function of the rotation of the embedded particles. In another study conducted by Zhao et al. (2017), the storage modulus of both soft and NdFeB-based MR plastomers was analyzed. They demonstrated that unlike soft MR plastomers, the storage modulus of hard MR plastomers kept increasing when the magnetic field decreased and in general the hard MR plastomers presented more complex viscoelastic behavior. Becker et al. (2018) investigated the material properties and vibration of magnetic hybrid elastomer beams fabricated using elastic composite with hard and soft particles aiming to evaluate the potential of magnetic hybrid elastomers to be used in acceleration sensor systems. Hybrid magnetoactive elastomers also have been studied by Borin et al. (2019). They compared the behavior of hybrid magnetoactive elastomers under dynamic axial loading with those of the conventional elastomers. It was shown that both the passive and active state properties of the hybrid elastomer can be tuned using the pre-magnetization of the magnetically hard particles and applied external magnetic field (up to B = 240 mT), respectively.

While there are a number of studies on H-MREs and their quasi-static modeling using the microscale approach, to the best of our knowledge, no studies have been reported on characterization of H-MREs under wide range of operating conditions, particularly on the effect of excitation frequency and negative magnetic field. Considering this, the present study aims to investigate the dynamic behavior of H-MREs operating in oscillatory shear mode. For this purpose, H-MRE samples with 15% volume fraction of hard magnetic particles, NdFeB, were fabricated and their storage and loss moduli were measured performing oscillatory shear tests. The moduli are evaluated under various loading conditions including shear strain amplitude, driving frequency as well as the applied magnetic flux density, and their influence on the shear properties of the H-MRE is investigated. A phenomenological model is then developed to predict the moduli as a function of applied magnetic flux density and excitation frequency. The proposed model is then validated comparing the predicted results with those obtained experimentally.

2. Experimental characterization of H-MREs

In this section, first the procedure to fabricate the isotropic H-MRE samples is briefly explained. This is then followed by the description of the experimental test setup and experimental tests that have been conducted.

2.1. Fabrication of isotropic H-MRE samples

The H-MRE samples with 15% volume fraction of hard magnetic particles were fabricated in the laboratory by mixing hard magnetic particles and silicone rubber matrix. The hard magnetic particles used were unmagnetized NdFeB with a mean particles size of approximately 50 μm. The silicone rubber (Ecoflex 00-50; Smooth On) was used as the matrix of H-MRE. First, the NdFeB particles and the silicone rubber were mixed thoroughly for about 5 min to obtain a homogeneous mixture. The mixture was then degassed in a vacuum chamber under −28 in-Hg for about 5 min. Afterward, the mixture was poured into a transparent plexi-glass mold and left for 24 h at room temperature to be cured. Finally, the cured material was cut into 20-mm diameter cylindrical samples. The samples were then magnetized using a double-yoke adjustable electromagnet under 2.7 T magnetic flux density with applied magnetic field along the axis of the cylindrical samples. The distance between the yokes is set according to the total thickness of H-MRE specimen (2 mm). Figure 1 shows the microstructure image of the H-MRE with two different scales. It is noted that fabricated H-MRE samples are isotropic (no external magnetic field has been applied during the curing process); therefore, the hard particles (white spots in microstructure images) were distributed randomly.

2.2. Test setup

The viscoelastic properties of the H-MRE with 15% volume fraction of hard particles were measured using an advanced rotary rheometer (Discovery HR-3, TA instrument) equipped with magnetorheology accessory. The accessory is capable of applying magnetic field ranging from −1.0 to 1.0 T along the central axis of the H-MRE sample. In the present study, the applied magnetic field is addressed as positive or negative when it is in the same or opposite direction compared to the direction of the magnetic poles inside the H-MRE sample, respectively. The test sample (20-mm diameter and 2-mm thickness) was placed between the rotating and fixed parallel geometries. The samples were initially subjected to 30 N axial compression load in order to ensure that there is no slippage between the geometries and the sample, and then the gap was maintained constant during all the tests. In other words, the tests have been done under constant pre-strain corresponding to 30 N axial load. The applied magnetic field provided by the magnetorheology accessory is perpendicular to the circular cross section of the sample. The temperature of the sample was kept constant during the tests at 20°C by circulating cooling fluid through the magnetorheology accessory.

