Application of Discrete Element Method to Study Mechanical Behaviors of Ceramic Breeder Pebble Beds Zhiyong An, Alice Ying, and Mohamed Abdou UCLA Presented at CBBI-4 Petten, The Netherlands September 6-8, 006
Being able to predict thermo-mechanical behaviors of a packed pebble bed system is one of the keys to the success of a solid breeder blanket Main issues of interest Thermo-mechanical behaviors of a packed pebble bed and their impacts on tritium release and temperature control Mechanical uncertainty related to thermal creep deformation and potential pebble cracking. Additional issues Can experimentally derived effective constitutive equations (Reimann correlations) applicable to estimate ceramic breeder pebble bed thermo-mechanical performance, in which the loading conditions of the reactor are different from those of experimental tests? Will the pebble bed reach a semi-equilibrium state under a pulsed operating condition? What best to describe such a state? What an initial compaction should it be for a blanket design? And why?
Thermo-mechanical Behaviors of Breeder Pebble Bed Systems Variables: Pebble materials Bed properties Boundary conditions Operation loadings Primary Reactants: Stress magnitude/distribution Particle breakage Thermal properties/temperature gradient Plastic/creep deformation Gap formation at breeder/structure interface Single/multiple effect experiments (Bed deformation and creep effect) Experimental Database (FZK, JAERI, CEA,UCLA) Thermo-physical and mechanical properties constitutive equations Finite Element Program (MSC.MARC) Discrete Element Model Design Guideline and Evaluation (ITER TBMs) ANSYS to replace MARC? 3
DEM Simulation relies on Contact Model Parameters Material parameters: (mechanical, geometric & boundary) Young s Modulus (E); Poisson ratio (v); Pebble size (R) and Creep model (n, c), etc. Direct parameters: Contact force (F), Overlapping distance ( δ). Indirect parameters: Stress ( σ) and Strain ( ε ). δ Hertz contact theory (Elastic state) δ = a R 9 = 6RE where * E * /3 F /3 ν ν = + E E and R = R + R 4
Derivation of Contact Model for Creep Deformation Material creep model for solid material observing a constant stress & ε = cσ n n = for diffusion creep; n > for power-law creep. Issues about particle contact problem Stress and strain are much large near the contact and highly non-uniform inside the particles; Displacement between two contact particles is a summation of the deformation of the materials in between; Contact neck size deforms as well, and can change the stress magnitude inside the particles; Two modeling schemes: Equivalent stress Equivalent volume/mass F R a F A typical contact between particles 5
) Equivalent stress model (Buler) F ( ε, t) β σ ~ ~ p( ε, t) ~ [ ε P] ε d ε ε δ σ, p and P are the local stress, equivalent contact stress and equivalent overall stress c n n( β ) n( β ) Creep Contact Models = α P Δt (4.5) ) Equivalent volume model & ε = ςε& a = cσ e δ H c Contact neck size y( a0 + y) where = cσ H 0 = ξ n e 0 a 0 n = s Δt n F y = ε& a a0 Δ t and s = c a0ς n π Equivalent overlap change n Δt ( δ = + ) } Basic δ e δ c equations H o is a virtual parameter and related to the contact neck size. ζ and ξare two unknowns. α,β is related to the pebble bed packing. δ = ςξ y (4.) c In both models, the coefficients need to be calibrated by experiments or FEA simulation. 6
Mechanical Behaviors (DEM simulation) Uniaxial compression & Isothermal heating 7 80 Axial Loading (MPa) 6 5 4 3 Loading process Unloading process 0 0.5.5 AxialStrain(%) Wall pressure (MPa) 60 40 0 0 00 400 600 800 Bed temperature ( o C) Figures: Mechanical behaviors of granular materials packed in a rectangular box. (Initial packing density is 60.3%; Total particle number 5,000; H L W is 35 30 30; Average radius of pebbles is.0.) Conclusion: DEM simulation results show that the stiffness of the packed pebble beds are nonlinearly dependent on the loadings. The loading processes can increase the stiffness of the packed pebble beds. Thermal expansion can potentially induce high stresses in the pebble bed structures. 7
DEM Model Verification DEM vs. FZK experimental data Mechanical behavior of a cylinder pebble bed under cycle loadings simulated by DEM program J. Reimann, et al., Fusion Engineering and Design, 6-6 (00) Conclusion: () DEM program has good capabilities to simulate the mechanical behaviors of granular materials. () DEM models can easily simulate different properties of pebble materials or loading conditions. However, contact models are important for the results and the simulation is limited by the computer power. 8
Example critical particle chain evolutions during compaction Pressure =.0MPa Pressure =.5MPa Pressure =.4Mpa Pressure =.9Mpa Figure: Particles having a contact force greater than 0 N are identified and its neighboring connecting particles with same magnitude or more contact forces are linked to one same pebble chain. Conclusion: DEM simulation is capable to reveal how the pebble interconnected structure plays a role on determining the 3D structural mechanical state of the packed pebble bed. A few pebbles have larger contact forces than others. Small segments of particles can attach or detach to a longer pebble chain. 9
Distribution probability 0-0 - 0-3 Internal/External (boundary condition) Connection Initial Packing Load = MPa Load = 4MPa Load = 6MPa Load = 8MPa Load = 0MPa 0 0-4 0 4 6 8 0 0 0 0 0 Loading pressure (MPa) Magnitude of contact forces (Compared with average contact froce at initial packing ) Left figure shows evolution of contact force magnitude during increasing loading process. Right figure shows the magnitude relationship between average contact force and overall external pressure. Average contact forces (N) 30 0 0 y = 3.43 x DEM results Conclusion: Contact forces at the particle/particle increase as the external loading increases. However, the increase rate of the average contact forces is 3.4 times faster than that of the external loading. 0
Studies on Contact Force Change Average value Increasing magnitude 8 6 4 < f c > < f c max > Particle A Particle B Particle C Change times Special group.8.6.4. Particle group : { f max c f max c > 4 < f c >} 0 0 4 6 8 0 External loading (MPa) Change in contact forces during an increase in external loading 0 50 00 50 Serial number Magnification of maximal contact forces during one interval of loading increase. Conclusion: The change of contact forces due to the external loading is an stochastic process. The average change is linear to the external loading.
