Cladding-strength analysis under the combined effect of creep and plasticity in fast-reactor environments

Paper C2/1 * NUCLEAR ENGINEERING AND DESIGN 18 ( 1972) 53-68 NORTH-HOLLAND PUBLISHING COMPANY

First Inte~

HB3U Ill Ilbl CLADDING-STRENGTH

ANALYSIS

~,i,.

UNDER THE COMBINED EFFECT OF CREEP AND PLASTICITY IN FAST-REACTOR

ENVIRONMENTS

M. GUYETTE

Belgonucl6aire, Brussels, Belgium Received 17 August 1971

A description is made of the computer program CRASH which allows the calculation of three-dimensional stresses and strains in fuel-cladding tubes under creep and/or plasticity. Input data axe easily introduced in the program by means of a quite flexible method which enables a realistic

representation of the complex loading conditions of the claddings. In particular, power and temperature cycling can be readily taken into account. Tentative cladding-strength criteria based on the cumulative damage concept axe then given. Typical results of calculations relative to the ratchetting of a can under combined creep and plasticity are also given. These results show, in particular, the marked influence of the steel swelling phenomenon. Their analysis points quite clearly to the present lack of reliable data on the properties of materials irradiated in fast-reactor environments.

1. Introduction The design of fuel pins for LMFBR's requires a thorough analysis of the clad stresses and strains. These have to be determined not only at the beginning of life but also during all the fuel irradiation lifetime. The fuel pin cladding is submitted to very complex loading conditions as the irradiation proceeds. A part of this loading is due to the inner gas pressure resulting from fission gas release. Moreover, an additional inner pressure is exerted by the fuel if the expansion of the latter is restrained by the cladding. The mechanical study of the cladding behaviour has therefore to take into account the behaviour of the fuel itself. For this reason, computer programs are being developed for years by Belgonucl6aire to predict the thermal and mechanical behaviour of the fuel and the clad under irradiation. In a first stage, two separate programs were developed: - a fuel behaviour program, called COMETHE II, in which the clad is simply assumed to behave elastically; - a clad mechanical analysis program called CRASH, in which the creep and the plasticity of the clad material are fully taken into account.

In a second stage, the two programs were linked together, the resulting program being called COMETHE III. However, due to the progress of knowledge in the field, these programs remain in continuous development and incorporate improved models and new material properties as more information becomes available. This is first carried out in the two separate programs; as soon as a modification is considered appropriate, it is then introduced in the comprehensive COMETHE III program. The purpose of the present paper is to present mainly the part of this program dealing with the clad stress and strength analysis and to show typical results obtained with it.

2. Description of the crash program 2.1. Basic requirements for a clad analysis program The first basic requirement which was set from the start, is to achieve by the program a degree of accuracy satisfactory for design purposes. Moreover, the computing speed was to be high enough to allow

54

M. Guyette, Cladding-strength analysis under combined creep and plasticity

intensive parametric studies. As presently the material properties for creep and plasticity are not very well known in most cases, the program must be able to handle several types of creep and plasticity laws. Besides the program has to calculate the evolution of the stresses and strains as the irradiation proceeds: the input data must thus allow to follow variations in time of the various loads acting on tire clad: temperature gradient, gas inner pressure and fuel dimensional changes; in particular, the program must be able to calculate the clad behaviour under power and temperature cycling conditions.

the permanent strains which are caused either by creep and/or by plasticity; the program takes into account both the irradiation induced and the thermal creep strain. In the whole paper, the swelling strain, although it has a permanent character, is never included in the permanent strain. This comprises only the creep and/ or plastic strains. The relations between stresses and strains have therefore the general form 13] : 1

e r = ~ [ o r - u ( o o+o z)] + a T + e s w + e p , r . 2.2. Geometrical and time-dependent models All these requirements led us to adopt a simple clad geometry, i.e. the axisymmetrical one. Although this constitutes only an approximation, it is considered sufficient for most cases of application. Moreover, the temperature and swelling gradients in the axial direction have been neglected as these are generally two or three orders of magnitude smaller than the radial ones. This allows to make use of the generalized plane strain hypothesis stating that planes perpendicular to the axis o f the tube remain plane after deformation. Under these assumptions, it is easy to show that the directions for the principal stresses are the radial, tangential and axial ones. For this reason, use is made of the cylindrical system of coordinates. In what concerns the time evolution, the stresses and strains are calculated at discrete values of the time, the averaged properties on the time interval being used to calculate the evolution for a given time step. When these averaged properties are not known, iterative processes are used. 2.3. Stress-strain relations The relations between the stresses and the total strains used in the program include: the elastic strains which are expressed by the classical Hooke's formulas [1,2] ; the thermal expansions assumed isotropic at any particular point; the thermal expansion coefficient can be any function of the temperature; the linear swelling expansion of the material due to fast flux irradiation. These swelling expansions can be calculated in the program with most of the current published correlations; they are also assumed to be isotropic; -

-

-

(la)

