Proceedings of the ASME 2013 Conference on Smart Materials, Adaptive Structures and Intelligent
Systems
SMASIS2013
September 16-18, 2013, Snowbird, UT, USA
SMASIS2013-3222
FOLDING ACTUATION AND LOCOMOTION OF NOVEL MAGNETO-ACTIVE
ELASTOMER (MAE) COMPOSITES
Paris von Lockette
Rowan University
Glassboro, NJ USA
Robert Sheridan
Rowan University
Glassboro, NJ US
comparison. Folding structures are also investigated as a
means to produce untethered locomotion across a flat surface
when subjected to an alternating field similar to scratch drive
actuators; geometries investigated show promising results.
ABSTRACT
Magneto-active elastomers (also called magnetorheological
elastomers) are most often used in vibration attenuation
application due to their ability to increase in shear modulus
under a magnetic field. These shear-stiffening materials are
generally comprised of soft-magnetic iron particles embedded
in a rubbery elastomer matrix. More recently researchers have
begun fabricating MAEs using hard-magnetic particles such as
barium ferrite. Under the influence of uniform magnetic fields
these hard-magnetic MAEs have shown large deformation
bending behaviors resulting from magnetic torques acting on
the distributed particles and consequently highlight their
ability for use as remotely powered actuators.
Using the magnetic-torque-driven hard-magnetic MAE
materials and an unfilled silicone elastomer, this work
develops novel composite geometries for actuation and
locomotion. MAE materials are fabricated using 30% v/v 325
mesh barium ferrite particles in Dow Corning HS II silicone
elastomers. MAE materials are cured in a 2T magnetic field to
create magnetically aligned (anisotropic) materials as
confirmed by vibrating sample magnetometry (VSM). Gelest
optical encapsulant is used as the uniflled elastomer material.
INTRODUCTION
Magneto-active elastomers (MAEs) are a class of smart
materials consisting of hard-magnetic particles such as barium
ferrite embedded in an elastomer matrix. Magneto-active
elastomers may be more well known as a sub-class magnetorheological elastomers (MREs) though these materials
traditional consist of soft-magnetic fillers. MREs are driven
by magnetic interactions between particles which are largely
magnetostrictive in nature (exclusively so for spherical
particles). The magneostritive phenomena leads to MREs’
ability show enhancements to shear modulus under a magnetic
field[1-10]. By contrast, MAEs are driven primarily by
magnetic torques, though interparticle forces necessarily
occur. These torques, when distributed coherently throughout
a sample, create a distributed, magnetically induced,
mechanical moment which can be expressed as bending
behavior[11-14].
Mechanical actuation tests of cantilevers in bending and of
accordion folding structures highlight the ability of the
material to perform work in moderate, uniform fields of
. Computational simulations are developed for
This work seeks to capitalize on the highly flexible nature of
MAEs in conjunction with their distributed moment response
to develop prismatic and composites structures capable of
un/folding, producing mechanical work, and achieving self-
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Copyright © 2013 by ASME
structural model. The Maxwell surface stress methods (MSS)
seeks to solve conservation of momentum coupled with
Maxwell’s equations using a finite element approximation.
The minimum energy structural (MES) model couples the
magnetic Zeeman energy in the MAE to the elastic energy
stored in the material at the fold itself treating the material as
rigid segments connected by hinges. The efficacy of the
models is tested by comparisons with experimental data
collected on samples of the two geometries. While the method
incorporating the Maxwell surface stress (MSS) is capable of
dealing with finite thickness effects along a fold line the
minimum energy structural model (MES) assumes zerothickness in terms of its geometry.
PREVIOUS NUMERICAL MODELING
While several constitutive models of MRE behavior exist, and
have been used in finite element simulations, these models
deal with soft-magnetic material response and hence focus on
magnetostrictive mechanisms of actuation [15-25].
Magnetostrictive phenomena are incapable or producing
bending in a homogenous MAE cantilever as has been shown
previously [11,12]. A previous work by the author uses the
MSS method to predict MAE response for an MAE in a
cantilever configuration[12], but does not include possibly
important aspects of the mechanics including large
deformation kinematics and hyper-elastic material behavior.
