A Multiscale-Based Micromechanics Model for Functionally Graded - - PowerPoint PPT Presentation

A Multiscale-Based Micromechanics Model for Functionally Graded Materials (FGMs)

• H. Yin, L. Sun
• Dept. of Civil and Environmental Engineering

The University of Iowa Acknowlegments: NSF

• G. H. Paulino
• Dept. 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 Ceramic Ceramic-

• Metal FGM

Metal FGM

}

THot

Ceramic matrix with metallic inclusions

}

}

}

Metallic matrix with ceramic inclusions Transition region

} Metallic Phase

TCold

Ceramic Phase

Microstructure

1-D 2-D 3-D

Functionally Graded Materials Functionally Graded Materials

ZrO2/SS FGM

Microstructure of FGM

10% ZrO2 / 90%SS 90% ZrO2 / 10%SS 40% ZrO2 / 50%SS

SEM Photographs courtesy of Materials Research Laboratory at UIUC

Civil Engineering

Fire Protection Blast Protection

Super heat-resistance

Thermal barrier coating for space vehicle components (SiC/C, TUFI)

Electro-magnetic & MEMS

Piezoelectric & thermoelectric devices Sensors & Actuators

Biomechanics

Artificial joints Orthopedic & Dental implants

Military

Military vehicles & body armor

Optics

Graded refractive index materials

Applications of FGMs

Other applications

Nuclear reactor components Cutting tools (WC/Co), razor blades Engine 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 / N

X t φ = 1 φ −

Transition zone Particle-Matrix Particle-Matrix t 100% C 0% SiC 0% C 100% SiC

3

X

2

X

1

X

Theoretical Preparation

• Eshelby’s equivalent inclusion method

( ) ( )

' = + ε r ε ε r ε

( ) ( ) ( ) ( ) ( )

* 1 2

' '     + = + −     C ε r ε r C ε r ε r ε r

( )

( ) ( )

' ' * ' '

,

ij ijkl kl

d ε ε

Ω

= Γ

∫

r r r r r

*

ε

*

ε

= + ε ε

2

C

2

C

2

C

1

C

Theoretical Preparation

• Pairwise interaction (Moschovidis and Mura, 1975)

Y Z

• 2

2

• 5
• 4
• 3
• 2
• 1

1 2

The difference of the averaged strain for two- particle solution and

• ne-particle solution

( ) ( )

1 2 1 2

, , , ,

ij ijkl kl

d a L a ε = r r r r

Micromechanics of FGMs

• RVE of particle-matrix zone

( ) ( )

1 2 3 3

? ? X X = = ε ε

3

X

2

X

1

X

2

x

1

x

3

x

( ) ( )

3 ,3 3

, X X φ φ

X

σ σ

( )

3

Given X φ

( )

( )

1 1 3

X <= ε ε

Micromechanics of FGMs

• Averaged strain in the central particle

( ) ( ) ( )

( )

1 2 1 1

: , ,

i i

a

∞ − =

= − ⋅∆ +∑ ε I P C ε d 0 x

( )

( ) ( ) ( ) ( ) ( )

1 2 3

, , | , , | , , :

i i 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

2

x

1

x

3

x

Micromechanics of FGMs

• Number density function P(r|0)

Homogeneous composite : Many-body system:

( ) ( )

3

| 4 /3 g x P a φ π = x 0

3

4 /3 N P V a φ π = =

2 4 6 8 10 0.0 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

( ) ( )

( ) ( )

/ 3 ,3 3 3 3

3 | 4

x

g x P X e X x a

δ

φ φ π

−

  = + ×   x 0

2

x

1

x

3

x

Neighborhood: Taylor’s expansion Far field: bounded Average: δ defines the size of the neighborhood

( )

3

X φ

( ) ( )

/ 3 ,3 3 3

0.74

r

X e X x

δ

φ φ

−

≤ + × ≤

Micromechanics of FGMs

• Averaged strain in the central particle

( ) [ ] ( ) ( ) ( ) ( ) ( )

