4.01.3.2.1 Irradiation creep: Macroscopic behavior

Under neutron irradiation, metals exhibit a high creep rate, much higher than the out-of-reactor ‘ther­mal’ creep rate, the creep rate increasing as the neu­tron flux increases. The behavior under irradiation of zirconium alloys, and particularly the creep behav­ior, has been studied extensively as pointed out by Franklin eta/.134 and Fidleris,150 because of the major importance of the prediction of the in-reactor defor­mation of the fuel assembly in the case of PWR and boiling-water reactor (BWR)169 or in-reactor struc­ture especially in the case of the CANDU

163,179

reactor.

It is usually assumed, for practical considerations, that the in-pile deformation consists of the sum of (i) the growth, (ii) the classical thermally activated out-of-pile creep, or so-called thermal creep, and (iii) the irradiation creep, strictly speaking.100,150,163,180 The ‘pure’ irradiation creep, subtracted from the two other components of the deformation, is the result of mechanisms which differ from the thermal creep and the growth. Nevertheless, these mechanisms are certainly coupled since they all imply disloca­tion loops, slip and climb of line dislocations, and point-defect bulk diffusion toward these defects. But very few authors have studied these potential

134,181

couplings.

The creep deformation under irradiation results, in fact, from two antagonistic phenomena. Indeed, while new deformation processes are activated, caus­ing the creep rate to increase, the thermal creep rate is strongly reduced by irradiation due to the irradiation-induced hardening. Indeed, it has been shown150 that a preirradiation reduces the thermal creep component of the deformation under irradia­tion. The effect of preirradiation on the reduction of the irradiation creep rate is particularly noticeable for RXA alloys. However, the hardening effect saturates at fluence of about 4 x 1024nm—2 and is followed by a steady-state creep rate. Concerning cold-worked materials, the effect of the preirradia­tion is much lower, according to Fidleris.150

As reported by several authors,134,150,153,182 the metallurgical state of the zirconium alloy has a sig­nificant effect on the in-reactor creep resistance. Indeed, while cold working may improve the thermal creep resistance of Zircaloy in certain test directions and stress range, it increases the in-reactor creep rate appreciably.150,153 Nevertheless, the creep sensitivity to the initial dislocation density is significantly lower than the growth sensitivity to the initial dislocation density.171 On the other hand, the grain size does not seem to have a significant effect on the creep strength in the range from 1 to 70 pm.

The in-reactor creep rate is very sensitive to irra­diation as well as loading conditions. The effects of flux, as well as the effect of stress, are usually described by a power correlation. The effect of tem­perature is usually described by an Arrhenius equa­tion.134 However, since it is in general very complex to distinguish between the ‘pure’ irradiation creep and the thermal creep, the authors usually use an overall creep constitutive law (eqns[1] and [2])163,180 and only growth is taken into account as a separate defor­mation component.

e ethermal—creep T ^irradiation—creep T egrowth

ecreep T egrowth Ц

with

ecreep = K s”ffexp(jT^

where e is the strain rate in s—1; s is the effective stress for thermal creep in MPa; n is the stress exponent; Tis the temperature in K; Qis the activation energy inJ; R is the gas constant, 8.31 J K—1 mol—1; ф is the fast neutron flux in n m—2 s— (E > 1 MeV); p is the flux exponent; and K is a constant for thermal creep in s—1 (MPa)—n(nm—2s—1)—p. According to various authors,134,150 the flux exponent (p) has been assigned values ranging from 0.25 to 1. A flux exponent ofp = 1 is commonly obtained for CANDU pressure tube deformation.163,183 For uniaxial creep tests per­formed at 280 °C on cold-worked Zy-2, Tinti184 has obtained a flux exponent increasing from 0.6 to 1.0 with increasing instant flux.