Figure 1: (top) Cantilever geometry showing field alignment,
, and cantilever dimensions. (bottom) Accordion composite
geometry with four MAE patches, silicone rubber substrate
(PDMS), remanent magnetization alignments,
, an field
alignment .
SIMULATION METHODOLOGY
Maxwell surface stress (MSS) modeling
The MSS method seeks to solve static conservation of
momentum,
, coupled with Maxwell’s equations for
magnetostatics,
where is magnetic field vector,
and
where is the magnetic induction. The MAE
materials were assumed to behave magnetically following
where
is the remanent magnetization.
Elastically, the MAE and PDMS materials were initially
assumed to behave as Mooney-Rivlin solids with strain energy
locomotion. Two geometries are examined, a simple cantilever
and an accordion-like structure, and are assessed for the ability
to produce large fold angles due to the magnetically induced
distributed moment and for their ability to perform
mechanical work (see Fig. 1). A third L-shaped geometry is
investigated for its ability to achieve self-locomotion (see
Fig. 8).
Along a single axis a traditional fold may be considered a case
where the opposing sides which are separated along the fold
line, rotate about the fold like a hinge. For materials with
finite thickness, however, this can never be the case and as
such the effect of material behavior, especially nearing the
fold line, may be of importance both kinematically and
elastically. In this work behavioral models of the folding
behavior of MAEs and MAE composite are generated using
two methods: a coupled hyperelastic-Maxwell surface stress
based finite element approach as well as a minimum energy
density function
,
(
)
(
)
where and are the first and second principle stretch
invariants and is the Jacobian matrix which was assumed
unity for the incompressibility assumption. The elastic and
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Figure 2: Accordion model geometry showing segment orientation angles , fold angles
, MAE patch
length
, free substrate length, , and magnetic alignment of MAE patches, north vs. south poles.
magnetic behaviors are coupled through application of the
Maxwell surface stress
on the boundary between all
magnetic and nonmagnetic mediums, i.e. MAE-air and MAEPDMS. The Maxwell stress tensor,
, is computed
numerically from the solution to the magnetic flux problem as
(
field (not shown). The size of the air box was scaled 14 times
larger than the corresponding dimensions in the MSS model.
This size was determine by a convergence study on the
predicted deformed shape to determine when variation in the
deformed shape at the maximum field strength simulated
became less than 2% for an increase air box scale, eg.
to
.
)
in which is the magnetic induction an
is the Kronecker
delta. This stress
is then used to compute the magnetically
induced Maxwell surface stress, or traction, on the boundary
.
The external magnetic field was applied using a magnetic
vector potential function,
where A was assigned
values on the left and right boundaries such that the desired
magnetic fiel was achieved.
A two dimensional plane strain geometry was implemented in
Comsol to incorporate the MSS method to model the
cantilever with dimensions the same as those from
experimentation. The Comsol model include Green-Lagrange
strains (large deformation kinematics) and the augmented
Eulerian-Lagrangian “moving mesh” formulation. The MSS
cantilever model was fixed at the base but allowed to deform
freely on all other faces. In addition the Maxwell surface stress
condition was applied to all boundaries of the MAE. The
MSS accordion model was also sized to the experimental
sample, fixed at its base, allowed to deform freely on all other
faces, and aligned similarly with respect to gravity. For the
accordion model, however, the Maxwell surface stress
condition was applied to the boundaries of all four MAE
patches and computed separately over each.
Solution of the MSS model required values for the Mooney
Rivlin constants,
and
, material densities
and
, and remanent magnetization
along the alignment
axis.
Minimum energy structure (MES) modeling
The MES method was used to model only the accordion
geometry, see Fig. 2. This method assumed the MAE patches
and corresponding segments of the substrate acted as rigid
links between the more compliant open sections of the
substrate only. These open sections were assumed to act as
torsional springs. The model was pursued following
observations that the MAE patches did not bend noticeably.
