Modelling of Pellet-Cladding Interaction Induced

1998:28

LULEL

UNIVERSITY OF TECHNOLOGY

Modelling of Pellet-Cladding Interaction Induced Failure of Light Water Reactor Nuclear Fuel Rods

Lars Olof Jernkvist

Department of Mechanical Engineering Division of Solid Mechanics 1998:28 • ISSN: 1402 - 1757 • ISRN: LTU - LIC - - 98/28 - - SE

Modelling of pellet-cladding interaction induced failure of light water reactor nuclear fuel rods

Lars Olof Jemkvist

Division of Solid Mechanics Luleå University ofTechnology SE-971 87 Luleå, Sweden September, 1998

Abstract Failures of nuclear reactor fuel are extremely rare occurrences, and most power plants are today operated without defective fuel elements in the core. This salutary reliability is a result of many years of targeted endeavours to reduce potential causes of failure. However, one remaining elusive mechanism for fuel failure is attributed to iodine-induced stress corrosion cracking of the thin-walled zircaloy cladding tubes that encapsulate the fissile fuel pellets. Mechanical and chemical interaction between the fuel pellets and the surrounding cladding are the underlying causes for this failure mechanism, which is the topic of the submitted thesis. A failure model is presented, in which the formation and propagation of stress corrosion cracks are treated by nonlinear fracture mechanics concepts in a finite element computational framework. The model is intended for predictive analyses of in-reactor pellet-cladding interaction (PCI) induced failures, and the propensity to failure is related to the cladding material properties and the in-reactor conditions, such as temperature, chemical environment and stress state. Since the predictive capacity of the proposed failure model is strongly conditioned by the accuracy by which the in-reactor behaviour of the pellet and cladding materials is represented, appropriate constitutive models have been developed for these materials. The constitutive models are focused on the material response under transient loading conditions, pertinent to the rapid increases of fuel rod power that are connected with the occurrence of PCI induced fuel failures. The introduced models have been verified and calibrated with experimental data, and conclusions regarding their applicability are presented.

List of contents This licentiate thesis concerns modelling of pellet-cladding interaction induced failure of light water reactor fuel rods. It comprises a short survey of the subject and three appended publications.

Survey 1

Introduction

2

Design and operational conditions of LWR fuel 2.1 Design of LWR fuel 2.2 Operational conditions

3

Mechanisms of pellet-cladding interaction 3. 1 Pellet-cladding mechanical interaction 3.2 Pellet-cladding chemical interaction

4

PCI-induced fuel rod failures 4.1 Failure mechanisms 4.2 Remedies

5

Models for the failure process 5.1 Existing approaches 5.2 Proposed approach

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Conclusions Acknowledgements References

Papers 1-III A model for predicting pellet-cladding interaction induced fuel rod failure Nuclear Engineering and Design, 156, 1995, pp 393-399. II

A continuum model for cracked UO2 fuel Nuclear Engineering and Design, 176, 1997, pp 273-284.

III

A model for inelastic deformation of irradiated zirconium alloy cladding under transient conditions Submitted to the 15th int conf on structural mechanics in reactor technology (SMiRT-15)

Appendices A-I

Supplementary information to paper I

A-il Supplementary information to paper II

II

1 Introduction The nuclear fuel in a typical light-water-reactor (LWR) core consists of some 40 000 fuel rods, each of which is made up of a thin hermetically sealed cladding tube which encapsulates uranium dioxide fuel pellets. The mechanical integrity of these cladding tubes is of outermost importance, since they provide a barrier between the radioactive fuel material and the reactor coolant circuit. The reliability of LWR fuel has improved immensely over the last decades, and the average fuel rod failure rate is currently below 2.10-5 failures per operational core cycle [1]. The low failure rate, which has largely been accomplished by reducing cladding failures due to manufacturing defects and debris fretting, means that most reactor cores are today operated without defective fuel rods and thus preserved from radioactive contamination of the coolant primary circuit. However infrequent, fuel failures lead to substantial costs in terms of lost capacity, core contamination and increased amounts of radioactive waste. These costs provide impetus for further reduction of the failure rate. A potential cause for fuel rod failures is due to the pellet-cladding interaction (PCI) phenomenon, a failure mechanism attributed to iodine-induced stress corrosion cracking of the zircaloy cladding. The phenomenon includes both mechanical and chemical interaction between the UO2 fuel pellets and the surrounding zircaloy cladding tube [2]. Under rapid increases of fuel rod power, the pelletcladding mechanical interaction induces high tensile stresses in the cladding. These stresses in combination with chemical attack by iodine and other aggressive fission products, emanating from the fuel pellets, may lead to failure of the zircaloy cladding tube. The failure results in leakage of radioactive material to the reactor coolant, and thus contamination of the coolant primary circuit. Despite the large effort spent on elucidating this phenomenon, many details of the failure mechanism remain unknown. This is mainly due to the difficulties in exploring the fuel in-reactor behaviour in detail. Conclusions regarding the fuel rod stress state and internal chemistry must usually be drawn from out-of reactor experiments, where these parameters are studied one at a time. This thesis presents an approach to model the pellet-cladding interaction failure mechanism. The objective has been to develop a computer model with ability to predict PCI-induced fuel rod failures. With this ability, apt operational restrictions for a certain fuel rod design may be formulated, such that in-reactor fuel failures can be avoided. Moreover, the computer model may serve as an effective tool in the search for new fuel rod designs with better resistance to PCI failures. Today, this search is accomplished by costly and time consuming in-reactor experiments. The idea behind the presented work has been to develop a number of submodels for the pelletclad mechanical and chemical interaction, to verify each of these models with well-defined separate effect experiments, and finally to combine all submodels in a fuel performance computer code. Submodels have consequently been developed for the UO2 fuel pellet and the zircaloy cladding mechanical behaviour, iodine induced stress corrosion cracking of zircaloy, and also for release, transport and mixing of corrosive fission products. The submodels have been compiled and incorporated in a computer code, intended for analyses of fuel rod thermomechanical behaviour under transient and off-normal operational conditions. By use of this computer code, the applicability of the models has finally been demonstrated. Following the sequence of this thesis, a brief description of the design and operational conditions of light water reactor nuclear fuel is first given in section 2. Section 3 is devoted to the phenomena behind pellet-clad interaction, whereas section 4 deals with PCI-induced fuel rod failures. Finally, the proposed approach to model this kind of failures is described and compared with existing methodologies in section 5.

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2 Design and operational conditions of LWR fuel 2.1 Design of LWR fuel The core of a LWR is made up of approximately 40 000 fuel rods, each with a length of 3.5-4 m. The fuel material in these rods is uranium dioxide (UO2), which is a ceramic fabricated in the form of cylindrical pellets, typically 9-10 mm long and 8-9 mm in diameter. These are stacked inside a thin-walled zirconium alloy cladding tube, which is filled with helium gas and hermetically sealed. The cladding tube has a wall thickness of 0.7-0.8 mm, and serves as a barrier between the radioactive fuel and the surrounding coolant; see figure 1. Uranium dioxide is used as fuel material mainly because of its high temperature stability and resistance to radiation, but also owing to its ability to retain gaseous fission products, that are created under fuel operation. Zirconium alloys (zircaloys) are used extensively as structural materials in reactor cores, since they have a high resistance to corrosion by water and also a small cross section for capture of thermal neutrons [3]. The alloys used for cladding tubes contain about 98 weight-% Zr, 1.5% Sn and small additions of Fe, Cr and Ni. Helium is used as a filler gas in order to provide best possible heat transfer across the pellet-cladding radial gap, which has an asfabricated width of 80 to 100 gm. In light water reactors, ordinary water is used as coolant [4]. Depending on the water pressure, a distinction is made between pressurised water reactors (PWR) and boiling water reactors (BWR). In a PWR, the water pressure and temperature is 15.5 MPa and 290325 °C respectively, which means that the coolant is in liquid phase throughout the core. In a PWR, the fuel rods are combined in assemblies, where 150-300 fuel rods are held together by a skeleton and 6-10 spacer grids; figure 1. Some of the assemblies in the core also contain empty guide tubes, into which neutron absorbing control rods can be inserted. The control rods are manipulated by a mechanism above the core. In a BWR, the water pressure and temperature is 7.0 MPa and 286 °C. The water is in liquid phase when entering the core from below, but then vaporises gradually as it passes by the heated fuel elements. The fuel assemblies in a BWR consist of 64100 fuel rods within a zircaloy channel of square cross section; figure 1. The channels are necessary in order to direct the steam and ensure a radially uniform vapourliquid distribution in the core. The control rods in a BWR have cruciform cross section and are inserted into spaces between the fuel Fig 1. Nuclear fuel rod, PWR and BWR fuel assemblies. channels from below. 2

2.2 Operational conditions The fuel operational conditions are fairly similar in PWRs and BWRs; the differences in coolant pressure and temperature do not significantly affect the fuel behaviour. In both reactor types, the fuel normally resides in the core under four years of operation, during which it reaches a bumup of 40 to 50 MWdays per kg uranium. The fuel rod average linear heat generation rate (LHGR) is approximately 20 kWm-1, and the large radial heat flux from the fuel pellets through the cladding to the coolant results in a steep temperature gradient from the pellet centre to its surface. The pellet centre temperature is typically 1500 °C, whereas the surface temperature is 500 °C. Under reactor operation, U-235 is fissioned, and fission products with lighter nuclei and larger specific volume than uranium are formed. The major part of these products is retained within the fuel pellets, which therefore gradually expand. The zircaloy cladding properties are also affected by the fission process. The exposure to high energy neutrons and fission fragments results in a pronounced hardening of the material, and also to an increased rate of inelastic deformation through irradiation enhanced creep. Moreover, the cladding material is affected by corrosion and hydrogen absorption from the waterside, and by interaction with the fuel pellets from the inside [5,6].

3 Mechanisms of pellet-cladding interaction The pellet-cladding interaction comprises both mechanical interaction of the expanding fuel pellets with the cladding, and chemical interaction between the cladding and aggressive fission products released from the fuel. The mechanical and chemical interaction mechanisms are described below.

