Proceedings of the ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems
SMASIS2012
September 19-21, 2012, Stone Mountain, Georgia, USA
SMASIS2012- 8143
ACTUATION BEHAVIOR IN PATTERNED MAGNETORHEOLOGICAL ELASTOMERS:
SIMULATION, EXPERIMENT, AND MODELING
Paris von Lockette,
Rutgers University, New Brunswick, NJ1
Rowan University, Glassboro, NJ2
1
Visiting Scientist, Mechanical and Aerospace Engineering Department
2
Associate Professor, Mechanical Engineering Department
1
Copyright © 2012 by ASME
ABSTRACT
Magnetorheological elastomers (MREs) are an emerging
class of smart materials whose mechanical behavior
varies in the presence of a magnetic field. Historically
MREs have been comprised of soft-magnetic iron
particles in a compliant matrix such as silicone elastomer.
Numerous works have experimentally cataloged the MRE
effect, or increase in shear stiffness, versus the applied
field. Several other researchers have derived constitutive
models for the large deformation behavior of MREs. In
almost all cases the arrays of embedded particles, and or
the particles themselves, are assumed magnetically
symmetric with respect to the external magnetic field, i.e.
the bulk materials exhibit magnetic symmetry in the given
experimental or analytical configuration.
In this work the author presents results of dynamic shear
experiments, Lagrangian dynamic analysis, and static
shear simulations on MRE material systems that exhibit
broken magnetic symmetry. These new materials utilize
barium hexaferrite powder as the magnetically anisotropic
filler combined with a compliant silicone elastomer matrix.
Simulations of representative laminate structures
comprised of varied arrays of magnetic particles exhibit
novel actuation behaviors including reversible shearing
deformation, variable magnetostriction, and most
surprisingly, piezomagnetism. Results of dynamic shear
experiments and analytical modeling support predicted
shearing actuation responses in MREs having broken
symmetry and only in those systems..
INTRODUCTION
Magnetorheological elastomers (MRE) are particulate
composites consisting of ferromagnetic particles in an
elastomer matrix that may be cured in a magnetic field to
produce some patterning of the embedded particles [1,2].
MRE behavior is driven by the interplay of magnetic
forces and torques coupled to the hyperelastic response
of the matrix. Magnetic forces, which are quadratic in
nature and hence not reversible, will exist between
particles regardless of particle shape, arrangement,
and/or magnetic behavior. These forces drive
magnetostrictive behaviors [3]. However, the shape of the
embedded particles and their magnetization response
determines whether or not magnetic torque is generated;
magnetic torque behavior is linear, and hence reversible,
in nature for hard-magnetic materials [4]. Magnetic
symmetry, symmetry between the internal magnetization
and the external applied field, must be broken at the
particle level to create reversible torque-driven behaviors
on a particle. It must also be broken at the aggregate level
to create macroscopic torque-driven behaviors distributed
throughout a sample.
Past works on MREs, theoretical, computational, and
experimental, have almost exclusively dealt with materials
exhibiting magnetic symmetry and thus have not taken
advantage of an entire class of possible behaviors that
result from torque generation(see [4] for a more thorough
discussion). Several models of MRE behavior exist which
link dipole energy formulations of various arrangements of
soft-magnetic particles with macroscopic deformation of a
hyperelastic matrix, such as an Ogden or Arruda-Boyce
material, to yield deformation- magnetic field-dependent
elastic responses (for example [5,6]). Other researchers
have formulated continuum models that link the
interaction of the external field with the particle’s
magnetization, sometimes including interparticle affects,
coupled with the mechanics of deformation to yield similar
deformationmagnetic
field-dependent
predictive
models(see for example [7,8]) . In all of these cases
magnetic symmetry is preserved at the particle and/or
aggregate level. Consequently, these works have
examined only responses available through interparticle
forces,
i.e.
quadratic
responses
driven
by
magnetostrictive effects.
Only recently have researchers begun examining the
effects of broken symmetry by analyzing MRE systems
comprised of ellipsoidal particles and/or particles of hardmagnetic material. A series of papers
lay the
homogenization framework [9] for studying such a
material, however the most recent paper in the series
[10], while addressing broken geometric symmetry
through particle aspect ratios, employs soft-magnetic
material behavior and still maintains aggregate symmetry.
