A new method for calculating the design strength of fuel cans for liquid metal cooled fast reactors

NUCLEAR ENGINEERING AND DESIGN 14 (1970) 117-135. NORTH-HOLLAND PUBLISHING COMPANY

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS FOR LIQUID METAL COOLED FAST REACTORS G. SCHMIDT

Nuklear-Ingenieur GmbH, Wolfgang b. Hanan, formerly lnstitut fi~r Reaktorentwicklung, Kernforschungszentrum Karlsruhe, Karlsruhe, Germany Received 4 June 1970

The method described shall provide a possibility for calculating the maximum design pressure and corresponding fission gas plenum length for cladding tubes of liquid metal cooled fast reactors with oxide fuel. The material properties - especially the creep behaviour - as well as temperature and power distributions must be given. The first part of the method deals with creep relaxation of a composite mechanical-thermal stress state on the basis of Norton's formula. Using integral relations for the strain rates the initial equations are transformed into a system of integrodifferential equations. These, after substitution of the integrals by easily calculated constant mean values for only few and relatively long periods of time, change into ordinary differential equations for which an approximate analytic solution can be quoted. This solution is further simplified and finally leads to the definition of a "limiting stress" analogous to the yield strength of the material. Thus the problem is reduced to the well-known case of an elasticplastic tube under internal pressure and intermittent heat flux; its solution can be generalized to yield a criterion for the limitation of the tube loading caused by intolerable cyclic growth, the so-called "thermal ratchetting". The method is applied to several examples with the aid of two small digital computer programs, and the results lead to conclusions for the appropriate design of fuel pin claddings.

1. Introduction In sodium-cooled fast breeder concepts, typical maximum cladding design temperatures are well above 600°C. In this temperature range the mechanical and thermal strength of the cladding tubes mainly depends on creep effects taking place in their walls, at least if they consist of materials like austenitic steels or Incoloy 800 which still form the basis of present design considerations. The most important process in this respect is caused by the combined effects of internal pressure (due to fission gas release and fuel swelling) and heat flux, the variations of the latter - according to the reactor power level - producing alternating thermal stresses which may lead to considerably increased creep strains. This effect is very similar to the socalled "thermal ratchetting" p h e n o m e n o n known from the literature for the elastic-plastic case [1, 2] and

will therefore be called just so in the following. This effect can limit the safe - that means gas-tight operation of the fuel pin canning insofar as recent irradiation experiments with pressurized cladding tubes [3] show that fine penetrating cracks may occur at circumferential creep strains as low as a few %0. A considerable fraction of such small amounts of creep may be produced by the mentioned ratchetting effect. Therefore, a detailed examination of this phenomenon seems to be necessary. An experimental verification of the ratchetting effect has been described by Shively [4] and confirms the anticipated order of magnitude ( 5 0 - 1 0 0 % increase in creep strain for the cases investigated). However, its theoretical treatment has been limited so far mainly to two studies both of which are based on greatly simplified model concepts. The first paper by J. Bree [5] results in an upper estimate of the effects on the basis of the highly conservative assumption that

118

G. SCHMIDT

a complete relaxation of thermal stresses by creep occurs in the first half of each operating cycle (full power operation); in the ensuing shutdown phase, elastic-plastic behavior of the material is assumed to prevail. However, in most cases this assumption will yield an overly pessimistic result, i.e. the permissible internal pressure at a given thermal load will be too low and, hence, the length of the end plenum, which is provided to reduce the fission gas pressure to an acceptable level, may be much larger than necessary. However, any addition to the length of fuel rods is most undesirable because of the resulting higher coolant pressure drop as well as for reasons of core design and fuel element handling. Thus, the upper limits obtained according to [5] may present too unfavorable a picture for many design cases. Also the other investigation (Hibbeler and Mura [6] ), published in this journal some time ago, is based on two greatly simplifying assumptions, i.e., the so-called "two-bar model" and a creep rate linearly depending on stress. Hence, it may often be difficult to .judge how well the actual situation can be described by this model in a specific case, especially with high Norton's exponents and large temperature differences in the tube wall. This study therefore tries to treat the problem in a way essentially free of restricting assumptions of this type. First of all, this requires a detailed investigation of the stress relaxation during the different operating phases of the reactor. Several numerical methods exist for this assessment [7-11 ] but all of them are rather complicated and hard to follow and, above all, have the disadvantage that their results are very difficult to be used as input data for subsequent calculations. For this purpose, the change of the stress field with time is investigated below by means of a semi-analytical approximate solution. A "limiting stress" is derived afterwards by simplification which, when used instead of the yield strength of the material, permits a treatment of the problem by extension of the solution quoted by Miller [1 ] for the elastic-plastic case. A relatively small digital program is sufficient to investigate the influence of the main design parameters and material data on dimensioning of the cladding tubes plus fission gas plenum.

2. Main conditions and assumptions 2.1. Geometry The fuel claddings considered are straight, smooth, circular-cylindrical and relatively thin walled tubes of great length. End effects can be neglected.

2.2. Mechanical stress The loading will be exerted by a hydrostatic internal pressure having the same value at every point of the fuel zone. (The pressure due to fuel swelling is assumed to be hydrostatic also.) The entire internal pressure is assumed to rise linearly with burnup and to remain constant during reactor shutdown; this is a conservative assumption. 2.3. Thermal conditions The temperature distribution in the tube wall is axisymmetric and the temperature difference between inner and outer surface corresponds to the local rod power which decreases from the center of the fuel zone towards the ends corresponding to the flux profile. A typical temperature distribution in the axial direction is shown in fig. i. The critical regime in this case extends axially from the center of the fuel zone to its hot end. The locations of maximum wall temperature (lowest material strength) and highest rod power (maximum thermal stresses) are in most cases relatively far away from each other. In a reasonable design the wall temperatures in this area may vary by a maximum of 100--150°C so that it is possible to use constant mean values for essential material properties, such as thermal conductivity X, thermal expansion coefficient ath, Young's modulus E, and Poisson's ratio v.

__ INL~

•

SURFACE TEt4P \x

[ ]I toSS,ON eAS~WERAX

rUEt

Fig. 1. Typical temperature distribution along the axis of a fuel pin.

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS

119

However, the temperature dependence of the creep parameters (cf. Section 2.5.4) always has to be taken into account. With respect to time, probably the most unfavorable assumption is used, i.e. the load factor f L is caused by periodical cycling between 100% and 0% power at constant coolant temperatures.

