micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Micromechanics-based magneto-elastic constitutive modeling of magnetostrictive composites

Lizhi Sun

Department of Civil & Environmental Engineering and

Department of Chemical Engineering & Materials Science

University of California, Irvine, CA, USA

Presented at

3rd IMR Symposium on Materials Science and Engineering

Institute of Metal Research, Shenyang, P. R. China

July 10-13, 2007

L. SunL. Sun

Magnetic Particulate Composites

Magnetic Particles: Iron, Nickel, Cobalt, Terfenol-D

Nonmagnetic Matrix: Elastomers, Polymers, Metals

L. SunL. Sun

Magnetic Particulate Composites

Damper

Sensor Actuator

Anjanappa M. and Wu Y.F. (1997), Smart Mater. Struct., 6, 393-402.Ginder, J.M., et al. (1999), Proc. Series of SPIE Smart struct. Mater. 1999, 3675, 131-138. Park, C. and Robertson, R.E. (1998), Dent. Mater., 14, 385-393

L. SunL. Sun

Magnetic particle filled composites

Magnetic particles(Solid, iron)

Nonmagnetic matrix (Liquid or Colloid, rubber)

H

High temperature

Random Structure

L. SunL. Sun

Magnetic particle filled composites

H

High temperature

Chain Structure

Halsey, T.C. (1992), Science,258, 761-766

L. SunL. Sun

Room temperature

H

Magnetostriction

Young’s modulus

Shear modulus

Mechanical Behavior

L. SunL. Sun

Modeling Framework

(b)

Composites

Microscale

RVE

Local fields

Test loading

Homogenization

Averaged fields

Macroscale

Constitutive law

RVE: Representative Volume Element

L. SunL. Sun

Equivalent Inclusion Method

* :Ω

C ε

0εD*C

C

Ω

D*C ε

C Ω

Eshelby, J.D. (1957 & 1959), Proc. Roy. Soc., A241, 376-396; A252, 561-569.

0εDC

C Ω

=

( )*:Ω−C ε εStress equivalent condition

+

L. SunL. Sun

Equivalent Inclusion Method

0 *

0 (0) *

, ( )

( ) , ( )ij ij kl

ij i

ijkl

klmnj ijkl klmn mn

S

SC I

ε ε ε

σ σ ε

= + ∈Ω

= + − ∈Ω

x

x

Ω

0

0( )ij

ij

σ

ε

0

0( )ij

ij

σ

ε

D

Strain/Stress inside the Particle

Strain/Stress inside the Matrix

0 *

0 (0) *

,( ( )

) , )( (

)ij ij kl

ij

ijkl

klmij ijkl mnn

D

C

G

G D

ε ε ε

σ σ ε

= + ∈ −Ω

= + ∈ −Ω

x

x

x

x

where the eigenstrain: * (1) (0) 1 (0) 1 0[ ( ) ]ij ijkl ijmn ijmn mnkl klS C C Cε ε− −= − + −

Eshelby’s tensor: (1) (2)

0

1 [ (0) (0)( )]4(1 )ijkl IK ij kl IJ ik jl il jkS S Sδ δ δ δ δ δ

ν= + +

−

L. SunL. Sun

Equivalent Inclusion Method

Ω

Exterior-Point Eshelby’s Tensor:

( ) ( ) , ( )

?

ijkl ijklG G d DΩ

′ ′= ∈ −Ω

=

∫x x - x x x

D

30

0

0

0 0

1( ) [ 158 (1 ) || ||

3 ( )

3 3(1 2 )

(1 2 ) (1 2 )( )]

ijkl i j k l

ik j l il j k jk i l il j k

ij k l kl i j

ij kl ik jl il jk

G n n n n

n n n n n n n n

n n n n

π νν δ δ δ δ

δ ν δ

ν δ δ ν δ δ δ δ

′ = −′−

+ + + +

+ + −

− − + − +

x - xx - x

L. SunL. Sun

Equivalent Inclusion Method

Exterior-Point Eshelby’s Tensor:

x

′x

nn̂

(1) (2)

0(3) (4) (5)

(6) (7)

1( ) [ ( )8(1 )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ]

ijkl IK ij kl IJ ik jl il jk

I ij k l K kl i j I ik j l il j k

J jk i l il j k IJKL i j k l

G S S

S n n S n n S n n n n

S n n n n S n n n n

δ δ δ δ δ δν

δ δ δ δ

δ δ

= + +−

+ + + +

+ + +

x

2 1i i

I

x xa′ ′

≤Inclusion:

Imaginary: 2 1i i

I

x xa λ′ ′

≤+

L. SunL. Sun

Two Mechanisms

Magnetostriction of particles:

Terfenol-D particles; Polymer matrixMagnetostrictive composites

H

ε εs

H

Magnetic force between particles:

Iron, Ni, Co particles; Rubber-like matrixFerromagnetic elastomers

Sandlund, L., et al. (1994), J. Appl. Phys. 75, 5656-5658.