The particles’ major axes are assumed collinear with the
field. The formidable problem of analytically quantifying
the role broken magnetic symmetry at the micro- and
macro-level plays in MRE response has yet to be
thoroughly studied. Recent work by the author begins to
address the role broken symmetry plays in MRE response
through experimentation [4]. The work quantitatively
characterizes the degree of magnetization-induced torque
behavior produced in various types of hard- and softmagnetic MREs. The work highlights how aligned hardmagnetic MRE materials can be well characterized by
magnetic torques while MRE comprised of soft-magnetic
material are better characterized by inter-particle forces
arising from demagnetizing fields.
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Copyright © 2012 by ASME
This work begins to address the problem by
investigating the effects of MRE microstructure on
macroscopic response using Comsol, a commercial
multi-physics finite element package, to solve the
coupled magneto-elastic problem. In addition,
experiments in dynamic shear response are
conducted and linked to a Lagrangian dynamic
analysis that highlights the need for breaking
magnetic symmetries in device application situations
to provide true actuation. Fundamentally, this work
argues for the importance of understanding magnetic
torque generation in MREs and how the manipulation
of distributed magnetic torques may be used to
develop novel material behaviors. Torque is
manipulated by varying the microstructure through
spatial variations in particle alignments and
magnetization behaviors. Finally, magnetic torques
are employed to generate actuation behavior that is
employed in dynamic shear.
(a)
(b)
(c)
Figure 2: MRE microstructure classifications (a)
uniform (b) alternating, (c) random. Blue arrows show
magnetization which is characterized by (a) orientation
with respect to horizontal, (b) orientation of first column
with respect to the horizontal with subsequent mirrored
columns, (c) orientation of individual magnetizations
with respect to he horizontal.
Figure 1: Schematic of MRE simulation architecture showing
material domains, boundary conditions, particle orientations,
and applied field orientation.
SIMULATION METHODOLOGY
The systems studied herein consist of a 5x5 array of
ferromagnetic particles arranged in a latticework structure
with defining particle orientation angle (measured with
respect to the horizontal axis) embedded in an elastic
matrix and surrounded by a volume of air (see Figure 1).
An aligned column of five particles results in over 90% of
the energy density of an infinite chain supporting this
initial sizing for these investigations [6,7].
Regular
arrayed structures have been assumed by previous
researchers developing constitutive theories therefore,
given the debate over the actual distribution of particles
and magnetizations in real systems [11], a similar
structure is justified here. The system studied is twodimensional as shown. The particle volume concentration
is fixed at 30% (here by area) recreating the optimal
mixture ratio shown repeatedly for soft-magnetic iron
based materials across a range of particle size and
particle size distributions [1,2,12,13] . Particle aspect ratio
and orientation, however, are variable.
Material properties of the composite’s constituents are
taken either from values in the literature or from
experimental values determined through in-house
magnetic and mechanical characterization(see [4] for
details). The hard-magnetic inclusions are modeled as
barium ferrite (BaM) with constant orthotropic remanant
magnetization. For small fields a constant valued easyaxis component of the remanant magnetization of BaM
and a negligible (zero) hard-axis component are
warranted. BaM at a 30% volume fraction in silicone
elastomer is chosen as a model constituent for
comparison with experimental studies using these same
material detailed later in this work. Model material
parameters are given in Table 1.
E
BaM
Air
Matrix
[]
1
1
1
[T]
0.5
-
-
[Pa]
200e9
-
866e3
[]
.29
-
.49
Table 1: Material Properties Used in Simulations
Multiphysics finite element simulations are developed in
COMSOL, a commercial package, to solve the magnetoelastic problem, which is coupled through the Maxwell
surface stress tensor, across three separate domains.
The three domains are the air box surrounding the
composite, the matrix material, and the set of twenty-five
inclusions. Maxwell’s equations are solved over all three
domains while equations of elasticity are solved only over
the matrix and inclusions. The formulation uses an
augmented Lagrangian-Eulerian description such that
field variables are calculated in the deformed
configuration as defined by the current solution to the
elasticity problem in an iterative fashion. Mesh points in
the air box are displaced smoothly between the air box
outer boundary and the matrix material’s outer boundary,
which conforms to the elasticity solution, using a
Laplacian operator.
