A dielectric elastomer whose edges are held fixed will buckle, given a sufficiently applied voltage, resulting in a nontrivial out-of-plane deformation. We study this situation numerically using a nonlinear elastic model which decouples two of the principal electrostatic stresses acting on an elastomer: normal pressure due to the mutual attraction of oppositely charged electrodes and tangential shear (“fringing”) due to repulsion of like charges at the electrode edges. These enter via physically simplified boundary conditions that are applied in a fixed reference domain using a nondimensional approach. The method is valid for small to moderate strains and is straightforward to implement in a generic nonlinear elasticity code. We validate the model by directly comparing the simulated equilibrium shapes with the experiment. For circular electrodes which buckle axisymetrically, the shape of the deflection profile is captured. Annular electrodes of different widths produce azimuthal ripples with wavelengths that match our simulations. In this case, it is essential to compute multiple equilibria because the first model solution obtained by the nonlinear solver (Newton's method) is often not the energetically favored state. We address this using a numerical technique known as “deflation.” Finally, we observe the large number of different solutions that may be obtained for the case of a long rectangular strip.

1.
A.
O'Halloran
,
F.
O'Malley
, and
P.
McHugh
,
J. Appl. Phys.
104
,
071101
(
2008
).
2.
R.
Pelrine
,
P.
Sommer-Larsen
,
R. D.
Kornbluh
,
R.
Heydt
,
G.
Kofod
,
Q.
Pei
, and
P.
Gravesen
,
Proc. SPIE
4329
,
335
349
(
2001
).
3.
B.
Tavakol
,
M.
Bozlar
,
C.
Punckt
,
G.
Froehlicher
,
H. A.
Stone
,
I. A.
Aksay
, and
D. P.
Holmes
,
Soft Matter
10
,
4789
(
2014
).
4.
B.
Tavakol
and
D. P.
Holmes
,
Appl. Phys. Lett.
108
,
112901
(
2016
).
5.
R.
Heydt
,
R.
Pelrine
,
J.
Joseph
,
J.
Eckerle
, and
R.
Kornbluh
,
J. Acoust. Soc. Am.
107
,
833
(
2000
).
6.
R.
Heydt
,
R.
Kornbluh
,
J.
Eckerle
, and
R.
Pelrine
,
Proc. SPIE
6168
,
61681M
(
2006
).
7.
F.
Carpi
,
S.
Bauer
, and
D.
De Rossi
,
Science
330
,
1759
(
2010
).
8.
S.
Vishniakou
,
B. W.
Lewis
,
X.
Niu
,
A.
Kargar
,
K.
Sun
,
M.
Kalajian
,
N.
Park
,
M.
Yang
,
Y.
Jing
,
P.
Brochu
,
Z.
Sun
,
L.
Chun
,
T.
Nguyen
,
Q.
Pei
, and
D.
Wang
,
Sci. Rep.
3
,
2521
(
2013
).
9.
K.
Jung
,
J. C.
Koo
,
Y. K.
Lee
, and
H. R.
Choi
,
Bioinspiration Biomimetics
2
,
S42
(
2007
).
10.
S.-I.
Son
,
D.
Pugal
,
T.
Hwang
,
H. R.
Choi
,
J. C.
Koo
,
Y.
Lee
,
K.
Kim
, and
J.-D.
Nam
,
Appl. Opt.
51
,
2987
(
2012
).
11.
R.
Pelrine
,
R. D.
Kornbluh
,
Q.
Pei
, and
J.
Joseph
,
Science
287
,
836
(
2000
).
12.
J.-S.
Plante
and
S.
Dubowsky
,
Int. J. Solids Struct.
43
,
7727
(
2006
).
13.
M.
Kollosche
,
J.
Zhu
,
Z.
Suo
, and
G.
Kofod
,
Phys. Rev. E
85
,
051801
(
2012
).
14.
J.
Zhu
,
M.
Kollosche
,
T.
Lu
,
G.
Kofod
, and
Z.
Suo
,
Soft Matter
8
,
8840
(
2012
).
15.
H.
Bense
,
M.
Trejo
,
E.
Reyssat
,
J.
Bico
, and
B.
Roman
,
Soft Matter
13
,
2876
(
2017
).
16.
N.
Goulbourne
,
E.
Mockensturm
, and
M.
Frecker
,
J. Appl. Mech.
72
,
899
(
2005
).
17.
M.
Wissler
and
E.
Mazza
,
Sens. Actuators, A
120
,
184
