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