The MES model of the accordion seeks to minimize the
combined Zeeman magnetic energy (e.g. sans demagnetizing
fields) of the MAE patches coupled with a linear torsional
spring energy that represents the elasticity of the fold. The
energy functional of the model is given as
The accordion and cantilever MSS models were further
surrounded by a nonmagnetic, passive air box in order to
simulate local flux density produced by an applied magnetic
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Copyright © 2013 by ASME
( )
∑[
(
e.g. perpendicular to the cast sheet. The accordion geometry
consisted of four active MAE patches,
, affixed to a passive poldymethylsiloxane (PDMS)
substrate
and equally spaced
apart. The MAE patches in the accordion structure
were aligned parallel to the length of the undeformed
accordion structure, e.g. in the plane of the cast sheet. The
accordion geometry was fixed at its base, as well. In both
experimental geometries the samples were aligned such that
gravity acted along the samples’ axis moving from the fixed to
the free ends.
) ]
where
,
is the magnetic permeability of
free space,
is the torsional spring constant, is the angle
the MAE patch makes with the horizontal (with
[
] , where is the
representing the fixed boundary),
applied field and
[( )
] for
oriented from south to north poles within the MAE patch,
respectively. The constants
an
represent the physical
volumes of the MAE patches and the elastic hinge sections,
respectively. Gravity is not considered in this model. In terms
of material parameters, the MES model requires knowledge of
the remanent magnetization of the MAE patches,
, and the
torsional elastic spring constant, . Fixed boundary conditions
were enforced by setting
at the fixed end; in addition
the elastic volume at the fixed end is
Experimental Configurations
Blocked force cantilever bending
The blocked force test was arranged in the configuration as
shown in Fig. 3. A non magnetic extension to a digital force
transducer was used to proscribe a displacement on the tip of
an MAE sample and simultaneously record the measured load
as field strength was varied through a C-shaped electromagnet.
The MAE in bending was extremely compliant with respect to
the extension in compression.
. The solution
of the MES model requires
yielding solutions for the four orientation angles, and thereby
the fold angles. MES method does not require solution of the
local flux density and therefore requires no air box.
EXPERIMENTAL METHODS
Sample Fabrication
Materials used in this work consist of the MAE patches and,
for composite geometries, a silicone rubber substrate. The
MAE material is comprise of 325 mesh, nominally 30% v/v
M-type barium ferrite (BaM) powder mixed with 70% v/v
Dow Corning HS II silicone rubber compound. The MAE
materials, cast in sheets, were cured in a magnetic field to
produce magnetic alignment of the samples. Two
magnetization alignments were fabricated, the parallel case
which was aligned in the plane of the cast sheet, and the
perpendicular case, which was magnetically aligned in the out
of plane orientation. Details of material fabrication may be
found in [11].
Figure 3: Schematic of blocked force test
in cantilever with load P and tip deflection
Experimental and MSS simulation results of blocked force
tests in the cantilever configuration are given in Fig. 4. A
representative deformed shape of the MSS finite element
model is given in Fig. 5. In these simulations, the elastic and
magnetic constants are found by fitting simulations to
experimental data for the
case, eg. purely elastic
proscribed tip displacement bending response, to determine
The two geometries, the cantilever and the accordion, were
examined in both numerical simulations and cut from the cast
and aligned sheets for magneto-mechanical experimentation.
The cantilever consisted of a
homogenous,
monolithic sample of MAE material fixed at its narrow end, its
base (see Fig. 1), and aligned through the
dimension,
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The model predicts the response well at low field strengths (an
zero tip deflection) , but begins to diverge at the higher field
strengths. This may be a result of non-uniform fields seen in
the area between the pole faces on the magnet. A perfectly
magnetically aligned material in a perfectly uniform magnetic
field would produce a uniformly distributed internal torque
which would in turn produce a bending moment
proportional to field strength. The model reflects this
behavior in the -propotional spacing between the simulation
curves with increasing . However, The fact that when the
sample is perfectly vertical, eg. no tip displacement, a
reduction is seen in the effective magnetization response also
suggests a degree of non-uniformity in magnetic alignment.
elastic parameters, or the
case, eg. the blocked
force test at zero tip displacement, to determine remanent
magnetization values. Best fits of these data yield
, and
for in units of . The magnetization in this
experiment is seen to vary with applied field strength. This
may occur due to the distribution of magnetic alignments of
the particles. The bulk of the particles are magnetically
aligned perpendicular to the field, however a fraction of the
particles have some magnetic component antiparallel to the
field. These particles are to some degree coerced and affect
overall magnetization.
