Characterization of Thermo-mechanical Performances related to Breeder Pebble Beds by Discrete Element Method Presented by Zhiyong (John) An With contributions from: A. Ying and M. Abdou Mechanical & Aerospace Engineering Department UCLA, Los Angeles, CA 90095 Thirteenth International Workshop on Ceramic Breeder Blanket Interactions Santa Barbara, CA, USA Nov. 30 th Dec. 2 nd 2005
Objectives Advantage of DEM Force magnitude and distribution Different boundary loadings Pebble bed geometry Different breeder pebbles (materials & size) Develop modeling capability to guide the design and to predict the thermo-mechanical performance Deformation characteristics Stress magnitudes and distribution Consequences of thermal creep
What is DEM? Discrete element method firstly applied by Cundall etc. (1979) To solve engineering problem which involves distinct interacting particles subjected to gross motion. Those large scale discontinuous behaviors cannot be solved by finite element method, which is based on continuum materials. Applications of Discrete Element Method Dynamic model: Movement ~ F(resultant forces) / M (inertia) (Cundall (1979), Fast movement ) Quasi-static model: Movement ~ F(resultant forces) / E(stiffness) (Serrano and Rodriguez-Ortiz (1973), Bojtár and K. Bagi(1993) & Bagi(1993): Slow movement )
Contact Model Elastic contact Hertz-Mindlin contact theory with Coulomb s law Creep deformation Following the power-law creep deformation and basic contact information Two Contact particles & ε = α & oc ε a ε& where α = 0.57a and is a α (& ε t)( 1 & ε t) a p 2n + a = c t 2 π a0 n The local contact creep deformation model.
Ceramic Breeder Pebbles Orthosilicate (FZK-Schott, 2003) d = 0.2-0.6 mm Metatitanate (CEA-CTI, 2001) d = 0.7-1.0 mm
Simulation Processes Initial stage 1. Randomly generate the pebbles and assign material properties; (Including pebble size and locations; packing density ~ 45%) 2. Packing process; (initial packing density ~ 60%) 3. Apply boundary loadings; (Check loading step and the equilibrium criterion by theory of probability.) 4. Output pebble bed information. (Stress-strain relation, force distribution map, and force magnitudes) Final result
Convergence and Stability 1E+08 Force at contact 8E+07 6E+07 4E+07 2E+07 Without ADR With ADR (h = 1.0) With ADR (h = 1.8) 0 50 100 150 200 Number of cycles Contact force as a function of the calculation cycles for instantaneous loading
Ngan s Model When a pebble bed reaches the equilibrium state, the distribution of contact force magnitude should follow Ngan s force distribution model, which is derived based on theory of probability and physics of granular materials. 10-1 DEM simulation Ngan's Model curve 10-1 Probability Density (%) 10-2 10-3 0 0.5 1 1.5 2 2.5 3 3.5 4 Normalized Contact Force DEM simulation results of contact force distribution (Contact forces are normalized by the mean value)
Force Distribution Map Figure on the right shows force distribution map of a 2D pebble bed reaching an equilibrium state, which is under a uniformly boundary loading of 2.0 MPa. Blue lines between neighboring particle centers represent inter-particle contact force, and the width reflects the contact force magnitude. In the force distribution map, the black dashed circles illustrate the contact areas with highest force values. The force distribution map shows that forces are emerged from the particle-wall contacts and transmitted through the particles. The lines are a kind of homogeneous distributed through the pebble bed, but the magnitude of contact forces are much different through the network. The critical contact forces may be higher than the crush load of breeder pebble.
Inelastic Result vs. Experiments Uniaxial compressing Mechanical behaviors of a cylinder pebble bed under cycle loadings J. Reimann, et al., Fusion Engineering & Design, 61-62 (2002) DEM simulation captures the irreversible pebble displacement which forms during the loading cycle. This plastic-like behavior is related to bed packing properties such as packing density, particle size distribution, container structure, etc.
Friction Effect 10 8 Axial stress (MPa) 6 4 2 Compress (μ =0) Depress (μ =0) Compress (μ =0.1) Depress (μ =0.1) Compress (μ =0.3) Depress (μ =0.3) 0 0.5 1 1.5 2 2.5 3 Axial strain (%) Mechanical behaviors of a cylinder pebble bed with different particle friction coefficients With the same simulation process, as the friction coefficient increases, the stiffness of the pebble bed has been effected. Figure on the top shows that the bed stiffness increases as the friction between the pebbles becomes significant, which is in accord with the experimental conclusion.
Contact Information Distribution probability 10-1 10-2 10-3 10-4 10 0 10 1 10 2 Magnitude of contact forces Initial Packing Load = 2MPa Load = 4MPa Load = 6MPa Load = 8MPa Load = 10MPa (Compared with average contact froce at initial packing ) Figure shows evolution of contact force magnitude during increasing loading process. (The insert figure shows the magnitude relation between average contact force and overall pressure.) Contact information: Contact force magnitude & the distribution probability
Associated Information Average contact forces (N) 30 20 10 y= 3.413x DEM results 0 0 2 4 6 8 10 Loading pressure (MPa) Relation between inside contact force and outside loading stress.
Creep Result vs. Experiments Creep deformation 6 Packing density ~ 60% Axial Stress (MPa) 4 2 R a =1.0mm 0 0 0.5 1 1.5 2 2.5 Axial Strain (%) J. Reimann, et al., Fusion Engineering & Design, 61-62 (2002) DEM program can simulate the creep deformation of a pebble bed under high temperatures. Our results show that the effective stiffness of a pebble bed can be decreased after creep deformation generates.
Size Effect 6 Packing density ~ 60% Axial Stress (MPa) 4 2 R a =1.0mm R a = 0.5~1.5mm 0 0 0.5 1 1.5 2 2.5 Axial Strain (%) The pebble size will be an important parameter and related to the stiffness changing.
Summary 1. Preliminary results show that DEM has capability to simulate the thermo-mechanical deformation of a packed pebble bed, especially under high temperatures. 2. Force distribution map provides the local details of the contact stresses, and the destructive locations can be identified. 3. Average contact force is linearly increased with the loading stress, and the coefficient is about 3.4 times. 4. Creep results show that the stiffness of the pebble bed drops after creep relaxation and pebble size can impact on the stiffness changing. 5. To further study the thermo-mechanical behaviors of ceramic breeder pebble beds, more experimental data are needed for us to understand the fundamental physical phenomena and to validate DEM simulation results.