thermomechanics of solid breeder and be pebble bed materials

Thermomechanics of solid breeder and Be pebble bed materials

J. Reimann a,�, L. Boccaccini a, M. Enoeda b, A.Y. Ying c

a Forschungszentrum Karlsruhe, Postfach 3640, D-76021 Karlsruhe, Germanyb Japan Atomic Energy Research Institute, Naka, Japan

c Mechanical and Aerospace Engineering Department, UCLA, Los Angeles, CA 90095-1597, USA

Abstract

The thermomechanical interaction of solid breeder and beryllium pebble beds with structural material (BSMI) has

been identified as a critical issue for solid breeder blanket designs. For example, the expansion of pebble beds restrained

by blanket structure exerts stresses on the pebbles as well as the blanket structure wall, which might cause the pebbles to

break and jeopardise the blanket operation. However, at elevated temperatures thermal creep will reduce these stresses

and might compensate for stress build-up due to irradiation-induced swelling. A significant influence of irradiation on

the pebble bed properties is expected. Computationally, the BSMI can be assessed in two ways: (i) by applying

appropriate finite element codes combined with the description of modules for the pebble beds. As input, these modules

require data on characteristic pebble bed properties determined in different standard-type tests; (ii) by numerical

simulations based on a discrete numerical model. Here, the stress profiles are calculated while the effective modulus and

bed thermal expansion coefficients are back estimated. In this paper, recent experimental results on thermomechanical

pebble bed properties for ceramic breeder (metatitanate and orthosilicate) pebble beds and beryllium pebble beds are

presented, including data on the moduli of deformation, thermal creep, inner friction angle, and thermal conductivity of

deformed pebble beds. Furthermore, modelling results of the BSMI for simple geometries are reported based both on

homogeneous and discrete models and are compared with experimental results.

# 2002 Published by Elsevier Science B.V.

Keywords: Thermomechanics; Pebble beds; Solid breeder; Beryllium

1. Introduction

The thermomechanics of solid breeder materials

have been identified as one of the key issues for

solid breeder blanket designs, particularly for

materials in the form of pebble beds. The problems

associated with ceramic blanket pebble bed ther-

momechanics are 2-fold: first, fundamental ther-

mal and mechanical property data have to be

quantified accurately to account for a narrow

design margin. Second, the changes of the packed

states through pebble/pebble and bed/clad interac-

tions during operation need to be well understood

because of their dominating effects on perfor-

mance. For example, the expansion of ceramic

breeder pebble beds restrained by blanket struc-

ture exert stresses on pebbles as well as the blanket

structure wall, which might cause the pebbles to

break and jeopardize the blanket operation. On

� Corresponding author. Tel.: �/49-7247-82-3498; fax: �/49-

7247-82-4837

E-mail address: joerg.reimann@iket.fzk.de (J. Reimann).

Fusion Engineering and Design 61�/62 (2002) 319�/331

www.elsevier.com/locate/fusengdes

0920-3796/02/$ - see front matter # 2002 Published by Elsevier Science B.V.

PII: S 0 9 2 0 - 3 7 9 6 ( 0 2 ) 0 0 2 1 4 - 4

the other hand, thermal creep might reduceremarkably these thermomechanical stresses and

might compensate for the stress build-up due to

irradiation induced swelling. However, gap forma-

tion during blanket shutdown and subsequent

heat-up phases are of concern. A significant

influence of irradiation on the pebble bed proper-

ties is expected which, at present, cannot be

quantified. Thus, thermomechanics efforts pre-sented in this paper focus on the pebble bed

data-base relevant for beginning of life (BOL).

Computationally, thermomechanical pebble

bed-structural material interaction can be deter-

mined by numerical simulations either based on

appropriate finite element codes combined with

the description of modules for the pebble bed, or

by a discrete numerical model. The first methodhas been extensively developed for soil mechanics

[1]; as input, these pebble bed modules require

data on characteristic pebble bed properties, such

as the modulus of deformation, E ; inner friction

angle, a and the wall friction coefficient, aw

determined in standard-type uniaxial compression

tests (UCTs), triaxial compression tests (TCTs)

and shear tests (STs). For fusion blankets withmaximum breeder and beryllium temperatures of

about 900 and 700 8C, thermal creep strain ocr and

the thermal conductivity k influenced by pebble

deformations must also be known. Again, UCTs

play an important role for the determination of

these properties. Contradictorily, the discrete nu-

merical model attempts to calculate the effective

properties (such as modulus of deformation) basedon the material propertie’s data of the pebble

itself.

