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Parallelogram-shaped dielectric elastomer generators: Analytical model and experimental validation

TL;DR: In this paper, a novel concept of parallelogram-shaped dielectric elastomer generator is presented and analyzed, and an analytical model for the electro-hyperelastic response of the generator is described and used to assess the maximum theoretical performances.
Abstract: Dielectric elastomers are smart materials that can be used to conceive solid-state electromechanical transducers such as actuators, sensors, and generators. Dielectric elastomer generators, in particular, are very promising for energy harvesting applications because they potentially feature large energy densities, good conversion efficiencies, good shock and corrosion resistance, and low cost. In this article, a novel concept of parallelogram-shaped dielectric elastomer generator is presented and analyzed. Parallelogram-shaped dielectric elastomer generators are rotary variable capacitance transducers, which are made by planar dielectric elastomer membranes that are covered with compliant electrodes and clamped along their perimeter to the links of a parallelogram four-bar mechanism. First, an analytical model for the electro-hyperelastic response of the parallelogram-shaped dielectric elastomer generator is described and used to assess the maximum theoretical performances of the device. Then, an experime...

Summary (2 min read)

Introduction

  • Dielectric elastomers (DEs) are highly deformable nonconductive polymeric materials that can be employed to conceive variable capacitance transducers.
  • In the last decade, several studies have demonstrated the possibility of using them for the implementation of generators, the so called dielectric elastomer generators (DEGs), which convert mechanical energy into direct electricity (Pelrine et al., 2001).
  • The choice of the specific architecture depends on the application requirements such as type of mechanical input, force–stroke characteristic, and encumbrance.
  • Compared to existing topologies, the PS-DEG is the first direct-drive rotary DEG that makes it possible to produce electricity from the motion of a rocker arm without the need of any additional mechanical transmission.

PS-DEG

  • This section describes the operating principle of the PSDEG and introduces an energy-based mathematical model that can be used for the representation of its electromechanical response.
  • The mathematical description of the electromechanical response of PS-DEGs can be obtained via an energy balance principle.
  • Using the abovementioned equations, the limit curves for PS-DEG operation are obtained as follows.
  • The mechanical rupture of the DE material poses a limitation on the maximum principal stretch that can be borne by the membrane.

Fitting of the stress–strain experimental results

  • The experimental data presented in Figure 2 have been used to determine the form and the parameters of the hyper-elastic strain-energy function,C(l1, l2).
  • This strain-energy function has been chosen since the more traditional hyper-elastic models (Gent, Ogden, Mooney–Rivlin, Yeoh) produced poor fitting results on the considered experimental data, principally because of the strong dependence of the stress on the transversal pre-stretch.
  • As shown, the use of different values for the relative dielectric constant only produces a slight variation in the shape of the loss-oftension curves.
  • The PS-DEG is more effective when operating away from the symmetric configuration.

Experimental validation

  • A prototype of a PS-DEG generator has been built and a set of experimental tests has been conducted with the aim of validating the proposed generator concept along with its model.
  • The prototype features a parallelogram mechanism with links having identical length l1 = l2 = 120 mm.
  • The experimental setup is composed by the following mechanical equipment: A high-voltage power amplifier (Trek 10/10BHS, with 610 kV and 610 mA voltage and current ratings);.
  • The charge and discharge of the active membrane can be synchronized with the movement of the linear motor in a way to accurately regulate the flows of electrical energy from and to the active membrane.

Mechanical response validation

  • A first set of measurements has been conducted with no electrical load in order to compare the experimental torque–angle characteristic of the PS-DEG with the theoretical response described by equation (10) using hyper-elastic model defined according to section ‘‘Experimental characterization of a DE material: mechanical properties and electrical tests.’’.
  • Before installation, the natural rubber membrane has been subjected to the pre-stretches l1p = 7.13 and l2p = 1.30.
  • The tests have been conducted by imposing saw-tooth position profiles with d ranging between 30 and 165 mm (corresponding to an angular position u ranging between 3 and 80 ) at a speed of nearly 30 mm/s, and by acquiring the force FDEG acting along the x-direction via the load cell.
  • As illustrated, the analytical model shows a good match with the experimental unloading curve, especially for the smaller values of u.
  • For larger values of u, the analytical model tends to underestimate the force response of the PS-DEG, which makes the considered hyper-elastic strain-energy function more conservative for the forecast of the loss-of-tension condition.

