Stability of Electroactive Polymers (EAP) and Compositesrudykh/2008presentationGraz.pdf · 2009-08-22 · Introduction Stability of Electroactive Polymers (EAP) and Composites Stephan

Introduction

Stability of Electroactive Polymers (EAP) andComposites

Stephan Rudykh and Gal deBotton

Department of Mechanical EngineeringBen-Gurion University

Israel* * * * *

Summer School on ”Modeling and Computation in Biomechanics”Graz, Austria

September 17, 2008

Stephan Rudykh and Gal deBotton Stability of Electroactive Polymers (EAP) and Composites

Introduction MotivationObjectives

Motivation

Electoactive polymers (EAPs) are polymers that can change theirshape in response to electrical stimulation.

Advantages - soft, light-weighted, fast responsive and resilient.

Limitations: the need for large electric field, relatively smallforce and energy density.

Typical polymers have small ratio of dielectric to elasticmodulus.

Motivation

Objectives

Electroactive polymer composites (EAPCs) are composites offlexible and high-dielectric-modulus materials.

Investigating the mechanical response of EAPCs undergoinglarge deformations due to electrostatic excitation.

Instability analysis at microscopic and macroscopic levels.

Objectives

ElectrostaticsElectromechanical Problem

Part I

Theoretical Background

Maxwell equationsMaxwell Stress Tensor

Maxwell equations

Maxwell equations for electrostatics:

∇ · D = 0,

whereD ≡ �0E + p

is electric displacement or electric induction, �0 is electricconstant,

E = −∇φ

is electric field, φ is a continuous scalar potential, and p ispolarization.

Maxwell equationsMaxwell Stress Tensor

Maxwell Stress Tensor

Maxwell stress tensor:

TM = E ⊗ D −�02

E · EI.

Electromechanical Problem

Equilibrium equations:

∇(σ + TM) + b = 0

Constitutive law: Cauchy stress and electric field in the elasticdielectric are

σ =1J∂Ψ

∂FFT and E =

∂Ψ

∂p,

where Ψ = Ψ (F,p).

Electromechanical ActuatorPlane Strain Electromechanical Problem. Results

Part II

Numerical Analysis

Electromechanical Actuator

Plane Strain Electromechanical Problem. Results

0 5x103 1x104 2x104 2x1041.00

1.01

1.02

1.03

1.04

1.05

m m

M Pa

h0 = 0.5

μ = 0.8 ε = 10

φ0, V

λ1

Figure: The circles correspond to numerical results and the continuous curve, toanalytical solution λ1 =

(1 − ��0

µ

(φ0h0

)2)−1/4.

Instability (Mechanics)Mechanical Instability of the Transversely Isotropic Materials

Part III

Instabilities

Instability (Mechanics)Mechanical Instability of the Transversely Isotropic Materials The Stability Criterion

The Stability Criterion

The stability criterion in terms of acoustic tensor

Qij(N)mimj = Aαiβj(F)NαNβmimj > 0,

whereAαiβj(F) is the tensor of elastic moduli, Nα and mi are aLagrangian and an Eulerian, respectively.

The strong ellipticity condition for Green-elastic materials

Qii(N) > 0 i = 1, 2, 3

Qii(N)Qjj(N) − Qij(N)2 > 0 j , i = 1, 2, 3

det Q(N) > 0,

for all N, no summation.

Instability (Mechanics)Mechanical Instability of the Transversely Isotropic Materials The Stability Criterion

The Stability Criterion

The stability criterion in terms of acoustic tensor

Qij(N)mimj = Aαiβj(F)NαNβmimj > 0,

whereAαiβj(F) is the tensor of elastic moduli, Nα and mi are aLagrangian and an Eulerian, respectively.

The strong ellipticity condition for Green-elastic materials

Qii(N) > 0 i = 1, 2, 3

Qii(N)Qjj(N) − Qij(N)2 > 0 j , i = 1, 2, 3

det Q(N) > 0,

for all N, no summation.

Instability (Mechanics)Mechanical Instability of the Transversely Isotropic Materials Analysis of a TIH Material

The Example of the Instability Analysis

We examine the transversely isotropic neo-Hookean (TIH) model ofthe strain energy density function suggested by deBotton et al. (2006).