3.1 Pellet-cladding mechanical interaction Due to the steep temperature gradient within the fuel pellet, thermal stresses will cause the ceramic material to crack when the fuel starts to produce power for the very first time. Furthermore, the nonuniform temperature distribution results in an hourglass shape of the pellet; see figure 2. After two to three years of operation, the combined effects of pellet fragment swelling and inward cladding creep will result in complete closure of the pellet-cladding gap, and onset of mechanical interaction. Further expansion of the fuel must thereafter be accommodated by outward creep of the surrounding cladding. Under steady-state reactor conditions, this accommodation takes place with a moderate contact pressure between pellet and cladding, since the fuel expansion from fission product swelling is slow. On the other hand, under a rapid t---3 t:" .7Th 2 power increase, the fast swelling and If thermal expansion of the fuel will 1 ; Cladding , cause a significant contact pressure, r, stress , since the time for relaxation through Zn-caloy -4,----e-orrosi-O-4,16. 7/ creep and other time dependent cladding crack (pi T = 1500 ° iv+ deformation mechanisms in the interit acting bodies is insufficient. This i T = 500 °C may result in high cladding stresses, 1i , \----_____.2---/ especially at pellet radial cracks and at pellet-pellet axial interfaces, as Fig 2. Mechanism of pellet-cladding interaction (PCI). depicted in figure 2.

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The intensity of the mechanical interaction is dictated by the fuel geometrical design, burnup, power level and the rate of power increase. Another important factor is the coefficient of pellet-clad friction, which plays an important role for the concentration of stress and strain to the above mentioned locations. Experimental investigations have shown that the dynamic coefficient of friction is in the range 0.5-0.7, fairly independent of contact pressure, temperature and irradiation [7]. However, the static coefficient of friction is strongly affected by these parameters, and also on the dwell time in contact. Due to bonding of the contacting surfaces by irradiation-enhanced creep and deposition of condensed fission products, the static coefficient of friction may reach values well above 3.0 [8]. Thus, the interaction is enhanced by long periods of steady-state reactor operation.

3.2 Pellet-cladding chemical interaction Approximately 80 fission products are generated by the fissioning of U-235. Most of them are unstable, and in average, they undergo four stages of decay before stable nuclei are formed. Consequently, roughly 300 isotopes are produced under the fuel lifetime, and the composition of the isotope inventory is dependent on fuel burnup and power. Stable species accumulate with burnup, whereas the proportion of unstable products increases with fuel fission rate and power. Many of the fission products are known to cause or aggravate cracking of zirconium alloys [9], but since most of these products are retained within the 1302 fuel, only volatile species with high release rates need be considered. Among them, iodine has been identified as the causative agent for pellet-cladding chemical interaction. This conclusion has been drawn from fractographic examinations of in-reactor cladding failures, which show the same characteristic features as cracks propagated in a pure iodine environment [10]. The release of iodine and other gaseous fission products is mainly controlled by diffusion and grain growth in the UO2 fuel, and therefore highly dependent on the fuel temperature [11]. Under rapid power excursions, the release exacerbates due to the fuel temperature increase. The pellet outer surface, which has a temperature of roughly 500 °C, does not significantly contribute to the release. Instead, the gas escapes from the fuel central part and is then transported to the cladding surface through radial cracks and through the gaps at the pellet-pellet axial interfaces. The axial transport of gas within the fuel rod is strongly dependent on the pellet-clad gap size. Under PCI conditions, a residual gap of only approximately 5 gm is available for axial gas flow, which leads to very slow equilibration of concentration gradients in the gas [12]. The released iodine easily reacts with other fission products, and therefore it is not immediately accessible for reactions with the zircaloy cladding. Most of the iodine is locked up by formation of CsI, a stable compound with only a limited capacity to promote zircaloy cracking [13,14]. Since CsI is in liquid phase for temperatures below 625 °C, it deposits at the clad inner surface, especially at pellet-pellet interfaces [15,16]. The deposited CsI is dissociated by radiolysis, which is a very complex radiochemical process related to the strong radiation field in the pellet-clad gap [17]. This dissociation is accelerated under power excursions, at which sufficient amounts of free iodine are made available for chemical reactions with the zircaloy cladding. However, the iodine-zircaloy reactions are inhibited by the cladding surface oxide layer [18]. The chemical reactions are therefore restricted to sites with high local straining, where the protective oxide film is cracked. This mechanism results in localisation of the chemical attack to cladding regions that are also exposed to mechanical interaction with the fuel pellets. At pellet cracks and/or at pellet-pellet interfaces, the cladding will thus be simultaneously subjected to high stress and a corrosive agent.

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4 PCI-induced fuel rod failures The combined effect of mechanical and chemical interaction with the fuel pellets may cause failure of the zircaloy cladding through iodine-induced stress corrosion cracking (SCC), see figure 3. This failure mechanism was recognised already in the late sixties, but some dispute about iodine being the causative chemical agent remained unsettled until the eighties. Remedies for the problem have evolved in parallel with exploration of various aspects of the failure mechanisms [2].

4.1 Failure mechanisms Three prerequisites are generally necessary for a crack to nucleate and propagate by SCC; a susceptible material, a detrimental chemical environment and a sufficient tensile stress. Zirconium and its alloys are susceptible for SCC by several substances [9], of which iodine provides the detrimental chemical environment found within a nuclear fuel rod. The third condition for SCC is met under increases in reactor power, since sufficient cladding stresses are then reached locally by pointwise contact with expanding fuel fragments. The formation of stress corrosion cracks is generally divided into two phases; initiation and propagation. For iodine-induced SCC in Zr-alloys, the propagation phase is fairly well known, whereas the mechanisms behind crack initiation are still wrapped in obscurity [19].

Fig 3. A PCI induced cladding crack [50].

Crack initiation According to the general consensus on the subject, the initiation of iodine-induced SCC in zircaloy is a question about chemical preconditioning, rather than formation of incipient cracks. The chemical preconditioning involves iodine-zirconium reactions, and a threshold iodine concentration exists, below which SCC of the cladding is never observed [20,21]. The fuel rod iodine inventory does not always exceed this critical threshold. The inventory consists of two stable iodine isotopes, that accumulate with fuel bumup, and five short-lived isotopes, whose contribution to the total inventory is governed mainly by the fuel fission rate. For low burnup fuel, the iodine inventory is made up solely of short-lived isotopes, and a considerable fuel power is in this case needed to reach the threshold for iodine-induced SCC. The chemical preconditioning is a time-dependent process, which manifests itself as a marked incubation period for the crack initiation. It has been experimentally shown that this incubation time is much shorter in a Zr114 environment than in pure iodine [22]. This indicates that formation of ZrI4 is a critical step of the crack initiation phase. Crack propagation In commercial zircaloy cladding tubes, pre-existing surface flaws and pits are abundant [23]. These defects, with depths up to 20-30 gm, are believed to serve as nucleation sites for crack propagation. Once the chemical preconditioning has been undertaken, immediate crack propagation can be attained at these defects, if the stress intensity exceeds a critical threshold [19]. 5

The crack propagates transgranularly, supposedly by an adsorption-induced mechanism that leads to a principally brittle failure of the cladding [2]. The crack propagation velocity is governed mainly by the stress intensity, iodine concentration and temperature, but it is also influenced by the zircaloy material properties. The crack growth is surprisingly fast, provided that a sufficient iodine concentration and stress exist. Perforation of the cladding is usually obtained within an hour, but through-wall cracks have also been observed after less than one minute in fuel rod power ramp experiments [24]. It should be particularly noticed, that the activation energy associated to the crack propagation coincides with the activation energy for ZrI4 formation [25]. The threshold stress intensity for onset of crack propagation is dependent on temperature, irradiation and material texture. The clad material is pilger-milled and thermomechanically processed in a hexagonally close-packed phase, which leads to a strong deformation texture in the cladding tube material [26]. The preferred basal pole directions are found in the radial-tangential plane of the tubes, with two concentration maxima at ± 30 0 to ±40° from the radial direction. Since the transgranular crack propagates preferentially on planes normal to the basal pole, a more radial texture yields a better resistance to cracking in the tangentially stressed cladding tube [21]. Variations in susceptibility to PCI failures between commercial zircaloy claddings from different fuel vendors can however not be explained solely by differences in the material texture. Experimental investigations of these variations in susceptibility have revealed a significant influence of pre-existing flaws at the cladding inner surface [27]. The propensity for PCI-failures for a certain Zr-based clad material is thus also related to the density and size distribution of surface defects, at which crack propagation can be initiated.

4.2 Remedies Cladding failures induced by PCI are more frequent in BWRs than in PWRs, due to the difference in reactivity control in these two reactors. Power uprating in a BWR is performed by successive withdrawal of control rods, which locally induces rapid increases in fuel rod power, and thereby also a strong pellet-clad interaction. In a PWR, the core power is controlled by slowly adding or reducing neutron absorbing boron in the reactor coolant [4]. Fast uprating of fuel rod power, that may lead to PCI failures, is therefore only obtained under off-normal power transients in PWRs. The PCI failure mechanism leads to a small, pinhole-looking, penetration of the cladding tube. The outleakage of radioactive material through this pinhole is moderate, and does not constitute a radiological hazard to the environment. More seriously, the pinhole failure leads to ingress of water into the fuel rod. Once inside, the water is dissociated by intense radiolysis in the pellet-cladding gap, and the radicals cause rapid oxidation and hydriding of the cladding inner surface [28]. The formation of voluminous Zr02 induces tensile cladding stresses, that will ultimately break the hydrogen-embrittled cladding. These secondary failures manifest themselves either as long axial splits, or as guillotine breaks [29]. In both cases, the failures result in excessive outleakage of radioactive material into the coolant. Fuel rod PCI failures are prevented by eliminating the prerequisites for iodine-induced SCC of the clad material, i.e. by reducing the material susceptibility, mitigating the detrimental chemical environment and reducing the cladding stress. In practise, this is done by modifying the fuel design and reactor manoeuvring procedures. The remedies described in the sequel have shown to be effective, but they have also involved increased expenditures for the power plant operators.