2.5.3. Dependence on stress and time According to various authors, e.g. Odqvist and Hult [13] or Mendelsohn, Hirschberg and Manson [9], this can be represented quite well by the time hardening relation

2.4. Influences to be neglected The following influences were assumed to be negligible: • Axial temperature gradient in the wall o f the cladding tube. • Heat production in the cladding material by radiation. • Decrease of wall thickness by corrosion. • Change in geometry due to deformation (i.e., deformations are small). • Pressure of the coolant and other forces exerted on the cladding tubes, e.g. by the spacers.

especially for relaxation problems. For secondary creep (m = 1), eq. (3) simplifies to the well-known Norton formula

2.5. Creep behavior o f the material 2.5.1. Creep under multiaxial stress conditions The material is assumed to be isotropic and incompressible (in case of pure creep) so that Soderberg's relations

(1)

can be assumed to hold between the principal stresses Oi,j, k and the corresponding creep r a t e s ei, j,k" ( ° e and e e indicate the effective stress and effective creep rate, respectively). 2.5.2. Choice of effective stress According to Kennedy, Harms and Douglas [ 12], the effective stress based on the Tresca criterion (max. shear theory) o e = o 1 - o 3 , (o 1 ) a 2 ) 03)

ee = K ' o n ,

(3)

(4)

where K and n are temperature dependent constants. 2.5.4. Temperature dependence Within the present problem only the creep parameters K and n of the material must be given in some way as a function of the temperature (m is assumed to be independent of temperature). If they are known for one temperature T 1 [°K] only, which happens quite frequently, conversion to a temperature T 2 within not too broad limits can be made best using the relation given by Larson and Miller [ 14]: T 1 [CL - l o g ~(T1) ] = T 2 [CL - l o g ~(T2) I. (5)

~e ei =~e [Oi - ½(o/+ Ok) ] , (i,/, k = 1,2,3; i ~ j ~ k )

ee = K" m" o n. t m - 1 ,

(2)

supplies results in creep tests most of which are at the upper limit of the scatter band, i.e., on the safe side. Hence, it will be used throughout the investigation below, also because it greatly facilitates the mathematical treatment of the problem.

If the material constant C L is not given, C L = 20 may be used as a reasonable estimate, according to [14].

3. Relaxation of a given initial stress distribution by creep During the period considered temperature and internal pressure in the cladding tube are assumed to be constant. The axisymmetric stress distribution in the wall then asymptotically tends towards a final steady state. This process will be investigated in greater detail below. The initial stress state is allowed to include plastic deformations; however, during the relaxation process only elastic and creep deformations are considered.

3.1. Basic equations Under these conditions the total strain rate ~ is taken as the sum o f creep rate ec and elastic strain

120 rate

G.SCHMIDT

6el, i.e. = ec + eel"

(6)

When this is applied to the three principal directions of strain in the tube wall (tangential, radial and axial, corresponding to the indices t, r and a), the combination of eqs. (1), (3) and Hooke's law results in the following system of equations: et = K m o n - 1 t m - 1 [o t _ 0 . 5 ( o a + Or) ]

1 + f f [O t -- P ( b a +

er = K m ° n - l t m - 1

hr) ] ,

(7)

[Or - 0'5(°t+ °a)]

+_1

E [ h r - v(ht + ha)] '

(8)

ea = K m ° ne - l t m - 1 [°a - 0"5(°r + at)]

1

+ ff [°a - v(b r + ht)] .

In addition, statements can be made about the distributions of i t and i a as functions of the radius r. Since the cross sections of the tube remain plane, the relation ea = const, e l ( r ) ,

(10)

must hold. Furthermore, it has been shown by the author in a more detailed publication [15] that it(r ) can be represented sufficiently well by a function of the form ~t - rC2 ,

(11)

although, strictly speaking, this relation holds only for steady state conditions [13]. After calculating the integral mean values of et and ea in the tube wall it is seen that the integrals of the elastic terms in eqs. (7) and (9) practically vanish. This follows f r o m ~ t e assumption of constant E and v and the equilibrium considerations

(9)

(12)

° bt d r = 0 ri

Presently, the total strain r a t e s c t ' e r ' ~a are unknown, aside from the principal stresses o t, o r, o a. (The dot denotes differentiation with respect to time.) The effective stress o e is given by eq. (2); all these quantities are functions of space and time.

and, for thin walled tubes ro

ro

f bardr=O~fhadr, ri

3.2. First simplifications The radial stress o r in the relatively thin-walled cladding tubes is small and undergoes but little changes with time. In addition, the changes in the mechanical and thermal fractions of o r counteract each other. Therefore, it can be assumed with very good approximation that hr=0

;

(13)

ri

(constant load in circumferential and axial direction, respectively; r i = inner radius, r o = outer radius). Introducing the dimensionless radial coordinate _r o - r ro

which is denoted by ~o (= dimensionless wall thickness) for r = r i, the integral mean values then turn out to be

consequently,

Or(r, t) ~. o r (r, 0), where ar(r, 0) is the given initial stress distribution. Since, moreover, the strain rate i r is not of interest, eq. (8) need not be considered any further.

1 ~o et d~ 6-t = ~o 0 -

(14)

mtm-1 ? K -n-1 oe

~o

0

[Ot - O . 5 ( a a+ Or) ] d~,

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS

ia = ~-°1 0

i a d~

(15)

f~o

-mtm-1 Kon-l[oa-O.5(Or+Ot )] d~. ~o 0 Because of relation (10), eq. (15) supplies the axial strain rate i a directly as a function of the stress components while, for the determination of i t, eq. (14) still has to be combined with (I 1). This yields m t m - l ( 1 - ~o) it:

~o(l=-~2

tl-m.E At - 2) ~-m v( 1

(it + v i a ) ,

(19)

- (i a + Vlt), Aa - m ( l _ v2 )

(20)

and

tl-m.E

f Kon-l[ot-O.5(Oa+Or )] d~. (16)

Substitution of (15) and (16) in the basic equations (7) and (9) results in a system of two partial integrodifferential equations for o t and o a. This system, when resolved for b t and b a, can be written in a general form as follows:

can be approximated by constant values, e.g., by their initial values which are easy to be determined analytically or numerically. Consequently, eqs. (17), (18) reduce to the ordinary differential equations

EK

bt = mtm_ 1 {A t

E [i t + Vi a -- (let 1 -

near the inner and outer surfaces, where the largest and hence most important changes in stress occur. (In case of primary creep, the strain rates have to be divided by the time hardening factor m-t m-1 before this approximation is made.) Under these circumstances, it is sufficient to subdivide the total period considered into few relatively large intervals; during these intervals the quantities

~o

0

bt -

121

+ Vlca)]

(17)

1 -

v 2

on_ l e

(21)

V2

E °a - 1 ~- v 2 [ia + vlt -- (ica+ vlct)] "

.l+v.. X [Ot(1 - ½v) -- Oa( ~ - v) -- O r ( - - ~ ) l )

(18)

ict and ica represent the local creep rates in the circumferential and axial directions (first terms on the right-hand sides of (7) and (9)). According to (16) and (15), the corresponding total strain rates i t and ea mainly depend on the integral mean values of these local creep rates.

