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Experimental characterization and viscoelastic modeling of isotropic and anisotropic magnetorheological elastomers
Polymer Testing 81 (2020) 106272
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Material Properties
Experimental characterization and viscoelastic modeling of isotropic and anisotropic magnetorheological elastomers �, Bohdana Marvalova � Tran Huu Nam *, Iva Petríkova Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Liberec, Studentsk� a 1402/2, 461 17 Liberec 1, Czech Republic
A R T I C L E I N F O
A B S T R A C T
Keywords: Magnetorheological elastomers Viscoelastic modeling Dynamic properties Double-lap shear test
The paper presents experimental research and numerical modeling of dynamic properties of magnetorheological elastomers (MREs). Isotropic and anisotropic MREs have been prepared based on silicone matrix filled by microsized carbonyl iron particles. Dynamic properties of the isotropic and anisotropic MREs were determined using double-lap shear test under harmonic loading in the displacement control mode. Effects of excitation frequency, strain amplitude, and magnetic field intensity on the dynamic properties of the MREs were examined. Dynamic moduli of the MREs decreased with increasing the strain amplitude of applied harmonic load. The dynamic moduli and damping properties of the MREs increased with increasing the frequency and magnetic flux density. The anisotropic MREs showed higher dynamic moduli and magnetorheological (MR) effect than those of the isotropic ones. The MR effect of the MREs increased with the rise of the magnetic flux density. The dependence of dynamic moduli and loss factor on the frequency and magnetic flux density was numerically studied using fourparameter fractional derivative viscoelastic model. The model was fitted well to experimental data for both isotropic and anisotropic MREs. The fitting of dynamic moduli and loss factor for the isotropic and anisotropic MREs is in good agreement with experimental results.
1. Introduction Magnetorheological elastomers (MREs) are considered as smart composites. Such composites are composed of micro-sized magnetic particles dispersed in a non-magnetic elastomeric matrix. MREs have attracted great interest, because their mechanical and rheological properties can be controlled rapidly and reversibly by the application of an external magnetic field which leads to the so-called magneto rheological (MR) effect [1–5]. Therefore, the dynamic mechanical properties of MREs are still under ongoing research. The magnetic interaction of the particles within the matrix of MREs leads to control lable changes in their elastic and damping properties. The stiffness of MREs increased after the application of an external magnetic field [6,7]. In addition, the possibility of controlling the properties of MREs in the real time has made them as excellent candidates for a wide range of applications, such as damping elements in the vibration absorbers, vi bration isolators, sensing devices, vehicle seat suspension, engine mounts, actuators to control the flow, and adaptive stiffness devices [8, 9]. Moreover, the MREs have recently been used to develop small-scale soft continuum robots with active steering and navigating capabilities
based on magnetic actuation [10–12]. MREs have been made from various types of matrix materials such as silicone rubber, natural rubber, polyurethane, and thermoplastic elas tomers [13]. Among them, silicone rubber is the most common matrix for the development of MREs [8]. Owing to their high saturation magnetization, micro-sized carbonyl iron particles (CIPs) are the most common type of fillers for the MRE development [14,15]. The fabrica tion process of MREs is generally conducted by thorough mixing of the three primary components (iron particles, silicone and its catalyst). The properties of MREs strongly depend on the distribution of iron particles [16]. The distribution of the iron particles in MREs can be either random or aligned in chains. The randomly distributed structures are obtained by mixing the iron particles throughout the matrix, while aligned chain-like structures are acquired by application of an external magnetic field during the cross-linking process [17–20]. The MREs with randomly distributed particles are referred to isotropic MREs and the MREs with chain-like arranged particles are referred to anisotropic MREs. It is acknowledged that MREs are considered as viscoelastic com posites that are composed of the elastomeric matrix and magnetosensitive particles [21]. Therefore, MREs inherit main properties of
* Corresponding author. E-mail addresses: [email protected], [email protected] (T.H. Nam). https://doi.org/10.1016/j.polymertesting.2019.106272 Received 9 September 2019; Received in revised form 24 November 2019; Accepted 1 December 2019 Available online 3 December 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) A SEM image of CIPs, (b) Diameter of CIPs as the functions of the volume fraction in overall distribution.
the elastomeric matrix such as large deformations, stress softening ef fect, amplitude and frequency dependency, reduction of stiffness at cy clic loading and viscoelastic time-dependent features [22,23]. The dynamic mechanical properties of MREs depend on the elastomeric matrix, the particle content and distribution, and external magnetic field applied [24]. The optimal volume fraction of the ferromagnetic particles in MREs for the largest change in the modulus at saturation was pre dicted to be 27% [6,23,25]. The magneto-dynamic characterization of MREs has been made by dynamic mechanical studies under oscillating forces. Most of the dynamic testing uses the double-lap shear technique, where two MRE pieces are adhesively bonded between three metallic sheets [26,27]. The magnetic field is applied in the perpendicular di rection of the shear stress and in the parallel direction of the aligned particle chains [16]. The dynamic mechanical properties of MREs are studied by the equations typically for viscoelastic materials. Viscoelastic modeling and understanding the dynamic behavior of MREs under various loading conditions are essential for engineering applications [28]. The visco elastic modeling of these materials is approached via different consti tutive models, such as classical differential models and fractional derivative models. Classical differential models consist of a combination of springs and dashpots [29], but they fail to fit well with experimental data in describing the mechanical behavior of viscoelastic materials [30]. In order to avoid this, fractional derivative models have been used to study the viscoelastic behavior of MREs [16–20,30,31]. Moreover, the fractional derivative models contain fewer material parameters than classical viscoelastic models. In addition, change in the properties of MREs under the application of an external magnetic field is the most important and is most extensively studied in recent years. The magneto-viscoelastic behavior using fractional derivatives has been modelled for isotropic [18] and anisotropic [19,29,30] MREs. For this paper, the isotropic and anisotropic MREs were prepared
from silicone matrix reinforced by 27% volume fraction of micro-sized CIPs. Shear stress and strain of the isotropic and anisotropic MREs were measured using double-lap shear test with frequencies from 2 to 20 Hz, strain amplitudes from 0.01 to 0.2, and magnetic flux densities from 0 to 0.65 T. The obtained experimental results were used to determine the dynamic properties of the isotropic and anisotropic MREs in terms of storage and loss moduli, and loss factor. The dependence of dynamic properties of the isotropic and anisotropic MREs on the fre quency, strain amplitude, and applied magnetic field was examined. A four-parameter fractional derivative model was applied to simulate the viscoelastic behavior of the isotropic and anisotropic MREs. The dependence of dynamic moduli and loss factor on the frequency and magnetic flux density was numerically calculated using investigated model. Scanning electron microscopy (SEM) ZEISS Ultra Plus (Germany) was used to investigate microstructural morphologies of the isotropic and anisotropic MREs. 2. Experimental study 2.1. Materials The components for fabricating MREs are micro-sized CIPs, liquid silicone ZA13 and its catalyst. The liquid silicone ZA13 and its catalyst are produced by Zhermack S.P.A (Italy) and are provided by Havel Composites Ltd. (Czech Republic). The micro-sized CIPs (type: 44890) are supplied by Sigma-Aldrich (USA). Fig. 1 shows a SEM image of micro-sized CIPs and their diameter as a function of the volume fraction in overall distribution. The CIPs had overall spherical shapes with 2–5 μm (�99.5%) in diameter.
