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

•Electromechanical coupling
•Large deformation, light weight, low cost…
•Soft robots, artificial muscles…
L
A
Dielectric
Elastomer
Compliant
Electrode
Reference State
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
l a
+
Q
−
Q
Current State
Φ
2

l
Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283 .
Φ
Φ
+
Q
−
Q thick thin
Q
Coexistent states: flat and wrinkled
Plante, Dubowsky,
Int. J. Solids and Structures 43 , 7727 (2006).
3

E i
= − Φ
, i
D i , i
= q
+Q +Q
In a vacuum , force between charges can be measured
Define electric field by E = F/Q
D i
= ε
E i
F
= i qE i
+Q
+Q
In a solid , force between charges is
NOT an operational concept
Historical work
•Toupin (1956)
•Eringen (1963)
•Tiersten (1971)
……
Recent work
•Dorfmann, Ogden (2005)
•Landis, McMeeking (2005)
• Suo, Zhao, Greene (2007)
……
4

E i
= − Φ
, i
D i , i
= q D i
= ε
0
E i
F i
= qE i
=
D j , j
E i
=
(
D j
E i
,
) j
−
D j
E i , j
D j
E i , j
= ε
0
E j
E j , i
=
ε
0
2
(
E k
E k
)
, i
F i
= ε
0
E j
E i
−
ε
0
2
E k
E k
δ ij
, j
+
Q
E i , j
= − Φ
, ij
=
E j , i
P
σ =
ε
0
2
E
2
σ ij
= ε
0
E j
E i
−
ε
0
2
E k
E k
δ ij
−
Q
5 P

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
σ
33
= −
ε
2
E
2
- - - - - - - - - - - -
+
-
+ + + + + + + + +
Electrostriction
•In dielectrics, electric force is not an operational concept.
•
ε varies with deformation in general.
•Why E 2 dependence?
•How about the sign of the Maxwell stress?
In solid, Maxwell stress has NO theoretical basis 6

Work done by the weight P
δ l
Work done by the battery
Φ δ
Q electrode
δ
Q ground
Φ dielectric
P δ l
A transducer
Suo, Zhao, Greene
JMPS, in press. http://imechanica.org/node/635
A system of elastic conductors and dielectrics is specified by a free-energy function U
( )
δ
U
=
P
δ l
+ Φ δ
Q
P
=
∂
U
∂ l
,
Φ =
∂
U
∂
Q
U
( ) is measurable
7

L
Reference State
A l
Current State a
+
Q
−
Q
Φ
Intensive Quantities
λ = l / L s
=
P / A
~
E
= Φ
/ L
~
D
=
Q / A
( E
= Φ
/ l )
( D
=
Q / a )
P
Work done by the weight P
δ l
=
( ) ( ) ( ) s
δλ
Work done by the battery
Φ δ
Q
=
( ) ( )
=
( ) ~
E
δ
~
D
δ
U
=
Free energy per unit volume W
P
δ l
+ Φ δ
Q
δ
W
= s
δλ +
~
E
δ ~
D
⎧
⎨
⎩ s
~
E
=
=
∂
W
∂
W
/
/
∂ λ
∂
~
D
8

Decouple elastic and dielectric energy
W
( )
=
W
0
( )
+
W
1
( )
Linear dielectric liquid W
1
( )
=
D 2
2
ε
D
=
Q a
=
Q
λ
1
λ
2
A
= λ
~
D
Arruda-Boyce elastomer:
W
0
= µ
⎡
⎢⎣
1
2
(
I
−
3
)
+
1
20 N
(
I
2
−
9
I
µ
: small-strain shear modulus
N: number of rigid units between neighboring crosslinks
= λ
2
1
+ λ
2
2
+ λ
2
3
)
+
11
1050 N
2
(
I
3
−
27
)
+
...
⎤
⎥⎦
9

l
+
Q
−
Q
~
E
= Φ
/ L
~
D
=
Q / A
( E
= Φ
/ l )
( D
=
Q / a )
Φ
10
Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283 .

Free energy of the system
G
=
L
1
L
2
L
3
W
(
λ
1
,
λ
2
,
~
D
)
−
P
1
λ
1
L
1
−
P
2
λ
2
L
2
− Φ
Q
Elastomer weights battery
δ
G
L
1
L
2
L
3
+
1
2
∂ 2
W
∂ λ
2
1
=
⎝
⎜⎜
⎛ ∂
W
∂ λ
1
δλ
2
1
+
1
2
− s
1
⎠
⎟⎟
⎞
δλ
1
+
⎝
⎜⎜
⎛ ∂
W
∂ λ
2
∂ 2
W
∂ λ
2
2
δλ
2
2
+
1
2
− s
2
⎠
⎟⎟
⎞
δλ
2
∂
∂
2
W
~
D
2
δ ~
D
2
+
⎝
⎜
⎛ ∂
∂
W
~
D
−
~
E
⎠
⎟
⎞
δ
~
D
+
∂ 2
W
∂ λ
1
∂ λ
2
δλ
1
δλ
2
+
∂
∂ λ
2
1
W
∂
~
D
δλ
1
δ
~
D
+
∂
∂ λ
2
2
W
∂
~
D
δλ
2
δ
~
D
Equilibrium state s
1
=
∂
W
∂ λ
1 s
2
=
∂
W
∂ λ
2
~
E
=
∂
∂
W
~
D det
( )
=
0
,
11

Experiment : Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory : Zhao, Suo http://imechanica.org/node/1456
12

A field of weights F iK
=
∂ x i
∂
X
K
∫ s iK
∂ ξ i
∂
X
K dV
= ∫
⎛
⎜⎜
~ b i
−
∂ 2
∂ t x i
2
⎞
⎟⎟
ξ i dV
+ ∫ t
~ i
ξ i dA
A field of batteries
~
E
K
= −
∂ Φ
∂
X
K
∫
⎛
⎜⎜
−
∂ η
∂
X
K
⎞
⎟⎟
~
D
K dV
= ∫ η q dV
+ ∫ η ω dA
Material laws
δ
W
= s iK
δ
F iK
+
~
D
K
δ ~
E
K s iK
=
∂
W
∂
F iK
~
E
K
•Linear PDEs
•Nonlinear material laws
=
∂
W
∂
F
~
D
( )
K
,
δ
Q
Φ
Suo, Zhao, Greene
JMPS, in press. http://imechanica.org/node/635 P δ l
13

Thick State
Transition
Thin State
Thin State
Transition
Thick State
14
Zhou, Hong, Zhao, Zhang, Suo http://imechanica.org/node/1447

• 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…
15