dynamic blocked transfer stiffness method of

© 2016 The Korean Society of Rheology and Springer 301

Korea-Australia Rheology Journal, 28(4), 301-313 (November 2016)DOI: 10.1007/s13367-016-0031-6

www.springer.com/13367

pISSN 1226-119X eISSN 2093-7660

Dynamic blocked transfer stiffness method of characterizing the magnetic field

and frequency dependent dynamic viscoelastic properties of MRE

Umanath R Poojary1,*, Sriharsha Hegde

2 and K.V. Gangadharan

1

1Department of Mechanical Engineering, National Institute of Technology Karnataka, Mangalore 575025, India2Department of Aeronautical and Automobile Engineering, Manipal Institute of Technology, Manipal 576104, India

(Received February 6, 2016; final revision received September 19, 2016; accepted September 22, 2016)

Magneto rheological elastomer (MRE) is a potential resilient element for the semi active vibration isolator.MRE based isolators adapt to different frequency of vibrations arising from the source to isolate the struc-ture over wider frequency range. The performance of MRE isolator depends on the magnetic field and fre-quency dependent characteristics of MRE. Present study is focused on experimentally evaluating thedynamic stiffness and loss factor of MRE through dynamic blocked transfer stiffness method. The dynamicstiffness variations of MRE exhibit strong magnetic field and mild frequency dependency. Enhancementsin dynamic stiffness saturate with the increase in magnetic field and the frequency. The inconsistent vari-ations of loss factor with the magnetic field substantiate the inability of MRE to have independent controlover its damping characteristics.

Keywords: MRE, direct stiffness method, viscoelasticity, dynamic stiffness, loss factor

1. Introduction

Base isolation is an effective way of minimizing the

threat to the structure's durability by isolating it from dam-

aging vibrations. Passive isolation through a flexible ele-

ment between the source and the disturbance is a commonly

adapted base isolation technique. The flexible element iso-

lates the structure by decoupling the transmitted vibration

from the source to the receiver. The passive isolators are

effective only at a particular operating frequency and

exhibit poor adaptability to different frequency of distur-

bances arising from the source (Eem et al., 2011; Li et al.,

2013; Stelzer et al., 2003). Limitations of passive isolation

technique are overcome by real time tunable smart mate-

rial based semi active isolators. The attributes possessed

by MRE analogues to a conventional viscoelastic resilient

element make it a potential smart material for semi active

isolation application (Eem et al., 2011; Kim et al., 2010;

Li et al., 2012; Opie and Yim, 2011).

MRE is a smart polymer composite comprising of fer-

romagnetic powdered ingredients embedded in a polymer

matrix. The ferromagnetic ingredients impart smart nature

to MRE by aligning along the flux lines of externally

applied magnetic field. The field induced characteristics of

MRE are assessed in terms of MR effect (Lokander and

Stenberg, 2003). MR effect is influenced by many pro-

cessing parameters like the type of matrix (Zhu et al.,

2013), type of filler particles (Koo et al., 2012; Lokander

and Stenberg, 2003; Padalka et al., 2010; Stepanov et al.,

2012), shape of the magnetic filler (Song et al., 2009), size

of the filler particle (Bose and Roder, 2009; Fan et al.,

2011; Hegde et al., 2014), volume percentage of filler par-

ticle (Boczkowska and Awietjan, 2009; Hegde et al.,

2015; Lokander and Stenberg, 2003; Qiao et al., 2012;

Tian et al., 2013), additives or plasticizers (Chen et al.,

2008a; Ge et al., 2013; Li and Sun, 2011), pre-cure vis-

cosity of the matrix (Lokander and Stenberg, 2003), and

the magnetic field applied during curing process (Lu et al.,

2012). The major factors influencing the MR effect are

type of filler particle, percentage content, and the type of

matrix. The optimum MR effect exists for the MRE with

a carbonyl iron particle (CIP) content of 25-30% by vol-

ume (Demchuck and Kuz'min, 2002; Hegde et al., 2015;

Lokander and Stenberg, 2003).

MRE also constitutes the group of filled elastomers due

to the embedded ferromagnetic fillers (Li and Sun, 2011).

Filler addition enhances the properties of elastomer by

inducing the reinforcing effect (Funt, 1988). The reinforc-

ing fillers are either nano sized or micron sized. In nano

size filler reinforced elastomers, the reinforcing effect is

dominated by the formation of filler aggregates (Funt,

1988; Ramier et al., 2006; Chazeau et al., 2000). How-

ever, the micron sized fillers impart reinforcing effect by

the formation of the matrix-filler interface (Stacer et al.,

1990; Stelandre et al., 2003; Gauthier et al., 2004). Unlike

elastic materials, the property of filled elastomer is neither

fully elastic nor fully viscous. This behavior is referred to

as viscoelastic, where the response includes the contribu-

tion of viscous and elastic effects (Lakes, 2009). The vis-

coelastic property of filled elastomer is influenced by the

operating parameters like, frequency, input strain, and the

temperature, which it inherited from the matrix (De La

Fuente et al., 2003; Leopoldes et al., 2004; Stacer et al.,

1990). Similar to the filled elastomer, the properties of*Corresponding author; E-mail: [email protected]

Umanath R Poojary, Sriharsha Hegde and K.V. Gangadharan

302 Korea-Australia Rheology J., 28(4), 2016

MRE under non-magnetized state are influenced by the

aforementioned operating parameters (Agirre-Olabide et

al., 2014; Chen et al., 2007; Gong et al., 2012; Li et al.,

2010a; Lu et al., 2012; Sun et al., 2008; Wang et al.,

2006). In addition, under magnetized state, MRE also

exhibits field dependent viscoelastic properties.

The viscoelastic properties of a filled elastomer are mea-

sured by transient method or dynamic rheological tech-

nique (Kumar et al., 2012). Transient method includes the

measurement of decaying rate of stress for the applied

constant instantaneous strain (Basdogon and Dikmen,

2011; Li et al., 2010b) or by measuring the rate of increase

in strain for the application of constant and instantaneous

stress (Lakes, 2009). The dynamic tests are preferred for

viscoelastic property characterization due to the inability

of transient method to effectively characterize the fre-

quency and the strain dependent response (Gunasekaran

and Ak, 2000). The dynamic test includes the measure-

ment of stress (strain) response of viscoelastic material for

input harmonic strain (stress) (Brown, 1996; Lakes, 2009)

