Theory of dielectric elastomers
Zhigang Suo
Harvard University
11:15 am – 12:00 noon, 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
Dielectric elastomer
Reference State
A
Current State
Dielectric
Elastomer
L
a
l
Compliant
Electrode
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
Q

Q
2
Parallel-plate capacitor
P
battery
a

Q
electrode
l
Q
vacuum
electrode
P
Electric field
Electric displacement field
stress field
force
E

l
Q
D
a
P

a
D  0E
0, permittivity of vacuum
1
2
   0 E 2 Maxwell stress
3
Trouble with Maxwell stress in dielectrics
-----------
+++++++++
Maxwell stress

 33   E 2
2
-----------+
-
+++++++++
Electrostriction
Our complaints:
•In general,  varies with deformation.
•In general, E2 dependence has no special significance.
•Wrong sign?
4
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 charge on the electrodes.
•Deformation: Distort the shape of the electron cloud.
5
A dipole in an electric field
------------

battery
Polar molecules
+++++++++
External electric field reorients dipoles.
•Polarization: Induce more charge on the electrodes.
•Deformation: Distort the shape of the sample.
6
Field equations in vacuum, Maxwell (1873)

Ei  
x i
Electrostatic field
E i
q

x i  0
Fi  qEi
A field of forces maintain equilibrium of a field of charges

Fi 
x j
Q
0



E
E

E
E

 0 j i
k k ij 
2


P

E
Q
0
2
E2
 ij   0 E j E i 
Maxwell stress
P
0
2
E k E k ij
7
Include Maxwell stress in force balance

1
0E2
2
“Free-body” diagram
h
gh
1
gh     0 E 2
2
1 2
E
2
8
James Clerk Maxwell (1831-1879)
“I have not been able to make the next step, namely, to
account by mechanical considerations for these stresses in
the dielectric. I therefore leave the theory at this point…”
A Treatise on Electricity & Magnetism (1873), Article 111
9
Trouble with electric force in dielectrics
In a vacuum,
external force is needed to maintain equilibrium of charges
+Q
+Q
P
Fi  qE i
P
In a solid dielectric,
force between charges is NOT an operational concept
+Q
+Q
Fi  qE i
10
The Feynman Lectures on Physics
Volume II, p.10-8 (1964)
“It is a difficult matter, generally speaking, to make a unique
distinction between the electrical forces and mechanical
forces due to solid material itself. Fortunately, no one ever
really needs to know the answer to the question proposed.
He may sometimes want to know how much strain there is
going to be in a solid, and that can be worked out. But it is
much more complicated than the simple result we got for
liquids.”
11
All troubles are gone if we use measurable quantities
Current State
Reference State
A
a
L
l
equilibrate elastomer and loads
Pl Q


AL AL LA
name quantities
equations of state
~ ~
W  s  ED
 
~
W  , D
s

Suo, Zhao, Greene,J. Mech. Phys. Solids 56, 467 (2008)
 
~
~ W  , D
E
~
D
Q
Q
P
F  Pl  Q
F
divide by volume

Nominal
W  F / AL 
  l/L
True
s  P/A
  P /a
~
E  /L
E   /l
~
D  Q/ A
D  Q/a
12
Dielectric constant is insensitive to stretch
VHB 4910
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787 (2003).
13
Ideal dielectric elastomer
Dielectric behavior is liquid-like, unaffected by deformation.
 
~
D2
W  , D  Wstretch   
2
Elasticity
Polarization
incompressibility
A
A 1  a
1

a
For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:
D
Q
a
~ Q
D
A
~
D  D
~
~
D 2 2
W  , D  Wstretch   
2
 
14
Ideal dielectric elastomer
~
~
D 2 2
W  , D  Wstretch   
2
 
In terms of nominal quantities
 
~
W  , D
s

~2
D 
dWstretch  
s

d

In terms of true quantities
~
D  D
  s
D  E
dWstretch  
 
 E 2
d
15
Maxwell stress represented in three ways
Reference State
A
Current State
Dielectric
Elastomer
L
a
l
Compliant
Electrode
Uniaxial stress
  E 2
biaxial stress
  E 2

Q
Q
triaxial stress
1
2
  E 2
16
For incompressible material, the 3 states of stress give the same deformation
Electrostriction
-----------
+++++++++
Maxwell stress

 33   E 2
2
-----------+
-
+++++++++
Electrostriction
17
Non-ideal dielectric elastomer
Dielectric constant
   1  c 1  2  3  3
Area ratio
deformation affects dielectric constant
18
Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).
Quasi-linear dielectric elastomer
    
