Mechanics of
Soft Active Materials (SAMs)
Zhigang Suo
Harvard University
Work with
X. Zhao, W. Hong, J. Zhou, W. Greene
1
Dielectric elastomers
Dielectric
Elastomer
A
L
a
l
Compliant
Electrode
Reference State
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
Q

Q
Current State
2
Dielectric elastomer actuators
•Large deformation
•Compact
•Lightweight
•Low cost
•Low-temperature fabrication
Kofoda, Wirges, Paajanen, Bauer
APL 90, 081916, 2007
3
Maxwell stress in vacuum (1873)
Q
P

E
0
2
 ij   0 E j Ei 
E2
0
2
Ek Ek  ij
Q
P
A field of forces needed to maintain equilibrium of a field of charges
Electrostatic field

Ei  
xi

Fi 
x j
Fi  qEi
Ei q

xi  0
0



E
E

E
E

 0 j i
k k ij 
2


4
Include Maxwell stress in
a free-body diagram

1
0E2
2
“Free-body” diagram
h
gh
1
gh     0 E 2
2
1 2
E
2
5
Trouble with Maxwell stress in dielectrics
-----------
-----------+
-
+++++++++
Maxwell stress

 33   E 2
2
+++++++++
Electrostriction
Our complaints:
•In general,  varies with deformation.
•In general, E2 dependence has no special significance.
•Wrong sign of the Maxwell stress?
In solid, Maxwell stress is not even wrong; it’s a bad idea.
Suo, Zhao, Greene, JMPS (2007)
6
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
7
Trouble with electric force in dielectrics
Ei q

xi  0

Ei  
xi
+Q
+Q
In a vacuum,
force is needed to maintain equilibrium of charges
Define electric field by E = F/Q
+Q
Historical work
•Toupin (1956)
•Eringen (1963)
•Tiersten (1971)
……
+Q
In a dielectric,
force between charges is
NOT an operational concept
Fi  qEi
Fi  qEi
Recent work
•Dorfmann, Ogden (2005)
•Landis, McMeeking (2005)
•Suo, Zhao, Greene (2007)
……
8
The Feynman Lectures on Physics
Volume 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.”
9
All troubles are gone if we use measurable quantities
Reference State
Current State
A
L
l
s  P/ A

Q
a
 l/L
~
E /L
~
D Q/ A
Q
P
Weight does work Pl
Battery does work Q
For elastic dielectric, work fully converts to free energy:
U
Pl Q


AL AL
LA
Material laws
Suo, Zhao, Greene, JMPS (2007)

~
W  , D
s

U  Pl  Q
~ ~
W  s  ED


~
~ W  , D
E
~
D

10
Game plan
•
•
•
•
Extend the theory to 3D.
Construct free-energy function W.
Study interesting phenomena.
Add other effects (stimuli-responsive gels).
11
3D inhomogeneous field
Linear PDEs
Suo, Zhao, Greene, JMPS (2007)
Q

P
A field of weights
xi X, t 


FiK X, t 
,
X K
 siK
~
 i
dV   bi i dV   ~
ti i dA
X K
siK X, t  ~
 bi X, t   0
X K
X, t 


X
,
t


,
K
X K
~
A field of batteries E
l
 
   X K
s X, t   s X, t N X, t   ~t X, t 

iK

iK
K
i
~
~dA
 DK dV  q~dV  

~
DK X, t  ~
 q X, t 
X K
D~

K
X, t   D~ K X, t N K X, t  12~X, t 
Material law
 
~
W
F
,
D
Elastic dielectric, defined by a free energy function
 
 
~
~
W F, D
W F, D ~
W 
FiK 
DK
~
FiK
DK
Q

P
Free energy
of dielectric
Free energy
of the system
A little algebra
Potential energy
of weights
~
Potential energy
of batteries
G   WdV   bixi dV   ~tixi dA   q~dV   ~dA
 
 
~
~
 W F, D

 W F, D ~  ~
G   
 siK FiK dV   
 EK DK dV
~
 FiK

 DK

Thermodynamic equilibrium: G  0
Material laws
l
 
~
~ W F, D
siK F, D 
,
FiK
 
~
for arbitrary changes FiK andDK
 
~
W F, D
~
~
E K F, D 
~
D K
 
13
Work-conjugate, or not
Reference State
Current State
A
L
Q
a
l

~
E /L
~
D Q/ A
(E   / l)
( D  Q / a)
Q
P
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
  
~
~
~ ~
Q  EL  DA   AL ED
True electric field and true electric displacement are NOT work-conjugate
Battery does work
Q  El  Da   la ED  EDla
14
True vs nominal
Reference State
Current State
A
L

Q
a
l
Q
P
s  P/ A
  P/a
 ij 
FjK
det F 
siK
~
E /L
E  /l
~
Ei  HiK EK
~
D Q/ A
D Q/a
Di 
FiK ~
DK
det F 
(H  F 1 )
15
Dielectric constant is insensitive to stretch
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787-793 (2003).
16
Ideal dielectric elastomers
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).
 
D2
~
W F, D  Ws F  
2
Stretch
Ws F  



2
2
1
Polarization

 22  32  3
Di 
FiK ~
DK
det F 
 
 
Di  Ei
 
~
~ W F, D
siK F, D 
FiK
~
W F, D
~
~
E K F, D 
~
D K
 
 ij 
FiK Ws F  
1

   Ei E j  Ek Ek  ij 17
det F  F jK
2


Electromechanical instability

Q
l
2 ~2

D
~ 
W  , D    21  3 
2

 
Q


l
Q



~
~ W  , D
E
~
D

Q
~
Ec ~

~
W  , D
s
0

~ 1/ 3
 D2 

  1 




~
~ D2
E

~
~
~ 2 / 3
E
D  D2 
1 





 /
 


106 N / m
~
 108V / m
10

10 F / m
Zhao, Suo, APL 91, 061921 (2007)
Stark & Garton, Nature 176, 1225 (1955).
18
Pre-stresses enhance actuation
3 L3
1L1
2 L2

Q
P2
P1
Experiment: Pelrine, Kornbluh, Pei, Joseph
Science 287, 836 (2000).
Theory: Zhao, Suo
APL 91, 061921 (2007)
19
Coexistent states

Q
l

Q
thick
thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Experiment: Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006).
Theory: Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)..
20
Elastomer: extension limit
 
D2
~
W F, D  Ws F  
2
Stretch
Polarization
Stiffening as each polymer chain approaches its fully stretched length
(e.g., Arruda-Boyce model)
1
1


Ws    I  3 
I 2  9   ...
20n
2

I  12  22  32
: small-strain shear modulus
n: number of monomers per chain
21
Coexistent states

Q
l
Q

Q
22
Zhao, Hong, Suo, Physical Review B 76, 134113 (2007).
Finite element method
Thick State
Transition
Thin State
Thin State
Transition
Thick State
23
Zhou, Hong, Zhao, Zhang, Suo, IJSS, 2007
Stimuli-responsive gels
Gel
•long polymers (cross-linked but flexible)
•small molecules (mobile)
reversible
collapsed
swollen
Stimuli
•temperature
•electric field
•light
•ions
•enzymes
Ono et al, Nature Materials, 2007
24
Applications of gels
Contact lenses
Artificial tissues
Drug delivery
25
Gates in microfluidics
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…)
26