Three separate array configurations, differentiated by the
orientation of the embedded particles, are simulated each
with its own set of displacement boundary conditions (see
Figure 2). The uniform array (Figure 2a) has orientation
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(a)
(b)
(c)
Figure 3: Representative finite element results of (a) random and (b) uniform microstructures showing
magnetizations (blue arrows), Maxwell traction on three particles (red arrows) and magnetic flux density
(background color, red=high, blue-low); (c) shear strain response vs. orientation angle for three field
strengths. The random orientation shows negligible response, <10% of maximum 15mT strain
response.
angle fixed for all particles. The alternating (Figure 2b)
array has orientations fixed by columns which are
mirrored. The random array (Figure 2c) has particle
orientations that are randomized. In all configurations,
however, particle centers adhere to the lattice structure
with a fixed distance on centers by row and column. The
uniform array is used to study the shear actuation
behavior of hard-magnetic MREs with symmetry broken
by particle aspect ratio in conjunction with orientation. The
alternating array is used to investigate how the roles of
interparticle forces and external-field induced torques vary
with broken symmetry in microstructure. The random
array highlights the effect of complete isotropy on MRE
response.
The displacement and applied field boundary conditions
vary by array configuration and are depicted in Figure 1.
In all three configurations external fields of
0 … 0.045 are applied in the
direction across all three
domains using a magnetic vector potential on the vertical
faces while the top and bottom faces of the air box are
perfect magnetic conductors; the magnetic boundary
conditions are the same for all three configurations. In
addition, for all array configurations the outer boundary of
the air box has fixed displacement. Inner matrix-air
boundary conditions on displacement vary to
accommodate varying deformation responses being
studied (See Figure 1). For the uniform and random
arrays, the bottom of the matrix-air boundary is fixed to
allow free shearing deformation while all other faces of
the matrix-air boundary are free. For the alternating
arrays, roller boundary conditions are used on the left and
bottom faces of the matrix-air boundary to recreate
quarter symmetry in displacement.
The system of equations governing the problem account
for linear elasticity, e.g. small deformation and linear
material behavior, as well as hard-magnetic behavior also
in a linear regime. A linear magnetization solution is
acceptable due to the small magnetic fields studied,
30% of the coercive field. The resulting small
deformations expected warrant linear elasticity theory
given appropriate initial moduli for the hyperelastic
material.
Solving the problem requires solution of Maxwell’s
equations for the magneto-static case with zero applied
electric field over all three domains. The governing
equations for the reduced magneto-static case can be
given as
where
is the applied magnetic
is the applied surface current density, and
is
field,
the magnetic flux density. Employing the potential
formulation
where
is a magnetic vector
potential defines as the solution variable.
The general constitutive equation for the magnetic
response of the air medium and the soft-magnetic
yielding the
particulate domains is given by governing equation for the air medium and soft-magnetic
particle domains in terms of the solution variable,
(1)
where
4
, is the relative permeability of the material and
10 is the permeability of free space.
The general constitutive equation of a hard magnetic
therefore,
material can be expressed by the governing equation for the hard-magnetic domains is
given by
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(a)
(c)
(b)
Figure 4: a) Finite element model of periodic MRE structure showing flux density streamlines and magnetization
45mT c) magnetostriction (strain) vs.
for
arrows b) magnetostriction (strain) response vs. orientation for
0 45°. In (b) the magnetostriction is shown to change sign with field for angles near 20 degrees. In (c) the
response is shown to transition from quadratic (concave down) near 0 degrees to nearly linear at 15 degrees back to
quadratic (concave up) at 45 degrees.
where is the initial modulus and Possion’s ratio. The
small strain tensor can be represented as
(2)
which in terms of the solution variable yields
(3)
The elastic and magnetic problems are directly coupled
through the Maxwell surface stress, given as
where , the displacement field, is the solution variable
sought. The resulting governing equation for the elasticity
problem in terms is given as,
(7)
(4)
where is the outward normal to the particle surface and
is a traction boundary condition applied to each particlematrix boundary.
The elasticity problem is defined by the conservation of
linear momentum in the static case without body forces,
where
is the Cauchy stress. For linear
or ·
material behavior generalized Hooke’s Law is
employed where
is the stress tensor, and
is the
stiffness matrix defined in 2D plane stress by
1
(5)
(6)
0
1
0
0 0 1
·
The system of equations is solved in an iterative fashion
with updates to the displacement field affecting particle
orientations and thereby the magnetic vector potential
solution which in turn necessitates recalculation of the
Maxwell surface stress boundary tractions on the
particles. Simulations require roughly 2M degrees of
freedom for a converged mesh solution with respect to
magnetostriction or shear strain.
