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Nonlinear Electroelastic Deformations of Dielectric Elastomer Composites
August 20162016 ICTAM (TS.SM08 – Multi-Component Materials and Composites), Montreal
Oscar Lopez-Pamies
Work supported by the National Science Foundation (DMS, CMMI)
In collaboration with: Victor Lefèvre
Dielectric elastomers
SRI (2009); Kofod et al. (2007); Carpi et al. (2010); Aschwanden & Stemmer (2006)
Energy Harvesters Actuators
Advantages: finitely deformable —short response times — light —inexpensive — biocompatible
Disadvantage: require very large electric fields for actuation (in the order of 100 MV/m)
Examples: (pretty much) any elastomer
…
Idea: add high-dielectric (or conducting) particles to make dielectric composites that can finitely deform under moderate electric fields
Dielectric elastomer composites
Huang et al. (2005)
c = 7%
c = 0%
Matrix: PUdielectric constant: 10
shear modulus: 10 MPa
Particles: PAA decorated o-CuPc dielectric constant: 104
shear modulus: 1 GPa
Competition of effects: adding high-dielectric particles increases the overall dielectric constant of the composite, but it may also make it stiffer, and hence less deformable
Microstructure: isotropic distribution of roughly spherical nano-particles at 7% vol. fraction
• Initial conjecture in the literature (Li, 2003; Tian et al., 2012): Enhancement is due to the heightened role of field fluctuations due to the nonlinearity of elastomers
• Alternative conjecture (Lewis, 2004; Lopez-Pamies et al., 2014): Enhancement is due to interphasial phenomena including free charges
Problem Formulation(elastic dielectrics)
Problem setting: 2-phase particulate material
Dorfmann & Ogden (2005) Toupin (1956)
KinematicsDeformed Undeformed Deformation
Gradient
Local constitutive behaviorwhere
LagrangianElectric Field
Total nominal stress
Lagrangian electric displacement Indicator function characterizing the initial microstructure
Problem setting: macroscopic response
Hill (1972) Lopez-Pamies (2014)
Definition: relation between the volume averages of and
Variational Characterization
Macroscopic nominal stress & electric displacement
Macroscopic deformation gradient & electric fieldBC’s
Problem setting: macroscopic responseDefinition: relation between the volume averages of and
Variational Characterization
Macroscopic nominal stress & electric displacement
Macroscopic deformation gradient & electric fieldBC’s
Euler-Lagrange equations
and
Two strategies to generate exact solutionsI. Iterated homogenization (Lopez-Pamies, 2014): for an infinitely poly-disperse
particulate microstructure is given implicitly by the nonlinearfirst-order Hamilton-Jacobi pde
subject to the initial condition
II. Hybrid finite elements (Lefèvre & Lopez-Pamies, 2016): periodichomogenization of an infinite medium made out of the repetition of a unit cellcontaining a random distribution of a finite number of particles
Solutions for Ideal Elastic Dielectrics
with Isotropic Microstructures
Isotropic ideal elastic dielectric compositesHere
• In view of the overall (constitutive and geometric) isotropy andincompressibility of these composites, their effective free-energy function is ofthe form
with
Solution from iterated homogenizationLet us rewrite the pde for in the classical form of H-J equations
with Hamiltonian
• H-J equations typically exhibit non-uniqueness of non-smooth (Lipschitz cont.) solutions
Crandall & Lions (1983, 1984)
• The physically correct solution is the so-called viscosity solution
“time”
“space”
Osher & Sethian (1988); Jiang & Shu (1996)
• Numerical scheme
“space” discretization over a Cartesian grid making useof a monotone scheme and WENO finite difference of fifthorder
explicit “time” integration of fifth order
IH sample results:
210 p
210 p
0 05.c
IH sample results:
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3 4 5 6 7 8 9 10
HJ ExactHJ Approx.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3 4 5 6 7 8 9
HJ ExactHJ Approx.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
HJ ExactHJ Approx.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3
HJ ExactHJ Approx.
-1
-0.5
0
0.5
1
1.5
2
2.5
3
8 9 10 11 12
HJ ExactHJ Approx.
Linear dependence on
independence of
for all
and all
Solutions via a hybrid FE formulationWith help of the partial Legendre transform
the effective free-energy function can be written as
for Y-periodic microstructures
We look for FE solutions making use of conforming Crouzeix-Raviart-type elements —stable and convergent!
Sample “isotropic” microstructures and meshes
Example: isotropic distribution of 5% vol. fraction of monodisperse spherical particles
Example: isotropic distribution of 5% vol. fraction of three families of spherical particles
FE sample results:
Linear dependence on
independence of
for all
and all
-0.5
-0.25
0
0.25
0.5
3 3.5 4 4.5
Sph. Mono. ExactSph. Mono. Approx.
-0.5
-0.25
0
0.25
0.5
3 3.5 4 4.5
Sph. Mono. ExactSph. Mono. Approx.
-0.5
-0.25
0
0.25
0.5
0.2 0.4 0.6 0.8 1
Sph. Mono. ExactSph. Mono. Approx.
-0.5
-0.25
0
0.25
0.5
0.4 0.6 0.8 1 1.2
Sph. Mono. ExactSph. Mono. Approx.
-0.5
-0.25
0
0.25
0.5
0.5 1 1.5 2
Sph. Mono. ExactSph. Mono. Approx.
An approximate closed-form solution
Here, denote the effective shear modulus, permittivity, and electrostriction constant in the limit of small deformations and moderate electric fields
where and are solutions of the one-way coupled linear pdes
and
Lefèvre & Lopez-Pamies (2014); Spinelli, Lefèvre, Lopez-Pamies (2015)Tian, Tevet-Deree, deBotton, Bhattacharya (2012)
Solutions for Non-Gaussian Elastomers
Isotropically Filled with
Non-Linear Elastic Dielectric Particles
Non-Gaussian elastomers; non-linear particles Here
Examples:(Lopez-Pamies, 2010)
Langevin (1904); Debye (1929)
• Numerically: exact solutions can be generated via the HJ equation & the hybrid FE formulation
• Analytically: approximate solutions can be generated via a nonlinear comparison medium method
Comparison with electrostriction experiments
Huang et al. (2005); Lefèvre & Lopez-Pamies (2016)
Matrix: PUdielectric constant: 10
shear modulus: 10 MPa
Particles: PAA decorated o-CuPc dielectric constant: 104
shear modulus: 1 GPaMicrostructure: isotropic distribution of roughly spherical
nano-particles at 7% vol. fraction
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15
Composite (Sph. FE)
Composite (Sph. Theory)
Matrix (Theory)
Matrix (Experiment)
Composite (Experiment)
Interphasial phenomena?
Sph. FE w/ charges
Sph. FE w/o charges
Experiment
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15
Lefèvre & Lopez-Pamies (2016)
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