A review of analytical criteria for fission gas induced fragmentation of oxide fuel in accident conditions

Progress in Nuclear Energy 119 (2020) 103188

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Review

A review of analytical criteria for fission gas induced fragmentation of oxide fuel in accident conditions Lars O. Jernkvist Quantum Technologies AB, Uppsala Science Park, SE-75183 Uppsala, Sweden

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Keywords: Uranium dioxide Fragmentation Fission gas release LOCA RIA

ABSTRACT Overpressurization of intergranular fission gas bubbles and pores in oxide fuel pellets may occur in nuclear reactor accidents that involve rapid overheating of the fuel. The overpressurization leads to fine fragmentation of the pellets and burst-type release of fission gas from the shattered material. Since these phenomena may worsen the radiological consequences of reactor accidents, they have been extensively studied in the past, both by experiments and modelling. A key element in the modelling is the formulation of an appropriate analytical criterion, by means of which fragmentation of the material can be predicted based on the size, shape, number density and internal pressure of the gas filled cavities. This work is a critical review and assessment of seven existing criteria of this kind. The assessment reveals significant differences among the criteria. Most of them exhibit unphysical trends with regard to key parameters and/or give results that contradict observed fuel behaviour. Only two of the assessed criteria are deemed suitable for use in computer programs intended for analyses of the thermal–mechanical behaviour of light water reactor fuel rods in accident conditions, such as RIA and LOCA. In addition, it is shown that one of the five deficient criteria can be turned useful by a slight re-formulation.

1. Introduction Cracking and fragmentation of oxide (UO2 or (U,Pu)O2 ) nuclear fuel pellets may occur by different mechanisms, depending on the operating conditions of the fuel. Under normal reactor operation, the pellets are subject to a steep radial temperature gradient that induces thermoelastic stresses in the material. Tensile stresses, predominantly in the hoop direction, arise at the cold outer surface of the pellet. These stresses cause radial cracks to form already at a linear heat generation rate (LHGR) of 5–6 kW⋅m−1 , and the cracking proceeds as the power is increased. Assuming that the pellet fragments deform by thermoelasticity alone (Eslami et al., 2013), the magnitude of local thermal stresses will be proportional to 𝛼1 𝐸𝑌 𝐿𝑟 |𝜕𝑇 ∕𝜕𝑟|, where 𝜕𝑇 ∕𝜕𝑟 is the radial temperature gradient, 𝛼1 and 𝐸𝑌 are the coefficient of linear thermal expansion and Young’s modulus of the fuel material, and 𝐿𝑟 is the radial extension of the considered fragment. Hence, the strength of the temperature gradient caused by the applied power dictates how small the fragments need to be, in order to keep the tensile stresses below the fracture threshold for the material (Mezzi, 1983; Oguma, 1983). Accordingly, the number of pellet fragments increases initially almost linearly with increasing LHGR, but the tendency for further cracking declines at high power, as a result of increased material plasticity at high temperature. In addition to the dependence on LHGR,

the number of cracks is also observed to increase slightly with fuel operating time or burnup (Walton and Husser, 1983). The reason is not clear, but it is likely that thermal stresses are particularly strong during fast and/or large changes of fuel power, leading to additional cracking for each reactor shutdown or fast power change. Build-up of internal stresses by differential swelling and weakening of the material by element transmutation and by accumulation of fission product gas along grain boundaries may also ease cracking at higher burnup. From ceramographic investigations of discharged light water reactor (LWR) fuel (Oguma, 1983; Walton and Husser, 1983; Coindreau et al., 2013), it is clear that the number of radial cracks rarely exceeds 15 in solid UO2 fuel pellets that have experienced normal operating conditions during their lifetime. Hence, fuel pellet fragments formed by thermal stresses under normal LWR conditions are generally larger than 2 mm. Significantly finer fragments may form in accident conditions that involve very steep temperature gradients in the fuel material. This is for example the case in the most challenging scenarios for reactivity initiated accidents (RIAs) in LWRs, where the fuel pellets are heated to high temperature within tens of milliseconds. Regulatory acceptance criteria for RIA generally postulate that fuel melting should be precluded or limited, meaning that local fuel temperatures up to about 3100 K can be reached momentarily (OECD/NEA, 2010). The steep

E-mail address: [email protected]. https://doi.org/10.1016/j.pnucene.2019.103188 Received 25 June 2019; Received in revised form 3 October 2019; Accepted 23 October 2019 Available online 1 November 2019 0149-1970/© 2019 Elsevier Ltd. All rights reserved.

Progress in Nuclear Energy 119 (2020) 103188

L.O. Jernkvist

temperature gradient that arises at the pellet periphery during the power surge induces a dense pattern of fine radial cracks close to the pellet surface (Lespiaux et al., 1997; Fuketa et al., 2000). Similar crack patterns may also form at the end of a loss-of-coolant accident (LOCA), when the fuel is re-wetted (quenched) from high temperature (Oguma, 1985). The peak fuel temperature at time of quenching is expected to be lower than 1700 K (OECD/NEA, 2009). The fuel fragments formed by thermally induced stresses during these accident conditions are typically 0.1–0.5 mm in size, and they are found predominantly at the pellet surface (Lespiaux et al., 1997; Fuketa et al., 2000; Oguma, 1985). Even finer fuel fragments may form under certain accident conditions by overpressurization of gas-filled bubbles and pores in the solid. This mechanism, which is fundamentally different from those mentioned above, has the potential to break up the material into fragments down to the size of individual grains, i.e. ≈10 μm for typical LWR UO2 fuel. Although recent experiments on very high burnup LWR fuel have revived interest in the phenomenon, it is by no means a new discovery. In fact, the first observations of fission gas induced fragmentation of oxide nuclear fuel were made in the early 1970s, more specifically in transient heating tests of liquid metal fast breeder reactor (LMFBR) UO2 and (Pu,U)O2 fuel. The phenomenon was initially called ‘‘fuel dust cloud breakup’’, alluding to the rapid and energetic disruption into fine fragments that was observed in out-of-reactor tests (Deitrich and Jackson, 1977). Nowadays, the gas induced fuel fragmentation is referred to as fuel powdering, fine fragmentation or pulverization, in order to discriminate it from the larger-scale cracking and fragmentation that result from thermally induced stresses in the solid. The gas induced fragmentation was immediately identified as a safety issue, and during the 1970s and 1980s, it was the subject of much research on LMFBR fuel safety: summaries of this work can be found in (DiMelfi and Kramer, 1983; Wright and Fischer, 1984; Matthews et al., 1990). Computational models, in the form of grain boundary rupture criteria, were formulated along with these early tests on LMFBR fuel (Gruber et al., 1973; Finnis, 1976; DiMelfi and Deitrich, 1979; Worledge, 1980). The criteria were implemented in computer programs for analyses of LMFBR fuel, which are no longer in use. As an example of these early studies, Fig. 1a shows the post-test appearance of an LMFBR (U0.75 Pu0.25 )O2 fuel sample with a pellet average burnup of 31 MW⋅d⋅(kgHM)−1 after heating to just under 3000 K with a rate of about 400 K⋅s−1 (Slagle et al., 1974). These early tests showed that the gas induced fracture mechanism occurred only at high temperature and at high heating rate, typically in the range 102 –105 K⋅s−1 . In addition, the fuel had to be ‘‘sensitized’’ by accumulating a sufficient amount of fission gas in grain boundary bubbles, which required a certain fuel burnup. The observed heating rate dependence was thought to result from a competition between instant pressure relief of the bubbles by grain boundary rupture and time-dependent mechanisms for pressure relaxation, such as bubble growth or venting of the overpressurized bubbles through tortuous flow paths (Worledge, 1980; Matthews et al., 1990). Hence, if the heating is fast enough, the thermally induced increase in bubble pressure will outrun the pressure relaxation by these time dependent mechanisms, and the overpressure will eventually break the grain boundaries if the material is heated to a sufficiently high temperature. From this understanding, it follows that grain boundary fracture will occur in parallel with time dependent relaxation mechanisms for certain heating rates. This has been confirmed experimentally, for example by Gehl (1982), who reported results from transient heating tests on UO2 fuel that had been pre-irradiated in a commercial pressurized water reactor (PWR) up to pellet average burnups of about 30 MW⋅d⋅(kgU)−1 . He concluded that the transient fission gas release (FGR) that was observed in these tests resulted from a combination of prompt burst release by grain boundary rupture and time dependent gas release by diffusion controlled bubble growth and interlinkage. He found that the relative importance of these gas release mechanisms depended largely on the

