On the creep characteristics of β-zircaloy

Journal of Nuclear Materials 78 (1978) 43-48 0 Nor-~oUand Publishing Company

ON THE CREEP CHA~CTERISTICS

OF fi -2IRCALOY

H.E. EVANS and G. KNOWLES Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, GL13 9PB, UK Received 3 April 1978

A recent model of creep deformation in pure materials is compared with published data on p-zirconium and p-zircaioy. It is shown that the zircaloys behave like the pure metal in this phase range and that the model is capable of predicting their creep rates to within a factor of 2 on tubing material. A condition is that the curved Arrhenius plot for diffusion in zirconium represents the intrinsic behaviour of point defects in the lattice and that solute atoms have an insignificant effect on the process. A corollary is that the contribution from dislocationshort-c~cuitingboth to diffusionand to creep is ne~i~ble.

1. Introduction

2. The creep model

During a postulated loss-of-coolant accident in a water reactor, the reduction in heat-transfer efficiency leads to a rapid increase in temperature of the zircaloy cladding. In addition, the pressure within the fuel pins results in a progressive increase in tensile stress in the cladding as the reactor core depressurises. The combined effect is to produce outward creep of the fuel pin with the accompanying risk of localised b~ooning or clad failure. The behaviour of the entire reactor system in such an accident is thus dependent to a large extent on the creep properties of the zircaloy cladding. This aspect has certainly been recognised in recent papers [l-4] which attempt to predict the deformation response and rupture temperatures of transiently heated fuel tubes. A difficulty with these has been the necessary use of creep equations obtained empirically from isothermal tube burst data which, although producing reasonable agreement with the experiments, cannot be extrapolated reliably, particularly into the high-temperature, P-phase region. Because of this, there has been much recent activity to determine the creep characteristics of this phase [S-8] and steady-state creep data are now available to 1773 K [8]. The purpose of this paper is to attempt to provide a physical basis for the creep of @zircaloy in a nonoxidising atmosphere by using a recent model of creep [9,10] known to apply to a diversity of pure, singlephase materials.

The model has been presented in detail in ref. [9]. Essentially, it envisages the potentia~y glissile dislocations as being arranged in a 3-dimensional network whose size X is inversely proportional to the applied, generalised stress u. The rate-controlling step is the release of a dislocation loop from the network by the climb of network links to an extent necessary either to operate a Frank-Read source or to break a network node. The migration of vacancies to and from climbing links can occur both through the lattice and along dislocation cores; the former is favoured by high temperatures and low stresses and the latter by the reverse conditions. An important feature of the model is that creep strain occurs entirely by the glide of released dislocations and that this glide step occurs instantaneously. It is also assumed that a gliding loop will be stopped after some critical distance s where, in the model’s most direct form, s is identified with the network spacing Y. This leads to the following equation for steady-state creep rate, is: is =

fi

4.2n03b a2p2kT

I

DL

n(q42a) ‘x

3Dpu2 I

where & is the Burgers vector of a unit dislocation, DL and Dp are respectively the lattice and dislocation-pipe 43

H.E. Evans and G. Knowles / Creep characteristics of p-zircaloy

44

diffusion coefficients, p is the shear modulus, v is Poisson’s ratio, (Yis a constant obtained from the relationship X = crd/o, k is Boltzmann’s constant and T is the absolute temperature. Here, the first term in the first square brackets reflects the contribution from the lattice diffusion path and the second term, the pipe diffusion path. The composite term in the curly brackets is the residue, after factoring, of the summation of forces favouring climb and involves, initially, terms incorporating the applied stress, the line energy of the network and the interaction energy between network links. The model also allows for perturbations to the slip distance x through extraneous factors such as a subcell size of order X or occasional second-phase particles or stringers produced during fabrication. The approach [9] was simply to accord a fixed value to s, thus modifying eq. (1) to:

I

1

I

1

10-11

s E

+

10-13

.-E .0 z 8 ” : ._

2 101: Ic 0

1 -.