Fig. 2. SEM micrographs showing the microstructural morphologies of (a) isotropic and (b) anisotropic MREs. 2
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Fig. 3. Double-lap shear test attached with a MRE specimen in the electromagnet system.
2.2. MRE fabrication
intensities from 0 to 0.65 T. The influence of magnetic field on the dy namic properties of the MREs was obtained using the electromagnet. The electromagnet was used to generate the desired external magnetic field in the perpendicular direction of the shear stress. In addition, the magnetic field was applied in the parallel direction to the aligned par ticle chains for anisotropic MREs. Images illustrating a MRE specimen, the double-lap shear test, and the electromagnet system are presented in Fig. 3.
Isotropic and anisotropic MRE samples are fabricated by following steps. To begin with, the liquid silicone ZA13 was mixed with its catalyst in the weight ratio of 1:1 and then the CIPs with a volume fraction of 27% were added. Afterward, the mixture was well blended in a glass cup and was placed in a vacuum chamber for about 15 min to eliminate the air bubbles trapped inside the material during mixing process. Next, the mixture was poured into a plastic mold and was placed in the vacuum chamber for about 10 min to remove throughout the air bubbles trapped inside the mixture. Finally, the mixture in the mold was cured for 24 h at room temperature to obtain the isotropic MRE. To prepare the aniso tropic MRE, the mixture inside an aluminum mold was exposed to magnetic flux density of about 450 mT in the thickness direction of the MRE sample using an electromagnet during the curing process. SEM micrographs showing microstructural morphologies of the isotropic and anisotropic MREs are depicted in Fig. 2. The isotropic MRE sample fabricated without a magnetic field (Fig. 2(a)) shows randomly distrib uted particles. Conversely, the anisotropic MRE sample produced under a magnetic field has a like-chain structure (Fig. 2(b)). The chain-like structure in the anisotropic MREs created in the curing process was modelled by Hossain et al. [32,33] in the presence of a magnetic load.
2.4. Characterizations The experimental study was conducted to characterize the dynamic properties of the MREs over the strain amplitude, excitation frequency, and magnetic flux density. The MRE samples are subjected to harmonic loading with static pre-displacement u0 ¼ 0.5 mm, the displacement amplitudes Δu from 0.05 to 1 mm corresponding to shear strain from 0.01 to 0.2, and the frequencies from 2 to 20 Hz. Displacement u(t) and force response F(t) that are obtained during each measurement are approximately sinusoidal harmonic functions. Therefore, shear stress τ(t) and shear strain γ(t) also vary sinusoidally with respective ampli tudes of τa and γa, but they will lag by a phase angle δ (loss angle or loss factor). The ratio of the amplitudes is called the dynamic complex modulus (G*). The amplitudes of shear stress and shear strain were determined from experimental recorded signals. The loss angle δ was determined using the discrete Fourier transform (DFT) at the main excitation frequency [23]. The loss tangent (tanδ) is the ratio of the loss modulus G00 to the storage modulus G’. The method to determine dy namic complex, storage, and loss moduli from amplitudes and loss angle was described in the standard ISO-4664, as follows:
2.3. Dynamic double-lap shear test A double-lap shear test setup was designed to characterize the me chanical behavior of MREs in the shear mode under dynamic loading. Rectangular pieces of pure silicone and MREs (20 � 20 � 5 mm3) were prepared for fabrication of the dynamic shear test specimens, as rec ommended in ISO-1827 [8,34]. Two pieces of pure silicone or MREs were sandwiched between the inner and outer aluminum strips to form a double-lap shear specimen [22,23]. The double-lap shear test for pure silicone and MRE specimens was conducted using Instron Electropuls testing system under harmonic loading in the displacement control mode. The dynamic double-lap shear tests were conducted for both isotropic and anisotropic MREs with the change of frequencies in steps from 2 to 20 Hz, strain amplitudes from 0.01 to 0.2, and magnetic field
τa
G* ¼ ; γa
G ¼ G* cosðδÞ; 0
G00 ¼ G* sinðδÞ:
(1)
The storage modulus indicated the ability of MREs to store the deformation energy, which contributes to the MRE stiffness. The loss modulus represents the ability of MREs to dissipate the deformation energy. Therefore, the loss modulus can be estimated from the energy dissipation, which corresponds to the area of the hysteresis loop [19]. The energy dissipation D per loading cycle is expressed as: 3
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Fig. 4. Typical shear stress-strain hysteresis loops of (a) isotropic and (b) anisotropic MREs at 20 Hz frequency for different strain amplitudes.
Fig. 5. Dependence of storage modulus and loss modulus on shear strain at frequency of 10 Hz for (a,b) isotropic and (c,d) anisotropic MREs.
4
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Fig. 6. Typical shear stress-strain hysteresis loops of (a) isotropic and (b) anisotropic MREs at 10% strain and 20 Hz frequency under different magnetic flux densities.
Z
T
D¼ 0
τðtÞ γ_ ðtÞdt ¼ πγ2a G* sin δ ¼ πγ2a G00 :
amplitude for the isotropic and anisotropic MREs was depicted in Fig. 4. The shear stress-strain hysteresis loops under low strain amplitudes are nearly elliptic and transform to non-elliptical shape under higher strain amplitudes. This result suggests that the MREs behave like a nonlinear viscoelastic material under high strain amplitudes, as presented by Dargahi et al. [8]. It is interesting that the increase in the strain ampli tude causes the hysteresis loops of the isotropic and anisotropic MREs to become larger and wider. In addition, the slope of stress-strain hysteresis loops for the isotropic and anisotropic MREs decreases with increasing the strain amplitude (Fig. 4), resulting in the reduction of the storage and loss moduli. The decrease in the storage and loss moduli with increasing the strain amplitude for the MREs both without and with the application of
(2)
2.5. Experimental results and discussion Experimentally measured data in the steady state of double-lap shear test for the MREs are used to describe the shear stress-strain character istics of the MREs and to study the influences of strain amplitude, excitation frequency, and magnetic flux density on the dynamic prop erties of the MREs. 2.5.1. Dependence of MRE dynamic properties on strain amplitude The dependence of stress–strain hysteresis loops on the shear strain
Fig. 7. Complex shear, storage and loss moduli, and loss angle of isotropic MREs as a function of frequency and magnetic flux density. 5
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Fig. 8. Complex shear, storage and loss moduli, and loss angle of anisotropic MREs as a function of frequency and magnetic flux density.
external magnetic field was described in Fig. 5. It is observed that the storage and loss moduli reduce rapidly at low strain amplitudes up to 5%, followed by a slow decrease at high strain amplitudes. The storage modulus approaches lower bound at the large strain amplitude of 20%. This tendency is more pronounced for the anisotropic MRE. The reduction in the storage modulus of the MREs with increasing the strain amplitude is attributable to the so-called Payne effect known in filled rubber. This phenomenon is caused by breakage and recovery of bonds linking the silicon rubber network and CPIs and by changes in the MRE microstructure [6]. In addition, the decrease of the storage and loss moduli with the rise of strain amplitude for the anisotropic MRE is greater than that for the isotropic MRE (Fig. 5). Therefore, the storage and loss moduli of anisotropic MRE display a more obvious dependence on the shear strain than those of isotropic MRE.