3.3. Step-wise approximation by ordinary differential equations An exact analytical solution of the system of eqs. (17), (18) seems to be hardly feasible. As explained in [ 15], it is possible, however, to introduce a major simplification in cases of higher Norton's exponents (n > 4, approximately). This is valid for practically all the cladding materials under consideration. Under these conditions the overall strain rates change much more slowly than the local creep rates, so that they can be assumed to be constant for relatively long periods of time. This is reasonable at least

ba=mtm_ I(A a-

EK_. n-1 v 2 ae

1 -

(22) l+v.. X [Oa(1 - -~v) - Ot( ½ -- V) -- Or( --~--)l } . It can be seen later that there is no strong coupling between these two equations.

3.4. Decoupling For this purpose, a differential equation for o e is derived from eq. (21) and/or (22). This is done in various ways, depending on whether the effective stress according to Tresca is represented by a) o e = o t b)

-

Or ,

Oe = O t -- O a ,

or C) O"e = O r - - O a ,

at the location considered.

122

G.SCHMIDT

Case a): Because of

b r = O,

The c o n s t a n t s A t , A a a n d A m which are just linear c o m b i n a t i o n s o f the total strain rates can be generally formulated as

we have

be = b t In addition, it was shown by the author [15] that 0a-

---0t

Or

A v =By" ~ v '

Oa(t=O)--O r

~

Or

Ot(t=O)--O

r

- a = const.

(23)

holds with sufficient accuracy. Then eq. (21) provides the relation

where the Ocv are " d u m m y " effective stresses introduced for reasons of convenience and thought to be c o n s t a n t t h r o u g h o u t the entire time interval considered. (For the case A v < 0, which is possible under extreme conditions, it must be assumed that Av=

be = m t m - 1 (A t _ B r a n ) ,

( v = t, m, a ) ,

Bv'ancv

")

(24) Consequently, the differential equations for the effective stress can be written as follows:

where Bt -

EK 1 --

(25)

(1-~v-½a+av).

v2

Range a)

be = mt m-1

.Bt(cr~Zt

on),

Range b)

be = mt m-I

"Bm(onm - o n ) ,

Range c)

b e = m t m - 1 . B (o n - o n ) .

(32) (33)

Case c): With the aid of be =

a"

ca

(34)

b~ ,

If, in addition, the dimensionless effective stress and

Ue

or - ot

Or - o t ( t = O )

Or -- 0 a

Or - O a ( t = 0 )

sv=(~-cv ) - 3 = const.

(26)

,

(v=t,m,a)

is introduced, it follows from (32) and (43) after t r a n s f o r m a t i o n and formal integration:

the same procedure can be applied yielding the relation b e = _ m r m - I (A a + B a o n ) ,

dst

(27)

- a n t 1 "B t • t m + const.

(35)

_ o n - 1 . B m . t m + const.

(36)

II

1

st

-

where Ba

EK ( 1 - ~ v - ~ 3 + 3 v ) . 1 -- v2

(28)

dsm 1

-

cm

s n

in ds a

C a s e b): Subtraction of eq. (22) from eq. (21) fur-

nishes directly and w i t h o u t simplifications b e = m t m - 1 (A m - B m o n ) ,

(29)

f

_

1-

sn

with the initial c o n d i t i o n s (index 0 for t = 0)

A m =A t -A a

(30)

EK Bm = ~ " 1 - v 2

(31)

Sm0 -

and 3

(37)

a

ot0 -- o r St0----oct

where

n - 1 . B a . t m + const.

(}'ca

,

atO -- Oa0 Ocm

(38)

(39)

A NEW METHOD FOR CALCULATING

THE DESIGN STRENGTH OF FUEL CANS

and, generally, Or - °a0 Sa0 = Oca

(40) B* P

3.5. In tegration

The integrals in eqs. (35) to (37) can be performed elementarily only for integer values of n and result in very complicated and unhandy terms if n > 4; in addition, they cannot be written explicitly in the desired form s = f(t). However, it has been shown in [15] that the replacement of the integrands in (35) to (37) by approximate functions of the form 1

1 x- en(s-l) gives good and conservative results, at least for the range of 0 ~< s <~ 2, which is practically the only range of interest. These functions can be integrated elementarily. One obtains f

•

_ 1 log l

ds

1 -+ e n(s-1)

n

n ' o n ~ -I " B v

n .A v Ocv

In the individual ranges the effective stresses at time t are given by (4~

a e = a cv" sv "

Since o r is assumed to be known, O t = Oe + O r is inamediately given in the range a) and o a = o r - o e in the range c). In case a) or c), the principal stress which is still missing can be now found by substituting of the solution ae(t ) into the corresponding equation (22) or (21). In case b), substitution in one of these equations is sufficient to determine o t and Oa •

This procedure results in a differential equation for the determination of the still missing principal stress o:

e n (s- 1) I • 1 + e n(s-1)

(49)

~r = G( t) - H ( t) • o ,

where G(t) and H ( t ) are known functions containing The substitution into eqs. (35) to (37), taking the initial conditions into account, results in the approximate relaxation functions for the nondimensional effective stresses. Range a): st

=

1

_

llog(l+C n

t'e-

B*

t'

tm

),

(41)

Range b): Sm= 1

_

llog(1 +Cm. e

-B*

m

.t m

),

(42)

s a = 1 ----1 log (--1 + C a" e+B*-~t m ) ,

(43)

n

Range c):

n

with the abbreviations Ct = e - n ( s t 0 - 1) _ 1 ,

(44)

Cm = e - n ( S m 0 - 1) _ 1 ,

(45)

Ca = e-n(Sa0 - 1) + 1 ,

(46)

%(0The general solution is t

-f

o(t) = e 0

r

H(r)dr

t

[o 0 +

f H(O)dO

f G(r) e°

dr]

0

(50)

where o 0 is the initial value of the stress and r and 0 are the current time coordinates. However, the integrations can no longer be performed elementarily but must be evaluated numerically, e.g. by Simpson's rule. In this case, already one subdivision of the integration interval in most cases supplies a sufficiently accurate result for not excessive values of t. The stresses so obtained at the end of the first time interval t I can be used directly as the initial values for the next interval, and so on, unless an iteration is performed in advance to improve the quantities A t and A a used. (In this case, the initial values for A t and A a are replaced by improved values obtained with advantage from the stresses at the time t = t l / 2 , see [15].)