A stress exponent of n = 1 is obtained at 300 °C for low applied stress (s < 100 MPa). As the stress increases, the stress exponent increases, reaching values up to n= 25 for 450 MPa applied stress for cold-worked Zr-2.5% Nb.183

The effect of temperature on the creep rate can be rationalized by plotting the creep rate in an Arrhenius plot (logarithm of the creep rate vs. inverse temperature). The activation energy is then the slope obtained in this plot. It can be seen in Figure 23 that for low temperatures, the creep activa­tion energy Q/R is very low, between 2000 and 5000 K.150,163 The irradiation creep at low temperature is therefore nearly athermal. At higher temperatures, the dependence increases rapidly toward values of Q/R of 25 000-30 000 K. These last values are close to the activation energy measured for thermal creep. These observations tend to prove that for low-temperature regime, mainly ‘pure’ irradiation creep mechanisms are activated. As the temperature increases, the ther­mal creep mechanisms become activated, yielding to activation energy close to the thermal creep values.

It has also been shown by several authors that while the thermal creep of zirconium alloys is aniso­tropic, the irradiation creep remains strongly anisotropic.150 According to Holt,171 the anisotropy of irradiation creep is nevertheless slightly lower than that of thermal creep.

4.01.3.2.2 Irradiation creep: Mechanisms

Various mechanisms for irradiation creep have been proposed in the literature as reviewed by Franklin et a/.,134 Holt,163,171 Matthews and Finnis,181 and Was.9 A nice history of the proposed mechanisms for both zirconium alloys and stainless steels is given by Franklin et a/.134 These mechanisms can fall mainly into two large categories:

1. The mechanisms based on stress-induced prefer­ential absorption (SIPA) of point defects by line dislocations arising from different fundamental phenomenon. These mechanisms lead to the climb of edge dislocations under applied stress, yielding a creep deformation.

2. The mechanisms based on climb-enhanced disloca­tion glide mechanisms, which are essentially a com­bination of climb of dislocations due the absorption of point defects under irradiation and glide resulting in a creep deformation. For this category of mechan­isms, the strain is essentially produced by glide but the strain rate is controlled by the climb.

Other mechanisms involving irradiation-induced loops have also to be added to these two categories of deformation mechanisms involving line disloca­tions. Indeed, the stress-induced preferential nucle — ation (SIPN) of loops or the stress-induced preferential growth of loops due to SIPA can lead to an additional creep strain.

The SIPA mechanism is based on the fact that under an applied stress, the bias of the dislocation becomes dependent on the orientation of the Burgers vector with respect to the direction of the stress.105,134,181 Indeed, as described previously, due to a higher relaxation volume, the sink strength of an edge dislocation toward SIAs is higher than toward vacancies. This difference in sink strength is the bias of the edge dislocation. It can be shown that a dislo­cation with a Burgers vector parallel to the applied stress exhibits a higher bias toward SIAs than a dislo­cation with a Burgers vector perpendicular to the applied stress. Therefore, under irradiation, the net flux of SIAs (SIA flux minus vacancy flux) toward dislocations, with Burgers vector parallel to the applied stress, is higher than the net flux of SIAs toward dislocations with Burgers vector perpendicular to the applied stress. This difference in the absorption of point defects by different types of dislocations leads to dislocation climb, resulting in a creep strain. The SIPA creep rate is insensitive to the grain size but is sensitive to the dislocation network.

However, it has been seen that for growth, the anisotropic diffusion of SIAs is believed to play an important role in the deformation mechanism. There­fore, any irradiation creep model proposed for zir­conium should also include anisotropic diffusion. The SIPA model that includes anisotropic diffusion is called the SIPA-AD model and has been reviewed by Matthews and Finnis.181

In the case of RXA zirconium alloys, the irradia­tion creep mechanisms are not clearly identified yet. Indeed, since the initial dislocation density is very low, another deformation mechanism has to be acti­vated. The creep strain could be partly due to the preferred nucleation and/or growth of the (a) type loops in the prismatic planes. Indeed, according to the SIPN or SIPA mechanism, the nucleation or growth of interstitial (a) loops can be favored in the prismatic planes perpendicular to the applied stress. For the same reason, the nucleation or growth of vacancy (a) loops can be favored in the prismatic planes parallel to the applied stress, leading to a resulting creep strain. According to Faulkner and McElroy,185 an applied stress increases the mean diameter of (a) loops without affecting the density, proving that the SIPA mechanism is efficient in their experiment. However, the growth of (a) loops under an applied stress can explain the measured creep strain only for low strain levels. Indeed, this creep strain should remain limited since the (a) loop den­sity and mean loop diameter saturate at relatively low doses. Since the initial dislocation density is very low in RXA zirconium alloys, creep mechanisms involv­ing climb of dislocations due to the SIPA mechanism or climb-plus-glide of dislocations require the gener­ation of a dislocation network. It is possible that (a) loops coalescence occurs, resulting in the creation of a dislocation network that is able to climb and glide under stress.181,186 However, this network is clearly observed only at 400 °C.67 Other types of dislocation sources, such as Frank-Read or Bardeen-Herring sources,147 can also be activated under both irradia­tion and applied stress, leading to the creation of a dislocation network that undergoes a SIPA or climb — enhanced glide mechanism.