To summarise, eqs. (1) and (2) predict stress exponents of creep rate in the range 3-4 provided that vacancy transport occurs predominantly through the lattice; the corresponding activation energy for creep would be that for lattice self diffusion. Stress exponents in the range 5-6 occur when vacancy transport takes place predominantly along dislocation cores; the corresponding activation energy for creep would then be that for pipe diffusion. It is recognised that the creep model was developed for a notionally pure material for which solute atoms do not produce a significant retardation of dislocation glide. Its application to the zircaloy alloys in the P-phase is justified on the grounds that comparison of existing creep data (see e.g. sect. 4) show no systematic differences between zircaloy-2 [7,8], zircaloy4 [S] and pure zirconium [ 1 l] over which range solute atom concentrations vary considerably.

3. Input parameters 3.1. Diffusion

10-l’

(2)

coefficients

Since the creep properties of P-phase zircaloys seem closely similar to those of pure zirconium it is probable

1

5.0

I

1

1

6.0

6.0

7.0

104 f.

9-O

-1 K

Fig. 1. Curved Arrhenius plot for self-diffusion in p-zirconium

[131.

that the diffusion characteristics are also similar. This surmise allows the use of the extensive self-diffusion data on the pure metal in this comparison. &zirconium is one of the classic examples of anomalous diffusion in the bee structure in that it exhibits a curved Arrhenius plot (fig. 1) and low frequency factors at lower temperatures. The various theories which have been proposed to account for this behaviour have been summarised by Douglass [ 121 and of these perhaps only two are now in contention. The first, probably attributable to Kidson (e.g. ref. [ 133 from which the mean line of fig. 1 was obtained) envisages the curvature as being due to an increasing contribution from dislocations with decreasing temperature. He calculates a reasonable lattice activation energy of -274 kJ/mole which applies at temperatures approaching the melting temperature (T,), decreasing to a pipe-diffusion value of N 116 kJ/mole applicable at temperatures approaching the phase transformation. The second theory, e.g. ref. [ 141, presumes a systematic change in elastic constants

Table 1 Values of diffusion coefficient and shear modulus for evaluation of creep rate ~I~ Temperature DL (from ref. [ 131) @(from ref. [17]) X 10’ 3 (m2/s) X 1O-1o (N/m*) G) ~^_ 1273 1.44 1.55 1323 2.12 1.48 1373 3.00 1.41 1423 4.25 1.34 1473 6.50 1.27 1523 9.00 1‘19 1573 13.8 1.12 1673 23.6 0.98 44.6 1773 0.83 -

as Ihe phase transfo~ation is approached, resulting in a decrease in both the energies of migration and formation af a vacancy. In the present context, the important difference from Kidson’s view is that diffusion occurs predominantly by lattice diffusion at ail temperatures. Reference to the Arrhenius plot of fig. 1 shows that significant departures from linearity occur at temperatures as high as 0.85T,, and for this to be associated with pipe diffusion a high dislocation density, say -1 07/mm2, would be required. This is the main weakness of the Kidson theory since the rapid recovery rates operating at these temperatures should soon reduce the density to much lower values. Indeed, tests incorporating annealing stages [15] have failed to produce the decrease in diffusion rates, reasonably to be expected if dislocation short-circuiting played an ~portant role. In a later paper [6], Kidson and Kirkaldy argue that lower dislocation densities do in fact apply but their argument requires implausibly low values of activation energy and frequency factor for lattice diffusion. It seems more likely that the curved Arrhenius plot (fig. 1) represents the intrinsic be~aviour of iattice diffusion (e.g. 1141) and this is the view adopted in this note. The numerical values used subsequently are given in table 1. Since the comparison with creep data is confined to temperatures >1273 K (Z&SOT,) it is considered that the contribution from pipe diffusion for likely dislocation densities will be negligible small and no attempt is made here to estimate the equivalent curve to fig. 1 for the pipe diffusion coefficient.