1 e o =-~ [o o - la(Or+Oz) ] + a T + esw + fp,0 ' (1 b) 1 % =~[o z-u(or+oo)]

+c~T+e~ w + % ; ,

lc)

where = total strains in each of the principal directions, o r, o o, o z = principal stresses, E = Young's modulus, /.t = Poisson's ratio, T = temperature difference between tire reference temperature (usually taken as 0°C) and the local temperature, a = linear thermal-expansion coefficient averaged over the temperature difference T, esw = local linear swelling strain, Cp,r, Cp,o, ep, z = permanent strains in the r-, 0- and z-directions caused by creep and/or plasticity. The calculation of the elastic and thermal strain components is classical and does not need further explanation. More attention will be paid to the swelling, creep and plastic strains. The swelling strains are calculated by the most recent empirical steel swelling correlations found in the literature [4 7]. These correlations give the relative steel volume increase by swelling as a function of the fast neutron doses and the temperature, under the assumption that the temperature remains constant as the dose increases which is not usually the case for current applications. The program calculates therefore er, e 0 , ez

M. Guyette, Cladding-strength analysis under c o m b i n e d creep and plasticity

the volume increase during each time interval as the difference, for the time averaged temperature over the interval, between the volume at the end of the interval and the volume at the start of the interval. The creep and plastic strains are derived from relations between an equivalent stress and an equivalent creep or plastic strain. The derivation of the permanent strains in each of the principal directions is based on the von Mises assumptions [3, 9] : ep,r--ep,O _ ep,r--ep,z _ ep,O--ep,z o r -- o 0

Or--O z

(2)

O0--O z

and on the volume conservation under creep or plasticity:

55

The thermal component of the equivalent creep strain can be calculated from various creep laws. Three types of laws have been incorporated in the CRASH program. There are: - The Norton creep law [11] expressec~ by: eeq = k a n

(9)

'

in which k and n are temperature-dependent material constants. The values of k and n are fed to the program in the form of tables as a function of the temperature. - An improved law taking the primary creep into account [8, 12] and expressed by: (10)

= kant m ,

ep, r + ep, 0 + ep, z = O .

(3)

where k, n and m are temperature-dependent material constants; - a law using the hyperbolic sinefunction [8, 12]"

This leads finally to the Soderberg equations [10]" ep,eq [o r ~p,r -. . Oe.q .

1 ~ (°o + °z)]

,

(4a)

[ ° o _ 1 ( o r + Oz)]

,

(4b)

ep,eq [o z _ I ~ (o r + o0) ] . Oe q

(4c)

_ ep,~

ep, 0 -

Oe q

- - -

Ei~,z -

The equivalent stresses or strains can then be derived from the stress or strain components in the principal directions using either the H e n c k y - v o n Mises criterion based on the deformation energy or the Tresca criterion based on the maximum shear [ 1, 2]. The first theory leads to the relations:

= A [sinh (ao)] n ,

A, a and n being temperature-dependent material constants. Concerning the irradiation-induced creep contribution, several formulae have been introduced in the program. Owing to the present lack of accurate knowledge concerning this phenomenon, the irradiationinduced creep strain is expressed as a polynomial which is a function of either the irradiation time or the fast flux or the fluence. Moreover, use can be made of the following correlation [14], very similar to that proposed by Hesketh [13]: e =Aam(1

Oeq = (1/X,,/2")4 ( 0

r-

00)2 +

(O r --

ep,eq = ~ x/r2X 4 ( e p , r - ep,0)2 + (ep,r - ep,z) 2 + (ep,0 - ep,z) 2 , (6) while the second gives: Oeq = Omax

- -

Omi n

,

2

Ep,eq = 3 ( e p , m a x -- ~ p , m i n ) •

- e -B*t)

+ can,t

,

(12)

Oz)2 + (O0 - az)2 ,

(5)

1

(l 1)

(7)

(8)

where A, B and C are temperature-dependent material constants. The plastic strains are calculated in turn by using either a simplified curve based on the yield stress as used in the well-known works of Miller [15] or a stress-strain power law. These two types of plasticity laws are shown on fig. 1 : - in the simplified relation, the actual stress-totalstrain curve is approximated by two straight lines as represented in fig. 1. The yield stress is assumed to be a function of the temperature;

56

M. Guyette, Cladding-strength analysis under combined creep and plasticity FIRST PLASTICITY

2.4. Equilibrium equations

LAW.

As mentioned earlier, one has assumed the axisymmetry of the tube and the generalized plane strain behaviour. Under these assumptions only the equation of equilibrium in radial direction is not trivial. Its wellknown expression is [ 1] :

ELASTIC

SECOND

÷

PLASTIC STRAIN

- O.

(14)

er = d u / d r ,

(15)

eo = u/r,

(16)

ez = C 3 ,

(17)

with C 3 being a constant to be determined with the help of boundary conditions.