1 2 2 2 ,3 ,3

: 0 : 0 : φ φ = − ⋅∆ + + ε I P C ε D ε F ε

( ) ( ) ( ) ( )

/ 2 3 3 3

3 3 , , ; , , 4 4

r D D

g r g r a d e a x d a a

δ

π π

−

= =

∫ ∫

D L 0 x x F L 0 x x ( )

[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 2 3 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 2

:

−

= ε C σ

( ) ( ) ( ) ( )

1 2 3 1 3 3 2 3

: 1 : X X X X φ φ = + −     σ C ε C ε

Boundary condition:

( ) ( ) ( ) ( )

1 1 3 3 2 2 3 3

: : X X X X = = ε T σ ε T σ

Solution:

( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 2 3 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

3

X

2

X

1

X

σ σ

Uniaxial loading

• Governing equations

( ) ( ) ( ) ( ) ( )

1 2 33 3 3 33 3 3 33 3

1 X X X X X ε φ ε φ ε = + −    

( ) ( ) ( ) ( )

1 1 3 3 2 2 3 3

: : X X X X = = ε T σ ε T σ

( ) ( ) ( ) ( )

11 3 33 33 3 13 33 3 33 3

; X E X v X X ε σ ε ε = = −

3

X

2

X

1

X

33

σ

33

σ

( ) ( ) ( ) ( ) ( )

1 2 11 3 3 11 3 3 11 3

1 X X X X X ε φ ε φ ε = + −    

Shear loading

• Governing equations

( ) ( ) ( ) ( ) ( )

1 2 13 3 3 13 3 3 13 3

1 X X X X X ε φ ε φ ε = + −    

( ) ( )

13 13 3 13 3

2 X X τ µ ε =

3

X

2

X

1

X

13

τ

13

τ

( ) ( ) ( ) ( )

1 1 3 3 2 2 3 3

: : X X X X = = ε T σ ε T σ

Averaged Fields

• Transition zone

( ) ( ) ( ) ( )

1 1 3 3 2 2 3 3

: : X X X X = = ε T σ ε T σ

( )

1 3 2

d X d φ < <

( ) ( ) ( ) ( ) ( )

3 3 3 3 3

1

I II

F 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: Matrix

3

X

2

X

1

X

σ σ

( ) ( )

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 2 3 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 0.5 2 4 6 8 10

EA=76.0GPa, vA=0.23, EB=3.0GPa, vB=0.4 Mori-Tanaka simulation Current simulation Young's modulus E(GPa) 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 1.0 1 10 100

Zone III Zone II Zone I (a) EA/EB=50 EA/EB=20 EA/EB=10 EA/EB=5 vA=vB=0.3 Effective Young's modulus E/EB Volume fraction φ

0.0 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 III Zone II Zone I (b) EA/EB=50 EA/EB=20 EA/EB=10 EA/EB=5 vA=0.2 vB=0.45 Effective Poisson'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 1.0 100 200 300 400 500

(a) φ(z)=(X3/t)

2

φ(z)=(X3/t) φ(z)=(X3/t)

1/2

ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi3Al=0.295 Young's modulus E (GPa) Location X3/t

0.0 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, vNi3Al=0.295 Poisson's ratio v Location X3/t

Results and Discussion

100% C 100% SiC

2

X

1

X

0.48t 0.52t

t

13

τ

13

τ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

EA=320GPa, vA=0.3, EB=28GPa, vB=0.3

FEM simulation (1997) Self-consistent method (1997) Current simulation

Averaged stress σ13/τ13

0 in Carbon

volume fraction φ

Results and Discussion

50 100 150 200 250 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 Young's modulus E (GPa) 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 1.0 100 200 300 400 500

(a) Experiment (1993) Simulation ETiC=460GPa, vTiC=0.19, ENi3Al=199GPa, vNi3Al=0.295 Young's modulus E (GPa) Volume fraction of Ni3Al φ

0.0 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, vNi3Al=0.295 Poisson's ratio 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