the shear modulus. These results suggest variations in the stiffness and damping characteristics of the MREs with changing the magnetic field. It is visible that maximum stress values of the anisotropic MREs are higher than those of the isotropic ones (Fig. 6). Moreover, there is a large change in the slope of shear stress-strain hysteresis loops of the aniso tropic MREs compared to that of the isotropic ones. Therefore, a com parison of shear stress-strain hysteresis loops between the isotropic and anisotropic MREs was conducted. The results showed that the slope of the shear stress-strain loops of the anisotropic MREs was higher than that of the isotropic MREs. As a result, the stiffness of the anisotropic MREs is greater than that of the isotropic MREs. The slope increase of the hysteresis loops for the MREs with the rise in the magnetic flux density is quantified by calculating the dynamic moduli and the loss angle for each of the applied magnetic fields and across the tested frequencies. The dynamic complex shear modulus, storage and loss moduli, and loss angle of the isotropic and anisotropic MREs over the frequency and magnetic flux density are portrayed in Fig. 7 and Fig. 8, respectively. The results showed that the anisotropic MREs exhibit higher dynamic moduli in comparison to the isotropic MREs, as presented in several reports [35,36]. This can be explained by the fact that the anisotropic MREs with the chain-like structure formed by aligned particles along the magnetic field direction acts as a rod-like filler [37]. For example, the mean percentage increases of storage modulus, loss modulus, and loss angle for three samples between the isotropic and anisotropic MREs under a magnetic flux density of 0.651 T are 33%, 73%, and 29%, respectively. In addition, the dynamic moduli and damping properties of the isotropic and anisotropic MREs enhance with the rise of the magnetic flux density. For instance, the mean percentage increases of storage modulus, loss modulus, and loss angle for three specimens between the zero and 0.651 T of magnetic field at the frequency of 20 Hz respectively are 20%, 34%, and 12% for the isotropic MREs and 30%, 64%, and 25% for the anisotropic MREs. It is interesting that the dynamic moduli and the loss angle increase sharply with the rise of the magnetic flux density to about 0.5 T, above 0.5 T they enhance slightly. The increase in the dynamic moduli and loss angle of the isotropic and anisotropic MREs
2.5.2. Effects of frequency and magnetic flux density on MRE dynamic properties A strain amplitude of 10% was used to investigate the dependence of MRE dynamic properties on the frequency and applied magnetic field. Typical shear stress-strain hysteresis loops of the isotropic and aniso tropic MREs with a strain amplitude of 10% at a frequency of 20 Hz under different magnetic flux densities are presented in Fig. 6. As observed in Fig. 6(a), the hysteresis loop of the isotropic MREs without magnetic field in the stress-strain coordinate system is approximately ellipse. Therefore, the isotropic MREs under zero magnetic field in tensity comply with the linear viscoelasticity. However, the character of hysteresis loops for the isotropic MREs becomes nonlinear as the mag netic field is applied (Fig. 6(a)). In addition, the nonlinear behavior is visible in the case of the anisotropic MREs under zero magnetic flux density (Fig. 6(b)). Particularly, the nonlinear response of the aniso tropic MREs is more pronounced after the magnetic field is applied. As Fig. 6 shows, the slope and area bounded by the shear stress-strain hysteresis loops of isotropic and anisotropic MREs increase considerably compared with those of pure silicone. In addition, the slope of the stressstrain hysteresis loops for both isotropic and anisotropic MREs enhance with the increase of the magnetic flux density, resulting in an increase in 6
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Fig. 9. Loss modulus of isotropic MREs as a function of frequency and magnetic flux density.
Fig. 10. Loss modulus of anisotropic MREs as a function of frequency and magnetic flux density.
Fig. 11. Relative MR effect of (a) isotropic MREs and (b) anisotropic MREs. 7
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magnetic flux density. Results show that the relative MR effect of the isotropic and anisotropic MREs increases with increasing the magnetic flux density. In addition, the relative MR effect decreases slightly with the rise of the frequency to 10 Hz, above 10 Hz it seems to be unchanged. This phenomenon is clearly visible in the case of high magnetic field strengths. Moreover, the relative MR effect of the anisotropic MREs is much larger than that of the isotropic ones, as presented by Sun et al. [39]. The maximal MR effects of the isotropic and anisotropic MREs are 20.6% and 30.6%, respectively. In general, the anisotropic MREs under magnetic stimulation have a larger MR effect than the isotropic MREs. 3. MRE viscoelastic modeling
Fig. 12. Four-parameter fractional derivative model.
3.1. Viscoelastic model for MREs
with increasing the frequency is also apparent (Figs. 7 and 8). However, the storage modulus of the MREs under dynamic test gains slightly at the high frequency range, especially for the anisotropic MREs at the fre quency higher than about 10 Hz (Fig. 8). Furthermore, effect of loss modulus on the frequency and magnetic flux density can be estimated from the dissipated energy according to Eq. (2). It is evident that the areas bounded by the shear stress-strain hysteresis loops in Fig. 6 correspond to the energy dissipated into the MREs. Density of dissipated energy was calculated as the mean value of the areas of twenty consecutive steady hysteresis loops in each mea surement. The loss moduli of the isotropic and anisotropic MREs at different frequencies under various magnetic flux densities determined from recorded signals according to Eq. (1) and estimated from the en ergy dissipation by Eq. (2) are presented in Figs. 9 and 10. Obviously, the values of loss modulus calculated from the energy dissipation are in very good agreement with those determined from recorded signals. In addi tion, the results indicated that the loss modulus increases with the rise of the magnetic flux density. Moreover, the anisotropic MREs show higher loss modulus compared to the isotropic ones.
