Electromechanical instability
of large deformation in
dielectric elastomers
Zhigang Suo
Harvard University
Work with
X. Zhao, W. Hong, J. Zhou, W. Greene
1
Pull-in instability
l
Dielectric
Elastomer
A
L
Compliant
Electrode
Q
polarizing
thinning

Q
a

l
Q
Q
Stark & Garton
Nature 176, 1225 (1955)
Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000)
Plante, Dubowsky
Int. J. Solids and Structures 43, 7727 (2006)
2
3 L3
s  P/ A
1L1
Theory
2 L2

~
E /L
Q
~
D Q/ A
Zhao, Suo, APL 91, 061921 (2007)
P1
P2
Free energy
of the system


System
Equilibrium
condition


~
~
G 1 , 2 , D  L1L2 L3W 1 , 2 , D  P11L1  P22 L2  Q
0
Dielectric
Weights
Battery
 W

 W

 W ~  ~
 
 s1 1  
 s2 2   ~  E D
L1L2 L3  1
 D


 2

G
Need free-energy function

~
W 1 , 2 , D

3
Dielectric constant is insensitive to deformation
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787-793 (2003).
4
Material law: Ideal dielectric elastomers
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Dielectric behavior is liquid-like, unaffected by deformation.


D2
~
W 1 , 2 , D  Ws 1 , 2  
2
Stretch
Polarization
Ideal electromechanical coupling is purely a geometric effect:
Q
D
a
~ Q
D
A
D
~
D
12
Choose a free energy of stretching. For example, neo-Hookean law:
Ws 1 , 2  



2
2
1

 22  32  3
incompressibility
3  1 / 12 
5
Theory of pull-in instability

Q
l
  



~
W  , D
s
0

Q
l
Q
polarizing
2 ~2

D
~  2
1
W  , D    2  3 
2

thinning

~
~ W  , D
E
~
D

~ 2 1/ 3
 D 

  1 

  
~
~
~ 2 2 / 3
E
D  D 
1 



 /
   

Q
c  0.63
~
Ec ~

106 N / m
~
 108V / m
10

10 F / m
6
Stark & Garton, Nature 176, 1225 (1955)
Zhao, Suo, APL 91, 061921 (2007)
Pre-stretch increases actuation strain
3 L3
1L1
2 L2

Q
P2
P1
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory: Zhao, Suo
APL 91, 061921 (2007)
7
stiffening
Coexistent states
thinning
polarizing
Q
l


Q
thick
thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Experiment: Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006)
Theory: Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
8
Stiffening due to extension limit
Stiffening as each polymer chain approaches its fully stretched length
(e.g., Arruda-Boyce model)
1
1


Ws    I  3 
I 2  9   ...
20n
2

I  12  22  32
: small-strain shear modulus
n: number of monomers per chain
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).

Q
l

Q
Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
9
Q
FEM: inhomogeneous deformation
Thick State
Transition
Thin State
Thin State
Transition
Thick State
10
Zhou, Hong, Zhao, Zhang, Suo, IJSS, 2007
Summary
• Ideal dielectric elastomers: dielectric behavior is
liquid like, unaffected by deformation.
• Pull-in instability: coupling large deformation and
electric charge.
• Pre-stretch increases actuation strain.
• Co-existent states: stiffening due to extension limit.
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Game plan
•
•
•
•
•
Inhomogeneous deformation.
Quantitative comparison with experiments.
Effect of viscosity on instability.
Nonideal electromechanical coupling.
Adding other effects (solvents, ions,
enzymes…): Soft Active Materials (SAMs)
http://imechanica.org/node/2280
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