Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 4, pp. 1061-1070, Warsaw 2014

VISCOPLASTIC PROPERTIES OF AN MR FLUID IN A DAMPER

Paweł Skalski

Institute of Aviation, Centre of New Technologies, Warsaw, Poland; e-mail: pawel.skalski@ilot.edu.pl

Robert Zalewski

Warsaw University of Technology, Institute of Machines Design Fundamentals, Warsaw, Poland

e-mail: robertzalewski@wp.pl

The aim of this paper is to mathematically describe and analyze the viscoplastic properties
of a magnetorehological (MR) fluid. The scope of the discussed research has been limited
to the T-MR SiMR 132 DG damper prototype. Laboratory tests have been performed on
a specially developed experimental stand with a kinematic excitation rule. On the basis
of the experimental data, conventional yield points and maximum shear stresses for the
analyzedMRfluid, includingvariable shear rate, current intensityflowing throughasolenoid,
liquid temperature and gap height have been analyzed. Basing on the acquired results, the
parameters of Bodner-Partom constitutive equations have been estimated. The identified
viscoplastic model has been used to carry out simulation tests that enabled verification of
the numerical results and real experimental data.

Keywords: MRfluid, modeling, Bodner-Partommodel, experiments, MR damper

1. Introduction

Magnetorheological fluids, beside ferromagnetic andelectrorheological fluids,belong to thegroup
of non-Newtonian rheostable fluids which are characterized by the yield point (Carlson and
Goncalves, 2008).Magnetic, ferromagnetic and electrorheological fluids are colloidal suspensions
ofmicroscopic solids ina liquidcarrier, andtheirmaincharacteristic is rapidgroupingofparticles
into a dense grid under the influence of an external stimulus (Carlson andWeiss, 1994).

Magnetorheological fluids (MRF) are very useful in solving damping problemswhich are one
of the biggest engineering dilemmas of the machines and devices construction and operation.
MRFs change their rheological properties under the influence of a magnetic field. These extra-
ordinary properties of MR fluids could not be fully utilized until the age of advanced computer
controlling equipment.

MRFs are used in dampers, shock absorbers, clutches and brakes (Goncalves, 2005). MR
dampers and shock absorbers are applied e.g. in damping control (Maślanka et al., 2007), in
operation of buildings and bridges (Bajkowski et al., 2012) or (Dyke et al., 1996) as well as in
damping of high-tension wires (Wu, 2006). Actually, MR fluids are used in large scale produc-
tion in the automotive and military industries. It is worth mentioning that MR fluids are also
successfully used as a part of various adaptive impact energy absorption systems (Mikułowski
and Holnicki-Szulc, 2007) or (Holnicki-Szulc et al., 2009).

In the development and production area of MR fluids, LORD Corporation is a dominating
figure on the global market manufacturing the majority of commercialized “smart” fluids and
devices. Although plenty of work is currently being led by universities and research centers,
there is still a need for a better and more extensive understanding of particular properties
of such liquids. Of special interest are investigations of their behavior in various operational
conditions, enabling better control of their changeable rheological properties.



1062 P. Skalski, R. Zalewski

Generally, we can distinguish two main approaches to the modeling of the MR dampers
response. In the first one, we are designing the mechanical model of the damper basing on
a system of fundamental rheological elements such as springs, dashpots or frictional sliders.
In this group of models, the simplest one and at the same time the most popular seems to
be the Bingham model (Sapiński and Filuś, 2003). Although it can not reflect the real force-
-velocity characteristic, it is very popular because of its simplicity and low numerical demand.
The second one involves sophisticated mathematical functions describing more accurately the
hysteresis loop of the investigated devices. In both considered cases, the models are identified
basing on phenomenological observations of the tested MR devices. In such an approach, the
model calibration is suitable only for selected constructions, andhas tobe repeatedwhenanother
type of MR device is under investigation.

The authors of the paper proposed another approach to themodeling ofMRdampers. They
attempted to describe the behavior of an MR fluid using viscoplastic constitutive equations
which are generally developed for metals. Unlike in the previously mentioned approaches, they
do not focus on designing parameters of specific MR dampers but consider only the main part
of the system which is the MR fluid. Before taking such a decision, one should ask and answer
the following questions: Can constitutive equations be used in the mathematical description of
properties of theMR fluid and its nonlinear behavior? Which ones are relevant, and why?

