A novel approach to characterize the magnetic field and frequency dependent dynamic properties of magnetorheological elastomer for torsional loading conditions

Journal Pre-proofs A novel approach to characterize the magnetic field and frequency dependent dynamic properties of magnetorheological elastomer for torsional loading conditions K. Praveen Shenoy, Umanath Poojary, K.V. Gangadharan PII: DOI: Reference:

S0304-8853(19)33293-7 https://doi.org/10.1016/j.jmmm.2019.166169 MAGMA 166169

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

17 September 2019 15 November 2019 16 November 2019

Please cite this article as: K.P. Shenoy, U. Poojary, K.V. Gangadharan, A novel approach to characterize the magnetic field and frequency dependent dynamic properties of magnetorheological elastomer for torsional loading conditions, Journal of Magnetism and Magnetic Materials (2019), doi: https://doi.org/10.1016/j.jmmm.2019.166169

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A novel approach to characterize the magnetic field and frequency dependent dynamic properties of magnetorheological elastomer for torsional loading conditions Praveen Shenoy K1,3*, Umanath Poojary2 and K.V. Gangadharan1,3 1Department

of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, Mangalore575025, India 2 Department of Mechanical and Manufacturing Engineering, Manipal Institute of Technology, Manipal-576104, India 3 Centre for System Design (CSD), National Institute of Technology Karnataka, Surathkal, Mangalore-575025, India *Corresponding author email: [email protected]

Abstract Magnetorheological elastomers (MRE) are potential resilient elements to improve the operating frequency range of a vibration isolator. The field-dependent characterization of MRE properties for varying input frequencies under lateral shear conditions has been well researched in past studies. In the present study, a novel approach to assess the magnetic field dependent rheological properties of magnetorheological elastomers under dynamic torsional loading is presented. Field and frequencydependent properties are estimated from the dynamic blocked transfer stiffness method specified by ISO 10846. Viscoelastic properties represented in-terms of complex torsional stiffness and loss factor are estimated from the Lissajous curves within the linear viscoelastic (LVE) limit. Experiments are performed at a frequency range of 10Hz to 30Hz under a constant input angular displacement. Magnetic field sensitive characteristics of MRE are evaluated under the field produced by a custommade electromagnet. The results reveal a strong influence of field dependent variations on the complex stiffness in comparison with the input frequency. Variations observed in the loss factor suggests a dominance of the imaginary part of the complex stiffness on the energy dissipation. The reduced field induced enhancements in the complex stiffness are interpreted from the Magneto-static and structural based numerical simulations using ANSYS 19.1.

Keywords:

Magnetorheological elastomer, Torsional shear, Blocked transfer stiffness method,

Lissajous plots, Complex stiffness, Loss factor

1

1.

Introduction

Magnetorheological elastomers (MRE) belong to a group of magnetically sensitive smart composites exhibiting field dependent characteristics [1]. MRE offers advantages over its fluid counterpart, Magnetorheological Fluid (MRF), particularly on the sedimentation and leakage issues [2]. The structure of the MRE consists of micro to nano-sized ferromagnetic fillers dispersed in a viscoelastic non-magnetic matrix medium. In the presence of the magnetic field, the filler particles realign themselves which results in the variation of the viscoelastic properties. The field sensitive characteristics of the MRE can be employed in the field of semi-active and active vibration isolation applications [3] [4] [5] [6]. The properties of the MRE depend on the nature of the constituent materials primarily the shape and size of the filler material, volume percentages of the matrix-filler composition, hardness of the viscoelastic material and the addition of plasticizers. Particle size predominantly affects the properties of the MRE due to its nature of bonding. Smaller sized fillers have a better matrix reinforcing effect compared to larger fillers due to agglomeration of particles [7]; largely seen in nanosized fillers [8]. With the increase in the particle size, the matrix-filler interaction dominates over the particle-particle interaction. Different ferromagnetic powders ranging from 5 to 100 m in diameter have been used as filler materials. Literature suggests that Carbonyl Iron Particles (CIP), with the average diameter in the range of 1 - 9 μm, form the ideal filler materials. The optimum particle volume concentration is in the range of 25-30% [9] [10]. Viscoelastic matrices include natural rubber and a wide range of synthetic rubbers such as nitrile and butyl have been used as the non-magnetic base of the MRE [7]. Of the available silicone rubbers, Room Temperature Vulcanized (RTV) based silicone rubber has been extensively used for its advantages over the natural rubber, primarily being its ease of handling and range of operating temperature. Depending on the stiffness provided by the silicone matrix, the silicone rubber is categorized as “Hard” and “Soft”. Hard matrix offers higher zero-field stiffness, but reduced damping capability compared to the soft matrix. Soft matrix offers a higher field-induced relative stiffness, however at the expense of reduced load-bearing capacity and reduced fatigue life [11] [12]. The selection of the suitable matrix largely depends upon the MRE applications. Further, the rheological properties of the MRE are also dependent on the addition of additives such as carbon black, carbon nanotubes, graphene and silicon carbide further influence [13][14]. Further, the effect of the application of magnetic field during the curing process has