The shear modulus, p, was obtained from the measurements of Young’s modulus for zirconium of Armstrong and Brown [ 171, using v = 0.33. Extrapolation beyond their upper limit of 1473 K was made assuming a temperature invariant dependence, dy/dT of - 14.2 MN/m2 * K. The numerical values used in the comparison are given in table 1. The Burgers vector, b, was taken as 3.1 I X IO-“m (=43/2n), where the lattice spacing a is that given by Aleksevsky et al. [IS] as 3.59 X lo-” m. The slip distance s [eq, (Z)] must be of the same order as the fikeiy network spacing because eqs. (2) and (1) should predict broadly similar values of creep rate. The value of 2 X 10m3mm chosen for s is the same as that used previosuly [9,10]; similarly, the value of (Yis set at 1.6. 3.3. The creep du fa Ardelf’s data [ 11] are used for pure zirconium, together with four other pieces of work on zircafoy f5-81 of which &lendenn~ng [5] used zircaloy-4 and the others [6-81, zircaloy-2. Ardeil fl 11 tested in uniaxiaf constant-stress conditions and his reported values of steady-state creep are used. Burton et al. [6] used uniaxial, constant-load conditions with laboratory-grade wire and report steady-state conditions after -2-3% strain. The reported rates have been used in the cornparison since the overestimates of true steady-state rates are generally
Figs. 2 and 3 contain all the reported creep data on P-zircafoy and encompass the temperature range

HJ. Evans and G. Knowles / Creep characteristicsof pzircaloy

46

Data

sources

to-2

l

Ctay and Stride

q

Burton

17 1

Reynolds and Barnes161

0 Clendenning f 5 I

Zircaloy-2

10-3

Zircaloy-4 to*

0.2

04

0.6

T=1323K

T= 7273K 103 -

/’

eqn. 2

Log.@ apf?iedO’Lgtre*s-(%N/rn2 ) O Fig. 2. Comparison of theory with data for the temperature range 1273-1373

sources [7,8) Clay and Stride a Burtan Reynolds and Barnes 161

K.

Data

l

0

Clendenning

x Ardell

1111

[51

Zircaloy-

2

Zircaloy - 4 Zirconium 10” c

10-I -

T= 1473 K

10-S -

10-4 -’ , 10-5 -, /

eqn. 2

to* _ -0-2

Loglo applied

stress (MN

’ ) P Fig. 3. Comparison of theory with data for the temperature range 1423-1473

K.

H.E. Evans and G. Knowles [Creep characteristicso~~zircaloy

Data

lo-’ -

T= 1773 K

sources

47

Tz1673K

fircaloy-2

k 1oQ -

,r Clay 8, Stride [81

1

0

I10-Zt

l

3 Burton Reynolds ,O_a

I

& Barnes

[61

cqn.2 0

0.2

0.4

,l _

“‘-

T= 1523 K 4_

l

T= 1573 K

3

/ r’

b ,

1

5,

-

0.2

0

0.2

0.4

O-6

Fig. 4. Comparison of theory with data for the temperature range 1523-1773 K.

1273-1773 K. Ardell’s data [ 111 on zirconium are shown at 1473 K at which temperature his experimental range of stress was most extensive. There is appreciable scatter amongst his results but, even so, the creep rates obtained are in close agreement with those of Clay and Stride [7,8] on zircaloy-2. Comparison of the data at 1273 and 1373 K (fig. 2) and again at 1473 K (fig. 3) also shows excellent agreement between Clay and Stride’s results on zircaloy-2 and Clendenning’s [S] on zircaloy-4. However, the temperature of 1523 K (fig. 4) is the only one at which the data of Burton et al. [6] can be compared directly with the others. Clearly, there is a discrepancy in creep rate of nearly an order of magnitude, the wire material [6] being weaker. There is also a tendency, apparent at 1323 and 1423 K for it to exhibit a lower creep index. The theoretical lines drawn on the figures represent the evaluation of eqs. (1) and (2) using the parameters given previously (sect. 3 and table 1) and assuming negligible contribution from pipe diffusion. it is clear that eq, (2) is capable of predicting the creep rates ob-

served by all workers, with the exception of Burton et al. [6], to an error consistently less than a factor of 2 over the whole temperature and stress range studied. On the other hand, the wire data can be predicted with similar accuracy by eq. (I). This discrepancy within the data probably reflects the difference in testing techniques. Thus, both Burton et al. and Clay and Stride used direct heating but the wire specimens of the former rapidly acquired a bamboo grain structure with concurrent necking of individual grains accompanied by a localised increase in temperature. This effect would lead to an enhanced creep rate. Such local ‘soft spots’ under biaxial conditions [7,8] would tend to be constrained and not to affect the overall deformation rate [19]. The magnitude of the error is uncertain but a systematic overestimation of creep rate will occur in the wire specimens. Because of this, it is simplest to consider the intrinsic creep properties of fl-zircaloy as being represented by eq. (2), which adequately describes the tube data.