Generator model validation

  • For completeness, the theoretical evolution in time of the PS-DEG voltage that is predicted by equation (29) is also reported with dashed line in Figure 10.
  • The comparison highlights a noticeable disagreement between theoretical and experimental performances, which can be attributed in part to the variation of dielectric constant with stretch and in part to charge dispersions.
  • Accordingly, the voltage ratio that could be expected from the capacitance measurements is h = 11 (that is significantly smaller than the theoretical value h = 15.2).

Conclusion

  • PS-DEGs are presented and investigated by means of theoretical models validated through experimental tests.
  • The maximum theoretical performances of PS-DEGs have been evaluated considering the limits imposed by both material properties and operational constraints (such as fixed stroke and end-stops).
  • A practical case study is presented that considers a PS-DEG prototype based on a natural rubber dielectric elastomer membrane (OPPO Band Red 8012).
  • This problem is almost eliminated with the use of stiffer materials such as natural rubber, the operation of which is marginally influenced by the loss-of-tension condition.
  • Second, a limiting value for the area expansion seems to exist beyond which the dielectric strength resistance is permanently degraded.

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Special Issue Article
Journal of Intelligent Material Systems
and Structures
2015, Vol. 26(6) 740–751
Ó The Author(s) 2014
Reprints and permissions:
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DOI: 10.1177/1045389X14563861
jim.sagepub.com
Parallelogram-shaped dielectric
elastomer generators: Analytical
model and experimental validation
Giacomo Moretti, Marco Fontana and Rocco Vertechy
Abstract
Dielectric elastomers are smart materials that can be used to conceive solid-state electromechanical transducers such as
actuators, sensors, and generators. Dielectric elastomer generators, in particular, are very promising for energy harvest-
ing applications because they potentially feature large energy densities, good conversion efficiencies, good shock and cor-
rosion resistance, and low cost. In this article, a novel concept of parallelogram-shaped dielectric elastomer generator is
presented and analyzed. Parallelogram-shaped dielectric elastomer generators are rotary variable capacitance transdu-
cers, which are made by planar dielectric elastomer membranes that are covered with compliant electrodes and clamped
along their perimeter to the links of a parallelogram four-bar mechanism. First, an analytical model for the electro-
hyperelastic response of the parallelogram-shaped dielectric elastomer generator is described and used to assess the
maximum theoretical performances of the device. Then, an experimental case study with a parallelogram-shaped dielec-
tric elastomer generator prototype featuring a natural rubber dielectric elastomer membrane and carbon conductive
grease electrodes is presented. Simulation and experimental results demonstrate the practical feasibility of the
parallelogram-shaped dielectric elastomer generator concept.
Keywords
Dielectric elastomer, generator, natural rubber, energy harvesting, rotary generator, four-bar mechanism
Introduction
Dielectric elastomers (DEs) are highly deformable non-
conductive polymeric materials that can be employed
to conceive variable capacitance transducers. DEs have
been extensively investigated for sensing and actuation
purposes (Carpi et al., 2008). In the last decade, several
studies have demonstrated the possibility of using them
for the implementation of generators, the so called
dielectric elastomer generators (DEGs), which convert
mechanical energy into direct electricity (Pelrine et al.,
2001). One of the most promising applications for
DEGs is in the field of wave energy harvesting (Chiba
et al., 2008; Jean et al., 2012; Moretti et al., 2014;
Vertechy et al., 2014a). In fact, in an application of this
kind, DEGs may provide the following advantages: (1)
light-weightiness, (2) good energy conversion efficiency
that is rather independent of operation frequency, (3)
easiness of manufacturing and assembly, (4) high shock
and corrosion resistance, (5) low noise, and (6) low cost.
The energy harvesting performance of DEGs is
highly dependent on their topology and size as well as
on the electromechanical properties of the employed
DE material (Koh et al., 2011). For DEGs made by
acrylic elastomer membranes and undergoing equi-
biaxial deformations, theoretical energy densities over
1 J/g have been predicted (Koh et al., 2011) and experi-
mental energy densities up to 0.56 J/g have been mea-
sured (Huang et al., 2013), although for a limited
number of cycles only. Despite DEGs under equi-
biaxial loading are very useful for theoretical studies
and for assessing the experimental performance of DE
materials, their usage in realistic energy harvesting sys-
tems is rather impractical due to the difficulty of imple-
menting kinematic linkages that make it possible to
impose this kind of loading constraints on the DE
membrane boundary. For practical energy harvesting
applications, different DEG architectures have been
proposed that feature conical shape (McKay et al.,
2010; Wang et al., 2012; Zhu et al., 2011), inflatable
diaphragms (Kaltseis et al., 2011; Rosati Papini et al.,
Scuola Superiore Sant’Anna, Pisa, Italy
Corresponding author:
Rocco Vertechy, PERCRO SEES, Scuola Superiore Sant’Anna, Piazza Dei
Martiri Della Liberta
`
33, Pisa, Italy.
Email: r.vertechy@sssup.it
at Scuola Superiore Sant'Anna on April 19, 2015jim.sagepub.comDownloaded from

