21
Nonlinear Electroelastic Deformations of Dielectric Elastomer Composites August 2016 2016 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

Nonlinear Electroelastic Deformations of Dielectric ...pamies.cee.illinois.edu/Presentations_files/Dielectric...deformations and moderate electric fields where and are solutions of

  • Upload
    others

  • View
    6

  • Download
    0

Embed Size (px)

Citation preview

  • 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)

    Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21