also been well explored in past studies [15][16][17]. Apart from its constituents and curing processes, MRE is classified based on the application of load, as shear mode [18], squeeze mode [19] or the mixed mode [20]. Apart from the field dependent variations, the properties of MRE are inherently dependent on operating frequency, input strain and operating temperature due to its viscoelastic matrix [21]. The effect of magnetic field on the performance of MRE has been extensively investigated for lateral shear 2

loading conditions. Due to the distinctive interaction between the iron particles and the matrix, the field dependent behaviour of the MRE is prominent. However, the effect is less pronounced beyond 0.4T [22]. Frequency-dependent study of the MRE reveals noticeable changes in the complex moduli of the MRE. Past studies show a visible change in the dynamic lateral shear properties of the MRE for a frequency up to 60 Hz [23] for a given configuration of MRE isolator. The strain dependence on viscoelastic materials has been well explored for lateral shear loading conditions. With the addition of filler materials, the strain dependency is significant under the effect of the magnetic field. Predominant variations in the dynamic properties, specifically in the complex stiffness are observed at higher amplitudes of displacement. Higher input displacement further results in the non-linear behaviour of the MRE [24]. Further, with MRE operating at higher amplitudes, the problem of delamination occurs [25]. Property characterization of the viscoelastic materials is carried out either by static tests or dynamic tests. Static tests represent the elastomers as “High Elastic” rather than the viscoelastic nature which is portrayed in dynamic tests [26]. The rheological properties under dynamic loading conditions are comprehended using the transient response method or the dynamic method [27]. Transient method involves obtaining the decaying output response for a given instantaneous input. The dynamic method measures the periodic response for a given harmonic input. Depending upon the region of operation, the dynamic method technique is further categorised as resonant and non-resonant methods [21]. Rheological property characterization using non-resonant method is carried out using Rheometer, DMA or through forced vibration tests. Rheometer [28] [29] and DMA [14] [15] are used widely in characterization using either controlled strain or controlled stress input. However, the above techniques are devoid of the geometrical aspects of the MRE and are used to understand the material properties of the elastomer. In general, for the characterization of the MRE isolator which involves evaluation of the geometry-dependent complex properties, the specimen is subjected to the forced vibration tests which are based on ISO 10846-1 [30]. This defines the testing procedures for vibroacoustic properties of resilient materials. ISO 10846 is further subdivided into direct stiffness method (ISO 10846-2) and indirect stiffness method (ISO 10846-3). The direct stiffness method which the input displacement, velocity or the acceleration with the response force, while the indirect stiffness method measures the transmissibility response of the resilient material. The direct method further categorized into drive point method and blocked transfer method [31]. Drive point method measures the force on the input side due to which the influence of the inertial component at higher frequencies is observed. Though the effect can be overcome with a reduced payload, the results are significantly affected by frequencies as low as 30Hz [32]. Blocked transfer method overcomes this issue by measuring the force at the blocked end, which is devoid of the effect of the inertial component. Further, the property estimation from the blocked transfer method is carried out either by the receptance method [33], transmissibility plot [34] or the Lissajous plots [26]. The Lissajous plots, also