Fig. 1. (a) Top, centre and bottom cross-sections of the FGR-10 test sample after ex-reactor transient heating in the Hanford Engineering Development Laboratory, USA (Slagle et al., 1974). (b) Fuel fragments after RIA simulation test FK-9 in the Nuclear Safety Research Reactor, Japan (Nakamura et al., 2002).

heating rate, and reported that prompt grain boundary rupture dominated the release for heating rates higher than about 50 K⋅s−1 (Gehl, 1982). Likewise, Matthews and co-workers noted that the nature of the grain boundary fracture changed at a heating rate around 200 K⋅s−1 (Matthews et al., 1990). They reported that the fuel fragmentation became energetic above this heating rate, and if unrestrained, the fuel was dispersed by the energy released on fragmentation. Fission gas induced fuel fragmentation received renewed interest in the 1990s when, in the aftermath of the Chernobyl accident, experimental programs on the behaviour of high burnup LWR fuel under RIA conditions were conducted in France, Japan and Russia (OECD/NEA, 2010). In particular, UO2 and (U,Pu)O2 fuel rodlets with burnups exceeding 40 MW⋅d⋅(kgHM)−1 were pulse irradiated in dedicated research reactors, such that fuel heating rates up to 105 K⋅s−1 were achieved. As a result of the rapid heating in combination with significant fuel burnup, the fuel material fragmented in a similar fashion as was observed for LMFBR fuel in the 1970s, and the fragmentation co-occurred with extensive gas release (Lemoine, 1997; Fuketa et al., 1997). As an example, Fig. 1b shows the fine fragments formed in a pulse reactor RIA simulation test on a boiling water reactor (BWR) UO2 fuel rod with a pellet average burnup of 61 MW⋅d⋅(kgU)−1 (Nakamura et al., 2002). The fission gas release during the test was nearly 17%. In the 1990s, a few new rupture criteria for the grain boundaries were developed, with the aim to better understand the phenomenon and to account for it in models for transient fission gas release (Olander, 1997; Lemoine et al., 2000; Likhanskii and Matveev, 1999). A decade later, the focus changed from RIA to LOCA conditions. The change was prompted by new findings from out-of-reactor fission gas release experiments (Une et al., 2006; Hiernaut et al., 2008) and integral-type LOCA simulation tests (Puranen et al., 2013; Oberländer and Wiesenack, 2014) on UO2 LWR fuel, which showed that fission gas induced fuel fragmentation may become effective also under fairly slow heating to moderate temperature, provided that the fuel burnup is sufficiently high. The aforementioned LOCA simulation tests suggested that a fuel pellet average burnup above 60-65 MW⋅d⋅(kgU)−1 is needed 2

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gaseous fission products caused by the fine fragmentation, will worsen the radiological consequences of an accident. In addition, the long term coolability and the risk for criticality of the fuel that is dispersed into the coolant are aspects that also warrant consideration (OECD/NEA, 2016). Thirdly, the hot fuel fragments ejected from failed fuel rods will interact thermally with the primary coolant water. The fuel coolant interaction (FCI) has the potential to cause coolant pressure pulses that may potentially lead to core damage in worst-case scenarios for LWR RIAs (OECD/NEA, 2010). Fine fragmentation of the fuel aggravates the FCI, since the ratio of the mechanical energy generated by coolant vaporization to the thermal energy in the dispersed fuel is inversely proportional to the fuel fragment size (OECD/NEA, 2010). In summary, fission gas induced fragmentation of oxide fuel is a phenomenon with recognized safety implications for RIA, while the potential safety implications for LOCA are still matter of research, mostly of experimental nature. This is also reflected in the amount of work reported on mathematical modelling and computational analyses of the phenomenon: while significant work has been done to mathematically model and numerically simulate fission gas induced fragmentation and/or transient gas release of oxide LWR fuel in RIA conditions (Suzuki et al., 2012; Moal et al., 2014; Khvostov, 2018; Guenot-Delahaie et al., 2018), very few attempts have hitherto been made to model these phenomena for LOCA conditions (Kulacsy, 2015; Jernkvist, 2019). A key element in the modelling, whether it is for RIA or LOCA, is the formulation of an appropriate analytical criterion, by which fragmentation of the material can be predicted based on the size, shape, number density and internal pressure of gas filled cavities. A variety of such criteria have been proposed over the years, from the early work on LMFBR fuel to more recent studies on high burnup LWR fuel. This paper is a critical review and assessment of these criteria. The objective is to identify suitable fragmentation criteria that are applicable to overpressurized intergranular bubbles and HBS pores, which are the gas-filled cavities of primary interest for modelling. The paper is organized as follows: Section 2 is a review of existing criteria for fission gas induced fragmentation of oxide fuel. Following an introductory part that deals with common attributes of the criteria, two criteria based on stress and five criteria based on linear elastic fracture mechanics are presented. A slight modification is also proposed for one of the existing criteria, to alleviate some of its limitations. In Section 3, the criteria are assessed and compared by use of a simple test case, designed to reproduce typical conditions in high burnup fuel under a LOCA. The results of the comparative assessment are discussed in Section 4, and the main conclusions are summarized in Section 5.