ShellFiber

n̂

qa

bDisturbing force

Analysis of a TIH Material

0 . 9 7 0 . 9 8 0 . 9 9 1 . 0 00 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

µ1/µ

2=100

λ

b / a = 3 b / a = 2 b / a = 1

u 2

Figure: The vertical component dependence of the central point displacement on theapplied stretch ratio. µ1 and µ2 are shear moduli of fibers and matrix, respectively.

Figure: The dependence of the critical stretch ratio on the fibers volume fraction c1.The results are consistent with the ones obtained by Triantafyllidis and Maker (1985)for laminated material.

Analysis of a TIH Material. Compression and Shear.

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0 00 . 0 50 . 1 00 . 1 50 . 2 00 . 2 50 . 3 0 s t a b l e

λ

γ

u n s t a b l e

Figure: The unstable domain under combined shear-compression deformations. Thevolume fraction of fibers c1 = 0.25, µ1/µ2 = 20.

Part IV

Future Research

To derive the stability criterion for an electromechanicalproblem.

To examine an electromechanical instability problem inhomogeneous and heterogeneous media.

Future Research

To derive the stability criterion for an electromechanicalproblem.

To examine an electromechanical instability problem inhomogeneous and heterogeneous media.

Future Research

(model2.avi)

model2.aviMedia File (video/avi)

MechanicsA 2-D Model. Assumptions.

Boundary ConditionsCOMSOL Finite Element Code Solution

Periodic Boundary Conditions(PBC)

Part V

Appendix

Formulae

Nanson’s formulads = JF−TdS.

Push-Forwardσ = J−1PFT

Increment of deformation

δF = F′ − F = (G − I)F

Coulomb’s lawF =

14πε0

q1q2r2

Instability condition

∫B0

(P′ − P) : (F′ − F)dV =∫

B0

(P′ − P)ij(F′ − F)ijdV =

=

∫B0

{[(P′ − P)ij(χ′ − χ)i],j − (P′ − P)ij,j(χ′ − χ)i

}dV =

=

∫∂B0

[(P′ − P)ijnj(χ′ − χ)i]dS −∫

B0

(P′ − P)ij,j(χ′ − χ)idV =

=

∫∂B0

[(t′ − t)i(χ′ − χ)i]dS −∫

B0

(b′ − b)i(χ′ − χ)idV .

That is for two different solutions of the same boundary valueproblem (Bifurcation)∫

B0

(P′ − P) : (F′ − F)dV = 0.

Any χ makes the integral negative for some χ′ we have negativecorresponds to unstable configuration∫

B0

(P′ − P) : (F′ − F)dV < 0.

The sufficent condition for uniqueness is that∫B0

(P′ − P) : (F′ − F)dV > 0,

for all kinematically admissible χ , χ′.

A stronger inequality

(P′ − P) : (F′ − F)dV > 0,for all pairs F , F′.

For Green-Elastic material we consider instead of previousexpression

W(F′) −W(F) − P : (F′ − F) > 0.

Note:W(F′) > W(F) +

∂W∂F

: (F′ − F) + O(F)2,

W(F′) > W(F) + P : (F′ − F) + O(F′ − F)2

W(F) > W(F′) + P′ : (F − F′) + O(F − F′)2

(P′ − P) : (F′ − F) > 0

Torus Topology

Equilibrium equation

∫∂B

tds =∫∂Bσn̂ds =

∫B

divσdv

with balance of linear momentum∫∂B

tds +∫

Bbdv = 0

∫B

(divσ + b)dv = 0

This relation supposed to hold for any volume vthe local form is

divσ + b = 0.

