Modeling and Simulation of Magnetic Shape- …numod.ins.uni-bonn.de/people/lenz/talks/2006-essen.pdf · Modeling and Simulation of Magnetic Shape–Memory Polymer Composites Martin

Modeling and Simulation ofMagnetic Shape–Memory Polymer Composites

Martin Lenz1

with Sergio Conti2 and Martin Rumpf1

1) Institute for Numerical SimulationUniversity Bonn

2) Department of MathematicsUniversity Duisburg–Essen

Supported by DFG Priority Program 1095Analysis, Modeling and Simulation of Multiscale Problems

GAMM Annual Meeting 2006on March 27th – 31st 2006 in Berlin

Martin Lenz (INS Bonn) Magnetic Shape–Memory Composites GAMM 2006 1 / 20

Introduction Table of Contents

Overview

Physical ProblemMagnetic Shape–Memory MaterialsCrystal–Polymer–Composites

ModelMicromagnetic–Elastic ModelRigid Particles and Linear Elasticity

SimulationEnergy DescentBoundary Element Method

Results


Physical Problem Magnetic Shape–Memory Materials

Magnetic Shape–Memory Effect

H

HHH

Magnetic Shape–Memory MaterialE.g. Nickel–Manganese–GalliumDiscovered in 1996 by Ullakko et al.

Crystal lattice aligns to magnetization

HHHH

In DetailExternal magnetic fieldMagnetization aligns to fieldMagnetic easy axis is short lattice axisMovement of domain boundariesMacroscopic deformation




H

H

HH



HHHH





HH

H

H



HHHH





HHH

H



HHHH





HHHH



HHHH





HHHH



H

HHH





HHHH



H

H

HH





HHHH



HH

H

H





HHHH



HHH

H




Properties of Magnetic Shape–Memory Materials

I Large deformations (≈ 10%)Compared to 0.2% for magnetostrictive, 0.1% for piezo ceramics,up to 10% for thermic shape–memory materials

I Moderate magnetic fields (≈ 1 T)Same order of magnitude as for magnetostrictive materials,less than for piezo ceramics

I High operating frequency (≈ 103 Hz)Faster than thermic shape-memory materials(which are limited by heat conduction)

I High work output (≈ 105 Pa)Larger than piezo ceramics and magnetostrictive materials,but less than for thermic shape–memory materials



Applications of Magnetic Shape–Memory Materials

Applications as Actuators, Dampers and Sensors

Experiment by O’Handley et al.

Promises toexhibit significant advantages over otheractive materials in different applications

Researchin fabrication, characterization, applicationconcepts, modeling and simulation

DFG Priority Program 1239Anderung vonMikrostruktur und Form fester Werkstoffedurch außere Magnetfelder


Physical Problem Crystal–Polymer–Composites

Deformation of Polycrystals

Deformations of 10% can be achievedonly for Single Crystals, but growing single crystalsof the size necessary for applications is difficult

In Polycrystalsthe incompatibilities at grain boundaries may lead to

I significantly smaller deformations,

I increase of the necessary external field,

I or even inhibit switching altogether.



Deformation of Polycrystals

Deformations of 10% can be achievedonly for Single Crystals, but growing single crystalsof the size necessary for applications is difficult

In Polycrystalsthe incompatibilities at grain boundaries may lead to

I significantly smaller deformations,

I increase of the necessary external field,

I or even inhibit switching altogether.



Embedding Crystals in a Polymer

Alternative ApproachEmbedding single crystals in a polymer bulk- Feuchtwanger et al. (2003)- Gutfleisch, Weidenfeller

Similar to an approachused with magnetostrictive Terfenol-D(Sandlund, Fahlander et al. 1994,McKnight and Carman 1999)





Configuration

I Small single crystal particlesin a polymer matrix

I Particles deform and possibly aligndue to applied magnetic field

I Deformation of polymer matrix,yellow to red encodes elastic energy density





Configuration

I Small single crystal particlesin a polymer matrix

I Particles deform and possibly aligndue to applied magnetic field

I Deformation of polymer matrix,yellow to red encodes elastic energy density

Here: 4.8% deformation of composite (11.6% in single crystal)with 30% volume fraction of active material


Model Micromagnetic–Elastic Model

Full Model for Micromagnetism and Elasticity

Q

p=1

p=0

ω

Ω

Ω ⊂ Rd area occupied by compositeω ⊂ Ω area occupied by particles

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Matrix ElasticityWmatr stored energy density

of polymer bulkv : Ω → Rd deformation

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Particle ElasticityWpart stored energy density

of particles, i.e. quadratic distance from strain to eigenstrainwith respect to crystal lattice orientation

p : ω → 1, . . . d phase parameter in particlesQ : ω → SO(d) lattice orientation in particlesv : Ω → Rd deformation

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Interaction with External Fieldm : ω → Rd magnetization

Hext ∈ Rd external magnetic field

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Demagnetizationm : ω → Rd magnetization

Hd : Ω → Rd demagnetization field

Hd = ∇ψ∆ψ = div m distributionally

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Anisotropym : ω → Rd magnetizationp : ω → 1, . . . d phase parameter in particlesϕp : Rd → R anisotropy in phase p

applied to magnetization in deformed latticeR ∈ SO(d) rotational part of deformation ∇u = RUQ : ω → SO(d) lattice orientation in particles