Viscoelastic models based on fractional derivatives have been used to deal with viscoelastic materials. These models include fractional deriv ative terms adding to viscous terms and elastic terms. They have fewer material parameters than classical viscoelastic models. In addition, the fractional derivative models can be used in both the time and frequency domain [16,19]. Therefore, to predict the dynamic viscoelastic response of the MREs, a four-parameter fractional derivative model, which was known as the four-parameter fractional derivative Zener model, was used in the frequency domain (Fig. 12). The four-parameter fractional derivative Zener model has solid theoretical basis, because it is related to the general fractional derivative constitutive equation of viscoelastic materials [40]. The model has successfully been fitted to experimental data on a wide variety of materials, especially polymers for vibration damping [41,42]. Moreover, this model is causal and satisfies the thermodynamic constraints [43]. The constitutive equation for the four-parameter fractional deriva tive model in the time domain is written as follows:
σ ðtÞ þ τα
2.5.3. Magnetorheological effect Changes in the MRE properties under the magnetic field, described as the MR effect, are related to the iron particles tendency to change their position under the application of an applied magnetic field. The mag netic field induces dipole moments in the ferromagnetic particles, which tend to obtain the positions of minimum energy state [24]. Movement of the particles introduces deformations in the elastomer matrix, resulting in the increase of shear modulus and stiffness of the MREs. In addition, the interactions between particles in a magnetic field bring them closer, resulting in increased stiffness of the material and shear moduli [38]. The MR effect can be described by both absolute and relative effect. The 0 absolute MR effect (ΔG ) is the difference between the maximum storage 0 modulus (Gmax ) achieved in the presence of a magnetic field and the storage modulus obtained without a magnetic field (G0). The relative MR effect is expressed by the following equation: � � 0 ΔG Relative MR effect % ¼ � 100 (3) G0
σ ðωÞð1 þ ðiτωÞα Þ ¼ εðωÞ½Ge ð1 þ ðiτωÞα Þ þ ðG∞
Ge [MPa]
G∞ [MPa]
τ (s)
α [rad]
0.368 0.382 0.407 0.416 0.423 0.426
1.686 1.791 1.895 1.959 2.004 2.034
0.009 0.012 0.015 0.015 0.015 0.015
0.281 0.290 0.298 0.298 0.298 0.298
Ge ÞðiτωÞα �
(5)
where ω is the angular frequency. Complex shear modulus of the four-parameter fractional derivative viscoelastic model in the frequency domain takes the form of Eq. (6): G* ðiωÞ ¼ G þ iG00 ¼ 0
Ge þ G∞ ðiωτÞα 1 þ ðiωτÞα
(6)
The complex shear modulus G* consists of a real part, the storage modulus or the rigidity, which characterizes the stiffness of the visco elastic material, and an imaginary part, is called loss modulus or the energy dissipation, which characterizes the viscous behavior. The ex pressions for the storage and loss modulus obtained for this model are as:
Table 1 Fitting parameters of the fractional derivative model for isotropic MREs under different flux densities. 0 0.201 0.373 0.478 0.580 0.651
(4)
where Ge is the static elastic shear modulus, G∞ ¼ Ge þ Gm is the high frequency limit value of the dynamic modulus, τ is the relaxation time, and α is the fractional parameter with value varying between 0 and 1 [18,40–42]. However, representing the model in the frequency domain is more useful and much easier than that in the time domain [16]. Therefore, Eq. (4) is represented in the frequency domain using Fourier transform as follows:
The relative MR effect of the MREs is dependent on the frequency and magnetic field intensity [37]. Fig. 11 presents the relative MR effect of isotropic and anisotropic MREs as a function of frequency and
Magnetic flux density B [T]
dα σ dα ε ¼ Ge εðtÞ þ G∞ τα α dtα dt
0
G ¼
G00 ¼
as:
8
Ge ½1 þ ðτωÞα cosðαπ =2Þ� þ G∞ ðτωÞα ½cosðαπ =2Þ þ ðτωÞα � 1 þ 2ðτωÞα cosðαπ =2Þ þ ðτωÞ2α ðG∞
Ge ÞðτωÞα sinðαπ =2Þ
1 þ 2ðτωÞα cosðαπ =2Þ þ ðτωÞ2α
(7) (8)
Therefore, the loss tangent in the frequency can be expressed domain
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nonlinear fit function was applied to simultaneous fitting of multiple data sets to identify the four parameters of the model. The dynamic properties for the isotropic and anisotropic MREs as a function of the frequency under magnetic flux densities were calculated using the fractional derivative viscoelastic model with the fitted parameters. The fitting and experimental results of the dynamic properties and loss factor as a function of frequency under different magnetic flux densities for the isotropic and anisotropic MRE samples are presented in Figs. 13 and 14. Calculated results in Figs. 13 and 14 showed that the fitting is ac curate for all magnetic flux densities in the investigated frequency band. Nevertheless, the fitting for the storage modulus is better than that for the loss modulus and loss angle. This is explainable that the storage modulus is one order of magnitude higher than the loss modulus [19]. The respective maximal fitting errors of the isotropic and anisotropic MREs does not exceed 0.7% and 2.2% for the storage modulus, and 3.1% and 5.3% for the loss modulus. In addition, the model parameters Ge and G∞ increase with the rise of the magnetic flux density, whereas the relaxation time τ and the fractional parameter α vary slightly or do not change (Tables 1–2). The increase in the Ge and G∞ proves that the stiffness of the MRE samples enhances with increasing the magnetic flux density. In general, the four-parameter fractional viscoelastic model was fitted well to experimental data in the investigated frequency band for both isotropic and anisotropic MREs.
Table 2 Fitting parameters of the fractional derivative model for anisotropic MREs under different flux densities. Magnetic flux density B [T]
Ge [MPa]
G∞ [MPa]
τ (s)
α [rad]
0 0.201 0.373 0.478 0.580 0.651
0.417 0.424 0.463 0.484 0.501 0.514
1.823 2.139 2.393 2.491 2.549 2.568
0.033 0.040 0.049 0.051 0.052 0.052
0.331 0.337 0.353 0.358 0.362 0.365
tan δðωÞ ¼
G00 ðG∞ Ge ÞðτωÞα sinðαπ =2Þ α 0 ¼ G Ge ½1 þ ðτωÞ cosðαπ =2Þ� þ G∞ ðτωÞα ½cosðαπ =2Þ þ ðτωÞα �
(9)
3.2. Viscoelastic modeling results The fractional derivative viscoelastic model in the frequency domain which takes the form of Eq. (6) (9) can be applied to viscoelastic modeling of the MREs. Fitting Eq. (6) to the experimental data, four parameters of the fractional derivative model for the isotropic and anisotropic MREs were obtained, with results presented in Tables 1–2. The fitting was done using nonlinear least squares regression of multiple data sets in Matlab, and the minimized error was calculated simulta neously for the storage modulus, loss modulus, and loss angle. The
Fig. 13. Dynamic properties of isotropic MRE samples as a function of frequency under different magnetic flux densities. 9
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Fig. 14. Dynamic properties of anisotropic MRE samples as a function of frequency under different magnetic flux densities.
4. Conclusions
study are available from the corresponding author on reasonable request.
The experimental and numerical studies of dynamic mechanical properties for isotropic and anisotropic MREs made of silicone matrix and micro-sized CIPs have been conducted in this paper. Dynamic me chanical properties of the isotropic and anisotropic MREs were measured using double-lap shear tests at various frequencies and strain amplitudes under different magnetic flux densities. The dynamic moduli of the MREs decreased with the rise of strain amplitude. The stiffness and damping properties of the isotropic and anisotropic MREs increased with increasing of excitation frequency and magnetic flux density. The magneto-induced dynamic mechanical properties of anisotropic MREs were better than the isotropic MREs. The relative MR effect of the isotropic and anisotropic MREs enhanced with increasing the magnetic flux density. The relative MR effect of the anisotropic MREs was greater than that of the isotropic ones. The dependence of dynamic properties of the isotropic and anisotropic MREs on the frequency and magnetic field intensity was numerically examined using the four-parameter fractional derivative viscoelastic model. The viscoelastic model was fitted well to experimental data in the studied frequency band for all magnetic flux densities for both isotropic and anisotropic MREs.
Author contribution statement Tran Huu Nam carried out the experiment with support from Iva � and Bohdana Marvalova �. Tran Huu Nam performed modeling Petríkova and numerical calculations with support from Iva Petríkov� a and Boh �. Tran Huu Nam wrote the manuscript with support dana Marvalova � and Bohdana Marvalov� � and Bohdana from Iva Petríkova a. Iva Petríkova � supervised the research work. All authors discussed the re Marvalova sults and contributed to the final manuscript. Acknowledgements This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic and the European Union European Structural and Investment Funds in the frames of Operational Program Research, Development and Education project Hybrid Materials for Hierarchical Structures (HyHi, Reg. No. CZ.02.1.01/0.0/0.0/16_019/ 0000843).
Data availability
Appendix A. Supplementary data
The datasets generated during and/or analyzed during the current
Supplementary data to this article can be found online at https://doi. 10
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org/10.1016/j.polymertesting.2019.106272.
