1

Mechanics of Soft Active Materials (SAMs)

Zhigang Suo

Harvard University

Work withX. Zhao, W. Hong, J. Zhou, W. Greene

2

Dielectric elastomers

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).

3

Dielectric elastomer actuators

Kofoda, Wirges, Paajanen, BauerAPL 90, 081916, 2007

•Large deformation•Compact•Lightweight•Low cost•Low-temperature fabrication

4

Maxwell stress in vacuum (1873)

ijkkijj

i EEEEx

F 20

0

ijkkijij EEEE 20

0

Q

Q

20

2E

P

P

ii x

E

0q

x

E

i

i

A field of forces needed to maintain equilibrium of a field of charges ii qEF

Electrostatic field

E

5

Include Maxwell stress in a free-body diagram

h

202

1E

gh

2

2

1E

202

1Egh

“Free-body” diagram

6

+ + + + + + + + +

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

+ + + + + + + + +

+-

Maxwell stressElectrostriction2

33 2E

Trouble with Maxwell stress in dielectrics

In solid, Maxwell stress is not even wrong; it’s a bad idea.

•In general, varies with deformation.•In general, E2 dependence has no special significance.•Wrong sign of the Maxwell stress?

Suo, Zhao, Greene, JMPS (2007)

Our complaints:

7

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

8

Trouble with electric force in dielectrics

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

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

In a vacuum, force is needed to maintain equilibrium of chargesDefine electric field by E = F/Q

+Q +Q

+Q +Q

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

ii qEF

0q

x

E

i

i

ii qEF

ii x

E

9

The Feynman Lectures on PhysicsVolume II, p.10-8 (1964)

“What does happen in a solid? This is a very difficult problem which has not been solved, because it is, in a sense, indeterminate. If you put charges inside a dielectric solid, there are many kinds of pressures and strains. You cannot deal with virtual work without including also the mechanical energy required to compress the solid, and 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.”

10

DW

s~,

D

DWE ~

~,~

Material laws

All troubles are gone if we use measurable quantities

L

A

Reference State

P

l

a Q

Q

Current StateLl /

APs /

LE /~

AQD /~

Weight does work lP Battery does work Q

QlPU

DEsW~~

LA

Q

AL

lP

AL

U

For elastic dielectric, work fully converts to free energy:

Suo, Zhao, Greene, JMPS (2007)

11

Game plan

• Extend the theory to 3D.

• Construct free-energy function W.

• Study interesting phenomena.

• Add other effects (stimuli-responsive gels).

12

3D inhomogeneous field

A field of weights ,

,,

K

iiK X

txtF

X

X

dAtdVbdVX

s iiiiK

iiK ~~

,

,,

~

KK X

ttE

X

X

dAdVqdVDX K

K

~~~A field of batteries

Linear PDEs

P

Q

l

0,~,

tb

X

tsi

K

iK XX tttNtsts iKiKiK ,~,,, XXXX

tqX

tD

K

K ,~,~

XX

ttNtDtD KKK ,~,,~

,~

XXXX

Suo, Zhao, Greene, JMPS (2007)

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Material law

dAdVqdAxtdVxbWdVG iiii ~~~~

Material laws

,~,~

,iK

iK F

Ws

DF

DF K

KD

WE ~

~,~

,~

DF

DF

P

Q

l

DF ~,WElastic dielectric, defined by a free energy function

K

K

iKiK

DD

WF

F

WW

~~

~,

~,

DFDF

Free energy of the system

dVDE

D

WdVFs

F

WG KK

K

iKiKiK

~~~

~,

~, DFDF

A little algebra

Potential energy of weights

Potential energy of batteries

Thermodynamic equilibrium: for arbitrary changes and 0G iKF KD~

Free energy of dielectric

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Work-conjugate, or not

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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True vs nominal

iKjK

ij sF

Fdet

KiK

i DF

D~

det F

KiKi EHE~

L

A

Reference State

P

l

a Q

Q

Current State

APs /

LE /~

AQD /~

lE /

aQD /

aP /

)( 1FH

16

Dielectric constant is insensitive to stretch

Kofod, Sommer-Larsen, Kornbluh, Pelrine Journal of Intelligent Material Systems and Structures 14, 787-793 (2003).

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2

~,

2DWW s FDF

ijkkjijK

siKij EEEE

F

WF 2

1

det

F

Fii ED

32

23

22

21

FsW

Stretch Polarization

Ideal dielectric elastomersZhao, Hong, Suo, Physical Review B 76, 134113 (2007).

iK

iK F

Ws

DF

DF~,~

, K

KD

WE ~

~,~

,~

DF

DF

KiK

i DF

D~

det F

18

l

Q

Q

Stark & Garton, Nature 176, 1225 (1955).

Electromechanical instability

3/12~

1

D

Q

Q

l

22

1~

322

~,

DDW

3/22~

1~

/

~

DDE

0

~,

DW

s

D

DWE ~

~,~

2~

~ DE

mVmF

mNEc /10

/10

/10~~

~ 810

6

Zhao, Suo, APL 91, 061921 (2007)

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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, SuoAPL 91, 061921 (2007)

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l

Q

Q

Theory: Zhao, Hong, SuoPhysical Review B 76, 134113 (2007)..

Coexistent states: flat and wrinkledExperiment: Plante, Dubowsky, Int. J. Solids and Structures 43, 7727 (2006).

Qthick thin

Top viewCross section

Coexistent states

21

2

~,

2DWW s FDF

...9

20

13

2

1 2In

IWs

Stiffening as each polymer chain approaches its fully stretched length (e.g., Arruda-Boyce model)

Stretch Polarization

Elastomer: extension limit

: small-strain shear modulusn: number of monomers per chain

23

22

21 I

22

Coexistent states

Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).

l

Q

Q

Q

23

Finite element method

Thick State

Transition

Thin State

Thick State

Transition

Thin State

Zhou, Hong, Zhao, Zhang, Suo, IJSS, 2007

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Gel•long polymers (cross-linked but flexible)•small molecules (mobile)

collapsed swollen

Stimuli

•temperature•electric field•light•ions•enzymes

Stimuli-responsive gels

Ono et al, Nature Materials, 2007

reversible

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Drug delivery

Applications of gels

Gates in microfluidics

Artificial tissuesContact lenses

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Summary

• A nonlinear field theory. No Maxwell stress. No electric body force.

• Effect of electric field on deformation is a part of material law.• Ideal dielectric elastomers: Maxwell stress emerges.• Electromechanical instability: large deformation and electric

field.• Add other effects (solvent, ions, enzymes…)