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2




Q

p=1

p=0

ω

Ω


Magnetic Exchangem : ω → Rd magnetization

E [v , m, p] =

+

E elastmatr =

+

ZΩ\ω

Wmatr(∇v)

+ E elastpart +

Zω

Wpart((∇v)Q, p)

+ Eext −Z

v(ω)

Hext · m

+ Edemag +

Zv(Ω)

1

2|Hd |2

+ Eanis +

Zv(ω)

ϕp((RQ)Tm)

+ Eexch +

Zv(ω)

1

2|∇m|2


Model Rigid Particles and Linear Elasticity

Reduction to Small, Rigid Particles

I Particles are small and hard particle deformations are affine

I Particles are single crystals lattice orientation Q constant on each particle

I Particles are single–domain phase p and magnetization m constant on particles Eexch = 0

I Deformations are small linearized elasticity


Model Homogenization

Reduction to Cell Problem

I Large numbers of small particles:Fully resolved simulation not feasible

I Homogenization: Study periodic configurations

I Consider unit square with some particles, periodic boundaryconditions for magnetic field, affine–periodic for deformation



Periodic Configuration

Keep in mind: Computational cell is part of periodic configuration


























Simulation Energy Descent

Numerical Relaxation

Minimize over internal variables

I Particle deformation

I particle magnetization

I particle phase

8 degrees of freedom per particle

For a given particle configuration,the energy now is a function of

I the external magnetic field and

I the macroscopic deformation.







I particle phase

6

1

1











I particle phase

6

1

1






Simulation Boundary Element Method

Computation of Energy

Energy minimization: Gradient DescentApproximate gradient by finite differences

Energy evaluation: Boundary element method

Elasticity in polymer matrixAffine-periodic cell boundaryDirichlet particle boundary

DemagnetizationComputation in deformed unit cellHd jumps on particle boundaries

Actual energy computed by partial integration




Dirichlet boundary

Affine−periodic boundary









Periodic boundary

Neumann valuesjumping







Results Exploring Configurations

Regular versus Random Configuration

5% Strain 2% Strain

Careful optimization necessary



Regular versus Random Configuration

5% Strain 2% Strain

Careful optimization necessary



Exploring Configurations: Particle Shape

Simple lattice, one particle in the periodic unit cellExternal field horizontal, compress lattice horizontally

Plot energy over stretch for different particle shapes

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.01 0.02

Ene

rgy

Den

sity

in M

Pa

Horizontal Stretch

4:11:11:4

Elongated particles better, but not very significant try softer polymerplot optimal stretch and work output for different aspect ratios

0.01

0.015

0.02

0.025

1:41:21:12:14:1

Str

etch

Aspect Ratio

E = 6 GPaE = 1.2 GPa

0.2

0.4

0.6

0.8

1:41:21:12:14:1

Wor

k O

utpu

t in

MP

a

Aspect Ratio




Exploring Configurations: Particle Shape

Simple lattice, one particle in the periodic unit cellExternal field horizontal, compress lattice horizontally

Elongated particles better, but not very significant try softer polymerplot optimal stretch and work output for different aspect ratios

0.01

0.015

0.02

0.025

1:41:21:12:14:1

Str

etch

Aspect Ratio


0.2

0.4

0.6

0.8

1:41:21:12:14:1

Wor

k O

utpu

t in

MP

a

Aspect Ratio




Exploring Configurations: Polymer Elasticity

Plot strain and work output for different polymer elastic moduli Longer particles, (somewhat) softer polymer(In comparison: For particles is E ≈ 100 000 M Pa)

0

0.01

0.02

0.03

0.04

0.05

0.06

1000 5000 25000

Str

ain

Polymer Elastic Modulus in MPa

1:12:13:14:1

0.2

0.4

0.6

0.8

1

1000 5000 25000

Wor

k O

utpu

t in

MP

a


1:12:13:14:1



Exploring Configurations: Particle Alignment

Do the same computation for particles that do not form chains

0

0.2

0.4

0.6

0.8

1

1000 5000 25000

Wor

k O

utpu

t in

MP

a


0 % Shift25 % Shift50 % Shift75 % Shift

100 % Shift



Exploring Configurations: Particle Orientation

Consider two particles that are not oriented in the direction of theexternal field, examine effect of misorientation

0

0.4

0.8

1.2

1.6

0 10 20 30 40

Wor

k O

utpu

t in

MP

a

Rotation


Results Conclusion

Conclusion

I Continuous Model

I Efficient Simulation

I Results identify important factors in composite design

I Elongated particles

I Softer polymer

I Orientation

I Alignment not necessary


Results Conclusion

Conclusion

I Continuous Model

I Efficient Simulation

I Results identify important factors in composite design

I Elongated particles

I Softer polymer

I Orientation

I Alignment not necessary


Results Conclusion

Outlook

Simulation of Polycrystals


Results Conclusion

Outlook

Simulation of Polycrystals

(Preliminarynumerics)

Thank you for your attention!Contact: [email protected]


Documents

Modeling and Simulation of Magnetic Shape- …numod.ins.uni-bonn.de/people/lenz/talks/2006-essen.pdf · Modeling and Simulation of Magnetic Shape–Memory Polymer Composites Martin