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Material Properties
Experimental characterization and viscoelastic modeling of isotropic and anisotropic magnetorheological elastomers �, Bohdana Marvalova � Tran Huu Nam *, Iva Petríkova Department of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Liberec, Studentsk� a 1402/2, 461 17 Liberec 1, Czech Republic
A R T I C L E I N F O
A B S T R A C T
Keywords: Magnetorheological elastomers Viscoelastic modeling Dynamic properties Double-lap shear test
The paper presents experimental research and numerical modeling of dynamic properties of magnetorheological elastomers (MREs). Isotropic and anisotropic MREs have been prepared based on silicone matrix filled by microsized carbonyl iron particles. Dynamic properties of the isotropic and anisotropic MREs were determined using double-lap shear test under harmonic loading in the displacement control mode. Effects of excitation frequency, strain amplitude, and magnetic field intensity on the dynamic properties of the MREs were examined. Dynamic moduli of the MREs decreased with increasing the strain amplitude of applied harmonic load. The dynamic moduli and damping properties of the MREs increased with increasing the frequency and magnetic flux density. The anisotropic MREs showed higher dynamic moduli and magnetorheological (MR) effect than those of the isotropic ones. The MR effect of the MREs increased with the rise of the magnetic flux density. The dependence of dynamic moduli and loss factor on the frequency and magnetic flux density was numerically studied using fourparameter fractional derivative viscoelastic model. The model was fitted well to experimental data for both isotropic and anisotropic MREs. The fitting of dynamic moduli and loss factor for the isotropic and anisotropic MREs is in good agreement with experimental results.
1. Introduction Magnetorheological elastomers (MREs) are considered as smart composites. Such composites are composed of micro-sized magnetic particles dispersed in a non-magnetic elastomeric matrix. MREs have attracted great interest, because their mechanical and rheological properties can be controlled rapidly and reversibly by the application of an external magnetic field which leads to the so-called magneto rheological (MR) effect [1–5]. Therefore, the dynamic mechanical properties of MREs are still under ongoing research. The magnetic interaction of the particles within the matrix of MREs leads to control lable changes in their elastic and damping properties. The stiffness of MREs increased after the application of an external magnetic field [6,7]. In addition, the possibility of controlling the properties of MREs in the real time has made them as excellent candidates for a wide range of applications, such as damping elements in the vibration absorbers, vi bration isolators, sensing devices, vehicle seat suspension, engine mounts, actuators to control the flow, and adaptive stiffness devices [8, 9]. Moreover, the MREs have recently been used to develop small-scale soft continuum robots with active steering and navigating capabilities
based on magnetic actuation [10–12]. MREs have been made from various types of matrix materials such as silicone rubber, natural rubber, polyurethane, and thermoplastic elas tomers [13]. Among them, silicone rubber is the most common matrix for the development of MREs [8]. Owing to their high saturation magnetization, micro-sized carbonyl iron particles (CIPs) are the most common type of fillers for the MRE development [14,15]. The fabrica tion process of MREs is generally conducted by thorough mixing of the three primary components (iron particles, silicone and its catalyst). The properties of MREs strongly depend on the distribution of iron particles [16]. The distribution of the iron particles in MREs can be either random or aligned in chains. The randomly distributed structures are obtained by mixing the iron particles throughout the matrix, while aligned chain-like structures are acquired by application of an external magnetic field during the cross-linking process [17–20]. The MREs with randomly distributed particles are referred to isotropic MREs and the MREs with chain-like arranged particles are referred to anisotropic MREs. It is acknowledged that MREs are considered as viscoelastic com posites that are composed of the elastomeric matrix and magnetosensitive particles [21]. Therefore, MREs inherit main properties of
* Corresponding author. E-mail addresses: [email protected], [email protected] (T.H. Nam). https://doi.org/10.1016/j.polymertesting.2019.106272 Received 9 September 2019; Received in revised form 24 November 2019; Accepted 1 December 2019 Available online 3 December 2019 0142-9418/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. (a) A SEM image of CIPs, (b) Diameter of CIPs as the functions of the volume fraction in overall distribution.
the elastomeric matrix such as large deformations, stress softening ef fect, amplitude and frequency dependency, reduction of stiffness at cy clic loading and viscoelastic time-dependent features [22,23]. The dynamic mechanical properties of MREs depend on the elastomeric matrix, the particle content and distribution, and external magnetic field applied [24]. The optimal volume fraction of the ferromagnetic particles in MREs for the largest change in the modulus at saturation was pre dicted to be 27% [6,23,25]. The magneto-dynamic characterization of MREs has been made by dynamic mechanical studies under oscillating forces. Most of the dynamic testing uses the double-lap shear technique, where two MRE pieces are adhesively bonded between three metallic sheets [26,27]. The magnetic field is applied in the perpendicular di rection of the shear stress and in the parallel direction of the aligned particle chains [16]. The dynamic mechanical properties of MREs are studied by the equations typically for viscoelastic materials. Viscoelastic modeling and understanding the dynamic behavior of MREs under various loading conditions are essential for engineering applications [28]. The visco elastic modeling of these materials is approached via different consti tutive models, such as classical differential models and fractional derivative models. Classical differential models consist of a combination of springs and dashpots [29], but they fail to fit well with experimental data in describing the mechanical behavior of viscoelastic materials [30]. In order to avoid this, fractional derivative models have been used to study the viscoelastic behavior of MREs [16–20,30,31]. Moreover, the fractional derivative models contain fewer material parameters than classical viscoelastic models. In addition, change in the properties of MREs under the application of an external magnetic field is the most important and is most extensively studied in recent years. The magneto-viscoelastic behavior using fractional derivatives has been modelled for isotropic [18] and anisotropic [19,29,30] MREs. For this paper, the isotropic and anisotropic MREs were prepared
from silicone matrix reinforced by 27% volume fraction of micro-sized CIPs. Shear stress and strain of the isotropic and anisotropic MREs were measured using double-lap shear test with frequencies from 2 to 20 Hz, strain amplitudes from 0.01 to 0.2, and magnetic flux densities from 0 to 0.65 T. The obtained experimental results were used to determine the dynamic properties of the isotropic and anisotropic MREs in terms of storage and loss moduli, and loss factor. The dependence of dynamic properties of the isotropic and anisotropic MREs on the fre quency, strain amplitude, and applied magnetic field was examined. A four-parameter fractional derivative model was applied to simulate the viscoelastic behavior of the isotropic and anisotropic MREs. The dependence of dynamic moduli and loss factor on the frequency and magnetic flux density was numerically calculated using investigated model. Scanning electron microscopy (SEM) ZEISS Ultra Plus (Germany) was used to investigate microstructural morphologies of the isotropic and anisotropic MREs. 2. Experimental study 2.1. Materials The components for fabricating MREs are micro-sized CIPs, liquid silicone ZA13 and its catalyst. The liquid silicone ZA13 and its catalyst are produced by Zhermack S.P.A (Italy) and are provided by Havel Composites Ltd. (Czech Republic). The micro-sized CIPs (type: 44890) are supplied by Sigma-Aldrich (USA). Fig. 1 shows a SEM image of micro-sized CIPs and their diameter as a function of the volume fraction in overall distribution. The CIPs had overall spherical shapes with 2–5 μm (�99.5%) in diameter.