Fig. 1. Plasticity laws as used in the CRASH program. - in the second case, the plastic equivalent strain is expressed by: pOOeq ,

r

PLASTICITY LAW

ELASTIC • PLASTIC STRAIN

=

crr - o 0

--+ dr

2.5. Compatibility equations The compatibility equations for generalized plane strain and axisymmetry in cylindrical coordinates are written as functions of the radial displacement u [ 1 ] :

o02.--ji,.°"-,............. i,,'~'

epl,e q

do r

2.6. Solution o f the equations It is not possible to derive an exact analytical solution of the system of equations comprising the stressstrains relations (1), the equilibrium and compatibility equations ( 1 4 - 1 7 ) , mainly because of the complexity of the stress-strain relations. However, an approximate solution of the problem can be derived if one assumes the permanent strains as well as the thermal and swelling strains to be known functions of the radial coordinate [3]. In this case, it is possible indeed to integrate the considered system of equations. The details concerning this integration are given in [ 16]. One gets finally for the expressions of the total strains:

1+___u __1rJ

(13)

with P and Q being material constants depending on the temperature. The values of P and Q are determined in such a way that the stress-strain curve goes through two points of known coordinates as the o0.2% - e = 0.2% point, the o1% - e = 1% point or the rupture point.

r

e° = 1--I~ r 2

( a T + esw)r dr

(1

1-2/1

(1

r (1

r ep,r -- ep,o + f a

r

~ C2 dr]+C l +-r2 '

(18)

57

M. Guyette, Cladding-strength analysis under combined creep and plasticity

vals. This has been programmed in such a way that the user does not need to be acquainted with the convergence method and the time-scale sub-division.

1 +la 1 -21a e r = - G0 + ~ _ ~ ( a T + esw) + 1 ------~

rE X ep, r + f

_ p,r r ep'° dr

) + 2C 1 .

(19)

a Using these equations (18) and (19), an iterative process is needed for the determination of the creep and plastic strains. The main steps in the calculation are as follows: assuming the states of stress and strain are known at the beginning of a time interval, one calculates first the temperature and swelling distributions in the clad at the end of the interval; - then one makes a first guess of the permanent strain increments during the interval; using the hereabove formulas, one determines the total strains compatible with the assumed state of permanent strains; - with the help of the relations between the stresses and the strains, one determines a state of stresses at the end of the time interval which is also compatible with the guessed state of permanent strains; - from this approximation of the state of stresses at the end of the interval, one calculates the corresponding permanent strains; the calculated state of permanent strains is then compared with the guessed one. If sufficient agreement is reached, the calculation can proceed to the next time interval'. If not, a new guess of the strains has to be made according to the method explained in the next paragraph. -

2.8. Boundary conditions Eqs. ( 1 7 - 1 9 ) hereabove contain three integration constants which have to be derived from boundary conditions. Two different cases have been considered according whether the clad is in contact or not with the fuel. In the case where no fuel-clad interaction takes place, the boundary conditions are as follows: the radial stresses at the inner and outer cladding surfaces are respectively equal to the inner gas pressure and the coolant pressure -

(°r)r=a = -Pgas ,

(20)

(Or)r=b = -- Pcool ;

(21)

-

-

2.7. Convergence process The convergence process is based on the N e w t o n Raphson method of solution of systems o f non-linear equations [17, 18]. A detailed analysis of the convergence method has been given in [ 16]. This method allows to reach an accuracy of 2 × 10 - 6 on the strain increments usually within three or four iterations. The number of iterations to be performed to reach the convergence depends largely on the size of the permanent strain increments. In order to avoid any convergence difficulties, the length of the time intervals and therefore the size of the strain increments are limited in the program. Special subroutines allow to determine automatically the length of the time inter-

- the axial force acting on the clad is due only to the gas inner pressure and the coolant pressure. This axial force is expressed as the resultant of o z on the cross-section of the tube: b F z = 2zr f

ozrdr = zr(a2pgas- b2pcool).

(22)

a In the case of contact between fuel and clad, an exact solution can only be obtained if the mechanical response of the fuel to the radial and axial interaction forces is known. This is performed in the COMETHE III program [ 19, 20]. However, under the assumption that, at each time of calculation, the fuel behaves linearly, an approximate solution can be obtained with the use of the CRASH program only. One assumes thus that the fuel behaves radially and axially according to the equations: rf = rf0 - pc/ar ,

(23)

ezf = ezf 0 - F z f / a z ,

(24)

where rf = actual hot radius of the fuel, rf0 = free hot radius of the fuel (without contact pressure),

M. Guyette, Cladding-strength analysis under combined creep and plasticity

58

Pc ar ezf ezf 0

= contact pressure, = inverse of the fuel radial stiffness constant, = actual axial strain of the fuel, = free axial strain of the fuel, Fzf = axial interaction force, az = inverse of the fuel axial stiffness constant. Moreover, as soon as the contact is established, one assumes that the actual fuel radius and the inner clad radius are the same and that the fuel facing a unit axial length of clad at the time of contact expands exactly of the same quantity as this slice o f clad as long as the contact is maintained. This last condition must, however, be satisfied only if the axial interaction force is positive (compression of the fuel). The boundary conditions become then:

