-
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.
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.
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.
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Introduction MotivationObjectives
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Introduction MotivationObjectives
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Introduction MotivationObjectives
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
ElectrostaticsElectromechanical Problem
Part I
Theoretical Background
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
ElectrostaticsElectromechanical Problem
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
ElectrostaticsElectromechanical Problem
Maxwell equationsMaxwell Stress Tensor
Maxwell Stress Tensor
Maxwell stress tensor:
TM = E ⊗ D −�02
E · EI.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
ElectrostaticsElectromechanical Problem
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).
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Electromechanical ActuatorPlane Strain Electromechanical
Problem. Results
Part II
Numerical Analysis
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Electromechanical ActuatorPlane Strain Electromechanical
Problem. Results
Electromechanical Actuator
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Electromechanical ActuatorPlane Strain Electromechanical
Problem. Results
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Instability (Mechanics)Mechanical Instability of the
Transversely Isotropic Materials
Part III
Instabilities
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
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
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Instability (Mechanics)Mechanical Instability of the
Transversely Isotropic Materials Analysis of a TIH Material
Analysis of a TIH Material
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Instability (Mechanics)Mechanical Instability of the
Transversely Isotropic Materials Analysis of a TIH Material
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Instability (Mechanics)Mechanical Instability of the
Transversely Isotropic Materials Analysis of a TIH Material
Analysis of a TIH Material
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Instability (Mechanics)Mechanical Instability of the
Transversely Isotropic Materials Analysis of a TIH 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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Part IV
Future Research
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Future Research
To derive the stability criterion for an
electromechanicalproblem.
To examine an electromechanical instability problem
inhomogeneous and heterogeneous media.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Future Research
To derive the stability criterion for an
electromechanicalproblem.
To examine an electromechanical instability problem
inhomogeneous and heterogeneous media.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Future Research
(model2.avi)
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
model2.aviMedia File (video/avi)
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Part V
Appendix
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Formulae
Nanson’s formulads = JF−TdS.
Push-Forwardσ = J−1PFT
Increment of deformation
δF = F′ − F = (G − I)F
Coulomb’s lawF =
14πε0
q1q2r2
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Instability condition
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′.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Instability condition
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
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Torus Topology
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
tetrahedron
Figure: Stress vector acting on a plane with normal vector n
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
KinematicsEquilibrium equationsConstitutive relation
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 .
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
KinematicsEquilibrium equationsConstitutive relation
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 .
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
KinematicsEquilibrium equationsConstitutive relation
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
KinematicsEquilibrium equationsConstitutive relation
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
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
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
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
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
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
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
Strong elipticity condition
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
Strong elipticity condition
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
Strong elipticity condition
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
MechanicsA 2-D Model. Assumptions.
Boundary ConditionsCOMSOL Finite Element Code Solution
Periodic Boundary Conditions(PBC)
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
Part VI
Heterogeneous materials
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
Periodic Two-Phase Domain
F̄ =1V
∫V F(x)dx = I + ∇u0, where ∇u0 is a constant matrix.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
Periodic Composite
a
b
2y
1y
µ1, µ2 are fiber and matrix shear moduli respectively.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
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.
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
Kinematics of Incremental Elasticity
0
F
Xx
1e 2e
3e
ReferenceconfigurationFixed Reference
configuration
′x
Infinitesimaldeformation
′
′F
G
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
-
Periodic Two-Phase DomainPeriodic Composite
Periodic Boundary Conditions. Shear.
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)
Stephan Rudykh and Gal deBotton Stability of Electroactive
Polymers (EAP) and Composites
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
Periodic Boundary Conditions(PBC)
Heterogeneous materialsPeriodic Two-Phase DomainPeriodic
CompositePeriodic Boundary Conditions. Shear.