in resonance (Gade et al., 1994; Medalia, 1978) or in non-

resonance region (Ooi and Ripin, 2011; Thompson et al.,

1998). The viscoelastic properties in non-resonance regions

are measured by DMA (Ramorino et al., 2003), Rheom-

eter (Li et al., 2010a) or from forced vibration tests (Lin

et al., 2003; Nadeau and Champoux, 2000; Ooi and Ripin,

2011; Thompson et al., 1998) as described in ISO 10846-

1. The forced vibration test method (ISO 10846-1) includes

direct stiffness measurement (ISO 10846-2) or indirect

stiffness measurement (ISO 10846-3). Direct method includes

two different types of dynamic stiffness estimations; dynamic

drive point stiffness and dynamic blocked transfer stiff-

ness. Susceptibility of dynamic drive point stiffness to the

mass between the force transducer and the test element

limits its applicability over higher frequencies (Nadeau

and Champoux, 2000; Ooi and Ripin, 2011). On the con-

trary, the dynamic blocked transfer stiffness overcomes

this limitation by estimation of stiffness with respect to the

blocked force (Nadeau and Champoux, 2000; Ooi and

Ripin, 2011). The viscoelastic property estimation from

dynamic blocked stiffness measurement is through the

receptance plot (Lin et al., 2003; Ooi and Ripin, 2011),

transmissibility plot (Koblar and Boltežar, 2016; Mallik et

al., 1999; Ramorino et al., 2003) or hysteresis loop (Ju et

al., 2012; Zhu et al., 2012). The hysteresis loop provides

the real time indication of linear viscoelastic response, and

it offers a simple procedure to extract dynamic stiffness

and loss factor (Brown, 1996).

The present study aims at experimentally evaluating the

magnetic field and the frequency dependent viscoelastic

properties of MRE through dynamic blocked transfer stiff-

ness method. Experiments are performed to assess the

effective operating range of RTV silicone based isotropic

MRE by harmonically exciting at frequencies below 80

Hz (representing the low frequency region as per ISO

10846-1) and magnetic field up to 318 kA/m. The dynamic

stiffness and loss factor are evaluated from the force-dis-

placement system hysteresis loops.

2. Experimental

2.1. Theoretical background

2.1.1. Dynamic stiffness

Dynamic stiffness represents the response of the visco-

elastic material in the linear viscoelastic region. It is

defined as the ratio of force response of MRE for the input

harmonic displacement (ISO 10846-1).

The viscoelastic property measurement approach of MRE

through direct stiffness method is schematically illustrated

in Figs. 1a and 1b. Input harmonic excitation to MRE

induces a force, F1 at the input end and a blocked force,

Fig. 1. (Color online) (a) and (b) Schematic representation of direct stiffness measurement approach for the characterization of MRE.

(c)-(e) Graphical representation of force-displacement system hysteresis loop.

Dynamic blocked transfer stiffness method of characterizing dynamic viscoelastic properties of MRE

Korea-Australia Rheology J., 28(4), 2016 303

F2 at the blocked end. The ratio, F2/X1 represents the

dynamic blocked transfer stiffness, which comprises of

viscous and elastic response of MRE (Lakes, 2009).

The mathematical representation of dynamic blocked

transfer stiffness is given by (Nadeau and Champoux, 2000),

(1)

where, K* represents the dynamic blocked transfer stiff-

ness. The in-phase component (K'), represents the stiff-

ness of MRE (real part of the dynamic blocked transfer

stiffness) and the out-of-phase component (K″), signifies

the energy dissipation capacity (imaginary part of the

dynamic blocked transfer stiffness). The in-phase and out-

of-phase components of the dynamic stiffness are evalu-

ated from the dynamic response of the system under sinu-

soidal input. In the linear viscoelastic region, the response

blocked force for the sinusoidal input displacement (Eq.

(2) and Fig. 1c) is also sinusoidal (Eq. (3) and Fig. 1d) in

nature but lags the displacement by a phase angle δ.

, (2)

. (3)

The in-phase and out-of-phase components of dynamic

blocked transfer stiffness are expressed as,

. (4)

The modified complex force-displacement response is

expressed as,

(5)

where, η =tanδ is the loss factor of MRE.

The viscoelastic response of MRE is represented in the

form of the force-displacement hysteresis loop by elimi-

nating the parameter t in Eq. (5). The mathematical expres-

sion to evaluate the dynamic stiffness and loss factor from

the force-displacement hysteresis loop (Fig. 1e) is given

by (Brown, 1996),

, (6)

Loss factor, . (7)

2.1.2. Energy dissipation

Energy dissipation of MRE under dynamic loading is

measured from the area under the force displacement hys-

teresis loop. The mathematical representation of deforma-

tion induced energy absorbed per unit volume by MRE at

time t is given by (Lakes, 2009),

. (8)

The energy dissipation under harmonic loading is ex-

pressed as,

.

(9)

The Ws represents the energy stored and Wd represents

the dissipated energy. Negligible inertial effect in dynamic

blocked transfer stiffness approach completely restores the

energy stored during stretching of the molecular configu-

ration (Luo et al., 2010).

The amount of energy dissipation estimated from the

dynamic blocked transfer stiffness for one complete cycle

of deformation is expressed as,

. (10)

The energy dissipation associated with viscous damping

in the material is given by (McConnel, 1995),

(11)

where, K″ = Cω. C represents the equivalent damping

capacity of MRE.

2.2. MRE synthesisThe fabrication process of isotropic MRE involves two

steps: mixing and curing. The MRE samples are prepared

by compounding 27% by volume CIP (diameter 5 μm;

BASF; Type CN) and 73% by volume of RTV silicone

matrix. The RTV silicone matrix is a two-part elastomer

(the weight ratio of base to crosslinker is 100:5) from

Dow Corning. MRE samples are prepared by blending the

particles and the matrix to form a homogenous mixture.

The mixture is then poured to an aluminium mould and

degassed in a vacuum chamber. Samples are cured at

room temperature under constant pressure for 15 h.

2.3. Dynamic property measurementThe schematic of dynamic viscoelasticity property mea-

surement experimental setup in SDOF configuration is

shown in Fig. 2. Sinusoidal signals generated from NI

PXI-5412 function generator are fed to the electrodynamic

shaker though a power amplifier system. MRE sample to

be tested in a single shear mode configuration (Li and

Sun, 2011) is fixed to the electrodynamic shaker through

a stinger. The magnetic field is provided by an electro-

magnet and field intensity is measured by Tesla gauge

(Lake Shore, Model 410). Force at the blocked end is

measured by a force transducer (KISTLER, type 9712).

The input strain to MRE is monitored by the signals from

an accelerometer (KISTLER, K-shear) attached to the

input end (ISO:10846-2). The sensed signals from accel-

erometer and force transducer are acquired through NI

PXI-4496 data-acquisition system.

