Fabrication and Analysis of Planar Dielectric Elastomer Actuators
Capable of Complex 3-D Deformation
William Lai, Ashraf F. Bastawros, Wei Hong, Soon-Jo Chung
Abstract— A new design for a dielectric elastomer
actuator with geometrically confining reinforcements
is presented. The resulting structures enable complex
3-dimentional motion without the need of the
membrane prestretch. An in situ imaging system is
used to capture the complex deformation pattern to
evaluate the surface curvatures. The deformation
mode is analyzed analytically using the bi-laminate
theory to explore the actuator performance and
further develop analytical model amenable for
control strategies. A finite element material model is
also developed to couple the applied electric field to
the resulting deformation. The model is used to
analyze more complex deformation patterns. The
proposed confining reinforcements would enable the
development of flexible wings for agile aerial robotics
and compliant continuum robotics, utilizing the
proposed deformation mechanisms to provide
controllable many degrees of freedom.
E
I. INTRODUCTION
LECTROACTIVE polymers (EAPs), one of the
common suitable materials for artificial muscle
applications [1]–[3], are polymers that can induce
deformation under electrical stimulation. These actuators
exhibit considerable large displacement, acceptable
response time provide EAPs and high energy to mass
ratio [1]–[3]. A group of the EAP family: the dielectric
elastomers have especially been considered due to their
high strain, comparably short response time, low cost,
and high electromechanical coupling efficiency [4]–[7].
Basically, dielectric elastomer actuators (DEAs) are
made of incompressible soft dielectric elastomer
membranes sandwiched between compliant electrode
layers to form a dynamic capacitors. When electric field
is applied across the electrodes, the columbic force
William Lai is with Dept. of Aerospace Engineering, Iowa State
University, Ames, IA 50011, USA (e-mail: william7@iastate.edu).
Ashraf F. Bastawros is Professor at Dept. of Aerospace Engineering,
Iowa State University, Ames, IA 50011, USA
(e-mail: bastaw@iastate.edu).
Wei Hong is Assistant Professor at Dept. of Aerospace Engineering,
Iowa State University, Ames, IA 50011, USA
(e-mail: whong@iastate.edu).
Soon-Jo Chung is Assistant Professor at Dept. of Aerospace
Engineering, University of Illinois, Urbana-Champaign, IL 61801,
USA (e-mail: sjchung@illinois.edu).
generates a stress called Maxwell stress: electrostatic in
nature [8] that attracts both electrodes together and
squeezes the sandwiched DEA layer. As a result, the inplane expansion of DEA arises due to elastomer
incompressibility. One of the main motivations of building EAP based
planar actuators is to contribute toward the broader
problem of developing a flexible-winged micro aerial
vehicle (MAV) capable of agile flight in constrained
environments [9], [10]. Birds and bats are natural role
models for designing tailless MAV wherein the
aforementioned attributes can be engineered. MAVs
typically fly in the range of 2-20 m/s, and in a Reynolds
number range of 103-105 which coincides with that of the
birds. Fig. 1 shows a schematic of one such MAV
concept inspired by small birds such as the Barn
Swallow. It would lack a vertical tail, and, instead, use
the wing dihedral and twist effectively for control.
Biomimetic gliding flight without conventional
aerodynamic control surfaces such as ailerons and
rudders is considered in this paper. It is shown in [9,10]
that the dihedral (up-and-down flapping) angles of both
wings can be varied symmetrically to control the flight
speed of gliding or perching flight independently of the
angle of attack and fight path angle, while an asymmetric
dihedral setting can be used to control yaw in the
absence of a vertical stabilizer to enable agile
maneuvers. Flexible wings are usually lighter than
geometrically similar rigid wings, and flexibility acts as
a natural actuation amplifier [9].
For a flexible wing model that undergoes a large
deformation, an effective dihedral angle of a flexible
wing can be computed. As shown in Fig. 1, the wing
dihedral primarily produces a side force, which is
actually a component of the total force produced by the
wing normal to its local plane. Let YA and ZA denote the
local forces produced by the wing along the body y and z
is
axes, respectively. Then, the effective dihedral,
defined as [9]
YA  y  dy
 eff  tan 1 ( 
Z  y  dy
)
(1)
A
This paper presents one potential method of generating
dihedral deflections by using 2-D planar DEAs as shown
Fig. 1 A schematic showing a tailless aircraft concept with flexible
EAP-based wings [9,10]. This aircraft is motivated by small, agile
animals like birds and bats. The bottom four figures show how
symmetric and asymmetric dihedral angles can be used for
longitudinal and directional control (Picture drawn by Aditya
Paranjape [10]).
in Fig. 2. Our eventual goal is to control the effective
dihedral generated by DEAs that can generate the
desired longitudinal and lateral/directional control
capability.
It has been shown that membrane prestretch