3

known as Hysteresis curves offer the real-time indication of the viscoelastic response and a simple technique to obtain the dynamic stiffness and loss factor. In the blocked transfer stiffness method, the fixed test specimen is subjected to a known harmonic displacement. One end of the specimen is rigidly fixed and the other end is subjected to a known lateral displacement. The Lissajous curves are obtained from the input displacement and the output force. From the plots, the variations in the complex stiffness and loss factor of the elastomer are assessed. Though the existing blocked transfer stiffness method is well suited for lateral shear, the rheological properties for torsional shear cannot be effectively characterized. In the present work, an experimental approach to characterize the dynamic properties of an RTV based isotropic MRE under torsional shear is studied. A novel test setup is developed for the characterization of the torsional properties based on the ISO 10846-2 standard. The MRE specimen is subjected to a known harmonic angular displacement and the blocked torque is measured at the fixed end (Figure 1 a). The along-theaxis magnetic field, perpendicular to the test specimen is generated using a specially developed electromagnet. Magnetic field and frequency dependence are studied for torsional frequencies in the range of 10 Hz to 30 Hz from the obtained Lissajous plots (Figure 1 b).

Figure 1: (a) Blocked transfer method for torsional shear (b) Lissajous plots for blocked torque and angular displacement

2. 2.1

Experimental Complex torsional stiffness

Under dynamic loading conditions, the input angular displacement for a frequency ω is given as, 𝜃(𝑡) = 𝜃0sin 𝜔𝑡

(1)

The corresponding torque generated at the output end is also harmonic in nature. For a perfectly elastic element, the generated torque is in-phase with the input angular displacement. However, for a viscoelastic material, the torque lags the displacement by an angle φ as represented in Figure 2 and is given by,

4

𝑇𝑜𝑢𝑡(𝑡) = 𝑇0sin (𝜔𝑡 + 𝜑)

(2)

The response of a viscoelastic material, under harmonic loading, is represented by the complex stiffness which represents the elastic and the loss modulus given by, 𝐾𝑡 ∗ = (𝐾𝑡′ + 𝑖𝐾𝑡")

(3)

The in-phase component, 𝐾𝑡′ represents the torsional stiffness of the MRE (real part of the complex stiffness) and the out-of-phase component, 𝐾𝑡" represents the energy dissipation of the MRE (imaginary part of the complex stiffness). The vector plot of the complex stiffness is represented as shown in Figure 2 where the in phase and out of phase components are emphasized. The input displacement (Equation 1) and the output torque (Equation 2) are used to obtain the Lissajous curves. As depicted in Figure 1 b, the slope of the Lissajous curves represents the complex torsional stiffness which by definition is the ratio of the blocked response torque at the output to the input angular displacement. 𝐾𝑡 ∗ =

𝑇0 𝜃0

(4)

From the complex stiffness 𝐾𝑡 ∗ , the stiffness 𝐾𝑡′ is obtained as,

𝐾𝑡′ =

2.2

𝑇𝑎 𝜃0

(5)

Loss Factor

The loss factor for a viscoelastic member is defined as the ratio of the imaginary part to the real part of the dynamic complex stiffness. It is the measure of energy lost to the energy stored by the material. For a viscoelastic material, the loss factor is obtained by the Lissajous plots and is given as,

𝜂=

𝐾𝑡" 𝐾𝑡′

(6)

From Figure 1 b, the loss factor [26] is also expressed as,

𝜂=

5

𝑇𝑏 𝑇𝑎

(7)

Figure 2: (a) Phase lag between applied angular displacement and the response torque (b) Vector plot of the real and imaginary component of the complex stiffness

2.3

Preparation of isotropic MRE

Carbonyl Iron particles (BASF, type CN; average diameter 5 μm) and a two-part RTV Silicone rubber (MoldSil 102LL, with CAT 9, Performance polymers) in the volume fraction ratio of 27:73, form the constituents of MRE. Silicone oil (10% by volume of the silicone matrix) is added to the above constituent mixture. The mixture is then poured into the mould followed by degasification in a desiccator to remove the air bubbles, to maintain the magnetic permeability of the MRE. The mixture is then cured for 24 hours under constant pressure at room temperature [35].

