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Graphite Component Stress Analysis
Barry J Marsden and Derek Tsang
Nuclear Graphite Research Group
School of Mechanical, Aerospace and Civil Engineering
University of Manchester
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Outline
1. Introduction
2. Constitutive equations
3. Numerical methods
4. Examples
5. Advance numerical method
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1. Introduction
The properties of nuclear graphite components are changed by
fast neutron irradiation and radiolytic oxidation. These irradiation induced changes can lead to build up of
significant stresses and deformation in the graphite components.
It is essential that the nuclear graphite components remainsufficiently strong and undistorted.
Hence we need to perform structural integrity assessments, i.e.stress analysis of nuclear graphite.
Apart from the elastic strain and thermal strain, graphiteexperiences additional strains due to fast neutron irradiation.
Irradiation creep and irradiation induced dimensional change are
two most important of these irradiation strains.
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1. Introduction
Finite element method has been used in structural integrity
assessments. Material properties of interest in stress analysis are:
Dimensional change rate
Youngs modulus Coefficient of Thermal Expansion (CTE)
Weight loss
Irradiation creep Irradiated graphite properties are based on the material test
reactor experiments.
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1. Introduction
There is more than one graphite material model in UK.
British Energy has been developed a graphite material model.
NGRG also has been developed a graphite material model.
In this lecture we will discuss how to implement a graphite material
model into a finite element program.
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1. Introduction
ABAQUS is a general finite element program, capable of
performing three-dimensional, time-integrated, non-linear stressanalysis.
ABAQUS finite element program has been chosen for graphitestress analysis.
Usually a user material (UMAT) subroutine is required to modelthe irradiated material properties.
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2. Constitutive equations
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2. Constitutive equations
total e pc sc dc th ith idc= + + + + + + Total strain is defined as
e Elastic strain
pc Primary creep strain
sc
Secondary creep straindc
Dimensional change strainth
Thermal strain
ith Interaction thermal strain due to creepidc
Interaction dimensional change strain due to creep
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2. Constitutive equations
Elastic strain
Elastic strain is related to the stress by means of Hookes law oflinear elasticity, i.e.
The material matrix D is a function of irradiation temperature andirradiation dose.
The incremental equation for Hookes law can be written as
e= D
e e
= + D D %
%
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2. Constitutive equations
Creep strain
0
2
4
6
8
10
12
14
16
0.E+00 2.E+21 4.E+21 6.E+21 8.E+21
Dose, EDND, n/cm2
Elastic
Strain
Units
Pluto 300C (Flux 4E13 n/cm2/s) BR2 300-650C (Flux 3E14 n /cm2/s)
Calder Hall 140-350C (Flux ~1E12 n/cm2/s) UK Creep law
Linear secondary creepNonlinear recoverable
primary creep
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2. Constitutive equations
Creep strain
UK creep law
The primary creep strain, which is also called transient creep, is a
recoverable strain
The secondary creep strain, which is also called steady state
creep, is an irreversible strain
creep pc sc= +
( ) 4 40
4 dpcc
T e e
= D
( )0
dsc cT
= D
and are function of tempeatureT
creep matrix, dosec = =D
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2. Constitutive equations
Creep strain
Primary creep equation can be written as
Hence the incremental equation for primary creep can be written
as
The incremental equation for secondary creep can be written as
( )d
4d
pcpc
c
=
D
( )4pc pcc D % % %
sc
c D % %
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2. Constitutive equations
Thermal strain
The thermal strain is defined as
The mean CTE is defined as
The instantaneous CTE can be found by using
( ) ( )refth
refT TT T
=
( ) ( )( )
1
ref ref
T
iT T Tref
dT T
=
( )20 120 , 1i i i i= + A B A
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2. Constitutive equations
Thermal strain
Differentiating the thermal strain equation, we have
Can be simplified to
Hence the incremental equation for thermal strain can be writtenas
( ) ( ) ( )ref ref th
refT T T T T T T
= +
( ) ( )20 120th
ref iT T T = +
( ) ( )20 120th ref iT T T +
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2. Constitutive equations
Dimensional change strain
The amount of dimensional change is assumed to be a function ofirradiation dose, temperature and weight loss.
The function h has been measured from Material Test Reactor. Different graphite model has different form of
The incremental equation for dimensional change strain can be
written as
( ), ,dcd
h Td
=
dc h =
( ), ,h T
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2. Constitutive equationsInteraction strains
The two interaction strains are assumed to be a function ofirradiation creep.
The interaction thermal strain is a correction in irradiated meanCTE.
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
-4.00 -3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Creep Strain %
Chang
einCTEx10-6K
-1
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2. Constitutive equationsInteraction strains
By using a linear approximation, the CTE correction can be writtenas
The effective creep strain is defined as
The parameter is the slope at the origin of the CTE/creep curve
and is lateral coefficient.