124

G.SCHMIDT

The stress distribution at the end of the last time step then represents the wanted final state after the entire period o f time considered. The mean overall tangential strain produced during relaxation is obtained in the simplest way by integration o f eq. (14) with respect to time. For practical calculations the actually time dependent quantity ( t l -m/m) ~-t is approximated by a constant mean value analogous to (19) and (20) for each time step, the corresponding strain being determined by integration and summed up afterwards over all time intervals. 3.6. Results of some examples Mainly to check the accuracy attainable with the method described a small F O R T R A N II digital program was set up to treat a number o f examples. To keep the expenditure in terms of programming low the period o f time considered is subdivided into sections o f equal lengths and, moreover, only the case m = 1 is studied (pure secondary creep). The latter restriction should not be unrealistic inasmuch as the processes of main interest occur at the end of the service life o f the fuel rod in the reactor, when, probably, there are no longer any significant contributions o f primary creep. However, in order to have an estimate of the error produced by the approximation used for integrating eqs. (35) and (37), an additional program version was set up * The first example deals with the change of an idealized linear initial stress distribution (with o r = 0) in a thin-walled tube of constant temperature with the input data K = 10 -12 , n = 6 (both constant over the cross section), E = 1.5 × 104 kp mm -2 , v = 0.3. Fig. 2 shows the stress curves obtained after 4, 12, 28, 56 and 100 hr for the exact solution and after 100 hr for the approximate solution. Both solutions are characteristic inasmuch as the circumferential stress in the upper left-hand area is almost parallel to the asymptotic final distribution ot=, i.e. it is almost independent o f the initial state, while, in the lower right-hand area, it mainly constitutes a parallel to the initial state. In between there is a transition range. * This program uses explicit values for s which are calculated by iteration from the implicit exact solution for n = 6 and thus provides a basis for comparison.

OO131

StLJ~ '

ii

I

i

I

I .$k_ ]

....

l

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

$

Fig. 2. Relaxation of an initially linear stress distribution in the tube wall (constant temperature; K = 10-12; n = 6). The shaded areas are almost identical, i.e., tile mean circumferential stress is retained almost unchanged for the exact solution; this indicates a rather good accuracy of the result. The approximate solution deviates from this result by only 3% and may thus be regarded as being almost equivalent, especially because the deviation consists chiefly of an easily correctable parallel shift. An even better result is obtained for the next, more realistic example o f a relatively thick walled tube (~0 = 0.184) with high local rod power (× = 500 W/cm) and the additional input d a t a K = 8.8 × 1 0 - 1 3 ; n = 6.0 at T O = 813°K (external wall surface temperat u r e ) ; C k = 16; k = 0.209 W c m - l ° C -1 ; E = 1.4 × l 0 4 k p m m - Z ; v = 0 . 3 ; a t h = l . 8 × 10 -s °C -1' p = 2 . 6 5 kp mm -2 (= pressure in the tube). After 100 hr the approximate solution is rather

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS

125

4. Simplification of the method The method described so far can be used to calculate the stress relaxation and the resulting overall strain after a given period of time for a specific cross section of the cladding tube. However, this is only a partial solution of the actual problem, i.e., the determination of the load carrying capacity of the cladding tube. What is still required is the investigation of quite a number of cross sections of the tube considered to find the most critical spot. If, in addition, different parameters (wall thickness, rod power, etc.) are varied for the determination of a particularly favourable design, the method would have to be applied very frequently, which requires untolerably high computation efforts. For this reason, a number of successive simplifications are introduced before proceeding any further. Most of these simplifications are on the safe side to approximate the actual stress distribution at the end of the period considered by a substitute stress distribution which, first of all, is relatively easy to be calculated. Secondly, this approach permits the specifi, cation of an upper limit of the creep strains produced by stress relaxation, but does not overestimate these strains as strongly as in the model used in [5].

Fig. 3. Stress redistribution in a relatively thick walled tube with high local rod power (Go = 0.184; × = 500 W/cm). close to the asymptotic stress distribution (fig. 3); 50 time intervals were used. The calculation of the asymptotic stress distribution is given in [15] and was done using eqs. (4) and (5); it can be determined also by a similar method presented by Odqvist and Hult [13]. For the case of a very thick walled tube (~o = 0.5) at constant temperature and under internal pressure it is shown in [15] that the convergence of the circumferential stress towards the steady state solution is also rather good. This is another indication of the usefulness of the approximate solution. The examples investigated show that this solution always provided results of sufficient accuracy within the range of parameters of interest to cladding tube design, at least for oxide fuel.

4.1. Simplified relation between stress relaxation and circumferential strain The total strain in the circumferential direction can be obtained in the way outlined in section 3.5, or by integrating eq. (7) with respect to time: Ae t = K ' m f o en - l t m - I X [o"t - 0.5(Oa+ Or) ] d t

1

+~f

[b t - v ( a a + o r ) ] dt.

(51)

Herein, the first integral entails some difficulties which can be avoided, analogous to 3.5, by substituting the stresses by mean values constant during a certain time interval. However, this requires too much effort. Much simpler is the use of an upper limit for the integrand. For all points of the wall cross section where, during relaxation, the stresses always remain below their asymptotic final values (i.e., certainly either on the inside or the outside of

126

G. SCHMIDT

the wall) such a limit is obtained by substituting these asymptotic final stresses. Strictly speaking, this applies only as long as o a and o t assume no excessive negative values, but in a reasonable design this would occur only in the very first phase, if at all, and thus need not be considered. Confined to secondary creep (m = 1), this causes replacement of the circumferential creep rate Jet' which was variable with time, by the (normally higher) asymptotic creep rate at this point, ~t=' Further it is always conservative to neglect in the second integral in eq. (51) the term vo a compared to b t, since °t and ~a always have the same sign. With br = 0 one thus obtains the upper limit of the circumferential strain after the time interval At as Ao t A e t = et= At + ~ -

,

(52)

where At Aot = f Ot dt = ot(At ) - ot0 , 0 at the inside and outside, respectively. This means that it is possible in a simple and, at the same time, conservative way to quote the overall circumferential strain as the sum of creep strain under pure internal pressure (asymptotic stress distribution) and the strain caused by the change in circumferential stress on the inside (or outside, respectively) of the tube wall, Ao t (see fig. 4). In this case the asymptotic creep rate i t = of thinwalled tubes can be calculated quite well by the

>~ 6 t

OUTSIDE

formula [ 15 ] et= = ~ K ° -en '

(53)

where K and ~ are the creep parameters pertaining to the average wall temperature T and P "(r o + ri) cre - 2(ro _ ri )