2.4

Experimental Setup

The experimental setup to characterize the rheological properties of the MRE under torsion is as shown in Figure 3. Figure 3(a-c) shows the schematic of the setup and Figure 3(e-g) shows the actual setup. It consists of a circular cross-sectional MRE sample (ø 50mm X 10mm thick) sandwiched between two aluminium shafts of negligible rotational inertia. One of the aluminium shafts (input shaft) is connected to the shaker and the other (output shaft) is blocked at the output end. Silicone rubber adhesive (Sil-Poxy), is used to bond the MRE to the two aluminium shafts. A constant harmonic input is given to the input shaft by the shaker (YMC MS-100) through a function generator (AGILENT 33220A) and an amplifier (YMC LA-200 Power amplifier) arrangement. A single-axis IEPE accelerometer (YMC 121A) is used to measure the input acceleration signals. The acceleration signals are converted to corresponding linear displacement using data acquisition methods [36]. Further, the linear displacement is converted to the angular displacement of the MRE. As shown, the input shaft is connected to the shaker via a force transducer (KISTLER 9712) attached to the stinger.

6

A second force transducer (KISTLER 9712) of identical specifications measures the blocked force at the output. The torque on the input and output side is calculated as the product of the force measured by the transducer and the distance of the force transducer from the shaft. To maintain geometrical similarity in the measurements, both the transducers are positioned equidistant from the axis of the shaft. Pure torsion is ensured on the shaft by maintaining the shafts and the MRE sample in-line. The signals are acquired via a NI 9234 data acquisition unit. A sampling rate of 25.6 kS/s is used. The acquired data is processed using LabVIEW 2014. The magnetic field is generated on the MRE sample using a custom in-house made electromagnet. Finite Element Method Magnetics (FEMM) is used to predict the 2D, homogenous magnetic field distributions across the MRE specimen. To enhance the field intensity across the MRE sample, an external ferromagnetic strip is provided to the electromagnet core. Figure 4 shows the contour plot of the FEMM predicted homogenous field distribution across the diameter of the MR elastomer. The magnetic field is measured using a Gaussmeter (Lake Shore 410). Figure 4 also shows the field generated for various input current. It is observed that a maximum magnetic field of 0.3T is obtained for a current of 5.5A The characteristics of the MRE sample are studied for a frequency range of 10 to 30 Hz in steps of 5 Hz. To exclude the influence of strain on the measurements, the experiments are conducted at a constant angular displacement of 0.008 rad. The influence of the magnetic field is studied by varying the input current to the electromagnet at various proportions (0A, 1A, 2A, 3A, 4A, 5A). To ensure consistency in the results, the readings are taken for a 10 sec time interval, once the results acquire steady state. To account for the Mullin’s effect seen in viscoelastic materials [37], the initial harmonic cycles are neglected. The steady-state response of the results is observed by taking 10 cycles. Each set of experiments is repeated 10 times and the average of the results are plotted. Between each trial, a recovery period is maintained to exclude the deformation history of the cyclic loading. All the tests are conducted at room temperature to exclude the effect of temperature.