Note the acceleration is assumed to be 0.Stephan Rudykh and Gal deBotton Stability of Electroactive Polymers (EAP) and Composites

Cauchy’s stress theorem

t = σn̂

T(n)dA − T(e1)dA1 − T(e2)dA2 − T(e3)dA3 = ρ(h3

ds)

a

dA1 = (n · e1) dA = n1dAdA2 = (n · e2) dA = n2dAdA3 = (n · e3) dA = n3dA

with h→ 0

T(n) = T(e1)n1 + T(e2)n2 + T(e3)n3 =3∑

i=1

T(ei)ni =(σijej

)ni = σijniej

T (n)j = σijni

tetrahedron

Figure: Stress vector acting on a plane with normal vector n

KinematicsEquilibrium equationsConstitutive relation

Kinematics

0

χ

Xx

1e 2e

3e

CurrentconfigurationReference

configuration

The deformation gradient is F = ∇X(χ), where ∇X( ) denotes thegradient operation in B0. J ≡ detF > 0 is the local current toreference volume ratio.

Equilibrium equations

Equilibrium equation in current configuration:

∇x · σ + b = 0,

where ∇x · ( ) denotes the divergence operation in B, b is thebody force in current configuration and σ is Cauchy stress.

Equilibrium equation in reference configuration:

∇X · P + b0 = 0,

where P is the 1st Piola-Kirchhoff (nominal) stress tensor andb0 = J−1b is the body force in reference configuration. Remindthat Cauchy stress is σ = J−1PFT .

Equilibrium equations

Equilibrium equation in current configuration:

∇x · σ + b = 0,

where ∇x · ( ) denotes the divergence operation in B, b is thebody force in current configuration and σ is Cauchy stress.

Equilibrium equation in reference configuration:

∇X · P + b0 = 0,

where P is the 1st Piola-Kirchhoff (nominal) stress tensor andb0 = J−1b is the body force in reference configuration. Remindthat Cauchy stress is σ = J−1PFT .

Constitutive relation

Green-elastic materials:There exists a scalar function Ψ(F) such that

P =∂Ψ(F)∂F

.

Incompressible Neo-Hookean model:

Ψ =µ12

(I1 − 3)

where µ1 is shear modulus and I1 is the first invariant of the rightCauchy-Green strain tensor C = FTF.

Constitutive relation

Green-elastic materials:There exists a scalar function Ψ(F) such that

P =∂Ψ(F)∂F

.

Incompressible Neo-Hookean model:

Ψ =µ12

(I1 − 3)

where µ1 is shear modulus and I1 is the first invariant of the rightCauchy-Green strain tensor C = FTF.

A 2-D Model. Assumptions.

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

1 The electrodes are flexible with a negligible elastic modulus.2 There are no external loads =⇒ traction BC is t = 0.3 The actuator is thin and long =⇒ fringing field effect can be

neglected (i.e.,T(0)M = 0).4 Plane strain loading conditions.

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

Boundary Conditions

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

The electrostatic boundary conditions:

n̂ · D = 0 on X1 = ±b/2,

φ = 0 on X2 = h0 and φ = φ0 on X2 = 0.

The mechanical boundary conditions:

u2 = 0 on X2 = 0 and u1 = 0 on X1 = b/2.

Boundary Conditions

b

h0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

X2

X1

The electrostatic boundary conditions:

n̂ · D = 0 on X1 = ±b/2,

φ = 0 on X2 = h0 and φ = φ0 on X2 = 0.

The mechanical boundary conditions:

u2 = 0 on X2 = 0 and u1 = 0 on X1 = b/2.

Strong elipticity condition

COMSOL Finite Element Code Solution

Multiphysics module

Structural mechanics module

Adding Maxwell stress term to mechanical stress balance

P =∂Ψ

∂F+ JTMF−T .

Electrostatics module: using moving geometry defined bydisplacement global variable ui.

Multiphysics module

P =∂Ψ

∂F+ JTMF−T .

Multiphysics module

P =∂Ψ

∂F+ JTMF−T .

Multiphysics module

P =∂Ψ

∂F+ JTMF−T .

Criterion for incremental stability

Elastic modulusA = ∂P

∂F.

The incremental 1st Piola-Kirchhoff stress tensor

δP = AδF.

Criterion for incremental stability∫B0

Tr{(AδF)δF}dV > 0,

for all δF.Stephan Rudykh and Gal deBotton Stability of Electroactive Polymers (EAP) and Composites

∂F.

δP = AδF.