( )20 120ec
=
creep
1
creep
2
creep3
creep
12
creep
13
creep
23
1 0 0 0
1 0 0 0
1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
ec
= +
+ +
0.5 =
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2. Constitutive equationsInteraction strains
Interaction dimensional change strain due to creep
Creep strain modified CTE. Dimensional change appears to be a function of CTE.
Therefore creep strain is expected to modify dimensional change
0
dd
d
idc T
c a
X
=
and are crystallite CTE in and directions respectivelyc a a c
is the shape factor for the graphite crystalliteT
X
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2. Constitutive equationsInteraction strains
is a function of irradiation temperature and dose
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
6.00E-01
7.00E-01
8.00E-01
0 50 100 150 200
Dose n/cm21020 EDND
dXT/d
(strain%p
erun
itdose)
530 EDT
384 EDT
d
dT
X
d
d
idc T
c a
X
%
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3. Numerical methods
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3. Numerical methods
ABAQUS has been chosen for the finite element analysis.
The material model has been implemented into ABAQUS via usermaterial subroutine (UMAT).
The UMAT subroutine has three functions:
1.The subroutine calculates all the different irradiationstrains, thermal strain and elastic strain.
2.The subroutine updates the stresses to their values at theend of the increment from the estimated total strain.
3.The subroutine provides the graphite material J acobianmatrix for the mechanical constitutive model.
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3. Numerical methods
ABAQUS method
Abaqus
UMAT
Abaqus
Estimated total strain
StressForce balance
Update total strain
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3. Numerical methods
The UMAT subroutine needs to calculate all the irradiation strains,
thermal strain and elastic strain. The thermal strain and dimensional change strain can be
calculated explicitly from the formulas.
The creep strain, elastic strain and interaction strains can be foundby using an explicit method or implicit method.
The explicit method is easier to implement but it requires verysmall time step.
Much larger time step can be used in implicit method. However itsnot so easy to program.
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3. Numerical methodsExplicit method
Using a predictor-corrector approach.
In a predictor step all the strains except the interaction strains areconsidered.
Once the creep strains have been determined. The interactionstrains can be found.
Hence in a corrector step all the strains can be updated.
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3. Numerical methodsExplicit method: predictor step
At a current time step i, it is assumed that the average value in anincrement can be approximately calculated by the centraldifference method:
The creep strains can be rewritten as
The total strain equation can be written as
2 2i i
= + = +
% %
( ) ( )2 2 41 2 1 2
e pc
c c c i c i ipc e
+ + =
+ +
D D D D D D D
% % % % % % %
2
4 2
pcsc e
c c c ic i +
= +
D D D D D D D
% % % % % %
%
totale pc sc dc th + + =
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3. Numerical methodsExplicit method: predictor step
Now we have three equations with three unknowns!
The solution of elastic strain is
The solutions for creep strains can be found.
Accordingly both interaction strains can be found.
( )( )
total
14
21 2 4 1 2
2
1 2 2
dc th
c i
pc
c i ie
c c
e
c i
= + + +
+ + + +
D
D I D D D D
D D
%
%
% % % %
% %
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3. Numerical methodsExplicit method: corrector step Update the total strain equation
A new approximation for elastic strain can be obtained.
Hence we have a better approximation for creep strains and interaction strains.
In theory, improved approximations to elastic strain can be obtained by iterating the
corrector step.
In practice, elastic strain converges to the actual approximation given by the
formula rather than to the true solution.
It is more efficient to use a reduction in the step size if improved accuracy is
needed.
totale pc sc dc th idc ith + + =
( ) ( )
total1
2 4 21 2 4
1 2 1 2 2
dc th idc ith
c i
e pcc c c i i e
c i
= + + + + + + +
D
I D D D D D
D D
%
% % % % %
% %
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3. Numerical methods Numerical solutions using explicit method
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3. Numerical methods
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3. Numerical methodsImplicit method
It is desirable to use an implicit time scheme for a set of stiffdifferential equation.
Consider a differential equation in the form
A first order approximation is
A second order approximation is
The first order approximation always gives stable numericalresults, however the solution only gives first order accurate.
The second order approximation gives more accurate numericalresults but it may produce an unstable solution for a strongstiffnessequation.
( ),dy
f y tdt
=
( )1 1 1 1,n n n ny t f f f y t+ + + + = =
( ) ( )1 2 ,n n n n ny t f f f f y t+ = + =
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3. Numerical methodsImplicit method
It is more convenient to rewrite as
First order approximation:
Second order approximation:
Different order of approximation can be used in different strains. Their orders are chosen according to their numerical behaviour.