greatly improves the DEAs performance [11], [12].
However, the prestretch is accompanied by several
technical setbacks. A supporting skeletal frame is
required to sustain the prestretched membrane and
thereby reducing the actuator stroke-to-weight ratio [13],
[14]. The total actuation strain is limited to the extent of
the prestretch; otherwise the actuator membrane would
wrinkle. Furthermore, non-uniform prestretch and stress
relaxation may affect subsequent actuation [15]. Our
preliminary work showed that the actuation strain of
DEAs is controlled by the biaxial prestretch ratio [16].
The membrane response could be greatly biased,
wherein direction with the larger prestretch tends to
induce lower actuation strain. Dielectric elastomer
minimum energy structures (DEMES) have been
introduced as a way to combine the role of prestretch
with compliant supporting structure to form a functional
3-D actuator [14], [17].
However, DEMES exhibit
several limitations for MAV with flapping wings. These
include: reduced energy to mass ratio, non-conforming
complex 3-D morphologies, incompatible with the
flapping aerodynamics, and finally the requirement of
complex control laws, derived from full scale finite
element simulation.
It has been shown that stacking planar expanding
actuator and inactive layer together can induce bending
and non-planar movement [13]. Expanding on the bimaterial laminate principles, we provided a different
method of constructing active/inactive layer combined
structure that demonstrated more complicated 3dimensional deformations with combined bending and
twisting. Our strategy is to manufacture laminates of
dielectric elastomers with geometrically confining
reinforcement
to
enable
complex
out-of-plan
deformation without the need of prestretch. The
stiffeners cover small area of the surface of the planar
actuators instead of full-area of the inactive layers. These
stiffeners would constrain the planar expansion
deformation along their axes, inducing a thickness
gradient of the expansion with thin DEA laminates. As a
result, a local curvature of the entire assembly would be
induced. In other words, these stiffeners guide the active
membrane deformation to form the targeted 3-D
complex shapes. Moreover, much thinner DEA
membranes have to be utilized to maximize the actuator
response under the applied electric potential.
In this study, we fabricated one-dimensional test
structures to analyze and understand the role of material
and geometric parameters on the actuator curvature and
the macroscopic stroke. Analytical model of the
deformation pattern was developed for further
integration into the control algorithm. Furthermore, we
fabricated test structures with different configuration of
stiffeners to generate complex 3-D configurations. For
such complex geometries and stiffeners, finite element
analysis was employed to optimize the structure for
given prescribed deformation pattern.
II. MATERIALS SELECTION
Several dielectric elastomers and silicones along with
compliant electrodes have been suggested in literatures
[3], [13], [18]–[20]. Here, in addition to acceptable
performance and geometric requirements, we also
considered materials availability and ease of actuator
fabrication and assembly into different reinforced
configurations.
A. Dielectric elastomer
A dielectric elastomer: 3M VHB (F9460PC) transfer
tapes were chosen in this study due to their suitable
electromechanical properties of the tape core material,
and their outer adhesive surface to retain the flexible
electrode coating, especially in powder form. Moreover,
the adhesive surface enabled multiple stacking of
fabricated unit cells for multilayer structures. The 3M
VHB tapes has been utilized in many DEAs prototypes
C. Stiffener
For stiffener, 3M Magic scotch tapes (~62.5μm thick)
were chosen in this study. This tape has good electrical
insulation, about the same thickness and several orders
of magnitude (3.2GPa) of the modulus of the DEA layer.
These features would make the stiffeners, electrically
inactive, constrain the DEA planar deformation, and
enable bending actuation.
III. FABRICATION PROCEDURE
Fig. 2 Sketches of DEA samples. (a) Top view of unit-cell DEA
structure and its measurement with (b) parallel reinforcement,
and (c) crossed reinforcement.
Fig. 3 Sketches of DEA samples fabrication with side view of
stack building unit and different sequence. (a) 2-stack/one unit
cell (b) 3-stack/two unit cells.
[5]–[7], [13], [14], [17], [20], due to their high dielectric
constant (4.7), low shear modulus (~0.042MPa), and the
ability of thicker VHB tapes to withstand very large
axial stretch, up to 6 times [21]. Furthermore, F9460PC
is 50μm thick, which is thinner than VHB 4910 tapes
after 4 times of biaxial prestretch. B. Compliant electrodes
For compliant electrodes, carbon black powders
(Super C65, TIMCAL Inc., USA) were chosen in this
study. In general, carbon-based powders (alone or
suspended in oil or grease) are very good compliant
electrode candidates for DEAs. They have outstanding
electrical conductivity whereas providing great
compliance and tolerance to large strains. Carbon black
