A nonlinear field theory of deformable dielectrics Zhigang Suo - - PowerPoint PPT Presentation

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A nonlinear field theory of deformable dielectrics

Talk 1 in Session 21-2-2, 10:00 am - 12:00 pm, Wednesday, 6 June 2007, McMat 2007, Austin, Texas Slides are available at

Zhigang Suo Harvard University

Work with

• X. Zhao, W. Greene, Wei Hong, J. Zhou

http://imechanica.org/node/635

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Dielectric elastomer actuators

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

Φ l a Q + Q −

Compliant Electrode Dielectric Elastomer

L

A Reference State 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

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

Φ

l Q + Q − Φ

Q

thick thin Coexistent states: flat and wrinkled

Plante, Dubowsky,

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

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Trouble with electric force

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

i i

E

,

Φ − =

i i

E D ε = q D i

i = , i i

qE F =

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

Historical work

• Toupin (1956)
• Eringen (1963)
• Tiersten (1971)

…… Recent work

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

……

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Maxwell stress in vacuum (1864?)

i i

E D ε =

i i

E

,

Φ − = q D i

i = ,

( )

j i j j i j i j j i i

E D E D E D qE F

, , ,

− = = =

i j ij j i

E E

, , ,

= Φ − =

( ) i

k k i j j j i j

E E E E E D

, , ,

2 ε ε = =

j ij k k i j i

E E E E F

,

2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = δ ε ε

ij k k i j ij

E E E E δ ε ε σ 2 − =

2

2 E ε σ =

P

Q + Q −

P

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Trouble with Maxwell stress in dielectrics

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

• - - - - - - - - - - -
• - - - - - - - - - - -

+ + + + + + + + +

+

• Maxwell stress

Electrostriction

2 33

2 E ε σ − =

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

In solid, Maxwell stress has NO theoretical basis

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All troubles are gone if we focus on measurable quantities

Φ P ground Q δ l δ electrode dielectric

Work done by the weight

l Pδ

Work done by the battery

Q δ Φ

Q l P U δ δ δ Φ + =

( ) ( ) ( ) ( )

Q Q l U Q l l Q l U Q l P ∂ ∂ = Φ ∂ ∂ = , , , , , A system of elastic conductors and dielectrics is specified by a free-energy function

( )

Q l U ,

A transducer

( )

Q l U ,

is measurable

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

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Intensive quantities

Reference State Current State Intensive Quantities L l / = λ

P Φ l a Q + Q − L

A A P s / = L E / ~ Φ = A Q D / ~ = ) / ( l E Φ = ) / ( a Q D =

( ) ( ) ( ) δλ

λ δ δ s AL L sA l P = =

Work done by the weight

( ) ( ) (

)

D E AL A D L E Q ~ ~ ~ ~ δ δ δ = = Φ

Work done by the battery

( )

D W ~ , λ

Free energy per unit volume

Q l P U δ δ δ Φ + = D E s W ~ ~δ δλ δ + = ⎩ ⎨ ⎧ ∂ ∂ = ∂ ∂ = D W E W s ~ / ~ / λ

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Ideal dielectric elastomer

Decouple elastic and dielectric energy

( )

( ) ( )

D W W D W

1

~ , + = λ λ

D A Q a Q D ~

2 1

λ λ λ = = =

( )

ε 2

2 1

D D W = Linear dielectric liquid

( )

( ) ( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − + − + − = ... 27 1050 11 9 20 1 3 2 1

3 2 2

I N I N I W µ

Arruda-Boyce elastomer: µ: small-strain shear modulus N: number of rigid units between neighboring crosslinks

2 3 2 2 2 1

λ λ λ + + = I

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Hysteresis and coexistent states

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

l Q + Q − Φ

L E / ~ Φ = ) / ( l E Φ = A Q D / ~ = ) / ( a Q D =

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Equilibrium & Stability

( )

Q L P L P D W L L L G Φ − − − =

2 2 2 1 1 1 2 1 3 2 1

~ , , λ λ λ λ

Elastomer weights battery Free energy of the system

2

P

1

P

2 2L

λ

3 3L

λ

1 1L

λ

Φ

Q

2

P

1

P

2 2L

λ

3 3L

λ

1 1L

λ

Φ

Q D D W D D W W D D W W W D E D W s W s W L L L G ~ ~ ~ ~ ~ ~ 2 1 2 1 2 1 ~ ~ ~

2 2 2 1 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 1 1 1 3 2 1

δ δλ λ δ δλ λ δλ δλ λ λ δ δλ λ δλ λ δ δλ λ δλ λ δ ∂ ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ ∂ =

1 1

λ ∂ ∂ = W s

2 2

λ ∂ ∂ = W s D W E ~ ~ ∂ ∂ =

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

,

( )

det = H

,

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Pre-stresses enhance actuation

2

P

1

P

2 2L

λ

3 3L

λ

1 1L

λ

Φ

Q

2

P

1

P

2 2L

λ

3 3L

λ

1 1L

λ

Φ

Q

Experiment: Pelrine, Kornbluh, Pei, Joseph Science 287, 836 (2000). Theory: Zhao, Suo http://imechanica.org/node/1456

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Inhomogeneous field

Φ P Q δ l δ

A field of weights

( ) ( )

K i iK

X t x t F ∂ ∂ = , , X X

∫ ∫ ∫

+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − = ∂ ∂ dA t dV t x b dV X s

i i i i i K i iK

ξ ξ ρ ξ ~ ~ ~

2 2

( ) ( )

K K

X t t E ∂ Φ ∂ − = , , ~ X X

∫ ∫ ∫

+ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − dA dV q dV D X

K K

ω η η η ~ ~ ~

A field of batteries Material laws

( ) ( )

iK iK

F W s ∂ ∂ = D F D F ~ , ~ ,

( ) ( )

K K

D W E ~ ~ , ~ , ~ ∂ ∂ = D F D F

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

• Linear PDEs
• Nonlinear material laws

K K iK iK

E D F s W ~ ~ δ δ δ + =

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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…