Viscoelastic property measurements of MRE are per-

K* = K′ iK″+( ) =

F2 t( )

X1 t( )-----------

X1 t( ) = X1,0 sinωt

F2 t( ) = F2,0 sin ωt δ+( )

K′ = F2,0

X1,0

--------cosδ and K″ = F2,0

X1,0

--------sinδ

F t( ) = K′ 1 iη+( )X t( )

K* =

F0

X0

-----

η = FB

FA

------

W = 0

2π

ω

------

∫ F t( )X·

t( )dt

W = ωF0X0 0

2π

ω

------

∫ cosωt sinωt cosδ + sin2ωt sinδ( )dt = Ws+Wd

Wd = πF2,0X1,0 sinδ = πK″X 1,0

2

Wd = πCωX 1,0

2



formed under steady state harmonic excitation to exclude

the errors associated with frequency sweep (Hegde et al.,

2015). The experiments are performed at frequencies: 6

Hz, 10 Hz, 20 Hz, 25 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz,

and 80 Hz. The influence of magnetic field is evaluated by

extending the viscoelastic property measurement studies

at different magnetic field strengths (0 kA/m, 159 kA/m,

239 kA/m, and 318 kA/m). Experiments are performed at

1.25% strain to ensure linear viscoelastic response. To

exclude the stress softening effect, MRE samples are sub-

jected to preconditioning cycles (Lion and Kardelky,

2004; Höfer and Lion, 2009).

Consistency and accuracy in the viscoelastic property

characterization are ensured by performing the experi-

ments on three identical MRE samples. Each set of exper-

iments is repeated 10 times, and the average value has

been considered for the analysis. The stable response of

the system is ensured by measuring the 10 cycles for each

loading case. The error bar representation is adapted to

indicating the uncertainty in dynamic stiffness and loss

factor estimation. MRE samples are allowed a recovery

period of 15 mins (Stacer et al., 1990) between successive

measurements to exclude the effect of deformation history

on the measured viscoelastic properties.

3. Results and Discussion

3.1. Steady state force-displacement relationshipThe steady state response of MRE is evaluated from the

force-displacement hysteresis loops. Figs. 3a and 3b rep-

resent the frequency dependent hysteresis loops recorded

for 0 kA/m and 318 kA/m. The variation in magnetic field

and frequency dependent hysteresis behavior of MRE is

assessed from the resistance force offered by the MRE at

the blocked end. The maximum resistance force offered

by MRE at 6 Hz and 0 kA/m magnetic field is 0.3416 N.

Fig. 2. (Color online) (a) Schematic representation of dynamic viscoelastic property measurement experimental setup and (b) an image

of the main apparatus.

Fig. 3. (Color online) (a) and (b) Steady state hysteresis loops of MRE at different frequencies corresponding to 0 kA/m and 318 kA/

m magnetic field.



Resistance force experienced by the MRE is increased to

0.429 N by increasing the frequency to 80 Hz. As evident

from the inset of Fig. 3, the effects of frequency on the

variation in hysteresis characteristics are not pronounced,

and it diminished above 50 Hz. The frequency dependent

saturation behavior of MRE can be correlated to the rigid

composite characteristics exhibited by the filled rubber

with the increase in frequencies (Chen et al., 2007).

The frequency dependency is a typical characteristic

observed in unfilled elastomer when it is subjected to har-

monic loading (Deshpande, 2010). This behavior is asso-

ciated with the relaxation of polymer chains under the

applied deformation. At low frequency, the chains have

sufficient time to rearrange themselves and therefore, the

viscous effect dominates the response of elastomer. With

the increase in frequency, the chains become rigid as the

time available for the molecular rearrangement is less,

which results in pseudo-solid behavior (Ahmed et al., 2006;

Chen and Xu, 2011; Ginic-Markovic et al., 2000). The

relaxation behavior of the unfilled elastomer is affected by

the addition of fillers, which influences its frequency

dependency characteristics. Embedded fillers occupy the

Fig. 4. (Color online) (a)-(f) Magnetic field dependent hysteresis characteristics of MRE at different frequencies.



space between the bulk polymer chains and offer restric-

tion to the mobility of the polymer chains under harmonic

loading (Jong, 2005; Osman and Atallah, 2006). This

reduced motion of polymer chains imparts pseudo-solid

behavior at lower frequencies compared to the unfilled

rubber (De La Fuente et al., 2003; Funt, 1988; Leopoldes

et al., 2004; Osman and Atallah, 2006). The above phe-

nomena can be evident from the saturation in hysteresis

characteristics of MRE with the increase in frequency

above 50 Hz.

Figures 4a-4f represent the variation in hysteresis behav-

ior of MRE with the increase in magnetic field at different

excitation frequencies. The hysteresis behavior of MRE

varies with the increase in magnetic field. The field induced

variation is pronounced up to a magnetic field of 239 kA/m

and the variation is not articulated for the increase in mag-

netic field from 239-318 kA/m. Field induced hysteresis

characteristics of MRE are associated with the interaction

between the ferromagnetic fillers, which functions as

magnetic dipoles under magnetic field (Dong et al., 2012).

These interactions vary the mechanical properties by alter-

ing the internal microstructure of MRE (Stepanov et al.,

2008).

Field dependent variation in hysteresis behavior of MRE

is unaffected by the increase in frequency. Corresponding

to 6 Hz, the maximum resisting force experienced by the

MRE is increased from 0.337 N at 0 kA/m to 0.571 N at

318 kA/m. The cumulative increase in resisting force is

53.33%. Under similar conditions, increasing the frequency

to 80 Hz, the resisting force is increased by 49.12%.

3.2. Dynamic stiffness of MREDynamic stiffness is referred to as the ratio of the resist-

ing force offered by the MRE for the given input displace-

ment. Fig. 5a represents the variation of dynamic stiffness

of MRE at different frequency and magnetic field. The

dynamic stiffness of MRE enhances with the increase in

magnetic field and the frequency.

Magnetic field induced enhancement in dynamic stiff-

ness of MRE is assessed in terms of MR effect (Jolly et

al., 1996; Lokander and Stenberg, 2003). The variation in

MR effect with the frequency and the magnetic field is

represented in Fig. 5b. MR effect is enhanced with the

increase in magnetic field which signifies the increased

interaction between the ferromagnetic fillers. Variation in

MR effect corresponding to 318 kA/m magnetic field is

listed in Table 1. The dynamic stiffness of MRE under non-

magnetized state and at 318 kA/m magnetic field is denoted

by Ko and ΔK0.318, respectively. The MR effect is denoted

by the ratio ΔK0.318/Ko. As evident from Table 1, frequency

dependent variation in MR effect with the increase in fre-

quency is not pronounced, which signifies the weak depen-

dency of frequency on the field induced property enhance-

ments of MRE.