(°r)r=a =

-Pgas

-

Pc =

-Pgas

~r [rfo - a ( l

+e0a)]

,

(25)

(°r)r=b =

Pcool,

(26)

b

Az = 2rr /

ozrdr

n(a2pa - b2pb)

:

+

%(ezf 0

a

(27)

These boundary conditions are only valid when Pc is positive. In the opposite case, use is made of the boundary conditions without contact. In the case where Pc is positive and F z is negative, use is made of eq. (22) instead of expression (27) for the axial boundary condition. 2.9. Schematic' flow chart o f the program A schematic flow chart of the program is given in fig. 2. This figure illustrates the iterative loop allowing to determine the permanent strains. In the extreme case where this calculation diverges, the program is able to reduce by itself the time interval length. The flow chart indicates also how the program makes the trials for the selection of the boundary conditions. At the beginning of a calculation, the program

ASSOMEBOUNDARY1 J D'V'SION IN

L

CONDITIONS w THOUTCONTACT I 7 SUBINTERVALS

I-

TEMPERATUREDISTRI1 BUTIONAT END OF | TIME SUBINTERVAL /

+"

REDUCTION OF THE ] TIME SUBINTERVAL LENGTH

FIRST GUESS OF

PERMANENTSTRAIN<-

~

'no

~_

Yes

l

I

PRINTING OF THE RESULTS

PERMANENTSTRAIN~ NEW GUESS OF /

ez) .

CALCULATIONOF TOTAL STRAINS AND STRESSES I : : : CMUALNAET INtONSTROAFI NSI

CHANGE THE 8OUN_ DARY CONDITIONS

no

Fig. 2. Schematic flow chart of the CRASH program.

Yes

M. Guyette, Cladding-strength analysis under combined creep and plasticity

assumes that there is no contact between the fuel and the can. As long as the fuel outer radius remains smaller than the clad inner radius, the program uses the first type of boundary conditions. When the fuel radius becomes larger than the clad radius, the program will use the boundary conditions for the contact case. These will prevail until a negative contact pressure is possibly obtained. 2.10. Cladding-strength criteria

The problem of strength criteria is very important from the design point of view but it is also very intricated. It is not intended to solve it completely here but only to show a possible way of solution. In a fuel element cladding, particularly for LMFBR's, the stresses and the temperature levels are usually high enough to cause creep and/or plasticity. The usual well known criteria, as the ASME code [21], which are based on an elastic analysis, are therefore not of direct application. Moreover, cycling of the stresses and the temperatures actually occur in most cases due to the power variations: in special circumstances (like scrams), conditions very similar to those of thermal shocks can also develop. An additional problem arises due to the irradiation effects which strongly modify the mechanical material properties. The problem is thus to work out strength criteria for materials under variable temperature and stress levels and for which the mechanical properties are dependent of the irradiation. In the approach presently used for the CRASH program, one has simply neglected the variations of the mechanical properties due to the irradiation: use is made during all the calculations of the constant properties either of irradiated or unirradiated materials. It must be pointed out that to account for the properties variations as a function of irradiation is by no means a severe problem in a computer program; the difficulty lies mainly in the lack of information in the literature about such properties changes. The strength criteria used in CRASH are based on the cumulative damage approach [22]. In this theory, the damage occurring during each time interval is added to the damage accumulated since the first loading of the material. Damage is considered to be produced either by creep or by plasticity or by both. Creep damage is handled in CRASH in terms of

59

life fractions, a concept introduced by Robinson [23, 24]. In this concept, one assumes that the damage caused in a material submitted during a time interval At to a given stress level at a given temperature, is equal to: F = At/tf,

(28)

where tf is the rupture time corresponding to the considered stress level and temperature. The rupture time is calculated in CRASH with the help of the Larson-Miller master curve for rupture [25]. This curve can be any function of the stress. Although the stresses and the temperatures vary continuously in most cases of application, the time intervals are in practice small enough to allow the use of the average stresses and average temperatures over the time interval without reducing appreciably the accuracy. The life consumption during a time interval for a given calculation point becomes then: F = A t × 10-(I"/T-C) ,

(29)

where f = value on the Larson-Miller master curve corresponding to the average stress during the interval, T = average absolute temperature, C = Larson-Miller constant. Plastic damage is due to the cyclic application of thermal and/or mechanical stresses. In a series of papers, Coffin has studied the resistance of materials against thermal fatigue [26-29]. He has shown that when a material is cyclically strained, either mechanically or thermally, the number of cycles to failure can be derived from the expression NaACp = C ,

(30)

where N = number of cycles to failure, Aep = plastic strain, a, C = material constants depending on the temperature. The life consumption per cycle can thus be expressed, using eq. (30), by: F=~ =

(31)

60

M. Guyette, Cladding-strength analysis under combined creep and plasticity

The values o f C and a, in this last expression, are determined for the average temperature during the cycle. The problem which arises now, is to combine the damage due to the creep and the one due to the plasticity. Taira [30] has produced a method, based on a limited number of experimental data, which consists simply in the addition of the life consumption due to the creep and that caused by cyclic plasticity.