Answering these questions,we shouldnote that in certain operational conditions anMRfluid
changes its density, transforming from a liquid to viscoplastic semi-solid, or even solid state. It
is one of the most significant features of the investigated MR fluids.

The previously mentioned phenomenon encouraged the authors to describe MR fluids and
their viscoplastic properties by using the Bodner-Partom constitutive law.

2. Research object

Presented in this paper experimental tests have been carried out for the T-MR SiMR 132 DG
damperprototype (Fig. 1),workingon thebasis of a classical LORDCorporationproduct–MRF
132DG. In the investigated device, the gas spring has been neglected. The research programhas
been carried out using a laboratory standwith the kinematic excitation rule specially developed
at the Faculty of Automobile andConstructionMachinery Engineering ofWarsawUniversity of
Technology (Zalewski et al., 2014).

Fig. 1. View of the T-MR SiMR 132DG damper prototype

TheMRfluid examined in this paper is based on hydrocarbons. Typical physical properties
of the MRF 132 DG fluid are presented in Table 1, while detailed information is available on
the producer’s website – www.lord.com.



Viscoplastic properties of an MR fluid in a damper 1063

Table 1. Fundamental properties of theMRF 132 DG fluid (www.lord.com)

Properties MRF 132 DG

Viscosity, temperature 40 [◦C] 0.092±0.015Pas
Density 2.98-3.18g/cm3

Solids content by weight 80.98%

Operating temperature −40◦C-130◦C
Flash point > 150◦C

Appearance Dark grey

3. Results of experimental research

The experimental stand consisted of appropriate force, temperature, speed and displacement
sensors.During laboratory tests, theparameters suchas forceacting onthepiston rod,movement
of the piston housing, temperature of the outer casing and rotational speed have been recorded.

On the basis of acquired experimental data, detailed analysis of dissipative properties of the
investigated prototype has been performed. In this paper, we would like to focus on the impact
of chosen operational parameters such as excitation frequency, current intensity, temperature
and the gap size.

The final results of our research have been the force-displacement characteristics. As has be-
en previously mentioned, the excitation rule was in a harmonic function form x(t) =Asin(ωt)
acting directly on the investigated damper piston. The amplitude was assumed to be
A=10mm.Various rotation speeds of the circular camwere taken into consideration ωi =100,
200, 300rpm. The recording time for a single experimental series was t=5s, and the sampling
frequency 400Hz. The applied speed resulted in three different values of the piston oscillation
frequencies: f =1.66, 3.33, 5.00Hz. Also three currents were taken into account: I =0.5, 1.0,
2.0A. Additionally, three different gap sizes (where a flow of MR fluid took place) were exa-
mined: h = 5, 7, 10 ·10−4m. Although the gap size is one of the most important operational
parameters of MR devices mainly because of the value of the magnetic flux density across the
gap influencing the viscous properties of the fluid, in this paper this problem is omitted. Some
information related to this topic can be found in former works by Bajkowski et al. (2008).

The important issue was to determine the impact of oscillation frequency on the experi-
mentally obtained dissipative characteristics of the tested devices. Varying the velocity of the
piston movement, changed the speed of the MR fluid flow through the gap, which caused the
observed variable damping force. The range of investigated oscillation frequencies was limited
by the laboratory stand (the engine power).

Investigating at least three different values of shear rates (oscillation frequencies) is themini-
mumeffort required for the identification process of theBodner-Partomviscoplastic constitutive
model.

Typical dissipative characteristics obtained during laboratory tests are depicted in Fig. 2.

Increasing the current flowing through the solenoid of the MR damper prototype increases
the magnetic field acting on the working fluid. According to assumptions, a higher magnetic
field transforms the MR fluid from a dense liquid consistency into a gel-like substance. This
phenomenon is visible in Fig. 2, better dissipative properties of theMRdamper are observed at
higher currents.



1064 P. Skalski, R. Zalewski

Fig. 2. Damping force characteristics for various currents; temperature 25◦C, frequency 1.66Hz

4. Analysis of the MR fluid

The results of cyclic experimental tests of the MR damper became a basis for the analysis of
viscoplastic properties of the MR fluid under the influence of a magnetic field in operational
conditions of the damper. To prepare suitable data for conducting the identification process of
the Bodner-Partommodel, yield point values have been first determined for various shear rates
(Kłosowski et al., 2004).