To illustrate the effects of microstructure, Figure 3a shows
deformed mesh results (not to scale) of simulations of
hard-magnetic particles with aspect ratio
1 embedded
in the silicone elastome matrix subjected to a
0.03 external field where
follows the + -axis for
varying orientation angle configurations. In Figure 3a the
particles have random magnetization orientation while in
Figure 3b the particles have uniform magnetization
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Magnet pole
face
Acrylic
vibrating
mass,
MRE
sample with
in-plane
poling
Magnet pole
face
Figure 5: Schematic of single lap shear experiment
showing magnet pole faces, poled MRE sample, and
acrylic vibrating mass.
orientation, both denoted by groupings of (blue) arrows.
In Figures 3a and 3b the three particles along the
northwest-southeast diagonal also have the Maxwell
surface stress vector depicted along the particle
boundaries by (red) arrows. Figure 3c shows the resulting
shear strain versus magnetization orientation for three
field strengths. The strain is zero when the array of
particles is symmetric with respect to the field (
90°)
and maximized, in magnitude, when the array is
orthogonal to the field (
180,0 °, 180°). In the
orthogonal microstructures the strain response is also
linear with respect to applied field and consequently not
driven by traditional quadratic magnetostrictive forces.
The strain direction is also seen to reverse as the
orientation traverses from 180 to 0 and back to -180
degrees which is expected as it gives the right hand rule
for magnetic torque actuation. Finally, the strains are
reversible with the sign of external field (not shown).
Figure 4 shows simulation results of an alternating array
of hard-magnetic particles with aspect ratio
2 and
subject to a varying -axis magnetic field. Figure 4a
shows a representative deformed mesh (not to scale) for
45° with remanent magnetizations depicted by
groupings of (blue) arrows within the particles and
magnetic flux density as streamlines. Figure 4b shows the
normalized -displacement, the magnetostriction, in the
sample versus first column orientation angle for two field
0.045 (intermediate values omitted
strengths,
5° … 20° the
for clarity). Within a range
magnetostricion is shown to change signs with field
strength. This is critically important because in classical
definitions magnetostriction is quadratic and hence not
reversible. Figure 4c highlights the microstructure in
which the response is reversible by recasting the data as
normalized -displacement vs. applied field strength
45 … 45 for six discrete values of
(
orientation angle
0°, 7.5°, 15°, 30°, 37.5°, 45° . The
simulation results clearly show transition from typical
quadratic magnetostrictive strain behavior to a linear and
reversible response at
15°. The results suggest that
microstructures described by the 15° alternating array
may show “piezomagnetic” behavior, e.g. reversible axial
strain vs. field response, which is uncommon in
traditional magneto-active materials.
DYNAMIC SHEAR ACTUATION EXPERIMENTS AND
MODELING
The ability of MREs with broken magnetic symmetry to
exhibit novel actuation mechanisms coupled with the
observed importance of the magnetic symmetry in the
particles and the aggregate suggests a reexamination
the use of MREs as shear-stiffening elements in typical
vibration absorbers. Specifically an examination of the
effect of broken symmetry in a typical shear setup
provides insight that may be useful in modeling the
response of H-MREs in application contexts.
Consider an acrylic oscillating mass, , attached in
single shear to a hard-magnetic MRE material bushing
with thickness , as shown in Figure 5. The material is
poled during curing such that the easy- (shown with block
and
arrows) and hard-magnetic (not shown) axes,
respectively are aligned as shown. The material is also
subjected to an external time varying field
aligned
vertically as shown. A Lagrangian, , description of the
system,
, where
is the kinetic energy of the
is the potential energy (elastic
mass and
), will allow direct determination of the
plus magnetic
equations of motion based on sample bulk magnetizations
and . The kinetic energy in small deformation is
given by
where is the sample height and
is the shear strain; the elastic energy by
where A is the cross sectional area and
is the purely
elastic shear modulus; and the magnetic potential energy
·
1/2 ·
where
is
by
the demagnetization field,
0,0; 0,1
is the
demagnetizing tensor for the macroscopic sample.
