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

A Multiscale-Based Micromechanics Model for Functionally Graded Materials (FGMs) G. H. Paulino Dept. of Civil and Environmental Engineering University of Illinois at Urbana-Champaign H. Yin, L. Sun Dept. of Civil and Environmental Engineering The University of Iowa Acknowlegments: NSF 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

Ideal Behavior of Material Properties in a Ideal Behavior of Material Properties in a Ceramic- -Metal FGM Metal FGM Ceramic Fracture Toughness High Temperature Resistance Thermal Conductivity Compressive Strength Metal Rich Ceramic Rich CrNi Alloy PSZ ( Ilschner , 1996 ) 500um FGMs Offer a Composite’s Efficiency w/o Stress Concentrations at Sharp Material Interfaces

Functionally Graded Materials Functionally Graded Materials } T Hot Ceramic Phase } Ceramic matrix with metallic inclusions } Transition region } Metallic matrix with ceramic inclusions } Metallic Phase 1-D T Cold 2-D Microstructure 3-D

Microstructure of FGM ZrO 2 /SS FGM 10% ZrO 2 / 90%SS 40% ZrO 2 / 50%SS 90% ZrO 2 / 10%SS SEM Photographs courtesy of Materials Research Laboratory at UIUC

Applications of FGMs 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 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 Macro-scale FGM Micro-scale Effective elasticity Local elastic field Averaged elastic fields Homogenization

φ Notation Two phases: 0% C Particle-Matrix 100% SiC Phase SiC: ( ) Transition zone N φ = X 3 / t t X 3 Phase Carbon: Particle-Matrix X 2 1 φ − 100% C X 1 0% SiC

Theoretical Preparation • Eshelby’s equivalent inclusion method ( ) ( ) = + 0 ε r ε ε r ' 0 0 ε ε ε * + = C ε * 1 C C C 2 2 2 ( ) ( ) ( ) ∫ ε = Γ ε 0 ' ' * ' ' r r r , r d r ε ij ijkl kl Ω ( ) ( ) ( ) ( ) ( )     + = + − 0 0 * C ε r ε r ' C ε r ε r ' ε r     1 2

Theoretical Preparation • Pairwise interaction ( Moschovidis and Mura, 1975 ) The difference of the 2 averaged strain for two- 1 particle solution and one-particle solution 0 -1 ( ) ( ) Z = a ε 1 2 1 2 0 d r r , , a L r r , , -2 ij ijkl kl -3 -4 -5 -2 0 2 Y

Micromechanics of FGMs • RVE of particle-matrix zone ( ) ( ) φ 0 φ 0 X , X 0 σ 3 ,3 3 x 3 x 2 X 3 x 0 X 1 X 2 ( ) ( ) 1 1 <= 0 ε X ε 0 3 0 σ X 1 ( ) φ Given X 3 ( ) ( ) 1 2 = = ε X ? ε X ? 3 3

Micromechanics of FGMs • Averaged strain in the central particle ( ) + ∑ ( ) ( ) ( ) ∞ − 1 1 2 = − ⋅∆ i ε 0 I P C : ε 0 d 0 x , , a 0 = i 1 x ( ) 3 ∑ ∞ i d 0 x , , a = i 1 x 2 ( ) ( ) ∫ = P x 0 d 0 x | , , a d x D x ( ) ( ) ( ) ∫ 2 = 1 P x 0 L 0 x | , , a : ε x d x 3 D

Micromechanics of FGMs • Number density function P ( r|0 ) φ Homogeneous composite : φ N = = P π 3 V 4 a /3 3.5 φ =0.1 Many-body system: φ =0.2 3.0 φ =0.3 ( ) φ =0.4 g x 2.5 ( ) = φ P x 0 | 2.0 π 3 4 a /3 g(r) 1.5 1.0 ( ) g x - radial distribution 0.5 0.0 Percus-Yevick solution 0 2 4 6 8 10 r/a

Micromechanics of FGMs • Number density function P ( r|0 ) for FGMs ( ) 3 g x ( ) ( ) ( )   = φ + − δ φ × 0 x / 0 P x 0 | X e X x   π 3 ,3 3 3 3 4 a x 3 x Neighborhood: Taylor’s expansion 2 Far field: bounded x 1 ( ) φ Average: 0 X 3 δ defines the size of the neighborhood ( ) ( ) ≤ φ + − δ φ × ≤ 0 r / 0 0 X e X x 0.74 3 ,3 3 3

Micromechanics of FGMs • Averaged strain in the central particle [ ] ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 = − ⋅∆ + φ + φ ε 0 I P C : ε 0 D 0 : ε 0 F 0 : ε 0 0 ,3 ,3 ( ) ( ) 3 g r 3 g r ( ) ( ) ∫ ∫ = = − δ r / 2 D L 0 x , , a d x ; F e L 0 x , , a x d x π π 3 3 3 4 a 4 a D D [ ] ( ) ( ) ( ) ( ) ( ) 1 2 2 = − ⋅∆ + φ ε X I P C : ε X X D X : ε X 3 0 3 3 3 3 ( ) ( ) ( ) 2 + φ X F X : ε X ,3 3 3 3 ,3

