a multiscale-based micromechanics model for functionally...
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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
Outline
• Introduction– FGMs– Micromechanics
• Micromechanical Analysis of FGMs• Examples• Conclusions and Extensions
Multiscale and Functionally Graded
Materials, 2006
Chicago, Illinois
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
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
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
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
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
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)
Multiscale Framework
FGM
Effective elasticity
Micro-scale
Local elastic field
Homogenization Averaged elastic fields
Macro-scale
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
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
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
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
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
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
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−≤ + × ≤
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
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
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= + −
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
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
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
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
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
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
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
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)
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
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)
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
http://cee.uiuc.edu/paulino