2013), and lozenge shape (Moretti et al., 2013). The
choice of the specific architecture depends on the appli-
cation requirements such as type of mechanical input,
force–stroke characteristic, and encumbrance.
This article introduces the parallelogram-shaped
dielectric elastomer generator (PS-DEG) and investi-
gates its energy harvesting performances via both theo-
retical arguments and experiments. Compared to
existing topologies, the PS-DEG is the first direct-drive
rotary DEG that makes it possible to produce electric-
ity from the motion of a rocker arm without the need
of any additional mechanical transmission.
In this article, sections ‘PS-DEG’ and ‘Analytical
model of PS-DEG operation and control’ introduce an
analytical electro-hyperelastic model that describes the
electromechanical response of PS-DEGs and that can
be used to study its energy conversion performances;
section ‘Experimental characterization of a DE mate-
rial: mechanical properties and electrical tests’ presents
the material properties of a reference DE material
(OPPO Band Red 8012) that has been considered for
the development of a PS-DEG prototype; section
‘Analysis of the energy harvesting performances of the
PS-DEG’ presents the procedure for the optimal selec-
tion of the design parameters of PS-DEGs subjected to
operational constraints; section ‘Experimental valida-
tion’ presents an experimental case study, which vali-
dates the feasibility of the proposed PS-DEG
architecture as well as the adequacy of the considered
electromechanical model.
PS-DEG
This section describes the operating principle of the PS-
DEG and introduces an energy-based mathematical
model that can be used for the representation of its elec-
tromechanical response. The PS-DEG is a planar vari-
able capacitor, which consists in a parallelogram
mechanism (namely, a four-bar mechanism with oppo-
site sides of equal length) that accommodates in its inte-
rior an active deformable membrane (ADM) made by
one or more DE films alternatively stacked with com-
pliant electrode layers. DE films are required to have
(1) large deformability, (2) large dielectric strength, (3)
large dielectric constant, (4) low viscosity, and (5) low
electrical conductivity. DE films can be made by sili-
cone elastomer, acrylic elastomer or natural rubber
(Carpi et al., 2008; Koh et al., 2011). Compliant electro-
des are required to (1) be conductive; (2) remain bonded
to the DE material during the cyclical deformation; and
(3) be highly compliant, in order to provide the system
with minimal extra-stiffness. Electrodes can be made by
carbon powder or grease, conductive rubber, or metal-
lic thin films (Rosset and Shea, 2013).
With reference to Figure 1(a), the PS-DEG has one
fixed link, with length l
1
lying along the j direction,
and one moving link, with length l
2
and being inclined
by the angle u with respect to the z-axis; l
1
and l
2
are
also the lengths of the ADM edges that, together with
u, define the ADM area A (A = l
1
l
2
cos u). Assuming
the DE material as being incompressible, as u increases,
the ADM area A reduces and its thickness t increases.
Since the capacitance of a planar condenser is given by
C = eA=t, where e is the absolute permittivity of the
dielectric medium, the PS-DEG capacitance decreases
monotonically with increasing u.
Functionally, the PS-DEG can be used as a rotary
generator with reciprocating motion to convert into
electricity the mechanical work spent in a cycle by a
torque applied to one of its oscillating links. The torque
output, T
DEG
, of the PS-DEG depends on the deforma-
tion and electrical states of the ADM and, thus, is func-
tion of the angular position, u, of the parallelogram
mechanism and of the electric field, E, acting across the
DE material.
To always be in a state of tension, the ADM is
required to be mounted on the links of the mechanism
with a certain amount of pre-stretch along the two
principal directions of strain (see Figure 1(b)) that lie
along the bisectors of the parallelogram angles. The
amount of pre-stretch needs to be referred to a specific
configuration identified by angle u
p
. Here, the pre-
stretches l
1p
and l
1p
are referred to the configuration
u
p
= 0. Note that this configuration may not always
be included in the angular operating range of the exam-
ined PS-DEG; however, it is always possible to virtu-
ally refer the pre-stretches to this angular position of
the parallelogram.
With the abovementioned assumption, the stretch
field within the ADM results as
l
1
= l
1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 sin u
p
, l
2
= l
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 +sinu
p
ð1Þ
where l
1
and l
2
are the first (along x-direction) and
second (along y-direction) principal stretches (see
Figure 1(b)). Due to DE material incompressibility, the
third principal stretch acting in the thickness direction
follows as
Figure 1. (a) Schematic of a parallelogram-shaped dielectric
elastomer generator and (b) scheme of the active deformable
membrane that shows the principal strain directions x-y.
Moretti et al. 741
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l
3
= l
1
l
2
ðÞ
1
ð2Þ
The mathematical description of the electromechani-
cal response of PS-DEGs can be obtained via an energy
balance principle. The state of the generator is
described by means of two independent variables. The
geometric configuration is represented by u and the
electrical state of activation is described by the value of
the electric field E in the DE membrane. The charge Q
residing on the electrodes or the electric potential dif-
ference V acting between the DEG terminals could also
be chosen as alternatives in place of E.
The energy balance for a DEG can be expressed in
differential form as it follows
dU
e
+ dU
m
= dW
e
+ dW
m
ð3Þ
The addenda appearing in the balance (3) represent,
respectively:
Electrostatic potential energy, U
e
, that is stored