for the gas induced fragmentation to occur at heating rates and temperatures that are expected in scenarios for LWR LOCA. Since these burnups are reached in many modern fuel designs, fission gas induced fuel fragmentation and its possible consequences to the fuel behaviour under LOCA have lately been in focus for much research. For example, a 2014 review and assessment of data from out-of-reactor experiments on LWR fuel with high or even very high (>75 MW⋅d⋅(kgU)−1 ) burnup (Yagnik et al., 2014; Turnbull et al., 2015) showed that the susceptibility to fission gas induced fragmentation is firmly correlated to the formation of a high burnup structure (HBS) at the pellet rim, but that the fragmentation occurs also outside the re-structured rim zone. The investigators concluded that pulverization is generally related to a high local population of overpressurized gas bubbles in the fuel material, and not to the HBS formation per se. They also concluded that the degree of pulverization, and hence, the resulting fragment size, depends on the heating rate and the temperature reached during the transient: the higher the heating rate and peak temperature, the smaller the fragments. The typical fragment size reported from the reviewed tests was in the range 20–200 μm. The review also showed that the pulverization and the fission gas release that it brings about can be substantially reduced if the fuel material is subject to hydrostatic pressure. The reviewed data suggest that a pressure of about 50 MPa is sufficient to suppress pulverization, which means that constraint from pellet-cladding mechanical interaction (PCMI) may limit pulverization and transient FGR in high burnup LWR fuel rods (Yagnik et al., 2014; Turnbull et al., 2015). This conclusion was corroborated by concurrent out-of-reactor heating tests, in which pulverization and transient fission gas release were measured for UO2 fuel rodlets that did or did not experience cladding ballooning and burst during the test (Bianco, 2015; Bianco et al., 2015). The results from these tests indicate that transient FGR is enhanced in high burnup fuel rods that experience cladding ballooning, leading to opening of the pellet-cladding gap and relaxation of the PCMI, and further enhanced in rods that also experience cladding burst with a sudden drop in internal gas pressure (Bianco, 2015; Bianco et al., 2015). Fragmentation of porous materials as a result of rapidly decreasing confining pressure is a phenomenon known from geophysics; see e.g. (Kameda et al., 2008) and references therein. Based on the aforementioned 2014 review of data, an empirical threshold for gas induced fuel fragmentation under LWR LOCA conditions was proposed. The threshold was formulated in terms of local fuel temperature versus local burnup in a first attempt to define combinations of these two parameters, for which gas induced fragmentation is practically negligible (Yagnik et al., 2014; Turnbull et al., 2015). Additional data, aimed to support more elaborate empirical thresholds, have been produced in a recent experimental program on high burnup LWR fuel under the auspices of the OECD Nuclear Energy Agency (OECD/NEA, 2019). Thresholds of this kind would be useful for defining conditions, in which fission gas induced fragmentation can be neglected in safety analyses. If it cannot be neglected, there are several aspects of the phenomenon that should be considered in analyses of typical design basis accidents, such as RIAs and LOCAs. Firstly, the fragmentation leads to rapid release of gaseous fission products from ruptured pores and bubbles in the fuel material. In high burnup fuel, this so-called burst release may significantly increase the fuel rod internal gas pressure, which may contribute to ballooning and burst of the cladding tubes. This is a potential failure mode for the fuel rods in both RIAs (OECD/NEA, 2010) and LOCAs (OECD/NEA, 2009). Secondly, the formation of very fine fuel fragments may increase the amount of fuel material that is ejected from failed rods into the coolant. This is partly a geometrical effect: small fuel fragments are more likely to pass through the cladding breach, and they are also more easily entrained in the stream of outflowing gas than large fragments. In addition, the small fuel fragments seem to move more easily within the cladding tube, a phenomenon known as fuel axial relocation, which may also increase the amount of ejected fuel (Jernkvist and Massih, 2015). The increased amount of ejected fragments, together with the extensive release of

2. Criteria for fission gas induced fragmentation of oxide fuel All existing criteria for fission gas induced fragmentation of oxide fuel, whether applied to LMFBR or LWR fuel, have been formulated and used with the aim to analyse the fuel behaviour under RIA conditions. Another common attribute is that they are focussed on bubbles on grain faces, i.e. the planar or near planar boundaries where two grains meet, and that a certain overpressure in these bubbles is assumed to be needed for breaking the material along the grain face. This means that a hydrostatic pressure imposed on the fuel pellet, e.g. by pelletcladding mechanical interaction, will have a constraining effect. More specifically, the difference 𝑃𝑔 − 𝑃ℎ , where 𝑃𝑔 is the bubble gas pressure and 𝑃ℎ is the hydrostatic pressure in the fuel material, is the key parameter in most existing rupture criteria: with a few exceptions, the criteria provide a threshold for this pressure difference, at which the grain faces are assumed to break. Another common feature of existing rupture criteria is that they make use of idealized representations of the grain face and gas bubble geometry. The grain face is without exception treated as a planar surface, while there is some variation in how the grain face bubbles are represented. The most common approach is to treat the bubbles as 3

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L.O. Jernkvist

on grain boundary stress: the overpressurized gas bubbles give rise to a certain tensile stress normal to the grain face, and when this stress exceeds the strength of the grain boundary, the grain face is assumed to break. The key material property in these criteria is the grain boundary strength or fracture stress. This is a local material property that is only loosely related to the bulk tensile strength of the material. The second type of rupture criteria is based on concepts from linear elastic fracture mechanics (LEFM) (Anderson, 2005): For a given overpressure in the bubble, a certain amount of elastic energy is available in the strained solid and the compressed gas. If a crack starts to grow along the grain face, some of this elastic energy will be released. When the gas overpressure exceeds a certain level, the released elastic energy will exceed the energy needed to create the new crack surface by tearing the material apart along the grain face. This critical overpressure defines the rupture criterion. The strength parameter of interest is the grain face fracture energy, or fracture toughness, which is a measure of the energy needed per unit crack area for tearing the grain face apart. This is, again, a local material property, which is expected to depend on temperature, fuel burnup and other parameters that define the state of the fuel material.

Fig. 2. A periodic array of identical lenticular bubbles along a planar grain face. 𝑟𝑐 : radius of curvature, 𝑟𝑏 : projected bubble radius, 2𝑙: bubble spacing, 𝜃: semi-dihedral angle.

lenticular, as shown in Fig. 2. For this geometry, the bubble volume is given by

2.1. Criteria based on stress

𝑉𝑏 = 4𝜋 𝜂𝑟 ̃ 3𝑏 ∕3,

Gruber et al. (1973). Probably the first criterion for fission gas induced grain boundary rupture of oxide fuel was proposed by Gruber and co-workers (Gruber et al., 1973; Worledge, 1980). They made the assumption that the grain face will break as soon as the local stress, normal to the grain face, at the periphery of the bubble reaches a critical value. The criterion can be stated as

(1)

where 𝑟𝑏 is the projected bubble radius and 𝜂̃ is a function of the semi-dihedral angle, 𝜃, viz. 1−

3 2

cos 𝜃 +

1 2

cos3 𝜃

. (2) sin3 𝜃 The semi-dihedral angle is reported to be typically in the range 35–55◦ for grain face bubbles in UO2 (Reynolds et al., 1971; Hodkin, 1980), leading to values for 𝜂̃ in the range of 0.25–0.43. Hence, the volume of a typical lenticular bubble is about one third of a spherical bubble with the same projected radius 𝑟𝑏 . In the assessment presented below, the original formulation of some rupture criteria has been generalized to this bubble geometry when needed. This was done to ease the comparison of the rupture criteria; see Section 3. The simplest rupture criteria consider isolated grain face bubbles, whereas the more advanced ones account for bubble-to-bubble interaction. This is done by assuming that the grain face is covered with a regular array of bubbles with equal size and shape; see Fig. 2. The bubble spacing, or alternatively, the grain face area fraction covered with bubbles, 𝜙2 , becomes important in these criteria. Ceramographic investigations of irradiated fuel show that the assumption of a lenticular shape for the grain face bubbles is justified as long as the area fraction covered with bubbles is moderate. However, when about 20% of the grain face is covered with bubbles, part of them start to coalesce and form elongated (vermicular) and multi-lobal bubbles (White, 2004). Most of the rupture criteria contains the capillary pressure in the gas bubble as an additional parameter. This pressure is calculated through the Young–Laplace equation ( ) 1 1 𝑃𝑠 = 𝛾 + , (3) 𝑅1 𝑅2 𝜂(𝜃) ̃ =

