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

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

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

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

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

6. FRACTURE OF FGM  Crack-Tip Fields in Isotropic 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 Both isotropic (Rahman & Rao; EFM ; 2003) and orthotropic (Rao & Rahman, CM ; 2004) FGMs can be analyzed Method I: Homogeneous Auxiliary Field Method II: Non-Homogeneous Auxiliary Field

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

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

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. SHAPE SENSITIVITY ANALYSIS W L  2b2b x1x1 2a2a 2b2b L W x2x2 2 L =2 W =20, 2 a =2, =0.3 Plane Stress Conditions  Example 2 (Plate with an Internal Crack) Gradation Direction

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

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

16.  Reliability Analysis FRACTURE RELIABILITY  Performance Function Approximations UnivariateBivariate Terms with dimensions 2 & higher Terms with dimensions 3 & higher

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

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

19. FRACTURE RELIABILITY  Example 3 (Results)

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

21. ONGOING WORK  Level-Cut Random Field for FGM Microstructure Translation Random FieldSecond-Moment Properties Filtered Non-Homogeneous Poisson Field Find probability law of Z(x) to match target statistics p 1 and p 11 Grigoriu (2003) Homogeneous microstructure volume fraction two-point correlation function

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.