Sharif Rahman The University of Iowa Iowa City, IA 52245

1. NSF Workshop on Probability & Materials: From Nano-to-Macro Scale STOCHASTIC FRACTURE OF FUNCTIONALLY GRADED MATERIALS Sharif Rahman The University of Iowa Iowa City, IA 52245 January 2005

2. OUTLINE • Introduction • Fracture of FGM • Shape Sensitivity Analysis • Reliability Analysis • Ongoing Work • Conclusions

3. INTRODUCTION • The FGM Advantage Fracture Toughness Thermal Conductivity Temperature Resistance Compressive strength Metal Rich CrNi Alloy Ceramic Rich PSZ Ilschner (1996) FGMs avoid stress concentrations at sharp material interfaces and can be utilized as multifunctional materials

4. Micro- Scale Macro- Scale Local Elastic Field Averaged Elastic Field Effective Elasticity Homogenization Volume fraction, Porosity, etc. Elastic Modulus, Poisson’s Ratio,etc. ceramic Eceramic metal Emetal INTRODUCTION • FGM Microstructure and Homogenization

5. Tensile Properties Applied Stress Fracture Toughness Crack Size and Shape Material Resistance Crack Driving Force > Geometry of Cracked Body Temperature Loading Rate Radiation Loading Cycles Fatigue Properties INTRODUCTION • Objective Develop methods for stochastic fracture-mechanics analysis of functionally graded materials Work supported by NSF (Grant Nos: CMS-0409463; DMI-0355487; CMS-9900196)

6. Crack-Tip Fields in Isotropic FGM FRACTURE OF FGM

7. FRACTURE OF FGM • J-integral for FGM • J-integral for Two Superimposed States 1 & 2 Superscript 1  Actual Mixed-Mode State Superscript 2  Auxiliary State with SIF = 1

8. FRACTURE OF FGM • New Interaction Integral Methods Method I: Homogeneous Auxiliary Field Method II: Non-Homogeneous Auxiliary Field Both isotropic (Rahman & Rao; EFM; 2003) and orthotropic (Rao & Rahman, CM; 2004) FGMs can be analyzed

9. FRACTURE OF FGM • Example 1 (Slanted Crack in a Plate) Plane Stress Condition L=2, W=1 =0.3 (N = 370) Gradation Direction

10. Velocity Field & Material Derivative • Governing and Sensitivity Equations Need a numerical method (FEM) to solve these two equations for V(x)   x  x  SHAPE SENSITIVITY ANALYSIS

11. SHAPE SENSITIVITY ANALYSIS • Performance Measure • Shape Sensitivity

12. SHAPE SENSITIVITY ANALYSIS • Sensitivity of Interaction Integral Method • Method I : Homogeneous Auxiliary Field • Method II : Non-Homogeneous Auxiliary Field Rahman & Rao; CM; 2004 and Rao & Rahman, CMAME; 2004

13. x2 L 2b 2b x1  2a L W W SHAPE SENSITIVITY ANALYSIS • Example 2 (Plate with an Internal Crack) 2L=2W=20, 2a=2, =0.3 Plane Stress Conditions Gradation Direction

14. Stochastic Fracture Mechanics FGM System Random Input Failure Probability Load Fracture initiation and propagation Material & gradation properties Geometry Failure Criterion FRACTURE RELIABILITY

15. FRACTURE RELIABILITY • Multivariate Function Decomposition • Univariate Approximation At most 1 variable in a term • Bivariate Approximation At most 2 variables in a term • General S-Variate Approximation At most S variables in a term

16. FRACTURE RELIABILITY • Reliability Analysis • Performance Function Approximations Univariate Bivariate Terms with dimensions 2 & higher Terms with dimensions 3 & higher

17. Lagrange shape functions FRACTURE RELIABILITY • Lagrange Interpolation • Monte Carlo Simulation Univariate Approximation Bivariate Approximation

18. 2 1 (a) FRACTURE RELIABILITY • Example 3 (Probability of Fracture Initiation) Performance Function (Maximum Hoop Stress Criterion) Gradation Direction

19. FRACTURE RELIABILITY • Example 3 (Results)

20. Stochastic Micromechanics Micromechanics Rule of Mixtures Mori-Tanaka Theory Self-Consistent Theory Eshelby’s Inclusion Theory Particle Interaction Gradients of Volume Fraction Nonhomogeneous Random Field Volume Fraction Porosity Nonhomogeneous Random Field Spatially-varying FGM Microstructure Stochastic Material Properties Elastic Modulus Poisson’s Ratio Yield Strength etc. ONGOING WORK

21. ONGOING WORK • Level-Cut Random Field for FGM Microstructure Grigoriu (2003) Homogeneous microstructure Translation Random Field Second-Moment Properties volume fraction two-point correlation function Filtered Non-Homogeneous Poisson Field Find probability law of Z(x) to match target statistics p1 and p11

22. ONGOING WORK • Multi-Scale Model of FGM Fracture

23. CONCLUSIONS • New interaction integral methods for linear-elastic fracture under mixed-mode loading conditions • Continuum shape sensitivity analysis for first-order gradient of crack-driving force with respect to crack geometry • Novel decomposition methods for accurate and computationally efficient reliability analysis • Ongoing work involves stochastic, multi-scale fracture of FGMs

24. REFERENCES • Rao, B. N. and Rahman, S., “A Mode-Decoupling Continuum Shape Sensitivity Method for Fracture Analysis of Functionally Graded Materials,” submitted to International Journal for Numerical Methods in Engineering, 2004. • Rahman, S., “Stochastic Fracture of Functionally Graded Materials,” submitted to Engineering Fracture Mechanics, 2004. • Xu, H. and Rahman, S., “Dimension-Reduction Methods for Structural Reliability Analysis,” submitted to Probabilistic Engineering Mechanics, 2004. • Rahman, S. and Rao, B. N., “A Continuum Shape Sensitivity Method for Fracture Analysis of Isotropic Functionally Graded Materials,” submitted to Computational Mechanics, 2004. • Rao, B. N. and Rahman, S., “A Continuum Shape Sensitivity Method for Fracture Analysis of Orthotropic Functionally Graded Materials,” accepted in Mechanics and Materials, (In Press). • Rahman, S. and Rao, B. N., “Continuum Shape Sensitivity Analysis of a Mode-I Fracture in Functionally Graded Materials,” accepted in Computational Mechanics, 2004 (In Press). • Rao, B. N. and Rahman, S., “Continuum Shape Sensitivity Analysis of a Mixed-Mode Fracture in Functionally Graded Materials,” accepted in Computer Methods in Applied Mechanics and Engineering, 2004 (In Press). • Rao, B. N. and Rahman, S., “An Interaction Integral Method for Analysis of Cracks in Orthotropic Functionally Graded Materials,” Computational Mechanics, Vol. 32, No. 1-2, 2003, pp. 40-51. • Rao, B. N. and Rahman, S., “Meshfree Analysis of Cracks in Isotropic Functionally Graded Materials,” Engineering Fracture Mechanics, Vol. 70, No. 1, 2003, pp. 1-27.