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Reducing material susceptibility As seen in section 4.1, the susceptibility to PCI failures is reduced in cladding tubes with a smooth inner surface and a radial material texture. Improved resistance to PCI failures is also provided by so called liner or barrier cladding, in which a soft inner layer of almost pure zirconium increases the resistance to SCC [30]. Liner cladding fuel is today used without exception in BWRs, and it is also gaining ground in PWRs. The soft inner layer increases the PCI failure resistance mainly by reducing the stress concentrations imposed by the pellet-clad mechanical interaction, but the liner material is also less susceptible to iodine-induced SCC than the zircaloy substrate. The effect of crack tip shielding should also contribute to the improved performance of liner cladding [31,32], but this effect has not yet been thoroughly analysed. Mitigating the detrimental chemical environment A less detrimental chemical environment is most simply obtained by preventing the release of corrosive fission products from the fuel pellets. This is accomplished by reducing the fuel temperature, either by reactor operational restrictions or by a modified fuel design. To this end, fuel assemblies with an increased number of fuel rods have been introduced, thereby allowing the heat generation rate of each rod to be reduced without decreasing the assembly power and core capacity. Furthermore, advanced assembly designs have been developed in order to secure a perfectly uniform power and temperature distribution among the fuel rods within the assembly. Another way of mitigating the hostile environment is to apply a chemically active coating to the cladding inside surface. The coating should either neutralise or immobilise iodine and other corrosive agents. This approach has not been pursued for LWR fuel, but silicone and graphite have been successfully used as coatings in CANDU heavy water reactor (HWR) fuel [2]. Reducing the cladding stress One way of reducing the cladding stress is to set restrictions on the rate of fuel rod power increase that could be allowed under reactor operation. By these restrictions, sufficient time is given to creep relaxation of stresses in both pellet and cladding, and severe stress concentrations are avoided. Moreover, it has recently been proven that the UO2 creep rate can be enhanced by modifying the material microstructure through additives [33]. The improved pellet ductility may alleviate some of the operational restrictions, and thus minimise capacity losses suffered by utilities. The pellet-clad mechanical interaction can also be mitigated by use of lubricants, that reduce the coefficient of pellet-cladding friction. Graphite has a pronounced lubricating effect, and is used as a surface coating in CANDU HWR fuel [34]. Graphite coated cladding has also been experimentally tested in LWR fuel, but was ousted by the zirconium liner cladding. Much effort has been spent on optimising the fuel pellet geometry in order to minimise the cladding stress concentrations at pellet-pellet interfaces. These stress concentrations arise from fuel thermal expansion, which lends an undesired hourglass shape to the originally cylindrical pellets. The effect is traditionally counteracted by chamfering the pellets and reducing their length to diameter ratio. These modifications of the pellet geometry are generally suggested by numerical analyses of the fuel deformation behaviour [35]. However, this optimisation is performed on rather flimsy grounds, since the deformation of the fuel pellets under reactor operation is extremely difficult to model. At high fuel burnup, the pellet configuration is strongly altered by cracking and swelling, and any traces of the original asfabricated chamfering disappear.

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5 Models for the failure process Much effort has been spent on experimentally elucidating the mechanisms behind PCI-induced fuel rod failures in contriving the above described remedies. The experiments have often been performed in the form of out-of-reactor separate effect tests, where the influence of single parameters on the failure process has been studied one at a time under well-controlled conditions. Analytical models have frequently been developed in order to understand and explain the observations made in these tests, and surprisingly simple models have often shown to accurately capture the observed phenomena. In contrast to these well-defined separate effect tests, the fuel rod conditions in a reactor core are very uncertain. Conclusions regarding stress state, temperature and chemical environment must generally be drawn from post-irradiation examinations of a limited number of fuel rods. These uncertainties make the formulation of predictive models for in-reactor PCI failures difficult, and the situation is further complicated by the wide variety of phenomena involved in the failure process. Predictive models are needed for several reasons, and the intended use of the models usually governs their scope and degree of complexity. They could be used to formulate appropriate operational restrictions for a certain fuel design, or as tools in the search for novel designs with reduced propensity for PCI failures. Other applications concern optimisation of core loading patterns [4], where the best possible fuel exploitation is sought with a sufficient safety margin to failures. A short survey of prevalent approaches to PCI failure modelling is given in the sequel. The methodology advocated in this thesis is also described and compared with the existing approaches.

5.1 Existing approaches In-reactor PCI failures are usually analysed by modelling the mechanisms that are believed to be significant to the failure process by use of fuel performance computer codes. These computer codes are intended for analyses of the fuel rod thermomechanical behaviour under various postulated operational conditions [36]. However, predictions of PCI failures can sometimes also be made merely based on empiricism. For a given fuel rod design, apt operational restrictions for avoidance of PCI failures may be established solely on the basis of operational experience and results from fuel rod ramp tests. Restrictions are generally formulated as upper limits on the LHGR and its rate of increase [37]. With the view to developing improved fuel designs or finding optimal core load patterns, computer analyses of the pellet-clad chemical and mechanical interaction and the resulting failure process are however necessary. The chemical part of the interaction is frequently neglected in these analyses, since the fuel rod chemical conditions are largely unknown and extremely difficult to capture by predictive models. Generally, a threshold LHGR is defined, above which the environment is assumed chemically preconditioned, and the clad material thus susceptible to SCC. This approach may be applicable to high burnup fuel under steady state operation, in which a significant iodine inventory is expected, but certainly not for all conceivable operational conditions. Analyses of the mechanical part of the interaction are usually confined to investigations of the cladding stress state, and the calculated hoop stress is often used as a measure for failure propensity by which different contemplated fuel rod designs or core loading patterns can be compared. Moreover, cladding failure criteria are as a rule formulated in terms of critical hoop stress. A major drawback with these stress-based failure criteria is that the calculated stress is strongly affected by the applied material models and the way stress concentrations arising from the localised pellet-clad mechanical interaction are handled. The pellet stack is generally modelled as a homogeneous cylindrical column, which is in uniform contact with the thin-walled cladding.

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Based on the uniform contact pressure obtained from such a model, the stress concentrations at pellet-pellet interfaces and pellet crack openings are then estimated by use of analytical or semianalytical solutions of the non-uniform contact problem [38-42]. As an alternative, the stress concentrations could be calculated for various postulated pellet crack patterns by explicit two- or three-dimensional finite element modelling of the interaction between the cracked pellet and the cladding [43,44]. Failure criteria based on the assumption of a critical cladding hoop stress are prevalent, but a few other criteria have been suggested. To this end, the strain energy density has taken the place of hoop stress as key parameter for cladding failure [45], and concepts from the field of fracture mechanics have also been used in formulations of failure criteria [46]. No matter how the failure criteria are formulated, all models are afflicted by the difficulties involved in accurately representing the in-reactor material behaviour of the cracked fuel pellets and the irradiated cladding. This applies particularly to the material response under transient loading, e.g. under fuel power uprating. Under these power excursions, the fuel rod internal chemistry and the frictional interaction between fuel pellet fragments and cladding are additional sources of uncertainty, that adds to the complexity of the problem. In conclusion, it should be mentioned that much attention has recently been paid to the behaviour of defective fuel rods, i.e. the evolution of secondary failures in fuel rods with perforated cladding. The secondary failure mechanism is dominated by cladding deterioration through oxidation and hydriding, and much of the modelling effort has been focused on the zircaloy-water reactions taking place in the pellet-clad gap [47,48].

5.2 Proposed approach The proposed failure model is intended for implementation in finite element based computer codes for fuel rod transient analyses. These codes are used for safety analyses, which are performed by evaluating the fuel rod thermomechanical behaviour under various postulated accidents and offnormal operational conditions. The primary output from the analyses comprises the fuel rod temperature distribution, deformations and stresses, fission gas release, cladding corrosion and other quantities, pertinent to the fuel rod mechanical integrity. The contemplated host codes are thus expected to provide these parameters to the failure model, which is therefore restricted to predicting initiation and propagation of PCI-induced cladding cracks. The proposed failure model, which is thoroughly described in paper I, is based on fracture mechanics concepts. The propagation of stress corrosion cladding cracks is explicitly modelled by a node release technique in the finite element framework, and clad failure is consequently attained when a crack reaches the clad outer surface. The explicit modelling of crack propagation makes it possible to account for the time-dependence of the failure process, and the possibilities of stable crack growth and crack arrest are tacitly considered. This is a major advantage over stress-based failure criteria, in which the time dependence of the failure process is neglected. Vindicated by experimental observations, the iodine-induced stress-corrosion cracks are supposed to nucleate at pre-existing manufacturing defects at the clad inner surface. These initial flaws are assumed to start propagating transgranularly, provided that the stress intensity exceeds a critical threshold and the clad material is sensitised through chemical preconditioning. The last condition is cast in the form of a threshold iodine concentration. The crack growth velocity is dictated by iodine concentration, stress intensity and temperature, and is calculated through correlations derived from well-defined separate effect tests.

