Elastic dielectrics
Zhigang Suo
suo@seas.harvard.edu
Room 309, Pierce Hall
Battery
Dielectric
Elastomer
A
a
l
Compliant
Electrode
Reference State
Q

Q
Current State
1
Dielectric elastomer actuators
•Large deformation
•Compact
•Lightweight
•Low cost
•Low-temperature fabrication
Kofod, Wirges, Paajanen, Bauer
APL 90, 081916, 2007
2
Parallel-plate capacitor
P
battery
a

Q
electrode
l
Q

Q

 0
a
l
vacuum
electrode
Q
P
force
Electric field
Electric displacement field
stress field
E

l
Q
D
a
P

a
D  0E
0, permittivity of vacuum
1
2
   0 E 2 Maxwell stress
3
An atom in an electric field
------------
p
e
p
e

battery
Hydrogen atom
+++++++++
External electric field displaces positive and negative charges somewhat.
•Polarization: Induce more change on the electrodes.
•Deformation: Distort the shape of the atom.
4
A dipole in an electric field
------------

battery
Polar molecules
+++++++++
External electric field reorients dipoles.
•Polarization: Induce more change on the electrodes.
•Deformation: Distort the shape of the sample.
5
thinning or thickening?
-----------
+++++++++
-----------+
-
Maxwell stress
+++++++++

   E2
Electrostriction
2
Subject to an electric field, many polymers thin, but many ceramics thicken.
Cheng, Zhang, MRS Bulletin 33, 183 (2008).
6
Thermodynamics of elastic dielectrics
Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
Reference State
Current State
A
l
s  P/ A

Q
a
L
 l/L
~
E /L
~
D Q/ A
Q
P
For elastic dielectrics, work fully converts to free energy:
Free-energy density
Equations of state
F
Pl Q


AL AL LA

~
W  , D
s


F  Pl  Q
~ ~
W  s  ED

~
~ W  , D
E
~
D

7
Ideal dielectric elastomers
Dielectric behavior is liquid-like, unaffected by deformation.
3 L3
1L1


2 L2
2

Q
D
~
W 1 , 2 , D  Ws 1 , 2  
2
P1
P2
Ideal electromechanical coupling is purely a geometric effect:
~
D
Q
~ Q
D
D
D
12
a
A
Choose a free energy of stretching. For example, neo-Hookean law:

Ws 1 , 2   12  22  32  3
incompressibility 3  1 / 12 
2




~
~
W 1 , 2 D
D 2  3 2
 3 2
s1 
  1  1 2  
1 2
1

~
~
W 1 , 2 , D
D 2  3 2
 3 2
s2 
  2  2 1  
2 1
2



~
~
~ W 1 , 2 , D D 2 2
E
 1 2
~

D
8
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Dielectric constant is insensitive to stretch
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787-793 (2003).
9
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
10
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)
11
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)
12