Field induced stiffness enhancement of MRE is attributed

to the localized compression of the matrix caused by the

magnetic force in the vicinity of the filler. The mechanism

of the magnetic field induced localized compression of the

matrix is schematically represented in Fig. 6. The phe-

nomenon of localized compression can be analysed on a

unit cell of MRE (Jolly et al., 1996). Unit cell represen-

Fig. 5. (Color online) (a) Magnetic field induced dynamic stiffness measured at different frequencies and (b) absolute increase in

dynamic stiffness at different magnetic field and frequency.

Table 1. MR effect (dynamic stiffness enhancement) of MRE at

different frequencies.

Frequency K0 K0.318 ΔK0.318 ΔK0.318/K0

(Hz) (kN/m) (kN/m) (kN/m) (%)

6 4.478 6.952 2.474 55.24

15 4.781 7.414 2.633 55.07

25 4.91 7.617 2.707 55.11

50 5.2 7.99 2.79 53.65

60 5.365 8.04 2.675 49.86

80 5.51 8.1 2.59 47.00



tation is an enlarged view of a section of the microstruc-

ture image in Fig. 6a, comprising of a pair of ferromagnetic

fillers separated by elastomer matrix. Schematic represen-

tation of MRE unit cell under non-magnetized and mag-

netized state is represented in Figs. 6b and 6c, respectively.

The dipole interactions induce a magnetic force of attrac-

tion and compress the matrix locally (Leng et al., 2015) as

illustrated in Fig. 6c. Localized compression increases the

stiffness of the matrix in the vicinity of the fillers and con-

tributes to the overall stiffness enhancement when bulk

effect is considered.

As evident from Fig. 5b, the field induced dynamic stiff-

ness enhancement of MRE is pronounced up to a mag-

netic field of 239 kA/m but the enhancement is not

significant between 239 kA/m and 318 kA/m. The dispar-

ity in dynamic stiffness enhancement is associated with

the variation in the energy interaction between the dipoles.

The interaction energy E, between a pair of dipoles in a

MRE unit cell is expressed as (Liao et al., 2012),

(12)

where, µ0 is the permeability of vacuum, µm is the relative

permeability of the elastomer matrix, d0 is the separation

distance of two adjacent particles, ζ = , and

m is the magnetic dipole moment of each iron particle.

As illustrated in the Fig. 6c, the magnetic force of attrac-

tion between the fillers varies the separation distance

between the dipoles which is expressed as (Guo et al.,

2014),

(13)

where, εmatrix is the localized compressive strain experi-

enced by the matrix. The εmatrix is increased with the

increase in magnetic field. The increase in εmatrix enhances

the energy interaction between the dipoles by reducing the

separation distance.

The compressive force arising from the field induced

interaction between the fillers increases the εmatrix, which

in turn enhances the localized stiffness of the matrix

(Fukushi et al., 2013; Martinelli, 2005). At lower mag-

netic field, the variation in εmatrix, is dominated by the

magnetic force of attraction. But, under higher magnetic

field, the contribution of matrix resistance force arising

from the localized stiffness enhancement results in reduc-

tion of variations in εmatrix. This phenomenon influences

the field induced energy interaction between the fillers and

imparts saturation behavior to MRE above 239 kA/m as

reported in past studies. (Chen et al., 2008a; Chen et al.,

2008b; Li and Gong, 2008).

3.3. Energy dissipation and equivalent dampingVibration isolation characteristics of a material depend

upon the ability to dissipate energy. Variation in energy

dissipated under different magnetic fields, and frequencies

are depicted in Fig. 7. The energy dissipation monotoni-

cally increases with the increase in magnetic field as well

as the frequency. Energy dissipation recorded for 6 Hz and

0 kA/m magnetic fields is 0.052 mN-m. Increasing the

magnetic field to 318 kA/m, the energy dissipation is

increased to 0.084 mN-m with a total increase of 0.028

mN-m. The field induced enhancement in energy dissipa-

tion is consistent with the increase in frequency. At 80 Hz

excitation frequency, the energy dissipation of MRE is

enhanced by 0.0292 mN-m for the increase in magnetic

field from 0 kA/m to 318 kA/m.

The energy dissipation (D) of MRE under dynamic

loading is the cumulative contribution of energy dissipa-

tion of the matrix (Dm), energy dissipation of the particles

(Dp), energy dissipation due to friction at matrix-filler

interface (Dpm), and energy dissipation due to ferromag-

E = 1

4πμmμ0

------------------–4ζm

2

d 0

3------------

Σk 0=

k=∞1/k

31.202≈

d = d0 1 εmatrix–( )

Fig. 6. (Color online) (a) Microstructure image of MRE, (b) schematic representation of a MRE unit cell under non-magnetized state,

and (c) unit cell of MRE under magnetized state representing the localized compression of the matrix.



netic filler-filler interaction (Dpp). The energy dissipation

capacity, D is expressed mathematically as (Chen et al.,

2008b; Fan et al., 2011; Ju et al., 2012),

. (14)

Energy dissipation of the micron sized filler particles is

minimum compared to the other effects (Ju et al., 2012).

The modified equation of D by excluding the effect of Dp

is written as,

. (15)

The relative slippage between the particles constitutes

the energy dissipation from the ferromagnetic filler-filler

interaction (Dpp). Contribution of Dpp on energy dissipation

in filled elastomers is significant at higher filler reinforce-

ment (Yang et al., 2012). The scope for filler-filler inter-

actions exists with the increase in number of filler particles

resulting from the increased filler particle content. As evi-

dent from the microstructure images shown in Fig. 6a, the

major contribution to Dpp in MRE arises from the agglom-

eration of CIP. The CIP particles are clustered at the sites

of agglomeration and surrounded by polymer matrix. The

fillers at the agglomeration experience relative slippage

under non-magnetized state, which is inhibited by the

presence of filler interactions under magnetic field. Mag-

netic force binds the particles at the agglomeration sites by

reducing the scope for relative slippage, which in turn

diminishes the contribution of Dpp on energy dissipation of

MRE.

The energy dissipation of the matrix, Dm represents the

intrinsic damping of the polymer matrix of MRE (Yang et

al., 2012). The Dm in MRE results from the polymer chain

friction at the molecular level of the matrix (Fay et al.,

1991). The chain friction is caused by the interaction

between the adjacent polymer chains during the coordi-

nated chain movement under harmonic shear loading (Tsai

and Whang, 2001; Tsai et al., 2007). The mechanism of

chain friction differs under magnetized as well as non-

magnetized state of MRE, which is attributed to the field

induced interactions between the fillers. The localized

compression caused by the magnetic force of attraction

brings the adjacent molecules of polymer chains closer, by

reducing the separation distance. This state can be envi-

sioned as the increase in chain density with the decrease

in inter-lock chain separation distance (Tsai and Whang,

2001). Figs. 8a and 8c represents the variation in number

of polymer chains per unit area in the vicinity of fillers

under non-magnetized and magnetized states. The unit

area under magnetized state is characterized by increased

number of polymer chains, which increases the interaction

between the chains and hence the chain friction. This dif-

ference in chain friction results increased contribution of

Dm on D of MRE.