Jolly, M.R., et al. (1996), Smart Mater. Struct., 5, 607-614.

L. SunL. Sun

Magnetostriction of Particles

Magnetostriction of particles:

Terfenol-D particles; Polymer matrixMagnetostrictive composites

H

ε εs

Eigenstrain Method: D* C ε

C Ω

L. SunL. Sun

Magnetic Forces between Particles

Magnetostatic problems:

( )2∇ = ∇ •U r MGoverning equation:

U= −∇H

( ) ( )1 0

0

m m

m

μ μμ−

=M r H r

Scalar magnetic potential:

Magnetization:

( ) ( ) ( )' ' ', ,= −∫ k kV

U G M dr r r r rSolution:

L. SunL. Sun

Magnetic Forces between Particles

( ) ( )1 0

0

m m

m

μ μμ−

=M r H r

( ) ( ) ( )' ' ',= Γ∫ mi ik kV

H M dr r r r r

( ) 0 1= + +i i ik kM M M rr

( ) ( ),, ' , 'Γ =mij ijGr r r r

L. SunL. Sun

Magnetic Forces between Particles

Y

Z

-2 -1 0 1 2-5

-4

-3

-2

-1

0

1

2

H3.40x10+00

3.13x10+00

2.87x10+00

2.60x10+00

2.33x10+00

2.07x10+00

1.80x10+00

1.53x10+00

1.27x10+00

1.00x10+00

4.80x10-04

4.74x10-04

4.69x10-04

4.63x10-04

4.58x10-04

4.52x10-04

4.47x10-04

4.41x10-04

4.36x10-04

4.30x10-04

0μ = 10−7 Η = (0,0,1)μ =7∗101−4

3

( ) ( ) ( )0 ,e

i k i kf M Hμ=r r r

ε ms

( )−f r

ε ms

( )f r

0σ

L. SunL. Sun

Green’s Function Technique

r

'r

D Source( )'δ r

Response( )'G −r r( ) ( )' ' 'G q d

Ω−∫ r r r r

Ω

( )'q r

Source (particles)

Non-mechanical strain

Magnetization

Magnetic body force

Response (anywhere)

Strain field

Magnetic field

Strain field

L. SunL. Sun

Mechanical Constitutive Modeling

( )1 0: 1 :D M

φ φΩ

= + −σ C ε C ε

( )1D M

φ φΩ

= + −ε ε ε

~MΩ

ε ε

Local solution in RVE:

Volume average of stress and strain:

~D D

σ ε

Constitutive law:

0σ

L. SunL. Sun

Equivalent Inclusion Method

Particle averaged strain:

*ε*ε*ε*ε

*ε*ε

*ε

( )'MΩ Ω

= +ε ε ε r ( ) ( )' 0 * '

0' :

p

e

pd

∞

Ω=

= − − ⋅∑∫ε r Γ r r C ε r

( ) ( ) ( ) ( ) ( )' ' ' ' ', , , ,

14

eijkl ik jl il jk jk il jl ikG G G G⎡ ⎤Γ − = − + − + − + −⎣ ⎦r r r r r r r r r rModified elastic Green’s function:

( ) ( )

2 ''

'

1 14 16 1

ijij e e

i j

Gv r r

δπμ πμ

∂ −− = −

− ∂ ∂−

r rr r

r rElastic Green’s function:

=

( )1 0 *: :Ω Ω= −C ε C ε εStress Equivalent Condition:

+

L. SunL. Sun

Particle’s Averaged Strain

( ) 11 1 0

1 0

:

with M

−− −Ω= Δ Δ − −

Δ = −

ε C C D D ε

C C C

( )=ijkl IK ij kl IJ ik jl il jkD R Tδ δ δ δ δ δ+ +

( ) ( )( )0

0 0

1= 4 530 1ijkl ij kl ik jl il jkeD v

vδ δ δ δ δ δ

μ⎡ ⎤− − +⎣ ⎦−

( ) ( )( ) ( )3

2 2 200 0 3 3 0 3 3

0 0

1.202 1 1.035 3 1-1.725 15 36.236 1IK I K I KeR

vρ ρ ρ δ δ ρ δ δ

μ⎡ ⎤= − − + + −⎣ ⎦−

( ) ( )( )3

2 200 0 0 0 3 3

0 0

1.202 1 2 1.035 3 -1.7256 1IJ I JeT v v

vρ ρ ρ δ δ

μ⎡ ⎤= − − + + +⎣ ⎦−

Particle averaged strain:

Eshelby tensor:

Interaction term:

Transversely isotropic effective elasticity: ( )( ) 10 1 01φ φ−

−⎡ ⎤= + Δ − − +⎣ ⎦C C C D D

Disregarding interaction term: ( ) 10 1 01φ φ−−⎡ ⎤= + Δ − −⎣ ⎦C C C D Mori-Tanaka Model

L. SunL. Sun

Young’s Modulus – Good Agreement

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

4

6

8

10

12

14

16

18

20

E0=3GPa, v0=0.45, E1=35Gpa, v1=0.25, ρ0=0.499

Voigt's upper bound The proposed model Carman's experiment Mori-Tanaka's model Reuss's lower bound

Effe

ctiv

e el

astic

mod

uli (

GPa

)

Volume fraction φ

Duenas T. A., Carman G.P. (2000), J. Appl. Phys.Yin, H.M., Sun, L.Z., Chen, J.S. (2007), J. Mech. Phys. Solids

L. SunL. Sun

Transversely Isotropic Elasticity

0.0 0.1 0.2 0.3 0.4 0.5

4

8

12

16

20

E0=3GPa, v0=0.45, E1=35Gpa, v1=0.25, ρ0=0.499

c33 c11 c12 c13 c44

Ef

fect

ive

elas

tic m

odul

i (G

Pa)

Volume fraction φ

11 12 13

11 13

33

44

44

11 12

:0 0 0

. 0 0 0

. . 0 0 0

. . . 2 0 0

. . . . 2 0

. . . . .

c c cc c

cc

cc c

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

σ C ε

C

2x

1x

3x

L. SunL. Sun

Magnetomechanical Loading

( ) ( ) ( )0 ,m e

i k i kf M Hμ=r r r

Particle magnetostriction:

Magnetic force between particles:

33 3 11 22 31;2

s sms ms ms

s sH HH Hλ λε ε ε

Ω Ω= = = −

*ε*ε*ε*ε

*ε*ε

*ε

= ,msε f

0 0,H σ

L. SunL. Sun

Strain Field Caused by Magnetic Body Force

Magnetic force between particles:

= msε

( ) ( ) ( )' ' ', ,

1, , ,2

fijk ik j jk iG G⎡ ⎤Γ = +⎣ ⎦r r r r r rThe second modified elastic Green’s function:

( ) ( )

2 ''

'

1 14 16 1

ijij e e

i j

Gv r r

δπμ πμ

∂ −− = −

− ∂ ∂−

r rr r

r rElastic Green’s function:

( ) ( ), ' ' 'fij ijk kV

f dε = Γ∫ r r r r

,msε f fmsεmsεmsε

msεmsεmsε

L. SunL. Sun

Magnetostriction of Composites

-0.4 -0.2 0.0 0.2 0.40

100

200

300

400

500

600

700

Effe

ctiv

e m

agne

tost

rictio

n (p

pm)

Flux density (T)

Experimental data Proposed model

Duenas, T.A., Carman, G.P. (2000), J. Appl. Phys.Yin, H.M., Sun, L.Z., Chen, J.S. (2007), J. Mech. Phys. Solids

H

L. SunL. Sun

Change of Shear Modulus of Composites

H

0.0 0.2 0.4 0.6 0.80

5

10

15

20

25

30

35

40

45

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

E0=1.8*106, v0=0.49, μ1m=2000μ0

m, E1=150*109, v1=0.3, ρ0=0.45

Cha

nge

in n

orm

aliz

ed s

hear

mod

ulus

%

Flux density (T)

φ=0.1 Experiment φ=0.2 Experiment φ=0.1 Prediction φ=0.2 Prediction

Jolly, M.R., Carlson, J.D., Munoz, B.C. (1996), Smart Mater. Struct.Yin, H.M., Sun, L.Z. (2005), Appl. Phys. Lett.