In this paper, recent experimental results for

thermomechanical properties of ceramic breeder

and beryllium pebble beds are summarised [2�/7].

A large variety of granular materials was investi-

gated, see Table 1, including orthosilicate (Li4-

SiO4) pebbles from FZK, various batches ofmetatitanate (Li2TiO3) pebbles differing in pebble

size, shape and surface roughness provided by

CEA and JAERI, lithiumoxide (Li2O) pebbles

from JAERI and different kinds of beryllium

pebbles manufactured either by Brush Wellman

or NGK. Fig. 1 shows photographs of some types

of pebbles.

Furthermore, modelling approaches are out-lined and calculated results for simple geometries

are presented.

2. Experimental data base of ceramic breeder and

Be pebble beds

2.1. Modulus of deformation during load increase/

decrease

The stress�/strain dependencies of pebble beds

allows the modulus of deformation to be deter-

mined by UCT’s, see e.g. [2�/6]. For these tests the

granular material is filled in a cylindrical cavity,

assisted by mechanical vibration in order to reach

high packing factors. Then, the bed is compressedby a piston; the piston pressure p (being identical

to the uniaxial stress s) and the axial strain o (ratio

of axial displacement to bed height H) are

measured. An important requirement for these

tests is to keep the ratio H /D small enough to

avoid an influence of wall friction, see [2]. Most of

the ceramic pebble bed experiments described in

the following were performed using a cylindricalcontainer with D�/60 mm and bed heights of H :/

10 mm [3�/6].

Fig. 2 shows characteristic UCT results from

[4,7,8]. The curve for the first increase of the

uniaxial stress is influenced by irreversible pebble

displacement forming a denser configuration and

elastic/plastic particle deformation. Decreasing the

uniaxial stress results in a steeper curve becauseonly elastic deformations are dominating. For

subsequent stress increases/decreases, the slopes

are close to that of the first stress decrease.

At ambient temperature, the stress�/strain de-

pendence for the first stress increase is similar for

metatitanate and beryllium pebble beds. However,

during stress decrease, the beryllium curve is stiffer

which indicates that during stress increase theberyllium pebbles were plastically deformed. At

elevated temperatures, the first stress increase

curve becomes softer due to an increased plastic

pebble deformation. Keeping the stress constant at

a given value, the strain increases with time due to

a thermal creep. Decreasing the load at the end of

J. Reimann et al. / Fusion Engineering and Design 61�/62 (2002) 319�/331320

the creep period results in an increased bed

stiffness due to the enlarged contact zones.

From the measured stress(s)/strain(o ) curves,

the uniaxial modulus of deformation E , defined by

E�/s /o ; is correlated by E�/Csm , where C and m

depend on the granular material. These moduli of

deformation are evaluated both for the first

compression and decompression phase. The results

show that for the stress increase period, the

modulus of deformation of most granular materi-

Table 1

Characteristics of investigated granular materials

Type Assoc./comp. Pebble diameter d (mm) Density ratio d (%) Grain size gs (mm) Sint. temp. Ts (8C)

Osi FZK 0.25�/0.6 98 50 �/

TiA CEA 0.8�/1.2 95 1.5�/2.5 1050

Ti-B CEA 0.8�/1.2 83 0.5�/1 950

Ti-D CEA 0.8�/1.2 90 1�/2 1050

Ti-D ltaa CEA 0.8�/1.2 92 50 1050

Ti-E CEA 0.8�/1.2 86 1�/5 1100

Ti-F CEA 0.8�/1.2 90 2�/7 1140

Ti-G CEA 0.8�/1.2 89 1�/5 1100

Ti-H CEA 0.7�/1.0 91 1�/3 1100

Ti-I JAERI 0.85�/1.15 92 10�/50 1200

Ti-J JAERI 2 84 1�/3 1200

Ti-K JAERI 1 83 1�/3 1200

Be Brush W. 2 98 20�/400 �/

Be Brush W. 2/0.1�/0.2 98/96 20�/400 �/

Be NGK 1 98 100�/300 �/

a Long term annealed.

Fig. 1. Examples of investigated pebbles.

Fig. 2. Stress�/strain dependence for metatitanate and beryl-

lium pebble beds.

