micromechanics-based magneto-elastic constitutive modeling

34
L. Sun L. 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 3 rd IMR Symposium on Materials Science and Engineering Institute of Metal Research, Shenyang, P. R. China July 10-13, 2007

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Page 1: 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

Page 2: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Magnetic Particulate Composites

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

Nonmagnetic Matrix: Elastomers, Polymers, Metals

Page 3: Micromechanics-based magneto-elastic constitutive modeling

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

Page 4: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Magnetic particle filled composites

Magnetic particles(Solid, iron)

Nonmagnetic matrix (Liquid or Colloid, rubber)

H

High temperature

Random Structure

Page 5: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Magnetic particle filled composites

H

High temperature

Chain Structure

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

Page 6: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Room temperature

H

Magnetostriction

Young’s modulus

Shear modulus

Mechanical Behavior

Page 7: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Modeling Framework

(b)

Composites

Microscale

RVE

Local fields

Test loading

Homogenization

Averaged fields

Macroscale

Constitutive law

RVE: Representative Volume Element

Page 8: Micromechanics-based magneto-elastic constitutive modeling

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

+

Page 9: Micromechanics-based magneto-elastic constitutive modeling

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= + +

Page 10: Micromechanics-based magneto-elastic constitutive modeling

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

Page 11: Micromechanics-based magneto-elastic constitutive modeling

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′ ′

≤+

Page 12: Micromechanics-based magneto-elastic constitutive modeling

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.

Page 13: Micromechanics-based magneto-elastic constitutive modeling

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

Page 14: Micromechanics-based magneto-elastic constitutive modeling

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:

Page 15: Micromechanics-based magneto-elastic constitutive modeling

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

Page 16: Micromechanics-based magneto-elastic constitutive modeling

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

Page 17: Micromechanics-based magneto-elastic constitutive modeling

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

Page 18: Micromechanics-based magneto-elastic constitutive modeling

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:

Page 19: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Equivalent Inclusion Method

Particle averaged strain:

*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:

+

Page 20: Micromechanics-based magneto-elastic constitutive modeling

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

Page 21: Micromechanics-based magneto-elastic constitutive modeling

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

Page 22: Micromechanics-based magneto-elastic constitutive modeling

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

Page 23: Micromechanics-based magneto-elastic constitutive modeling

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

= ,msε f

0 0,H σ

Page 24: Micromechanics-based magneto-elastic constitutive modeling

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

Page 25: Micromechanics-based magneto-elastic constitutive modeling

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

Page 26: Micromechanics-based magneto-elastic constitutive modeling

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.

Page 27: Micromechanics-based magneto-elastic constitutive modeling

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

Page 28: Micromechanics-based magneto-elastic constitutive modeling

L. SunL. Sun

Magnetostrictive Compositeswith Randomly Dispersed Particles

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

( ) ( ) ( ) ( ) ( )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

= +

0ε 0ε

2C2C 2C1C

Page 29: Micromechanics-based magneto-elastic constitutive modeling

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

Page 30: Micromechanics-based magneto-elastic constitutive modeling

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

Page 31: Micromechanics-based magneto-elastic constitutive modeling

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

Page 32: Micromechanics-based magneto-elastic constitutive modeling

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)

Page 33: Micromechanics-based magneto-elastic constitutive modeling

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)

Page 34: Micromechanics-based magneto-elastic constitutive modeling

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