The equation of motion is then given from the well known
where in this instance the generalized
form
displacement, , follows
velocity , follows
.
and the generalized
Next consider the components of the magnetization of a
, poled in plane such that
bulk sample,
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which can be approximated by
0
for small fields. Under a shear strain the magnetization
′
will
rotate
following
where
cos , sin ; sin , cos . Given these assumptions
the equation of motion can be shown to be
(8)
γ
γ
This equation has the form of standard spring-massdamper configurations used to model traditional MRE
field-dependent shearing behavior but includes an
additional forcing term on the right hand side representing
a magnetic actuation which Is not present in soft-magnetic
MRE models. In eq. (8) is the damping coefficient ,
is proportional to the elastic spring stiffness and
is proportional to a constant valued magnetic stiffness.
The magnetic forcing term highlights the expected ability
of this material in this configuration to produce active
shearing behavior that is reversible in
and is
independent of deformation. This behavior stands in
contrast to traditional soft-magnetic MRE materials in
which the elastic stiffness term (and only that term) shows
quadratic field dependence, and therefore is not
reversible, and which is linear in displacement and
therefore provides no response at zero deformation.
Figure 6: Measured peak velocity of vibrating mass
vs. frequency of applied magnetic field. The magnetamplifier system is tuned to provide 150 mT DC; the
field decays with frequency.
amplifier-electromagnet system; the applied
nominally 150 DC, diminishes with frequency
field,
An experimental setup was developed to test for the
ability of H-MREs poled in plane to produce shearing
actuation. The test configuration follows the arrangement
show in Figure 5 using an MRE bushing, 20 50 3 in
single lap shear attached to one pole face of a C-shaped
electromagnet; the other side of the bushing was affixed
to a 13g acrylic mass. The material was fabricated using
nominally 40
BaM powder at 30% v/v in Dow Corning
2
HS II silicone elastomer and cured in a
magnetic field to produce magnetic alignment. A
sinusoidal magnetic field was provided by an amplifier
driven electromagnet. The field was varied from 2.5 – 100
Hz while a laser Doppler velocometer was used to
measure vibrations of the mass. Baseline measurements
of the mass’s vibration taken with no field and the
magnet’s vibration taken with an oscillating field but no
sample present show negligible background vibration in
the system compared to subsequently measured
oscillations of the magnetic-bushing-mass system.
CONCLUSIONS
This work has presented results of simulations of the
effect of varying microstructure on the actuation behavior
of MRE materials. Specifically, the role of
broken
symmetry in magnetization has been explored. Shear
simulations of uniformly ordered particles show that hardmagnetic particles having an easy-axis misaligned with
the external field will produce actuation driven by particle
level torques; this is in addition to interparticle forces
which are also present in soft magnetic materials.
Furthermore, magnetic isotropy (symmetry), either from
alignment with the external field or from completely
random configurations precludes this actuation.
Additional simulations of alternating microstructures show
the ability to transition from magnetic torque dominated to
magnetic force dominated behavior by highlighting the
transition from linear to quadratic magnetostritive
behavior.
Figure 6 shows the measured peak velocity of the mass
versus frequency of the applied magnetic field. The figure
clearly shows the ability of the system to actuate the mass
in a shearing mode. It must be noted that the
characteristic shape of the curve stems from the inherent
dynamics of the system described by equation (8) in
conjunction with the impendence characteristics of the
Finally, a Lagrangian dynamic analysis of a single lap
shear oscillator based on coupled elastic and magnetic
energies predicts the occurrence of shearing actuation
for materials with the easy-axis lying in plane,
perpendicular to the applied field. This geometry would
produce mislignment (broken symmetry) between the
external field and the internal magnetization of the sample
thereby producing a distributed torque throughout similar
to the results of simulated misaligned microstructures.
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Copyright © 2012 by ASME
Single lap dynamic shear experiments conducted to test
this hypothesis were successful in showing macroscopic
shear actuation resulting from an alternating magnetic
field.
These results strongly suggest that distributed torque
generation via careful patterning of hard magnetic MRE
microstructures provides a means of producing reversible
shear actuation beyond the well known stiffness increase
of traditional soft-magnetic MREs and possibly a
reversible magnetostriction or “piezomagnetism”.
ACKNOWLEDGEMENTS
The author would like to acknowledge the support of NSF
CMMI – 0927326 and Rutgers University, New Brunswick,
NJ where the experimental work was conducted during
the author’s tenure as a visiting scientist. Any opinions,
findings, and conclusions or recommendations expressed
in this material are those of the author(s) and do not
necessarily reflect the views of the National Science
Foundation
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