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Theory of dielectric elastomers
Zhigang SuoHarvard University
11:15 am – 12:00 noon, 11 January 2010, Monday
PhD Winter SchoolDielectric Elastomer Transducers
Monte Verita, Ascona, Switzerland, 11-16 January 2010http://www.empa.ch/eap-winterschool
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Dielectric elastomer
Compliant Electrode
Dielectric Elastomer
L
A
Reference State
l
a Q
Q
Current State
Pelrine, Kornbluh, Pei, Joseph High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000).
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Parallel-plate capacitor P
P
lQ
Qbattery
force
electrode
vacuumelectrode
lE
aQ
D
aP
Electric field
Electric displacement field
stress field
a
ED 0
0, permittivity of vacuum
202
1E Maxwell stress
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+ + + + + + + + +
- - - - - - - - - - - - - - - - - - - - - - - -
+ + + + + + + + +
+-
Maxwell stressElectrostriction2
33 2E
Trouble with Maxwell stress in dielectrics
•In general, varies with deformation.•In general, E2 dependence has no special significance.•Wrong sign?
Our complaints:
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An atom in an electric field
e
Hydrogen atom
e
+ + + + + + + + +
- - - - - - - - - - - -
p
External electric field displaces positive and negative charges somewhat.•Polarization: Induce more charge on the electrodes.•Deformation: Distort the shape of the electron cloud.
p
battery
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A dipole in an electric field
Polar molecules
+ + + + + + + + +
- - - - - - - - - - - -
External electric field reorients dipoles.•Polarization: Induce more charge on the electrodes.•Deformation: Distort the shape of the sample.
battery
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Field equations in vacuum, Maxwell (1873)
ijkkijj
i EEEEx
F 20
0
ii x
E
0q
xE
i
i
A field of forces maintain equilibrium of a field of charges ii qEF
Electrostatic field
ijkkijij EEEE 20
0
Q
Q
20
2E
P
P
E
Maxwell stress
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Include Maxwell stress in force balance
h
202
1E
gh
2
2
1E
202
1Egh
“Free-body” diagram
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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
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Trouble with electric force in dielectrics
In a vacuum, external force is needed to maintain equilibrium of charges
+Q +Q
+Q +Q
In a solid dielectric,force between charges is NOT an operational concept
P Pii qEF
ii qEF
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The Feynman Lectures on PhysicsVolume 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.”
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All troubles are gone if we use measurable quantities
L
A
Reference State
P
l
a Q
Q
Current State
Ll /
APs /
LE /~
AQD /~
DEsW~~
equilibrate elastomer and loads
DWs
~,
D
DWE ~
~,~
equations of state
QlPF
LAQ
ALlP
ALF
divide by volume
name quantities
aP /
lE /
aQD /
ALFW /
Nominal
True
Suo, Zhao, Greene,J. Mech. Phys. Solids 56, 467 (2008)
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Dielectric constant is insensitive to stretch
Kofod, Sommer-Larsen, Kornbluh, Pelrine Journal of Intelligent Material Systems and Structures 14, 787 (2003).
VHB 4910
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2
~,
2DWDW stretch
Elasticity Polarization
Ideal dielectric elastomer
aQ
DAQ
D~
DD~
Dielectric behavior is liquid-like, unaffected by deformation.
For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:
2
~~
,22D
WDW stretch
aA 1
incompressibility
1A
a
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Ideal dielectric elastomer
DWs
~,
In terms of nominal quantities
In terms of true quantities
2
~~
,22D
WDW stretch
2~
Dd
dWs stretch
2Ed
dWstretch
sDD~
ED
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Maxwell stress represented in three ways
Compliant Electrode
Dielectric Elastomer
L
A
Reference State
l
a Q
Q
Current State
2E 2E 2
21E
For incompressible material, the 3 states of stress give the same deformation
Uniaxial stress biaxial stress triaxial stress
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+ + + + + + + + +
- - - - - - - - - - - - - - - - - - - - - - - -
+ + + + + + + + +
+-
Maxwell stressElectrostriction2
33 2E
Electrostriction
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Non-ideal dielectric elastomer
31 321 c
Die
lect
ric
con
stan
t
Area ratio
Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).