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The proposed modelling approach makes heavy demands upon the fuel performance host code, since accurate predictions of failure implies that the iodine concentration, stress intensity and temperature provided by the code as input to the failure model are impeccable. The iodine concentration can preferably be determined by use of the non-equilibrium model presented in paper I. A precise determination of stress intensity necessitates not only a detailed modelling of the pelletcladding mechanical interaction with its build-up of stress concentrations, but also realistic representation of the pellet and cladding in-reactor material behaviour. The cladding stress concentrations at pellet irregularities can be determined by finite element modelling of the interaction, where individual pellet cracks are represented in detail. The stress concentrations are strongly dependent on pellet-clad friction, and reliable in-reactor data for the friction coefficient is found in [7,8]. With regard to the material properties, specific models have been developed for the in-reactor behaviour of both pellet and cladding. These constitutive models, which are thoroughly described in papers IT and III, are focused on the inelastic deformation under transient loading conditions. The pellet constitutive model accounts for the material mechanical behaviour under cracking, fragment relocation and pellet-clad mechanical interaction. An essential part of the model is the representation of pellet cracks, which significantly affect both the mechanical and thermal behaviour of the fuel under operation. Cracking is here modelled in a continuum context, where cracks are represented by nonelastic strains in the material. The continuum representation is convenient, since cracking can be treated in the same manner as plasticity and creep. The model is derived in the form of a nonlinear constitutive relation for the fuel material, that may be implemented in either two- or three-dimensional finite element computer codes. The cladding constitutive model belongs to the class of generalised standard materials, and it is based on the theory of thermodynamics with internal state variables. From this theory, an anisotropic viscoplastic law with combined isotropic and kinematic hardening is derived, in which effects of irradiation and static recovery are included. The model is intended for prediction of cladding creep and viscoplasticity under transient reactor conditions, and has been calibrated with experimental in-reactor and out-of-reactor data taken from literature. Together with a detailed finite element modelling of pellet-cladding mechanical interaction, these constitutive models create the necessary conditions for accurate predictions of cladding stresses under the rapid increases of power that are generally connected with PCI induced fuel rod failures.

lo

6 Conclusions A novel approach to model and predict pellet-cladding interaction induced failures of LWR fuel rods has been developed. The methodology is based on fracture mechanics modelling of formation and propagation of iodine-induced stress corrosion cracks in the zircaloy cladding. The predictive capacity of this model is largely conditioned by the accuracy by which the in-reactor behaviour of the pellet and cladding materials can be represented. For this reason, constitutive models have been developed for these materials. Moreover, the SCC failure model presented in paper I and the constitutive models in papers II and III have been implemented in the ABB Atom STAV-T computer program. STAV-T is a twodimensional finite element code, which is intended for thermomechanical analyses of fuel rod behaviour under transient and off-normal operational conditions [49]. The introduced submodels have been individually verified, and extensive data bases from both in- and out-of-reactor experiments have been used for their calibration. The proposed modelling approach has proven to be particularly useful when studying PCIfailures under short transients, where the time-dependence of the failure process is significant. However, for events with duration longer than an hour, the crack propagation modelling seems over-ambitious. In these cases, a simple clad failure criterion based on critical hoop stress is believed to be appropriate. The modelling of iodine gas release under rapid power excursions is a foible of the proposed methodology, and the presented model, in which iodine release is thought of merely as a diffusion process, should thus be replaced by a more elaborate counterpart. Finally, the applied probabilistic treatment of crack initiation deserves some attention. Since the stress corrosion cladding cracks are supposed to nucleate at pre-existing surface defects, an accurate prediction of the probability for clad failure must be founded on cognisance of the actual defect distribution in the considered material. With this knowledge, a truly probabilistic approach to the PCI failure phenomena is feasible.

Acknowledgements The major part of the work presented in this thesis was conducted while the author held a position as research engineer at the R&D department in the Nuclear Fuel Division of ABB Atom AB, Västerås, Sweden. The thesis was completed at the division of solid mechanics, department of mechanical engineering, Luleå University of Technology, under the supervision of Professor Per Ståhle, to whom I am most grateful for helpful discussions and suggestions. The work was initiated by Professor Ali R. Massih, whose power of initiative and inexhaustible energy have been invaluable assets in the course of this project. I sincerely thank you Ali, not only for your daily technical guidance and inspiration, but also for your recognition of the need for lifelong learning, long before these words came onto everybodys lips. The author is also indebted to both present and former colleagues at ABB Atom for sharing their knowledge and constructive ideas. Special thanks are due to Tero Manngård and Magnus Limbäck, whose friendship and support have significantly contributed to the completion of this work. I am also grateful to Professor Sören Östlund and all other members of the department of solid mechanics at the Royal Institute of Technology, Stockholm, for allowing me admission to a number of excellent post graduate courses taught at their department. Last but not least, I wish to express my appreciation to all colleagues at the division of solid mechanics in Luleå, for contributing to a friendly, creative and most enjoyable working atmosphere.

11

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R.L. Yang Meeting the challenge of managing nuclear fuel in a competitive environment Proc. ANS int, topical meeting on LWR fuel performance, Portland, OR, March 2-6, 1997, pp 3-10. B. Cox Pellet-clad interaction (PCI) failures of zirconium alloy fuel cladding - a review J. Nucl. Mat. 172, 1990, pp 249-292. C. Lemaignan, A.T. Motta Zirconium alloys in nuclear applications, Material Science and Technology, Vol 10B, 1994, pp 1-51. VCH Verlagsgesellschaft mbH, Weinheim, Germany. S. Glasstone, A. Sesonske Nuclear reactor engineering, 3rd ed. Krieger Publishing Company, Malabar, FL, 1981. B. Cox Oxidation of zirconium and its alloys In Advances of Corrosion Science and Technology, Eds. M.G. Fontana, R.W. Staehle Plenum Press, New York, 1975, pp 173-391. Corrosion of zirconium alloys in nuclear power plants Int. Atomic Energy Agency report IAEA-TECDOC-684, IAEA, Vienna, 1993. V.M. Shchavelin, A.V. Kostochka, A.A. Kuznetsov, I.S. Golovnin, Y.K. Bibilashvili. In-reactor study of the friction characteristics of reactor materials Atomnaya Energiya, 61 (3), 1986, pp 175-178. (In Russian) Y.V. Bozhko, A.M. Bolobolichev, A.V. Kostochka, V.M. Shchavelin. Coefficient of static friction of the uranium dioxide - zirconium alloy pair under irradiation Atomnaya Energiya, 71 (5), 1991, pp 463-466. (In Russian) B. Cox Environmentally-induced cracking of zirconium alloys - a review J. Nucl. Mat. 170, 1990, pp 1-23. B. Cox, B.A. Surette, J.C. Wood Proc. conf. on environmental degradation of engineering materials in aggressive environments, Virginia Polytechnic Institute and State University, Blacksburg, VA, September 1981, p 193. K. Forsberg, A.R. Massih Diffusion theory of fission gas migration in irradiated nuclear fuel UO2 J. NucL Mat. 135, 1985, pp 140-148. S.J. Dagbjartsson, B.A. Murdock, D.E. Owen, P.E. MacDonald. Axial gas flow in irradiated PWR fuel rods US Nuclear Regulatory Commission Report TREE-NUREG-1158, 1977. 0. Götzmann Thermomechanical evaluation of PCI failures in LWR fuel pins J. Nucl. Mat. 107, 1982, pp 185-195. B. Cox, B.A. Surette, J.C. Wood Stress corrosion cracking of zircaloys in unirradiated and irradiated CsI J. Nucl. Mat. 138, 1986, pp 89-98. G. Lysell, D. Schrire Fission product distribution at different power levels IAEA meeting on fuel performance at high burnup for water reactors, Studsvik, Sweden, June 5-8, 1990, Report IWGFPT/36, pp 132-139.

12

[16] C.T. Walker, C. Bagger, M. Mogensen Observations on the release of cesium from UO2 fuel J. Nucl. Mat. 240, 1996, pp 32-42. [17] K. Konashi, Y. Shiokawa, H. Kayano Simulation of CsI decomposition in fuel-cladding gap J. Nucl. Mat. 232, 1996, pp 181-185. [18] K. Une Stress corrosion cracking of zircaloy-2 cladding in iodine vapour J. NucL ScL Technol. 14, 1977, pp 443-451. [19] B. Cox, R. Haddad Recent studies of crack initiation during stress corrosion cracking of zirconium alloys Proc. 7th int. symp. on zirconium in the nuclear industry, Strasbourg, France, June 1985 ASTM STP 939, pp 717-733. [20] W.S. Ryu, Y.H. Kang, J-Y. Lee Effects of iodine concentration on iodine-induced stress corrosion cracking of zircaloy-4 tube J. Nucl. Mat. 152, 1988, pp 194-203. [21] M. Peehs, H. Stehle, E. Steinberg Out-of-pile testing of iodine stress corrosion cracking in zircaloy tubing in relation to the pellet-cladding interaction phenomenon Proc. 4th int, symp. on zirconium in the nuclear industry Stratford-upon-Avon, UK, June 1978, ASTM STP 681, pp 244-260. [22] S. Shimada, T. Matsuura, M. Nagai Stress corrosion cracking of zircaloy-2 by metal iodides J. Nucl. Sci. Technol. 20, 1983, pp 593-602. [23] R.P. Tucker, P.H. Kreyns, J.J. Kearns The effect of internal surface flaws, iodine concentration and temperature on SCC behaviour of zircaloy-4 tubing Bettis Atomic Power Laboratory, 1976, Report WAPD-TM-1248. [24] H. Mogard, D.A. Howl, M. Grounes The international Trans-Ramp II fuel project: A study of the effects of rapid power ramping on the PCI resistance of PWR fuel, Proc. ANS int, topical meeting on LWR fuel performance, Williamsburg, VA, April 17-20, 1988, pp 232-244. [25] R.E. Williford The iodine-induced strain rate sensitivity of zircaloy fuel rod cladding NucL Engrg. Des. 78, 1984, pp 23-36. [26] E. Tenckhoff Deformation mechanisms, texture and anisotropy in zirconium and zircaloy ASTM STP 966, 1988. [27] B.C. Syrett, D. Cubicciotti, R.L. Jones The origin of variations in the iodine stress corrosion cracking susceptibility of commercial zircaloy-2 tubings J. Nucl. Mat. 92, 1980, pp 89-102. [28] D.R. Olander et al. Investigations of the roles of corrosion and hydriding of barrier cladding and fuel pellet oxidation in BWR fuel degradation Proc. ANS int, topical meeting on LWR fuel performance, Portland, OR, March 2-6, 1997, pp 149-156.