Fig. 2. SEM micrographs showing the microstructural morphologies of (a) isotropic and (b) anisotropic MREs. 2
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Fig. 3. Double-lap shear test attached with a MRE specimen in the electromagnet system.
2.2. MRE fabrication
intensities from 0 to 0.65 T. The influence of magnetic field on the dy namic properties of the MREs was obtained using the electromagnet. The electromagnet was used to generate the desired external magnetic field in the perpendicular direction of the shear stress. In addition, the magnetic field was applied in the parallel direction to the aligned par ticle chains for anisotropic MREs. Images illustrating a MRE specimen, the double-lap shear test, and the electromagnet system are presented in Fig. 3.
Isotropic and anisotropic MRE samples are fabricated by following steps. To begin with, the liquid silicone ZA13 was mixed with its catalyst in the weight ratio of 1:1 and then the CIPs with a volume fraction of 27% were added. Afterward, the mixture was well blended in a glass cup and was placed in a vacuum chamber for about 15 min to eliminate the air bubbles trapped inside the material during mixing process. Next, the mixture was poured into a plastic mold and was placed in the vacuum chamber for about 10 min to remove throughout the air bubbles trapped inside the mixture. Finally, the mixture in the mold was cured for 24 h at room temperature to obtain the isotropic MRE. To prepare the aniso tropic MRE, the mixture inside an aluminum mold was exposed to magnetic flux density of about 450 mT in the thickness direction of the MRE sample using an electromagnet during the curing process. SEM micrographs showing microstructural morphologies of the isotropic and anisotropic MREs are depicted in Fig. 2. The isotropic MRE sample fabricated without a magnetic field (Fig. 2(a)) shows randomly distrib uted particles. Conversely, the anisotropic MRE sample produced under a magnetic field has a like-chain structure (Fig. 2(b)). The chain-like structure in the anisotropic MREs created in the curing process was modelled by Hossain et al. [32,33] in the presence of a magnetic load.
2.4. Characterizations The experimental study was conducted to characterize the dynamic properties of the MREs over the strain amplitude, excitation frequency, and magnetic flux density. The MRE samples are subjected to harmonic loading with static pre-displacement u0 ¼ 0.5 mm, the displacement amplitudes Δu from 0.05 to 1 mm corresponding to shear strain from 0.01 to 0.2, and the frequencies from 2 to 20 Hz. Displacement u(t) and force response F(t) that are obtained during each measurement are approximately sinusoidal harmonic functions. Therefore, shear stress τ(t) and shear strain γ(t) also vary sinusoidally with respective ampli tudes of τa and γa, but they will lag by a phase angle δ (loss angle or loss factor). The ratio of the amplitudes is called the dynamic complex modulus (G*). The amplitudes of shear stress and shear strain were determined from experimental recorded signals. The loss angle δ was determined using the discrete Fourier transform (DFT) at the main excitation frequency [23]. The loss tangent (tanδ) is the ratio of the loss modulus G00 to the storage modulus G’. The method to determine dy namic complex, storage, and loss moduli from amplitudes and loss angle was described in the standard ISO-4664, as follows:
2.3. Dynamic double-lap shear test A double-lap shear test setup was designed to characterize the me chanical behavior of MREs in the shear mode under dynamic loading. Rectangular pieces of pure silicone and MREs (20 � 20 � 5 mm3) were prepared for fabrication of the dynamic shear test specimens, as rec ommended in ISO-1827 [8,34]. Two pieces of pure silicone or MREs were sandwiched between the inner and outer aluminum strips to form a double-lap shear specimen [22,23]. The double-lap shear test for pure silicone and MRE specimens was conducted using Instron Electropuls testing system under harmonic loading in the displacement control mode. The dynamic double-lap shear tests were conducted for both isotropic and anisotropic MREs with the change of frequencies in steps from 2 to 20 Hz, strain amplitudes from 0.01 to 0.2, and magnetic field
τa
G* ¼ ; γa
G ¼ G* cosðδÞ; 0
G00 ¼ G* sinðδÞ:
(1)
The storage modulus indicated the ability of MREs to store the deformation energy, which contributes to the MRE stiffness. The loss modulus represents the ability of MREs to dissipate the deformation energy. Therefore, the loss modulus can be estimated from the energy dissipation, which corresponds to the area of the hysteresis loop [19]. The energy dissipation D per loading cycle is expressed as: 3
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Fig. 4. Typical shear stress-strain hysteresis loops of (a) isotropic and (b) anisotropic MREs at 20 Hz frequency for different strain amplitudes.
Fig. 5. Dependence of storage modulus and loss modulus on shear strain at frequency of 10 Hz for (a,b) isotropic and (c,d) anisotropic MREs.
4
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Fig. 6. Typical shear stress-strain hysteresis loops of (a) isotropic and (b) anisotropic MREs at 10% strain and 20 Hz frequency under different magnetic flux densities.
Z
T
D¼ 0
τðtÞ γ_ ðtÞdt ¼ πγ2a G* sin δ ¼ πγ2a G00 :
amplitude for the isotropic and anisotropic MREs was depicted in Fig. 4. The shear stress-strain hysteresis loops under low strain amplitudes are nearly elliptic and transform to non-elliptical shape under higher strain amplitudes. This result suggests that the MREs behave like a nonlinear viscoelastic material under high strain amplitudes, as presented by Dargahi et al. [8]. It is interesting that the increase in the strain ampli tude causes the hysteresis loops of the isotropic and anisotropic MREs to become larger and wider. In addition, the slope of stress-strain hysteresis loops for the isotropic and anisotropic MREs decreases with increasing the strain amplitude (Fig. 4), resulting in the reduction of the storage and loss moduli. The decrease in the storage and loss moduli with increasing the strain amplitude for the MREs both without and with the application of
(2)
2.5. Experimental results and discussion Experimentally measured data in the steady state of double-lap shear test for the MREs are used to describe the shear stress-strain character istics of the MREs and to study the influences of strain amplitude, excitation frequency, and magnetic flux density on the dynamic prop erties of the MREs. 2.5.1. Dependence of MRE dynamic properties on strain amplitude The dependence of stress–strain hysteresis loops on the shear strain
Fig. 7. Complex shear, storage and loss moduli, and loss angle of isotropic MREs as a function of frequency and magnetic flux density. 5
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Fig. 8. Complex shear, storage and loss moduli, and loss angle of anisotropic MREs as a function of frequency and magnetic flux density.
external magnetic field was described in Fig. 5. It is observed that the storage and loss moduli reduce rapidly at low strain amplitudes up to 5%, followed by a slow decrease at high strain amplitudes. The storage modulus approaches lower bound at the large strain amplitude of 20%. This tendency is more pronounced for the anisotropic MRE. The reduction in the storage modulus of the MREs with increasing the strain amplitude is attributable to the so-called Payne effect known in filled rubber. This phenomenon is caused by breakage and recovery of bonds linking the silicon rubber network and CPIs and by changes in the MRE microstructure [6]. In addition, the decrease of the storage and loss moduli with the rise of strain amplitude for the anisotropic MRE is greater than that for the isotropic MRE (Fig. 5). Therefore, the storage and loss moduli of anisotropic MRE display a more obvious dependence on the shear strain than those of isotropic MRE.