𝑐𝑟 𝜎𝑙 = (𝑃𝑔 − 𝑃𝑠 )𝐹1 − 𝑃ℎ 𝐹2 ≥ 𝜎𝑔𝑏 ,

(4)

where 𝑃𝑔 is the bubble internal gas pressure, 𝑃ℎ = −trace(𝝈)∕3 is the hydrostatic pressure in the fuel material with a three-dimensional stress state given by the Cauchy stress tensor 𝝈, and 𝑃𝑠 is the capillary pressure in the bubble, given by Eq. (3). Moreover, 𝜎𝑙 is the local 𝑐𝑟 is its critical value, i.e. the tensile stress at the bubble periphery, 𝜎𝑔𝑏 strength of the grain boundary, and 𝐹1 and 𝐹2 are stress concentration factors that depend on the bubble geometry. For a spherical bubble, 𝐹1 = 1∕2 and 𝐹2 = 3∕2, which can be easily derived from Lamés equations (Timoshenko and Goodier, 1969). For lenticular bubbles, 𝐹1 and 𝐹2 will depend on both 𝑟𝑏 and 𝜃; see Fig. 2. No closed-form solutions are available for 𝐹1 and 𝐹2 in this case. In terms of the bubble gas pressure, the rupture criterion by Gruber and co-workers reads 𝑃𝑔 ≥ 𝑃𝑔𝑐𝑟 = 𝑃𝑠 +

𝑐𝑟 + 𝑃 𝐹 𝜎𝑔𝑏 ℎ 2

𝐹1

,

(5)

where 𝑃𝑔𝑐𝑟 is the critical gas pressure for grain boundary rupture. It should be remarked that the stress concentration factors 𝐹1 and 𝐹2 given in Gruber et al. (1973), Worledge (1980) are in error. They pertain to cavities in a large body subjected to uniaxial tension, which is not the loading situation considered here. Also, we note that Kulacsy used the rupture criterion in Eq. (5) in her recent study on rim zone fragmentation during LOCA (Kulacsy, 2015). She assumed that the overpressurized rim zone pores were spherical, but obviously used 𝐹1 = 1 instead of the correct value 𝐹1 = 1∕2. A drawback with the local stress criterion by Gruber and co-workers is that the critical gas pressure obtained through Eq. (5) will depend strongly on the bubble shape and size for small values of 𝜃, i.e. when the bubble is similar in shape to a planar crack. Another drawback is that the criterion considers only a single bubble, and does not account for bubble spacing or fractional coverage of grain face bubbles. Finally, 𝑐𝑟 , is a we note that the local tensile strength of the grain boundary, 𝜎𝑔𝑏 material property that will be difficult to determine by conventional materials testing. It is expected to be only loosely related to the bulk tensile strength of the fuel material, for which some data are available from tests on centimetre-size specimens. In the aforementioned work by Kulacsy (2015), bulk tensile strength data for UO2 were used as an 𝑐𝑟 . approximation for 𝜎𝑔𝑏

where 𝛾 is the fuel/gas specific surface energy and 𝑅1 and 𝑅2 are the principal radii of curvature for the bubble. For the case of lenticular grain face bubbles as shown in Fig. 2, 𝑅1 = 𝑅2 = 𝑟𝑐 . Moreover, 𝛾 is an effective quantity for these bubbles, given by 𝛾𝑒 = 𝛾𝑓 𝑠 − 𝛾𝑔𝑏 ∕2, where 𝛾𝑓 𝑠 is the true fuel/gas specific surface energy and 𝛾𝑔𝑏 is the grain boundary energy (Worledge, 1980). These energies are related through 𝛾𝑔𝑏 = 2𝛾𝑓 𝑠 cos 𝜃 (Hodkin, 1980), which means that 𝛾𝑒 = 𝛾𝑓 𝑠 (1 − cos 𝜃). This apparent reduction in surface energy for lenticular grain face bubbles has been observed experimentally (Hall et al., 1987), but is not always properly recognized in models. Existing criteria for fission gas induced grain boundary rupture may be divided into two principally different types. The first type is based 4

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crack area) is considered as a material property, which is referred to as fracture energy or fracture toughness (Anderson, 2005). In general, LEFM is used for analysing large-scale structures containing cracks or sharp defects, and in most cases, for determining the critical load at which the cracks or defects will start to grow. This critical load depends on the size of the existing cracks or defects, and a typical feature of LEFM is that this size dependence, in a first-order −1∕2 approximation, can be written as 𝑃 𝑐𝑟 ∝ 𝑙𝑑 , where 𝑃 𝑐𝑟 is the critical load and 𝑙𝑑 is the size of the defect (Anderson, 2005). An example with relevance to our study is the elementary case with a large (infinite) body, containing an internally pressure-loaded, planar penny-shaped circular crack with radius 𝑟𝑏 . For this simple case, the critical gas pressure for crack growth is given by Irwin (1957) √ 1 𝜋𝐸𝒢 𝑐𝑟 𝑃𝑔 = 𝑃ℎ + . (8) 2 (1 − 𝜈 2 )𝑟𝑏

Fig. 3. Grain face geometry considered by Olander (1997), Lemoine et al. (2000). Gas bubbles with an internal gas pressure 𝑃𝑔 cover a fraction 𝜙2 of the grain face, and the average normal stress in the ligaments between the bubbles is 𝜎𝑔𝑏 . 𝑃𝑠 is the capillary pressure in the bubble.

where 𝐸, 𝜈 and 𝒢 are the elastic modulus, Poisson ratio and fracture energy of the material. Eq. (8) is derived for a sharp circular crack in an infinite body, which means that the configuration corresponds to a lenticular gas bubble for the special case of 𝜃 = 0 and 𝜙2 = 0; see Fig. 2. As will be seen further on, Eq. (8) provides critical gas pressures that compare quite well with those obtained from more elaborate rupture criteria, in which the bubble geometry is modelled with higher fidelity and the bubble fractional coverage is accounted for. Some of the rupture criteria dealt with in this section define the critical gas pressure at which cracks are assumed to start growing from the periphery of the gas bubble into the ligaments that connect the bubbles. This critical pressure is then assumed to define the bubble pressure at which complete grain boundary rupture occurs. Hence, the possibility of stable crack growth and crack arrest halfway through the ligaments is neglected in these criteria. The simplification is probably justified as long as the material is brittle, which is the case for temperatures below about 1900 K in unirradiated UO2 fuel (Evans and Davidge, 1969; Canon et al., 1971). Other criteria presented below consider the possibility of stable crack growth along the grain face.