13

[29] J.E. Harbottle et al. The behaviour of defective BWR barrier- and non-barrier fuel Proc. ANS int, topical meeting on LWR fuel performance, West Palm Beach, FL, April 1994, pp 391-397. [30] H.S Rosenbaum, R.A. Rand, R.P. Tucker, B. Cheng, R.B. Adamson, J.H. Davies, J.S. Armijo, S.B. Wisner. Zirconium-barrier cladding attributes Proc. 7th int, symp. on zirconium in the nuclear industry, Strasbourg, France, June 1985 ASTM STP 939, pp 675-699. [31] T.S. Cook, F. Erdogan Stresses in bonded materials with a crack perpendicular to the interface Int. J. Engrg. Sci. 10, 1972, pp 677-697. [32] Y. Sugimura, P.G. Lim, C.F. Shih, S. Suresh Fracture normal to a bimaterial interface: effects of plasticity on crack-tip shielding and amplification. Acta metall. mater. 43 (3), 1995, pp 1157-1169. [33] M. Toba, S. Kobayashi, Y. Yuasa, T. Matsuda Characteristics of fuel pellet with additive of Al and Si Proc. IAEA technical committee meeting on advances in pellet technology for improved performance at high burnup, Tokyo, Japan, Oct. 28 - Nov. 1, 1996, paper 1/1. [34] S.C. Wood, B.A. Surette, I. Aitchison, W.R. Clendening Pellet-cladding interaction - evaluation of lubrication by graphite J. Nucl. Mat. 88, 1980, pp 81-94. [35] A. Bengtsson Finite element analysis of UO2 inelastic deformation in a nuclear fuel rod, Master thesis (In Swedish) Luleå University of Technology Report LTU-EX-1997/8, 1997. [36] K. Lassmann, H. Blank Modelling of fuel rod behaviour and recent advances of the Transuranus code Nucl. Engrg. Des. 106, 1988, pp 291-313. [37] A.R. Massih, T. Rajala, L.O. Jernkvist Analyses of pellet-cladding mechanical interaction behaviour of different ABB Atom fuel rod designs Nucl. Engrg. Des. 156, 1995, pp 383-391. [38] J.H. Gittus Theoretical analysis of the strains produced in nuclear fuel cladding tubes by the expansion of cracked cylindrical fuel pellets NucL Engrg. Des. 18, 1972, pp 69-82. [39] G. Roberts The concentration of stress in cladding produced by the expansion of cracked fuel pellets NucL Engrg. Des. 47, 1978, pp 257-266. [40] G.V. Ranjan, E. Smith Determination of stress within zircaloy cladding due to pellet-cladding interaction NucL Engrg. Des. 56, 1980, pp 263-272. [41] M. Nakatsuka Theoretical and experimental analyses of cladding strain produced by expansion of cracked fuel pellets Nucl. Engrg. Des. 65, 1981, pp 197-204. [42] P.A. Jackson The effect of pellet wheatsheaf growth during power ramps on cladding stress concentration Nucl. Engrg. Des. 101, 1987, pp 225-232.

14

[43]

[44]

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[46]

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[49]

[50]

B.D. Nobrega Iodine induced stress corrosion cracking of zircaloy-2 cladding under near plane strain and localized stress-strain conditions Ph.D. thesis, The University of Michigan, 1984. L. Caillot, B. Linet, C. Lemaignan Pellet clad interaction in PWR fuel: Analytical irradiation experiment and finite element modelling Trans. 12th int, conf, on structural mechanics in reactor technology, Stuttgart, Germany, Aug 15-20 1993, paper C04/2. C.K. Chao, C.C. Tseng A power-rate-dependent model for pellet/cladding mechanical interaction NucL TechnoL 101, 1993, pp 202-211. A.K. Miller, K.D. Challenger, A. Tasooji SCCIG: a phenomenological model for iodine stress corrosion cracking of zircaloy Electric Power Research Institute Report EPRI NP-1798, 1981. C. Gräslund, N. Kjaer-Pedersen, G. Lysell INTERPIN.DF modeling of secondary hydriding in Studsvik fuel tests Trans. 14th int, conf. on structural mechanics in reactor technology, Lyon, France, Aug 17-22 1997, paper C01/1. R.0 Montgomery, V.R. Rashid, 0. Ozer Evaluation of post-defect fuel behavior Proc. ANS int, topical meeting on LWR fuel performance, West Palm Beach, FL, April 1994, pp 447-453. L.O. Jernkvist, M. Limbäck Analysis of fuel rod behaviour under loss-of-coolant accidents using the STAV-T fuel performance code Trans. 13th int, cont.. on structural mechanics in reactor technology, Porto Alegre, Brazil, Aug 13-18 1995, paper CO2/2. C. Lemaignan Water reactor fuel damage mechanisms: A review Trans. 14th int, conf. on structural mechanics in reactor technology, Lyon, France, Aug 17-22 1997, paper CW/1.

15

Nuclear Engineering and Design 156 (1995) 393-399

Nuclear Engineering and Design

A model for predicting pellet—cladding interaction-induced fuel rod failure Lars Olof Jernkvist Asea Brown Boveri Atom, S-721 63 Västerås, Sweden

Abstract A model for predicting pellet—cladding mechanical-interaction-induced fuel rod failure is presented. Cladding failure is predicted by explicitly modelling the formation and propagation of radial cladding cracks by the use of non-linear fracture mechanics concepts in a finite element computational framework. The failure model is intended for implementation in finite element fuel performance codes in which local pellet—clad interaction is modelled. Crack initiation is supposed to take place at pre-existing cladding flaws, the size of which is estimated by simple probabilistic concepts, and the subsequent crack propagation is assumed to be due to either iodine-induced stress corrosion cracking or ductile fracture. The novelty of the outlined approach is that the development of cladding cracks which may ultimately lead to fuel rod failure can be treated as a dynamic and time-dependent process. The influence of complex or cyclic loading, ramp rates and material creep on the failure mechanism can thereby be investigated. The presented failure model has been incorporated in the ABB Atom STAV-T transient fuel performance code. Numerical results from some applications of the code are used to illustrate the usefulness of the model.

1. Introduction

Fuel rod failure due to mechanical and chemical interaction between fuel pellets and cladding (PCI) has been the topic of many investigations during the last two decades. The effort spent on elucidating the initiation and propagation of cladding cracks under the combined effects of mechanical straining and exposure to corrosive fission products bears witness to the complexity of this failure process. Three properties of in-reactor PCI-induced fuel rod failures which considerably complicate any approach to numerically analyse the failure process can be identified.

(1) Cladding stresses and strains are extremely localized owing to both radial cracking and axial hour-glassing of the fuel pellet. The vast majority of PCI-induced cracks are usually found near pellet radial cracks and/or at pellet—pellet interfaces. This pronounced spatial localization of damage can be explained by the concentration of stresses and the abundance of released fission products in these areas. (2) Fuel rod failures due to PCI usually occur following sudden increases in power. This may be due to both intense fission gas release and insufficient relaxation of cladding stresses during rapid power excursions.

0029-5493/95/509.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0029-5493(94)00961-9

394

L. 0. Jernkvist / Nuclear Engineering and Design 156 (1995) 393-399

(3) In-reactor PCI-induced fuel rod failures usually show strong variability. Out of maybe a thousand similar fuel rods subjected to the same loading, only one or two will fail. This random occurrence of in-reactor failures is in glaring contrast with laboratory failure tests, generally exhibiting a fair reproducibility. These three features have been borne in mind when developing the failure model. The localization of failure in both space and time is tackled by detailed finite element modelling of the pellet— clad interaction in a transient fuel performance code, whereas the stochastic nature of in-reactor failures is approached by simple concepts from probabilistic fracture mechanics. In Sections 2 and 3 the models for initiation and propagation of cladding cracks are presented. Section 4 is devoted to the calculation of crucial crack tip parameters. Finally, the implementation and application of the failure model are briefly described in Sections 5 and 6. 2. Crack initiation The mechanisms governing the formation of iodine-induced stress corrosion cracks in zircaloy are not yet fully understood, although much experimental work has been done to clarify the matter (Brunisholz, 1987; Cox, 1987). The experiments, usually performed by pressurizing short pieces of zircaloy tubes in the presence of iodine, suggest an initial slow intergranular mode of propagation, followed by a sudden transition to rapid transgranular crack growth as soon as a critical stress intensity is reached at the crack tip. In the failure model presented here, the first stage of slow intergranular crack growth is disregarded. Instead, cladding cracks are assumed to initiate at pre-existing internal flaws sufficiently large to generate the stress intensity needed for immediate onset of transgranular crack growth. This approach is vindicated only if there is a non-vanishing probability of finding a large enough surface defect in the material. Since this probability obviously increases with the volume of material under consideration, the approach is

suited for studies of in-reactor failures where large quantities of cladding are subjected to similar loading. Following the above assumption, a condition for initiation of in-reactor PCI cracks could be based on the probability of finding surface flaws with lengths exceeding a critical threshold dictated by the current mechanical loading. This probability may be quantified using Weibull's theory for probabilistic fracture mechanics in brittle materials (Weibull, 1951). According to this theory, the probability F of finding a body with area A having a strength less than cf is

F(o-)---- 1 — exp[ — Af(a)] where f(o) is the probability of failure per area. Usually f can be written as

f(a)

=(" 21 0

Cro

where A0 , cro and m are constants specific to the material. Applying these relations to PCI-induced cladding failure, we first have to assume that crack propagation occurs when the stress intensity at a surface flaw of length a reaches a critical threshold value J

J ka Js,

(3)

In Eq. (3) J has been assumed proportional to the crack length, which is true for brittle materials. sce is the threshold stress intensity for transgranular crack growth. Assembling Eqs. (1)—(3), we arrive at a distribution function for the surface defects: )m ] (4) F(a)= 1 — exp[ -A (c22 a A', Here F expresses the probability of finding a surface flaw longer than a in a body with area A. It should be noticed that the form of F has been deduced merely by assuming that zircaloy subjected to PCI behaves as a brittle material highly sensitive to surface flaws. It has been shown by Miller et al. (1981), however, that the actual distribution of surface defects in zircaloy is well described by Eq. (4), where A'0 , a, and m' may be established

L. 0. Jernkvist Nuclear Engineering and Design 156 (1995) 393-395

by investigating the distribution of flaws in a moderate number of samples of the clad material. In the failure model, Eq. (4) is used to determine the longest pre-existing surface flaw which, with any extent of confidence, is expected in the cladding area under study. This probabilistic treatment of the crack initiation process is justified by the stochastic nature of in-reactor PCI failures.