 
~
D2
W  , D  Wstretch   
2  
 
~
W  , D
s

~2
dWstretch   D d  2 
s



d
2 d    
Zhao, Suo, JAP 104, 123530 (2008)
19
The nominal vs. the true
Reference State
Current State
A
L
Q
a
l

~
E /L
~
D Q/ A
(E   / l)
( D  Q / a)
Q
P
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
  
~
~
~ ~
Q  EL  DA   AL ED
True electric field and true electric displacement are NOT work-conjugate
Battery does work
Q  El  Da   la ED  EDla
20
Extend the theory to general loads
21
Ideal dielectric elastomer
Current State
Reference State
3
1
2

2
1
1
1
Incompressibility
Elastic energy density
3
1
1
2
3
1 23  1
W 1 , 2 
Equations of state
 1   3  1
W 1 , 2 
 E 2
1
 2   3  2
W 1 , 2 
 E 2
2
22
Neo-Hookean model
1 23  1
Incompressibility
W
Elastic energy density
W 1 , 2  
Equations of state

2

2
1

2

2
1
 22  23  3
 22  12 22  3
 1   3   21  12 22  E 2
 2   3   22  12 22  E 2
23
Extend the theory to inhomogeneous field
24
A field of markers: stretch

l
L
Reference state
X
l
L
Current state
x(X, t)
X+dX
x(X+dX, t)
FiK
x i X ,t 

X K
x i X  dX,t   x i X,t 
dX K
25
A field of batteries: electric field
Reference state
L
Current state
l
X
x(X, t)

~ 
E
L
x(X+dX, t)
X+dX
X,t 
X  dX,t 
X  dX ,t   X ,t 
dX K
ground
 X ,t 
~
EK  
X K
26
3D inhomogeneous field
Condition of equilibrium


 2 xi
WdV   Bi   2
t



x i dV  Tix i dA  qdV  dA




Need to specify a material model
 
~
W , D
 ~

~
W F, D
~

~
W F, D  siK FiK  E K DK
Q

siK

~
W F, D

FiK

 
~
W F, D
~
EK 
~
DK
P
l
27
PDEs
FiK
 X ,t 
~
EK  
X K
x X ,t 
 i
X K
siK X, t 
 x i X, t 
 Bi X, t   
X K
t 2
2
siK

~
W F, D

FiK

~
DK X , t 
 qX , t 
X K
 
~
W F, D
~
EK 
~
DK
Toupin (1956), Eringen (1963), Tiersten (1971),
Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005)…
28
Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
The nominal vs the true
Reference State
Current State
A
L

Q
a
l
Q
P
s  P/ A
  P/a
~
E /L
E  /l
~
D Q/ A
D Q/a
 ij 
F jK
det F 
siK
~
FiK E i  E K
Di 
FiK ~
DK
det F 
29
Ideal dielectric elastomers
Liquid-like dielectric behavior, unaffected by deformation
 
~
D2
W F, D  Ws F  
2
Stretch
Polarization
Ideal electromechanical coupling is purely a geometric effect:
Di 
 
D i  E i
 
~
~ W F, D
siK F, D 
FiK
~
~
~ W F, D
E K F, D 
~
DK
 
FiK ~
DK
det F 
 
 ij 
FiK Ws F  
1

   E i E j  E k E k ij 
det F  F jK
2


30
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Finite element method
x i X , t   X , t 
Solve for
FiK 
 X ,t 
~
EK  
X K
x i X ,t 
X K
 
~
~ ~
ˆ
W F, E  W  D K E K
Legendre transformation
Conditions of thermodynamic equilibrium

ˆ 

 2 xi
W
i
dV   Bi  2
FiK X K
t



 i dV  Ti i dA



ˆ 
W
dV  qdV   dA
~
E K X K


31
States of equilibrium
P B
 
P
A
L
R

Simple Layer
Zhao and Suo, APL 93, 251902 (2008)
Ring
Sphere
32
Oscillating Balloon
Excitation by sinusoidal voltage
Oscillation of the balloon
Zhao and Suo, APL 93, 251902 (2008)
Zhu, Cai, Suo, http://www.seas.harvard.edu/suo/papers/216.pdf
33
Summary
• Ideal dielectric elastomer: dielectric behavior is
liquid-like.
• Only for ideal dielectric elastomer, the Maxwell
stress is valid.
• A more general model can represent both the
Maxwell stress and electrostriction.
• Inhomogeneous, time-dependent field can also
be modeled.
34