J. Reimann et al. / Fusion Engineering and Design 61�/62 (2002) 319�/331 321

als is well fitted by m :/0.45, see e.g. [4]. Theconstant C for dense pebble beds (maximum

packing factors 625/g5/64.5%) at ambient tem-

perature was determined to be between 150 and

250, depending on material properties and geome-

trical parameters (diameter distribution, pebble

shape, surface condition). The values are fairly

independent of temperature up to 600 8C, at

higher temperatures, C decreases. A correlationC as a function of temperature was developed [2]

for orthosilicate pebble beds, and metatitanate

pebble beds of type Ti-D. However, it was

recommended in later papers [4,5] to neglect this

temperature influence for E but to take it into

account in the description of thermal creep, see

below.

The stress strain behaviour for lithium oxidediffers significantly: at ambient temperature, the

bed behaves significantly softer (smaller values of

C ) and at temperatures T �/500 8C it looses

completely its compressive strength [3,5].

Pebble bed mechanics in more blanket relevant

geometries were investigated [2] using a test section

with filling through a vertical pipe at one corner,

Fig. 3a. The test section was placed during thevibration assisted filling on different sides. For

filling, one steel plate was replaced by a glass plate

and the filling could be controlled by visualisation.

After filling, the test section was turned to the

horizontal and UCTs were performed. For an

optimised filling technique a homogeneous stress

distribution existed in the bed (visualised by using

a pressure sensitive film at the container bottom)and the same thermomechanical results were

obtained as with conventional UCTs. Without

controlling visually the filling process, always

non-homogeneous stress distributions were ob-

tained, see Fig. 3b. This indicates that filling of

large blanket bed structures is not trivial.

2.2. Thermal creep at elevated temperatures

Thermal creep of pebble beds behaves differ-

ently compared with solid materials because (i)

stresses are not uniform but concentrate in the

particle contact zones, and (ii) contact zones

increase with time. It was shown [9,10] that for a

creep rate of the solid material ocr/dt �/sb , the

pebble bed creep rate and creep strain, respec-

tively, become: ocr/dt �/sb /(1�b )t1/(1�b ), ocr�/

sb /(1�b )t1/(1�b ). The stress dependence of ceramic

materials varies in a wide range, for orthosilicate

pellets, b varied between 1 and 5 [11].

Thermal creep experiments using the UCT set-

up were performed at constant temperature by

first increasing the stress to a given value and then

keeping it constant for time periods of up to 10

days. It proved [5,6] that for granular materials

with grain sizes of gs:/25�/50 mm (example: Ti-D

lta, see Fig. 4) thermal creep strain increased

linearly with creep time t in a log�/log plot if

thermal creep occurring during stress increase was

taken into account by a creep hardening rule. A

stress exponent factor of p�/0.65 was observed

and an exponent n�/0.2, indicating that the stress

exponent for the solid material is larger than 2.

Fig. 3. Experiments with blanket relevant test section: (a) test

section: (b) pressure distribution.

Table 2 contains the thermal creep correlations for

these large grain materials.

The pebbles with small grain sizes (example: Ti-

D with gs�/1�/2 mm, see Fig. 4) showed for the

first 100 min a similar behaviour as the large grain

pebbles but then exhibited significantly larger

creep rates.

Different batches of metatitanate pebbles were

provided by CEA [12] and JAERI [13] in order to

investigate in more detail the effects of grain size,

sintering temperature Ts, and pebble density d

(expressed as percentage of theoretical density).

Fig. 5 (from [4]) shows the largest creep strains for

Ti-B, characterised by the lowest Ts and gs, and a

low pebble density. It was concluded [4] that the

grain size influences creep the most; but this

influence becomes small for gs�/5 mm. Creep

also increases as pebble density decreases. How-

ever, there might be other parameters, not suffi-

ciently considered yet (e.g. impurity contents and

pebble sphericity and surface roughness).

The present results have confirmed that the

correlations from Table 2 are suited to describe

thermal creep of the batches Ti-F, Ti-G, Ti-H, and

For the group of materials characterised by an

increase of creep rate after about 2 h (Ti-B, Ti-D,

Ti-E and Ti-J), the correlation according to Table

2 can be applied up to creep times of about 2 h.

For the following creep regime, again a stress

dependence of ocr�/s0.65 was found; however, the

exponent n increases with temperature; values are

given in [4].