Olander (1997). Olander (Olander, 1997; Lemoine et al., 2000) took another approach in deriving a stress-based rupture criterion for the grain boundaries. In contrast to Gruber and co-workers, he considered the average normal stress in the ligaments that connect the grain face bubbles. By considering force balance across the grain face for a piece of material that experiences a far-field tensile stress 𝜎∞ normal to the grain face as shown in Fig. 3, he concluded that ( ) 𝑃𝑔 − 𝑃𝑠 𝜙2 + 𝜎∞ = 𝜎𝑔𝑏 (1 − 𝜙2 ), (6) where 𝜎𝑔𝑏 is the average stress across the grain face ligament, and 𝜙2 is the fractional coverage of grain face bubbles. Olander then assumed that the grain face breaks when 𝜎𝑔𝑏 reaches the grain boundary strength 𝑐𝑟 . From Eq. (6), it follows that the critical gas pressure in the bubble 𝜎𝑔𝑏 is 𝑐𝑟 (1 − 𝜙 ) − 𝜎 𝜎𝑔𝑏 2 ∞ . (7) 𝑃𝑔𝑐𝑟 = 𝑃𝑠 + 𝜙2 Olander’s rupture criterion in Eq. (7) is principally different from the one by Gruber and co-workers in Eq. (5) in that it considers the fractional coverage of grain face bubbles and that it is independent of bubble shape. Both criteria depend on bubble size through 𝑃𝑠 = 2𝛾∕𝑟𝑐 . Olander’s rupture criterion, as defined by Eq. (7), is applied in the SCANAIR computer program, which is used for analysing the thermal–mechanical behaviour of LWR fuel rods under RIA conditions (Moal et al., 2014). It is also used in similar computer programs, FALCON (Khvostov, 2018) and RANNS (Suzuki et al., 2012), although the implementation of Eq. (7) seems to be somewhat in error in the 𝑐𝑟 , is in SCANAIR treated as latter code. The grain boundary strength, 𝜎𝑔𝑏 a scalar input parameter, which has to be defined as a constant value by the user. In FALCON, an adapted correlation for the bulk fracture 𝑐𝑟 (Khvostov, 2018), and in RANNS, strength of the fuel is used for 𝜎𝑔𝑏 √ 𝑐𝑟 = 𝐶 it is estimated through the relation 𝜎𝑔𝑏 𝑈 𝐸𝛾𝑔𝑏 ∕𝜆𝑜 , where 𝐸, 𝛾𝑔𝑏 and 𝜆𝑜 are the fuel material elastic modulus, grain boundary surface energy and lattice parameter, and 𝐶𝑈 is a non-dimensional user-defined tuning parameter (Suzuki et al., 2012). We note that a typical value √ of 𝐸𝛾𝑔𝑏 ∕𝜆𝑜 for UO2 fuel is 15–18 GPa, which is about two orders of magnitude higher than the bulk tensile strength of the material (Evans and Davidge, 1969; Canon et al., 1971).

Harwell (1976–1990). The earliest grain boundary rupture criterion based on LEFM that can be found in the open literature is due to Finnis, who published his work in a 1976 Harwell research report (Finnis, 1976). Eight years later, the criterion was published by other researchers at Harwell (Matthews and Wood, 1984), and in 1990, a slightly revised and corrected version appeared (Matthews et al., 1990). According to this version, the critical gas pressure for initiation of crack growth along the grain boundary is √ 𝜋𝐸𝒢𝑔𝑏 1 𝑃𝑔𝑐𝑟 = 𝑃ℎ + 𝑃𝑠 + , (9) 𝐹3 (1 − 𝜈 2 )𝑟𝑏 where 𝒢𝑔𝑏 is the grain boundary fracture energy and 𝐹3 is a nondimensional function that depends on the bubble geometry. According to Matthews et al. (1990), √ 𝐹3 ranges from 2 for a planar pennyshaped circular crack to 3𝜋∕2(1 − 𝜈) for a spherical bubble with a circumferential incipient crack. While the lower limit for 𝐹3 follows from Eq. (8), the upper limit is not explained in Matthews et al. (1990). With 𝜈 = 0.31, the upper limit for 𝐹3 is about 2.61. Hence, the rupture criterion given by Eq. (9) is very similar to that for a pennyshaped circular crack in Eq. (8). The capillary pressure term 𝑃𝑠 appears in Eq. (9), since the bubble has a certain curvature, in contrast to the planar crack, for which 𝑟𝑐 = ∞. Moreover, the Harwell criterion suggests that the critical gas pressure is lower for a bubble than for a planar penny-shaped crack with the same radius, in cases where the capillary pressure term 𝑃𝑠 can be neglected. This is probably because the Harwell rupture criterion accounts for the contribution of elastic energy from the overpressurized gas in the bubble: the bubble has a finite volume of gas, which is not the case for the planar crack, and the elastic energy release from the gas will contribute to crack growth. Finally, it should be remarked that the geometry dependent factor 𝐹3

2.2. Criteria based on linear elastic fracture mechanics As already mentioned, rupture criteria based on concepts from linear elastic fracture mechanics provide a critical threshold for the bubble gas pressure by considering the elastic energy released by stress and pressure relaxation when a crack grows along the grain face. The critical gas pressure is reached when the released elastic energy exceeds the energy needed to create the new crack surface by tearing the material apart along the grain face. The latter energy (per unit 5

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in Eq. (9) contains no explicit dependence on bubble spacing or bubble fractional coverage. Such a dependence could in fact be included in the geometry dependent factor 𝐹3 , but no information is available on this issue in Matthews et al. (1990), Finnis (1976), Matthews and Wood (1984). Chakraborty, Tonks and Pastore (2014). Chakraborty, Tonks and Pastore (Chakraborty et al., 2014), henceforth referred to as C-T-P, used LEFM in combination with two-dimensional finite element (FE) analyses to formulate a criterion for crack initiation at internally pressurized grain face bubbles with lenticular shape. They studied the particular case of bubbles with 𝜃 = 50◦ , and the results that they present in Chakraborty et al. (2014) are restricted to this angle. However, the bubble radius and fractional coverage are treated as free parameters in their study. For lenticular bubbles with 𝜃 = 50◦ , C-T-P state that the gas pressure for initiation of crack growth along the grain face is √ 𝜋𝐸𝒢𝑔𝑏 1 𝑃𝑔𝑐𝑟 = 𝑃ℎ + 𝑃𝑠 + , (10) 𝐹4 (1 − 𝜈 2 )𝑟𝑏

Fig. 4. Vermicular bubble geometry considered by DiMelfi and Deitrich in their analysis of grain face fracture (DiMelfi and Deitrich, 1979).

where 𝐹4 is a non-dimensional function that depends on the fractional coverage of grain face bubbles through1 ( ) 𝐹4 (𝜙2 ) = 𝜋 0.568𝜙22 + 0.059𝜙2 + 0.5587 . (11)

Fig. 5. Critical condition for grain boundary rupture, as defined in the rupture criterion by Likhanskii and Matveev (1999).

