A nonlinear field theory of
deformable dielectrics
Zhigang Suo
Harvard University
Work with
X. Zhao, W. Greene, Wei Hong, J. Zhou
Talk 1 in Session 21-2-2, 10:00 am - 12:00 pm, Wednesday, 6 June 2007, McMat 2007, Austin, Texas
Slides are available at http://imechanica.org/node/635
1
Dielectric elastomer actuators
•Electromechanical coupling
•Large deformation, light weight, low cost…
•Soft robots, artificial muscles…
Dielectric
Elastomer
A
L
a
l
Compliant
Electrode
Reference State
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
Q

Q
Current State
2
Electromechanical instability
Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.

Q
l

Q
thick
thin
Q
Coexistent states: flat and wrinkled
Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006).
3
Trouble with electric force
Ei   ,i
+Q
Di  Ei
+Q
In a vacuum,
force between charges can be measured
Define electric field by E = F/Q
+Q
Di ,i  q
+Q
In a solid,
force between charges is
NOT an operational concept
Fi  qEi
Historical work
•Toupin (1956)
•Eringen (1963)
•Tiersten (1971)
……
Recent work
•Dorfmann, Ogden (2005)
•Landis, McMeeking (2005)
•Suo, Zhao, Greene (2007)
……
4
Maxwell stress in vacuum (1864?)
Ei   ,i
Di   0 Ei
Di ,i  q
Fi  qEi  D j , j Ei  D j Ei , j  D j Ei , j
D j Ei , j   0 E j E j ,i 
0
2
0


Fi    0 E j Ei  Ek Ek  ij 
2

, j
Ek Ek ,i
Ei , j   ,ij  E j ,i
P
Q

 ij   0 E j Ei 
0
2
Ek Ek  ij
0
2
E2
Q
P
5
Trouble with Maxwell stress in dielectrics
Maxwell said that he didn’t make progress with dielectrics, but his qualms have
not prevented people from using his name anyway…
------------
-----------+
-
+++++++++
Maxwell stress

 33   E 2
2
+++++++++
Electrostriction
•In dielectrics, electric force is not an operational concept.
•  varies with deformation in general.
•Why E2 dependence?
•How about the sign of the Maxwell stress?
In solid, Maxwell stress has NO theoretical basis
6
All troubles are gone
if we focus on measurable quantities
Work done by the weight Pl
Work done by the battery Q
electrode
A system of elastic conductors and dielectrics
is specified by a free-energy function U l , Q
dielectric
Q
U  Pl  Q

ground
P
l
A transducer
Suo, Zhao, Greene
JMPS, in press. http://imechanica.org/node/635
Pl , Q  
U l , Q 
,
l
l , Q  
U l , Q 
Q
U l , Q is measurable
7
Intensive quantities
Reference State
Current State
Intensive Quantities
 l/L
A
Q
a
L
l
Q
P

s  P/ A
~
E /L
~
D Q/ A
(E   / l)
( D  Q / a)
Work done by the weight Pl  sA L   ALs
~   ~ 
Work done by the battery Q  EL  DA   AL ED

~
Free energy per unit volume W  , D
U  Pl  Q
~ ~

~ ~
W  s  ED
s  W / 
~
~8
 E  W / D
Ideal dielectric elastomer
Decouple elastic and dielectric energy
 
~
W , D  W0    W1 D
Linear dielectric liquid
Arruda-Boyce elastomer:
D2
W1 D  
2
D
Q
Q
~

 D
a 12 A
1
11
1

3



W0    I  3 
I 2  9 
I

27

...

20 N
1050 N 2
2
: small-strain shear modulus
N: number of rigid units between neighboring crosslinks
I  12  22  32
9
Hysteresis and coexistent states
Q
l

Q
~
E /L
~
D Q/ A
(E   / l)
( D  Q / a)
10
Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.
Equilibrium & Stability
Free energy of the system


~
G  L1L2 L3W 1 , 2 , D  P11L1  P22 L2  Q
3 L3
Elastomer
1L1
2 L2
 W

 W

 W ~  ~
 
 s1 1  
 s2 2   ~  E D
L1 L2 L3  1
 D


 2

1  2W 2 1  2W 2 1  2W ~ 2

1 
2 
~ D
2 12
2 22
2 D 2
Equilibrium state
 2W
 2W
~  2W
~

12 


D



D
~ 1
~ 2
1 2
1 D
2 D
P1
s1 
battery
G

Q
P2
weights
W
1
W
2
s2 
~ W
E ~
D
Equilibrium state becomes unstable when the, Hessian ceases to be positive-definite
det H  0
,
11
Pre-stresses enhance actuation
3 L3
1L1
2 L2

Q
P2
P1
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory: Zhao, Suo
http://imechanica.org/node/1456
12
Inhomogeneous field
x X, t 
A field of weights FiK X, t   i
X K
 ~ ~  2 xi 
i
~


s
dV

b



dV

 iK X K
  i t 2  i
 tii dA
X, t 
~
A field of batteries E
K X, t   
X K
Material laws
~
 
   X K
~
~dA
 DK dV  q~dV  

~
W  siKFiK  DK EK
siK
 
~
~ W F, D
F, D 
FiK
 
 
~
W F, D
~
~
E K F, D 
~
D K
 
•Linear PDEs
•Nonlinear material laws
Q

13
Suo, Zhao, Greene
JMPS, in press. http://imechanica.org/node/635
P
l
Finite element method
Thick State
Transition
Thin State
Thin State
Transition
Thick State
14
Zhou, Hong, Zhao, Zhang, Suo http://imechanica.org/node/1447
Summary
• A nonlinear field theory. No Maxwell
stress. No electric body force. No
polarization vector.
• Electromechanical instability.
• Hysteresis and coexistent states.
• Finite element method.
These slides are available at http://imechanica.org/node/635.
iMechanica get together. Wednesday, 5.45pm-9:00pm, Room 2.120. Beer, snacks…
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