It should also be pointed out that in order to explain the observed creep rate, some mechanisms must be activated that allow the dislocations to over­come the high density of dislocation loops during their climb and glide motion, even for low applied stress. It is possible, as pointed out by MacEwen and Fidleris187 in the case of Zr single crystal, that the gliding dislocations are able to clear the loops during in-pile deformation, leading to the dislocation chan­neling mechanism. All these mechanisms probably occur in series, as proposed by Nichols,188 explaining the evolution of the stress dependency as the stress increases. Indeed, according to this author, for zero applied stress, growth of zirconium occurs, and then as the stress increases, (a) loop alignment occurs (SIPA on loops). For higher stress, the climb of line dislocations via SIPA takes place, and then the dislo­cation climb and glide processes occur at even higher stress. For very high stress, close to the YS, disloca­tion channeling occurs.

For cold-worked zirconium alloys, such as SRA Zircaloy or cold-worked Zr-2.5Nb alloy,163 the SIPA mechanism on the initial dislocations is a likely mech­anism for irradiation creep. However, according to Holt,171 the creep anisotropy of cold-worked zirco­nium alloys computed from the SIPA mechanism assuming only (a) type dislocations is not in agree­ment with the experimental anisotropy. The anisot­ropy computed from the climb-plus-glide mechanism assuming 80% prism slip and 20% basal slip is in good agreement with the experimental anisotropy, demon­strating that climb-plus-glide mechanism is probably the effective mechanism. It should also be pointed out that, since dislocations climb toward grain boundaries or toward other dislocations, recovery of the initial dislocation network occurs. In order to maintain a steady-state creep rate, multiplication of dislocations should also occur either via loop coalescence or via dislocation sources, as discussed previously.

It should also be pointed out that, as there is a coupling between swelling and irradiation creep in stainless steel,181 we could assume a coupling between growth and irradiation creep to occur in zirconium alloys due to the effect of the stress on the partitioning of point defects.134,162 Nevertheless, the simple assumption of two separable deformation components has proved to hold correctly for the results given in the literature.163,180

4.01.3.2 Outlook

Concerning damage creation and point-defect clus­ter formation, improvement in the knowledge of anisotropic diffusion of SIAs as well as better under­standing of the microstructure of vacancy and inter­stitial (a) loops and basal (c) vacancy loops (origin of the loop alignment, origin of the corduroy contrast for instance) has to be aimed at. Multiscale modeling approaches coupled with fine experimental analyses of the irradiation microstructure (high-resolution TEM, synchrotron radiation analyses, tomography atom probe, etc.) should bring new insight concerning the previous points mentioned but also elements in order to propose modeling of the microstructure evolution during irradiation: for instance, origin of the alignments of Nb precipitates, stability of p-Nb pre­cipitates, etc.

Concerning the mechanical behavior of Zr alloys after irradiation, multiscale modeling of the postirra­diation deformation with a better understanding of the dislocation channeling mechanism and under­standing of its effects on the postirradiation mechan­ical behavior are needed.

Moreover, better understanding of the postirradi­ation creep deformation mechanisms is also needed using multiscale modeling.

The last point concerns the deformation mechan­isms under irradiation. In that field, the basic questions are still without answers: What are the irradiation creep deformation mechanisms? What are the coupling between the deformation under irradiation and the thermal creep and growth? Progress has to be made especially using in situ deformation devices under irradiation, coupled with modeling approaches. (See also Chapter 1.01, Fundamental Properties of Defects in Metals; Chapter 2.07, Zirconium Alloys: Properties and Characteristics and Chapter 5.03, Corrosion of Zirconium Alloys).