in the DEG and given by
U
e
=
eOE
2
2
ð4Þ
where O is the total volume of DE material.
Elastic potential energy, U
m
, that describes the
mechanical state of tension of the DE membrane.
Similar to other rubber-like materials, DEs can
be considered as hyper-elastic solids admitting a
strain-energy function C that governs their
mechanical stress–strain response (Holzapfel,
2000). A review of the principal hyper-elastic
models for rubber-like materials is presented by
Steinmann et al. (2012). For PS-DEGs, the elastic
potential energy assumes the following form that
only depends on the variable u
U
m
= OC uðÞ ð5Þ
Electrical work, W
e
, that is performed on the
DEG by the external electric circuit, namely
dW
e
= VdQ ð6Þ
Using u and E as state variables, and considering
that the DEG capacitance is
C =
Q
V
=
el
2
1
l
2
2
cos
2
u
O
ð7Þ
the differential of the electrical work results as
dW
e
= eOEdE eOE
2
tan udu ð8Þ
Mechanical work, W
m
, that is performed by the
external torque applied to the parallelogram
mechanism. Let T
DEG
be the torque provided by
the DEG, which has same module and opposite
sign of the externally applied torque. Then, for
an infinitesimal rotation of the PS-DEG
dW
m
= T
DEG
du ð9Þ
In the considered energy balance (3), the following
contributions have been neglected: (1) gravitational
potential energy that is related to the orientation in
space of the PS-DEG, (2) inertial terms that are related
to the kinetic energy of the PS-DEG components, (3)
visco-elasto-plastic losses that are due to the mechanical
hysteresis of the DE material, and (4) electrical losses
due to leakage currents through the DE material and to
the resistivity of the deformable electrodes.
The combination of equation (3) with equations (4),
(5), (8), and (9) yields an analytical expression for the
torque provided by the PS-DEG. In particular, T
DEG
can be split into two addenda: an electrostatic contribu-
tion and an elastic mechanical term, namely
T
DEG
= T
DEG, e
+ T
DEG, m
,
T
DEG, e
= eOE
2
tan u,
T
DEG, m
= O
dC
du
ð10Þ
The knowledge of the torque characteristic of the
PS-DEG is important in order to assess its influence on
the dynamic response of the external source from which
mechanical energy is harvested. If required by the appli-
cation, the mechanical component, T
DEG,m
, can be com-
pensated or corrected by means of passive mechanical
components (for instance, linear or torsional springs)
that can be properly mounted between the links of the
parallelogram mechanism (Vertechy et al., 2010).
The amount of electrical energy that can be pro-
duced or spent for an infinitesimal displacement du of
the PS-DEG is
dU
e
dW
e
= eOE
2
tan udu ð11Þ
When this expression is positive, electricity is gener-
ated. This happens when udu . 0 and E 0. In this
article, it is assumed that electric charges are present on
the PS-DEG electrodes only during one half of the
oscillation cycle, which corresponds to when the device
capacitance decreases. In this phase, the applied exter-
nal torque performs mechanical work against both the
mechanical stresses of the membrane, which are due to
DE material elasticity, and the electrically induced
stresses, which are caused by the electrostatic attraction
of the DEG electrodes. During the other phases, when
udu 0 and E = 0, the work performed by the
applied external torque is balanced only by the varia-
tion of the elastic energy that is stored in the DE
material.
742 Journal of Intelligent Material Systems and Structures 26(6)
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Analytical model of PS-DEG operation
and control
In this section, the electromechanical limits for PS-
DEG operation are described and modeled analytically.
The identification of the failure modes for a DEG pro-
vides important indications for generator operation
and control.
DEGs perform a cyclical sequence of electromecha-
nical transformations that is referred to as energy con-
version cycle (ECC). ECCs can be represented on two-
axis diagrams by referring to a couple of independent
state variables.
In scientific literature, ECCs are usually represented
on the u-T
DEG
or the Q-V planes (Koh et al., 2011). In
this article, the Q-V plane is used.
The range of admissible physical states for the PS-
DEG is bounded by a series of failure conditions (Koh
et al., 2011) and operative constraints. On the Q-V
plane, these conditions are curves and enclose an opera-
tional surface, whose area is equal to the maximal
energy (E
n
) that can be converted in a cycle by the PS-
DEG.
In the following, the fundamental practical con-
straints for the PS-DEG are described and modeled
analytically. In order to simplify the mathematical anal-
ysis, the dielectric constant of the DE material is taken
as independent of the stretch.
The combination of equations (1) and (7) provides a
relation between the stretch field of the material and the
state variables Q and V, namely
l
1
= l
1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
O
el
2
1
l
2
2
Q
V
s
v
u
u
t
,
l
2
= l
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
O
el
2
1
l
2
2
Q
V
s
v
u
u
t
ð12Þ
Equations (12) indicate that iso-stretch transformations
(i.e. transformations that occur with the parallelogram
mechanism in a fixed position) are straight lines passing
through the origin of the Q-V plane.
As regards the DE stress field, electric activation is
responsible for an electrostatically induced stress
s
em
= eE
2
= QV =O that acts along the first and second
principal directions of the DE membrane (Pelrine et al.,
2001). Thus, the total principal stresses in the plane of
the DE membrane follow as
s
1
= l
1
C l
1
, l
2
ðÞ
l
1
s
em
,
s
2
= l
2
C l
1
, l
2
ðÞ
l
2
s
em
ð13Þ
Using the abovementioned equations, the limit
curves for PS-DEG operation are obtained as follows.
Geometric constraint. In equations (12), the condi-
tion of the positive square root argument yields the
limit curve
V =
OQ
el
2
1
l
2
2

