Finite Element Analysis of Radiation Damage Stresses in the Graphite of Matrix Fuel Elements EUR 4472

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EUR 4472 e

FINITE-ELEMENT ANALYSIS OF RADIATION DAMAGE STRES-SES IN THE GRAPHITE OF MATRIX FUEL ELEMENTS by J. DONEA

Commission of the European Communities

Joint Nuclear Research Center — Ispra Establishment (Italy) Physical Chemistry Department

Luxembourg, May 1970 — 18 Pages — 1 Figure — FB 40,—

A finite-element procedure is presented for the determination of the radiation damage stresses in the graphite of matrix fuel elements. The assumption of a generalized plane strain situation is made. The graphite is assumed to be transversely isotropic. Irradiation induced creep as well as radiation damage and thermal strains are taken into account by using an incremental type of analysis.

A computer program based on the developments presented in this paper is being prepared.

EUR 4472 e

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(5)

EUR 4472e

COMMISSION OF THE EUROPEAN COMMUNITIES

FINITE-ELEMENT ANALYSIS OF

RADIATION-DAMAGE STRESSES IN THE

GRAPHITE OF MATRIX FUEL ELEMENTS

by

J. Donea

1970

(6)

ABSTRACT

KEYWORDS

POWER

REACTOR CORE

GAS COOLED REACTORS PRESSURE

COOLANT LOOPS HELIUM

FUELS

(7)

1. Introduction *)

It has been shown that the impregnation of graphite with suitable molten metals leads to an impervious material with excellent neutronio characteristics (ref. J_ 1_/ ).

A new concept of matrix fuel element for thermal reaotors has been developed based on this material. The cross-seotion of such an element is shown for illustrative purposes in figure 1 with fuel pellets and cooling channels arranged in a sauare pattern.

The fabrication of the fuel element takes plaoe durin,? the impregnation process of the graphite /~2_7. The fuel and the graphite oanning are thus brought into contaot across a thin layer of impregnating metal. The purpose of the present paper is to describe a method of analytic for the determination of the radiation-damage stresses in the graphite of such matrix fuel elements.

Steady and transient loads are considered including the interaction

between the fuel and graphite matrix (swelling and fission gas pressure), the ooolant pressure, the axial load as well as radiation-damage and thermal strains.

The graphite is assumed to be transversely-isotropic and irradiation induced oreep is acoounted for. The assumption of a generalized plane strain is made, i.e. the total axial strain is a constant which in general is non-zero.

The finite-element method is used to solve numerically the variational problems associated with the determination of temperatures and stresses in the graphite. The inclusion of non-linear strains in the finite-ele-ment method is handled by using an increfinite-ele-mental type of procedure, i.e.

(8)

the life of the fuel element is divided into suitable intervals.

The method presented in this paper is quite general and is independent

of the type of creep law used.

2. Assumptions.

A oertain number of assumptions are made in order to bring the problem

of predicting the stresses in the graphite matrix into a tractable range.

2.1 Repetitive stress field.

The fuel element is regarded as the assembly of basio cells and it is

assumed that the stress field is repetitive from oell to oell, except

near the edge of the element. Henoe, for the determination of the

repetitive stress field, the problem reduoes to the analysis of a single

oell.

2.2 Generalized plane strain.

The fuel element can be considered as being very long with respect to

its transverse dimensions. A condition of generalized plane strain oan thus be assumed, i.e. the total axial strain is a oonstant which in

general is non zero. The solutions obtained are thus valid over the

central region of the element.

?..λ Trannvprne isotropy of graphite.

The axial direction of the element is supposed to correspond with the

extrusion direction of graphite. It is assumed that a rotational symmetry

(9)

- 5

element axis. This assumption combined with that of plane strain reduces to four the independent elastic constants that are neoessary

to define the stress-strain relations,

2.4 Interaction between fuel and graphite matrix.

As already mentioned, the fuel and the graphite canning are brought into oontaot aoross a thin layer of impregnating metal.

Sinoe Young's modulus of both possible fuels (UO and UO ) is very large as oompared to the graphite modulus, it will be assumed that the fuel expands freely.

At the start of the life of the element, radial displacements due to differential thermal expansion are thus imposed along the fuel-graphite interfaoe. During the life of the element, these boundary conditions have to be adjusted to take into aocount the irradiation effeots on

the fuel (swelling or fission gas release, or both).

3. Method of analysis.

The life of the fuel element is divided into suitable intervals, At the start of the first interval the temperature distribution in the element is oaloulated. To simplify the analysis, the thermal con-ductivity of graphite is taken as independent of temperature.

