Large deformation and instability
in dielectric elastomers
Zhigang Suo
School of Engineering and Applied Sciences
Harvard University
1
Dielectric elastomer actuators
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
Φ
−Q
Current State
2
Parallel-plate capacitor
P
battery
a
Φ
−Q
electrode
l
+Q
Φ
Q
Φ
= ε0
a
l
vacuum
electrode
Q
P
force
Electric field
Electric displacement field
stress field
E=
Φ
l
Q
D=
a
P
σ=
a
D = ε0E
ε0, permittivity of vacuum
1
2
σ = ε 0 E 2 Maxwell stress
3
Field equations in vacuum, Maxwell (1873)
∂Φ
Ei = −
∂x i
Electrostatic field
∂E i
q
=
∂x i ε 0
Fi = qE i
A field of forces maintain equilibrium of a field of charges
Fi =
−Q
∂
∂x j
ε0
⎛
⎞
ε
E
E
E
E
δ
−
⎜ 0 j i
k k ij ⎟
2
⎝
⎠
P
σ=
E
+Q
ε0
2
E2
σ ij = ε 0 E j E i −
Maxwell stress
P
ε0
2
E k E kδ ij
4
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
5
Trouble with Maxwell stress in dielectrics
-----------±
+++++++++
Maxwell stress
ε
σ 33 = − E 2
2
-----------+
-
+++++++++
Electrostriction
Our complaints:
•In general, ε varies with deformation.
•In general, E2 dependence has no special significance.
•Wrong sign?
In solids, the Maxwell stress is a bad idea.
6
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
7
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.”
8
All troubles are gone if we relate measurable quantities
Reference State
Current State
A
+Q
a
L
l
λ =l/L
Φ
s = P/ A
~
E =Φ/L
~
D =Q/ A
−Q
P
Equilibrium condition
Free-energy density
Equations of state
δF = Pδl + ΦδQ
δF
Pδl ΦδQ
=
+
AL AL LA
( )
~
∂W λ , D
s=
∂λ
~ ~
δW = sδλ + EδD
( )
~
~ ∂W λ , D
E=
~
∂D
9
Extend the theory to 3D inhomogeneous field
Historical work
•Toupin (1956)
•Eringen (1963)
•Tiersten (1971)
……
Recent work
•Dorfmann, Ogden (2005)
•McMeeking, Landis (2005)
……
Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
10
Two ways of doing work
A field of weights do work through displacements:
∫ Biδxi dV + ∫ Tiδxi dA
A field of batteries do work through changing:
δQ
∫ ΦδqdV + ∫ ΦδωdA
Φ
P
δl
Equilibrium condition
∫ δWdV = ∫ Biδxi dV + ∫ Tiδxi dA + ∫ ΦδqdV + ∫ ΦδωdA
11
A field of markers: stretch
L
λ=
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
12
A field of batteries: electric field
Reference state
L
Current state
l
X
x(X, t)
Φ
~ Φ
E=
L
x(X+dX, t)
X+dX
Φ(X ,t )
Φ (X + dX ,t )
Φ (X + dX ,t ) − Φ (X ,t )
dX K
ground
∂Φ (X ,t )
~
=
−
EK
∂X K
13
A field of weights: stress
Define the stress siK, such that
∫ siK
∂ξ i
dV = ∫ Biξ i dV + ∫ Tiξ i dA
∂X K
holds for any test function ξi (X)
Apply divergence theorem, one obtains that
∂siK (X ,t )
+ Bi (X ,t ) = 0
∂X K
in volume
(s
−
iK
(X ,t ) − siK+ (X ,t ))N K (X ,t ) = Ti (X ,t )
on interface
14
A field of charges: electric displacement
Define the electric displacement D~K, such that
+
∂ζ ~
−∫
DK dV = ∫ ζqdV + ∫ ζωdA
∂X K
-
+
holds for any test function ζ(X).