7

Figure 3: (a) Block diagram of the experimental setup (b) Blocked end (c) Input end (d) Blocked transfer stiffness method for torsional systems (e) Experimental setup (f) Input and output force transducer g) MRE sandwiched between the input and output shaft

8

(a)

(b)

(c) Figure 4: (a) Electromagnet and magnetic path arrangement (b) Predicted field generated using FEMM and (c) actual field generated across the developed electromagnet

3. 3.1

Results and Discussions Blocked Torque-Input angular displacement relationship

Figure 5 shows the variations in the output torque at 0A, 2A and 5A at the input frequency of 10Hz. It is observed that the output torque increases with an increase in the input current. For 0A, the maximum amplitude of torque observed is 0.0409 N-m. With an increase in the magnetic field, the amplitude increases and a maximum value of 0.0449 N-m is obtained at 5A. Similar trend is observed for the rest of the tested frequencies (Figure 6). It is seen that the obtained Lissajous curves are a symmetrical ellipse, which signifies that the MRE is within the Linear Viscoelastic Limit [38]. Further, with the increase in the field, the Lissajous curves show an increase in the slope as observed in the inset of Figure 6. For a viscoelastic material like MRE, the slope of the Lissajous curves represent the complex stiffness. The variations in the complex stiffness highlight the field induced stiffening of the MRE (Sec. 3.2). Also, the complex stiffness is affected by the input frequency (Sec. 3.3). 9

Further, the area under the Lissajous curves represents the energy dissipation capacity of the MRE. it is observed that the area of the Lissajous plots vary with the input parameters. These variations in the energy dissipation capacity are also expressed in terms of Loss factor which is the ratio of the energy dissipating term (Kt”) to the stiffness term (Kt’). The field dependent and frequency dependent variations in the loss factor as explained in sec 3.4.

Figure 5: Output torque variations with varying input current

10

Figure 6: Variations in field dependent Lissajous curves for different input frequencies (a) 10Hz (b) 15Hz (c) 20Hz (d) 25Hz (e) 30Hz at constant angular displacement of 0.008 rad

3.2

Complex stiffness variations with the magnetic field

The complex stiffness values are calculated from the Lissajous curves (Figure 6 a-e) as per equation 4 and the corresponding values are plotted in Figure 7 and are listed in Table 1 for varying input fields and frequencies. It is observed that the complex stiffness of the MRE sample increases with an increase in the magnetic field. A similar trend is observed in the complex stiffness values of MRE for translatory shear conditions[39]. This enhancement in the complex stiffness across varying magnetic fields is expressed using the MR effect. MR effect is defined as, the percentage ratio of the difference in complex stiffness between a given magnetic field and zero-field to the complex stiffness at zerofield. The dynamic torsional stiffness at 10Hz, 0A is found to be 5.194 N-m/rad. Corresponding value at 5A is 5.65 N-m/rad with an overall enhancement of 8.87%. A similar trend is observed for other frequencies.

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Figure 7: Field dependent complex stiffness values for different frequencies Table 1: Complex stiffness for varying current and absolute MR effect variations 𝐾𝑡 ∗ (Hz)

Absolute MR Effect

(N.m rad-1)

Frequency

(

)

𝐾 ∗ 𝑡5 ― 𝐾 ∗ 𝑡0

0A

1A

2A

3A

4A

5A

(0T)

(0.06T)

(0.12T)

(0.18T)

(0.21T)

(0.28T)

10

5.194

5.316

5.419

5.518

5.581

5.655

8.87 %

15

5.277

5.424

5.524

5.61

5.698

5.741

8.79 %

20

5.353

5.466

5.564

5.653

5.721

5.795

8.26 %

25

5.419

5.481

5.576

5.677

5.732

5.812

7.25 %

30

5.437

5.613

5.701

5.761

5.821

5.871

7.98 %

𝐾 ∗ 𝑡0

× 100

Increase in the complex stiffness is attributed to the field enhanced interactions between the ferromagnetic fillers. In the absence of a magnetic field, the particles are embedded into the matrix due to which the zero-field stiffness of the elastomer is increased in comparison with the unfilled elastomer. In the presence of magnetic field, the iron particles act as magnetic poles, which further enhances the filler-filler interactions. This force of attraction between the iron particles causes a localized compression of the matrix. Figure 8 (a) shows the SEM image of the MRE with CIP entrapped between the silicone matrix under zero magnetic fields. Figure 8 (b) shows the enlarged view of a typical cross-section of the elastomer in the absence and presence of the magnetic field under an external strain. The iron