First order approximation is used in primary creep strain.
Second order approximation is used in secondary creep andinteraction strains.
y
( )1
1 2
n n
y t f f
+
= +1 21, 0 = =
1 2 1 2 = =
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3. Numerical methodsImplicit method
The change in dimensional change strain
The change in thermal strain
The total change in strains
The change in primary creep strain
total 0e pc sc dc th idc ith + + + + + + =
( ) ( )20 120th ref iT T T = +
dc h =
( )( ) ( )
1 1
1 1
1 1
1 2 1 2
4 1 4
4 0
n n
pc e pc
n n n n n n
pc pc e pc
+ +
+ + + +
+ + =
D D
D D D D
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3. Numerical methodsImplicit method
The change in secondary creep strain
The change in interaction thermal strain
The change in interaction dimensional change strain( )( )
11 0
2
n nn n
ith ref T T T+
+ + =
( )1 1 1 1
1 1 2 0n n n n n n n
sc e sc sc sc e + + + +
+ + =D D D D D D
1
1 2
1
0
n n
T Tidc
c a c an n
dX dX
d d
+
+
=
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3. Numerical methodsImplicit method
Five nonlinear equations with five unknowns:
These strain equations can be rewritten as a set of equation
Can be solved by Newtons method! This Newtons solver within UMAT is to determine strains only.
, , , ,pc sc ith idc e
( ) ( ), , , ,e pc sc ith idc= = W X 0 X
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3. Numerical methodsImplicit method
The J acobian matrix J used in Newtons method can be foundanalytically
T T T T
=
+ +
I I I I I
H I 0 0 0
J G 0 I 0 0
0 F F I F F
0 F F F I F
3
2n ref
T T T T
= +
11 4 = + 11 T
c a
dX
d
=
14 pc = H D D1 sc = G D D1 1 1 1n n n n
pc sc ith idc
+ + + + = = = =
F
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3. Numerical methods
Numerical solutions using implicit method
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3. Numerical methods
Once the elastic strain is known, the change in stress can bedetermined:
The UMAT subroutine needs the J acobian matrix of theconstitutive model.
Using approximation for J acobian matrix can increase the number
of Newtons iterations in each time increment.
However it doesnt affect the accuracy of the solution.
As a first approximation, the J acobian matrix of the constitutive
model is assumed to be
e e
= + D D %
%
total e
=
D
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3. Numerical methods
It is a poor approximation.
About 13 iterations are required in each time step.
A second approximation:
Examples on a 3D brick analysis
1
1
total
4
1 4pc sc
+ + +
D D D
Approximation Computational times (s) Iterations
First 71808 (19 hours 56 mins) 774
Second 15049 (4 hours 10 mins) 130
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3. Numerical methods It is possible to reduce the computational time by running the
ABAQUS job on multi-processors computer.
ABAQUS supports Message-Passing Interface (MPI).
By including MPI extension in UMAT, stress analysis can beperformed much quicker.
0
2000
4000
6000
8000
10000
12000
14000
16000
0 2 4 6 8 10 12 14 16
Number of CPU
Com
putationaltime(s)
4 hours 10 mins
31 mins
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4. Examples
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4. Examples
A typical ABAQUS analysis using the UMAT subroutine involvesthe following three steps:
1. Unirradiated start up step: This step is used to model the reactor firststart up. Only elastic strain and thermal strain is considered, thevirgin Youngs modulus and CTE are used.
2. Irradiated history step: This step is normal irradiation step for thenuclear graphite when the reactor is at power. All the seven strainsare considered.
3. Irradiated shutdown/start-up step: This step represents the reactor
during a shutdown period or start-up period. Only elastic strain andthermal strain are considered. The temperature within the reactorreduces to the room temperature. The irradiated Youngs modulusand CTE are used in this step.
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4. Examples
Two sets of data are required by the UMAT to perform materialmodelling of irradiated graphite.
They are irradiated material data and field variables.
All the irradiated material data are read from a UMAT input file
The field variables are read from the ABAQUS input file. The field variables are used to define the time-dependent
properties required by the UMAT.
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4. Examples
The field variables must be defined at every node on eachgraphite element in the model, and they must cover the total timeof the analysis.
Four field variables are required by the UMAT. They are operatingtemperature, irradiation dose, irradiation temperature and weight
loss. For the majority of analysis the irradiation temperature and
operating temperature will be identical.
However, at shutdown the operating temperature will decrease tothe shutdown temperature but the irradiation temperature willremain unchanged, since it is used to define the materialproperties.