ð14Þ
This condition physically means that the DEG capa-
citance reaches its maximum value, C
max
= el
2
1
l
2
2
=O,
when u =0.
Electrical breakdown condition. During PS-DEG
operation, the electric field acting across the active
membrane must be lower than the dielectric strength,
E
BD
, of the considered DE material. Previous works
(Koh et al., 2011; Tro
¨
ls et al., 2013) have employed
either constant values for E
BD
or analytical relations
expressing it as a function of DE membrane stretches.
Here, a constant E
BD
is assumed. However, based
on experimental observations (Vertechy et al., 2014b),
according to which the dielectric strength of certain DE
materials may experience a drastic irreversible reduc-
tion in the case of excessive area expansion, the electri-
cal breakdown condition takes the usual form
E E
BD
with the further constraint l
1
l
2
G. Such
conditions yield to the following limit curves
VQ = eOE
2
BD
ð15Þ
V =
Ol
2
1p
l
2
2p
el
2
1
l
2
2
G
2
Q ð16Þ
On the Q-V plane, the electrical breakdown curve
(15) is a hyperbola, while constraint (16) represents a
straight line.
Angular range constraints. For PS-DEGs operating
with a fixed stroke, the endpoints of the angular opera-
tional range may be imposed, namely u u
max
and u
u
min
. These constrains may represent mechanical
end-stops or other external limiting positions. Using
equations (1) and (12), the resulting limit curves are
V =
O
e l
1
l
2
cos u
max
ðÞ
2
Q,
V =
O
e l
1
l
2
cos u
min
ðÞ
2
Q
ð17Þ
which are iso-capacitance curves consisting in straight
lines passing through the origin of the Q-V plane.
Mechanical rupture condition. The mechanical rup-
ture of the DE material poses a limitation on the
maximum principal stretch that can be borne by the
membrane. The simplest fracture criterion is
the K awabata’s condition (Hamdi et al., 2006), which
asserts that rupture in the i-th principal direction does
not occur if l
i
\ l
max
(for i =1,2,3),wherel
max
is
a constant only d ependent on the considered elasto-
meric material.
Since the DE membrane is pre-stretched in the plane
of the DEG electrodes, the stretch in thickness
Moretti et al. 743
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direction, l
3
, is always lower than 1. It is then sufficient
to impose l
1
\ l
max
and l
2
\ l
max
. Using equation
(12), the resulting limit curves are
V =
O
el
2
1
l
2
2