(54)

is the mean effective stress in the tube wall. What is required now is a simple approximation for the determination of Ao t which is obtained by the method outlined below. 4.2. Construction o f a substitute stress distribution The investigation of various examples [ 15 ] shows that practically independent o f the initial state the stress relaxation will result in a typical circumferential stress distribution as for example shown in fig. 4; this holds for high n-values (n > 4) characteristic for austenitic materials. The curve is composed of three parts of different extension: a section I which can be represented largely by const. × Ot~ , a parallel II to the initial distribution ot0, and a transition section IIl in between. (An exception has to be noted in those cases only where the initial distribution extends relatively far into the compressive stress region. Then, section It is no longer parallel everywhere to this distribution; however, as will be shown later, this need not be considered in the construction of the substitute distribution.) Since the stress obtained is of interest practically only near the inside and outside of the wall, whereas it is of hardly any interest in the central region, the actual stress distribution can quite well be replaced by a curve consisting only o f the two branches I, resp. Ia (const. × ot= ) and II resp. IIa (parallel to ot0), cf. fig. 4. Since, moreover, the equilibrium condition in the circumferential direction must be fulfilled, the knowledge of the upper branch const. × at= is sufficient to construct the substitute stress distribution and, hence, to be able to determine Ao t *. How to obtain this

IVSIDE

Fig. 4. Typical resulting circumferential stress distribution consisting of 3 sections; construction of a substitute distribution.

* The direct determination of Aot by calculating the lower branch II of fig. 4 would yield too inaccurate results, at least with the simplifications of the next section, as shown in [ 151.

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS upper branch in a simple way and independent o f the initial state is explained in the following section. 4.3. Derivation of a "Limiting Stress" analogous to

the yield point o f the material," elastic-plastic equivalent system. For this purpose, the method of calculating the stress relaxation outlined in section 3 is greatly simplified and then applied to the region above ot~ considering only secondary creep (m = 1). The simplification comprises these points: a) Treatment of the entire relaxation process in one single step (without subdivision of the time inter-

val). b) Replacement o f the quantity oct in eqs. (32) and (47) by the always smaller and relatively easily determined value a e . c) Neglecting the axial stresses, i.e. the quantity a in eq. (25) (which is always positive in the respective region) from which follows Bt ~ E.K

(25a)

(assuming (1 - ½u)/(1 - v 2) ~ 1), inste a d o f (25). All these assumptions are conservative, i.e. the rate and thus also the final amount of stress relaxation are sometimes greatly overrated so that the distribution of the ultimate stress obtained from eqs. (41) and (48) is closer to ot~ than the actual one. For reasons of simplicity, always o e = o t - o r will be considered below instead o f o t and treated like a uniaxial circumferential stress; the difference is slight within thin-walled tubes and, moreover, the relation Aa t = Ao e holds. The conservative approximation Oe(t) thus obtained can be regarded, at each point ~, as the result of a decrease of the dimensionless effective stress from its initial value st0 to the final value s t according to eq. (41). This time dependent process is represented in fig. 5 for n = 6 as a function of the non-dimensional time (eq. (41))

it can be supposed to be constant over the cross section with a sufficiently good approximation. It is seen that large st0 values are reduced very quickly, while initial values which are but slightly above unity hardly undergo any change. This results in the typical shape of the section o f the curve above o t . in fig. 4. Nearly the same curve would be obtained also with a material exhibiting a usual elastic-plastic behavior with a curved characteristic and deviating greatly from Hookes' straight line above Oe~ and s = 1, respectively. In this case, instead of the theoretical stress Oe0 (according to Hooke's law) the actual stress will be Oe(t), which is smaller by the difference Ao and is accompanied by a permanent strain Ae. However, in such a case usually the actual stress-strain curve of the material is replaced by a linear elastic-perfect plastic characteristic. This concept necessitates the introduction o f a distinct yield stress Oy, which is defined as that stress corresponding to a prescribed permissible or just negligible permanent strain. Applied to the present case this means that all stresses initially higher than the yield point are reduced to the yield stress while all stresses below this level remain completely unchanged. In an analogous way it is possible now to define a "Limiting Stress" t7L of the material with respect to the relaxation by creep occurring during a specific period of time. o L will be represented by that effective stress which reduces during the time considered by an amount Ao corresponding to a given - permissible or just negligible - circumferential strain. (Although this strain will occur as a creep strain only in the range above o e ~, a consideration on the basis of figs. 7 and 8 - see section 4.4 - yet shows that it represents equally well the overall permanent strain of the cross section.) If this strain Ae is known, one obtains the permissible amount of reduction Ao under the assumptions cited above simply as Ao = E . A e ,

(56)

and with As = st0 - s t = AO/Ae, insertion in eq. (41) resolved for st0 provides the wanted Limiting Stress.

t* = B~'. t m . In the present case, t* is calculated to be

t* = n E K . ~ z l . t

127

~,

(55)

OL = ° e "

1 fie (n'A~+t*) -1 +nl°gC e-~--T

11)

.

(57)

128

G.SCHMIDT

St I

1.4

[J

~

~

~

~ ~. / S l , -

Z.O tEXACT SOLUTIONJ

~

~

i. . . .

0.1 0.75

0,5

0.75

1,0

t~ 8~.fm

Fig. 5. Decrease of dimensionless effective stress with nondimensional time. The value o f the bracket in (57) still depends on to some extent, but for practical purposes it should be sufficient to use the value at ~ = 0 as a constant factor over the entire wall thickness. I n this way the wanted upper branch of the equivalent stress distribution is now given, despite of the slightly different form O L = const. × Crew,used here because Oe® is easier to be determined than o t . Now, only a suitable value o f the permissible additional circumferential creep strain Ae still has to be found. It is useful to proceed in such a way that, initially, a specific tolerable fraction of the permissible permanent overall growth ect is reserved for effects of this kind. (This fraction will be greater than zero in any case.) Then the amount thus obtained - etl is distributed to the individual operating cycles; it has to be noted that only those cycles supply a major contribution during which the internal pressure is sufficiently high. Since a limiting value of this effect is hard to be quoted it is recommended to proceed as outlined in greater detail in ref. [15].

Following this procedure the time-dependent internal pressure (linear increase with burnup) will be substituted by its final value Pc' which, however, will not act during the entire lifetime t s but only during the effective time t s [1 - (p0/Pe)if+l] te = (1 - p o / P e ) ( K + 1)

'

(58)

(P0 = internal pressure at the time t = 0). This type of loading generates the same creep strain as the actual time dependent internal pressure. If this model is applied to the present case, the whole amount of/xe is equally distributed to the operating cycles occurring during the "effective time" re; for the sake of simplicity, all cycles will be assumed to have the same lengths t z. (A reasonable lower limit of t z can be derived from minimum requirements for a meaningful reactor operation; in the examples treated the very conservative value of t z = 125 hr was assumed.)