∂F.

δP = AδF.

Criterion for local incremental stability

Tr{(AδF)δF} > 0,

for all δF.Strong elipticity condition

AαiβjNαNβmimj > 0,

for all Nα and mi, a Lagrangian vector and a Eulerian vector,respectively.

Criterion for local incremental stability

Tr{(AδF)δF} > 0,

for all δF.Strong elipticity condition

AαiβjNαNβmimj > 0,

for all Nα and mi, a Lagrangian vector and a Eulerian vector,respectively.

Transversely Isotropic Neo-Hookean (TIH) model

Strain energy-density function:

WTIH =µ̃

2

(I1 − 3

)+µ̄ − µ̃

2

(I4 + 2

√1I4− 3

)where I4 = n̂ · Cn̂ is the 4th invariant of right Cauchy-Greenstrain tensor,

µ̃ = µ2(1 + c1)µ1 + (1 − c1)µ2(1 − c1)µ1 + (1 + c1)µ2

is the in-plane and out-of-plane shear modulus and

µ̄ = c1µ1 + c2µ2

is the deviatoric shear modulus with the volume fraction of fibersc1 and shell c2 = 1 − c1, shear moduli of fiber and shell µ1, µ2,respectively.

Periodic Boundary Conditions

a

b

2y

1y

Periodic boundary conditions

u1(b2, y2

)= u1

(−b

2, y2

)+ ∇u011 · b,

u2(b2, y2

)= u2

(−b

2, y2

)+ ∇u021 · b,

u1(y1,

a2

)= u1

(y1,−

a2

)+ ∇u012 · a,

u2(y1,

a2

)= u2

(y1,−

a2

)+ ∇u022 · a.

Periodic Boundary Conditions

a

b

2y

1y

Periodic boundary conditions

u1(b2, y2

)= u1

(−b

2, y2

)+ ∇u011 · b,

u2(b2, y2

)= u2

(−b

2, y2

)+ ∇u021 · b,

u1(y1,

a2

)= u1

(y1,−

a2

)+ ∇u012 · a,

u2(y1,

a2

)= u2

(y1,−

a2

)+ ∇u022 · a.

Periodic Two-Phase DomainPeriodic Composite

Periodic Boundary Conditions. Shear.

Part VI

Heterogeneous materials

Periodic Two-Phase Domain

F̄ =1V

∫V F(x)dx = I + ∇u0, where ∇u0 is a constant matrix.

Periodic Composite

a

b

2y

1y

µ1, µ2 are fiber and matrix shear moduli respectively.

Periodic Boundary Conditions(PBC). Shear

We will use a periodic boundary condition. To illustrate this type ofboundary value problem we make the example

Figure: The simple shear deformation with PBC.

∇u0ij =(

0 γ0 0

), µ1 = 200 · 106 Pa, µ2 = 20 · 106 Pa, γ = 10−3.

Kinematics of Incremental Elasticity

0

F

Xx

1e 2e

3e

ReferenceconfigurationFixed Reference

configuration

′x

Infinitesimaldeformation

′

′F

G

Future work

The experimental work of the investigators from MIT illustrate thistype of problem. There are two sheets of flexible electrodes and a filmof elastomer between them. The instability occurs when the criticalapplied potential is applied. It is shown in the right pictures.

Circular Actuator (Jean-Sébastien Plante and Steven Dubowsky,2006)

IntroductionMotivationObjectives

Theoretical BackgroundElectrostaticsMaxwell equationsMaxwell Stress Tensor

Electromechanical Problem

Numerical AnalysisElectromechanical ActuatorPlane Strain Electromechanical Problem. Results

InstabilitiesInstability (Mechanics)The Stability Criterion

Mechanical Instability of the Transversely Isotropic MaterialsAnalysis of a TIH Material

Future ResearchAppendixMechanicsKinematicsEquilibrium equationsConstitutive relation

A 2-D Model. Assumptions.Boundary ConditionsCOMSOL Finite Element Code SolutionStrong elipticity condition

Heterogeneous materialsPeriodic Two-Phase DomainPeriodic CompositePeriodic Boundary Conditions. Shear.