3. Crack propagation In our failure model, cladding cracks are assumed to propagate owing either to iodine-induced transgranular stress corrosion cracking or ductile fracture. Iodine-induced stress corrosion (I-SCC) is considered as the leading mechanism for propagation under moderate loading provided that there is a sufficient quantity of iodine available in the neartip region. Ductile fracture may occur, however, under intense mechanical straining of the cladding or when I-SCC is hindered by either a depleted chemical environment or a pronounced radial texture of the cladding material (Ryu, 1988; Schuster, 1992). Another assumption that we have made is that the two modes of crack growth at any instant can be treated as non-interactive. This is generally not true, but in the vast majority of cases PCI-induced cracks will propagate mainly via brittle transgranular stress corrosion cracking until the remaining ligament fails, in some cases under unstable ductile fracture. In such a case it is justified to believe that the incremental crack growth under a very short time is due solely to either I-SCC or ductile fracture. The failure mechanism yielding the dominant contribution to crack growth under this time step is supposed to be the current mode of propagation. In either of these failure mechanisms the crack propagation velocity at any instant is assumed to be controlled by the current conditions at the crack tip, with particular respect to stress intensity, temperature and iodine concentration. These crack tip parameters, supplied to the failure model from the finite element host code,

395

are used to determine the current crack growth velocity from material correlations described in the following sections. The crack propagation velocity is finally returned to the host code, where the finite element model of the cladding geometry is modified to accommodate the crack extension during the current time step. This explicit modelling of crack propagation is performed using a finite element node release technique (Shih, 1979). Both stress intensity and availability of iodine at the crack tip are time-dependent parameters that have to be repeatedly determined during a prescribed power history. This time dependence is partly due to changes in heat generation rate, but the major contribution in many cases arises from the crack growth itself. By the explicit finite element modelling of the propagating crack, these effects can be taken into account. 3.1. Propagation due to I-SCC The crack propagation veloclity da/dt under iodine-induced transgranular stress corrosion cracking can be expressed by a correlation of the form

da

0

if J < Jscc

CF(I2)(-j exp( — --Q—) if J J scc (5) Jscc RT where C (m s- 1) is a constant, n is a non-dimensional constant, R (J mo1-1 K -1) is the molar gas constant, Q (J mol- l ) is the activation energy, J. (Nm-1) is the I-SCC threshold stress intensity, F is a non-dimensional function of the iodine concentration, J (N m -') is the stress intensity and T (K) is the temperature. The parameters Q and C are dependent on the material under consideration, whereas n is approximately the same for both Zircaloy-2, Zircaloy-4 and unalloyed zirconium and can be found in the range 1.35-1.50 (Nagai, 1985). The threshold stress intensity for transgranular stress corrosion, Jscc, is dependent mainly on the texture of the material but is also influenced by temperature and irradiation (Brunisholz, 1987; Schuster, 1993). The threshold value for Zircaloy4 under normal operating conditions can be found

396

L. 0. Jernkvist 1 Nuclear Engineering and Design 156 (1995) 393-399 45 1,0 -

0,8 -

0,6 › 0,4 -

0,2 -

0,01-0-5

10-4

3.0

10.2 «

o

10

Iodine Concentration mole/m2.1 Fig. 1. The non-dimensional function F(12 ) from Eq. (5).

in the range 250-1000 N m -1. The non-dimensional function F(12 ) is shown in Fig. 1 and the crack propagation velocity in unirradiated Zircaloy-4 as a function of stress intensity at two different temperatures is shown in Fig. 2. 3.2. Propagation due to ductile fracture At extremely high stress intensities the crack growth is assumed to be controlled by the J

.

20 0

2

4

8

6

10

Ductile Crack Extension I 1.1n1 Fig. 3. An idealized JR curve of zircaloy. ./1c is the material fracture toughness, which is dependent on temperature and irradiation.

resistence (JR ) curve of the cladding material (Shih, 1979). Since no JR measurements have been reported on irradiated ziracaloy, the idealized curve shown in Fig. 3 is used to determine the incremental crack growth under ductile fracture. Given the current stress intensity J(a) and the accumulated crack growth due to ductile fracture, ad , the incremental crack growth Aa can be found by imposing the crack propagation to follow the JR curve:

. 1,00

0,80 -

Aa = 0 if J(a) 0.

(7)

Since eq. (7) has to be satisfied for arbitrary changes in ee and T, we have

ap

• = p -. ee

dIP .

s

and

(8)

Thus, eq. (7) can be written

1

▪ eze + U — Aá — — q •VT > ()

(9)

where

ap

A = p --

(10)

are thermodynamic variables, associated to the internal state variables a. Eqs. (8) and (9) constitute general conditions that have to be fulfilled by any constitutive model, formulated on the basis of internal state variables. Now, considering the irreversible power input from irradiation, U, we make the assumption that it is proportional to the fast neutron flux,

(11) U = ku/i where p. is the average linear energy transfer to the irreversible microstructural damage processes. With a minor loss of generality, we may now separate eq. (9) into a thermal and a mechanical (intrinsic) dissipation inequality: 1 — q •VT

(12)

0

a:ine +

—A•a>0 .

(13)

If we apply Fouriers law as the constitutive relation for heat conduction,

q = — k • VT ,

(14)

the thermal dissipation inequality in eq. (12) will be fulfilled, provided that the thermal conductivity tensor k is positive definite. Fulfilment of the intrinsic dissipation inequality in eq. (13) is secured by applying the concept of a generalised standard material (Maugin,1992), wherebz ine , 1.1 and the evolution of internal state variables are derived from a convex function co through the normality rule ne _

a9* dCf

.39* =

•

dO

. d9* a= — dA

(15)

(a, ,A) is the dual dissipation potential, and has to be a non-negative convex function, which is zero at the origin (a = Ø = A = 0).

3

3. MODEL FORMULATION The concept of generalised standard materials provides a basis upon which physically admissible material models can be formulated. Such a formulation firstly requires that a set of state variables, a, and associated thermodynamic variables, A, are identified. Secondly, analytical expressions for the Helmholtz free energy and the dual dissipation potential must be specified in terms of these variables, so that changes in material properties under inelastic deformation, heating and irradiation are properly represented by the state- and evolution laws in eqs. (10) and (15). These steps are described in the sequel, but a restriction of the general theory to the case of axisymmetric conditions and orthotropic materials will first be made. 3.1

Axial symmetry

Due to the axial symmetry of the cladding tubes, only four components of the stress and strain tensors need be considered. Henceforth, a four-component Voigt-notation will be used: (16) a = ( ar, az 'cry 0-8) and e = ( Sr, Yrz, ee ) • Other tensorial quantities can be written in the same fashion, e.g. for the deviatoric stress, = a — tr(cr)I/3, we have (17) = ( 0-'7-, , Try 0-'19 ) • The texture induced anisotropy of the cladding material is considered by introducing an equivalent stress through a- eq (

=

3 a' • M

(18) , 2 where the symmetric rank 2 tensor M is dependent on the material texture and the preference for slip in certain crystallographic directions. For an isotropic material, aeg in eq. (18) degenerates to the von Mises effective stress (Baltov,1964). For a certain cladding material, M can be determined experimentally. However, approximate analytical expressions for M are also available, an example of which is given in appendix. In the model, the texture induced anisotropy is assumed to be unaffected by temperature and inelastic deformation, which restricts the applicability to small inelastic strains and temperatures below 1100K. 3.2

Selection of internal state variables

The anisotropy connected with inelastic deformation of zirconium alloy cladding is not only due to the material texture, but also to kinematic hardening (Bauschinger effect). This flow-induced anisotropy is considered by introducing a tensorial internal state variable, x, which corresponds to kinematic hardening. In the same way, two scalar state variables, rd and rr, are selected for representation of isotropic hardening due to inelastic deformation and irradiation respectively. The thermodynamic variables associated to rd, r,. and x will be denoted Rd, Rr and X. They represent an isotropic increase of the yield strength and a back stress respectively, and are derived from eq. (10) Rd

(9T = p--,-urd ,

(91 dr

R , = p—1

Al X = .13—d T '

and

(19)

The evolution of the state variables rd, r, and x is given by eq. (15) . _ (99* rd — — dR d '

dcp* l-r =

and

i=

—

dcp* . dX

The above equations are combined by introducing analytical expressions for xP and 9*.

4

(20)

The Helmholtz free energy function

3.3

The Helmholtz free energy may be written in a partitioned analytical form,

= IPe (ee , T) + where the elastic part

`P' (rd, x,T) ,

(21)

Te is given by

ee

ee P-(T—T0 )se (22) P Here, C is the rank 2 isotropic elasticity tensor and ß is a scalar constant. From eq. (8), we see that eq. (22) yields the Duhamel-Neumann form of Hookes law of elasticity. The inelastic part is chosen according to of 2 2 21/2 0.2 pne rd + 7.7(23) 2p 9p eq` 2p

where 112 is a model parameter. Moreover, we have from eq. (19) Rd = rd ,

R,. = r,

and X=

2H 3

2 M•x'

(24)

The dual dissipation potential

3.4

A partitioned analytical form is assumed also for the dual dissipation potential,

9* = Slp (up, T) + Qc(crc, T,) + Cla(R d ,R,,T) + 2r(R,,I;) ,

(25)

where the subscripts p, c, a and r allude to effects of plasticity, creep, annealing and radiation. This partitioning is convenient, since it allows separate but mutually interacting models to be developed for these four dissipative processes. The arguments crp and cr, are o.P

-=

a, =

(1+12,.1 k)e (Teq (0-- X) d + er ) + 1+ O (k+Rd +12,) (o-eq (cr)+k)(k+Rd +Rr) (k+Rd +Rr) cseq (cr—X) 7

(1+ [3R,./ k)

1+

130r \ (1+Rr Ik ) 0 + (cseq(a)+k)(1+ PR,/ k) (k+ fiRr),

(26)

(27)

where the functions Od(R d, rd) = Rd — rd

(28)

er(R r, rr ) = Rr — Tr

(29)

2 0(X,x) = D fir2 aeq(X) —

DH M. M ' x'

3 2 x'

(30)

are identical to zero, according to eq. (24). The purpose of introducing these trivial functions is that they give rise to appropriate expressions for dynamic recovery of the hardening variables through eq. (20). Although the procedure is a bit artificial, it is compatible with the concept of generalised standard materials. In the above equations, k represents the yield stress of the material in its fully recrystallised state, whereas ß and D are model parameters. From eqs. (15), (20) and (25)-(30), we find the laws of evolution for the inelastic strain and the internal state variables. By splitting the nonelastic strain into a viscoplastic part, e", and one part from creep, ec, we find after some algebra E

=

315 -(0j -X')

(31)

2cseq(a— X)

5

M • ( -X') 2o-eq (cr - X)

(32)

an, aRd

=

(33)

Mi. aR r

Ma

rr

dR r

(34)

3D(p+i)(1-FR,1 k)

x =

(35)

2112( 0-eq (Cr) -1- k)

where /3 and C are the effective strain rates from viscoplasticity and creep:

d52

1

P = (k+Rd +R„) dap

=

3.5

1 df2, (1+ f3R,/k) dcs,

= .12 \ 3E

m--1

(36)

2 .c .1 \ 3 e • 1U- • e .