Matrix-filler interface friction energy dissipation (Dpm)

depends on the interaction between the particle and matrix

at the interface (Yang et al., 2012). The Dpm is pronounced

in weakly bonded interface (Chen et al., 2008a; Ge et al.,

2013; Li and Gong, 2008; Li and Sun, 2011) which is

characterized by unstable chemical crosslink between the

matrix and the filler. This results in frictional energy dis-

sipation under the imposed deformation. The microstruc-

ture image of MRE shown in Fig. 6a reveals the existence

of a weakly bonded interface between the matrix and the

fillers as reported in past studies (Chen et al., 2008a; Qiao

et al., 2012).

The frictional energy dissipation, Dpm in MRE with weakly

bonded interface is mathematically expressed as (Chen et

al., 2008b; Li and Gong, 2008),

D = Dm + Dp + Dpm + Dpp

D = Dpp + Dm + Dpm

Fig. 7. (Color online) Variation in magnetic field dependent

energy dissipation of MRE at different frequencies.Fig. 8. (Color online) (a-b) MRE unit cell under non-magnetized

and magnetized states, (c) number of polymer chains per unit

area in the vicinity of ferromagnetic filler under non-magnetized

state of MRE unit cell, and (d) number of polymer chains per

unit area in the vicinity of ferromagnetic filler under magnetized

state of MRE unit cell.



(16)

where, F is the sliding friction force between the particle

and the matrix, n is the number of interfaces, and s is the

relative displacement at the interface.

The frictional force at the interface is mathematically

expressed as a function of normal force N and the friction

factor µ as (Chen et al., 2008b),

. (17)

The expression for the frictional energy dissipation Dpm

in terms of normal force N is given by,

. (18)

The normal force at the interface is a function of mag-

netic field, which causes the disparity in interface friction

energy dissipation. Under non-magnetized state of MRE,

the normal force is constant and it increases with the

increase in magnetic field. Increased normal force increases

the interface friction force and promotes interface friction

energy dissipation.

The contributions of Dm, Dpm, and Dpp on D are different

under magnetized as well as non-magnetized state. The

modified expression for energy dissipation in MRE under

magnetized state is expressed as,

. (19)

The cumulative contribution of Dm and Dpm increases the

energy dissipation with the increase in magnetic field. The

energy dissipation is pronounced between 0 kA/m and 239

kA/m, but above 239 kA/m, the energy dissipation is not

pronounced. This disparity in field induced energy dissi-

pation is attributed to the saturation behavior of MRE as

discussed in section 3.2. The saturation behavior of MRE

is characterized by the reduction in field induced localized

compression. This diminishes the field induced enhance-

ments in Dm and Dpm and contributes to reduction in

enhancements of D above 239 kA/m.

3.4. Loss factorLoss factor is defined as the ratio of imaginary to the

real part of the complex stiffness function (Lakes, 2009)

and it is estimated from the hysteresis loop (Brown, 1996).

Fig. 9 represents the variation in loss factor of MRE with

the magnetic field and the frequency. The loss factor of

MRE decreases with the increasing frequency, which is

attributed to the dominance of real part of the complex

stiffness over the imaginary part (Osman and Atallah,

2006; Yurkeli et al., 2001).

As evident from Fig. 9, MRE exhibits a unique trend in

loss factor variation with the increase in magnetic field.

The loss factor is decreased with the increase in magnetic

field up to 239 kA/m and it demonstrated an increasing

trend for the increasing the magnetic field from 239 kA/

m to 318 kA/m. The unique nature of loss factor variation

is analysed from the field induced variation in equivalent

viscous damping and stiffness.

For a SDOF system, the loss factor η is given by (Ungar

and Kerwin Jr., 1962),

(20)

where, C and K′ are the equivalent viscous damping capac-

ity and stiffness of the MRE, respectively, and ω is the

angular frequency of excitation. Variation of C and K′ with

respect to the magnetic field is represented in Fig. 10. The

equivalent damping capacity C of MRE is estimated from

Eq. (11) and the stiffness component, K′ is evaluated from

the resultant dynamic stiffness.

As represented in Fig. 10, MRE exhibits both C and K′

variations with respect to frequency and magnetic field.

Frequency dependent variation of C is more pronounced

compared to the stiffness. The variation of C with fre-

quency is more articulated for the frequencies up to 50 Hz

which is attributed to the dominance of viscous effect

(Deshpande, 2010; Payne, 1966). The contribution of vis-

cous component is diminished with frequency above 50

Hz indicating the dominance of elastic component (Jong,

2005; Osman and Atallah, 2006).

Modified expression for loss factor by incorporating the

field induced changes in C and K′ is expressed as,

(21)

where, Cref and K′ref are the corresponding equivalent damp-

ing and stiffness measured at reference magnetic field

strength. The incremental change in C and K′ with the

increase in magnetic field is denoted as ΔCincr and ΔK′incr,

respectively. The incremental change represents the increase

corresponding to the value measured at the next higher

Dpm = nFs

F = nμN

Dpm = nμN

D = Dm + Dpm

η = Cω

K′--------

η = Cref ΔCincr+( )ω

Kref′ ΔKiner′+-----------------------------------

Fig. 9. (Color online) Variation of magnetic field dependent loss

factor observed at different frequencies.



incremental magnetic field relative to the value registered

for reference magnetic field. For example, the C value

measured at 239 kA/m is taken as reference to evaluate

the incremental change in C between 239 kA/m to 318

kA/m.

The magnetic field dependent variation in ΔCincr and

ΔK′incr are represented in Fig. 11. As evident from the

graph, the increase in ΔCincr and ΔKincr is pronounced for

the increase in magnetic field between 0 kA/m-239 kA/m.

The variation in ΔCincr and ΔKincr is minimum for the mag-

netic field range of 239 kA/m-318 kA/m. For the magnetic

field below 239 kA/m, the field induced variations in

ΔKincr is pronounced over ΔCincr. But, ΔCincr increase is

pronounced compared to ΔKincr increase for an increase in

magnetic field from 239 kA/m to 318 kA/m. The differ-

ence in ΔKincr and ΔCincr variations with respect to the

magnetic field contributes to the field induced increase

trend and decreasing trend in loss factor.