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS

129

Then the number o f "effective" cycles is given by te

z =-- .

(59)

tz

The permissible amount of strain due to each cycle as a result o f stress relief is the fraction etl

Ae c = - z

,

1

(60)

which then must be shared - equally, to be meaningful - among the full power and shutdown phases. The value needed for substitution in eq. (56) then reads

02~

PHASE l 01

etl

Ae = -~- .

A~t~

:0 FO

/l

(61)

After determination o f a e ~ the curve of o L for both phases can be calculated using eq. (57). This being done, the whole problem reduces to the treatment o f an equivalent elastic-perfect plastic system with OL as the material yield strength. 4.4. Thermal ratchetting criterion for the general case The amount o f the stress difference Aa t (section 4.1) and the excess strain (exceeding the permitted value Ae) can now be found for each phase by generalizing a method mentioned first by D.R.Miller [ 1 ] for a tube with constant yield strength across the wall. The processes occurring in this case during the individual operational phases o f the reactor are shown in fig. 6. Each part represents the distribution o f the stress o e = o t - o r * from the inside to the outside of the tube wall with O 1 - O 2 as the reference line. In the first full power phase the initial state O1BEO 2 is composed of the elastic stress distribution resulting from internal pressure (Oem) and from heat flux (o'e). Its fraction exceeding the limiting stress aLl (shaded area F 1) is removed by creep and added to the portion below a L l parallel to BE at a distance of Ao2, thus producing the final distribution O1ACDO 2. The areas F 1 and F 2 must be equal for reasons of equilibrium. At the beginning of the second phase, i.e., when the reactor is shut down, the thermal stresses o'e are subtracted from the stress curve so far obtained, which results in the new distribution O1FIJO 2. At * (which is not everywhere identical with the actual effective stress!)

j/

j~,]

I

o~

K PHASE 2 O01"~OE

o~'

H

p

l~/OE

OHASE 3

Fig. 6. Stress redistribution in the tube wall during the individual operating phases of the reactor.

the same time, all parts of the wall assume the coolant temperature, thus causing the limiting stress in this phase to rise to the constant value of OL2. If the new stress distribution exceeds this value, another rearrange ment takes place and the curve O1GHKO2 is obtained. After a new start-up o f the reactor, so that the thermal stresses are added again, practically the same rearrangement occurs in the third phase, but in the reverse direction. This results in almost the same stress distribution as at the end o f the first phase; the small deviation is caused merely by the fact that the section

130

G.SCHMIDT

/dr o f the curve in phase 2 in general is not parallel to OL2. All the following cycles are only repetitions o f the phases 2 and 3, each producing a permanent expansion o f the cladding tube; this effect is the thermal ratchetting mentioned above. It should be noted that the depicted changes in stress Ao2, AÜ4 and Ao 5 always correspond to the quantity Aa t. Eq. (52), in each phase. This can be used to determine the permanent strain which exceeds the permissible amount of Ae used in the derivation of o L in section 4.3. Since the individual stresses in figs. 6 - 8 are known as functions of the wall coordinate ~, the stress relaxation wanted can be determined by calculation. Use of the equilibrium condition in the circumferential direction and of the symbols b, c, d, e for the indicated distances yields the following relations:

1st phase." Ao 2 = OLl(b ) - Oem(b) - oe(b)

o.~ •

o.Ia3

om

o.~5s

IVlECH + THEIg~I4L S T R E ' S ~ : ~ r

"~o.~ o~25 o.25 o.m

o.Jo o.s~

[

o.~

olin ~ , o

¥

n

Fig. 7. Permissible final pressure versus dimensionless wall thickness for different rod powers Xmax (K = 10-x2, n = 6 at 650°C; ect = 0.15%).

(62)

b

f (°Ll-

uolO.O~

mcl O.~'/

o.~

12123

0.139

QI55

°em -- o ; ) d~

0

+

Ao2(~0

--

(63)

b) = 0

1

(Oem and Oe denote the initial mechanical and thermal stresses, respectively.) This result is used to determine b and Ao 2 simultaneously.

2nd phase." AO 3 = O e m ( ~ 0 ) + A o 2 - O L 2 ( ~ 0 ) ,

I

I

i

(64) (65)

OL2(¢) = OL1 (c) -- Oe(e ) .

5Wlcmi I

! oLo.2o

i 2

i

o.~5

o.~5

L o., 5

o.~o

0.5

o.35

(Equation determining c.)

¢t~v~ Qd0 x ~

Fig. 8. Corresponding minimum length of the fission gas plenum. AO4 = OL2(d ) -- OLl(d ) + O'e(d),

(66)

3rd phase."

= F 3 , =

AO 5 = OLl(e ) -- OL2(e ) -- O'e(e) .

C

= F4

OL2 -- Oe) d~ + AO5(~0

e)

C

F 3 =f (OLI--O e-OL2 )d~ 0 + ( O e m + A O 2 -- OL2 ) d ~

P

f (OL1

b

=f(oL2- OL1 + Oe) d~ + A o ' d .

(67)

(68) (69)

This results in e and Ao 5 .

d Eqs. (66) and (67) together result in d and Ao 4.

The calculation of the quantities b, Ao 2, etc. generally requires a considerable computational el-

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS fort (iterations). However, this can be greatly reduced for the present application where the permissible amount of strain caused by ratchetting is limited to relatively small amounts. In this case it is reasonable to calculate the limiting stress o L taking for Ae the total amount o f the permissible additional strain. This procedure can give rather high values for OL, but for compensation no further additional permanent strain - proper "thermal ratchetting" - is allowed to occur. If the strain in the initial phase is neglected, then the conditions above request that the change in stress Ao 4 or equivalently the area F 3 = F 4 (fig. 4) vanishes. This also can be expressed in practice by the requirement Ao 3 ~< 0 , which is easier to be met; the check of this condition requires only eqs. (62), (63), (64). Thus, a criterion has been found which indicates whether for a given combination of internal pressure and rod power thermal ratchetting, that means untolerably high permanent strain, may occur. If ratchetting is indicated by this criterion, it can be avoided by an appropriate reduction of internal pressure. In this way, the permissible upper limit of the internal pressure can be evaluated, and applying the gas equation this can easily be used to derive the required minimum length of the fission gas plenum, if the quantity of gas released is given [ 15].