Data depicted in Fig. 3a shows an example characteristic illustrating the full change in force
values acting on the piston rod of the tested damper in a rod displacement function. For further
identification only the initial part of the force-displacement diagramwas taken into consideration
(Fig. 3b).

Fig. 3. Damping force in function of the displacement; (a) full cycle, (b) data necessary for the
identification process

Prior to the relevant experimental research, the authors estimated the impact of the dry
friction force on the acquired results. To evaluate its influence, a series of laboratory tests
without theMR fluidwere conducted. The experimentally observed friction was less than 10N,
which is approximately 2% of the total value of the resistance force acting on the piston rod.
This negligible dry friction was consequently neglected in further laboratory tests.

On the basis of data depicted inFig. 3b, the shear stress τ, fluid shear strain γ, plastic shear
strain γI, may be written in a form

τ =
F

A
γ =
∆x

h
(4.1)

γI = γ−
τ

G
(4.2)

r= r1+
2

3
(r2−r1) A=2πrl (4.3)



Viscoplastic properties of an MR fluid in a damper 1065

where A is the shear surface area, h– gapheight, ∆x –displacement increment, G–Kirchhoff’s
modulus, r1 – internal radius of the gap, r2 – outer gap radius, r – calculation radius, l – length
of the gap.

The scheme of theMR damper head, including the shape of the gap, is illustrated in Fig. 4.

Fig. 4. Scheme of the damper with the selected working gap

Todetermine the shear rate value, itwas necessary to prepare an additional shear strain-time
characteristic. The shear strain in the loading was calculated from (4.1). As has been previously
mentioned in the experimental part, the sine excitation rule has been taken into account. The
authorsmade an assumption that the initial part of the applied in laboratory tests sine function
(circled area inFig. 5a) canbeapproximatedusinga linear function.Subsequently, the strain rate
value of interest was determined by using the slope of the line to approximate the experimental
data (Zalewski and Wolszakiewicz, 2011). The discussed stage of the identification process is
depicted in Fig. 5b.

Fig. 5. (a) Sine excitation rule; (b) determination of the shear rate value

Shear rate values for all the experimentally applied excitation frequencies have been deter-
mined. For example, for the gap height h=7 ·10−4m, the calculated shear rate values for the
oscillation frequencies 1.66, 3.33, 5.0Hz were 73, 144 and 2171/s, respectively.

The next step of the identification process was determining the yield point values of theMR
fluid (Pyrz and Zairi, 2007). This stage is graphically presented in Fig. 6.

The shear stress values were calculated from (4.1). It was assumed that the yield point value
was defined as the intersection point of two lines, approximating the elastic and plastic parts of
the shear stresses diagram (Fig. 6).

Another parameter which describes the MR fluid properties in the magnetic field is Kirch-
hoff’s modulus, also called the shear modulus. Its values were determined according to the
graphical scheme depicted in Fig. 6 (Pyrz and Zalewski, 2010).



1066 P. Skalski, R. Zalewski

Fig. 6. Evaluation of the yield point τ0 andKirchhoff’s modulus G of theMR fluid

Figure 7 illustrates theyield point variations obtained for threevalues of the current intensity
0.5, 1.0, 2.0A and various shear rate values 73, 144, 2171/s. Increasing the shear rate or current
intensity results in increasing the yield point values.

Fig. 7. Impact of the shear rate and current intensity on the yield point values; h=0.0007m

In the next step of the identification strategy, a plot of the shear stress against the inelastic
shear strain has been found (Fig. 8).

Fig. 8. Exemplary shear stress – inelastic shear strain diagram

5. Bodner-Partom model

TheBodner-Partommodel is a typical law formetals. Equations of thismodel allow description
of viscoplastic behavior of a material. This particular law has been selected from the wide list
of potential viscoplastic formulationsmainly due to its universality (Skalski, 2010). For example
in Zalewski (2010) or Zalewski and Pyrz (2010), the authors used a similar formulation in
the modeling process of an other innovative structure – Vacuum Packed Particles. The same
model was also used for description of the non-linear properties of solid propellants (Zalewski



Viscoplastic properties of an MR fluid in a damper 1067

et al., 2010), polymers (Zairi et al., 2007) or a metal alloy (Rowley and Thornton, 1996). The
constitutive formulation of Bodner-Partom can be expressed in the following form (Bodner and
Partom, 1975)

ε̇′ij =
3

2
ṗ
σ′ij

J(σ′rs)
(5.1)

where ε̇′ij is the deviatoric plastic strain rate tensor, ṗ – accumulated inelastic strain rate,
σ′ij – deviatoric stress tensor and J(σ′rs) – invariant of the plastic strain rate tensor.