2.3. Design of experiments

The viscoelastic properties of H-MREs depend on the various operating conditions including the strain amplitude, excitation frequency, and applied magnetic flux density. In the present study, two sets of experiments were performed in order to investigate the dynamic viscoelastic behavior of H-MREs operating in different loading conditions. First, the storage and loss moduli of the H-MRE sample were evaluated by sweeping the shear strain amplitude between 0.01% and 5%, while the driving frequency was kept steady. The tests were conducted under different levels of applied magnetic flux density, and the linear viscoelastic region in which the H-MRE operates independent from the shear strain amplitude was identified. Then, the influence of the excitation frequency on the storage and loss moduli of the H-MRE was investigated through sweep frequency tests. For this purpose, the oscillatory shear test was performed at linear shear strain amplitude of 0.01% while the excitation frequency was swept from 0.1 to 50 Hz. It should be noted that all the tests have been performed under both positive and negative applied magnetic flux densities and the effect of negative magnetic field on the storage and loss moduli of the H-MRE samples was evaluated.

3. Experimental results and discussion

The obtained data were analyzed in the following sections to investigate the effect of different loading conditions including the shear strain amplitude, excitation frequency, and magnetic flux density on the viscoelastic properties of H-MREs.

3.1 Effect of shear strain amplitude

In order to investigate the influence of shear strain amplitude on the storage and loss moduli of H-MRE’s, the oscillatory shear test was performed by sweeping the strain amplitude in the range of 0.01%–5% while the driving frequency and the temperature maintained steady at 1 Hz and 20°C, respectively. The tests were implemented under various levels of positive and negative applied magnetic flux density varied from B = −0.2 T to B = 1 T to study the effect of applied magnetic field on the mechanical properties of the H-MRE. The variation of the storage modulus with respect to the shear strain amplitude is shown in Figure 2. As it can be realized, the storage modulus is independent of strain amplitude for excitations with strain amplitudes below 0.1% and further increase in the strain amplitude decreases the modulus demonstrating the nonlinear viscoelastic behavior. The region in which the storage modulus is independent of the strain amplitude is called linear viscoelastic region, and the associated strain amplitude is represented as $γ_{lin .}$ . Figure 2 also demonstrates that the storage modulus increases by increasing the applied magnetic flux density. It is noted that the rate of increase in the storage modulus decreases as the magnetic field increases due to the magnetic saturation. Moreover, unlike the soft MREs, applying magnetic field in the direction opposite to the orientation of the magnetization weakens the magnetic bonding between the hard particles and consequently reduces the storage modulus of the H-MRE. Thus, it is possible to make the H-MRE softer or stiffer by changing the direction of magnetic field. This behavior offers a unique opportunity to utilize use H-MREs in different vibration control applications in which both negative and positive variation of the stiffness is required.

Figure 3 shows the variation of the loss modulus with respect to the strain amplitude. Results show that the loss modulus is generally less sensitive to strain amplitude compared with the storage modulus. Figure 3 also shows that regardless of the orientation of the magnetization, applying magnetic field causes the loss modulus of the H-MRE to increase. Increasing the applied magnetic field in positive direction increases the loss modulus; however, this increment is very slight compared with that of storage modulus. On the other hand, the effect of negative field on the loss modulus is in contrary with that on the storage modulus as presented in Figure 2. This is mainly due to the fact that applying negative magnetic field causes the magnetized particles to rotate in the matrix which enhances the energy dissipation capacity of the H-MRE and, subsequently, its loss modulus. Moreover, it is clear in Figure 3 that the loss modulus is almost strain-independent for the strain amplitude range of 0.01%–5% regardless of the applied magnetic flux density.