Eq. (11) is based on results obtained from axisymmetric FE analyses of a representative volume element that contained a lenticular bubble with 𝜃 = 50◦ . The dependence on 𝜙2 was determined by varying the bubble size relative to the radius of the representative volume element, 𝑟𝑒 . The fractional coverage was calculated through 𝜙2 = (𝑟𝑏 ∕𝑟𝑒 )2 , which follows directly from the axisymmetric geometry used in the FE analyses. To model the effects of neighbouring gas bubbles, C-T-P used zero displacement boundary conditions for the cylindrical surface of the representative volume element, i.e. for the surface given by 𝑟 = 𝑟𝑒 . This approximation is rather simplistic, but probably the best that can be achieved with a two-dimensional model. In order to model the interaction of cracks that are arranged periodically along the grain face more accurately, one has to resort to three-dimensional FE analyses combined with LEFM methods. This is not trivial (Anderson, 2005). It is interesting to compare the critical gas pressure obtained through Eqs. (10) and (11) with the results from Eqs. (8) and (9). Since the latter are derived for a single gas bubble, we set 𝜙2 = 0 in Eq. (11) to allow a fair comparison. Likewise, we use 𝐹3 = 2.61 in Eq. (9), which is claimed to be an upper bound value (Matthews et al., 1990), and 𝜈=0.31. We also set 𝑃ℎ = 𝑃𝑠 = 0 for simplicity. With these assumptions, the critical gas pressure calculated through Eq. (10) is 14% higher than that for a planar penny-shaped crack, and 49% higher than that obtained with the lower bound for the Harwell rupture criterion. Hence, the C-T-P criterion is the least restrictive of the three for the case with 𝜙2 = 0. Additional comparisons are presented in Section 3.

dependence on 𝑟𝑏 that is expected for LEFM-based criteria. Secondly, it contains only one material property, and thirdly, when 𝜃 approaches zero, 𝑃𝑔𝑐𝑟 tends towards infinity. This trend is opposite to what is expected for a crack-like defect as it becomes increasingly sharper. The trend also contrasts to that of other rupture criteria, for which 𝑃𝑔𝑐𝑟 approaches 𝑃ℎ or 𝑃ℎ + 𝑃𝑠 as 𝜃 decreases2 . As will be shown in Section 3, the critical gas pressure calculated through Eq. (12) is very low in comparison with that from other criteria. The criterion by DiMelfi and Deitrich was applied in early analyses of results from rapid heating tests on LMFBR fuel (DiMelfi and Kramer, 1983; DiMelfi and Deitrich, 1979; Kramer et al., 1982; Rest, 1982). More specifically, it was used to discriminate between brittle grain boundary fracture and ductile deformation by fission gas induced grain boundary swelling. The fundamental assumption was that grain face bubbles can reduce their overpressure either by propagating as cracks along the grain face or by expanding as bubbles via vacancy diffusion. Likhanskii and Matveev (1999). Likhanskii and Matveev presented a grain boundary rupture criterion that makes use of Eq. (8), i.e. the fracture criterion for an internally pressurized penny-shaped crack in an infinite body (Likhanskii and Matveev, 1999). The paper by Likhanskii and Matveev is very brief; only the most important assumptions and simplifications that form the basis of the criterion are presented, followed by the final results. Here, the criterion is derived in some more detail. The grain boundary geometry considered by Likhanskii and Matveev is shown in Fig. 5. A plane grain face is assumed to be covered with a regular array of lenticular bubbles with identical size and shape. The centre-to-centre spacing of the bubbles is 2𝓁. This parameter can be calculated from the fractional coverage of grain face bubbles through √ 2𝓁 = 𝜋∕𝜙2 𝑟𝑏 . Sharp cracks are assumed to initiate at the periphery of each bubble and grow along the grain face. In contrast to the rupture criteria presented above, it is assumed that the cracks grow in a stable manner. The reason for the assumed stability is that the gas pressure that drives the crack growth will drop as the crack grow, since the volume of the original bubble is increased by the void volume of the growing cracks. The cracks are assumed to grow in a stable manner until they reach a critical size for unstable crack growth or until they interconnect. The latter condition is illustrated in Fig. 5.

Dimelfi and Deitrich (1979). An early application of LEFM to grain face gas bubbles is due to DiMelfi and Deitrich (1979). They took a somewhat different approach, in that they viewed the grain face bubbles as crack nuclei rather than existing cracks. By applying a crack nucleation and growth model to an elongated, vermicular, bubble as illustrated in Fig. 4, they arrived at the following relation for the critical gas pressure that would lead to brittle and unstable crack growth from the vertices of the bubble: 𝒢𝑔𝑏 sin 𝜃 . (12) 𝑃𝑔𝑐𝑟 = 𝑃ℎ + 2𝑟𝑏 (1 − cos 𝜃) Eq. (12) is much different from other LEFM-based crack initiation criteria in this review. Firstly, it does not have the inverse square root

1 Note that the first two terms in the expression given in the paper by C-T-P (equation (5) in Chakraborty et al. (2014)) have incorrect sign.

2

6

Note that 𝜃 → 0 implies that 𝑟𝑏 → ∞ for a bubble with a given volume.

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The critical condition for grain boundary rupture in Fig. 5 is defined by a set of three equations. Firstly, the fracture criterion for a sharp circular crack with radius 𝓁 follows from Eq. (8): √ 𝜋𝐸𝒢𝑔𝑏 1 𝑐𝑟 , (13) 𝑃̃𝑔 = 𝑃ℎ + 2 (1 − 𝜈 2 )𝓁 where 𝑃̃𝑔𝑐𝑟 is the critical gas pressure in the deformed bubble geometry defined in Fig. 5, consisting of the original bubble volume 𝑉𝑜 and the crack volumes. Secondly, by assuming that the crack grows fast enough that variations in temperature and number of gas atoms trapped in the bubble can be neglected during the crack growth, 𝑃̃𝑔𝑐𝑟 can be related to the sought critical gas pressure in the original bubble volume, 𝑃𝑔𝑐𝑟 , through Boyle’s law3 𝑃̃𝑔𝑐𝑟 = 𝑃𝑔𝑐𝑟

𝑉𝑜 , 𝑉̃

Fig. 6. Critical condition for grain boundary rupture, as defined in the rupture criterion by Worledge (1980).

crack. As shown in Fig. 6, he assumed that the crack opened to a width ℎ𝑐 , which he treated as a constant model parameter with an assumed value of 2 nm. He also assumed that the original bubble geometry and volume, defined by Eq. (1) did not change as a result of the cracking. Hence, according to Worledge (1980), the change in gas volume resulting from crack growth from 𝑟 = 𝑟𝑏 to the critical value 𝑟 = 𝓁 is ( ) 𝛥𝑉 = 𝜋ℎ𝑐 𝓁 2 − 𝑟2𝑏 . (17)

(14)

where 𝑉̃ is the total bubble + crack gas volume when the crack has grown stably to its critical radius 𝓁. Thirdly, Likhanskii and Matveev estimated this volume through the relation 4𝓁 3 ̃ 𝑐𝑟 (𝑃 − 𝑃ℎ ), 𝑉̃ = 𝑉𝑜 + 3𝜇 𝑔

(15)