the shear modulus. These results suggest variations in the stiffness and damping characteristics of the MREs with changing the magnetic field. It is visible that maximum stress values of the anisotropic MREs are higher than those of the isotropic ones (Fig. 6). Moreover, there is a large change in the slope of shear stress-strain hysteresis loops of the aniso tropic MREs compared to that of the isotropic ones. Therefore, a com parison of shear stress-strain hysteresis loops between the isotropic and anisotropic MREs was conducted. The results showed that the slope of the shear stress-strain loops of the anisotropic MREs was higher than that of the isotropic MREs. As a result, the stiffness of the anisotropic MREs is greater than that of the isotropic MREs. The slope increase of the hysteresis loops for the MREs with the rise in the magnetic flux density is quantified by calculating the dynamic moduli and the loss angle for each of the applied magnetic fields and across the tested frequencies. The dynamic complex shear modulus, storage and loss moduli, and loss angle of the isotropic and anisotropic MREs over the frequency and magnetic flux density are portrayed in Fig. 7 and Fig. 8, respectively. The results showed that the anisotropic MREs exhibit higher dynamic moduli in comparison to the isotropic MREs, as presented in several reports [35,36]. This can be explained by the fact that the anisotropic MREs with the chain-like structure formed by aligned particles along the magnetic field direction acts as a rod-like filler [37]. For example, the mean percentage increases of storage modulus, loss modulus, and loss angle for three samples between the isotropic and anisotropic MREs under a magnetic flux density of 0.651 T are 33%, 73%, and 29%, respectively. In addition, the dynamic moduli and damping properties of the isotropic and anisotropic MREs enhance with the rise of the magnetic flux density. For instance, the mean percentage increases of storage modulus, loss modulus, and loss angle for three specimens between the zero and 0.651 T of magnetic field at the frequency of 20 Hz respectively are 20%, 34%, and 12% for the isotropic MREs and 30%, 64%, and 25% for the anisotropic MREs. It is interesting that the dynamic moduli and the loss angle increase sharply with the rise of the magnetic flux density to about 0.5 T, above 0.5 T they enhance slightly. The increase in the dynamic moduli and loss angle of the isotropic and anisotropic MREs
2.5.2. Effects of frequency and magnetic flux density on MRE dynamic properties A strain amplitude of 10% was used to investigate the dependence of MRE dynamic properties on the frequency and applied magnetic field. Typical shear stress-strain hysteresis loops of the isotropic and aniso tropic MREs with a strain amplitude of 10% at a frequency of 20 Hz under different magnetic flux densities are presented in Fig. 6. As observed in Fig. 6(a), the hysteresis loop of the isotropic MREs without magnetic field in the stress-strain coordinate system is approximately ellipse. Therefore, the isotropic MREs under zero magnetic field in tensity comply with the linear viscoelasticity. However, the character of hysteresis loops for the isotropic MREs becomes nonlinear as the mag netic field is applied (Fig. 6(a)). In addition, the nonlinear behavior is visible in the case of the anisotropic MREs under zero magnetic flux density (Fig. 6(b)). Particularly, the nonlinear response of the aniso tropic MREs is more pronounced after the magnetic field is applied. As Fig. 6 shows, the slope and area bounded by the shear stress-strain hysteresis loops of isotropic and anisotropic MREs increase considerably compared with those of pure silicone. In addition, the slope of the stressstrain hysteresis loops for both isotropic and anisotropic MREs enhance with the increase of the magnetic flux density, resulting in an increase in 6
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Fig. 9. Loss modulus of isotropic MREs as a function of frequency and magnetic flux density.
Fig. 10. Loss modulus of anisotropic MREs as a function of frequency and magnetic flux density.
Fig. 11. Relative MR effect of (a) isotropic MREs and (b) anisotropic MREs. 7
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magnetic flux density. Results show that the relative MR effect of the isotropic and anisotropic MREs increases with increasing the magnetic flux density. In addition, the relative MR effect decreases slightly with the rise of the frequency to 10 Hz, above 10 Hz it seems to be unchanged. This phenomenon is clearly visible in the case of high magnetic field strengths. Moreover, the relative MR effect of the anisotropic MREs is much larger than that of the isotropic ones, as presented by Sun et al. [39]. The maximal MR effects of the isotropic and anisotropic MREs are 20.6% and 30.6%, respectively. In general, the anisotropic MREs under magnetic stimulation have a larger MR effect than the isotropic MREs. 3. MRE viscoelastic modeling
Fig. 12. Four-parameter fractional derivative model.
3.1. Viscoelastic model for MREs
with increasing the frequency is also apparent (Figs. 7 and 8). However, the storage modulus of the MREs under dynamic test gains slightly at the high frequency range, especially for the anisotropic MREs at the fre quency higher than about 10 Hz (Fig. 8). Furthermore, effect of loss modulus on the frequency and magnetic flux density can be estimated from the dissipated energy according to Eq. (2). It is evident that the areas bounded by the shear stress-strain hysteresis loops in Fig. 6 correspond to the energy dissipated into the MREs. Density of dissipated energy was calculated as the mean value of the areas of twenty consecutive steady hysteresis loops in each mea surement. The loss moduli of the isotropic and anisotropic MREs at different frequencies under various magnetic flux densities determined from recorded signals according to Eq. (1) and estimated from the en ergy dissipation by Eq. (2) are presented in Figs. 9 and 10. Obviously, the values of loss modulus calculated from the energy dissipation are in very good agreement with those determined from recorded signals. In addi tion, the results indicated that the loss modulus increases with the rise of the magnetic flux density. Moreover, the anisotropic MREs show higher loss modulus compared to the isotropic ones.