3.2. Effect of excitation frequency

Sweep frequency tests have been performed in order to investigate the effect of excitation frequency on the behavior of H-MREs operating in shear mode. For this purpose, the oscillatory shear test was conducted under constant strain amplitude (0.01%), while the frequency was swept in the range of 0.1–60 Hz. The test was performed under various magnitudes of applied magnetic flux densities ranging from B = −0.2 T to B = 0.85 T in order to investigate the effect of both negative and positive magnetic field on the dynamic behavior of H-MREs. Figures 4 and 5 show the variation of the storage and loss moduli with respect to the excitation frequency for different levels of applied magnetic flux densities. Results presented in Figure 4 show that the storage modulus increases by increasing driving frequency. Results show that the storage modulus increases almost exponentially at low frequencies between 0.1 and 10 Hz (region (a)) and the rate of increase in storage modulus reduces noticeably in this region. Further examination of results reveals that the rate of increase in storage modulus becomes constant and the modulus increases almost linearly beyond 10 Hz (region (b)), so that one can assume linear variation of the storage modulus with respect to the excitation frequency in this region. This behavior could be due to the fact that by increasing the driving frequency, the measurement duration becomes less than the lifetime of the bonding between the matrix chains as well as the silicone rubber matrix and hard magnetic particles. Therefore, the effective bonding increases at higher frequencies which contributes to higher storage modulus of the H-MRE. In addition, Figure 4 shows that apart from the driving frequency, the storage modulus increases by increasing the magnetic flux density and the trend of increment is quite similar over the observed frequency range.

Figure 5 shows the variation of the loss modulus versus frequency for different magnetic flux densities. Increasing the excitation frequency rises the intermolecular heat generated inside the H-MRE sample, and consequently the energy dissipation in the H-MRE increases by increasing the excitation frequency. This phenomenon appears as the growth in the loss modulus with respect to the frequency. As shown in Figure 5, the trend of variation in loss modulus with respect to the driving frequency is similar to that of the storage modulus. The loss modulus increases almost linearly in frequencies higher than 10 Hz (region b), whereas the loss modulus increases almost exponentially for frequencies lower than 10 Hz (region (a)). With respect to the influence of the applied magnetic flux density, as discussed before unlike the storage modulus, applying magnetic field in the opposite direction as the direction of the magnetization does not reduce the loss modulus. Results show that the level of the loss modulus is almost the same for −0.2 and 0.2 T applied magnetic flux density. For instance, for 60 Hz excitation frequency, the loss modulus is 23.68 and 23.2 kPa in presence of −0.2 and 0.2 T applied magnetic flux density, respectively. Moreover, the influence of the applied magnetic field on the loss modulus is not significant compared to its effect on the storage modulus and only applying significant amount of magnetic field (e.g. 0.85 T) causes a slight enhancement in the loss modulus.

3.3. Effect of applied magnetic field

Variation of the storage modulus of the H-MRE with respect to the applied magnetic flux density is shown in Figure 6. Results are generated at various levels of shear strain amplitudes ranging from 0.01% to 5% at driving frequency of 1 Hz. Results clearly show that as expected the storage modulus increases with an increase in the applied magnetic flux density up to saturation limit of almost 0.8 T and further enhancement of the field does not noticeably affect the modulus. This can be attributed to the magnetic saturation of the particles above 0.8 T. Moreover, results again confirm that by applying magnetic field in the opposite direction to that of magnetization direction of hard particles, storage modulus reduces.

4. Model formulation

Phenomenological-based mathematical models are developed to predict the viscoelastic properties of the H-MRE with respect to the applied magnetic flux density and excitation frequency. The model is developed for H-MREs operating in shear mode at linear viscoelastic region ( $γ = 0.01 %$ ). The proposed models are capable of predicting the storage and loss moduli of H-MREs in presence of both negative and positive magnetic field (−0.2 to 0.85 T) and wide range of operating frequency (1–50 Hz).