The importance of this second creep regime

decreases if stress relaxation processes in blanket

structures are very fast. The results [8] shown in

Fig. 6 imply that this might be the case: after

reaching the maximum uniaxial stress syy of 8

MPa, the pebble bed volume was kept constant by

manual control and the stress decrease was mea-

sured. For 770 8C, the uniaxial stress drops to :/

25% of its initial value in only 2 h.

At present, creep data for beryllium pebble beds

were restricted to maximum temperatures of :/

480 8C [8,9] which are too low to establish reliable

correlations. Experiments have started with a test

facility [14] designed for a maximum temperature

of 650 8C.

Fig. 4. Thermal creep strains of CEA metatitanate pebble beds

(Ti-D and Ti-D lta).

Table 2

Thermal creep correlations (from [5])

Granular material ocr(1)�/A exp(�/B /T (K)) s (MPa)p

t (s )n

A B p n

FZK�/Li4SiO4 12.12 10 220 0.65 0.2

CEA�/Li2TiO3 (Ti-D lta) 0.67 7576 0.65 0.18

JAERI�/Li2TiO3 (Ti-J) 0.37 6947 0.65 0.19

Fig. 5. Thermal creep strain at large creep times of metatitanate

pebble beds at 750 8C.

2.3. Thermal conductivity of deformed pebble beds

Many data exist on thermal conductivity mea-

surements of ceramic breeder and beryllium pebble

beds for uncompressed beds [15,16] or beds with a

small degree of compression [17�/19]. In [20],

experiments are described where the pebble bed

conductivity k at ambient temperature was inves-

tigated as function of an external load (varied up

to 1.4 MPa), however, without recording bed

strains.

Only recently, conductivity measurements were

presented for strongly deformed breeder and

beryllium pebble beds as a function of temperature

and bed strain and stress, respectively [21,22]. The

experiments were carried out in a UCT set-up with

the pulsed hot wire technique, already used by

[15,16].

For ceramic breeder beds the conductivity

increase with increasing bed deformation is ex-

pected to be small compared with beryllium pebble

beds because of the much smaller difference

between pebble conductivity kp, and surrounding

gas conductivity kg. For orthosilicate pebble beds,

k increases at ambient temperature Ta only by :/

15% for a strain of :/1.8%; for T�/800 8C, the

conductivity increase is :/25% for the maximum

strain of :/4.5%, see Fig. 7 (from [22]). For

metatitanate pebble beds in air at ambient tem-

perature this conductivity increase is more ex-

pressed but this increase becomes smaller with

increasing T (because kp decreases and kg in-

creases). A weak linear dependence between con-

ductivity and strain is observed. For non-

deformed pebble beds in helium the present

measurements confirm

. for orthosilicate beds the correlation of Dalle

Donne et al. [17];

. for metatitanate pebble beds the SBZ-model

[23] with rk2�/0.049 as used by Enoeda [15].

Fig. 7 contains also results for a binary metati-

tanate pebble bed (2 and 0.2 mm) in air atmo-

sphere at ambient temperature. Compared with

the 2 mm monosized pebble bed, the bed con-

ductivity is higher by a factor of :/2. However, for

blanket relevant conditions, this difference will

reduce significantly. According to the SBZ-model

this factor becomes :/1.3 for helium at 600 8C.

The same factor was also measured by [15] for

binary and monosized Al2O3 pebble beds in

helium at 600 8C.

Corresponding experiments with 1 and 2 mm

beryllium pebble beds were performed in helium

atmosphere at TaB/T B/485 8C [21]. Fig. 8 con-

tains the stress�/strain dependence and corre-

sponding values of measured bed conductivities.

The influence of bed deformation on conductivity

is very expressed (the conductivity of beryllium is

larger by a factor of 30 than that of ceramic

materials). The linear dependency between k and o

is clearly seen, see Fig. 9. The increase of

conductivity with strain is most expressed for Ta,

Fig. 6. Stress relaxation of orthosilicate beds. Fig. 7. Thermal conductivity of deformed metatitanate and

orthosilicate pebble beds.

this effect becomes again smaller with increasing T

due to the increasing gas conductivity and decreas-

ing beryllium conductivity.

Comparing the dependence of conductivity on

pressure, the measurements from [21] agree well

with those from [20]; whereas, the correlation

proposed by [19] predicts too high values at high

pressures.The difference between the results from [21,19]

is very expressed in respect to the dependence of

conductivity on volumetric strain: the correlation

presented in [19] predicts values which are by far

too high at small volumetric strains: e.g. for zerostrain (uncompressed bed) the predicted value is

too high by a factor of :/2.