Q
1 1 l
2
max
.
l
2
1p

2
,
V =
O
el
2
1
l
2
2

Q
1 1 l
2
max
.
l
2
2p

2
ð18Þ
that represent straight lines crossing the origin of the Q-
V plane.
As a matter of fact, more sophisticated criteria exist
in the literature for representing the mechanical rupture
of elastomers (Hamdi et al., 2006), and thus for
DEs. Nonetheless, the combination of the constraint
l
i
\ l
max
with the one described by l
1
l
2
G (which
has been introduced as part of the electrical breakdown
condition) provides a conservative limitation that can
protect the employed DE material from mechanical
rupture.
Loss-of-tension (or buckling) condition. For a correct
operation, the principal stresses within the DE material
must be positive. Otherwise, the DE membrane may
wrinkle and the efficacy of the transduction process
could be compromised. The loss-of-tension conditions
for the principal directions yield to the following limit
curves
l
1
C l
1
, l
2
ðÞ
l
1
=
QV
O
, l
2
C l
1
, l
2
ðÞ
l
2
=
QV
O
ð19Þ
which highlight that the loss-of-tension limit is the only
condition that depends on the hyper-elastic constitutive
model chosen for the considered DE material.
In order to generalize the discussion and to adapt
it to different choices of PS-DEG size and propor-
tions, the following dimensionless charge and voltage
variables
Q
=
Q
eE
BD
l
1
l
2
, V
=
l
1
l
2
OE
BD
V ð20Þ
are introduced, and conditions (14) to (19) are reformu-
lated by means of them.
With reference to the new variables, the mentioned
constraints can be expressed in the following dimen-
sionless form:
Dimensionless geometric constraint
V
= Q
ð21Þ
Dimensionless electrical breakdown condition
Q
V
= 1, V
=
l
2
1p
l
2
2p
G
2
Q
ð22Þ
Dimensionless angular range constraints
V
=
Q
cos u
max
ðÞ
2
, V
=
Q
cos u
min
ðÞ
2
ð23Þ
Dimensionless mechanical rupture condition
V
=
Q
1 1 l
2
max
.
l
2
1p