Through examination of results presented in Figures 4 and 5, one can realize that variation of the storage and loss moduli of the H-MRE with respect to the driving frequency can be represented by power functions. This observation stems from the fact that the storage and loss moduli increase almost linearly versus excitation frequency in the logarithmic scale as shown in these figures. The mathematical models for the storage and loss moduli are, thus, proposed to be in the following form

G' (f) = a_{B} f^{b_{B}}

(1)

G ″ (f) = c_{B} f^{d_{B}}

(2)

where $G' (f)$ and $G ″ (f)$ are storage and loss moduli in Pa, respectively, and f represents the excitation frequency in Hz. $a_{B}$ , $b_{B}$ , $c_{B}$ , and $d_{B}$ are field-dependent parameters which should be determined for each applied magnetic flux density using the experimental results. For this purpose, the error function between the results obtained using the developed model and those obtained experimentally are minimized using least square method. The error function can be expressed as

J = \sum_{i = 1}^{M} {({G'}_{Model} (i) - {G'}_{Exp .} (i))}^{2}

(3)

J = \sum_{i = 1}^{M} {({G ″}_{Model} (i) - {G ″}_{Exp .} (i))}^{2}

(4)

in which subscripts $Exp .$ and Model represent the experimental results and those obtained using the developed model, respectively. For each magnetic flux density, two parameters $a_{B}$ and $b_{B}$ for storage modulus and $c_{B}$ and $d_{B}$ for loss modulus are identified through the minimization of the error functions in equations (1) and (2) and are listed in Table 1. It should be noted that the power functions in equations (1) and (2) are capable of predicting the moduli in the range of 1–50 Hz accurately.

Table 1.

Field-dependent parameters in the developed models at various applied magnetic flux densities.

Magnetic flux density (T)	$a_{B}$	$b_{B}$	$c_{B}$	$d_{B}$
–0.2	7.787e+04	0.09197	9452	0.2263
–0.1	8.069e+04	0.08805	9463	0.2229
0.0	8.421e+04	0.08321	9575	0.22
0.2	8.975e+04	0.07845	9845	0.2151
0.4	9.397e+04	0.07717	1.008e+04	0.212
0.6	9.73e+04	0.075	1.028e+04	0.2098
0.8	1.004e+05	0.07579	1.061e+04	0.2096
0.85	1.01e+05	0.07573	1.065e+04	0.2092

Figure 7 shows the variation of the obtained parameters with respect to the applied magnetic flux density. Examination of Figure 7 suggests that a second-order polynomial function can be effectively utilized to quantify these parameters as a function of applied magnetic flux density as follows

a_{B} = a_{B_{2}} B^{2} + a_{B_{1}} B + a_{B_{0}}

(5)

b_{B} = b_{B_{2}} B^{2} + b_{B_{1}} B + b_{B_{0}}

(6)

c_{B} = c_{B_{2}} B^{2} + c_{B_{1}} B + c_{B_{0}}

(7)

d_{B} = d_{B_{2}} B^{2} + d_{B_{1}} B + d_{B_{0}}

(8)

where B is the applied magnetic flux density in T. $a_{B_{0, 1, 2}}$ , $b_{B_{0, 1, 2}}$ , $c_{B_{0, 1, 2}}$ , and $d_{B_{0, 1, 2}}$ are the parameters which are identified by minimizing the error function obtained as the difference between parameters presented in Table 1 and those calculated using equations (5)–(8). Table 2 provides the identified parameters.

Table 2.

Identified parameters in equations (5)–(8).