12

particles displace with a change in separation distance (r0 to r). In the presence of an external magnetic field, the interaction energy between two iron particles of radius ‘a’, is given as [40],

𝐸12 =

[

]

𝑚1𝑚2 3 1 ― 5(𝑚1𝑟)(𝑚2𝑟) 4𝜋𝜇0𝜇1 𝑟3 𝑟

(8)

Where 

m1 and m2 represent the dipole moment of the two iron particles



r0 and r represents the separation distance between the two particles without and with the presence of field



µ0 and µ1 represent the permeability of vacuum and relative permeability of the elastomer.

Further, the force between the two particles (F) is given as, 𝐹=

𝑑𝐸 sin 𝛼 𝑑𝑟

(9)

Where 𝛼 is the angle between the displaced position with the un-displaced position of the CIP. Assuming constant dipole moments, 𝑚 = 𝑚1 = 𝑚2. Hence, (10)

(5𝑟02 ― 𝑟2)𝑚2

3 ⟹𝐹 = 𝐶 2

𝑟2 ― 𝑟02

𝜋𝜇0𝜇1𝑟7

Where 𝐶 = 1.2 for a larger number of filler particles

Figure 8: (a) SEM image of isotropic MRE (b) Dipole interactions during torsional shear The effect of magnetic force on the embedded matrix is further substantiated through the magnetostatic and static structural numerical simulation. Figure 9 depicts the 2D model developed in ANSYS 19.1, APDL. A unit cell of MRE with iron particles of diameter 4µm and a separation 13

distance of 6µm is considered. The magnetic field is induced between the poles using two permanent magnets (Alnico; Coercivity 109300 A/m). To focus the field lines on the iron particles, a C-shaped path is generated using a permeable material. The magnetic force generated in unstrained and strained cases of the iron particles is obtained from the magneto-static analysis. The magnetic force generated is further used to obtain the mechanical strain and displacement on the iron particles by performing static structural analysis.

Figure 9: 2D model of the setup The simulation results of a unit cell in the unconstrained and constrained conditions are shown in Figure 10. In the unconstrained position of the MRE, the iron particles are aligned in the direction of the magnetic field. The magnetic flux generated exerts a force of attraction between the iron particles (Figure 10, a). In the presence of an external torque, the iron particles attain new positions which increase the separation distance between the dipoles. This results in the reduction of magnetic force on the iron particles as observed from the magnetic force contour plots (Figure 10 b). The developed magnetic force of the iron particles exerts a compressive force on the embedded matrix which results in the increase of the localized complex stiffness of the MRE. Figure 10 c shows the mechanical strain of the iron particles which indicates the compression caused by the iron particles on the matrix. It is observed that the absolute variations in the MR effect are limited to a maximum of 8.87%. The variations are attributed to the influence of the thickness of the MRE sample on the intensity of the magnetic field. The torsional stiffness, 𝐾𝑡, for the elastomer sample of thickness t, radius r and shear modulus G, is given as [41], 𝐾𝑡 =

14

𝜋𝐺𝑟4 2𝑡

(11)

In the absence of a magnetic field, the stiffness of the sample for the unstrained and strained conditions is dependent on the geometrical parameters as represented in the above equation. Under the influence of magnetic field, the complex stiffness increases with an increase in the magnetic field. However, this increase is inversely dependent on the thickness of the MRE. To comprehend the influence of the sample thickness on the magnetic force, two samples of variable thickness are considered. In both the cases, the separation distance between the MRE sample and the permanent magnets is maintained at 2 mm on either side. The increase in sample thickness is represented by the increase in the number of iron particles as shown in Figure 10 d. The simulation results shown in Figure 10 d suggest that, for a fixed distance between the sample and the magnetic poles, the magnetic force generated on the individual iron particles reduce with the increase in thickness of the sample. The reduced magnetic force signifies the reduction in the displacement of the iron particles in comparison with the thinner samples as was shown in Figure 10 c. Due to this, the interaction energy (Equation 8) between the dipole decreases and this leads to the reduction of the compressive force exerted on the embedded matrix. This further leads to the reduced MR effect of the test sample. Thus, the overall effect of the geometry and the intensity of the magnetic field influence the field induced variations on the MRE. This can be addressed by increasing the magnetic field across the MRE. Though this does increase the MR effect, however, the relative shift in the field dependent absolute stiffness reduces with increase in the thickness of the MRE [42]. Further, from Table 1, it is also observed that the trend remains the same for all the tested frequencies.