4. Conclusion

This study is focused on experimentally investigating

the influence of magnetic field and frequency on dynamic

viscoelastic properties of MRE through blocked transfer

stiffness method. The experimental results show that the

dynamic viscoelastic properties of MRE are frequency

and magnetic field dependent. The Influence of magnetic

field on the dynamic viscoelastic property is stronger com-

pared to the frequency. Viscoelastic properties of MRE

saturate with the magnetic field as well as with the fre-

quency. The saturation limit for viscoelastic properties of

tested MRE sample is around 50 Hz and 239 kA/m mag-

netic field. The field induced property enhancement in

MRE has least influence on the frequency. The loss factor

of MRE cannot be controlled independently as it is a func-

tion of field induced enhancement in stiffness and equiv-

alent damping.

Fig. 10. (Color online) (a) Magnetic field and frequency dependent stiffness variation of MRE and (b) magnetic field and frequency

dependent equivalent viscous damping variation of MRE.

Fig. 11. (Color online) (a) Magnetic field induced incremental change in stiffness and (b) magnetic field induced incremental change

in equivalent damping.



Acknowledgements

The authors acknowledge the funding support from

SOLVE: The Virtual Lab@NITK (Grant number: No.

F.16-35/2009-DL Ministry of Human Resources Develop-

ment) (www.solve.nitk.ac.in) and experimental facility

provided by Centre for System Design (CSD): A Centre

of excellence at NITK-Surathkal.

References

Agirre-Olabide, I., J. Berasategui, M.J. Elejabarrieta, and M.M.

Bou-Ali, 2014, Characterization of the linear viscoelastic

region of magnetorheological elastomers, J. Intell. Mater. Syst.

Struct. 25, 2074-2081.

Ahmed, J., H.S. Ramaswamy, and P.K. Pandey, 2006, Dynamic

rheological and thermal characteristics of caramels, LWT-Food

Sci. Technol. 39, 216-224.

Basdogon, I. and E. Dikmen, 2011, Modeling viscoelastic response

of vehicle door seal, Exp. Tech. 35, 29-35.

Boczkowska, A. and S.F. Awietjan, 2009, Smart composites of

urethane elastomers with carbonyl iron, J. Mater. Sci. 44, 4104-

4111.

Bose, H. and R. Roder, 2009. Magnetorheological elastomers

with high variability of their mechanical properties, J. Phys.-

Conf. Ser. 149, 012090.

Brown, R.P. 1996, Physical Testing of Rubber, 3rd ed., Chapman

& Hall, London.

Chazeau, L., J.D. Brown, L.C. Yanyo, and S.S. Stenstein, 2000,

Modulus recovery kinetics and other insights into the Payne

effect for filled elastomer, Polym. Compos. 21, 202-222.

Chen, L., X.L. Gong, and W.H. Li., 2008a, Effect of carbon black

on the mechanical performances of magnetorheological elas-

tomers, Polym. Test 27, 340-345.

Chen, L., X.L. Gong, and W.H. Li., 2008b, Damping of magne-

torheological elastomers, Chin. J. Chem. Phys. 15, 271-274.

Chen, L., X.L. Gong, W.Q. Jiang, J.J. Yao, H.X. Deng, and W.H.

Li, 2007, Investigation on magnetorheological elastomers

based on natural rubber, J. Mater. Sci. 42, 5483-5489.

Chen, Y. and C. Xu, 2011, Specific nonlinear viscoelasticity

behaviors of natural rubber and zinc dimethacrylate composites

due to multi-crosslinking bond interaction by using rubber pro-

cess analyzer 2000, Polym. Compos. 32, 1593-1600.

De La Fuente, J.L., M. Fernández-García, and M.L. Cerrada,

2003, Viscoelastic behavior in a hydroxyl-terminated polybu-

tadiene gum and its highly filled composites: Effect of the type

of filler on the relaxation processes, J. Appl. Polym. Sci.

88,1705-1712.

Demchuck, S.A. and V.A. Kuz'min, 2002, Viscoelastic properties

of magnetorheological elastomers in the regime of dynamic

deformation, J. Eng. Phy. Thermophys. 75, 396-400.

Deshpande, A.P., 2010, Oscillatory shear rheology for probing

nonlinear viscoelasticity of complex fluids: Large amplitude

oscillatory shear, In: Deshpande, A.P., J.M. Krishnan, and

P.B.S. Kumar, eds., Rheology of Complex Fluids, Springer,

New York, 87-110.

Dong, X., N. Ma, M. Qi, J. Li, R. Chen, and J. Ou, 2012, The

pressure-dependent MR effect of magnetorheological elasto-

mers, Smart Mater. Struct. 21, 075014.

Eem, S.H., H.J. Jung, and J.H. Koo, 2011, Application of MR

elastomers for improving seismic protection of base-isolated

structure, IEEE Trans. Magn. 47, 2901-2904.

Fan, Y.C., X. Gong, S. Xuan, W. Zhang, J. Zheng, and W. Jiang,

2011, Interfacial friction damping properties in magnetorheo-

logical elastomers, Smart Mater. Struct. 20, 035007.

Fay, J.J., C.J. Murphy, D.A. Thomas, and L.H. Sperling, 1991,

Effect of morphology, crosslink density, and miscibility on

interpenetrating polymer network damping effectiveness,

Polym. Eng. Sci. 31, 1731-1741.

Fukushi, T., S.H. Kim, S. Hashi, and K. Ishiyama, 2013, Mag-

netic silicone rubber: Fabrication and analysis with application,

J. Korean Phys. Soc. 63, 686-690.

Funt, J.M., 1988, Dynamic testing and reinforcement of rubber,

Rubber Chem. Technol. 61, 842-865.

Gade, S., K. Zaveri, H. Konstantin-Hansen, and H. Herlufsen,

1994, Complex modulus and damping measurements using

resonant and non-resonant methods, In: Zaveri, K., eds., Tech-

nical Review: Damping Measurements - From Impulse Response

Functions - From Resonance and Non-resonance Excitation

Techniques, Brüel & Kjær A/S, Naerum, 28-44.

Gauthier, C., E. Reynauda, R. Vassoillea, and L. Ladouce-Ste-

landreb, 2004, Analysis of the non-linear viscoelastic behavior

of silica filled styrene butadiene rubber, Polymer 45, 2761-

2771.

Ge, L., X. Gong, Y. Fan, and S. Xuan, 2013, Preparation and

mechanical properties of the magnetorheological elastomer

based on natural rubber/rosin glycerin hybrid matrix, Smart

Mater. Struct. 22, 115029.

Ginic-Markovic, M, N.K. Dutta, M. Dimopoulos, N. Roy Choud-

hury, and J.G. Matisons, 2000, Viscoelastic behaviour of filled,

and unfilled, EPDM elastomer, Thermochim. Acta 357-358,

211-216.