$a_{B_{0}} = 84, 068$	$b_{B_{0}} = 0.0843$	$c_{B_{0}} = 9596.1$	$d_{B_{0}} = 0.22$
$a_{B_{1}} = 29, 271$	$b_{B_{1}} = - 0.0307$	$c_{B_{1}} = 956.6$	$d_{B_{1}} = - 0.0275$
$a_{B_{2}} = - 11, 136$	$b_{B_{2}} = 0.0249$	$c_{B_{2}} = 352.97$	$d_{B_{2}} = 0.0177$

As it can be seen in Figure 7, the proposed quadratic functions in equations (5)–(8) can accurately predict the variation of the parameters in the whole range of applied flux densities. Finally, by substituting equations (5) and (6) into equation (1) and equations (7) and (8) into equation (2) and using the parameters presented in Tables 2, the following equations are obtained for the storage and loss moduli of the H-MRE

G' (B, f) = (a_{B_{2}} B^{2} + a_{B_{1}} B + a_{B_{0}}) f^{(b_{B_{2}} B^{2} + b_{B_{1}} B + b_{B_{0}})}

(9)

G ″ (B, f) = (c_{B_{2}} B^{2} + c_{B_{1}} B + c_{B_{0}}) f^{(d_{B_{2}} B^{2} + d_{B_{1}} B + d_{B_{0}})}

(10)

In the following section, it has been shown that the proposed models can accurately predict the storage and loss moduli of the H-MRE under wide range of applied excitation frequency and magnetic flux densities.

5. Validation of the developed models

In this section, the results obtained using the developed phenomenological models are compared with those measured experimentally to evaluate the performance of the proposed models. Figures 8 and 9 show the results for the storage and loss moduli with respect to the applied magnetic flux density at various excitation frequencies, respectively. Results clearly show that the results obtained from the proposed models agree very well with those obtained experimentally and thus the proposed model can accurately predict the storage and loss moduli under different loading conditions. Comparing Figures 8 and 9 also confirms that that the dependency of the loss modulus on the applied magnetic flux density is negligible compared with the influence of the field variation on the storage modulus.

The variation of the storage and loss moduli with driving frequency are presented in Figures 10 and 11, respectively, in which the results obtained using proposed models are compared with those measured experimentally. As it can be realized, the proposed models can accurately capture the behavior of the H-MRE versus excitation frequency for both positive and negative magnetic fields. The model prediction is completely in accordance with the experimental results especially in the range of 1–20 Hz while slightly deviated from the experiential results at higher frequencies.

For further evaluation of the developed model, the coefficients of determination ( $R^{2}$ ) are also calculated as the difference between the storage or loss modulus predicted by the proposed models and those obtained from the experiments to quantify the error as

R^{2} = [1 - \frac{\sum {(G_{Model} - G_{Exp})}^{2}}{{\sum [G_{Model} - mean (G_{Exp})]}^{2}}] \times 100

(11)

where G represents the storage or loss modulus of the H-MRE. It is noted that ( $R^{2}$ ) varies between 0 and 100% in which 1 (100%) represents perfect fit. Table 3 shows the results for the coefficient of determination for different applied magnetic flux densities in the frequency range of 1–50 Hz. As it can be realized, for all cases, $R^{2}$ is generally 95% suggesting that the proposed models can accurately capture the dynamic viscoelastic behavior of the H-MRE.

Table 3.

The coefficient of determination, $R^{2}$ , for both the storage and the loss moduli under different magnetic flux densities.

Magneticfield (T)	R² (for the storagemodulus)	R² (for the lossmodulus)
B = −0.2	96.51	98.24
B = −0.1	95.80	98.79
B = 0.0	96.35	99.15
B = 0.2	97.59	99.09
B = 0.4	96.52	99.38
B = 0.6	95.84	99.45

The percentage error is also employed to find the relative error between simulation and experimental results at different applied magnetic flux density and driving frequency. It is found that for majority of cases the average percentage error between the experimental and model results is below 3%.