15

Figure 10: Magnetic force generated on iron particles in (a) unstrained condition and (b) strained condition; (c) Mechanical strain on iron particles under strained condition; (d) Influence of increase in thickness on the magnetic force of the particles

3.3

Complex stiffness variations with frequency

Figure 11 shows the Lissajous plots of blocked torque and angular displacement for 10 Hz, 20 Hz and 30 Hz at 0A and 5A respectively. The plots reveal a small shift in the slope with an increase in the frequency from 10 Hz to 30 Hz. This is further illustrated by the maximum amplitude of torque as shown in Figure 5. At 0A magnetic field and 10 Hz frequency, the maximum torque experienced by the MRE is 0.0409Nm. Corresponding value at 30 Hz frequency is 0.0436 Nm, with an increase in torque of 6.67%. The shift in the complex stiffness is attributed to the frequency dependent characteristics of the viscoelastic material under dynamic loading conditions. MRE, by virtue of its polymer matrix, exhibits frequency dependent characteristics [43]. At lower frequencies, the polymer chain molecules have adequate time to return to their original positions, allowing it to behave like viscous material. At higher frequencies, these polymer chains have less time to relax, consequently, the elastic part dominates over the viscous part. The overall effect is visible in terms of enhancement in the stiffness with the input frequency. The addition of filler into the matrix, however, induces a composite behaviour into the MRE sample which increases the resistance to the cyclic loading at lower frequencies. The micro-nano sized particles embedded within the matrix results in the coexistence of filler-matrix interaction alongside the matrix-matrix interaction. This enhances the stiffness of the elastomer at lower frequencies due to which the increase 16

in the frequency dependent increment in the complex stiffness is not as dominant as seen in unfilled elastomers. The frequency dependent stiffness variation is evaluated in the form of the absolute enhancement in the torsional stiffness from 10 Hz to 30 Hz and is given as, Absolute enhancement in the frequency dependent stiffness

=

(

) × 100

𝐾 ∗ 𝑡(30Hz) ― 𝐾 ∗ 𝑡(10Hz) 𝐾 ∗ 𝑡(30Hz)

(12)

As evident from Table 1, it is observed that, under the absence of magnetic field, the absolute enhancement in the frequency dependent stiffness from 10 Hz to 30 Hz is restricted to 4.68%. It is also seen that, with an increase in the magnetic field to 5A, the enhancement observed is 3.82%. Further, it is observed that the frequency induced enhancement of the complex stiffness is relatively less when compared with the field induced variations seen in the stiffness. While the maximum shift in the field dependent complex stiffness observed is 8.87%, the frequency dependent enhancement is limited to 5.59%. This shows the weak frequency dependent enhancement in the complex stiffness of the MRE in the absence as well as in the presence of a magnetic field.

Figure 11: Frequency dependent complex torsional stiffness variations at (a) 0A (0T) and (b) 5A (0.28T)

3.4

Loss factor variations with field and frequency The loss factor represents the energy dissipation capacity of the material. The variations in the

loss factor of the MRE under torsional loading at different input current and excitation frequencies are as shown in Figure 12. As evident from the graph, the loss factor increases with an increase in the frequency [38]. However, the field induced variations are not significant on the variations of the loss factor. At 10 Hz, the loss factor varies from 0.238 to 0.258 with an increase in input current from 0A to 5A. Correspondingly, for 30 Hz, the loss factor varies from 0.299 to 0.316. This typical characteristic of the MRE is attributed to the field dependent variations observed in the storage stiffness and loss stiffness of the complex stiffness.