(10)

- 6

During the interval, damage to the graphite oauses a decrease of its thermal conductivity and changes of dimensions. The next step in the calculation is thus the derivation of~the thermal and radiation-damage

strains at the ond of the interval. The boundary conditions between fuel and graphite matrix are then adjusted to take into account the

irradiation effects on the fuel.

The creep strain increments during the interval are evaluated by an iterative proooss. A. first estimate of the stresses at the end of the interval is obtained using the oreep strain increments of the pro ceeding interval (zero for the first step). A first approximation of the creep strain increments is then obtained by assuming that the

stresses remain constant, having a mean value between the known stresses at the start of the interval and the first estimate of the final stresses. A second estimate of the final stresses can then be obtained and the

prooess is continued until adequate convergenoe is attained.

The same steps in the calculation are repeated until either the final neutron dose is reached or until the stress distribution does not change anymore, i.e., a steady-state condition is established.

4. Expressions for strains and stresses.

4.1 Total strains.

Suppose the z-coordinate corresponds to the axial direotion of the fuel element and let u,v,w be the disnlaoement components along 0 . 0 and 0 .

χ y ζ

(11)

and they are defined in terms of displacements by wellknown relations«

Ox.

_{¡ S}

(1)

It is assumed that during a time interval the total change in strain can be separated into elastio and nonlinear parts

[U]

.

[hi]

Λ [&Ì]

(2)

where [ Δ ε ] = ( Δ £« ,, Δ t3 , Δ ^ , Δ £2) (3)

4·2 Stressstrain relations.'

In view of the assumption of transverse iRotropy the inorements of the inplane stresses are related to the ohanges of the elastio strains by

tA<r]

- U * ] C Δ^ 3

(4)

where _Τ

(Aer] = ( Δ ^ ,

ACj ,

Δτ„ )

[]**

-Ο + » Ί ) (Ί-ο, - ζ κ ^ ;

(5)

lvC»-KV*) /n^,^hVÍ) O MVt(H\>,)

° J

(6) and Ά. = E -i j £t

In the elasticity matrix (6) the constants E, V^ are associated with

(12)

8

the behaviour parallel to that direotion.

Substituting eq. (2) into eq. (4) yields the relation between the

incremental stress, total strain and non-linear strain.

Uri

-

U*3 (uh -{Δε}} W

The expression for the axial stress inorement is given by Hooka's lav

The relations (7) and (8) for the stress increments involve the ohange

of the total axial strain. This last oan be eliminated by equating the

total load inorement on a transverse seotion of the element to the

ohange of the applied axial load Δ Pz during the interval

A f

z

+ jf Δ<τ

2

A» Aj « 0 (9)

S «ft «tv

4.3 Non-linear strains.

The non-linear part of the s t r a i n inorement i s due to thermal d i l a t a t i o n , radiation damage and oreep

[ Δ ΐ ΐ - { * Α τ ] f [Ltì * [Li] (10)

4.3.I Thermal strains.

A finite-element prooedure is used to derive the two-dimensional

temp-erature distribution in the fuel and in the graphite matrix. The

con-tinuous body is replaoed by a system of disorete triangular elements

(13)

9

-with the oorner point temperatures as parameters /~3__7· Boundary values of temperature and heat flux, internal heat generation and heat-transfer coefficients may be a funotion of position.

The nodal values of the temperature are obtained by minimizing the functional associated with Poissons's equation. The thermal

strain veotor is then constructed for each finite element by multiply-ing the appropriate coeffioient of thermal expansion by the arithmetio mean of the three nodal temperatures.

4·3·2 Radiation-damage strains.

Irradiation induced dimensional ohanges in graphite are funotions of both the neutron dose and the irradiation temperature. Sinoe the

thiokness of graphite oanning is small, the assumption of a uniform Wigner strains distribution can be made. Nevertheless, any expression

giving the Wigner strains as a function of dose and temperature oould easily be incorporated in the present analysis.

4.3.3 Oreep strains.

The transient oreep of graphite is usually negleoted and the steady oreep rate assumed to be independent of the neutron dose and direotly proportional to the applied stress.

The multiaxial creep laws for this particular oase are readily obtained by an analogy to linear elastioity.

(14)

10

It has been shown £^J to bo fruitful to introduce the speed of

energy dissipation during the creep process and to consider this speed as a single valued function of the state of stress.

This oan be written as

M

T

Í¿

C

3 - jV) ..yo

₍₁₁₎

where \ic\ is the creep strain rate veotor, S$\ the stress vector

and + (ff) the speed of energy dissipation function.