Table 2 Power per unit length and coolant temperature at core midplane and core outlet. Location

Power [W/cml

Core mid-plane Core outlet

580 330

515 652

purpose, the irradiation history has been represented by 25 cycles of 14 days (336 hrs), each cycle being composed o f a 13 days period at full power and a shut-down period of 1 day (fig. 3). This schematic irradiation history corresponds to 325 EFPD with a peak burn-up o f about 80 000 MWd/t.ox. During the power periods, the coolant temperature is equal to the values mentioned in table 2, while during the shut-down phases, the sodium temperature is equal to the inlet temperature of 380°C. The coolant pressure remains constant at 3 kg/cm 2 during all the time. The inner pressure increases from 3 kg/cm 2 at the start of life up to 103 kg/cm 2 at the end of life as shown on fig. 3. Fig. 4 shows the distribution of the stresses during the power and the shut-down periods. The full-line curves on the left side of fig. 4 give the tangential stress distribution at the end o f the power periods (points B in fig. 3) while the d o t t e d lines correspond to the beginning o f the power periods (points A in fig. 3). The curves on the right part of fig. 4 are relative to the state at the end of the shut-down period

3. Typical applications of the program

3.1. Introduction In order to show the usefulness of the program, some typical applications are presented hereafter. The fuel cladding considered in the calculations has an inner diameter of 5.24 mm and an outer diameter of 6.00 mm, leading thus to a clad thickness of 0.38 mm. The cladding material is the German steel WN 4988. A summary of the main properties used in the calculations is given in table 1. The calculations described in this paper are relative to a highly rated pin. The power per unit length and the coolant temperature at the core mid-plane and core outlet are given in table 2. These values include the hot channel and hot-spot factors contributions. 3.2. Cladding cyclic growth due to creep and plasticity The first calculation is relative to effects of the cycling on the stress and strain distribution. For this

Table 1 Main properties of the WN 4988 steel as used in the calculations. Temperature [°Cl

Coolant temperature [°C]

Creep properties k n [-1 [-I

Plasticity properties 00.2% o1% [kg/mm2 ] [kg/mm2 ]

400

1.03 × 10-is

3.02

17

23.5

500

1.64

× 10 -12

2.68

16

22.4

600

4.81 × 10-l°

2.41

14

20.1

650

5.2 × 10-9

2.30

12.5

18.2

700

4.4 × 10-8

2.20

11.5

16.4

750

1.28 × 10-5

1.90

9.3

12.9

M. Guyette, Cladding-strength analysis under combined creep and plasticity ago

. .

r---

.

.

CYCLE 1

~

.

.

CYCLE

2

.

.

~

61

j..|8oo

.

CYCLE 3

./~.~_.

J F-. . . . . . . . . . . . . . . . . . POINTBj

600 i

~ ' ~POINTA

~ POINTB / '

F. . . . . .

,

/

:~-"

............

r I !

~6oo

,N.ER GAS PRESSURE _ '' 10

5OO

J

u

o i i

F~)INTc.-.-.---L_

L-J

L_J

400

/--"~P(~VER PER UNITLENGTH ~A 300 ~

6

3°° I .-;/ 200~__~_J"

200

'~

...

OUTERPRESSURE 100

0

100

J

200

300

tOO

50

600

700

800

900

1000

0 1100

TIME(hrs)

Fig. 3. Power, temperature and pressure history during cycling (core outlet). (points D in fig. 3). At the beginning of the first cycle, the stresses are due only to the cladding differential thermal expansion. The curve in fig. 4 corresponding to that time indicates that there exists already a small stress relaxation by plasticity at the inner and outer surfaces of the clad. At the end of the first power period, some creep has already occurred. The curve in fig. 4 shows clearly that the relaxation by creep is much larger on the inner surface due to the higher temperature at this location. This can also be noticed in fig. 4 from the trend of the stress curve for the first shut-down period. This relaxation phenomenon by creep during the power periods increases with the number of cycles as shown by comparing the curves relative to the cycles 1,5 and 10. This effect causes the stress diagram during the shut-down periods to become more and more close to a trapezium. Moreover as the inner pressure increases with the cycle number, the average stress level increases. This causes the stresses to reach high values at the inner side of the clad during the shut-down periods. After about 10 cycles, the stress diagram reaches an equilibrium in that sense that its general shape does not vary appreciably. This can be observed on the fig. 4 by noting

that the distance between the curve relative to the beginning of the power period and that relative to the end of the same period does not vary with the radius at least up to cycle 18. From this cycle, a new phenomenon appears: during the shut-down phases, the stresses at the inner side of the tube reach a level high enough to cause plastic strains. This causes a smaller increase of the stress at the inner side of the tube at the beginning of the following power period leading to a larger stress at the outer side o f the tube. The net effect is that the tube grows more rapidly when cycled as when submitted to a constant temperature gradient and an increasing inner pressure. This effect is similar to that of ratchetting described by Miller where growth was observed in a pressure vessel submitted to a constant inner pressure and a cycled thermal gradient. However, in our case, the permanent strains are mainly due to creep during the power phases and to plasticity during the shut-down phases. This can be observed from the curves of fig. 5 where one has plotted the inner and outer tangential permanent strain reached at point B in each cycle for three calculation cases: - a calculation where creep only is taken into account;

62

M. Guyette, Cladding-strength analysis under combined creep and plasticity

POWER PERIODS __

......