Apply divergence theorem, one obtains that
~
∂DK (X ,t )
= q(X ,t )
∂X K
in volume
(
)
~+
~−
DK (X , t ) − DK (X , t ) N K (X ,t ) = ω (X , t )
on interface
15
Material model
( )
~
W = W F, D
~ ~
δW = siK δFiK + E K δDK
siK
~
(
~ ∂W F, D )
(F, D) = ∂F ,
iK
( )
~
~
~ ∂W F, D
E K F, D =
~
∂D K
( )
16
A field theory
A field of markers
FiK (X , t ) =
A field of weights
∂x i (X , t )
∂X K
∂siK (X , t )
+ Bi (X , t ) = 0
∂X K
A field of batteries
A field of charges
~
∂Φ (X , t )
E K (X , t ) = −
∂X K
Equations of state
( )
~
~ ∂W F, D
siK F, D =
∂FiK
( )
~
∂DK (X , t )
= q(X , t )
∂X K
( )
~
~
~ ∂W F, D
E K F, D =
~
∂D K
( )
17
Construct a specific material model:
( )
~
W F, D
18
The nominal vs. the true
Reference State
Current State
A
L
+Q
a
l
Φ
~
E =Φ/L
~
D =Q/ A
(E = Φ / l)
( D = Q / a)
−Q
P
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
( )( )
~
~
~ ~
ΦδQ = EL δ DA = ( AL )EδD
True electric field and true electric displacement are NOT work-conjugate
Battery does work
ΦδQ = (El )δ (Da ) = (la )EδD + EDlδa
19
The nominal vs the true
Reference State
Current State
A
L
Φ
+Q
a
l
−Q
P
s = P/ A
σ = P/a
~
E =Φ/L
E = Φ/l
~
D =Q/ A
D =Q/a
σ ij =
F jK
det(F )
siK
~
FiK E i = E K
Di =
FiK ~
DK
det (F )
20
Dielectric constant is insensitive to stretch
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787-793 (2003).
21
Ideal dielectric elastomers
Dielectric behavior of an elastomer is liquid-like, unaffected by deformation
( )
~
D2
W F, D = Ws (F ) +
2ε
Stretch
Polarization
Ideal electromechanical coupling is purely a geometric effect:
Di =
( )
D i = εE i
( )
~
~ ∂W F, D
siK F, D =
∂FiK
~
~
~ ∂W F, D
E K F, D =
~
∂D K
( )
FiK ~
DK
det (F )
( )
σ ij =
FiK ∂Ws (F ) ⎛
1
⎞
+ ε ⎜ E i E j − E k E kδ ij ⎟
det(F ) ∂F jK
2
⎝
⎠
22
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007)
Neo-Hookean, incompressible dielectric elastomers
~2
D
~ µ 2
W λ1 , λ2 , D = λ1 + λ22 + λ23 − 3 +
2
2ελ21 λ22
(
λ3 L3
λ1 L1
λ2 L2
) (
)
λ1 λ2 λ3 = 1
~2
∂W
D
s1 =
= µ λ1 − λ1−3λ2−2 −
λ1−3λ2−2
∂λ1
ε
(
Φ
Q
)
~2
∂W
D
= µ λ2 − λ2−3λ1−2 −
λ2−3λ1−2
s2 =
∂λ2
ε
(
~
~ ∂W D −2 −2
E = ~ = λ1 λ2
∂D ε
P1
P2
)
(
)
D = εE
(
)
λ1 λ2 λ3 = 1
σ 1 = µ λ21 − λ23 − εE 2
σ 2 = µ λ22 − λ23 − εE 2
23
An application: Electromechanical instability
24
Electromechanical instability
Φ
+Q
l
2 ~2
D
λ
~ µ 2
W λ , D = λ + 2λ−2 − 3 +
2
2ε
( ) (
)
−Q
l
(
)
(
)
~
∂W λ , D
s=
=0
∂λ
Q
~
~ ∂W λ , D
E=
~
∂D
Φ
~ −1/ 3
⎛ D2 ⎞
⎟
λ = ⎜⎜1 +
⎟
⎝ εµ ⎠
~
~ Dλ2
E=
ε
~
~
~ −2 / 3
E
D ⎛ D2 ⎞
⎟
⎜1 +
=
µ /ε
εµ ⎜⎝ εµ ⎟⎠
Q
~
Ec ~
Stark & Garton, Nature 176, 1225 (1955)
Zhao, Suo, APL 91, 061921 (2007)
106 N / m
µ
= 108V / m
~
−10
10 F / m
ε
λc ≈ 0.63
25
Pre-stretch increases actuation strain
λ3 L3
λ1 L1
λ2 L2
Φ
Q
P2
P1
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory: Zhao, Suo
APL 91, 061921 (2007)
26
Coexistent states
stiffening
thinning
polarizing
+Q
l
Φ
Φ
−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)
27
Stiffening due to extension limit
Non Gaussian statistics (e.g., Arruda-Boyce model): Ws = µ ⎡ 1 (I − 3) + 1 (I 2 − 9 ) + ...⎤⎥
⎢2
20n
⎦
⎣
I = λ12 + λ22 + λ32
µ: small-strain shear modulus
n: number of monomers per chain
Φ
+Q
l
Φ
−Q
Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
Q 28
Finite element method
Thick State
Transition
Thin State
Thin State
Transition
Thick State
29
Zhou, Hong, Zhao, Zhang, Suo, Int. J. Solids and Structures 45, 3739 (2008)
Summary
• A field theory. No Maxwell stress. No electric body force.
• Effect of electric field on deformation is a part of material
model.
• Ideal dielectric elastomers: Maxwell stress emerges.
• Electromechanical instability.
• Add other effects (solvent, ions, enzymes…)
30