2
,
V
=
Q
1 1 l
2
max
.
l
2
2p

2
ð24Þ
Dimensionless buckling (loss-of-tension) condition
l
1
C
l
1
= eE
2
BD
Q
V
, l
2
C
l
2
= eE
2
BD
Q
V
ð25Þ
These equations make it possible to compare PS-
DEGs with different geometries but exploiting the same
DE material. In fact, even if dimensionless variables
are used, equation (25) presents a dependence on the
electromechanical properties of the specific DE mate-
rial considered for the application.
The surface enclosed by ECCs on the dimensionless
Q
*
-V
*
plane represents the dimensionless energy that
can be converted by the PS-DEG in a cycle
E
n
=
E
n
eE
2
BD
O
ð26Þ
Experimental characterization of a DE
material: Mechanical properties and
electrical tests
The experimental case study described in the following
sections employs a natural rubber DE (OPPO Band
Red 8012). To determine the hyper-elastic strain-energy
function, C(l
1
, l
2
), of the considered material, three
cyclic tensile tests on pure-shear DE membrane speci-
mens (Holzapfel, 2000) with different transversal pre-
stretches, l
2p
(along the second principal direction),
have been performed on a custom made tensile stage.
Detailed explanation of experimental testing proce-
dures and obtained results is provided by Vertechy
et al. (2014b). The first specimen features initial (i.e. in
its undeformed state) longitudinal length, transversal
length and thickness equaling 15, 150, and 0.175 mm,
and a transversal pre-stretch l
2p
= 1. The second spe-
cimen features initial longitudinal length, transversal
length and thickness equaling 15, 105, and 0.175 mm,
and a transversal pre-stretch l
2p
= 2. The third speci-
men features initial longitudinal length, transversal
length and thickness equaling 15, 70, and 0.175 mm,
and a transversal pre-stretch l
2p
= 3. Specimens are
tested in the longitudinal (first principal) direction with
744 Journal of Intelligent Material Systems and Structures 26(6)
at Scuola Superiore Sant'Anna on April 19, 2015jim.sagepub.comDownloaded from

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References
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Journal ArticleDOI

2,782 citations


"Parallelogram-shaped dielectric ela..." refers background or methods in this paper

  • ...Similar to other rubber-like materials, DEs can be considered as hyper-elastic solids admitting a strain-energy function C that governs their mechanical stress–strain response (Holzapfel, 2000)....

    [...]

  • ...…the hyper-elastic strain-energy function, C(l1, l2), of the considered material, three cyclic tensile tests on pure-shear DE membrane specimens (Holzapfel, 2000) with different transversal prestretches, l2p (along the second principal direction), have been performed on a custom made tensile…...

    [...]

  • ...To determine the hyper-elastic strain-energy function, C(l1, l2), of the considered material, three cyclic tensile tests on pure-shear DE membrane specimens (Holzapfel, 2000) with different transversal prestretches, l2p (along the second principal direction), have been performed on a custom made tensile stage....

    [...]

Book
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TL;DR: In this paper, the authors introduce the concept of stress and balance principles for tensors and invariance of tensors in the context of Vectors and Tensors, and present a survey of the main aspects of objectivity.
Abstract: Introduction to Vectors and Tensors. Kinematics. The Concept of Stress. Balance Principles. Some Aspects of Objectivity. Hyperelastic Materials. Thermodynamics of Materials. Variational Principles. References. Index.

2,082 citations

BookDOI
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TL;DR: In this paper, the authors provide a comprehensive and updated insight into dielectric elastomers; one of the most promising classes of polymer-based smart materials and technologies, which can be used in a broad range of applications, from robotics and automation to the biomedical field.
Abstract: This book provides a comprehensive and updated insight into dielectric elastomers; one of the most promising classes of polymer-based smart materials and technologies This technology can be used in a very broad range of applications, from robotics and automation to the biomedical field The need for improved transducer performance has resulted in considerable efforts towards the development of devices relying on materials with intrinsic transduction properties These materials, often termed as "smart or "intelligent , include improved piezoelectrics and magnetostrictive or shape-memory materials Emerging electromechanical transduction technologies, based on so-called ElectroActive Polymers (EAP), have gained considerable attention EAP offer the potential for performance exceeding other smart materials, while retaining the cost and versatility inherent to polymer materials Within the EAP family, "dielectric elastomers , are of particular interest as they show good overall performance, simplicity of structure and robustness Dielectric elastomer transducers are rapidly emerging as high-performance "pseudo-muscular actuators, useful for different kinds of tasks Further, in addition to actuation, dielectric elastomers have also been shown to offer unique possibilities for improved generator and sensing devices Dielectric elastomer transduction is enabling an enormous range of new applications that were precluded to any other EAP or smart-material technology until recently This book provides a comprehensive and updated insight into dielectric elastomer transduction, covering all its fundamental aspects The book deals with transduction principles, basic materials properties, design of efficient device architectures, material and device modelling, along with applications * Concise and comprehensive treatment for practitioners and academics * Guides the reader through the latest developments in electroactive-polymer-based technology * Designed for ease of use with sections on fundamentals, materials, devices, models and applications

605 citations


"Parallelogram-shaped dielectric ela..." refers background in this paper

  • ...DEs have been extensively investigated for sensing and actuation purposes (Carpi et al., 2008)....