END OF PERIOD BEGINNING OF PERE)D

F

ONLY

. . . . .

CREEP

........

PLASTICITY ONLY COMBINED

/,

CREEP

AND PLASTICITY

8

E E

~6

,/

// /

...........

11)// / /

2

-6 i

2.62

270

2180 2190 RADIUS (mm)

300

262

220

2.80 290 RADIUS(ram)

300

0

Fig. 4. Tangential stress distribution during cycling (first example). a calculation where plasticity only is taken into account; - a calculation where both creep and plasticity are taken into account. This effect can still be better observed in fig. 6 where a comparison of the stresses during the 25th cycle has been made for the three calculation cases mentioned hereabove.

10OO

,

2000

3000

4000 TIME (hrs)

5000

6000

7 00

8 00

Fig. 5. Permanent strain evolution when taking into account creep only, plasticity only or both.

-

3.3. Analysis o f some effects o f the steel swelling The effect of the steel differential swelling on the ratchetting phenomenon described hereabove has also been examined. The data of the problem are the same as for the previous one except that the clad swelling has been taken into account. This has been calculated according to the recent WADCO correlation for annealed steel [7]. The fast flux ( E > 0.1 MeV) is considered equal to 3 X 1015 n/cm 2 sec in the cross-section considered for the calculation, i.e. at the core outlet. This leads

t 14

T i i POWER PERIOOS .

__

I

12~_. . . . . .

14

SHUT DOWN PERIODS

CREEP ONLY 12

PLASTICITY ONLY

/

COMBINED CREEP AND PLAST C TY

J

/ E t0k

m

10

¢I 6

z

/ 4

2

262

21'~

2L80 2190 RADIUS(ram)

3.0

0

2.62

2~0

L

21.80 2.90 RADIUS(mm)

3,0

Fig. 6. Stress distribution during the 25th cycle when taking into account

creep only, plasticity

only or both.

63

M. Guyette, Cladding-strength analysis under combined creep and plasticity

/ POWER PERIODS

'

'

'

1.4

END OF PERIOD ......

BEGINNINGOF PERIOD 1.2 ......................... SWELLING STRAIN I0!

~.. 8 E E o~

L

T H E R M ASTRAIN L

o8

~s

== OZ

0;

-2

0 2 62

o_Z.J,/

-4 -

ii

[ i/i I II /I

(5) 2 '70

280 290 RADIUS ( mm )

300

Fig. 8. Thermal and swelling strain distribution for various cycles.

/

-6

2.62

2170

i

i

280 Z90 RADIUS(mm)

3.00

2 62

2.70

.2.80 290 RADIUS(mm)

300

Fig. 7. Tangential stress distribution during cycling when taking into account creep, plasticity and swelling. to a fluence of 8.5 × 1022 n/cm 2 at the end of the irradiation time. Fig. 7 shows the evolution of the tangential stress distribution as the irradiation proceeds. This figure can be compared with fig. 4 relative to the case without swelling. At the beginning of the irradiation, the stress distributions are of course the same in both cases (with and without swelling). Due to the high level of the clad temperature in the considered cross-section, the strain gradient produced by the swelling is of opposite sign to the one produced by the thermal expansion. Fig. 8 illustrates how the swelling strains develop in the clad as irradiation proceeds and how they compare with the thermal strain. The effect of the swelling is mainly to prevent to reach an equilibrium in the shape of the curves as was observed for the case without swelling. The fact that the equilibrium is not

reached can be observed at fig. 7 where the distance between the curves at the beginning and at the end of the same power period decreases when the radius increases. As the swelling strain increase is larger at the outer side than at the inner side of the clad (see fig. 8), the stresses at the outer side tend to decrease leading to a larger loading of the inner side of the tube. This effect is counterbalanced by the creep effect which is much larger at the inner side due to the higher temperature level at this location. The global effect is to give stress distribution curves during the power periods which are flatter than those for the case without swelling. This, in turn, leads to more tilted curves during the shut-down phases. The stresses at the inner side will thus be larger during the shut-down periods in the case of swelling. This will thus lead to a more severe ratchetting effect and a larger plastic fatigue. Fig. 9 allows to compare the tangential permanent strain at the clad inner and outer sides in the cases with and without swelling. This figure shows clearly that the strain at the inner side is much larger in the case where swelling takes place due to the larger straining by creep during the power periods and by plasticity during the shut-down periods.