After analysing the experimental results, the authors suggested some modifications to the
original Bodner-Partom viscoplastic model for the description of theMR fluid behavior. These
are related to omitting the hardening functions. The simplified constitutive law consists of three
material parameters: n,R0,D0, dependent on four operational variables: plastic shear rate γ̇

I,
current intensity i, temperature t, and gap height h. The constitutive equation of the Bodner-
Partom law was finally written in the form (Skalski, 2010)

γ̇I =2D0exp

[

−
1

2

(

R0√
3τ

)2n
n+1

n

]

sgn(τ) (5.2)

where n,R0,D0 are model parameters.

A large number of parameters describing the constitutive equations cause considerable pro-
blems with their identification. Identification methods of the Bodner-Partommodel parameters
arewell describedbyPyrzandZairi (2007). In this paper, themethodology and the identification
of parameters of the B-P law are based on the scheme given by Skalski (2010). The parameters
estimation has been performed using theMarquardt-Levenberg algorithm.

The yield stress values obtained for different shear rates values allow determination of the
first two parameters of the considered viscoplastic model (n,R0).

It can be assumed that for small values of inelastic deformation, the impact of the isotropic
hardening is R = R0, and the kinematic hardening in negligible. The following equation (5.3)
can thus be treated as the Bonder-Partom contractual definition of the yield point and can be
presented in the following form

τ0 =
R0√
3

(

2n

n+1
ln
2D0
γ̇I

)−
1

2n

(5.3)

Figure 9 shows the initial yield point values τ0 corresponding to the three experimentally in-
vestigated shear rate values. Then, using the least squares method and condition (5.3), n and
R0 parameters can be estimated based on the plot in Fig. 9. The parameter D0 is arbitrarily
chosen (Kłosowski et al., 2004).

Fig. 9. Evaluation of n and R0 parameters of the B-Pmodel



1068 P. Skalski, R. Zalewski

The final version of the B-Pmodel for the investigated MR fluid can be written in the form

γ̇I =2D0exp

[

−
1

2

(

R0(t,i,h)√
3τ

)2n(t,i)
n(t,i)+1

n(t,i)

]

sgn(τ) (5.4)

where

n(t,i)= 0.18+0.10i−0.0001t R0(t,i,h) =
1

h
(6.8−0.07i−0.01t) (5.5)

The value of the D0 parameter was selected 10
61/s according to (Skalski, 2010).

6. Numerical simulations and conclusions

The identified viscoplastic model was used to carry out numerical simulations to verify the
proposed mathematical model with experiments and to describe the behavior of the MR fluid
in the working gap of the head according to the experimental data. Selected verification results
are depicted in Fig. 10.

Fig. 10. Comparison of numerical simulation results of the Bodner-Partommodel with experimental
data; I =1.0A and shear rate: (a) 731/s, (b) 1441/s, (c) 2171/s

In this paper, the viscoplastic behavior of a typicalMRfluid has been studied. The inelastic
part of the constitutive law is governed by a unifiedmodel using internal variables representing
all aspects of viscoplastic deformation during the loading process of theMRdamper. In order to
acquire realisticmaterial parameters form the experimental data, an analyticalmethodology has



Viscoplastic properties of an MR fluid in a damper 1069

beenapplied.The initial resultshavebeenverifiedbynumerical simulationof the laboratory tests
using the identified material parameters. In all investigated cases, a relatively good coincidence
of the experimental and numerical curves can be observed. This confirms the universality of the
Bodner-Partom model and its capacity to capture the real behavior of anMR fluid.
A hybridmethodology, coupling both analytical and numerical identification (Zalewski and

Pyrz, 2013) and incorporating knowledge of the specific problem is worth investigating in future
studies.

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Manuscript received April 14, 2014; accepted for print June 3, 2014