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A Multiscale-Based Micromechanics Model for Functionally Graded Materials (FGMs)
H. Yin, L. SunDept. of Civil and Environmental Engineering
The University of Iowa
Acknowlegments: NSF
G. H. PaulinoDept. of Civil and Environmental Engineering
University of Illinois at Urbana-Champaign
US-South America Workshop: Mechanics and Advanced Materials Research and Education
Rio de Janeiro; 08/05/2004
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Outline
• Introduction– FGMs– Micromechanics
• Micromechanical Analysis of FGMs• Examples• Conclusions and Extensions
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Multiscale and Functionally Graded
Materials, 2006
Chicago, Illinois
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High Temperature Resistance Compressive Strength
Fracture Toughness Thermal Conductivity
Ceramic Rich PSZ
Metal Rich CrNi Alloy
( Ilschner, 1996 )
FGMs Offer a Composite’s Efficiency w/o Stress Concentrations at Sharp Material Interfaces
500um
Ideal Behavior of Material Properties in a Ideal Behavior of Material Properties in a CeramicCeramic--Metal FGMMetal FGM
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THot
Ceramic matrix with metallic inclusionsMetallic matrix with
ceramic inclusions
Transition region
Metallic PhaseTCold
Ceramic Phase
Microstructure
1-D
2-D
3-D
Functionally Graded MaterialsFunctionally Graded Materials
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ZrO2/SS FGM
Microstructure of FGM
10% ZrO2 / 90%SS
90% ZrO2 / 10%SS40% ZrO2 / 50%SS
SEM Photographs courtesy of Materials Research Laboratory at UIUC
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Civil EngineeringFire ProtectionBlast Protection
Super heat-resistanceThermal barrier coating for space vehicle components (SiC/C, TUFI)
Electro-magnetic & MEMSPiezoelectric & thermoelectric devices Sensors & Actuators
BiomechanicsArtificial jointsOrthopedic & Dental implants
MilitaryMilitary vehicles & body armor
OpticsGraded refractive index materials
Applications of FGMs
Other applicationsNuclear reactor components Cutting tools (WC/Co), razor bladesEngine components, machine parts
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Introduction - Micromechanics
• Analytical composite models:Mori-Tanaka, Self-Consistent, Hashin-Shtrikman bounds, etc(Zuiker, 1995; Gasik, 1998)
1. Volume fraction => effective elasticity: unrelated to gradient of volume fraction
2. Non-interaction between particles
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Introduction - Micromechanics
• Numerical methods
FEM: 2D problem(Reiter, Dvorak, et al, 1997, 1998)(Cho, Ha, 2001)
Higher-order cell model: 3D problem(Aboudi, Pindera, Arnold, 1999)
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Multiscale Framework
FGM
Effective elasticity
Micro-scale
Local elastic field
Homogenization Averaged elastic fields
Macro-scale
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Notation
Two phases:
Phase SiC:
Phase Carbon:
φ
( )3 / NX tφ =
1 φ−
Transition zone
Particle-Matrix
Particle-Matrix
t
100% C0% SiC
0% C100% SiC
3X
2X
1X
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Theoretical Preparation
• 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
= +
0ε 0ε
2C2C 2C1C
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Theoretical Preparation• 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 2 1 2 0, , , ,ij ijkl kld a L a ε=r r r r
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Micromechanics of FGMs
• RVE of particle-matrix zone
( ) ( )1 23 3? ? X X= =ε ε
3X
2X
1X
2x
1x
3x( ) ( )0 0
3 ,3 3, X Xφ φ
0X
0σ
0σ( )3 Given Xφ
( ) ( )1 103X <=ε ε 0
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Micromechanics of FGMs
• Averaged strain in the central particle
( ) ( ) ( ) ( )1 210 1
: 0 , ,ii
a∞−
== − ⋅∆ +∑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
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Micromechanics of FGMs
• Number density function P(r|0)Homogeneous composite :
Many-body system:
( ) ( )3|
4 / 3g x
Pa
φπ
=x 0
34 / 3NPV a
φπ
= =
0 2 4 6 8 100.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
φ=0.1 φ=0.2 φ=0.3 φ=0.4
g(r)
r/a
φ
( )g x - radial distribution
Percus-Yevick solution
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Micromechanics of FGMs
• Number density function P(r|0) for FGMs
( ) ( ) ( ) ( )0 / 03 ,3 3 33
3|
4xg x
P X e X xa
δφ φπ
− = + × x 0
2x
1x
3x
Neighborhood: Taylor’s expansion
Far field: bounded
Average:
δ defines the size of the neighborhood
( )03Xφ
( ) ( )0 / 03 ,3 3 30 0.74rX e X xδφ φ−≤ + × ≤
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Micromechanics of FGMs
• Averaged strain in the central particle
( ) [ ] ( ) ( ) ( ) ( ) ( )1 2 2 20 ,3 ,3
: 0 0 : 0 0 : 0φ φ= − ⋅∆ + +ε 0 I P C ε D ε F ε
( ) ( ) ( ) ( )/ 233 3
3 3, , ; , ,
4 4r
D D
g r g ra d e a x d
a aδ
π π−= =∫ ∫D L 0 x x F L 0 x x
( ) [ ] ( ) ( ) ( ) ( )( ) ( ) ( )