The results presented in this section demonstrate

that for fusion blankets the influence of deforma-

tion on thermal conductivity can be neglected for

ceramic breeder materials but is very significant

for beryllium pebble beds.

2.4. Pebble bed friction

An important quantity for the description of the

mobility of granular materials (see Section 3.1) is

the inner friction angle, determined by TCTs. In

contrast to UCTs the bed can expand duringvertical axial deformation also in the horizontal

direction acting against a constant horizontal

pressure p2; for details, see [7].

Fig. 10 shows that the axial strain becomes

increasingly larger with increasing axial stress due

to the bed deformation in the horizontal direction.

The maximum of the curve is characteristic for the

‘state of perfect plasticity’ and the internal frictionangle is determined at this position defined by

sin a�/(p1�/p2)/(p1�/p2).

In Fig. 11, the slope of the curve is representa-

tive for the friction angle. The inner friction angles

of beryllium and metatitanate pebbles is larger

compared with that of orthosilicate pebbles, be-

Fig. 8. Stress�/strain dependence and conductivities of beryl-

lium pebble beds (helium; T�/25 8C).

Fig. 9. Thermal conductivity of 1 mm beryllium pebbles as a

function of strain.

Fig. 10. Characteristic TCT results.

cause of their more irregular shape and rougher

surfaces.

3. Modelling efforts on evaluation of pebble bedthermomechanics performance

3.1. Finite element code approach

The macroscopic mechanical behaviour of peb-

ble beds can be described by constitutive equations

commonly used in soil mechanics analyses con-

sidering the granular material as a continuum

medium which can undergo reversible elastic

deformations, inelastic volume compactions (con-solidation) and pressure dependent shear failures.

To account for these properties different models

have been developed which are implemented in

structural computational programs; examples are:

the ‘Elastic porous/Drucker�/Prager/Cap’ model

[9] implemented in ABAQUS [24], the modified

Cam�/Clay model (Hujeux model) used with the

CASTEM 2000 code [25], and the NRG approachused with the MARC code [26].

The first model is briefly outlined: the material

is assumed to have an elastic behaviour as long as

the stress state lies within a volume in stress space,

the surface of which is called the yield surface. Fig.

12 shows a representation of this surface in the p �/

t plane, where p is the equivalent pressure stress

and t a deviatoric stress measure (if the depen-

dence on the third deviatoric stress is neglected, t

becomes the von Mises stress). A typical yield

surface is built up by a linear part, the Drucker�/

Prager shear failure surface Fs, and the so called

Cap, Fc. Shear failure occurs when the deviatoric

stress exceed values that depend linearly on the

pressure stress. Such a behaviour is closely related

to Coulomb friction; the parameter b is known as

the material’s friction angle. For cohesionless

particle bed materials for fusion applications the

parameter d vanishes, d�/0. The material is about

yielding by compression (consolidation) if the

stress state lies on Fc. In the region bounded by

the two yield surfaces, pebble beds show elastic

behaviour. The hardening/softening law has been

introduced to model the inelastic response of the

granular material. It causes hardening during

consolidation (when yielding on the Cap) and

softening during volumetric plastic dilatation

(when yielding on the shear failure).

Creep laws for Drucker�/Prager�/Cap materials

are also implemented in ABAQUS. However, as

these laws are not able to reproduce the observed

creep behaviour, a modified compaction creep law

has been developed and implemented in a user

defined subroutine [9].

For using these models, the material parameters

have to be calibrated by UCTs and TCTs; for

details, see [27]. In the following some compar-

isons between computational and experimental

results are shown.

Fig. 11. State of perfect plasticity of different granular

materials.

Fig. 12. Modified Drucker�/Prager/Cap model: representation

in the p �/t plane.

Fig. 13 shows a thermal creep experiment at

800 8C performed in two steps [10]: up to 5600

min, the stress was kept constant at 4.3 MPa. This

period is used for calibration of the creep model.

Then, the stress was increased in about 3 min to a

value of 8.58 MPa and then kept constant again.

There is an excellent agreement between calcula-

tions and experiment for this period.

In other creep experiments, different load ramps

during pressure increase were used, followed by a

period with s�/constant. Again, the calculation

predicted well the pressure increase and subse-

quent creep period [10]. The code was also used to

calculate the stress relaxation shown in Fig. 6. The

calculations agree fairly well for T�/770 8C.