The scalar funotion JfV) can only define a veotor by means of its partial derivatives with respeot to the components of the stress veotor, i.e.

\CC\ m.\ -1 { f (12)

1 J 1 -bici 3

Energy dissipation functions can be obtained from monoaxial tensile experiments by defining the creep rate for a given stress level. Por the steady oreep of graphite one has

C - Bc*,;

(-V.ÍTÍ-O) (ΐ3ββ)

-Ν -

'AGÌ (I3.b)

To transform this experimental result into a multiaxial state of stress, an equivalent stress is introduced, acoording to von Mises oriterion

f* * °* + S + ^2 - ^ » S -CVJCJ - C * » ^ t Ua j (13.O)

For the monoaxial experiment

(15)

11

heno e

i 1$) Β C £χ + Ç3 v Vi C* Cj _ ς-j (Tj .. (TÍ Cx ·* 3Tx3) . .

The individual components of the oreep rate can then be obtained

from eq. (12). The result for the steady oreep of graphite is

5· Finiteelement formulation of the stress problem.

In the present analysis the strain and stress distributions are derived

from the oaloulus of variations by minimizing the potential energy of

the loaded structure.

The finiteelement technique known as the displacement method is used

to solve numerically the variational problem associated with the

prinoiple of minimum potential energy. For a complete description of

the method the reader is referred to Zienkiewicz ■ s basic text J_ 3_/·

The continuous body is divided into triangular elements interconnected

at their corner points. These elements have the advantage of being able

to fit any boundaries and, also, permit the use of a graded mesh.

5.1 Characteristics of a triangular finite element.

Within a typical triangular element (fig. 2) with nodes i,j,m numbered

in anticlockwise order the inpl^ne components u and ν of the dis

placement vector are expressed by linear polynominals with the dis

(16)

12

lvi..« l ' L · » " » i '

f c J l

J

(15.a)

where

(15.1))

( . . . c y o l i o o r d e r i , j , m )

C¿ s X » *¿

Κ' β area .{ fcrUnjW ¿¿'** ■

The c o n t i n u i t y r e q u i r e m e n t s on t h e displacements a l o n g t h e edges of

a d j a o e n t elements a r e thus a u t o m a t i c a l l y s a t i s f i e d .

The i n p l a n e components of t h e t o t a l s t r a i n inorement a r e d e f i n e d i n

terms of t h e ohange i n d i s p l a c e m e n t s by e q . ( 1 ) . I n view of eq. (I5«a·)

t h i s oan be w r i t t e n a s

t .T

( Δε« , juj , Δ ^ ) .

Lai

[W]

(16)

w i t h

C B l *

-1

Iff-te o Ci o Ci

\>¡ o

0

Η

o c.

o

C MA

0 %

The t o t a l a x i a l s t r a i n inorement i s given by e q . (9)» i . e .

h

Γ&ιΖλ^-^")]*-^**

- \

(17)

wkere

(17)

13

Combining eqs. (7) and (16) the inorements of the inplane stresses

oan be expressed as

τ Τ

Uc.MTj.Zlt.j) = D O t U ^ Ü * ] +D*(A£it)/ltifc/0) [>*] I At J

where the matrix ƒ D_/ oonsists of the first three columns of the

matrix J_ D J given in (5) and where

(18)

The axial stress inorement is

L\^2

- E

fc

U c | - Δε?) - ^ (.ACx +

Ü ^ J

(19)

The next step in the finiteelement analysis is to define element

nodal foroes whioh are statically equivalent to the inplane boundary

stresses.

Using the theorem of virtual work it is easy to show that these

foroes are

"Τ" Τ

Í A F ' i

= t K

e

H k U « } + U Í î>**A

e

(ûi

2fc

lAcî.oj L&3Ci,*]A

c

lûiJ

(20)

with

U f f

* C ûF

x

. , AF

b¿

, ÄF^ , Λ ty , AF.« , & F ^ )

L K» J β À^.U\\tiS 'Knitr·:*, of tkt t*¿Av>.t*l«A. «leluful.

(18)

14

-5.2 Overall equilibrium equations.

The structure is considered to be loaded inorementally by external

foroes ίΑ.ΛΛ applied at the nodes.

The equilibrium conditions of a typical node i are established by

equating eaoh component of {AR^to the sum of the component foroes

of type (20) contributed by the elements meeting at node i.

Provided the non-linear strains and the total axial strain oa.n be

considered to be known, the resulting linear equilibrium equations

U M -

hûF

t

]

₍₂₁₎

oontain the nodal displacements as unknowns.

