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A nonlinear field theory of deformable dielectrics

Zhigang Suo

Harvard University

Work withX. 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, TexasSlides are available at http://imechanica.org/node/635

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Dielectric elastomer actuators•Electromechanical coupling•Large deformation, light weight, low cost…•Soft robots, artificial muscles…

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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Electromechanical instability

l

Q

Q

Q

Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.

Coexistent states: flat and wrinkled

Plante, Dubowsky, Int. J. Solids and Structures 43, 7727 (2006).

thick thin

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Trouble with electric force

Historical work•Toupin (1956)•Eringen (1963)•Tiersten (1971)……

Recent work•Dorfmann, Ogden (2005)•Landis, McMeeking (2005)•Suo, Zhao, Greene (2007)……

In a vacuum, force between charges can be measuredDefine electric field by E = F/Q

+Q +Q

+Q +Q

In a solid,force between charges is NOT an operational concept

iiE ,

ii ED

qD ii ,

ii qEF

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Maxwell stress in vacuum (1864?)

jijjijijjii EDEDEDqEF ,,,

ijijji EE ,,, ikkijjjij EEEEED ,0

,0, 2

jijkkiji EEEEF

,

00 2

ijkkijij EEEE 20

0

iiE , ii ED 0qD ii ,

Q

Q

20

2E

P

P

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Trouble with Maxwell stress in dielectrics

+ + + + + + + + +

- - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + +

+-

Maxwell stressElectrostriction2

33 2E

In solid, Maxwell stress has NO theoretical basis

Maxwell said that he didn’t make progress with dielectrics, but his qualms have not prevented people from using his name anyway…

•In dielectrics, electric force is not an operational concept.• varies with deformation in general.•Why E2 dependence?•How about the sign of the Maxwell stress?

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All troubles are goneif we focus on measurable quantities

Pground

Q

l

electrode

dielectric

Work done by the weight lPWork done by the battery Q

QlPU

Q

QlUQl

l

QlUQlP

,,,

,,

A system of elastic conductors and dielectricsis specified by a free-energy function QlU ,

A transducer QlU , is measurable

Suo, Zhao, Greene

JMPS, in press. http://imechanica.org/node/635

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Intensive quantities

L

A

Reference State

P

l

a Q

Q

Current State

Ll /

Intensive Quantities

APs /

LE /~

AQD /~

)/( lE

)/( aQD

Work done by the weight sALLsAlP

Work done by the battery DEALADLEQ~~~~

Free energy per unit volume DW~

,

QlPU DEsW~~

DWE

Ws~

/~

/

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Ideal dielectric elastomer

DWWDW 10

~,

Decouple elastic and dielectric energy

...27

1050

119

20

13

2

1 32

20 I

NI

NIW Arruda-Boyce elastomer:

: small-strain shear modulusN: number of rigid units between neighboring crosslinks

23

22

21 I

DA

Q

a

QD

~

21

Linear dielectric liquid 2

2

1

DDW

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Hysteresis and coexistent states

Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.

l

Q

Q

LE /~

AQD /~

)/( lE

)/( aQD

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Equilibrium & Stability

2P 1P

22L

33L

11L

Q

2P 1P

22L

33L

11L

Q

QLPLPDWLLLG 22211121321

~,,

Free energy of the system

Elastomer weights battery

DD

WD

D

WW

DD

WWW

DED

Ws

Ws

W

LLL

G

~~

~~

~~

2

1

2

1

2

1

~~~

2

2

2

1

1

2

2121

2

2

2

2222

2

2212

1

2

222

111321

0det H

Equilibrium state becomes unstable when the Hessian ceases to be positive-definite

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W

s2

2 W

sD

WE ~~

,

,

Equilibrium state

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Pre-stresses enhance actuation

2P 1P

22L

33L

11L

Q

2P 1P

22L

33L

11L

Q

Experiment: Pelrine, Kornbluh, Pei, JosephScience 287, 836 (2000).

Theory: Zhao, Suohttp://imechanica.org/node/1456

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Inhomogeneous field

P

Q

l

A field of weights K

iiK X

txtF

,

,X

X

dAtdVt

xbdV

Xs iii

ii

K

iiK ~~~

2

2

K

K X

ttE

,

,~ XX

dAdVqdVDX KK

~~~A field of batteries

Material laws

iK

iK F

Ws

DF

DF~

,~,

K

KD

WE ~

~,~

,~

DF

DF

Suo, Zhao, Greene

JMPS, in press. http://imechanica.org/node/635

•Linear PDEs•Nonlinear material laws

KKiKiK EDFsW~~

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Finite element method

Thick State

Transition

Thin State

Thick State

Transition

Thin State

Zhou, Hong, Zhao, Zhang, Suo

http://imechanica.org/node/1447

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Summary

• 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…