1. Size Effect on the Fracture Properties of Nuclear Graphite Gyanender Singh, Haiyan Li and Alex Fok University of Minnesota

2. Weibull Model Based on the Weakest link theory. Quantifies size effect on strength: m is the Weibull modulus, σ fi is the failure stress for the Component i with effective volume V i, i=1,2.

3. Viability of Weibull model George Quinn, J. Am. Ceram. Soc., 73(8), 1990 Characteristic strength of bend bar as a function of effective volume RBSN: Reaction-bonded silicon nitride

4. Viability of Weibull model George Quinn, J. Am. Ceram. Soc. 74(9), 1991 Studies on three different grades of silicon nitride reviewed. Strength decreases with increase in the effective volume.

5. Deviations from the Weibull Model - Graphite Experimental study conducted by Brocklehurst (1977) Specimens of IM1-24 graphite subjected to tension and four-point bending Flexural strength increases with increase in specimen volume for smaller volumes Tensile strength increases with increase in specimen volume and the trend levels off for specimens with greater volumes.

6. Deviations from the Weibull Model - Graphite Mitchell et al., Journal of Nuclear Materials (2003)

7. Deviations from the Weibull Model – IG-11 Graphite Li et al., Carbon (2013)

8. Summary for Graphite Strength can increase with increase in volume/strain gradient. Weibull modulus can also increase with increase in volume/strain gradient.

9. Previous Work Conducted Li and Fok, Journal of Nuclear Materials (2009) Deterministic model with material strain-softening (post-peak stress retention) was employed to predict the failure process of quasi-brittle materials subjected to different strain gradients. Considered rectangular beams under bending and L-shaped beams under tension.

10. Previous Work Conducted before material damage during material damage but prior to cracking during crack propagation Stress/strain distributions under pure bending Li and Fok, Journal of Nuclear Materials (2009)

11. Previous Work Conducted Without strain-softeningWith strain-softening Li and Fok, Journal of Nuclear Materials (2009)

12. Previous Work Conducted Li and Fok, Journal of Nuclear Materials (2009)

13. Previous Work Conducted Li and Fok, Journal of Nuclear Materials (2009)

14. Previous Work Conducted Results from Monte Carlo analysis with 3000 specimens per test. Properties were assumed to follow a 2-parameter Weibull distribution. Li and Fok, Journal of Nuclear Materials (2009)

15. Previous Work Conducted Nemeth et al., Carbon (2013) Li and Fok, Journal of Nuclear Materials (2009)

16. Previous Work Conducted Li and Fok, Journal of Nuclear Materials (2009) Mitchell et al., Journal of Nuclear Materials (2003)

17. Previous Work Conducted Conclusions: Model correctly predicts the failure behavior of specimens with different strain gradients. Strain softening (post-peak stress retention) leads to higher flexural strength than tensile strength. Increase in Weibull modulus with increasing strain gradient correctly predicted. Limitations of the study: Less successful for L-shaped specimens. All specimens are assumed homogeneous, even though different from each other. Size effect not considered. Li and Fok, Journal of Nuclear Materials (2009)

18. Li et al., Carbon (2013) Conducted 3-point bend test on notched NBG-18 graphite specimens of three different sizes. Evaluated critical stress intensity factor (fracture toughness) and the corresponding Weibull modulus. Employed Digital Image correlation to measure effective crack length.

19. Bazant’s size-effect model was used to study the effect of size on fracture toughness. The width of the specimen is represented by d and fracture toughness by K IC. K If and c f are the size independent parameters of the Bazant Law. These parameters were evaluated. Li et al., Carbon (2013)

20. Results: With a decrease in specimen size the fracture toughness as well as Weibull modulus decreased. Size effect is due to different effective crack lengths: effect is smaller in bigger specimens. Li et al., Carbon (2013)

21. Current Work Extends Li and Fok (2009) Addresses size effect with more sophisticated FE models Perform Monte Carlo analyses for Tensile test for 3 different sizes of specimens Flexural test for 4 different sizes of specimens Notched-beam flexural test for 2 sizes of specimens Corresponding experimental studies: Brocklehurst (1977) and Li et al. (2013)

22. Current Work σ δ G IC Addresses heterogeneity in the material.

23. Current Work: Simulation Heterogeneous interface

24. Current Work: Approach mσmσ σcσc mGmG G IC Tensile (rectangular beams)CCPP BendingNotchedPPCC Un-notchedPPPP Tensile (L-shaped specimen)PPPP C C P Determine m σ and σ c through midsize specimens Determine m G and G IC through midsize specimens Obtain through simulation (30 specimens) using the calibrated values

25. Current Work: Results

27. Tension Bending (un-notched) Brocklehurst (1977) obtained Weibull modulus as 16 for 4-point bending specimens with volumes greater than 1 cm 3. Volumes of the specimens in current study were: 250, 31, 2, 0.03 cm 3. Price (1976) performed tensile tests and obtained Weibull modulus of 8.6 and 9.7 for small and large tensile specimens with volumes 724 and 9847 mm 3. Volumes of the specimens in current study were 250, 2 and 0.03 cm 3

28. Current Work: Results Weibull moduli comparison Small SizeMedium SizeLarge Size Current study12.118.320.3 Experiment (Li et al.)18.135.541.1

29. Why does Strength Depend on Strain Gradient? Stress Strain

30. Why Does Weibull Modulus Increase With Strain Gradient? Single die Two dices... Many dices

31. Why Does Weibull Modulus Increase With Size? Probability of a large specimen containing all-sized defects is high. Fracture is therefore controlled by large defects only, which gives a higher Weibull modulus. Probability of small specimens containing all-sized defects is low. Therefore, fracture distribution depends on all the defects, which gives a lower Weibull modulus.

32. Acknowledgement Funded by the DOE Office of Nuclear Energy’s Nuclear Energy University Programs (NEUP)