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4. Examples
A three-dimensional brick analysis
Height = 410mm
fuel brick height = 820mm
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4. Examples
A three-dimensional fuel brick analysis30 full power year
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4. Examples
A three-dimensional brick analysisUnirradiated
irradiated
About 3% decrease in heightand bore radius
top (z=0)
middle (z=410)
As the ir radiation induced shrinkage is ini tially
highest in the high dose region near the bore, the
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-10
-8
-6
-4
-2
0
2
4
6
-5 0 5 10 15 20 25 30 35
fpy
Hoopstress(MPa)
4. Examples
A three-dimensional brick analysis
Compressive stress is
caused by the
temperature variation.
Quickly relieved by the
primary creep
g g g ,
stress changes sign with increasing dose and
tensile stress is developed
Compressive stress develops again
at the bore due to expansion of the
graphite after turnaround
Shut down stress is higher than theoperating stress. It is because the init ial
thermal stress is crept out during operation,
but return in the opposite sense at shut-
down and add to the stress
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-8
-6
-4
-2
0
2
4
6
8
10
12
-5 0 5 10 15 20 25 30 35
fpy
Hoopstress(MPa)
4. Examples
A three-dimensional brick analysis
The initial keyway stress is
tensile at the keyway.
The stress profiles at the keyway are
in opposite sign to those in the bore
The stress changes sign and
compressive stress developed
The stress changes sign again due todimensional change turn around
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4. Examples
A three-dimensional brick analysis
Strains history in hoop direction at z=410
Bore location Keyway location
The shrinkage is biggerat the bore than at the
keyway.
Turn around
Tensile stress givestensile secondary
creep strain
IDC and SC are
in opposite signs
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4. Examples
A three-dimensional brick analysis
Strains history in hoop direction at z=410
Bore location Keyway locationBigger change in
thermal strainhistory
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4. Examples
A three-dimensional brick analysis
Strains in hoop direction along a path from the inner bore to the outside of the brick
Before turnaround (15fpy) After turnaround (30fpy)
Biggest DC strainoccurs at the bore
Biggest DC strainoccurs within the brick
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4. Examples
A three-dimensional brick analysis
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4. ExamplesFull or reduced integration?
Analysis can be performed much quicker when using reduceintegration.
However numerical difficult may occur! Example:
Full height 3D fuel brick analysis Results along the bore height
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4. ExamplesFull or reduced integration?
Analysis can be performed much quicker when using reduceintegration.
However numerical difficult may occur! Example:
Full height 3D fuel brick analysis Results along the bore height
full integration reduced integration
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4. Examples
Full or reduced integration?
The combination of poor mesh and the use of reduced integrationcause the inaccurate results.
full integration reduced integration
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5. Advance numerical method
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5. Advance numerical method
Finite element analyses can provide detail displacement andstress solution for each individual component in a reactor core.
The effects of irradiation, thermal expansion and radiolyticoxidation are included in the analyses.
However conventional finite element method is not suitable for
whole core modelling.
Very computational demanding!
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5. Advance numerical method
A superelement technique has been developed to model thewhole core.
A superelement can be reused for representing identical structure.
Significant effort can be saved in the procedure of finite elementdata input.
The computation efficiency can be improved considerably due tolarge reduction in the degrees of freedom.
Superelements have been implemented into ABAQUS via User
Element Subroutine (UEL)
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5. Advance numerical method
Finite element model has 15360 nodes and 4768 elements. Superelement model has 1465 nodes and 8 superelements. Each superelement has 217 nodes
The superelement model is about 40% faster than the FE model
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5. Advance numerical method
5. Advance numerical method Crack can be easily modelled with superelement by combining
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Crack can be easily modelled with superelement by combining
superelement and conventional finite element Pre-existing crack analysis Contact elements have been used for crack faces interactions.
5 Ad i l th d
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5. Advance numerical method
Four core arrays have been analysed by using Superelementmethod and compared with finite element method.
The array sizes are 2x2 3x3 4x4
5x5 ABAQUS gap elements have been used for interactions between
each brick components.
5 Ad i l th d
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Array size
Superelement model Finite element model
No. of elements No. of nodes No. of elements No. of nodes
2x2 144 6785 20084 64606
3x3 432 16433 46348 149077
4x4 876 30409 83564 2687775x5 1476 48713 131732 423705
5. Advance numerical method
5 Ad i l th d
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5. Advance numerical method
Array sizes
1 CPU
Superelement Finite element
2x2 1490s 3063s
3x3 3335s 7386s
4x4 6307s 14179s5x5 10525s 23051s
15 CPUs
Superelement Finite element
539s 740s
754s 1847s
1326s 3424s2165s 7746s
5 Advance numerical method
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5. Advance numerical method
Superelement model is more efficient than the full finite elementmodel.
Less computer memory
Faster computational time