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Probabilistic 3D Fracture

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Arindam Chakraborty

The document describes a stochastic multiscale fracture analysis method for functionally graded materials. The method uses polynomial dimensional decomposition (PDD) to efficiently calculate crack-driving forces and fracture reliability in three-dimensional functionally graded media with random microstructure and material properties. It was applied to an edge-cracked silicon carbide-aluminum functionally graded material with spatially varying particle distribution and material properties modeled as random fields. The method required fewer simulations than Monte Carlo and efficiently generated stress intensity factor distributions and probability of fracture initiation along the crack front. Future work will include modeling crack growth and additional failure mechanisms.

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Sharif Rahman The University of Iowa Iowa City, IA 52245 Stochastic Multiscale Fracture Analysis of 3D Functionally Graded Media 2009 ASME PVP Conference, Prague, Czech Republic, July 2009 Work supported by NSF (CMS-0409463) Arindam Chakraborty Structural Integrity Associates San Jose, CA 95138
OUTLINE Introduction Moment-Modified Polynomial Dimensional Decomposition (PDD) Example Conclusions & Future Work
INTRODUCTION Two Challenging Problems Modeling random microstructure Predicting tail probabilities of fracture response W/Cu FGM (Zhou et al ., JNM, 2007) FGM Fracture Mosaic or Level-Cut Poisson random field (Grigoriu, JAP, 2003; Rahman, IJNME, 2008) crack
INTRODUCTION Objective: Develop a probabilistic, concurrent, multiscale model for calculating crack-driving forces in 3D FGM under mixed-mode deformations Various Multiscale Analyses (Chakraborty & Rahman, EFM, 2008) Sequential Invasive Concurrent
PDD METHOD A Crack in a Two-Phase FGM Output Crack-driving forces (SIFs) Fracture reliability Crack-propagation path random particle vol. fraction random microstructure random constituent properties Mosaic or level-cut RF Input weak form elasticity tensor
PDD METHOD Polynomial Dimensional Decomposition NONLINEAR SYSTEM S -variate PDD of y (Rahman, IJNME, 2008)
PDD METHOD Expansion Coefficients by MCS/CV Two sets of coefficients needed for two distinct crack-tip conditions
PDD METHOD Moment-modified PDD (each crack-tip cond.) microscale elements (microstructure) macroscale elements (no microstructure)
PDD METHOD 2D Verification Edge-cracked SiC-Al FGM Random microstructure and constituent properties Det. crack location and size/BCs Univariate PDD requires five times fewer FEA than crude MCS (2000 vs. 10,000 FEA)
EXAMPLE Edge-Cracked SiC-Al FGM Particle vol. fraction 1D, inhomogeneous, Beta RF Particle location Mosaic RF, spatially-varying Poisson intensity Constituent properties of SiC & Al indep. LN variables (Means of E : 419.2, 69.7 GPa; Means of : 0.19, 0.34) = i =1 kN/cm 2 ; o = 0.6 kN/cm 2 Part. rad. = 0.48 cm Nearly 700 RVs ( = 0.4) A B 16 cm 16 cm 8 cm 4 cm C Crack
EXAMPLE Global Responses (Two Samples) Sample 1 Sample 2
EXAMPLE Mode-I SIFs (Univariate) A B 16 cm 16 cm 8 cm 4 cm C Crack
EXAMPLE Modes-II and III SIFs (Univariate)
EXAMPLE Conditional Probability of Fracture Initiation A B 16 cm 16 cm 8 cm 4 cm C Crack
CONCLUSIONS/FUTURE WORK A moment-modified polynomial dimensional decomposition method was developed Fourier-polynomial expansions MCS/control variate moment-modified random output Efficiently generates SIF distributions Probability of fracture initiation varies significantly along the crack front Future work: Crack growth, cohesive zone models, particle-matrix debonding, dynamic & thermal fracture, etc .

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Probabilistic 3D Fracture

  • 1. Sharif Rahman The University of Iowa Iowa City, IA 52245 Stochastic Multiscale Fracture Analysis of 3D Functionally Graded Media 2009 ASME PVP Conference, Prague, Czech Republic, July 2009 Work supported by NSF (CMS-0409463) Arindam Chakraborty Structural Integrity Associates San Jose, CA 95138
  • 2. OUTLINE Introduction Moment-Modified Polynomial Dimensional Decomposition (PDD) Example Conclusions & Future Work
  • 3. INTRODUCTION Two Challenging Problems Modeling random microstructure Predicting tail probabilities of fracture response W/Cu FGM (Zhou et al ., JNM, 2007) FGM Fracture Mosaic or Level-Cut Poisson random field (Grigoriu, JAP, 2003; Rahman, IJNME, 2008) crack
  • 4. INTRODUCTION Objective: Develop a probabilistic, concurrent, multiscale model for calculating crack-driving forces in 3D FGM under mixed-mode deformations Various Multiscale Analyses (Chakraborty & Rahman, EFM, 2008) Sequential Invasive Concurrent
  • 5. PDD METHOD A Crack in a Two-Phase FGM Output Crack-driving forces (SIFs) Fracture reliability Crack-propagation path random particle vol. fraction random microstructure random constituent properties Mosaic or level-cut RF Input weak form elasticity tensor
  • 6. PDD METHOD Polynomial Dimensional Decomposition NONLINEAR SYSTEM S -variate PDD of y (Rahman, IJNME, 2008)
  • 7. PDD METHOD Expansion Coefficients by MCS/CV Two sets of coefficients needed for two distinct crack-tip conditions
  • 8. PDD METHOD Moment-modified PDD (each crack-tip cond.) microscale elements (microstructure) macroscale elements (no microstructure)
  • 9. PDD METHOD 2D Verification Edge-cracked SiC-Al FGM Random microstructure and constituent properties Det. crack location and size/BCs Univariate PDD requires five times fewer FEA than crude MCS (2000 vs. 10,000 FEA)
  • 10. EXAMPLE Edge-Cracked SiC-Al FGM Particle vol. fraction 1D, inhomogeneous, Beta RF Particle location Mosaic RF, spatially-varying Poisson intensity Constituent properties of SiC & Al indep. LN variables (Means of E : 419.2, 69.7 GPa; Means of : 0.19, 0.34) = i =1 kN/cm 2 ; o = 0.6 kN/cm 2 Part. rad. = 0.48 cm Nearly 700 RVs ( = 0.4) A B 16 cm 16 cm 8 cm 4 cm C Crack
  • 11. EXAMPLE Global Responses (Two Samples) Sample 1 Sample 2
  • 12. EXAMPLE Mode-I SIFs (Univariate) A B 16 cm 16 cm 8 cm 4 cm C Crack
  • 13. EXAMPLE Modes-II and III SIFs (Univariate)
  • 14. EXAMPLE Conditional Probability of Fracture Initiation A B 16 cm 16 cm 8 cm 4 cm C Crack
  • 15. CONCLUSIONS/FUTURE WORK A moment-modified polynomial dimensional decomposition method was developed Fourier-polynomial expansions MCS/control variate moment-modified random output Efficiently generates SIF distributions Probability of fracture initiation varies significantly along the crack front Future work: Crack growth, cohesive zone models, particle-matrix debonding, dynamic & thermal fracture, etc .