(
2005
).
18.
M.
Wissler
and
E.
Mazza
,
Sens. Actuators, A
134
,
494
(
2007
).
19.
B.-X.
Xu
,
R.
Mueller
,
M.
Klassen
, and
D.
Gross
,
Appl. Phys. Lett.
97
,
162908
(
2010
).
20.
T.
Li
,
C.
Keplinger
,
R.
Baumgartner
,
S.
Bauer
,
W.
Yang
, and
Z.
Suo
,
J. Mech. Phys. Solids
61
,
611
(
2013
).
21.
P.
Sommer-Larsen
and
A. L.
Larsen
,
Proc. SPIE
5385
,
68
77
(
2004
).
22.
J.-S.
Plante
and
S.
Dubowsky
,
Smart Mater. Struct.
16
,
S227
(
2007
).
23.
Z.
Gao
,
A.
Tuncer
, and
A. M.
Cuitiño
,
Int. J. Plast.
27
,
1459
(
2011
).
24.
H. S.
Park
,
Z.
Suo
,
J.
Zhou
, and
P. A.
Klein
,
Int. J. Solids Struct.
49
,
2187
(
2012
).
25.
R.
Vertechy
,
A.
Frisoli
,
M.
Bergamasco
,
F.
Carpi
,
G.
Frediani
, and
D.
De Rossi
,
Smart Mater. Struct.
21
,
094005
(
2012
).
26.
K. A.
Khan
,
H.
Wafai
, and
T.
El Sayed
,
Comput. Mech.
52
,
345
(
2013
).
27.
S.
Klinkel
,
S.
Zwecker
, and
R.
Müller
,
J. Appl. Mech.
80
,
021026
(
2013
).
28.
H. S.
Park
,
Q.
Wang
,
X.
Zhao
, and
P. A.
Klein
,
Comput. Methods Appl. Mech. Eng.
260
,
40
(
2013
).
29.
D. L.
Henann
,
S. A.
Chester
, and
K.
Bertoldi
,
J. Mech. Phys. Solids
61
,
2047
(
2013
).
30.
F.
Vogel
,
S.
Göktepe
,
P.
Steinmann
, and
E.
Kuhl
,
Eur. J. Mech. A-Solids
48
,
112
(
2014
).
31.
S.
Seifi
and
H. S.
Park
,
Int. J. Solids Struct.
87
,
236
(
2016
).
32.
S.
Wang
,
M.
Decker
,
D. L.
Henann
, and
S. A.
Chester
,
J. Mech. Phys. Solids
95
,
213
(
2016
).
33.
X.
Zhao
,
W.
Hong
, and
Z.
Suo
,
Phys. Rev. B
76
,
134113
(
2007
).
34.
X.
Zhao
and
Z.
Suo
,
J. Appl. Phys.
104
,
123530
(
2008
).
35.
Z.
Suo
,
X.
Zhao
, and
W. H.
Greene
,
J. Mech. Phys. Solids
56
,
467
(
2008
).
36.
J.
Zhou
,
W.
Hong
,
X.
Zhao
,
Z.
Zhang
, and
Z.
Suo
,
Int. J. Solids Struct.
45
,
3739
(
2008
).
37.
X.
Zhao
and
Z.
Suo
,
Appl. Phys. Lett.
93
,
251902
(
2008
).
38.
B.
O'Brien
,
T.
McKay
,
E.
Calius
,
S.
Xie
, and
I.
Anderson
,
Appl. Phys. A
94
,
507
(
2009
).
39.
G. R.
Mirams
,
C. J.
Arthurs
,
M. O.
Bernabeu
,
R.
Bordas
,
J.
Cooper
,
A.
Corrias
,
Y.
Davit
,
S.-J.
Dunn
,
A. G.
Fletcher
,
D. G.
Harvey
,
M. E.
Marsh
,
J. M.
Osborne
,
P.
Pathmanathan
,
J.
Pitt-Francis
,
J.
Southern
,
N.
Zemzemi
, and
D. J.
Gavaghan
,
PLoS Comput. Biol.
9
,
e1002970
(
2013
).
40.
S.
Balay
,
S.
Abhyankar
,
M. F.
Adams
,
J.
Brown
,
P.
Brune
,
K.
Buschelman
,
L.
Dalcin
,
V.
Eijkhout
,
W. D.
Gropp
,
D.
Kaushik
,
M. G.
Knepley
,
D. A.
May
,
L. C.
McInnes
,
K.
Rupp
,
P.
Sanan
,
B. F.
Smith
,
S.
Zampini
,
H.
Zhang
, and
H.
Zhang
, “
PETSc users manual
,” Technical Report ANL-95/11 - Revision 3.8, Argonne National Laboratory,
2017
.
41.
C.
Geuzaine
and
J.-F.
Remacle
,
Int. J. Numer. Methods Eng.
79
,
1309
(
2009
).
42.
P. E.
Farrell
,
Á.
Birkisson
, and
S. W.
Funke
,
SIAM J. Sci. Comput.
37
,
A2026
(
2015
).
43.
R.
Díaz-Calleja
,
P.
Llovera-Segovia
,
J. J.
Dominguez
,
M. C.
Rosique
, and
A. Q.
Lopez
,
J. Phys. D: Appl. Phys.
46
,
235305
(
2013
).
44.

Note that since we do not require each end to deform to the same height in the z-direction, the enforced periodicity is equal to twice the length of the simulation domain. Fully periodic solutions may be obtained by a reflection at either end.

You do not currently have access to this content.