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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

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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.