Viscoelastic models based on fractional derivatives have been used to deal with viscoelastic materials. These models include fractional deriv ative terms adding to viscous terms and elastic terms. They have fewer material parameters than classical viscoelastic models. In addition, the fractional derivative models can be used in both the time and frequency domain [16,19]. Therefore, to predict the dynamic viscoelastic response of the MREs, a four-parameter fractional derivative model, which was known as the four-parameter fractional derivative Zener model, was used in the frequency domain (Fig. 12). The four-parameter fractional derivative Zener model has solid theoretical basis, because it is related to the general fractional derivative constitutive equation of viscoelastic materials [40]. The model has successfully been fitted to experimental data on a wide variety of materials, especially polymers for vibration damping [41,42]. Moreover, this model is causal and satisfies the thermodynamic constraints [43]. The constitutive equation for the four-parameter fractional deriva tive model in the time domain is written as follows:
σ ðtÞ þ τα
2.5.3. Magnetorheological effect Changes in the MRE properties under the magnetic field, described as the MR effect, are related to the iron particles tendency to change their position under the application of an applied magnetic field. The mag netic field induces dipole moments in the ferromagnetic particles, which tend to obtain the positions of minimum energy state [24]. Movement of the particles introduces deformations in the elastomer matrix, resulting in the increase of shear modulus and stiffness of the MREs. In addition, the interactions between particles in a magnetic field bring them closer, resulting in increased stiffness of the material and shear moduli [38]. The MR effect can be described by both absolute and relative effect. The 0 absolute MR effect (ΔG ) is the difference between the maximum storage 0 modulus (Gmax ) achieved in the presence of a magnetic field and the storage modulus obtained without a magnetic field (G0). The relative MR effect is expressed by the following equation: � � 0 ΔG Relative MR effect % ¼ � 100 (3) G0
σ ðωÞð1 þ ðiτωÞα Þ ¼ εðωÞ½Ge ð1 þ ðiτωÞα Þ þ ðG∞
Ge [MPa]
G∞ [MPa]
τ (s)
α [rad]
0.368 0.382 0.407 0.416 0.423 0.426
1.686 1.791 1.895 1.959 2.004 2.034
0.009 0.012 0.015 0.015 0.015 0.015
0.281 0.290 0.298 0.298 0.298 0.298
Ge ÞðiτωÞα �
(5)
where ω is the angular frequency. Complex shear modulus of the four-parameter fractional derivative viscoelastic model in the frequency domain takes the form of Eq. (6): G* ðiωÞ ¼ G þ iG00 ¼ 0
Ge þ G∞ ðiωτÞα 1 þ ðiωτÞα
(6)
The complex shear modulus G* consists of a real part, the storage modulus or the rigidity, which characterizes the stiffness of the visco elastic material, and an imaginary part, is called loss modulus or the energy dissipation, which characterizes the viscous behavior. The ex pressions for the storage and loss modulus obtained for this model are as:
Table 1 Fitting parameters of the fractional derivative model for isotropic MREs under different flux densities. 0 0.201 0.373 0.478 0.580 0.651
(4)
where Ge is the static elastic shear modulus, G∞ ¼ Ge þ Gm is the high frequency limit value of the dynamic modulus, τ is the relaxation time, and α is the fractional parameter with value varying between 0 and 1 [18,40–42]. However, representing the model in the frequency domain is more useful and much easier than that in the time domain [16]. Therefore, Eq. (4) is represented in the frequency domain using Fourier transform as follows:
The relative MR effect of the MREs is dependent on the frequency and magnetic field intensity [37]. Fig. 11 presents the relative MR effect of isotropic and anisotropic MREs as a function of frequency and
Magnetic flux density B [T]
dα σ dα ε ¼ Ge εðtÞ þ G∞ τα α dtα dt
0
G ¼
G00 ¼
as:
8
Ge ½1 þ ðτωÞα cosðαπ =2Þ� þ G∞ ðτωÞα ½cosðαπ =2Þ þ ðτωÞα � 1 þ 2ðτωÞα cosðαπ =2Þ þ ðτωÞ2α ðG∞
Ge ÞðτωÞα sinðαπ =2Þ
1 þ 2ðτωÞα cosðαπ =2Þ þ ðτωÞ2α
(7) (8)
Therefore, the loss tangent in the frequency can be expressed domain
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nonlinear fit function was applied to simultaneous fitting of multiple data sets to identify the four parameters of the model. The dynamic properties for the isotropic and anisotropic MREs as a function of the frequency under magnetic flux densities were calculated using the fractional derivative viscoelastic model with the fitted parameters. The fitting and experimental results of the dynamic properties and loss factor as a function of frequency under different magnetic flux densities for the isotropic and anisotropic MRE samples are presented in Figs. 13 and 14. Calculated results in Figs. 13 and 14 showed that the fitting is ac curate for all magnetic flux densities in the investigated frequency band. Nevertheless, the fitting for the storage modulus is better than that for the loss modulus and loss angle. This is explainable that the storage modulus is one order of magnitude higher than the loss modulus [19]. The respective maximal fitting errors of the isotropic and anisotropic MREs does not exceed 0.7% and 2.2% for the storage modulus, and 3.1% and 5.3% for the loss modulus. In addition, the model parameters Ge and G∞ increase with the rise of the magnetic flux density, whereas the relaxation time τ and the fractional parameter α vary slightly or do not change (Tables 1–2). The increase in the Ge and G∞ proves that the stiffness of the MRE samples enhances with increasing the magnetic flux density. In general, the four-parameter fractional viscoelastic model was fitted well to experimental data in the investigated frequency band for both isotropic and anisotropic MREs.
Table 2 Fitting parameters of the fractional derivative model for anisotropic MREs under different flux densities. Magnetic flux density B [T]
Ge [MPa]
G∞ [MPa]
τ (s)
α [rad]
0 0.201 0.373 0.478 0.580 0.651
0.417 0.424 0.463 0.484 0.501 0.514
1.823 2.139 2.393 2.491 2.549 2.568
0.033 0.040 0.049 0.051 0.052 0.052
0.331 0.337 0.353 0.358 0.362 0.365
tan δðωÞ ¼
G00 ðG∞ Ge ÞðτωÞα sinðαπ =2Þ α 0 ¼ G Ge ½1 þ ðτωÞ cosðαπ =2Þ� þ G∞ ðτωÞα ½cosðαπ =2Þ þ ðτωÞα �
(9)
3.2. Viscoelastic modeling results The fractional derivative viscoelastic model in the frequency domain which takes the form of Eq. (6) (9) can be applied to viscoelastic modeling of the MREs. Fitting Eq. (6) to the experimental data, four parameters of the fractional derivative model for the isotropic and anisotropic MREs were obtained, with results presented in Tables 1–2. The fitting was done using nonlinear least squares regression of multiple data sets in Matlab, and the minimized error was calculated simulta neously for the storage modulus, loss modulus, and loss angle. The
Fig. 13. Dynamic properties of isotropic MRE samples as a function of frequency under different magnetic flux densities. 9
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Polymer Testing 81 (2020) 106272
Fig. 14. Dynamic properties of anisotropic MRE samples as a function of frequency under different magnetic flux densities.
4. Conclusions
study are available from the corresponding author on reasonable request.
The experimental and numerical studies of dynamic mechanical properties for isotropic and anisotropic MREs made of silicone matrix and micro-sized CIPs have been conducted in this paper. Dynamic me chanical properties of the isotropic and anisotropic MREs were measured using double-lap shear tests at various frequencies and strain amplitudes under different magnetic flux densities. The dynamic moduli of the MREs decreased with the rise of strain amplitude. The stiffness and damping properties of the isotropic and anisotropic MREs increased with increasing of excitation frequency and magnetic flux density. The magneto-induced dynamic mechanical properties of anisotropic MREs were better than the isotropic MREs. The relative MR effect of the isotropic and anisotropic MREs enhanced with increasing the magnetic flux density. The relative MR effect of the anisotropic MREs was greater than that of the isotropic ones. The dependence of dynamic properties of the isotropic and anisotropic MREs on the frequency and magnetic field intensity was numerically examined using the four-parameter fractional derivative viscoelastic model. The viscoelastic model was fitted well to experimental data in the studied frequency band for all magnetic flux densities for both isotropic and anisotropic MREs.
Author contribution statement Tran Huu Nam carried out the experiment with support from Iva � and Bohdana Marvalova �. Tran Huu Nam performed modeling Petríkova and numerical calculations with support from Iva Petríkov� a and Boh �. Tran Huu Nam wrote the manuscript with support dana Marvalova � and Bohdana Marvalov� � and Bohdana from Iva Petríkova a. Iva Petríkova � supervised the research work. All authors discussed the re Marvalova sults and contributed to the final manuscript. Acknowledgements This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic and the European Union European Structural and Investment Funds in the frames of Operational Program Research, Development and Education project Hybrid Materials for Hierarchical Structures (HyHi, Reg. No. CZ.02.1.01/0.0/0.0/16_019/ 0000843).
Data availability
Appendix A. Supplementary data
The datasets generated during and/or analyzed during the current
Supplementary data to this article can be found online at https://doi. 10
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org/10.1016/j.polymertesting.2019.106272.
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