    [...]

  • ...DE films can be made by silicone elastomer, acrylic elastomer or natural rubber (Carpi et al., 2008; Koh et al., 2011)....

    [...]

Proceedings ArticleDOI
16 Jul 2001
TL;DR: In this article, the authors discuss the fundamentals of dielectric elastomer generators, experimental verification of the phenomenon, practical issues, and potential applications, and discuss the operating conditions and materials required for high efficiency.
Abstract: Dielectric elastomers have shown great promise as actuator materials. Their advantages in converting mechanical to electrical energy in a generator mode are less well known. If a low voltage charge is placed on a stretched elastomer prior to contraction, the contraction works against the electrostatic field pressure and raises the voltage of the charge, thus generating electrical energy. This paper discusses the fundamentals of dielectric elastomer generators, experimental verification of the phenomenon, practical issues, and potential applications. Acrylic elastomers have demonstrated an estimated 0.4 J/g specific energy density, greater than that of piezoelectric materials. Much higher energy densities, over 1 J/g, are predicted. Conversion efficiency can also be high, theoretically up to 80-90%; the paper discusses the operating conditions and materials required for high efficiency. Practical considerations may limit the specific outputs and efficiencies of dielectric elastomeric generators, tradeoffs between electronics and generator material performance are discussed. Lastly, the paper describes work on potential applications such as an ongoing effort to develop a boot generator based on dielectric elastomers, as well as other applications such as conventional power generators, backpack generators, and wave power applications.

474 citations


"Parallelogram-shaped dielectric ela..." refers background in this paper

  • ...In the last decade, several studies have demonstrated the possibility of using them for the implementation of generators, the so called dielectric elastomer generators (DEGs), which convert mechanical energy into direct electricity (Pelrine et al., 2001)....

    [...]

  • ...As regards the DE stress field, electric activation is responsible for an electrostatically induced stress sem = eE(2) =QV=O that acts along the first and second principal directions of the DE membrane (Pelrine et al., 2001)....

    [...]

  • ...=O that acts along the first and second principal directions of the DE membrane (Pelrine et al., 2001)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors compared different technologies used to make compliant electrodes for DEAs in terms of: impact on DEA device performance (speed, efficiency, maximum strain), manufacturability, miniaturization, integration of self-sensing and self-switching, and compatibility with lowvoltage operation.
Abstract: Dielectric elastomer actuators (DEAs) are flexible lightweight actuators that can generate strains of over 100 %. They are used in applications ranging from haptic feedback (mm-sized devices), to cm-scale soft robots, to meter-long blimps. DEAs consist of an electrode-elastomer-electrode stack, placed on a frame. Applying a voltage between the electrodes electrostatically compresses the elastomer, which deforms in-plane or out-of plane depending on design. Since the electrodes are bonded to the elastomer, they must reliably sustain repeated very large deformations while remaining conductive, and without significantly adding to the stiffness of the soft elastomer. The electrodes are required for electrostatic actuation, but also enable resistive and capacitive sensing of the strain, leading to self-sensing actuators. This review compares the different technologies used to make compliant electrodes for DEAs in terms of: impact on DEA device performance (speed, efficiency, maximum strain), manufacturability, miniaturization, the integration of self-sensing and self-switching, and compatibility with low-voltage operation. While graphite and carbon black have been the most widely used technique in research environments, alternative methods are emerging which combine compliance, conduction at over 100 % strain with better conductivity and/or ease of patternability, including microfabrication-based approaches for compliant metal thin-films, metal-polymer nano-composites, nanoparticle implantation, and reel-to-reel production of μm-scale patterned thin films on elastomers. Such electrodes are key to miniaturization, low-voltage operation, and widespread commercialization of DEAs.

451 citations


"Parallelogram-shaped dielectric ela..." refers background in this paper

  • ...Electrodes can be made by carbon powder or grease, conductive rubber, or metallic thin films (Rosset and Shea, 2013)....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "Parallelogram-shaped dielectric elastomer generators: analytical model and experimental validation" ?

In this article, a novel concept of parallelogram-shaped dielectric elastomer generator is presented and analyzed. Then, an experimental case study with a parallelogram-shaped dielectric elastomer generator prototype featuring a natural rubber dielectric elastomer membrane and carbon conductive grease electrodes is presented.