1. Random Finite Element Modeling of thermomechanical behavior of AGR bricks Jose David Arregui Mena, Louise Lever, Graham Hall, Lee Margetts, Paul Mummery

2. Introduction AGR Reactors Random Finite Element Method -Young’s Modulus Random Field Compression Tests Preliminary Results Random Thermoelastic Analysis

3. AGR Graphite Moderated Reactors

4. Fast Neutron Damage Neutron bombardment of graphite Radiolytic Oxidation Chemical reaction between irradiated CO 2 and graphite Damage in nuclear reactors

5. Safety Requirements Requirements during normal and fault conditions: Unimpeded loading and unloading of control rods and fuel rods An adequate flow of coolant gas Provide neutron moderation and thermal inertia

6. Hypothesis Initial, pre-operation spatial variation in the values of the material properties of nuclear graphite have an effect on stress and strain distribution in graphite bricks, which in turn determines the safe operation of a nuclear graphite core

7. Random Finite Element Method and Nuclear Graphite

8. The Finite Element Method Numerical technique to solve differential equations Transforms differential equations to a set of algebraic equations Externalforces Material properties and geometry Displacements

9. s Probability of failure

10. Young’s Modulus Random Field

11. Top-Down Approach, Local Average Method Process Adapted from (Vanmarcke, 1983) 2D Local Average Method Process

12. Scale of fluctuation 10 mm 1 mm The average of a portion of the random field of 1x1 mm will return the mean value of the Young’s Modulus μ 1 mm μ Scale of fluctuation of 1 mm

13. Random Fields for Young’s Modulus +Young’s Modulus -Young’s Modulus Mean Value Correlation length 0.1 Correlation length 1.0 Correlation length 100.0

14. Calibration of the random field Grey Scale Density and Young’s Modulus CT X-Ray Tomography Porosity

15. 3D Random Fields from 2D Images Young’s Modulus Porosity

16. Compression Tests

17. Boundary Conditions for Axial Compression tests Fixed in x,y,zFixed in z Uniform axial Displacement of 4.2 mm

18. Deterministic Realization

19. Random Simulation with a scale of fluctuation (100, 100, 100) Maximum Value – 82.495

20. Maximum Value – 64.324 Random Simulation with a scale of fluctuation (500, 500, 500)

21. Maximum Value – 70.894 Random Simulation with a scale of fluctuation (1000, 1000, 1000)

22. Preliminary Results Random Thermoelastic Analysis

23. Preliminary Thermoelastic Analysis Octant of an AGR brick Free to expand Thermal strains α – Coefficient of Thermal expansion T f – Final temperature T 0 – Reference temperature

24. Temperature profile for the simulations - ΔT

25. Random Material Properties for Young’s Modulus Random Properties Deterministic Properties

26. Displacements Random simulationDeterministic simulation

27. Random simulationDeterministic simulation Stress analysis

28. Road Map Compression test Calibration of the Random fields and Creation of a random Field for CTE Thermomechanical Analysis Creep

29. Acknowledgements

30. Thank you!