Electromechanical instability
Zhigang Suo
Harvard University
2:00 pm – 2:45 pm, 11 January 2010, Monday
PhD Winter School
Dielectric Elastomer Transducers
Monte Verita, Ascona, Switzerland, 11-16 January 2010
http://www.empa.ch/eap-winterschool
1
Theory of pull-in instability

Q
l
 


~
W  , D
s
0

Q
l
Q
polarizing
2 ~2
~  2

D
W  , D    21  3 
2
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
2
Stark & Garton, Nature 176, 1225 (1955)
Zhao, Suo, APL 91, 061921 (2007)
Free energy
3 L3
1L1
L2
2 L2

Q
L3
L1
P1
P2




~
~
G 1 , 2 , D  L1 L2 L3W 1 , 2 , D  P1 1 L1  P2 2 L2  Q
System
Dielectric
Weights
Battery
3
Free energy minimized at a state of equilibrium
 W

 W

 W ~  ~
 
 s1 1  
 s2 2   ~  E D
L1 L2 L3  1
 D


 2

G
~  2W
~
1  2W 2 1  2W 2 1  2W ~ 2  2W
 2W






D






D



D
1
2
1
2
1
2
~
~
~
2 21
2 22
2 D 2
1 2
1 D
2 D
Equations of state




~
~
W 1 , 2 , D
W 1 , 2 , D
s1 
, s2 
,
1
2
Critical condition for onset of instability
  2W

2
 1
  2W
det 
 1 2
  2W

~
 1 D
 2W
1 2
 2W
22
 2W
~
2 D

~
~ W 1 , 2 , D
E
~
D

 2W 
~
1 D 
 2W 
~  0
2 D 
 2W 
~ 
D 2 
4
Zhao, Suo, APL 91, 061921 (2007)
Pre-stretch
increases deformation of actuation
3 L3
1L1
2 L2

Q
P2
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
P1
Theory: Zhao, Suo
APL 91, 061921 (2007)
5
stiffening
Coexistent states
thinning
polarizing
Q
l


Q
thick
thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Observation: Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006)
Interpretation: Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
6
When stretched, a polymer stiffens
n links
released
b
Langevin
b n
P
Gauss
stretched
P

P
n
fully stretched
P
bn
P
7
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)
8
Q
FEM: inhomogeneous deformation
Thick State
Transition
Thin State
Thin State
Transition
Thick State
9
Zhou, Hong, Zhao, Zhang, Suo, IJSS, 2007
Failure Modes
Rupture
Loss of Tension
  R
Electrical Breakdown
+ + + + + + + + + +
s1  0
or
s2  0
Electromechanical
Instability
+
+
+
+ + + + +
–
–
stable
– – – – –
unstable
– – – – – – – – – –
E  EBD
–
10
Allowable states of a transducer
allowable states
Pei, Pelrine, Stanford, Kornbluh, Rosenthal, Meijer, Full
Proc. SPIE 4698, 246 (2002)
Moscardo, Zhao, Suo, Lapusta
JAP 104, 093503 (2008)
11
Energy harvesting
Generate electricity from walking
Generate electricity from waves
12
Pelrine, Kornbluh…
Maximal energy that can be converted
by a dielectric elastomer
6 J/g, Elastomer (theory)
0.4 J/g, Elastomer (experiment)
0.01 J/g ceramics
E=0
EMI
R
EB
LT
13
Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)
A “practical” design

2 J/g
Q
14
Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)
Summary
• Pull-in instability: coupling large deformation
and electric charge.
• Pre-stretch increases actuation strain.
• Co-existent states: stiffening due to
extension limit.
• Allowable states: as defined by various
modes of failure
15