1

Application of Discrete Element Method to Study Mechanical Behaviors of Ceramic

Breeder Pebble Beds

Zhiyong

An, Alice Ying, and Mohamed AbdouUCLA

Presented at CBBI-14

Petten, The NetherlandsSeptember 6-8, 2006

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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?

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Thermo-mechanical Behaviors of Breeder Pebble Bed Systems

Finite Element Program (MSC.MARC)

Discrete Element Model

Design Guideline and Evaluation (ITER TBMs)

Experimental Database

(FZK, JAERI, CEA,UCLA)

Thermo-physical and mechanical properties constitutive equations

Single/multiple effect experiments(Bed deformation and creep effect)

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

ANSYS to replace MARC?

4

DEM Simulation relies on Contact Model

•

Parameters

Direct parameters:Contact force (F), Overlapping distance ( ).

Indirect parameters:Stress ( ) and Strain ( ).σ

δ

ε

Material parameters: (mechanical, geometric & boundary)Young’s Modulus (E); Poisson ratio (v);Pebble size (R) and Creep model (n, c), etc.

3/23/1

2*

2

169 FERR

a⋅⎟

⎠⎞

⎜⎝⎛==δ

2

22

1

21

*

111EEEνν −

+−

=21

111RRR

+=where and

•

Hertz contact theory (Elastic state)δ

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Derivation of Contact Model for Creep Deformation

ncσε =&

•

Material creep model for solid material observing a constant stress 1=n

1>nfor diffusion creep;

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

F

a

R

A typical contact between particles

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Creep Contact Models

α,β

is related to the pebble bed packing.

2) Equivalent volume model1) Equivalent stress model (Buler)

In both models, the coefficients need to be calibrated by experiments or FEA simulation.

tsyay n Δ⋅=+ 20 )(

tay a Δ⋅= 0ε& 10

−⋅⋅= ςπ

aFcs n

n

where and

•

Contact neck size

•

Equivalent overlap change

nea cσεςε == &&

tHc nec Δ⋅⋅= 0σδ

00 aH ξ=} Basic

equations

][1~),(1~),(~ 2 Ptpd

tF βεε

εεε

εσ

, p and P

are the local stress, equivalent contact stress and equivalent overall stressσ

)1(11

)1(1 −−−− Δ= ββαδ nnn

c tP (4.5)

yc ⋅= ςξδ (4.22)

Ho

is a virtual parameter and related to the contact neck size. ζ

and ξare two unknowns.

( )ce δδδ +=

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Mechanical Behaviors (DEM simulation)

•

Uniaxial compression & Isothermal heating

Bed temperature (oC)

Wal

lpre

ssur

e(M

Pa)

200 400 600 8000

20

40

60

80

Axial Strain (%)

Axi

alLo

adin

g(M

Pa)

0 0.5 1 1.5

2

3

4

5

6

7

Loadingprocess Unloading

process

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 1.0.)

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.

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DEM Model Verification

Conclusion:(1) DEM program has good capabilities to simulate the mechanical behaviors of

granular materials.(2) 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.

•

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, 61-62 (2002)

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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.

Pressure = 1.9MpaPressure = 1.5MPa

Pressure = 1.4MpaPressure = 1.0MPa

Example critical particle chain evolutions during compaction

Figure: Particles having a contact force greater than 10 N are identified and its neighboring connecting particles with same magnitude or more contact forces are linked to one same pebble chain.

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Magnitude of contact forces

Dis

tribu

tion

prob

abili

ty

100 101 10210-4

10-3

10-2

10-1

Initial PackingLoad = 2MPaLoad = 4MPaLoad = 6MPaLoad = 8MPaLoad = 10MPa

(Compared with average contact froce at initial packing )

Loading pressure (MPa)Ave

rage

cont

actf

orce

s(N

)

0 2 4 6 8 100

10

20

30

DEM results

y = 3.413 x

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.

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.