(37)

Empirical correlations

The effective strain rates, together with the derivatives of Oa and 2, in eqs. (33)-(34), are given by empirical correlations. The nonelastic strain is split into contributions from viscoplasticity and creep, since different deformation mechanisms are believed to be operative at high and low stress, and the effect of annealing on inelastic deformation rate is more easily considered by this separation. The effective viscoplastic strain rate is given by

Ij(o-p,T) = Pi(e

P2 _1)

(38)

where are the McCauley brackets, and P1 and P2 are temperature dependent parameters. The effective creep strain rate is calculated from an additive creep law

=

e-Qc /T sinhC3 (C 2 o- c.)

e 4 4p.85 0.

(39)

where Ci are model parameters. The first term is related to thermal creep, where Q, is the creep activation energy divided by the molar gas constant. The second term gives the athermal creep contribution from irradiation. The annealing involves high-temperature recovery of lattice defects, which annihilates prior isotropic and kinematic hardening. The defects caused by irradiation are generally smaller and more easily dissolved than dislocation loops induced by inelastic deformation (Torimaru,1996), and the irradiation hardening is therefore much faster recovered under annealing than the deformation hardening. The recovery rates are correlated to temperature by

dS2, _ dRa

e

_Qd/T(Rd 2 (il 1 )

(40)

d a Rr e-9r/T dR, = In(R,IR,) A 2

(41)

where AI, A2, Qd, Qr and R, are model parameters.

6

Finally, the following expression is used for the irradiation hardening

l-R 1/R

(42)

.

Here, R1 is a constant. To summarise, by use of these correlations and eqs. (24), (33)-(35), we get the following equations of evolution for inelastic deformation and hardening e =

=

3/3 M • ( — X ')

(43)

2 creq ( cr — X)

M • ( — X')

(44)

2 aeq ( — X) D 2 - e-Qd/2"(111 A/

(45)

R, e-Qrn. ln(R„../R,) A2

= = 2H,s m 3

ip

ic )

D(13 + «1+ R I k) (0eq (0")+k)

MM•X

(46) (47)

where we have made use of the fact that both x and X will be deviatoric tensors, provided that their initial values are deviatoric. Together with the correlations for effective strain rates from viscoplasticity and creep, the above system of equations constitutes the material model. The comparatively simple model belongs to the class of generalised standard materials, and thus satisfies the thermodynamic constraint of intrinsic dissipation. As can be seen from eq. (45), isotropic deformation hardening is neglected, and only softening due to static recovery is considered. This is in accordance with experimental observations on zirconium alloy behaviour under moderate inelastic deformation and normal operating temperatures (Robinet,1995). From the expression in eq. (26) for the reduced stress

0-eq(cr — X) P

(48)

(k +Rd +R,)

applied in the viscoplastic correlation, we see that the contributions to isotropic hardening from deformation and irradiation are assumed to be additive. Moreover, these two forms of material hardening are in the model treated as independent phenomena. On the other hand, the kinematic deformation hardening is strongly coupled to irradiation. As can be seen from the last term of eq. (47), the dynamic recovery rate of X increases with the irradiation hardening variable R, 4. MODEL CALIBRATION The introduced model parameters have been fitted to experimental data taken from literature, and Table 4.1 specifies the in- and out-of-reactor experiments that were used for calibration. The considered experiments include several investigations, in which materials with differing heat treatments are compared with respect to their inelastic deformation behaviour. Most of the tests were performed under steady-state conditions, i.e. under constant temperature and stress. The static recovery experiments performed under stress-free conditions were included in order to verify the models ability to simulate annealing. All the experiments were performed on tubular cladding specimens, which in most cases were subjected to a fixed biaxial stress state (ce / crz ) =2 by internal pressurisation. Since the in-reactor tests were exclusively of this kind, the influence of irradiation on cladding anisotropy could not be characterised.

7

For this reason, the anisotropy tensor M has been assumed to be unaffected by irradiation, and was determined from the expressions given in appendix. This assumption is in accordance with analyses of in-reactor creep by Clevinger et al. (1979), although there is evidence that irradiation attenuates the material anisotropy in the high stress regime (Nakatsuka,1987). The model parameters were determined from experimental data in three steps, where out-ofreactor, in-reactor, and recovery tests were used in the respective step. For the creep tests, the model parameters were optimised with respect to relative error in predicted creep strain. For the tensile tests, the relative error in predicted stress was considered, and the recovery tests were evaluated with respect to changes in yield stress. Table 4.2 presents the resulting best estimate parameter values, which give a reasonable predictability of the material behaviour within the operational domain covered by the experiments, i.e. for temperatures from 570 to 800 K, inelastic strain rates between 10-13 and 10-4 s-1 and fast neutron fluxes up to 6.3.1017 nm-2s-1. The experimental database for the upper part of this domain is meagre, and consequently, the accuracy of the model is expected to decrease with increasing temperature and strain rate. Experimental data used for calibration.

Table 4.1

Investigator Frenkel, 1974 Kim, 1997 Limbäck, 1996 Robinet, 1995 Matsuo, 1989 Steinberg, 1984 Franklin, 1983 McGrath, 1996 Torimaru, 1996 Bauer,1978

Clad material Zr4 Zr-4 Zr-2/4 Zr-4 Zr-4 Zr-4 Zr-2/4 Zr-2 Zr-2 Zr-4

Heat treatment 3 different 2 different 3 different 2 different CWSR CW 2 different RX RX CWSR

CW: Cold worked

* RX: Recrystallised

Stress state Temperature Test [KI conditions Geksz 673 2.0 Steady-state 623-773 0.5,1.0,2.0 Steady-state 603-673 2.0 Steady-state 573-673 Several S-s/transient 603-693 2.0 Transient 733-883 Recovery 671-682 Steady-state 2.0 2.0 573-593 Transient 773-973 Recovery 700-978 Recovery

CWSR: Cold worked stress relieved

Model parameter best estimate values.

Table 4.2

Creep model: • = 1. 43108

Annealing model: s-I

C2 = 1. 06.10-8 Pa-I C3 = 2.13 = 1. 00 • 10-33 (pas)-i(m2s)0.85 •

Irradiated material? No No No No No No Yes Yes Yes Yes

= 24175

A1 = 3.1410 - (Pas)" A2 = 7.43.10-11 S K Qd = 40000 K Q,. = 23700

K

Irradiation hardening model:

• R

= 1.12-10-26 m2 Pa = 3.90.108

ß

=0.40

Deformation hardening model: H2 = 1.50.1010 Pa D = 4.50.1010 Pa

Viscoplastic model: • = 6. 60 -10-7 s-1 P2

=

T 500K 0.14e3210/T T > 500K

86.0 {

k=

8

6.02.108 -1.44.106 T+1.01.103 T2 T 713K l8.8710

5. RESULTS AND DISCUSSION The model was calibrated to experimental data pertaining to materials with disparate chemical composition and thermomechanical treatment, and a common set of model parameters was determined without discriminating the materials. Differences in inelastic behaviour between the materials in their unirradiated state are thus reflected only in the initial values of the hardening variables Rd and X, and also in the anisotropy tensor M. In a fully recrystallised material, Rd and X are both initially zero. For other materials, the initial values can be derived from tensile tests by use of the presented model (inverse problem). In the experimental database, differences in measured steady-state out-of-reactor creep rates of 50% or more were found also for materials with identical chemical composition, heat treatment, yield strength and texture. Such differences cannot be accounted for by the model, and thus the model parameters have to be tuned to a specific material, if accurate predictions of steady-state creep rates are needed. However, the presented model is primarily intended for analysis of creep and viscoplasticity under reactor transients, and should be judged with respect to its capacity to capture the inelastic deformation under time varying conditions. As a first example of the model predictability, we use the out-of-reactor creep experiment by Matsuo (1989), in which cold-worked stress-relieved Zr-4 cladding was subjected to varying temperature and stress. The tubular specimens had an inner diameter of 9.48 mm and a wall thickness of 0.62 mm. The specimens were loaded by internal/external overpressure, resulting in a fixed biaxial stress = 2. state ((se / In figure 1, the measured hoop creep strain is compared with model predictions for a cyclic stress history under constant temperature. The histories of applied hoop stress (MPa) and temperature (K) are given at the top of the figure. Figure 2 shows the result of another test case, in which both stress and temperature were changed with time. As can be seen from these figures, the very fast transient creep following stress reversal is fairly well predicted by the model. The effect is accounted for solely by the kinematic hardening variable.

148

-78

147

-78

664 K 1,5

e->

8

Eg, 1,0

OD

8 0,5

(Matsuo,1989)

•

Predicted 0,0 0

200

1000

800

400 600 Time [ h ]

Fig 1. Isothermal stress reversal creep test.

98

-98 633

1,5

0

145 663

I

118 633

£1

(Matsuo,1989)

-0,5 0

200

400

600 Time [ h)

Predicted . . 1200 1000 800

Fig 2. Variable stress and temperature creep test.