Fold angle in accordion geometry
For the accordion model, the composite was placed between
the pole faces of a commercial electromagnet and subjected to
a transverse magnetic field as shown in Fig. 6a.
40
30
20
10
0
10
10
20
30
40
50
Figure 4: Comsol MSS simulations (lines) and
experimental data (points) for the blocked force
cantilever geometry.
Figure 6: (top left) representative experimental deformed
geometry in accordion structure; (top center) representative
MSS model of deformed shape; (top right) representative
MES model of deformed shape; (bottom) comparison of MES
(dotted line ) and MSS (solid line) models to experimental
data (points).
Figure 5: Comsol model of deformed
geometry.
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relative field orientations, however these large deformation
simulations encountered substantial numerical difficulties.
Numerical issues centered on difficulty in generating accurate
predictions for the local flux density near the corners of the
MAE patches. Vortecies in magnetic flux lines developed
near these corners which generated erroneous local Maxwell
surface stress vectors that increased in magnitude as field
strength was increased, limiting the average fold angle in
simulations to a maximum of roughly 40 degrees regardless
material parameters; the results of this model effectively
assymptoted to a particular deformed geometry for all
parameter values.
Figure 7: Results of dead load tests on accordion geometry.
Measurements of fold angles were taken from digital images
at varying field strengths and use to compute the average fold
angle across all three fold lines. Representative deformed
configurations are given and simulations are compared to
experimental data in Fig. 6. Fold angle results show reversible
behavior and large fold capability.
Average fold angle results using the MES method are also
compared to MSS and the experimental data in Fig. 6. The
MES method was not limited in terms of maximum achievable
fold angle and shows excellent agreement with the data both
qualitatively and quantitatively. The MES model, however, is
perfectly symmetric and hence reversal of field gives perfectly
symmetric behavior while the experimental results vary
slightly due to model geometry asymmetry.
The four-segmented accordion geometry was simulated using
both the MSS method and the MES approach. In the MSS
model, the elastic and magnetic constants were determined for
the MAE and PDMS substrate from direct experimentation,
vibrating sample magnetometry to determine
in plane for the MAE patches and tensile tests to determine
an
in the substrate. Tensile tests
on the in-plane MAE material showed it to be roughly
less compliant than the substrate, therefore little to no
deformation of the MAE patches was expected (nor was it
observed experimentally). Consequently, to ease numerical
complexity a linear elastic material model was used in the
MAE sections with
Accordion model dead load test
The accordion geometry was subject to a dead load test using
varying masses over a range of field strengths. The composite
was fixed at its top while various masses were attached to the
free end and the external magnetic field was varied. In these
tests the amount of work performed by the composite
( ), was calculated as
( )
( ) where
structure,
is the weight of the mass attached to the free end of the
composite and ( ) is the translation of the mass against the
force of gravity at a given applied field strength. Results
neglected the weight of the composite. Results given in Fig. 7
showed that the composite structure was most effective
against lower loads and required an initial minimum field
strength of approximately 35 mT to produce work. The curves
for the highest loads suggest the behavior is dominated by the
effects of the magnetic torque over the elastic response as load
increases, the elastic stiffness is inconsequential when
compared to the load.
In the MES method simulations, the torsional spring constant
that models the elastic behavior of the substrate was found by
best fit to the maximum fold angle data point from
experiments at
in Fig. 6, yielding a fixed value
of
while remanent magnetization
of the MAE patches was given the value found experimentally
and used in the MSS accordion model,
. A
representative deformed configuration is shown in Fig. 6.
