graphite course

7/30/2019 Graphite Course

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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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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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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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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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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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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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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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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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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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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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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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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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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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ABAQUS method

Abaqus

UMAT

Abaqus

Estimated total strain

StressForce balance

Update total strain


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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 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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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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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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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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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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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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Numerical solutions using implicit method


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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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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


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


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


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


Strains history in hoop direction at z=410

Bore location Keyway locationBigger change in

thermal strainhistory


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4. Examples


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



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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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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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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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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 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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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 Ad i l th d


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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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Superelement model is more efficient than the full finite elementmodel.

Less computer memory

Faster computational time

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