131

They are based on a design which largely corresponds to the reference design of the Karlsruhe Na 2Study [16]. The respective data are: Coolant inlet temperature Coolant outlet temperature Hot channel factor for coolant temperature rise Heat transfer coefficient wall to coolant Corresponding hot channel factor Hot channel factor for temperature difference in the wall Maximum nominal rod power Peaking factor for axial power distribution Height o f fuel zone Height of one axial blanket each Density of fuel and fertile material Average fuel temperature Fission gas volume produced per cm 3 of fuel (at an axially averaged burnup of 10 s MWd/to) Load factor

T c l = 380°C Tc2 = 574°C (nominal ]'1 = 1.34 a w = 14.5 Wcm-2 °C-I f3 = 1.19

f2 = 1.22 Xmax= 456 W/cm Cax = 0.80 L c = 95 cm L B = 40 cm PF = 0.87 "Pth

TFm = 1,500 °C

VG = 28 ncm 3 fL = 0.80

In addition, there are the values of quantities already explained:

5. Discussion of the most important results For the application of the criterion described another relatively small digital program was set up which singles out among a number of cross sections the most critical ones and, in addition, permits the variation of the wall thickness in the same computer run [15]. The temperature distribution in the fuel rod cladding is calculated in a subprogram. The pressure due to fuel swelling is not considered. Instead, however, a 100 percent release o f the fission gases is assumed. A number o f parameter investigations were carried out with this program and the most important results will be discussed below.

ect = 0.15%,

v=0.3

r o = 3.35 mm

X = 0.21 Wcm -a °C-1

G0 = 0.123

ath = 16× 10-6 °C -1

(corresponding to 0.41 mm wall thickness) K = 10 -12

Oy = 22 kpmm -2 at 650°C

n=6.0

t s = 2 × 104 hr

(for 16/13 CrNi steel)

t z = 125 hr

CL= 15 E = 1.6 X 10akpmm -2

132

G.SCHMIDT

The fission gas plenum is located on the coolant inlet side, i.e., it has the temperature Tcl just as the lower axial blanket, while the upper one is at the temperature Tc2. The cross sections of all zones are identical. For the reference design the calculation results in a permissible internal overpressure at the end of the lifetime of

is shifted towards lower values with increasing rod power. Figs. 9 and 10 show the influence of modified creep parameters on the conditions corresponding to fig. 7. (The cited values can be taken to be fairly representative of the material Incoloy 800, which has been repeatedly discussed as a cladding material; at

Pe = 99.6 atm which requires a length of the fission gas plenum under the circumstances given of Lp = 50 cm,

0.106

o. 123

Gm

o,m

MECH.,~ , , THERMAL , s-sl, MECH. STRESSES

b

~0

456 W / c m _ _

i.e., slightly more than half the length of the fuel zone. Assuming that this plenum should be not longer than the entire fuel zone, this result indicates that there is still a sufficient safety margin which would permit, e.g., the accommodation of an additional fuel swelling pressure of some 3 0 - 4 0 atm or an increase in temperature by somewhat more than 30°C. Fig. 7 shows the influence of wall thickness and rod power on the permissible final pressure Pe, Fig. 8 shows the same influence for the length of the gas plenum Lp. As a measure of the relative wall thickness in this case, the value ~0 = (ro r l ) / r o, and, in addition, the ratio x = cross section of cladding tube/cross section of fuel, are plotted on the abscissa. Fig. 7 for comparison also shows the permissible internal pressures without taking thermal stress effects into account. However, it should be noted that in this case for programming reasons only 2/3 of the permissible total creep strain was used in the calculation, i.e., the fraction of strain provided for ratchetting remained unused. On the other hand, the increase in the allowable amount of ratchetting with decreasing internal pressure (deviation from the dashed curve) was not taken into account. This makes the results even more conservative, but their characteristic behavior is essentially preserved. It is seen that the thermal stresses show no significant effects if the wall thickness is below a certain minimum value; but for very thick walled tubes they cause a decrease of the permissible internal pressure. Hence, there exists an optimum wall thickness which

To

,,,i

120

I00 r

~O W/em

80 ~

!/IsO w,e m "--I 6Co.2o o.2~

~2s

~

03o

o.Jas o.~

o..r~ -o.L 1

Fig. 9. Influence of modified creep parameters (K = 10-]5, n = 9) on the relations from fig. 7.

Lpg~]

~o

,

I

1

O.2O O.225 0.2S

l. . . . . O.275

I O.JO

O325

~i

~

X~

Fig. 10. Corresponding gas plenum length.

,

0.4

A NEW METHOD FOR CALCULATING THE DESIGN STRENGTH OF FUEL CANS 10 kpmm -: they result in the same creep rate as does the austenitic steel taken as basis for figs. 7 and 8.) It is seen that the individual curves remain basically unchanged. Because of the more rapid stress relaxation due to the higher Norton exponents the influence of thermal stress is stronger and starts earlier. The optimum wall thickness is shifted towards higher values as a result of the dominating influence of the mechanical stresses which are inversely proportional to ~0' The effect o f an increased or decreased permissible circumferential creep strain, ect, is shown in figs. 11 and 12. It clearly turns out that this quantity is of considerably greater significance if thermal stress effects are taken into account, compared to pure internal pressure loading. The comparison with fig. 7 shows that a factor o f 2 or 0.5 of ect has a much stronger influence on the maximum sustainable internal pressure than would have to be expected from pure application of Norton's formula (solid curves against dashed curves), Especially, for ect-values > 0.3% ratchetting effects do not occur unless the wall thickness or rod power is near the upper limit o f the probable design range. For ect-values less than 0.075%, there is practically no margin left for design; therefore the situation becomes rather critical. From this the conclusion may be drawn that maintaining a sufficient ductility under high neutron doses is almost more important for the choice and development of cladding materials than increasing the creep-rupture strength which had been the primary goal up to now. Finally, it should be pointed out that the results discussed here and obtained by the simplified process of calculation derived in section 4 can be checked more thoroughly in critical cross sections by means of the original method described in section 3. As an example, fig. 13 shows the stress distribution calculated by this method for the most critical tube cross section of the cited reference design and the corresponding substitute distribution which obviously overrates the additional strain produced per phase.

~,~,.~

d aaur

o.~zJ

ato~

I

.~CH..

-

MECH. SI"RESSE$

r ~

133

o.~J

e. ,.

o. s

srt~ss~ I

[

I

I

~.,~

;.-22. I'

II

' ~o ~2

0.~

~T~

t 0.3~ X

i

O~

Fig. 11. Effect of increased permissible creep ,strain ( ~ = 0.3%),

aoe7

O.2

~25

a~e r

a25

r

o.~J t

¢~,"rJ 0 ~

]

~m -

O.325 0 , ~

o.m

0..~ X

0.,~ =

Fig. 12. Effect of reduced permissible creep strain (ect = 0.075%). I

..! e ~ / , , , . 2 1

6. Conclusion The information obtained by the method outlined should be regarded as being mainly qualitative in view of the numerous uncertainties which still exist in the

Fig. 13. Actual against substitute stress distribution after 100 hr for the reference design.