Internal/External (boundary condition) Connection

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Studies on Contact Force Change

External loading (MPa)

Incr

easi

ngm

agni

tude

0 2 4 6 8 100

2

4

6

8< fc >< fc

max >Particle AParticle BParticle C

Serial number

Cha

nge

times

0 50 100 1501

1.2

1.4

1.6

1.8

2

{ fcmax | fc

max > 4 < fc >}Particle group :

Magnification of maximal contact forces during one interval of loading increase.

Change in contact forces during an increase in external loading

•

Average value •

Special group

Conclusion: The change of contact forces due to the external loading is an stochastic process. The average change is linear to the external

loading.

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Stress Distribution MapStress Map

1fr

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.

Force Map

∑=

=CN

jiij rfV 1

1α

αασ

cNfr

2fr

3fr

αfr

1fr

...

1rr

2rrcNrr

3rr...

αrrA

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DEM model for pebble bed Poisson ratio derivation

P ~ 0 -10MPa Model description:Applying uniform loading (P) on a

rectangular pebble bed from top-side. Pebble size is normally distributed in 0.5~1.5mm. The contact model includes normal and shear forces. The loading can be considered as a static process.

Parameters:Particle number: 1,000Young’s modulus of pebble: 101GPaPoisson ratio of pebble: 0.24Young’s modulus of wall: 206GPaPoisson ratio of wall: 0.3

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Evolution of Stress Maps

Initial State(P = 0.5MPa)

* The color of pebbles stands for same value of stress magnitude.

Midst State(P = 1.25MPa)

Final State(P = 2.5MPa)

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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.

Poi

sson

Rat

io

0

0.5

1

1.5

2

Poisson ratio

Loading steps

Ave

rage

stre

sses

onw

all(

MP

a)

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3σxσy

Numerical value of pebble bed Poisson ratio is needed to address

thermomechanics

behavior along the longitudinal direction

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1

1

FEA Analysis of ITER TBM ---

Coupled thermal & pebble bed mechanics

analysis

Stress profile

Numerical data:~ 770oC (max. T in Breeder); ~ 540oC (max. T in Beryllium)~ < 2.0MPa ( max. σv

in Breeder); ~ 50MPa ( max. σv in Beryllium)

Temperature profile

A

B

CA`

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?

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Evolution of Max. Bed Temp

The figure shows the evolution of maximal temperatures inside the pebble beds. 1)

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;

2)

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 (350oC).

Time (s)

ΔT(o C

)

0 1000 2000 3000 4000 5000

400

500

600

700

800

Tmax in solid breeder pebble bedTmax in Beryllium pebble bed

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Temperature evolutions at different locations

A A’ A”Time (s)

Tem

pera

ture

(o C)

0 1000 2000 3000 4000 5000

400

500

600

700

800

T at coolant/breeder interfaceTmax in solid breeder pebble bed

Time (s)

ΔT(o C

)

0 1000 2000 3000 4000 50000

50

100

150

200

250

300

350

Time (s)

Ttem

pera

ture

(o C)

0 1000 2000 3000 4000 5000350

375

400

425

450

On breeder sideOn coolant structural side

AA’

A’A”

Time (s)

Con

tact

Forc

e(N

)0 1000 2000 3000 4000 50000

5

10

15

Contact Force @ A’

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Stress Distribution & Evolution

Distance from front wall (mm)

Von

Mis

esS

tress

es(M

Pa)

100 200 300 4000

0.4

0.8

1.2

1.6

1st Cycle2nd Cycle3rd Cycle4th Cycle5th Cycleridge

plain

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.

1) 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;

2) The plain part

covers nearly 3/4 parts of the pebble beds and the Von Mises

stress decreased after every cycle.

500

550

600

650

700

750

Temperature

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Equivalent Strain (%)

Equ

ival

entS

tress

(MP

a)

0.2 0.4 0.6 0.8

0.4

0.8

1.2

1.6

Point A'

Equivalent Strain (%)

Equ

ival

entS

tress

(MP

a)

0.2 0.4 0.6 0.8

0.4

0.8

1.2

1.6

Point C

Stress-Strain Behaviors

Equivalent Strain (%)

Equ

ival

entS

tress

(MP

a)

0.2 0.4 0.6 0.8

0.4

0.8

1.2

1.6Point A

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.)

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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.