5. Conclusion

A fundamental study has been conducted to investigate the viscoelastic behavior of the H-MRE under wide range of excitation and applied magnetic field. H-MRE sample with 15% fraction of NdFeB particles was fabricated. Oscillatory shear tests have been performed under different loading conditions including the various shear strain amplitudes, excitation frequencies, and applied magnetic flux densities. The storage and loss moduli were measured under both positive and negative magnetic fields, and the influence of field direction and magnitude on the moduli was investigated. The results illustrated that the storage modulus of the H-MREs is a function of the magnetic field and the strain amplitude as well as the excitation frequency. Enhancing the applied magnetic flux density increases the storage modulus; however, unlike the soft MREs, applying magnetic flux density in the direction opposite to the orientation of the magnetization reduces the storage modulus of the H-MRE. It was also shown that the loss modulus is almost independent of the shear strain amplitude and applied magnetic field. In addition, the influence of driving frequency on the viscoelastic properties of the H-MREs was investigated and it was shown that both moduli are frequency dependent and increasing the excitation frequency rises the moduli. Finally, phenomenological-based mathematical models were developed to predict the linear viscoelastic behavior of the H-MRE under varying operating conditions. Results show that the proposed models can accurately predict the variation of storage and loss moduli under wide range of excitation frequencies and in presence of both positive and negative magnetic fields.

Footnotes

Acknowledgements

Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been funded by NSERC, Grant No. RGPIN/6696-2016.

ORCID iDs

Masoud Hemmatian

Ramin Sedaghati

References

Anderson

Bravoco

Hargrave

, et al. (2015) Dynamic shear response of hard versus soft magnetic magnetoactive elastomers. Smart Materials and Structures 24(2): 094003 025022.

Asadi Khanouki

Sedaghati

Hemmatian

(2019) Experimental characterization and microscale modeling of isotropic and anisotropic magnetorheological elastomers. Composites Part B: Engineering 176: 107311.

Becker

Zimmermann

Borin

, et al. (2018) Dynamic response of a sensor element made of magnetic hybrid elastomer with controllable properties. Journal of Magnetism and Magnetic Materials 449: 77–82.

Bogdanov

Borin

Stepanov

, et al. (2009) Usage of magneto-active elastomers in a bumper of a vehicle for front impact protection. Journal of Physics: Conference Series149(1): 012089.

Borin

Stepanov

Dohmen

(2019) Hybrid magnetoactive elastomer with a soft matrix and mixed powder. Archive of Applied Mechanics 89(1): 105–117.

Borin

Stepanov

Odenbach

(2013) Tuning the tensile modulus of magnetorheological elastomers with magnetically hard powder. Journal of Physics: Conference Series412(1): 012040.

Chen

Song

, et al. (2019) A spherical actuator-based hand-held haptic device for touch screen interaction. IEEE Access 7: 15125–15139.

Dargahi

Sedaghati

Rakheja

(2019) On the properties of magnetorheological elastomers in shear mode: design, fabrication and characterization. Composites Part B: Engineering 159: 269–283.

Chen

(2012) MEMS magnetometer based on magnetorheological elastomer. Measurement 45(1): 54–58.

10.

Zhang

(2011) Semi-active variable stiffness vibration control of vehicle seat suspension using an MR elastomer isolator. Smart Materials and Structures 20(10): 105003.

11.

Farshad

Le Roux

(2004) A new active noise abatement barrier system. Polymer Testing 23(7): 855–860.

12.

Ginder

Nichols

Elie

, et al. (1999) Magnetorheological elastomers: properties and applications. Smart Structures and Materials 3675: 131–138.

13.

Jonsdottir

Gudmundsson

, et al. (2015) Preparation and characterization of a prototype magnetorheological elastomer for application in prosthetic devices. In: 7th ECCOMAS Thematic Conference on Smart Structures and Materials, Ponta Delgada, Portugal, 3–6 June.

14.