9

A similar transient creep experiment has recently been performed in-reactor at the Halden test reactor in Norway (McGrath,1996). A test specimen of fully recrystallised Zr-2 material was taken from a water rod, irradiated in a commercial reactor to a fast neutron fluence of 6.1025nm-2 (>1 MeV). The tubular specimen, with an inner diameter and wall thickness of 13.48 and 0.76 mm, was placed in the test reactor and subjected to a varying internal pressure history under more than 14000 hours. On-line diameter measurements were made throughout the test, in which the cladding temperature was held between 573 and 593 K, and the average fast neutron flux was 3.2.1017 nm-2s-1. Figure 3 shows the measured inelastic deformation together with the predicted hoop creep strain. In this test, only a moderate transient creep is observed upon stress reversal. This is accurately captured by the model, since the kinematic hardening is assumed to decrease due to the irradiation. As a last example, we study the loss of material strength obtained under a short term temperature excursion, thought to be representative of a dryout event in a boiling water reactor. The considered material is recrystallised Zr-2, which has obtained a fast neutron fluence of 2.5.1025nm-2. The material is exposed to a fast neutron flux of 4.5.1017 nm-2s-1 when the temperature is suddenly raised from 563 to 873 K under 30 seconds.

58

-159

0,20

128 585

596

-64 -24 28 581

84 575

(McGrath,1996) 0,00 Predicted -0,05

•

0

10000

5000

15000

Time [ h j

Fig 3. In-reactor stress reversal creep test.

80 MPa

— 225 22

563

873

563

200 .= 175

Figure 4 shows the predicted -2 150 evolution of the irradiation under and hardening variable Rr 125 after the temperature excursion. Under the transient, the material 100 yield strength decreases with 2 3 4 5 6 7 8 -1 0 100 MPa, which is in good Log(Time) [ s ] agreement with experimental observations by Torimaru et al. Fig 4. Simulation of irradiation annealing. (1996). Furthermore, it is predicted that approximately four months of full power operation is needed for the clad material to regain its prior yield strength. This long period of operation with a low strength material may have significant importance to the fuel post-dryout performance.

10

6. CONCLUSIONS A model for combined creep and viscoplasticity in zirconium alloy cladding tubes has been derived within the framework of thermodynamics with internal state variables. The model is intended for analysis of cladding inelastic deformation under transient conditions, and has been calibrated to experimental data for recrystallised as well as cold worked cladding materials. Comparisons with both out-of-reactor and in-pile experiments confirm that the transient creep behaviour under varying stress and temperature histories is captured by the model. However, the accuracy in predicted steady-state creep rate is generally poor, unless the model is recalibrated to the specific cladding material under consideration. The concept of internal state variables leads to a system of coupled equations for the inelastic strain and a set of hardening variables, which are related to the material microstructure and lattice defect density. This modular structure makes it fairly easy to modify and extend the model, so that additional effects can be accounted for within the same framework. To this end, dynamic strain ageing and transient creep under variable fast neutron flux are two phenomena suggested for future introduction into the model. 7. REFERENCES A. Baltov, A. Sawczuk, A rule of anisotropic hardening, Acta Mechanica, Vol 1, No 2 (1964) 81-92. A.A. Bauer, L.M. Lowry, Nucl. Tech. 41(1978) 359. G.S. Clevinger, B.L. Adams, K.L. Murty, Analysis of irradiation growth and multi-axial deformation behavior of nuclear fuel cladding, Proc. 4th int, symp. on zirconium in the nuclear industry, ASTM STP 681, American Society for Testing and Materials, 1979, 189-201. C.C. Dollins, R.P. Tucker, Irradiation-induced primary creep, J. Nucl. Mat. 52 (1974) 277-288. V. Fidleris, R.P. Tucker, R.B. Adamson, An overview of microstructure and experimental factors that affect growth behavior of zirconium alloys, ASTM STP 939, American Society for Testing and Materials, 1987, 49-85. D.G. Franklin, G.E. Lucas, AL. Bement, Creep of zirconium alloys in nuclear reactors, ASTM STP 815, American Society for Testing and Materials, 1983. J.M. Frenkel, M. Weisz, Effect of the annealing temperature on the creep strength of cold-worked zircaloy-4 cladding, ASTM STP 551, American Society for Testing and Materials, 1974, 140-144. J. Gittus, Irradiation effects in crystalline solids, Applied Science Publishers Ltd, London, 1978. Y.S. Kim, Generalized creep model of zircaloy-4 cladding tubes, J. Nucl. Mat. 250 (1997) 164-170. D. Lee, F. Zaverl, E. Plaza-Meyer, Development of constitutive equations for nuclear reactor core materials, J. Nucl. Mat. 88 (1980) 104-110. C. Lemaignan, AT. Motta, Zirconium alloys in nuclear applications, Materials Science and Technology, Vol 10B, 1994, 1-51, VCH Verlagsgesellschaft mbH, Weinheim, Germany. M. Limbäck, T. Andersson, A model for analysis of the effect of final annealing on the in- and out-of-reactor creep behavior of zircaloy cladding, Proc. 11th int, symp. on zirconium in the nuclear industry, ASTM STP 1295, American Society for Testing and Materials, 1996,448-467. M.A. McGrath, In-reactor creep behaviour of zircaloy-2 under variable loading conditions in IFA-585, OECD Halden Reactor Project report HWR-471, 1996.

11

Y. Matsuo, Creep behavior of zircaloy cladding under variable conditions, Proc. 8th int, symp. on zirconium in the nuclear industry, ASTM STP 1295, American Society for Testing and Materials, 1989, 678-691. G.A. Maugin, The thermomechanics of plasticity and fracture, Cambridge University Press, Cambridge, UK, 1992. K.L. Murty, Biaxial creep behavior of textured zircaloy tubing, JOM - J. of the Minerals, Metals & Materials Society, 44 (2) (1992) 49-55. M. Nalcatsuka, M. Nagai, Reduction of plastic anisotropy of zircaloy cladding by neutron irradiation, (I) Yield loci obtained from Knoop hardness, J. Nucl. Sci. Techn. 24 (10) (1987) 832-838. S. Oldberg, A.K. Miller, G.E. Lucas Advances in understanding and predicting inelastic deformation in zircaloy, Proc. 4th int, symp. on zirconium in the nuclear industry, ASTM STP 681, American Society for Testing and Materials, 1979, 370-389. P. Robinet, Etude experimentale et modelisation du comportement viscoplastique anisotrope du zircaloy 4 dans deux etats metallurgiques, PhD thesis no 442, 1995, Universit6 de Franche-Comt6, France (In French). B.N. Singh, S.J. Zinkle, Influence of irradiation parameters on damage accumulation in metals and alloys, J. Nucl. Mat. 217 (1994) 161-171. E. Steinberg, H.G. Weidinger, A. Schaa, Analytical approaches and experimental verification to describe the influence of cold work and heat treatment on the mechanical properties of zircaloy cladding tubes, Proc. 6th int, symp. on zirconium in the nuclear industry, ASTM STP 824, American Society for Testing and Materials, 1984, 106-122. E. Tenckhoff, Deformation mechanisms, texture and anisotropy in zirconium and zircaloy, ASTM STP 966, American Society for Testing and Materials, 1988. T. Torimaru, T. Yasuda, M. Nakatsuka, Changes in material properties of irradiated Zircaloy-2 fuel cladding due to short term annealing, J. Nucl. Mat. 238 (1996) 169-174.

12

APPENDIX: According to Hill's theory for plasticity in orthotropic materials, the equivalent stress in the axisymmetrically loaded cladding tube may be written =

3F 2 3G 2 3H -2- (az - ae ) + (a, - az ) + (ae - a,.)2 +3M

With our Voigt notation, where 3

cseq (o-) = •\I 7 a' •

=

(Al)

"1",, CYO, this expression corresponds to

(A2)

• a'

where the anisotropy tensor M is

-G+ H -G

0 -H 0 -F 0 0 0 2M -H -F 0 F -i- H

-G F+G

M=

(A3)

The parameters F, G, H and M are F=G=H=M13=113 for an isotropic material, and the deviation from these values indicates the degree of anisotropy for a certain material. Zirconium alloy cladding tubes exhibit texture induced anisotropy under creep and viscoplastic deformation, since the manufacturing process introduces a preferential orientation of the hcp crystals. Due to asymmetric distribution of slip systems within the crystals, inelastic deformation is less likely along the crystal basal pole as in other directions (Tenckhoff,1988). In a polycrystal, the average orientation of basal poles is characterised by Kearn's texture factors, fr, fe, fz, which can be determined by X-ray diffraction analysis of the material. Oldberg et al. (1979) presented approximate expressions, from which the parameters F, G, H and M can be derived from the cladding texture factors. In the proposed model, these expressions are used in a somewhat extended form

F= G= H=

( 1 - fz )(f r + afe) (1+ a )(1- )2 ± (f r + afe)(fe + afr)

(A4)

( 1- f)(fe afr)

(A5)

(14- a )(i- fz )2 +(fr + af e )(fe + afr )

(fe + afr)(fr + afe) (1+ a )(1- fz )2 (f r + afe)(fe -F afr)

2M = F+ H+ 4G .

(A6) (A7)

The parameter a is found in the range from zero to unity, and it defines the relative propensity for inelastic deformation along the basal poles as compared with other directions in the hcp crystal; a= 1 thus corresponds to isotropy. For a certain material and operating temperature, a can easily be determined from a tension test in the tube axial direction. The tangential/radial contractile plastic strain ratio, R, can thereby be measured. From the flow rule, we have P

se F R = — = — = fr+ afe G

(A8)

fe afr

and thus from the experimentally determined parameters, we have a = fr- Rfe Rfr - fe •

(A9)

Al

Appendix A-I: Supplementary information to paper I The correlation used for estimating the probability of finding a surface flaw longer than a in a body with area A is given by eq. (4) in paper I,

F(a) = 1 - exp[

A (a° ni a) j

(Al)

A

where the coefficients a«, A

and m 'have been fitted to the experimental data presented in [23], (A2)

= 2.50.10-i m

A = 7. 28.10-2 m2

(A3)

m' = 2.591 .

(A4)

These coefficients have been derived from optical inspection of inner surfaces from 19 samples of commercial zircaloy-4 cladding tubes. The crack propagation velocity da/dt under iodine-induced stress corrosion cracking is calculated from eq. (5) of paper I,

da _ dt -

0

if

n e —Q/RT

C F(I2) \,JScc

J