1 Mechanics of Soft Active Materials (S AMs) Zhigang Suo Harvard University rk with Zhao, W. Hong, J. Zhou, W. Greene
Dec 16, 2015
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Mechanics of Soft Active Materials (SAMs)
Zhigang Suo
Harvard University
Work withX. Zhao, W. Hong, J. Zhou, W. Greene
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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:
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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
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
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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.”
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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)
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Game plan
• Extend the theory to 3D.
• Construct free-energy function W.
• Study interesting phenomena.
• Add other effects (stimuli-responsive gels).
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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
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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
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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
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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
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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
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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
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Coexistent states
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).
l
Q
Q
Q
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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…)