ElectroMechanical Modeling of Dielectric Elastomers

1
Electro-Mechanical Modeling of Dielectric
Elastomers

• Zhan Gao and Alberto Cuitino
• Department of Mechanical and Aerospace
  Engineering, Rutgers University
• New Brunswick, New Jersey, USA

2004 ASME International Mechanical Engineering
Congress and RDD Expo Session MAT-10 Advanced
Applications of Heterogeneous Material Systems
November 14 - 19, 2004 Anaheim, California USA
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Introduction- comparison of the properties of
EAC, SMA, and EAP

• Materials that generate large mechanical
  actuation induced by external electric field
• Electroactive Ceramic (EAC)
• High response speed (?sec to sec), low strain
  levels (0.1), high actuation force (30-40MPa),
  drive voltage (50-800V), fragile
• Shape Memory Alloys (SMA)
• low speed (msec to min), large strain (short fatigue life), high force (200MPa), low
  drive voltage (5V), resilient
• Electroactive Polymer (EAP)
• large strain ( to 300), low force, resilient,
  lightweight
• Ionic EAP (gels, polymer-metal composites,
  conductive polymers)
• Low speed (sec to min), Low drive voltage
  (1-5V), need to be wet
• Electronic EAP (ferroelectric polymer,
  dielectric elastomers)
• High speed (?sec to msec), high drive field
  (150V/?m).

3
Comparison of high-speed actuator technologies
(R.E. Pelrine, et al, 1998)
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Applications of EAP
The first commercial EAP product a fish robot
that made by EAMEX Corporation, Japan, 2002.
The first EAP/Human Arm-wrestling Contest at San
Diego, Mar. 2005. Three challengers are scheduled
to take part in.
5
Application of Dielectric Elastomer
An small walking robot called Skitter
Q. Pei, R. Pelrine, M. Rosenthal, S. Stanford, R.
Kornbluh, H. Prahlad SRI
International, Menlo Park, Calif.
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Finite Elastic-Dielectric Model

• The weak forms for momentum balance and
  quasi-electrostatic of Maxwell equation in
  dielectrics,

Total Piola-Kirchhoff stress Pt is defined by
M is the mixed frame stress tensor of m, the
Maxwell electrostatic stress tensor in spatial
frame,
? is permittivity of linear dielectric.
E is the spatial-frame electric field, ?
material-frame electric field,
D is electric displacement in material frame,
while d is in spatial frame,
In above equations, F is deformation gradient, C
is right Cauchy-Green tensor, ?, electric
potential, ?, virtual displacement, ? virtual
electric potential.
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Constitutive Equations

• Electric displacement for isotropic linear
  dielectric,

Elastic strain energy for compressible Ogdens
model,
Where ?1, ?2, ?3 are stretch ratio, ?k and ?k are
material constants, ? is Lame constant.
Elastic Cauchy stress
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Actuation Response of Polyacrylate Dielectric
Elastomers Experiment and Previous Model
VBHTM 4910 is an acrylic rubber, available as 1mm
thick adhesive film. The film is pre-strained in
x direction, then fix on frames (along x
direction). Force is applied and measured along y
direction. Compliant electrodes are on the front
and back plane of film (in z direction).
Film pre-strained 500 (in x direction), was
stretched in y direction. The force-strain curve
measured was used to fit a two-term Ogden model.
The material constants for Ogden model are
?168.5kPa, ?1 0.7, ?2767Pa, ?23.441. ?4.7?0.
A 1-D incompressible Ogden model (along y
direction) with maxwell stress effect was
proposed by Kofod to model the dielectric
elastomer and explain the improvement of
actuation strain by pre-stretching the film.
Ph.D. thesis of
G. Kofod
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Simulation of Electrical-Mechanical Actuation

• Three steps simulation,
• Pre-strain 300 in x direction
• Apply force in y direction till it reach 0.12N,
  then keep the force constant
• Apply electric voltage till the film breakdown
  (no solution in calculation). The electrodes
  cover the entire top and bottom z planes.

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Simulation Results Electro-Mechanical Fields
Central X Plane during Actuation
Electric Potential
Elastic stress in y and z direction.
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Comparison of Results for VBHTM 4910 Actuator
(pre-strain 300)
Simulation
Experiment
25mm?5mm?1mm VBH 4910 film pre-strained 300 in x
direction, applied constant force in y direction,
then actuated. The plot show change of length of
film in y direction with force and voltage.
Ph.D. thesis of G. Kofod
Simulation results for 5mm ?5mm?0.5mm film.
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Comparison of Results for VBHTM 4910 Actuator
(pre-strain 300)
Simulation
Experiment
Ph.D. thesis of G. Kofod
Actuation strain in y direction vs force voltage
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Prediction of Electrical Breakdown (No
Mechanical Constrains)
Assume polymer is incompressible,
Assume t01mm, using experimental material
constants, to keep the elastic and electric
stress balance, the relation between voltage V
and stretch ratio ? is,
Conclusion from above analysis, for 1mm thick VBH
4910 film, the breakdown voltage is 17.6kV,
breakdown Field is 28.6MV/m.
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Prediction of Electrical Breakdown(For Isotropic
Pre-strains)
Experiment
Model
Assume film is incompressible, the stretch ratios
are uniform.
Due to pre-strain, when voltage is low, over-all
stress ?x 0. Once voltage increase to the point
that ?x 0, the film start buckling, which can
be treated as the breakdown point for VHB. Set ?x
0, assume t01mm, and solve above equations, we
get the relation of breakdown voltage (or
breakdown field) with thickness.
VHBTM 4910 were pre-stretched an equal amount in
both X and Y direction, and glued to frames. The
film was clamped between two metal electrodes,
the distance between them can be read from a
thickness gauge. During actuation, there were no
measurable thickness changes observed. G.
Kofod, P. Sommer-Larsen, R. Kornbluh, R. Pelrine
J. Intel. Mater. Sys. Struc, 14 (2003).
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Comparison of Experimental Results with
Prediction for Electrical Breakdown (For
Isotropic Pre-strains)
Experiment results G.
Kofod, P. Sommer-Larsen, R. Kornbluh, R. Pelrine
J. Intel. Mater. Sys. Struc, 14 (2003).
Analytical Results
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Summary

• An electro-mechanical model based on FEM is
  developed and successfully simulate the actuation
  response of one of the most promising EAP
  material, VBHTM 4910. This new method
  incorporates finite elastic Ogden and linear
  dielectric model to solve the electro-mechanical
  coupled large deformation non-linear strain and
  electric fields. The results agree with those
  from the experiment in two aspects
• show as documented by G. Kofod that the
  actuation strain is increased by pre-stretching
  of film
• accurately predict the breakdown strength
  systematically for the first time, allowing to
  explore the mechanism of electric breakdown for
  the material