Modelling and Simulation of Electroactive Materials and Structures

TUD Division of Solid Mechanics | ACMFMS 2012 |

Modelling and Simulation of Electroactive Materials and Structures

D. Gross1, B.X. Xu2, R. Müller3

1 Division of Solid Mechanics, TU Darmstadt 2 Dept. of Material Science, TU Darmstadt 3 Chair of Mechanics, TU Kaiserslautern


Hillenbrand & Sessler, 2000

Ferroelectrics Ferroelectrics

Dielectric Dielectric Elastomers Elastomers

FerroelectretFerroelectrets s

+

-

P0E

E


FerroelectricsFerroelectrics

electro-mechanical coupling principle

BaTiO3, PZT

T>TCurie: cubic

+

-

P0

T<TCurie: tetragonal

piezoelectric effectE<Ecoerc

polarization switchE>Ecoerc

remanent polarization, remanent strain

switching due to electrical and mechanical loads

+

-

P0

+

-

E


micro structural aspectsmicro structural aspectsdomains polycrystal

Schmitt & Kleebe, 2006 Jaffe, 2001

macroscopic behaviourmacroscopic behaviour

Boehle, 1999

dielectric hystersis butterfly hystersis

damaged domains

-4 -3 -2 -1 0 1 2 3 4

-1

0

1

2

Str

ain

, s

[10

-3]

Electric Field, E [kV/mm]


Lupascu, 2002

electric fatigue under cyclic loading

point defects (oxygen vacancies?)

agglomeration, interaction with domain walls

cracking

Utschig, 2005

ProblemsProblems

Deluca, 2010


Basic equations

Balance laws

Constitutive relations

Kinematics and electric potential

: remanent polarization

: remanent strain

: elasticity tensor

: piezoelectric tensor

: dielectric tensor

Phase field model

TUD Division of Solid Mechanics | AK 2012 |

Phase field model

remanent polarization P is considered as order parameter

extended electric enthalpy

: local (classical) electric enthalpy

: domain separation energy

: interface energy

evolution of P (Ginzburg-Landau)


interface energy

separation energy

local (classical) electric enthalpy

Numerical implementation

FEM, arbitrary geometry & boundary conditions

nodal degrees of freedom:

implicit time integration of Ginzburg-Landau equation

nonlinearity: Newton iteration required

damping and stiffness matrix required

symmetric system matrix

direct computation of configurational forces



Example: microstructure evolution at a crack tip

Microstructure at stationary crack tip


J integral and configurational forces


configurational forces


Dependence on applied electric field

positive electric field inhibits crack initiation

negative electric field promotes crack initiation

Wang & Singh 1997


Crack face boundary conditions

permeable:

impermeable:

semipermeable:

energy consistent:


Influence of boundary conditions


Open questions

influence of 2d/3d character

boundary conditions for order parameter (polarization)

conditions at grain boundaries in poly-crystals

initial conditions (virgin state of the material)

origin of the characteristic spacing of domains

importance of surrounding fields

motion of point defects in the „phase field continuum“


TUD Division of Solid Mechanics | Dresden 2011 |

piezo actuators vs. dielectric elastomer actuators

piezo actuator soft actuator

coupling through constituive laws

small displacements

large actuation force

coupling through electrostatic volume forces: MAXWELL stresses

large displacements

small actuation force

Dielectric Elastomer ActuatorsDielectric Elastomer Actuators


Balance laws

Constitutive relations (actual configuration)

Kinematics and electric potential

Basic equations

MAXWELL stress

Neo Hooke


Analysis of electro-mechanical stability

initial dimension:

deformed dimension:


Constitutive equations

Equilibrium


Equilibrium configuration

Stability (Lagrange multiplyer strategy)


Neo-Hooke model

special case:

Zhao & Suo, 2007

Numerics

FEM, arbitrary geometry & boundary conditions

perfect agreement between analytical and numerical results

simulation and optimization of composites possible

direct computation of configurational forces possible

simulation of dynamic response



homogeneous

inhomogeneous

barium titanate particle


Charge Density Evolution of FerroelectretsCharge Density Evolution of Ferroelectrets

piezoelectric 2 phase material (composite)

specific interface properties

(macroscopic) coupling through constitutive law

(microscopic) coupling through electrostatic volume forces: MAXWELL stresses

large displacements

diel 2

diel 1

interface


Basic equations

no breakdown: 2 dielectric materials

charge density, 1st breakdown

breakdown criterion

backfield evolution

charge density evolution

Numerics incremental time integration of internal variables

similar to plasticity with kinematic hardening

nonlinear FEM implementation

specific (embedded) interface element


ePTFE FEP

Example: FEP – ePTFE unit



B

B

C

C


ConclusionsConclusions

modeling & simulation offers a deeper insight in coupled problems

allows to replace (some) experiments

allows to analyze and optimize structures

nonlinear FEM and use of configurational forces often advantageous


interface energy

separation energy


Configurational forces

configurational force balance

Eshelby stress tensor, config body force

inhomogeneities (singularities) cause configurational forces

resultant configurational force:

configurational force on a point defect:

configuartional traction acting on an interface:

driving force at a domain wall:

overall driving force:

Microstrucure evolution, Poling

phase 1 : from random distribution towards equilibrium

phase 2 : application of an external field



3D microstructure evolution

potential free b.c.


3D electric poling


Influence of domain wall energy


Domain wall thickness

ϵ = 8.0 e-7 m



ϵ = 5.0 e-7 m



ϵ = 2.0 e-7 m


Size effect: thin films

Robin b.c.

Electric b.c.

Documents

Modelling and Simulation of Electroactive Materials and Structures