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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?
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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.