As will be shown later, this is a questionable approximation that has a strong impact on the resulting rupture criterion. The fundamental assumption behind Worledge’s total energy criterion is that the grain boundary will break if the work done by expanding the gas through the grain face crack exceeds the energy required to create a crack that interlinks with its neighbour. This energy follows directly from ( ) 𝐸𝑐𝑟𝑎𝑐𝑘 = 𝒢𝑔𝑏 𝜋 𝓁 2 − 𝑟2𝑏 , (18)

where 𝜇 is the shear modulus of the fuel material. The second term on the right hand side is an approximation for the volume of a pennyshaped crack with internal overpressure. This crack is thus assumed to extend all the way from 𝑟 = 0 to 𝑟 = 𝓁, meaning that the existence of a bubble from 𝑟 = 0 to 𝑟 = 𝑟𝑏 is not accounted for in the approximation. By introducing the parameter 𝑃𝑒 = 3𝑉𝑜 𝜇∕(4𝓁 3 ) and combining Eqs. (13) to (15), we find the sought critical gas pressure related to the original bubble configuration √ 𝜋𝐸𝒢𝑔𝑏 𝜋𝐸𝒢𝑔𝑏 ( ) + 1 + 𝑃ℎ ∕𝑃𝑒 . (16) 𝑃𝑔𝑐𝑟 = 𝑃ℎ + 4𝓁𝑃𝑒 (1 − 𝜈 2 ) 4𝓁(1 − 𝜈 2 )

where 𝒢𝑔𝑏 is the fracture energy of the grain boundary and the remaining factor is the crack area created in the process; see Fig. 6. The work done by the expanding gas is 𝑉𝑏 +𝛥𝑉

This expression contains the hydrostatic pressure 𝑃ℎ , geometric parameters and material properties. The rupture criterion given by Eq. (16) is quite handy, since it captures the influence of material properties, bubble geometry and bubble fractional coverage or spacing in a general closed-form formulation. However, as will be shown in Section 3, the criterion gives an unphysical trend for 𝑃𝑔𝑐𝑟 with regard to bubble fractional coverage.

𝐸𝑔𝑎𝑠 =

𝑃𝑔 (𝑉 )𝑑𝑉 ,

∫𝑉𝑏

(19)

where the gas pressure 𝑃𝑔 will decrease as the total gas volume increases from the original bubble volume 𝑉𝑏 due to cracking. By use of the ideal gas law and assuming that the crack grows fast enough that variations in temperature and number of gas atoms trapped in the bubble can be neglected during the crack growth, we may write Eq. (19) as ( ) 𝑉𝑏 +𝛥𝑉 𝑑𝑉 𝛥𝑉 𝐸𝑔𝑎𝑠 = 𝑛𝑅𝑇 = 𝑛𝑅𝑇 ln 1 + . (20) ∫𝑉𝑏 𝑉 𝑉𝑏

Worledge (1980). In 1980, Worledge carried out a study of fission gas induced grain boundary rupture and swelling with the aim to interpret and understand results of heating tests on LMFBR fuel (Worledge, 1980). As part of this study, he reviewed and discussed grain boundary rupture criteria that existed at the time. He also derived a new criterion for grain boundary rupture by equating the energy needed for breaking all the ligaments between the grain face bubbles with the energy released from the overpressurized bubbles when the fracture occurs. He called the result a ‘‘total energy criterion’’, since it addressed the total energy consumed in breaking all the ligaments across the grain face (Worledge, 1980). In the sequel, a somewhat generalized form of this criterion is presented and some improvements to Worledge’s original formulation are proposed. While Worledge considered a periodic array of spherical bubbles with a centre-to-centre spacing 2𝓁 at the grain face, we will study the more general case with lenticular bubbles, as shown in Fig. 6. Worledge assumed that planar circumferential cracks would initiate at the bubble periphery and grow stably along the grain face until neighbouring cracks interconnect. The interconnection leads to a complete grain boundary breakdown. These assumptions are identical to those made by Likhanskii and Matveev (1999), but Worledge applied a different approximation for the additional void volume created by the growing

Considering that 𝛥𝑉 ∕𝑉𝑏 ≪ 1 for the bubble geometries of practical interest, we may approximate the logarithm in Eq. (20) by using the first two terms in a Maclaurin expansion ( ( )2 ) ( ) 𝛥𝑉 1 𝛥𝑉 𝛥𝑉 𝐸𝑔𝑎𝑠 ≈ 𝑛𝑅𝑇 − = 𝑃𝑔𝑐𝑟 𝛥𝑉 1 − , (21) 𝑉𝑏 2 𝑉𝑏 2𝑉𝑏 where 𝑃𝑔𝑐𝑟 = 𝑛𝑅𝑇 ∕𝑉𝑏 is the critical gas pressure in the bubble before any cracking occurs. The last factor within brackets is a measure of the relative drop in gas pressure caused by the crack opening. Finally, the critical gas pressure for grain boundary rupture is found by equating 𝐸𝑔𝑎𝑠 and 𝐸𝑐𝑟𝑎𝑐𝑘 : ( ) 𝒢𝑔𝑏 𝜋 𝓁 2 − 𝑟2𝑏 𝑃𝑔𝑐𝑟 = (22) ( ). 𝛥𝑉 𝛥𝑉 1 − 2𝑉 𝑏

With 𝛥𝑉 from Worledge’s approximation in Eq. (17) and 𝑉𝑏 from Eq. (1), we get 𝑃𝑔𝑐𝑟 =

𝒢𝑔𝑏

( ℎ𝑐

1−

3ℎ𝑐 (𝓁 2 −𝑟2𝑏 )

),

(23)

8𝑟3𝑏 𝜂(𝜃) ̃

where 𝜂(𝜃) ̃ is defined in Eq. (2). The rupture criterion given by Eq. (23) has been applied in two different stand-alone computer models for

3 Boyle’s law is used here for simplicity. Likhanskii and Matveev make use of the Van der Waals equation of state for the gas.

7

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transient fission gas release; SANDPIN (Wright and Fischer, 1984) and KFGR-T (Sim et al., 1992). However, these models were restricted to spherical bubbles (𝜃 = 90◦ ) and the mathematical expressions presented for the rupture criterion in both Wright and Fischer (1984) and Sim et al. (1992) contain some errors. To our knowledge, the SANDPIN and KFGR-T fission gas release models are no longer in use. Present work (improved Worledge criterion). The rupture criterion by Worledge in Eq. (23) differs from all other criteria by being independent of the hydrostatic pressure. This unphysical result is caused by the simplistic approximation that Worledge made for the gas volume increase related to crack growth. From Eq. (17) and Fig. 6, we recall that the cracks are assumed to attain a constant width, ℎ𝑐 , which is independent of the gas pressure in the bubble and the hydrostatic pressure in the fuel material. In reality, the crack width should depend on radial position 𝑟, 𝑃𝑔 , 𝑃ℎ and the elastic properties of the fuel material. A far better approximation to the gas volume increase is the relation proposed by Likhanskii and Matveev in Eq. (15). More precisely, by using the relation 𝛥𝑉 =

4𝓁 3 𝑐𝑟 (𝑃 − 𝑃ℎ ) 3𝜇 𝑔

Fig. 7. Initial equilibrium gas pressure in grain face bubbles with different assumed geometries, defined by 𝑃𝑜 = 𝑃ℎ𝑜 + 𝑃𝑠 , where 𝑃ℎ𝑜 is 20 MPa and 𝑃𝑠 is given by Eq. (3), using 𝛾 = 𝛾𝑓 𝑠 (1 − cos 𝜃) and the UO2 /gas specific surface energy in Table 1.