deformation affects dielectric constant
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Quasi-linear dielectric elastomer
Zhao, Suo, JAP 104, 123530 (2008)
2~
,2D
WDW stretch
DWs
~,
22
2
~
ddD
ddW
s stretch
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The nominal vs. the true
L
A
Reference State
P
l
a Q
Q
Current State
LE /~
AQD /~
)/( lE
)/( aQD
Battery does work DEALADLEQ~~~~
True electric field and true electric displacement are NOT work-conjugate
aEDlDElaDaElQ
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
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Extend the theory to general loads
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Ideal dielectric elastomer
1
11
12
1
2
2
1
Reference State Current State
Incompressibility 1321
Elastic energy density 21,W
Equations of state 2
1
21131
,E
W
2
2
21232
,E
W
3
3
3
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Neo-Hookean model
Elastic energy density
32
, 22
21
22
2121 W
Equations of state 222
21
2131 E
222
21
2232 E
32
23
22
21
W
Incompressibility 1321
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Extend the theory to inhomogeneous field
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A field of markers: stretch
Reference state Current state
X x(X, t)
K
iiK X
txF
,X
X+dX x(X+dX, t)
L lLl
K
ii
dXtxtdx ,, XXX
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A field of batteries: electric field
t,X
X
x(X, t)
X+dXx(X+dX, t)
td ,XX
ground
Reference state Current state
L l LE
~
KdX
ttd ,, XXX
K
K Xt
E
,~ X
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3D inhomogeneous field
dAdVqdAxTdVxtx
BWdV iiii
i
2
2
P
Q
l
Need to specify a material model
DF~
,W DW~
,
KKiKiK DEFsW~~~
, DF
iK
iK FW
s
DF~
, K
KD
WE ~
~,~
DF
Condition of equilibrium
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PDEs
K
iiK X
txF
,X
KK X
tE
,~ X
iK
iK FW
s
DF~
, K
KD
WE ~
~,~
DF
Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
2
2 ,,
,t
txtB
Xts i
iK
iK
XX
X tqX
tD
K
K ,,
~X
X
Toupin (1956), Eringen (1963), Tiersten (1971), Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005)…
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The nominal vs the true
iKjK
ij sF
Fdet
KiK
i DF
D~
detF
KiiK EEF~
L
A
Reference State
P
l
a Q
Q
Current State
APs /
LE /~
AQD /~
lE /
aQD /
aP /
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2
~,
2DWW s FDF
ijkkjijK
siKij EEEE
FWF
21
detF
Fii ED
Stretch Polarization
Ideal dielectric elastomers
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
iK
iK FW
s
DFDF
~,~
, K
KD
WE ~
~,~
,~
DF
DF
KiK
i DF
D~
detF
Liquid-like dielectric behavior, unaffected by deformation
Ideal electromechanical coupling is purely a geometric effect:
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dATdVtx
BdVXF
Wiii
ii
K
i
iK
2
2ˆ
Conditions of thermodynamic equilibrium
dAdVqdVXE
W
KK
~ˆ
Finite element method
KKEDWW~~~
,ˆ EF
Solve for txi ,X t,X
Legendre transformation
K
iiK X
txF
,X
KK X
tE
,~ X
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States of equilibrium
L
Simple Layer
A
BP
Ring
P
R
Sphere
Zhao and Suo, APL 93, 251902 (2008)
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Oscillating Balloon
Zhao and Suo, APL 93, 251902 (2008)Zhu, Cai, Suo, http://www.seas.harvard.edu/suo/papers/216.pdf
Excitation by sinusoidal voltage
Oscillation of the balloon
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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.