Stress Distribution Map Force Map 3 f r f r f r r r r c r N A 3 r...... r r α f r N c α f r Stress Map f r σ ij = N C V α = f α i r α j Black arrow represents the stress on each particle. The particle color and the arrow size are related to the stress magnitude. Red and blue colors respectively mean high and low stresses, and green is the stress magnitude in between.
DEM model for pebble bed Poisson ratio derivation P ~ 0-0MPa Model description: Applying uniform loading (P) on a rectangular pebble bed from top-side. Pebble size is normally distributed in 0.5~.5mm. The contact model includes normal and shear forces. The loading can be considered as a static process. Parameters: Particle number:,000 Young s modulus of pebble: 0GPa Poisson ratio of pebble: 0.4 Young s modulus of wall: 06GPa Poisson ratio of wall: 0.3 3
Evolution of Stress Maps Initial State (P = 0.5MPa) Midst State (P =.5MPa) Final State (P =.5MPa) * The color of pebbles stands for same value of stress magnitude. 4
Calculated Poisson ratio suggests a frictionless pebble bed acting like a fluid * DEM simulation results without considering friction effect. The loading is pressed from y-direction. Average stresses on wall (MPa) 3.5.5 0.5 σ x σ y Poisson ratio 0 0 0 4 6 8 0 Loading steps.5 0.5 Poisson Ratio Numerical value of pebble bed Poisson ratio is needed to address thermomechanics behavior along the longitudinal direction 5
FEA Analysis of ITER TBM --- Coupled thermal & pebble bed mechanics analysis Numerical data: ~ 770 o C (max. T in Breeder); ~ 540 o C (max. T in Beryllium) ~ <.0MPa ( max. σ v in Breeder); ~ 50MPa ( max. σ v in Beryllium) A` A C B Temperature profile Stress profile A: Center of max. T in breeder bed; A`: Interface between breeder bed and coolant structure; B: Near the end of breeder pebble bed; C: Center of max. T in Beryllium. How will the gravity effect be taken into account in the FEA analysis? 6
Evolution of Max. Bed Temp 800 T max in solid breeder pebble bed T max in Beryllium pebble bed 700 ΔT ( o C) 600 500 400 0 000 000 3000 4000 5000 Time (s) The figure shows the evolution of maximal temperatures inside the pebble beds. ) At the location (A point) of max. temperature in breeder beds, the highest temperature value does not change during cycles; however, the lowest value is increased with cycles; ) At C point in Beryllium pebble beds, the highest and lowest value of temperatures are repeatable during cycles. The lowest temperature is close to the coolant temperature (350 o C). 7
Temperature evolutions at different locations A A A Temperature ( o C) Ttemperature ( o C) 800 700 600 500 400 450 45 400 375 T at coolant/breeder interface T max in solid breeder pebble bed 0 000 000 3000 4000 5000 Time (s) On breeder side On coolant structural side 350 0 000 000 3000 4000 5000 Time (s) A A Contact Force (N) A A ΔT ( o C) 5 0 5 350 300 50 00 50 00 50 0 0 000 000 3000 4000 5000 Time (s) Contact Force @ A 0 0 000 000 3000 4000 5000 Time (s) 8
Stress Distribution & Evolution Von Mises Stresses (MPa).6. 0.8 0.4 0 ridge plain st Cycle nd Cycle 3rd Cycle 4th Cycle 5th Cycle Temperature 00 00 300 400 Distance from front wall (mm) 750 700 650 600 550 500 The figure shows Von Mises stresses along the center line of solid breeder pebble bed when the bed temperature reaches maximum in different pulsed cycles. The curves show that the stress distributed inside the bed can be divided into two parts: ridge part and plain part. ) The ridge part is appeared near the front wall and is corresponded to the highest temperature region of the solid breeder pebble beds. The stress only in the forefront part increased with each cycle; ) The plain part covers nearly 3/4 parts of the pebble beds and the Von Mises stress decreased after every cycle. 9
Stress-Strain Behaviors Equivalent Stress (MPa).6. 0.8 0.4 Point A Point A' Point C 0. 0.4 0.6 0.8 Equivalent Strain (%) Conclusion: The stress-strain behavior of breeder material is varied with location, which is highly interacted with temperature. > Point A : High temperature area (Stress is low; deformation is large.) > Point A : Near contact interface (Stress is high; deformation is large.) > Point C : Low temperature area (Stress is low; deformation is small.) 0
Future Works It is unclear that whether the pebble bed can reach an equilibrium thermo-mechanical state after a number of thermal cycles. Further research need to study dynamic loading conditions, and to determine the impacts on its packing structure and subsequent effects on blanket performance. The material properties of the pebble materials still need to study and develop. For instance, the friction coefficient of the pebble surface, crack properties of different pebbles and the deformation map of candidate breeder materials under different stresses and temperatures. The computing efficiency of the DEM program has not yet fully optimized. During a numerical packing process, it should only focus on the pebbles with larger unbalance statues, which need to be relocated to achieve minimum contact force states. Also, it d better add the breaking judgment to improve the current DEM program.