In order to investigate particle flow in shallow

beds [27], the biaxial compression test (BCT) was

used shown in Fig. 14 which consists of a

rectangular cavity with a horizontal cross section

L �/W and a height H . The bed is loaded

vertically by a central force Fv resulting in an

uniaxial stress sv. Using several vertical displace-

ment transmitters, both the mean vertical strain ov

and piston inclination can be measured. One side-

wall consists of a movable piston (displacement sh,

strain oh), preloaded with a force Fh (stress sh).

Fixing the horizontal piston, the test set-up can be

used for UCTs. These UCTs are used to calibrate

the elasticity model and cap hardening model.

Fig. 14 contains also characteristic results for

orthosilicate pebbles and H�/10 mm. The vertical

load was increased up to the maximum value sv:/

5 MPa in a given time period (2 h for the

experiments with sh�/0.058, 0.12 and 0.4 MPa,

and 16 h for sh�/0.075 MPa. Then, the load was

kept constant for 2 h and, finally, the force was

reduced to zero during 15 min. When the vertical

force is increased above a certain value, the

horizontal piston starts to move, depending on

sh. It moves more or less linearly over a wide range

with increasing vertical force.

The important result is that the values oh are

very small: for sh�/0.4 MPa the maximum

horizontal displacement of particles becomes at

the highest vertical load only oh:/10�2 (corre-

sponding to 0.1 mm). Even the maximum displa-

cement of about oh:/4�/10�2 for sh�/0.075 MPa

is not very large considering the very small

horizontal load compared with the maximum

vertical one. The experimental results are com-

pared with ABAQUS calculations using an inter-

nal friction angle of b�/448. The agreement is

satisfactory and deviations are within the range of

Fig. 13. Thermal creep experiments with orthosilicate pebble

Fig. 14. Biaxial experiments: experimental set-up and results.

experimental accuracy. Similar experiments withbed heights of 20 and 30 mm are described in [28].

The Drucker�/Prager/Cap model with the ABA-

QUS code was used by [29] to predict the thermo-

mechanical performance of the ITER breeding

blanket and the NRG approach was used for the

benchmark experiment SCATOLA where a cera-

mic pebble bed constrained by a metallic box was

isothermally heated up.

3.2. Direct simulation approach

The aforementioned approach utilised effective

continuum properties for estimating thermome-

chanical states of packed bed materials. This

transformation, from a complicated discrete sys-

tem to a simpler continuum system, results in the

loss of certain information. To remedy this defi-

ciency and to better understand bed effective

mechanical properties and thermomechanics be-haviour, a 3D discrete numerical model has been

developed at UCLA [30�/32].

In a direct numerical simulation, the elastic

phenomenon of a particle bed is modeled as a

collection of rigid particles interacting via Mind-

lin�/Hertz type contact interactions. Under a

quasi-static condition, the force acting on any

particle leading to the equation of motion reducesto the following equilibrium conditions

Fc�0 (1)

where F is the externally imposed force and the

summation is performed over the contacts of the

particle. If the particle is not at equilibrium, it is

subjected to displacement (DD ) according to the

active force and bed stiffness (k ) in both the

normal (n ) and shear (s) directions at the contact.

Numerically, the contact forces (normal, shear orfriction forces) are decomposed into x , y , z

components based on the unit vector evaluated

from the centers of the two contacted particles.

The incremental displacement of the particle in the

x -direction is derived based on the net active force

along the x -axis according to:

DDx�Fx

kn � ks

kn � ksjX

FxcjBkf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Xc

½Fyc½

��X

½Fzc½

otherwise,

DDx�Fx � kf ½Ft½

Fxc 9 kf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Xc

½Fyc½

��X

½Fzc½

where Ff is the friction force at an interface and is

described using Coulomb’s friction law. The 9/

signs in the above equation should be chosensuch that the friction forces are opposite to the

active force. Similar expressions can be written for

the displacements in the y and z directions.

Recently, the model was used to describe the

experiments of SCATOLA, a cylindrical pebble

bed thermomechanical test assembly [33]. The

ceramic lithium orthosilicate pebbles are enclosed

between two plates that are fixed at their circum-ference. In the experiment, the relative movements

of the top and bottom plates, with respect to

temperature rises, are measured. Numerically, two

iterative loops are set up to calculate a self-

consistent force and displacement relationship.