Gong, X., Y. Xu, S. Xuan, C. Guo, L. Zong, and W. Jiang, 2012,

The investigation on the nonlinearity of plasticine-like magne-

torheological material under oscillatory shear rheometry, J.

Rheol. 56, 1375-1391.

Gunasekaran, S. and M.M. Ak, 2000, Dynamic oscillatory shear

testing of foods-selected application, Trends Food Sci. Technol.

11, 115-127.

Guo, F., C.B. Du, and R.P. Li, 2014, Viscoelastic parameter

model of magnetorheological elastomers based on abel dash-

pot, Adv. Mech. Eng., 629386.

Hegde, S., K. Kiran, and K.V. Gangadharan, 2015, A novel

approach to investigate effect of magnetic field on dynamic

properties of natural rubber based isotropic thick magnetorhe-

ological elastomers in shear mode, J. Cent. South Univ. 22,

2612-2619.

Hegde, S., U.R. Poojary, and K.V. Gangadharan, 2014, Experi-

mental investigation of effect of ingredient particle size on

dynamic damping of RTV silicone base magnetorheological

elastomers, International Conference on Advances in Manu-

facturing and Materials Engineering (ICAMME 2014), Man-

galore, India, 2301-2309.



Höfer, P. and A. Lion, 2009, Modelling of frequency-and ampli-

tude-dependent material properties of filler-reinforced rubber,

J. Mech. Phys. Solids 57, 500-520.

ISO 10846-1, 2008, Acoustics and vibration-Laboratory mea-

surement of vibro-acoustic transfer properties of resilient ele-

ments-Part 01: Principles and guidelines, International Standard

Organization, Geneva.



ments-Part 02: Dynamic stiffness of elastic supports for

translator motion-direct method, International Standard Orga-

nization, Geneva.



ments-Part 3: Indirect method for determination of the

dynamic stiffness of resilient supports for translatory motion,

International Standard Organization, Geneva.

Jolly, M.R., J.D. Carlson, and B.C. Muñoz, 1996, A model of the

behaviour of magnetorheological materials, Smart Mater.

Struct. 5, 607-614.

Jong, L., 2005, Viscoelastic properties of ionic polymer compos-

ites reinforced by soy protein isolate, J. Polym. Sci. Pt. B-

Polym. Phys. 43, 3503-3518.

Ju, B.X., M. Yu, J. Fu, Q. Yang, X.Q. Liu, and X. Zheng, 2012,

A novel porous magnetorheological elastomer: Preparation and

evaluation, Smart Mater. Struct. 21, 035001.

Kim, Y.K., J.H. Koo, K.S. Kim, and S. Kim, 2010, Vibration iso-

lation strategies using magneto-rheological elastomer for a

miniature cryogenic cooler in space application, 2010 IEEE/

ASME International Conference on Advanced Intelligent

Mechatronics, Montreal, Canada, 1203-1206.

Koblar, D. and M. Boltežar, 2016, Evaluation of the frequency-

dependent Young's modulus and damping factor of rubber

from experiment and their implementation in a finite-element

analysis, Exp. Tech. 40, 235-244.

Koo, J.H., A. Dawson, and H.J. Jung, 2012, Characterization of

actuation properties of magnetorheological elastomers with

embedded hard magnetic particles, J. Intell. Mater. Syst. Struct.

23, 1049-1054.

Kumar, A., A.D. Stickland, and P.J. Scales, 2012, Viscoelasticity

of coagulated alumina suspensions, Korea-Aust. Rheol. J. 24,

105-111.

Lakes, R., 2009, Viscoelastic Materials, Cambridge University

Press, New York.

Leng, D., L. Sun, F. Gordaninejad, A. Bayat, and Y. Lin, 2015,

The dynamic performance of magnetic-sensitive elastomers

under impact loading, Smart Mater. Struct. 24, 045023.

Leopoldes, J., C. Barres, J.L. Leblanc, and P. Georget, 2004,

Influence of filler-rubber interactions on the viscoelastic prop-

erties of carbon-black-filled rubber compounds, J. Appl. Polym.

Sci. 91, 577-588.

Li, J.F. and X.L. Gong, 2008, Dynamic damping property of

magnetorheological elastomer, J. Cent. South Univ. 15, 261-

265.

Li, R. and L.Z. Sun, 2011, Dynamic mechanical behavior of

magnetorheological nanocomposites filled with carbon nano-

tubes, Appl. Phys. Lett. 99, 131912.

Li, W., X. Zang, and H. Du, 2012, Development and simulation

evaluation of a magnetorheological elastomer isolator for seat

vibration control, J. Intell. Mater. Syst. Struct. 23, 1041-1048.

Li, W.H., Y. Zhou, and T.F. Tian, 2010a, Viscoelastic properties

of MR elastomers under harmonic loading, Rheol. Acta 49,

733-740.

Li, W.H., Y. Zhou, T. Tian, and G. Alici, 2010b, Creep and recov-

ery behaviors of magnetorheological elastomers, Front. Mech.

Eng. China 5, 341-346.

Li, Y., J. Li, T. Tian, and W. Li, 2013, A highly adjustable mag-

netorheological elastomer base isolator for applications of real-

time adaptive control, Smart Mater. Struct. 22, 095020.

Liao, G., X. Gong, S. Xuan, C. Guo, and L. Zong, 2012, Mag-

netic-field-induced normal force of magnetorheological elasto-

mer under compression status, Ind. Eng. Chem. Res. 51, 3322-

3328.

Lin, T.R., N.H. Farag, and J. Pan, 2003, Evaluation of frequency

dependent rubber mount stiffness and damping by impact test,

Appl. Acoust. 66, 829-844.

Lion, A. and C. Kardelky, 2004, The Payne effect in finite vis-

coelasticity: Constitutive modelling based on fractional deriv-

atives and intrinsic time scales, Int. J. Plast. 20, 1313-1345.

Lokander, M. and B. Stenberg, 2003, Performance of isotropic

magnetorheological rubber materials, Polym. Test 22, 245-251.

Lu, X., X. Qiao, H. Watanabe, X. Gong, T. Yang, W. Li, K. Sun,

M. Li, K. Yang, H. Xie, Q. Yin, D. Wang, and X. Chen, 2012,

Mechanical and structural investigation of isotropic and aniso-

tropic thermoplastic magnetorheological elastomer composites

based on poly(styrene-b-ethylene-co-butylene-b-styrene) (SEBS),

Rheol. Acta 51, 37-50.

Luo, W., X. Hu, C. Wang, and Q. Li, 2010, Frequency- and

strain-amplitude-dependent dynamical mechanical properties

and hysteresis loss of CB-filled vulcanized natural rubber, Int.

J. Mech. Sci. 52, 168-174.

Mallik, A.K., V. Kher, M. Puri, and H. Hatwal, 1999, On the

modelling of non-linear elastomeric vibration isolators, J.