134

G.SCHMIDT

data and formulae for the creep behavior o f the material, especially u n d e r irradiation. (However, the results are very likely to be on the safe side.) The necessity o f extensive irradiation experiments yet remains unchanged, However, the m e t h o d is to supply the basis for better defining the objectives of such experiments and to integrate the results into an overall picture which facilitates their interpretation, The basis so o b t a i n e d will be useful in particular to compare different c o m b i n a t i o n s o f design and operating parameters, thereby facilitating the o p t i m i z a t i o n of fuel rod designs even w i t h o u t accurate knowledge o f all relevant data.

x z a ath /3 e ec ee

ect eel etl

7. Notation /)

A

B C CL E F G,H K Lp T

rcl re2 b,c,d,e

TL m n

P Po Pe r ri ro s t

tl t* te ts tz

= abbreviation for a special f u n c t i o n o f total strain rates, taken as a c o n s t a n t = modified creep c o n s t a n t = c o n s t a n t (general) = material c o n s t a n t (Larson-Miller) = Young's m o d u l u s (kp m m -2) = area of redistribution in stress diagram = f u n c t i o n names = creep c o n s t a n t ( N o r t o n ) = length of fission gas p l e n u m = temperature (general) = = = = = = = = =

coolant inlet temperature coolant outlet temperature special distances in stress diagram load factor o f reactor operation time hardening e x p o n e n t creep e x p o n e n t ( N o r t o n ) internal pressure (kp cm -2) initial internal pressure (kp cm -2) permissible int. pressure at end o f lifetime (kp cm -2) = radial coordinate ( m m ) = inner radius o f tube ( m m ) = outer radius o f tube ( m m ) = dimensionless effective stress = time (hr) = length o f first time i n c r e m e n t (hr) = n o n d i m e n s i o n a l time = effective lifetime o f fuel pin (hr) = entire lifetime o f fuel pin (hr) = d u r a t i o n of one reactor power cycle (hr)

~o O" 0c 0e 0L

OL1 OL2 Oem t oe Oy 7

x

= = = =

cladding cross section/fuel cross section n u m b e r o f " e f f e c t i v e " cycles stress ratio thermal expansion coefficient of tube material (°C -1 ) = stress ratio = total strain = creep strain = effective strain = permissible total tangential creep strain = elastic strain = permissible tangential strain due to "ratchetting" = current time coordinate = thermal conductivity o f tube material (W • (cm ° C ) - l ) = Poisson's ratio = dimensionless radial coordinate = dimensionless wall thickness = stress (general) (kp m m - 2 ) = formally generated effective stress (kp m m -2) = effective stress (kp m m -2) = limiting stress analogous to Oy (general) (kp m m -2 ) = limiting stress at full power (kp m m -2 ) = limiting stress at zero power (kp ram-2 ) = effective stress due to mechanical loading (kp m m -2 ) = = = =

effective stress due to heat flux (kp m m -2 ) yield stress (kp m m -2) current time coordinate rod power (W/cm)

Indices." = current index = indices (general) t = in tangential direction (or corresponding to o t as "leading" stress) r = in radial direction m = corresponding to crr as m e a n principal stress a = in axial direction (or corresponding to o a as "leading" stress) 0 =att=0 1,2,3,4,5, = c o u n t i n g index for consecutive stress changes o~ = for t ~ co (asymptotic steady state) 1)

i,j,k

G.SCHMIDT Pref~: A

= difference or increment, resp.

Superscripts: - (bar) = denotes m e a n value over wall thickness • ( d o t ) = denotes differentiation with respect to time

References [1] D.R.Miller, Thermal-stress ratchet mechanism in pressure vessels, Trans. ASME, J. Basic Engng, June 1959, 190196. [2] K.M.Horst, Fast oxide breeder-stress considerations in fuel rod design, GEAP-3347 (1960). [3] H.B6hm, H.J.Hauck, H.J.Laue, Multiaxial in-reactor stress rupture strains of austenitic stainless steels and a nickel alloy, Int. Symp. on Effects of Radiation on Struct. Metals, San Francisco, June 1968. [4] J.H.Shively, Thermal gradient effects on stress-rupture behaviour of thin-walled tubing, AI-AEC-12 696, June 1968. [5] J.Bree, Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with applications to fast-nuclear-reactor fuel elements, J. of Strain Analysis, vol. 2, no. 3 (1967).

135

[6] R.Hibbeler, T.Mura, Viscous creep ratchetting of nuclear reactor fuel elements, Nucl. Eng. Design 9 (1969) 131143. [7] L.F.Coffin, P.R.Shepler, G.S.Cherniak, Primary creep in the design of internal pressure vessels, Trans. ASME, J. Appl. Mechanics 16 (1949) 229-241. [8] J.F.Besseling, Investigation of transient creep in thickwalled tubes under axially symmetric loading, IUTAM Colloquium on Creep in Structures, Stanford 1960 (Proc. Berlin: Springer, 1962). [9] A.Mendelson, M.H.Hirschberg, S.S.Manson, A general approach to the practical solution of creep problems, Trans. ASME, J. Basic Eng., Dec. 1959, 585-598. [ 10] H.Poritsky, F.A.Fend, Relief of thermal stresses through creep, Trans. ASME, J. of Appl. Mechanics, Dec. !958, 589-597. [11] J.P.Yalch, J.E.McConnelee, Plain strain creep and plastic deformation analysis of a composite tube, Nucl. Eng. Design, Febr. 1967, 52-62. [ 12] C.R.Kennedy, W.O.Harms, D.A.Douglas, Multiaxial creep studies on lnconel at 1500°F, Trans. ASME, J. Basic Eng., Dec. 1959, 599-609. [ 13] F.K.G.Odqvist, J.Hult, Kriechfestigkeit metallischer Werkstoffe (Springer, 1962). [14] F.R.Larson, J.Miller, A time-temperature relationship for rupture and creep stresses, Trans. ASME, July 1952, 765-775. [15] Ein Rechenverfahren zur festigkeitsmassigen Auslegung der Brennstabhiillrohre bei fliissigmetallgekiihlten schnellen Reaktoren, KFK 808, EUR 3968 d, Juli 1968. [ 16 ] K.Gast, E.G.Schlechtendahl, Schneller Natriumgekilhlter Reaktor Na 2, KFK 660, EUR 3706d, Okt. 1967.