Averaged Fields • Solve the averaged strain 0 σ [ ] ( ) ( ) ( ) ( ) ( ) 1 2 2 = − ⋅∆ + φ ε X I P C : ε X X D X : ε X 3 0 3 3 3 3 ( ) ( ) ( ) 2 + φ X F X : ε X ,3 3 3 3 ,3 X 3 ( ) ( ) ( ) ( ) 1 2 = φ +  − φ  0 σ X C : ε X 1 X C : ε X   3 1 3 3 2 3 X 2 ( ) 2 Boundary condition: = − 1 0 ε 0 C : σ 0 σ X 2 1 Solution: ( ) ( ) 1 = 1 0 ε X T X : σ 3 3 ( ) ( ) 2 = 2 0 ε X T X : σ 3 3

Uniaxial loading • Governing equations σ 0 33 ( ) ( ) 1 = 1 0 ε X T X : σ 3 3 ( ) ( ) 2 = 2 0 ε X T X : σ 3 3 X ( ) ( ) ( ) ( ) ( ) 3 1 2 ε = φ ε +  − φ  ε X X X 1 X X   33 3 3 33 3 3 33 3 X 2 ( ) ( ) ( ) ( ) ( ) 1 2 ε = φ ε +  − φ  ε X X X 1 X X   11 3 3 11 3 3 11 3 X 1 σ 0 33 ( ) ε σ 0 X ( ) = = − 11 3 E X 33 ; v ( ) ( ) 33 3 13 ε ε X X 33 3 33 3

Shear loading • Governing equations τ 0 13 ( ) ( ) ( ) ( ) ( ) 1 2 ε = φ ε +  − φ  ε X X X 1 X X   13 3 3 13 3 3 13 3 ( ) ( ) 1 = 1 0 ε X T X : σ X 3 3 3 ( ) ( ) X 2 = 2 0 ε X T X : σ 2 3 3 X 1 τ 0 13 τ 0 ( ) µ = X 13 ( ) 13 3 ε 2 X 13 3

Averaged Fields ( ) • Transition zone < φ < d X d 1 3 2 0 σ Phase 1: Particle Phase 2: Particle Phase 2: Matrix Phase 1: Matrix X 3 ( ) ( ) ( ) ( ) ( ) = +  −  I II F X f X F X 1 f X F X   3 3 3 3 3 X 2 Transition function: 0 σ X (Hirano et al 1990, 1991; Reiter, Dvorak, 1998) 1 ( ) ( ) ( ) 1 = E , v v 1 0 ε X T X : σ 33 13 23 3 3 ( ) µ µ ( ) ( ) 2 = 2 0 ε X T X : σ 13 23 3 3

Results and Discussion • Interaction • Drop last two terms => Mori-Tanaka • Gradient of volume fraction [ ] ( ) ( ) ( ) ( ) ( ) 1 2 2 = − ⋅∆ + φ ε X I P C : ε X X D X : ε X 3 0 3 3 3 3 ( ) ( ) ( ) 2 + φ X F X : ε X ,3 3 3 3 ,3

Results and discussion E A =76.0GPa, v A =0.23, E B =3.0GPa, v B =0.4 10 Mori-Tanaka simulation Current simulation 8 Young's modulus E(GPa) 6 4 2 0 0.0 0.1 0.2 0.3 0.4 0.5 Volume fraction φ

Results and discussion v A =0.2 v B =0.45 v A =v B =0.3 0.5 100 E A /E B =50 E A /E B =50 E A /E B =20 E A /E B =20 Effective Young's modulus E/E B 0.4 E A /E B =10 E A /E B =10 Effective Poisson's ratio v E A /E B =5 E A /E B =5 0.3 10 0.2 0.1 Zone II Zone I Zone III (a) (b) Zone II Zone III Zone I 0.0 1 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 0.6 0.7 0.8 0.9 1.0 Volume fraction φ Volume fraction φ

Results and discussion E TiC =460GPa, v TiC =0.19, E Ni 3 Al =199GPa, v Ni 3 Al =0.295 E TiC =460GPa, v TiC =0.19, E Ni 3 Al =199GPa, v Ni 3 Al =0.295 500 0.5 Young's modulus E (GPa) 400 0.4 Poisson's ratio v 300 0.3 200 0.2 2 1/2 φ (z)=(X 3 /t) φ (z)=(X 3 /t) 100 0.1 φ (z)=(X 3 /t) φ (z)=X 3 /t 2 1/2 φ (z)=(X 3 /t) φ (z)=(X 3 /t) (a) (b) 0 0.0 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 0.6 0.7 0.8 0.9 1.0 Location X 3 /t Location X 3 /t

Results and Discussion τ 0 13 E A =320GPa, v A =0.3, E B =28GPa, v B =0.3 1.4 FEM simulation (1997) Self-consistent method (1997) 1.2 Current simulation 0 in Carbon 100% SiC 1.0 0.52t 0.8 Averaged stress σ 13 / τ 13 0.48t t 0.6 0.4 X 2 0.2 100% C 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X 1 τ 0 volume fraction φ 13