H.E. Evans and G. Knowles / Creep characteristics of pzircaloy

48

5. Concluding

References

remarks

It has been shown that negligible difference exists between the creep properties of zirconium, zircaloy-2 and zircaloy-4 in the P-phase. It is concluded that solutes have no significant effect on the creep process. This, in turn, is considered to be one of recovery creep within the meaning of a recent model [9]. Good fit is obtained with this model (with a predictive error consistently less than a factor of 2 in creep rate for tubing material) provided that the curved Arrhenius plot of zirconium self-diffusion represents the intrinsic behaviour of lattice diffusion. The contribution to diffusion from dislocations is considered negligible for the whole of the P-phase. A corollary of this agreement is that the self-diffusion characteristics of the alloys are well represented by those of the pure metal. The creep behaviour of the zircaloys, at least over the temperature range 1273-l 773 K, is adequately given by eq. (2) of the text which is, in simplified form: P, = 14.1 -

su4

p3kT

DL ln(0.8p/o)

[I j K.R. Merckx, Proc. 3rd Int. Conf. Structural [ 21 [3]

[4]

[5] [6] [7] [S] [Y] [lo] [ll]

[ 121 [13] [ 141

’

where e, is in s -‘, u is the generalised stress (N/m*), k = 1.38 X1O-23 J/K, s = 2 X 10e6 m, DL is given by fig. 1 or in table 1 and p is given in table 1.

Mechanics in Reactor Technology, London (1975) paper C3/7. T. Healey, H.E. Evans and R.B. Duffey, J. Br. Nucl. En. Sot. 15 (1976) 247. P.M. Jones, J.H. Gittus and E.D. Hindle, Proc. Specialist Meeting, Spatind, Norway, CSNI Report No. 13 Part 1 OECD/Nuclear Energy Agency, (1976). B.D. Clay, T. Healey and G.B. Redding, Proc. Specialist Meeting, Spatind, Norway. CSNI Report No. 13 Part 1. OECD/Nuclear Energy Agency (1976). W.R. Clendenning, Proc. 3rd Int. Conf. Structural Mechanics in Reactor Technology, London (1975) paper C2/6. B. Burton, G.L. Reynolds and J.P. Barnes, J. Nucl. Mat. 73 (1978) 70. B.D. Clay and R. Stride, CEGB report RD/B/N3782 (1977). B.D. Clay and R. Stride, CEGB report RD/B/N3950 (1977). H.E. Evans and G. Knowles, Acta Met. 25 (1977) 963. H.E. Evans and G. Knowles, Acta Met. 26 (1978) 141. A.J. Ardell, Ph.D. Thesis, University of Stanford, California (1964). D.L. Douglass, The Metallurgy of Zirconium (IAEA, Vienna) (1971) p. 293. C.V. Kidson, Electrochem. Techn. 4 (1966) 193. H.I. Aaronson and PG. Shewmon, Acta Met. 15 (1967) 385.

[15] G.V. Kidson 1146. [16] G.V. Kidson 1057.

and J.F. M&urn,

Can. J. Phys. 39 (1961)

and J.S. Kirkaldy,

Phil. Mag. 20 (1969)

[ 171 P.E. Armstrong 230 (1964)

[ 181 N. Ye Aleksevsky,

Acknowledgement This paper is published by permission Electricity Generating Board.

of the Central

and H.L. Brown,

Trans.

Met. Sot. AIME

962.

O.S. Ivanov, 1.1. Rayevsky and M.V. Stepanov, Fiz. Metal. Metalloved 23 (1957) 28. [ 191 T. Healey, B.D. Clay and R.B. Duffey, CEGB report RD/B/N4154 (1977).