M. Guyette, Cladding-strength analysis under combined creep and plasticity

64

0.45

STEEL SWELLING NOT TAKEN INTO ACCOUNT

040

/

t

,/

STEEL SWELLING TAKEN INTO ACCOUNT

/ 0

35

/

/

t

i

/

/

i

/

030

025 z

0.20

o~s ¢/

//

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0.10

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OUTER SIOE

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INNER SIDE

I"

I

L

i

i

0

1000

2000

3000

i

4000

i

5000

i

6000

I

7000

i

8000

TIME ( h r s )

Fig. 9. Comparison of the permanent strains when taking the the steel swelling into account or not.

3.4. Cladding axial ratchetting due to the interaction

with the fuel When the fuel comes in contact with the cladding, an axial ratchetting phenomenon can be initiated under certain circumstances. The physical mechanism of this ratchetting can be explained as follows (fig. 10): Let us consider a unit length of fuel column and an associated equal length of cladding in cold state (fig. 10a). When going hot and subsequently at power, the fuel and clad radii and their axial lengths increase. Let us assume, as this is usually the case, that the fuel expansions are larger than the clad ones. At some time, the fuel will just come in contact with the clad as shown on fig. 10b. Let us further assume that, at this

stage, the axial friction coefficient between the fuel and the clad is large and that no sliding between the fuel and the clad can occur. If the fuel continues to expand due either to a further power increase or to swelling, the clad will be stretched both axially and radially. If the expansion is sufficiently large, a permanent axial deformation by creep or by plasticity or by both will occur (fig. l Oc). At the next power shut-down, the fuel column shrinks and becomes smaller than its associated length of cladding (fig. 10d). When going again at power the contact will be reestablished (fig. lOe) between the fuel and the clad. At the moment of contact however, as the clad has been permanently stretched, the unit length of fuel column does no more face the same length of clad as at the previous time of contact. An axial ratchetting of the clad will thus take place when the fuel comes in contact with the clad. This has been simulated using the CRASH program again for the highly rated fuel rod considered in the previous calculations. In this problem, one has considered the core mid-plane cross-section of the fuel pin. The pin has been submitted to the same type of cycles as those shown on fig. 3. The fuel behaviour itself has been calculated with the COMETHE II program. Only the results concerning the clad will be presented here. Contact between the fuel and the clad occurs at first during the 19th cycle. Fig. 11 shows the stress distribution at the 18th cycle just before the contact and for a number of cycles after the contact. When one compares these curves with those of the previous examples, it must be recalled that in the present case, the cladding temperatures are lower and the power per unit length higher (mid core plane cross-section instead of core outlet). Fig. 11 illustrates the large change occurring in the stress distribution when contact occurs between fuel and clad. A new equilibrium is reached at about the 22th cycle. This can be seen by observing that the curves at the beginning and at the end of each power period have the same shape for subsequent cycles. During the shut-down periods at the end of the 19th and 20th cycles, the stresses at the inner side of the clad reach for the first time a very high level leading to a plastic elongation which is clearly observed in the bottom diagram of fig. 12. This is no more the case for the cycles above the 20th, because the stresses are high enough during the power periods to give a sufficient

M. Guyette, Cladding-strength analysis under combined creep and plasticity POWER=O

POWER>O

POWER = 100 °/.

POWER = 0

65

POWER > 0

7-/777

_J

~//// / / / / /

d

/ / / / / /

/

/

/

/

"/////

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b

d

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Fig. 10. Scheme of the axial ratchetting mechanism.

20 .

k

strain hardening to prevent any plastic strain during the shut-down periods. The middle and top diagrams of fig. 12 show the evolution of the contact pressure, the clad inner radius, the axial force (due to the inner pressure and the interaction effects) and the total axial strain. This figure shows clearly an axial ratchetting effect: - the axial force at the beginning of each cycle is smaller than at the end of the power period of the previous cycle; - the total axial strain of the clad has the same shape as the axial force. The change of the contact pressure between the end of the power period of a cycle and the beginning of the next one is due to a well known elastic effect: when the cladding tube is less stressed in the axial direction, its radius increases (according to Hooke's law) leading to a reduction of the contact pressure.

SHUT DOWN PERIOOS ( ) CYCLE NUMBER

t

(20)

1/.

4. Conclusions

2.62

2.?0

2~eO 2-qO RADIUS(ram)

3.00

2.62

270

2.80 2.90 RADIUS (ram)

Fig. l 1. Stress distribution in axial ratchetting.