Once the nodal displacements have been obtained, the element strains

and stresses are readily oomputed using equations (16) to (19).

5.3 Solution teohnique.

The life of the fuel element is divided into suitable intervals and in eaoh interval the total inorement of strain is divided into its

elastio, thermal, radiation-damage and oreep components. If the

inore-ment of non-linear strains during an interval oan be considered to be known, the ohange of stress oan be found as an ordinary elastioity

problem in presenoe of initial strains. The solution teohnique is thus

as follows t

1) At the start of the life of the fuel element suitable boundary

(19)

15

-with the thermal strains considered as initial strains and -with the total axial strain oonstrained to satisfy eq. (17)«

2) The inorements of the nodal forces required to balanoe the non-linear strains that developed during a time interval are then oomputed. For this purpose the creep strain ohanges are evaluated from eq. (Ah ) by the iterative prooess desoribed in seotion 2 and are added to the ohanges of the thermal and radiation-damage

strains.

3) After adjustment of the boundary conditions, the ohange in nodal point displacements are evaluated from eq. (21) with the inorement of the total axial strain oomputed from eq. (17)·

The total ohange in strain is determined by eq. (1.6) and the in-cremental stresses are obtained from eqs. (18) and (19)·

4) Steps 2 and 3 are repeated for the subsequent intervals until either the maximum neutron dose has been reached or until the stress

distribution does not change anymore, i.e. a steady-state oondition is established.

6. Conclusions.

A method has been presented for predicting the radiation-damage stresses in the graphite of matrix fuel elements.

The proposed method of solution is based on the analysis of generalized plane strain situations by a finite-element procedure. The use of an incremental type of approach makes it easy to incorporate in the analysis arbitrary radiation-damage and oreep laws.

(20)

16

References

^/~1_7 Η. Burg, F. Lanza, G. Marón go t Obtention d'un g r a p h i t e

impermeable par imprégnation de métaux fondus. EUR.2988 f,

f¡ 2_J J . Donea, F. Lanzas I n f l u e n c e du gonflement de l'UC sur l e oomportfìmont mécanique d'une gaine en g r a p h i t e imprégné.

Communication 1.3 1928 (1968) (Not a v a i l a b l e ) .

]_ 3_7 O.C. Zienkiewioz. The f i n i t e element method i n s t r u c t u r a l and oontinuum mechanics. Mc GrawHill ( 1 9 6 8 ) .

(21)

17

I d e a l i z e d

FIG. 1

ero ss - sect ion of a matrix fuel element

(22)

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Finite Element Analysis of Radiation Damage Stresses in the Graphite of Matrix Fuel Elements EUR 4472

Ill

'Aw'T

f a l l i i

tå

m

peni

mm

§

COMMISSION OF THE EUROPEAN COMMUNITIES

Í E S

WÍ

wmßmm

msvsammm

ηκ. *

| i [ : FINITE­ELEMENT ANALYSIS OF

« g

RADIATION­DAMAGE STRESSES IN THE

ρ

'il

ms

ms

i»

LEGAL NOTICE

'}M¿

mi

ftSISI

3.-!Ë*illill

måa&im

ü¡

It

EUR 4472e

COMMISSION OF THE EUROPEAN COMMUNITIES

FINITE-ELEMENT ANALYSIS OF

RADIATION-DAMAGE STRESSES IN THE

GRAPHITE OF MATRIX FUEL ELEMENTS

by

J. Donea

- 6

¡ S

[U]

.

[hi]

Λ [&Ì]

(2)

tA­<r]

- U * ] C Δ^ 3

(Aer] = ( Δ ^ ,

ACj ,

Δτ„ )

[**]

° J

Uri

-

U*3 (uh -{Δε}} W

A f

+ jf Δ<τ

A» Aj « 0 (9)

M

Í¿

3 - jV) ..yo

C - Bc*,;

-Ν -

'AGÌ (I3.b)

lvi..« l ' L · » " » i '

f c J l

J

(15.a)

( Δε« , juj , Δ ^ ) .

Lai

[W]

C B l *

Η

h

Γ&ιΖλ**^-*^"*)]*-^

The axial stress inorement is

- E

U c | - Δε?) - ^ (.ACx +

Ü ^ J

(19)

| i [ : FINITEELEMENT ANALYSIS OF

RADIATIONDAMAGE STRESSES IN THE

_{¡ S}

tA<r]

[]**

Γ&ιΖλ^-^")]*-^**

lAcî.oj L&3Ci,*]A

tø!'

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