17

Figure 12: Loss factor under different magnetic field and frequency As depicted in Figure 13, the storage stiffness (Kt’) and the loss stiffness (Kt”) enhance with the increase in the magnetic field. However, the relative enhancements vary with the intensity of the input current. It is observed that for the 10 Hz input frequency, the absolute enhancement of the Kt’ is 7.84% with an increase in the magnetic field from 0A to 5A. Similarly, it is observed that the loss stiffness, Kt” has an absolute increment of 15.06%. Consequently, these variations influence the field dependent variations in the loss factor.

Figure 13: Field dependent variations in (a) Kt' (b) Kt"

4.

Conclusions

The study presented a novel approach to the evaluation of the dynamic stiffness of MRE under torsional loading conditions for varying magnetic fields and varying frequencies. Tests were carried 18

out for frequencies in the range of 10 Hz to 30Hz. A custom-made in-house built electromagnet was developed to provide the in-line magnetic fields perpendicular to the torsional shear of the MRE. An experimental setup based on the blocked transfer stiffness method was developed to characterize the MRE under torsional shear. The symmetrical shape in the ellipse of the Lissajous plots suggests that the MRE is operated within the LVE limit. The dynamic viscoelastic properties of MRE under torsion are sensitive to changes observed in the magnetic field and input frequency. Magnetic field dependence is stronger compared to the frequency dependent characteristics. The field sensitive characteristics expressed by the absolute MR effect increased by 8.87%. Compared to the rheometric studies, the field induced enhancements are lower as the thickness of MRE plays a prominent role. With an increase in the sample thickness, the intensity of the magnetic field for a fixed sample-pole distance is affected and the increase in the torsional stiffness of the MRE is reduced. The cumulative contributions from the geometry and the magnetic field define the overall property enhancements of MRE. Numerical simulations were carried out to justify the same. Further, the field sensitive properties can be enhanced by a stronger magnetic field which could create an increased interaction between the fillers embedded in the matrix. The ferromagnetic fillers inside the matrix diminish the frequency dependent characteristics of MRE and the present MRE sample could show a maximum enhancement of 5.59% stiffness for the frequency variation from 10Hz to 30Hz. Variations in the loss factor suggest the dominance of the field-induced changes in the imaginary part of the complex stiffness which explains its variations with the magnetic field.

Acknowledgements The authors acknowledge the support from SOLVE: The Virtual Lab @ NITK (Grant number: No.F.16-35/2009-DL, Ministry of Human Resources Development) and experimental facility provided by Centre for System Design (CSD): A Centre of Excellence (http://csd.nitk.ac.in/) at National Institute of Technology Karnataka, India. The authors also wish to acknowledge Mr Susheel Kumar, CSD, for his valuable inputs.

Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Highlight Magnetorheological elastomers (MRE) are potential resilient elements to improve the operating frequency range of a vibration isolator. The field-dependent characterization of MRE properties for varying input frequencies under lateral shear conditions has been well researched in past studies. In the present study, a novel approach to assess the magnetic field dependent rheological properties of magnetorheological elastomers under dynamic torsional loading is presented. Field and frequency-dependent properties are estimated from the dynamic blocked transfer stiffness method specified by ISO 10846. Viscoelastic properties represented in-terms of complex torsional stiffness and loss factor are estimated from the Lissajous curves within the linear viscoelastic (LVE) limit. Experiments are performed at a frequency range of 10Hz to 30Hz under a constant input angular displacement. Magnetic field sensitive characteristics of MRE are evaluated under the field produced by a custommade electromagnet. The results reveal a strong influence of field dependent variations on the complex stiffness in comparison with the input frequency. Variations observed in the loss factor suggests a dominance of the imaginary part of the complex stiffness on the energy dissipation. The reduced field induced enhancements in the complex stiffness are interpreted from the Magnetostatic and structural based numerical simulations using ANSYS 19.1.

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