1 2 23 0 3 3 3 3
2,3 3 3 3,3
: :
:
X X X X X
X X X
φ
φ
= − ⋅∆ +
+
ε I P C ε D ε
F ε
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Averaged Fields
• Solve the averaged strain
( )2 1 020 :−=ε C σ
( ) ( ) ( ) ( )1 203 1 3 3 2 3: 1 :X X X Xφ φ= + − σ C ε C ε
Boundary condition:
( ) ( )( ) ( )
1 1 03 3
2 2 03 3
:
:
X X
X X
=
=
ε T σ
ε T σ
Solution:
( ) [ ] ( ) ( ) ( ) ( )( ) ( ) ( )
1 2 23 0 3 3 3 3
2,3 3 3 3,3
: :
:
X X X X X
X X X
φ
φ
= − ⋅∆ +
+
ε I P C ε D ε
F ε
3X
2X
1X
0σ
0σ
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Uniaxial loading
• Governing equations
( ) ( ) ( ) ( ) ( )1 233 3 3 33 3 3 33 31X X X X Xε φ ε φ ε= + −
( ) ( )( ) ( )
1 1 03 3
2 2 03 3
:
:
X X
X X
=
=
ε T σ
ε T σ
( ) ( )( )( )
011 333
33 3 1333 3 33 3
;X
E X vX X
εσε ε
= = −
3X
2X
1X
033σ
033σ
( ) ( ) ( ) ( ) ( )1 211 3 3 11 3 3 11 31X X X X Xε φ ε φ ε= + −
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Shear loading
• Governing equations
( ) ( ) ( ) ( ) ( )1 213 3 3 13 3 3 13 31X X X X Xε φ ε φ ε= + −
( ) ( )013
13 313 32
XX
τµε
=
3X
2X
1X
013τ
013τ
( ) ( )( ) ( )
1 1 03 3
2 2 03 3
:
:
X X
X X
=
=
ε T σ
ε T σ
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Averaged Fields
• Transition zone
( ) ( )( ) ( )
1 1 03 3
2 2 03 3
:
:
X X
X X
=
=
ε T σ
ε T σ
( )1 3 2d X dφ< <
( ) ( ) ( ) ( ) ( )3 3 3 3 31I IIF X f X F X f X F X= + −
Transition function:(Hirano et al 1990, 1991; Reiter, Dvorak, 1998)
Phase 1: Particle
Phase 2: Matrix
Phase 2: Particle
Phase 1: Matrix3X
2X
1X
0σ
0σ
( )( )
33 13 23
13 23
,E v v
µ µ
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Results and Discussion
• Interaction• Drop last two terms => Mori-Tanaka• Gradient of volume fraction
( ) [ ] ( ) ( ) ( ) ( )( ) ( ) ( )
1 2 23 0 3 3 3 3
2,3 3 3 3,3
: :
:
X X X X X
X X X
φ
φ
= − ⋅∆ +
+
ε I P C ε D ε
F ε
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Results and discussion
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10EA=76.0GPa, vA=0.23, EB=3.0GPa, vB=0.4
Mori-Tanaka simulation Current simulation
Yo
ung'
s m
odul
us E
(GP
a)
Volume fraction φ
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Results and discussion
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01
10
100
Zone IIIZone IIZone I (a)
EA/EB=50 EA/EB=20 EA/EB=10 EA/EB=5
vA=vB=0.3
Effe
ctiv
e Yo
ung'
s m
odul
us E
/EB
Volume fraction φ0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
Zone IIIZone IIZone I (b)
EA/E
B=50
EA/EB=20 EA/EB=10 EA/EB=5
vA=0.2 vB=0.45
Effe
ctiv
e P
oiss
on's
ratio
v
Volume fraction φ
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Results and discussion
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
100
200
300
400
500
(a)
φ(z)=(X3/t)2
φ(z)=(X3/t) φ(z)=(X3/t)
1/2
ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi
3Al=0.295
Youn
g's
mod
ulus
E (G
Pa)
Location X3/t0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
(b)
φ(z)=(X3/t)1/2
φ(z)=X3/t φ(z)=(X3/t)
2
ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi
3Al=0.295
Poi
sson
's ra
tio v
Location X3/t
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Results and Discussion
100% C
100% SiC
2X
1X
0.48t0.52t
t
013τ
013τ
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4EA=320GPa, vA=0.3, EB=28GPa, vB=0.3
FEM simulation (1997) Self-consistent method (1997) Current simulation
Aver
aged
stre
ss σ
13/τ
130 in
Car
bon
volume fraction φ
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Results and Discussion
0 50 100 150 200 2500
1
2
3
4
5
6
7
Experiment with polyester matrix (2000) Simulation with Polyester matrix Experiment with polyester-plasticizer matrix (2000) Simulation with polyester-plasticizer matrix
Ep-p=2.5GPa, vp-p=0.33, Ep=3.6GPa, vp=0.41, Ec=6.0GPa, vc=0.35
Youn
g's
mod
ulus
E (G
Pa)
Location X3 (mm)
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Results and discussion
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
100
200
300
400
500
(a) Experiment (1993) Simulation
ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi
3Al=0.295
Youn
g's
mod
ulus
E (G
Pa)
Volume fraction of Ni3Al φ0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
Experiment (1993) Simulation (b)
Volume fraction of Ni3Al φ
ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi
3Al=0.295
Poi
sson
's ra
tio v
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Conclusions and Extensions
• Micromechanics-based FGM model • Effective elastic property estimates• Pairwise interaction• Gradient of volume fraction• 2-scale model (Multiscale)• Extension to Nano-FGMs (additional scale)
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V=1m/s V=15m/s
1m/s, LD 04 Apr 2003 2-D ELASTODYNAMIC PROBLEM 15m/s, LD 04 Apr 2003 2-D ELASTODYNAMIC PROBLEM
Extension – Dynamic Fracture/Branching
v
v
a0=0.3mm
3mm
3mm
10m/s, LD 04 Apr 2003 2-D ELASTODYNAMIC PROBLEM
V=10m/s
Poster Presentation Tomorrow:Ms. Zhengyu (Jenny) Zhang
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http://cee.uiuc.edu/paulino