Presented by: Nassia Tzelepi

1. Progress on the Graphite Crystal Plasticity Finite Element Model (CPFEM)J F B PayneL Delannay, P Yan (University of Louvaine) Presented by: Nassia Tzelepi

2. Overview • Background • Basic principles • Inputs and assumptions • Model description • Results • Future work

3. Background • It is clear that the graphite microstructure is extremely important to its behaviour under irradiation. • The mechanical properties and the response to irradiation of polycrystalline graphites differ greatly from single graphite crystals • Most notably polycrystalline graphite is much more nearly isotropic than the highly anisotropic graphite crystal • Strains well beyond the elastic limit of the crystal are accommodated in the polycrystalline material. • The anisotropy of individual crystals and void formation affects the mechanical behaviour of many polycrystalline materials • The effect of crystal anisotropy has been studied in these materials.

4. Basic principles • The first piece of physics in the model is that the crystallites making up the polycrystalline agglomerate are anisotropic and that the crystal symmetry axes of adjacent crystallites are not aligned. • As pointed out by Mrozowski this misalignment can cause cracking because of the mismatch between thermal expansion of adjacent crystallites along the crystallite boundary. Therefore… • The second piece of physics in the model is the possibility of cracking when a threshold strength is exceeded. • The final piece of physics in the model is the behaviour of isolated graphite crystals. • Elasticity, thermal and dimensional change data for an isolated crystal are taken from published experimental (MTR) results.

5. Model inputs Constitutive behaviour of single graphite crystals Total strain: The only inputs to the model are: • Single crystal elasticity • Single crystal CTE • Single crystal response to irradiation induced dimensional change • Irradiation creep only by shear between basal planes

6. Model Inputs: Elastic Anisotropy stiff in-plane directions : ‘1’ ‘3’ compliant along the c-axis : Isotropic in the basal plane ‘2’ direction taken to coincide with Z

7. Model inputs: Anisotropic thermal expansion Contraction of basal plane is negligible Other

8. Model inputs: Irradiation induced dimensional change Volume is preserved per neutron

9. Model inputs: Irradiation induced creep Macroscopically: At the crystal level: Creep occurs by shear between basal planes only:

10. Simulation of manufacturing followed by irradiation Cooling (manufacturing): The anisotropy of the graphite crystal is such that it shrinks almost exclusively in the direction of the c-axis Irradiation: After the 4th step the c-axis recovers its initial length Dimensional change preserves the volume Manufacturing

11. Model description (1) • 100 crystallites included as Voronoï cells • Their seeds are positioned randomly inside a square section and then periodically reproduced outside the square • Min distance between seeds: dmin = 0.08L • Uniform crystallite sizes and realistic crystallite shapes • The crystal c-axis lies within the 2D plane • Random orientation of crystallites in 2D

12. Model description (2) • Periodicity of the model microstructure • Modified UMAT used for elastic-plastic deformation of individual crystallites in metals. y Y x

13. Model description (3)

14. Background • Basic principles • Model description • Inputs and assumptions • Results • Future work

15. Preliminary investigations What is the influence of the internal stresses and cracks due to cooling from manufacturing temperatures on the CTE and dimensional change of the polycrystal under irradiation?

16. Crack closure midway through irradiation After Step 4 After cooling

17. Opening of new cracks upon further irradiation After full irradiation After cooling

18. Preliminary findings Manufacturing

19. Preliminary findings Without creep With creep Colored according to the shear stress on the basal plane

20. Results 1° Calculate the elastic modulus of the polycrystal without cracks • Which crystallite configuration is more likely to lead to cracking at the boundary? 2° CTE change of the polycrystal with cooling from manufacturing temperatures and irradiation

21. Results – Pure elastic modelling • Pure elastic modelling (no cracking) • Average elastic modulus of the polycrystal

22. Results – Pure elastic modelling • Sensitivity of the macroscopic YM to the crystal stiffness

23. Results – Pure elastic modelling • Stress distribution at the crystal boundary • Which crystallite configuration is most likely to lead to cracking? • The interface stresses (normal and shear) depend on the two angles between the interface and the two c-axes of the two grains connected by the interface θ1 varies from -90˚to 90 ˚, while θ2 varies from 0˚to 90˚

24. Results – Pure elastic modelling

25. Results – Pure elastic modelling

26. Results – Model with cracks • Evolution of cracking

27. Results – Model with cracks • Variation of shear strain (in local coordinates of individual crystals)

28. Results – Model with cracks 2000˚C 1000˚C c-axis recovery ambient Manufacturing Irradiation

29. In summary • Model input data are crystal properties only.  • Crystal orientation of each crystallite is chosen randomly in modelling plane. • The model predicts that: • cracks occur on cooling from graphitisation temperature • cracks close with irradiation • new cracks open as irradiation proceeds. • The model shows that this cracking behaviour explains • the reduction of CTE of as-manufactured graphite • the irradiation induced dimensional change.

30. In summary • The model without cracks has provided some useful insights on: • the modulus of elasticity of the polycrystal • the crystallite configurations that are most likely to lead to cracking. • The model predicts CTE change during manufacturing and during irradiation.

31. Further work • The existing 2D model will be refined and its application to thermal and irradiation conditions expanded. • A 3D model is being developed, based upon the refined 2D model. • Sensitivity of the models to known features of manufactured nuclear graphite, including raw materials, crystallite size, porosity and degree of anisotropy. • Both models will be compared with historical graphite irradiation data.

32. Thank you for your attention Thank you for your attention

33. Macroscopic dimensional changes during cooling: 2300 K 1300 K dArea dVoid Isolated -3 %0 % FE (no cracks) -2.4 % N/A FE -2.4 % 0.1 % dLin = -1.2 x 10-2 2300-1300 = 1000K Average CTE over cooling step = -1.2 x 10-5/K

34. Second cooling step: 1300 K  300 K dArea dVoid Isolated -3 %0 % FE (no cracks) -2.3 % FE -1.5 % 1.6 % • The model predicts that the presence of cracks reduces CTE of as-manufactured graphite and affects the temperature dependence of CTE • CTE is lower at lower T, where more cracks are open Average CTE over cooling step= -0.75 x 10-5/K

35. Estimated fluence at crack closure - Irradiation preceded by 2000 K cooling during manufacturing. - Two irradiation “steps” needed to recover the c-axis length Manufacturing