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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
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Room temperature
H
Magnetostriction
Young’s modulus
Shear modulus
Mechanical Behavior
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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= + +
−
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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
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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
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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
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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
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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:
+
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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)
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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)
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Summary and Conclusion
Micromechanics-based framework for magneto-mechanical behavior
Composites responses:
Anisotropic/isotropic effective elasticity
Effective magnetostrictive deformation
Magnetic field dependent elasticity