was selected in this study due to: (i) ease of handling
compared to grease, (ii) better packing (percolation) and
uniform area coverage, especially during actuation. We
noticed that graphite powder with flake-like structure
formed discontinuous electrode especially upon
activation and thereby limited the performance. (iii)
Carbon grease tended to remain viscous after
application, making it prone for smearing and short
circuits; whereas powders were always dry and would
not be spread around. And (iv) carbon grease got to
squeezing out during the fabrication of multilayer
structure. Also, since it remained in the viscous form, the
ease of sliding increases the difficulty of peeling off
back paper of a VHB tape that was adhered on another
layer with electrodes in between.
The manufactured device has a square shape of
25×25mm. It contains 22×22mm active area with 1.5mm
inactive border for sealing and preventing short circuit
(see Fig. 2a). A building stack is composed of a pair of
DE tapes, sandwiching the electrode. A carbon black
powder electrode is brushed uniformly through a
window mask on one of sections. A long and narrow
aluminum foil is attached to the edge of the electrode to
form the external terminal. The other layer of the tape is
added to seal the electrode. Two stacks are combined
together to form a full unit cell of 100m thick, active
section, and 50m cover on each sides (Fig. 3a).
Assembly of three building stacks would result in two
unit cells as shown in Fig. 3b. The attachments of the
aluminum terminal positions are alternated at each stack
to separate the positive and negative electrodes. The
stiffeners are cut from the 3M Magic scotch tapes into
long-narrow 3mm strips. Two configurations of stiffener
reinforcement were addressed, as shown in Fig. 2b, c.
From experiences, we noticed that local non-uniform
electrodes would increase the possibility of failure.
Defects were easily forming at these non-uniform areas
during the assembly of multilayer structures. In addition,
short circuits were more probable due to local variation
of the applied electric field. Therefore, we had to make
sure that carbon black particles were uniformly
deposited and completely attached on VHB tapes with
homogenous thickness and particle amount. IV. EXPERIMENTS
In the experiments, DEA samples are hung from the
attachment electrodes and connected to the electric
terminal. A voltage conversion circuit [22] is devised
with a DC-DC converter (Q-80, EMCO Inc.). The
converter has a linear high DC output range of 0-8kV for
a 0-5V input range. A step voltages in the range of 23.2kV are employed in the current experiment to avoid
DE breakdown. The actuator movements were captured
by high-resolution CCD camera (2448×2048 pixel,
Grasshopper, Point Grey Inc.) for curvature analysis and
further analysis of the in-plane surface displacement and
the corresponding deformation mechanisms (beyond the
scope of this work).
V. RESULTS
In this section, we will first discuss the experimental
measurements of the actuator response. Bending
curvatures were calculated from the recorded images
under different activation voltage, and for different stack
configuration. Also, curvatures were also calculated
analytically and numerically for comparison. Next, we
further used the experimental results for the simple
rotational degree of freedom configuration to calibrate
the FEM model. The model was used to analyze other
complex deformation patterns.
A. Measured actuator response
The experimentally evaluated bending curvatures
under different level of applied electric fields are shown
in Fig. 4. The curvature of the entire actuator was
assumed to be approximately uniform for ease of
numerical analysis. Measurements were conforming to
the active region only. The experimental results showed
that the stack curvature increased with increasing the
number of active unit cells. Despite the increase of the
beam thickness, the curvature increased, due to the
increase of the driving forces that have to remain in
balance with the forces generated within the stiffening
layer. Moreover the role of the inactive cover layer on
each side of the stack has to be considered. In our
design, the actuator stack is covered with an additional
VHB layer to protect the electrodes on the top and
bottom surfaces (Fig. 3a, b). As a result, increasing the
number of stacked cells would dilute the effect of the
external capping layers. B. Analytical model
To understand the observed trend and provide an
analytical model amenable for control strategy, we
utilized the Timoshenko’s analysis of bi-metal
thermostats [23] and introduced the resulting strain from
Maxwell stress, instead of the thermal strains to arrive at
a general deformation representation under the applied
electric field. For a dielectric strip, subjected to an
electric field along the z-direction, the components of
the Maxwell stress are given by [24]
1
2
 zz   0 2
1
2
 xx   yy  0 2
(2)
Here, is the dielectric constant and is the vacuum
permittivity. The electric field is defined as the applied
electric potential V over the DE thickness ;