3.00

The newly completed additions to the CRASH and COMETHE programs have significantly expanded their capabilities to handle the prediction, interpretation and to some extent design problems associated with fuel pins irradiations. These additions deal with the introduction of the steel swelling and the irradiation induced component of the creep strain, the combined calculation of the clad creep and plasticity, as well as the calculation of strength criteria for creep and plasticity. Calculations carried out with the improved programs have pointed out some interesting phenom-

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M. Guyette, Cladding-strength analysis under combined creep and plasticity e

ena:

- clad radial d e f o r m a t i o n by thermal ratchetting may occur even at low power w h e n one takes simult a n e o u s l y into account clad creep and plasticity; the creep p h e n o m e n o n is more effective during the power periods while plasticity occurs mainly during the shut-down phases; - differential clad swelling and the subsequent thermal ratchetting can, under some circumstances, further increase the radial d e f o r m a t i o n ; - axial ratchetting may occur when the fuel comes in contact with the clad if the p o w e r and temperature are cycled. The magnitude o f all these effects is strongly dep e n d e n t on the assumed properties o f materials. The comparative analysis o f predicted and experimental data for fuel pins p e r f o r m a n c e reveals a severe lack o f knowledge of a n u m b e r o f material properties under b o t h irradiated and unirradiated conditions. Data scarcity is o f particular concern for the strength properties which are very i m p o r t a n t for the fuel design. It is believed that in the future most o f the imp r o v e m e n t s in predicting the fuel pin behavionr will be obtained by a better knowledge o f the materials rather than by further refinements brought to the m a t h e m a t i c a l models of the programs.

Nomenclature

a A b B C E F k m n p P Q r t T u

= clad inner radius = constant = clad outer radius = constant = constant = Young's m o d u l u s = force = constant in the creep law = constant in the creep law = constant in the creep law = pressure = constant in the plasticity law = constant in the plasticity law = current radius = time = temperature = radial displacement = thermal expansion coefficient

a

= = = =

67

strain Poisson's ratio stress neutron flux

Subscripts cool eq f gas 0 p pl r sw z 0

= = = = = = = = = = =

relative relative relative relative relative relative relative relative relative relative relative

to to to to to to to to to to to

the coolant equivalent stress or strain the fuel the inner gas a free state the p e r m a n e n t strain the plasticity the radial direction the steel swelling the axial direction the tangential direction

References

[ 1] S. Timoshenko, Strength of Materials, Part II, Advanced Theory and Problems (Van Nostrand Reinhold Company, New York) 1958. [ 2] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity (Mac Graw Hill Book Company, New York) 1970. [3] S.S. Manson, Thermal Stress and Low Cycle Fatigue (Mac Graw Hill Book Company, New York) 1966. [41 H. Mayer, Volumszunahme yon Stahl unter Neutronenbestrahlung (lnteratom, Q-B 1 Protokoll) 1969. [5] T.T. Claudson et al., Nuclear Applications and Technology 9, 1 (1970) 10. [6] S. Oldberg et al., Trans. A.N.S. 12, 2 (1969) 588. [7] T.T. Claudson, Proceedings of the International Meeting on Fast Reactor Fuel and Fuel Elements (Karlsruhe, 1970) p. 637. [8] Garofalo, Fundamentals of Creep and Creep Rupture in Metals (The Macmillan Company, New York) 1966. [9] L. Finnie and W.R. Heller, Creep of Engineering Materials (Mac Graw Hill Book Company, New York) 1959. [10] C.R. Soderberg, Trans. ASME 58 (1936) 733. [ 11 ] F.H. Norton, Creep of Steel at High Temperature (Mac Graw Hill Book Company, New York) 1929. [ 12] Y.N. Rabatnov, Creep Problems in Structual Members (North-Holland Publishing Company, Amsterdam) 1969. [13] R.V. Hesketh, Phil. Mag. 8 (1963) 1321. [ 14] A. Boltax et al., Nuclear Applications and Technology 9, 3 (1970) 326. [ 15] D.R. Miller, Trans. ASME, J. Basic Engineering 81 (1959) 190.

68

M. Guyette, Cladding-strength analysis under combined creep and plasticity

[ 16] M. Guyette, CRASH - A Computer Program for the Analysis of Creep and Plasticity in Fuel Pin Sheaths, KFK-1050 (1969). [ 17] A. Rabston and H.S. Wilf, Mathematical Methods for Digital Computers, Volume II (John Wiley and Sons, New York). [18] A.S. Householder, Principles of Numerical Analysis (Mac Graw Hill Book Company, New York) 1963. [ 19} R. Godesar, M. Guyette and N. Hoppe, Nuclear Applications and Technology 9, 2 (1970) 205. [20] J. Dewandeleer et al., Proceedings of the International Meeting on Fast Reactor Fuel and Fuel Elements (Karlsruhe) 1970, p. 29. [21] ASME Boiler and Pressure Vessel Code Section III "Nuclear Vessels", American Society of Mechanical Engineers (1968).

[22] M.A. Miner, Tranactions ASME J. Applied Mechanics 12 (1945) A 159. [23] E.L. Robinson, Transactions ASME, 60, 3 (1938) 253. [24] E.L. Robinson, Transactions ASME 74 (1952) 777. [25] F.R. Larson and J. Miller, Transactions ASME 74 (1952) 765. [26] L.F. Coffin, Transactions ASME 76 (1954) 931. [27] L.F. Coffin, Transactions ASME 78 (1956) 527. [28] L.F. Coffin, Transactions ASME 79 (1957) 1637. [29] L.F. Coffin, Product Engineering 28, 6 (1957) 175. [30] S. Taira, High Temperature Structures and Materials, Third Symposium on Naval Structural Mech. (New York) January 23-25, 1963.