Kalina

Brummund

Metsch

, et al. (2017) Modeling of magnetic hystereses in soft MREs filled with NdFeB particles. Smart Materials and Structures 26(10): 105019.

15.

Koo

Dawson

Jung

(2012) Characterization of actuation properties of magnetorheological elastomers with embedded hard magnetic particles. Journal of Intelligent Material Systems and Structures 23(9): 1049–1054.

16.

Kramarenko

Chertovich

Stepanov

, et al. (2015) Magnetic and viscoelastic response of elastomers with hard magnetic filler. Smart Materials and Structures 24(3): 035002.

17.

Lee

Jung

, et al. (2014) Modulus-tunable magnetorheological elastomer microcantilevers. Smart Materials and Structures 23(5): 055017.

18.

, et al. (2013) Development and characterization of a magnetorheological elastomer based adaptive seismic isolator. Smart Materials and Structures 22(3): 035005.

19.

Qiao

Watanabe

, et al. (2012) Mechanical and structural investigation of isotropic and anisotropic thermoplastic magnetorheological elastomer composites based on poly (styrene-b-ethylene-co-butylene-b-styrene) (SEBS). Rheologica Acta 51(1): 37–50.

20.

Qian

Xin

Bai

, et al. (2017) State observation–based control algorithm for dynamic vibration absorbing systems featuring magnetorheological elastomers: principle and analysis. Journal of Intelligent Material Systems and Structures 28(18): 2539–2556.

21.

Stepanov

Borin

Storozhenko

(2017a) Rotation of magnetic particles inside the polymer matrix of magnetoactive elastomers with a hard magnetic filler. Journal of Magnetism and Magnetic Materials 431: 138–140.

22.

Stepanov

Borin

Bakhtiiarov

, et al. (2017b) Magnetic properties of hybrid elastomers with magnetically hard fillers: rotation of particles. Smart Materials and Structures 26(3): 035060.

23.

Stepanov

Borin

Kramarenko

, et al. (2014) Magnetoactive elastomer based on magnetically hard filler: synthesis and study of viscoelastic and damping properties. Polymer Science Series A 56(5): 603–613.

24.

Stepanov

Chertovich

Kramarenko

(2012) Magnetorheological and deformation properties of magnetically controlled elastomers with hard magnetic filler. Journal of Magnetism and Magnetic Materials 324(21): 3448–3451.

25.

Sun

Yang

Deng

, et al. (2015) Horizontal vibration reduction of a seat suspension using negative changing stiffness magnetorheological elastomer isolators. International Journal of Vehicle Design 68(1–3): 104–118.

26.

Wang

Gordaninejad

Calgar

, et al. (2009) Sensing behavior of magnetorheological elastomers. Journal of Mechanical Design 131(9): 091004.

27.

Wen

Wang

Gong

(2017) The magnetic field dependent dynamic properties of magnetorheological elastomers based on hard magnetic particles. Smart Materials and Structures 26(7): 075012.

28.

Woods

BKS

Wereley

Hoffmaster

, et al. (2007) Manufacture of bulk magnetorheological elastomers using vacuum assisted resin transfer molding. International Journal of Modern Physics B 21(28–29): 5010–5017.

29.

Yang

Sun

, et al. (2014) A novel magnetorheological elastomer isolator with negative changing stiffness for vibration reduction. Smart Materials and Structures 23(10): 105023.

30.

Ying

(2013) Stochastic micro-vibration suppression of a sandwich plate using a magneto-rheological visco-elastomer core. Smart Materials and Structures 23(2): 025019.

31.

Zhao

Pang

Gong

(2017) Novel magnetorheological plastomer filled with NdFeB particles: preparation, characterization, and magnetic–mechanic coupling properties. Industrial & Engineering Chemistry Research 56(31): 8857–8863.

32.

Zhou

Wang

(2005) Magnetorheological elastomer-based smart sandwich beams with nonconductive skins. Smart Materials and Structures 14(5): 1001.