V
V

x  y
ha hao
(3)
Fig. 4 Deformation sequence of 4-stack configuration under
applied electric field of 20-32MV/m of (a) experimental results
and (b) numerical FEM results.
Taking the DE to be incompressible,
/  ,
where  is the stretch ratio in x- and y-directions. For a
narrow and long DE strip, the equivalent mechanical
force per unit area in the z-direction is given by
 xx   zz   0  2
(4)
Assuming small deformation and the DE can be
approximately represented by an elastic isotropic
response, the corresponding strain along the axis of the
strip is given by
 xx 
 0  2
2 Ea
(5)
Here, Ea is the young’s modulus of the DE material. To
account for the inactive capping layer, the resulting
lateral stretch as given by Eq. 4 will be reduced by a
factor that depends on the number of active unit cells.
Since a symmetric lay-up is implemented, only axial
deformation will commence, wherein the cover layer
would retard the lateral stretch. Solving for the new
equilibrium lateral strain, the geometric factor is found
to be 1/2, 2/3, 3/4, for the 1, 2, and 3 active unit cell
stacks. Therefore, the effective Maxwell stress induced
lateral strain for the whole DEA stack as a function of
applied electric field is given by
 xx  
 0  2
2 Ea
(6)
This effective lateral strain is used in Timoshenko bimetal thermostat analysis [23], by replacing the thermal
strains. The resulting composite actuator curvature  is
given by

2-stack(1
3-stack(2
4-stack(3
2-stack(1
3-stack(2
4-stack(3
0.12
0.1
0  2
2 Ea
 ha  hs   2  Ea I a  Es I s   1  1 
2
 ha  hs   Ea Aa Es As 
(7)
Curvature (1/mm)

0.14
unit
unit
unit
unit
unit
unit
cell)
cells)
cells)
cell) - Analytical
cells) - Analytical
cells) - Analytical
0.08
0.06
0.04
Here, E is stiffness, h is thickness, is cross-section
area, and is the section second moment of area. The
subscripts a and s represent the DEA layer and stiffener
respectively. The predicted curvature using Eq. (7) is
plotted on Fig. 5 along with the experimental results for
one, two and three active unit cells under different
applied voltage. The nominal applied electric field is
estimated, noting that in bending, small strain is

1. The analytical results concur
prevailed and 
with the experimentally measured trends, especially at
low applied voltages. Deviation at higher voltages arises
from geometrical (finite deformation) and material
(hyper-elastic response) nonlinearities. Despite such
deviation, these nonlinearities can be further
characterized and establish the bounds for control
strategies. These issues are further addressed in the
numerical framework.
C. Numerical FEM results
ABAQUS finite element software is utilized. A user
material subroutine is developed within the software to
handle the electromechanical coupling for EAP
materials. The electric field is imposed similar to an
external thermal field with the resulting thermal strains.
A constitutive material utilizing incompressible NeoHookean material description is employed due to its
simple structure to couple the mechanical deformation
with the applied electric. In this model, the strain energy
density W is given in terms of a single material
parameter μ, the shear modulus, and the principal
stretches  ;
W

2

2
1
 22  32  3 ;
12 3  1
(8)
The Cauchy stress differences are given in terms of the
principal stretches  by
 i   j  i
W
W
 j
   i 2   j 2 
i
 j
(9)
0.02
0
20
22
24
26
28
30
32
E0 (MV/m)

Fig. 5 Comparison of curvature of experimental and analytical
results.
Fig. 6 (a) Sketch, (b) experimental, and (c) FEM result (quarter)
of saddle-shape deformation of DEA with cross configuration
reinforcements.
quadratic brick element is used in constructing each
individual layer. The surface-based tie constraint is
utilized to link the individual layers to form the
laminated actuator. An additional layer of 30μm thick is
added to represent the elctrode layer. The electrode
thickness is dervied from the experimentally measured
equivalent stiffness of a uniaxial tensile test of the
stacked array. Figure 4 shows a comparison of the FEM
simulation results with the experimental measurements
of the deformed actuator under different applied electric
field for a 4-stack actuator with 3-active cells. The FEM
predictions capture the experimental trend reasonably
well. The bending and rotation of a complex crossstiffened section is shown in Fig. 6. The FEM model
well captures the details of the deformed shape. Such
capability of deformation modes combining bending and
twisting would enable further development of flapping
wing construction.
VI. CONCLUSION
A 3-D geometric model for the laminated cantilever is
A
new
design
for
a 2-D dielectric elastomer actuator
constructed. One edge is completely anchored, and the
other edge was left free to move. A hybrid 20-node with geometrically confining reinforcements, capable of
complex 3-dimentional actuation and generating dihedral
deflections is presented. The actuator is free standing
laminates with no supporting skeleton or prestretch, and
thereby compatible with aerodynamics of flapping
wings. An analytical model, utilizing Timoshenko bimaterial theory, is implemented to study the role of
geometric parameters on the actuator performance. The
model provides a base for devising control strategies.
The experimentally calibrated finite element model
enables the development of actuator configuration to
deliver the required bending and twist combination for
flapping flight control. The presented experimental and
modeling framework is a step towards controlling the
DEA effective dihedral motion that can generate the
desired longitudinal and lateral/directional control
capability for MAV with flapping wings.
ACKNOWLEDGEMENT
This work is supported by U.S. Army Research Office
under Award No W911NF-10-1-0296.
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