Sound Vibr. 219, 239-253.

Martinelli, A.E.B., 2005, Rubber Bearings for Precision Positioning

Systems, M.S. Thesis, Massachusetts Institute of Technology.

McConnel, K.G., 1995, Vibration Testing: Theory and Practice,

John Wiley & Sons, New York.

Medalia, A.I., 1978, Effect of carbon black on dynamic proper-

ties of rubber vulcanizates, Rubber Chem. Technol. 51, 437-

523.

Nadeau, S. and Y. Champoux, 2000, Application of the direct

complex stiffness method to engine mounts, Exp. Tech. 24, 21-

23.

Ooi, L.E. and Z.M. Ripin, 2011, Dynamic stiffness and loss fac-

tor measurement of engine rubber mount by impact test, Mater.

Des. 32, 1880-1887.

Opie, S. and W. Yim, 2011, Design and control of a real-time

variable modulus vibration isolator, J. Intell. Mater. Syst. Struct.

22, 113-125.

Osman, M.A. and A. Atallah, 2006, Effect of the particle size on

the viscoelastic properties of filled polyethylene, Polymer 47,

2357-2368.

Padalka, O., H.J. Song, N.M. Wereley, J.A. Filer II, and R.C.



Bell, 2010, Stiffness and damping in Fe, Co, and Ni nanowire-

based magnetorheological elastomeric composites, IEEE Trans.

Magn. 46, 2275-2277.

Payne, A.R., 1966, Physical Properties of Natural Rubber, M.S.

Thesis, Durham University.

Qiao, X., X. Lu, W. Li, J. Chen, X. Gong, T. Yang, W. Li, K. Sun,

and X. Chen, 2012, Microstructure and magnetorheological

properties of the thermoplastic magnetorheological elastomer

composites containing modified carbonyl iron particles and

poly(styrene-b-ethylene-ethylenepropylene-b-styrene) matrix,

Smart Mater. Struct. 21, 115028.

Ramier, J., C. Gauthier, L. Chazeau, L. Stelandre, and L. Guy,

2006, Payne effect in silica filled styrene butadiene rubber:

Influence of surface treatment, J. Pol. Sci. Part B Pol. Phy. 45,

286-297.

Ramorino, G., D. Vetturi, D. Cambiaghi, A. Pegoretti, and T.

Ricco, 2003, Developments in dynamic testing of rubber com-

pounds: Assessment of non-linear effects, Polym. Test 22, 681-

687.

Song, H.J., O. Padalka, M. Werely, and R.C. Bell, 2009, Impact

of nanaowire versus spherical microparticles in magnetorheo-

logical elastomer composites, 50th AIAA/ASME/ASCE/AHS/

ASC Structures, Structural Dynamics, and Materials Confer-

ence, Palm Springs, USA, AIAA 2009-2118.

Stacer, R.G., C. Hübner, and D.M. Husband, 1990, Binder/filler

interaction and the nonlinear behavior of highly-filled elasto-

mers, Rubber Chem. Technol. 63, 488-502.

Stelandre, L.L., Y. Bomal, L. Flandin, and D. Labarre, 2003,

Dynamic mechanical properties of precipitated silica filled rub-

ber: Influence of morphology and coupling agent, Rub. Chem.

Technol. 76, 145-160.

Stelzer, G.J., M.J. Schulz, J. Kim, and R.J. Allemang, 2003, A

magnetorheological semi-active isolator to reduce noise and

vibration transmissibility in automobiles, J. Intell. Mater. Syst.

Struct. 14, 743-765.

Stepanov, G.V., A.V. Chertovich, and E.Y. Kramarenko, 2012,

Magnetorheological and deformation properties of magneti-

cally controlled elastomers with hard magnetic filler, J. Magn.

Magn. Mater. 324, 3448-3451.

Stepanov, G.V., D.Y. Borin, Y.L. Raikher, P.V. Melenev, and N.S.

Perov, 2008, Motion of ferroparticles inside the polymeric

matrix in magnetoactive elastomers, J. Phys.-Condes. Matter

20, 204121.

Sun, T.L., X.L. Gong, W.Q. Jiang, J.F. Li, Z.B. Xu, and W.H. Li,

2008, Study on the damping properties of magnetorheological

elastomers based on cis-polybutadiene rubber, Polym. Test 27,

520-526.

Thompson, D.J., W.J. van Vliet, and J.W. Verheij, 1998, Devel-

opments of the indirect method for measuring the high fre-

quency dynamic stiffness of resilient elements, J. Sound Vibr.

213, 169-188.

Tian, T.F., X.Z. Zhang, W.H. Li, G. Alici, and J. Ding, 2013,

Study of PDMS based magnetorheological elastomers, J.

Phys.-Conf. Ser. 412, 012038.

Tsai, M.H., S.L Huang, P.C. Chiang, and C.J. Chen, 2007, Mor-

phology, dynamic mechanical properties, and gas separation of

crosslinking silica-containing polyimide nanocomposite thin

film, J. Appl. Polym. Sci. 106, 3185-3192.

Tsai, M.H. and W.T. Whang, 2001, Dynamic mechanical prop-

erties of polyimide/poly(silsesquioxane)-like hybrid films, J.

Appl. Polym. Sci. 81, 2500-2516.

Ungar, E.E. and E.M. Kerwin Jr., 1962, Loss factors of visco-

elastic systems in terms of energy concepts, J. Acoust. Soc. Am.

34, 954-957.

Wang, Y., Y. Hu, L. Chen, X. Gong, W. Jiang, P. Zhang, and Z.

Chen, 2006, Effects of rubber/magnetic particle interactions on

the performance of magnetorheological elastomers, Polym.

Test 25, 262-267.

Yang, J., X. Gong, H. Deng, L. Qin, and S. Xuan, 2012, Inves-

tigation on the mechanism of damping behavior of magneto-

rheological elastomers, Smart Mater. Struct. 21, 125015.

Yurkeli, K., R. Krishnamoorti, M.F. Tse, K.O. McElrath, A.H.

Tsou, and H.C. Wang, 2001, Structure and dynamics of carbon

black-filled elastomers, J. Polym. Sci. Pt. B-Polym. Phys. 39,

256-275.

Zhu, J.T., Z.D. Xu, and Y.Q. Guo, 2012, Magnetoviscoelasticity

parametric model of an MR elastomer vibration mitigation

device, Smart Mater. Struct. 21, 075034.

Zhu, J.T., Z.D. Xu, and Y.Q. Guo, 2013, Experimental and mod-

eling study on magnetorheological elastomers with different

matrices, J. Mater. Civ. Eng. 25, 1762-1771.

dynamic blocked transfer stiffness method of

Documents