The first iterative loop searches for an equilibrium

pebble bed configuration with respect to particle

relocation due to a rise in temperature. The seconditerative defines the new container boundary based

on the calculated plate deformation value caused

by the stress exerted on the plate. The thermo-

mechanical behaviour of SCATOLA experiments

was simulated using 26 010 1 mm particles.

Numerical estimations indicate that deforma-

tion depends on boundary conditions. Deforma-

tion is higher under a simply supported boundarycondition as compared with that of the fixed

boundary condition. Comparison of the results

shows that the experimental deformation data falls

within the range of that of the calculated results

with simply supported and with fixed boundary

conditions as shown in Fig. 15 and, that it is much

closer to the predictions based on the fixed

boundary condition [32]. This appears reasonable

since the imposed boundary condition in the

experimental set-up lies somewhat between the

aforementioned two boundaries, while the circum-

ference of the plate was fixed. The benefit of the

discrete numerical simulation is that it provides

detailed information concerning the thermome-

chanical state of the particle bed. Specifically, the

stress profile distribution, the contact force profile,

and its subsequent peak stress location, can be

identified. This provides insight into the weak

points of the design, which designers can take into

consideration. In a previous calculation, about

1.5% of particles were found to be break, when

subjected to a load greater than the crushing load

This direct simulation code has been extended

recently to model the evolution of bed thermal

creep deformation. In the present attempt, the rate

change of creep deformation was modeled at the

particle contact based on Coble diffusion creep

mechanism. The model has been applied to study

strain evolutions of a ceramic breeder pebble bed

subjected to an externally applied stress. When

compared with the experimental results, calcula-

tions based on the Coble creep mechanism predict

smaller strains than that observed in the experi-

ments [34]. This may be due to a significantincrease in the stress magnitude when the applied

moderate stress transmitted to the stress at the

contact, which could then induce different creep

mechanisms (such as power-law creep) and result

in a larger strain rate. The stress evolutions are

also simulated for stress generated from a ther-

mally induced breeder-structure interaction. The

calculation shows that the average stress exertedon the wall drops from an initial value of 18.75

MPa to less than 6.3 MPa at around 1000 min and

to less than 1 MPa at around 2000 min after the

initiation of creep due to the increase of the

contact area caused by creep. This appears to be

desirable since the goal is for stress to be reduced

to a significantly lower value at a relatively short

period of time after the development of creep. Thisshould prevent further creep deformation and an

undesirable sintering formation.

4. Conclusions

Significant progress has been achieved in the last

2 years with respect to both the experimental

pebble bed data base and the modelling develop-ment of pebble bed-structural material thermo-

mechanics interactions.

Concerning the data base, mainly generated by

UCT set-ups, extensive data have been obtained

for ceramic breeder pebble beds including relation-

ships for the modulus of deformation, thermal

creep, thermal conductivity and inner friction

angle. For beryllium pebble beds, data on thermalcreep and deformation dependent conductivity are

still missing for temperatures above 500 8C.

The question remains: How relevant does the

data obtained using UCT methods describe blan-

ket conditions where the stress states may be

different? Experiments involving stress as the out-

come of the material thermomechanics interac-

tions, which resemble prototypical operatingconditions should be considered for comparisons

with numerical calculations.

Concerning the models, the ‘finite element

approaches’ (FEA) should be ready now for high

temperature applications (including thermal creep

and deformation dependent thermal conductiv-

Fig. 15. Results from SCATOLA experiments.

ities) in rather complex geometries. Improvementsare still required in respect to the description of

pressure decrease mechanisms (relevant for inter-

mittent blanket operation). The ‘direct simulation

approach’ (DSA), at present, is not as advanced as

the FEA; here, e.g. more work is still required in

modelling thermal creep for individual pebbles.

The potential of the DSA is in areas where the

continuity model fails, such as describing theinteraction of individual pebbles with walls or in

bed geometries where the ratio of pebble diameter

to bed dimension is no longer negligible, such as

constrained pebble beds in irradiation capsules.

It should be mentioned that the present efforts

on pebble bed thermomechanics focus on non-

irradiated materials. This provides information

necessary for a proper design. However, in orderto predict a complete lifetime performance, experi-

mental and modelling investigations including

irradiation effects are needed.

References

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thermomechanics of solid breeder and be pebble bed materials

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