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

3x

1x

2x

Ω0

a

(b)

( ) ( )11 1 01

: ,− ∞− −

Ω == Δ ⋅ Δ − +∑ K

M Kε C C D ε d 0 r

( ) ( ) ( )1, , |∞

== =∑ ∫K

ij ij ijK Dd d a d a P dr r r 0 r

Conditional probability function:

( ) 3

3 2| 4

0 2

φπ

⎧ ≥⎪= ⎨⎪ <⎩

aP a

a

rr 0

r

( ) 11 1 0 1:−− − −

Ω= Δ ⋅ Δ − + Δ ⋅

M Mε C C D ε C L ε

Pair-wise interaction ( ), Kd 0 r

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

• Eshelby’s equivalent inclusion method( ) ( )0 '= +ε r ε ε r

0ε

( ) ( ) ( ) ( ) ( )0 0 *1 2' '⎡ ⎤ ⎡ ⎤+ = + −⎣ ⎦ ⎣ ⎦C ε r ε r C ε r ε r ε r

( ) ( ) ( )' ' * ' ',ij ijkl kl dε εΩ

= Γ∫r r r r r

*ε

*ε

= +*ε

*ε

= +

0ε 0ε

2C2C 2C1C

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

• Pairwise interaction (Moschovidis and Mura, 1975)

Y

Z

-2 0 2-5

-4

-3

-2

-1

0

1

2The difference of the averaged strain for two-

particle solution and one-particle solution

( )( )

1 2pair single

1 2 0

, , | |

, ,

ij ij ij

ijkl kl

d a

L a

ε ε

ε

= −

=

r r

r r

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

( ) ( ) ( ) ( )1 210 1

: 0 , ,ii

a∞−

== − ⋅Δ +∑ε 0 I P C ε d 0 x

( )( ) ( )

( ) ( ) ( )

1

23

, ,

| , ,

| , , :

ii

D

D

a

P a d

P a x d

∞

=

=

=

∑∫∫

d 0 x

x 0 d 0 x x

x 0 L 0 x ε x

2x

1x

3x

2x

1x

3x

2x

1x

3x

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

0.0 0.1 0.2 0.3 0.4 0.5

4

6

8

10

12

(a)

E0=3GPa, v0=0.45, E1=35Gpa, v1=0.25

Chain-structured Random Periodic

Effe

ctiv

e Yo

ung'

s m

odul

i (G

Pa)

Volume fraction φ

0.0 0.1 0.2 0.3 0.4 0.51.0

1.5

2.0

2.5

3.0

3.5

(b)

E0=3GPa, v0=0.45, E1=35Gpa, v1=0.25

Chain-structured Random Periodic

Effe

ctiv

e sh

ear m

odul

i (G

Pa)

Volume fraction φ

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

Chain-structured composites Random composites

0 100 200 300 400 500

2.5

3.0

3.5

4.0φ=0.15

φ=0.1

φ=0.05

φ=0.0

H30 (kA/m)

μ0m=4π*10-7, E0=2.1*106, v0=0.49, μ1

m=1000μ0m, E1=1.1*105E0, v1=0.3, ρ0=0.4

Effe

ctiv

e Yo

ung'

s m

odul

us (M

pa)

(a)

0 100 200 300 400 5002

3

4

5

6

7

8

9

10

φ=0.15

μ0m=4π*10-7, E0=2.1*106, v0=0.49, μ1

m=1000μ0m, E1=1.1*105E0, v1=0.3

φ=0.50

φ=0.40

φ=0.30

φ=0.05

H30 (kA/m)

Effe

ctiv

e Yo

ung'

s m

odul

us (M

pa)

(a)

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

Chain-structured composites Random composites

0 100 200 300 400 5000.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

φ=0.15

φ=0.1

φ=0.05

φ=0.0

H30 (kA/m)

μ0m=4π*10-7, E0=2.1*106, v0=0.49, μ1

m=1000μ0m, E1=1.1*105E0, v1=0.3, ρ0=0.4

Effe

ctiv

e sh

ear m

odul

us (M

pa)

(b)

0 100 200 300 400 500

1.0

1.5

2.0

2.5

3.0

φ=0.15

φ=0.50

φ=0.40

φ=0.30

φ=0.05

H30 (kA/m)

μ0m=4π*10-7, E0=2.1*106, v0=0.49, μ1

m=1000μ0m, E1=1.1*105E0, v1=0.3

Effe

ctiv

e sh

ear m

odul

us (M

pa)

(b)

L. SunL. Sun

Summary and Conclusion

Micromechanics-based framework for magneto-mechanical behavior

Composites responses:

Anisotropic/isotropic effective elasticity

Effective magnetostrictive deformation

Magnetic field dependent elasticity