There are different types of models to describe the behavior of pebble beds (see, e. g., Reimann et a/.97): finite-element models based on continuum mechanics, also called finite-element modeling (FEM), and the so-called discrete-element models, DEM, for description of mechanics at micromecha­nics level (i. e., individual pebble-pebble interactions). The development of computational tools at these two different length scales (macro — and microscales) allows for a better description of the thermomecha­nics of the pebble beds.

Figure 28 Bi-axial pebble-bed deformation tests: experimental results and calculations with ABAQUS code system. Reproduced from Hermsmeyer, S.; Reimann, J. Fusion Eng. Des., 2002, 61-61, 367-373.

4.15.4.5.1 Continuum models

The macroscopic behavior of pebble beds is de­scribed by constitutive equations commonly used in soil mechanics, considering the granular material as a continuum material that can undergo reversible elastic deformations, inelastic volume compaction (consolidation), and pressure-dependent shear fail­ure. To account for these properties, different models have been developed, which are implemented in structural computational programs.91,115,117,120-125

Pebble-bed data (see sections above) had to be implemented, and user-defined subroutines had to be written, for example, for the thermal creep laws.115

The codes were first validated with fairly simple experiments.104,123,126,127 Figures 28 and 29 show examples of results from the biaxial experiment126 and the SCATOLA experiment for calculational benchmarking127 (Figure 29).

These codes were later validated with more com­plex mock-up experiments such as HELICA and HEXCALIBUR91,108,114,128,129 and, as outlined below, aimed to be validated by the pebble bed assembly (PBA) experiment.130 Figures 30(a)117 and 30(b)91 show a comparison between measured and calculated HELICA temperatures.

Figure 30 (a) Measured temperatures in the HELICAexperiment, reprinted from Dell’Orco, G.; di Maio, P. A.; Giamusso, R.; Tincani, A.; Vella, H. Fusion Eng. Des., 2007, 82, 2366-2374. (b) Calculated temperatures for the HELICA experiment, reprinted with courtesy from Gan, Y. Ph. D. Thesis, Universitat Karlsruhe, 2008.

1st cycle (PD=68.29%) 2nd cycle (PD=60.95%) 3rd cycle (PD=61.10%) Loading process Unloading process

6

30 25 20

ro CL u> c T5 ro _o "ro і

5

ГО CL

U)

4

~o ro _o "го I

Loading process

Unloading process

3

2

Axial strain (%)

(a)

Figure 31 Stress-strain behavior of granular materials in (a) a rectangular box under uniaxial compaction and (b) packing density effect. Reproduced from An, Z.; Ying, A.; Abdou, M. Fusion Eng. Des. 2007, 82, 2233-2238.

The final goal of these codes is to determine the thermomechanical behavior of pebble beds in TBMs for ITER and DEMO blanket modules. At present, the codes are set up for this task. Up to now, only small portions of a DEMO blanket have been modeled. In these calculations, maximum stresses (very loca­lized) of 6 and 2 MPa were obtained for the ceramic breeder and beryllium pebble beds, respectively. Because of thermal creep, these values were re­duced after 2 h to 75% and 25%, respectively, of the initial value.