Results of the MSS method are plotted as normalized average
bend angle (right hand axis) vs. field strength and show good
qualitative agreement with the experimental data, however the
magnitude of the simulated fold angles is approximately 1/2 of
the experimental result. The results show the ability of the
model to capture the characteristic response of the system, the
dependence of the magnetic torque on deformed geometry and
UNTETHERED LOCOMOTION
Composites consisting of active MAE patches bonded to a
passive PDMS substrate in an L-configuration were
investigated for possible untethered self-locomotion behavior,
see Fig. 8. These composite geometries were derived from
scratch drive actuators[26,27]. Motion is produced as the
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Copyright © 2013 by ASME
Figure 8: L-shaped geometry for untethered locomotion
consisting of MAE patches bonded to PDMS layer (in-plane
remanent magnetization orientation shown in black).
MAE patches cause the L to expand with the initial half-cycle
of the field, reaching forward, and contract with the ending
half cycle of the field, drawing up its trailing edge. The Lshaped geometry shown in Fig. 8 was tested under an
oscillating field of
and for varying silicon
strip length, shown in Table 1. Results of observed,
unidirectional translation versus design parameters are given
in Table 1 under Performance. Results show the structure of
Designs 1 and 2 are capable of unidirectional motion under a
transverse magnetic field and that shorter silicone strip lengths
promote translation.
This work has investigated varying geometries of MAE
composites for their ability to fold/unfold under the influence
of an external field, their ability to perform mechanical work,
and their ability to produce unidirectional motion.
Additionally, methods of modeling MAE behavior using a
Maxwell surface stress (MSS) method and a minimum energy
structure (MES) approach were develop and compared.
Blocked force experiments showed the ability of MAE
cantilevers to perform work against load. Simulations using
the MSS method for the cantilever, however, diverged from
experiments at increasing field strengths when deformed.
Since variation in magnetization at higher field strengths was
incorporated into the model in the sample’s undeformed state,
this divergence suggests a geometry- (deformation-)
dependent phenomena and may be a result of non-uniform
field effects of the magnet used.
Table 1: L-Shape geometry parameters and
untethered locomotion results
Design (1)
Design (2)
Design (3)
Front magnet length (in.)
.815
.83
.83
Front magnet with (in.)
.49
.47
.47
Silicon strip length (in.)
1.8
2
2.75
Silicon strip width(in.)
.5
0.5
0.5
Back magnetic length (in.)
.875
.875
.875
Back magnet width (in.)
.37
0.47
0.47
0.138
0.0625
0
Performance
(travel per cycle /composite
length)
The accordion geometry was also shown capable of un/folding
behavior and large fold angles, over 70 degrees, in
experiments. While the MSS simulation method was unable
to quantitatively capture the response of the structure,
qualitatively the behavior was predicted quite well. Errors in
simulations arose from difficulty in accurately modeling the
local magnetic field through transitions from magnetic to nonmagnetic domains near corners in the geometry. This problem
was solved by carful mesh creation in the MAE cantilever, but
similar methods were not able to overcome the use of multiple
discrete MAE patches and subsequent large translations of the
composite accordion structure. The minimum energy structure
(MES) simulation method, however, proved capable of
CONCLUSIONS
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capturing the fold angle behavior very well and had no issues
with large deformation. It is important to note that the MES
method does not require calculation of the local magnetic flux
density. The accordion composite geometry was also shown
capable of producing mechanical work in dead load
experiments. The accordion structure was most efficient
against lower loads. Finally, an L-shaped composite structure
was found capable of untethered, unidirectional locomotion
transverse to an oscillating magnetic field. The L-shape was
more effective as the length of the long side of the L was
reduced.
This work has investigated several aspects of MAE and MAE
composite response to magnetic fields. These initial
quantifications of their ability to produce large fold angles,
mechanical work, and untethered locomotion highlight a need
to conduct further research on MAE materials and MAE
composite geometries for use as actuators, untethered
machines, and in folding applications. Furthermore,
observations on the use of MSS and MES simulation methods
highlight a need for further understanding of the numerical
issues that arise between coupled magneto-elastomechanics
simulations as well as begin to question the need for direct
calculation of local magnetic flux in simulations altogether
when only predictions of macroscopic response are desired.
ACKNOWLEDGMENTS
This work was sponsored by NSF EFRI grant # 1240459.
The authors would also like to thank the work of the Penn
State an Rowan University Origami Engineering Capstone
Design Teams, Fall 2012.
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