(24)

in Eq. (22), we get a non-linear equation for the critical gas pressure 𝑃𝑔𝑐𝑟 . This equation has to be solved numerically, e.g. by the Newton– Raphson method. A good starting approximation for the Newton– Raphson solution can be obtained by setting 1 − 𝛥𝑉 ∕(2𝑉𝑏 ) ≈ 1 in the combined Eqs.(22) and (24) and solving for 𝑃𝑔𝑐𝑟 . The result is √ 𝑃ℎ2 3𝜇𝒢𝑔𝑏 𝜋(𝓁 2 − 𝑟2𝑏 ) 𝑃ℎ 𝑐𝑟 𝑃𝑔 ≈ + + . (25) 2 4 4𝓁 3

Table 1 Material properties applied in the calculations.

Hence, by combining Eqs. (22) and (24), and using 𝑃𝑔𝑐𝑟 from Eq. (25) as a starting point for Newton–Raphson iterations, it is straightforward to calculate 𝑃𝑔𝑐𝑟 numerically. The formulation is general with regard to bubble geometry, and we note that the bubble spacing 2𝓁 can be calculated from the fractional coverage of grain face bubbles. Notwithstanding its simplicity, this rupture criterion reproduces the expected trends with regard to key parameters. It is compared with other rupture criteria in the following section.

Property/parameter:

Unit:

Source:

𝐸 = 𝜈 = 𝒢𝑔𝑏 =

190 0.31 2.0

[GPa] [–] [J⋅m−2 ]

𝑐𝑟 𝜎𝑔𝑏 =

55

[MPa]

𝛾𝑓 𝑠 =

0.80

[J⋅m−2 ]

Siefken et al. (2001) Siefken et al. (2001) Evans and Davidge (1969), Solomon (1972), Kutty et al. (1987), Gatt et al. (2015) Evans and Davidge (1969), Canon et al. (1971), Oguma (1982), Tachibana et al. (1979) Kutty et al. (1987), Hall et al. (1987), Matzke et al. (1980), Nikolopoulos (1974)

𝑐𝑟 are based on data from mechanical testing of macro-scale strength 𝜎𝑔𝑏 specimens of un-irradiated UO2 . As we will see in Section 3.2, these data have limited applicability to the local strength and toughness of grain boundaries in irradiated UO2 fuel. When the simulated accident initiates, the hydrostatic pressure on the fuel is assumed to vanish because of cladding expansion in response to heating and loss of coolant pressure.4 At the same time, the fuel temperature starts to increase, as illustrated in Fig. 8. The fuel heatup is assumed to be sufficiently fast that changes in the number of gas atoms in the grain face bubbles as well as diffusion-controlled bubble growth can be precluded. To a first approximation5 , the evolution of the bubble gas pressure follows Amontons’ law, 𝑃𝑔 (𝑡)∕𝑃𝑜 = 𝑇 (𝑡)∕𝑇𝑜 . In the following, we calculate these ratios at time of grain boundary rupture, using the eight different rupture criteria presented in Section 2. From LOCA simulation experiments on very high-burnup UO2 fuel (Oberländer and Wiesenack, 2014; Flanagan et al., 2013), we know that a ratio 𝑇 ∕𝑇𝑜 of about 2–3 is sufficient to cause grain boundary rupture when the fuel is unconstrained.

3. Comparative assessment of criteria 3.1. Considered test case With the aim to assess and compare the eight grain boundary rupture criteria from Section 2, we consider here a simple test case that resembles a loss-of-coolant accident. The simulated accident is supposed to initiate from steady-state operational conditions, defined by an arbitrary fuel temperature 𝑇𝑜 and a hydrostatic pressure 𝑃ℎ𝑜 . The latter is supposed to result from pellet-cladding mechanical interaction. It is assumed that the fuel has been operated under these steady-state conditions for a sufficiently long time and at sufficiently high temperature to achieve equilibrium pressure conditions in the intergranular bubbles. Hence, at start of the simulated LOCA, the gas pressure in the bubbles is 𝑃𝑜 = 𝑃ℎ𝑜 + 𝑃𝑠 where 𝑃ℎ𝑜 is the pre-accident hydrostatic pressure and the bubble capillary pressure 𝑃𝑠 is given by Eq. (3), using the effective surface energy 𝛾 = 𝛾𝑓 𝑠 (1 − cos 𝜃). The capillary pressure depends on the assumed geometry of the bubble, and for most criteria, lenticular bubbles with 𝜃 = 50◦ are assumed in the comparison. Spherical bubbles are assumed for the criterion by Gruber and co-workers and the Harwell criterion, since the geometry dependent functions in these criteria are unknown for other bubble shapes. For the rupture criterion proposed by DiMelfi and Deitrich, the vermicular bubble geometry in Fig. 4 is assumed. The initial pressure for these bubble geometries is plotted versus the projected bubble radius in Fig. 7 for a case with 𝑃ℎ𝑜 = 20 MPa. The material properties applied in the calculations are defined in Table 1. The applied values for fracture energy 𝒢𝑔𝑏 and grain boundary

3.2. Influence of bubble radius Let us first consider a fairly representative case with a pre-accident hydrostatic pressure (𝑃ℎ𝑜 ) of 20 MPa and a grain face bubble fractional

4

Likewise, it is assumed that the far-field stress 𝜎∞ in Eq. (7) vanishes. The relative error in Amontons’ law is less than 15% for gas pressures below 400 MPa and temperatures from 1000 to 2000 K. 5

8

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Fig. 8. Change in fuel temperature and hydrostatic pressure from steady-state to simulated accident conditions.

coverage (𝜙2 ) of 0.4. For this case, the calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture are shown in Figs. 9 and 10. Two of the rupture criteria shown in Fig. 9 predict grain boundary rupture for reasonable values of 𝑇 ∕𝑇𝑜 . However, the criterion by DiMelfi and Deitrich (1979) predicts rupture for 𝑇 ∕𝑇𝑜 < 1, which means that no heating at all is needed: the grain boundaries are calculated to break as soon as the pre-accident hydrostatic pressure of 20 MPa is relaxed. This result contradicts the fuel behaviour observed in experiments, and the rupture criterion must therefore be discarded. The other two criteria shown in Fig. 9 are based on critical grain boundary stress, and they exhibit similar trends versus 𝑟𝑏 : the small grain face bubbles will be the first to cause grain boundary rupture when the material is heated. The reason is the higher initial gas pressure in the small bubbles; see Fig. 7. The LEFM-based rupture criteria in Fig. 10 show a similar trend for small bubbles, but with the exception of Worledge’s criterion, this trend is reversed as the bubbles grow larger: the other four criteria in Fig. 10 predict that large bubbles are the most detrimental. As already mentioned in Section 2.2, this trend is expected from the inherent defect size dependence of LEFM-based rupture criteria. All the rupture criteria shown in Fig. 10 suggest that the fuel temperature has to increase by at least a factor 20 for the grain boundaries to break. This is not at all in line with the aforementioned LOCA simulation experiments on very high-burnup UO2 fuel (Oberländer and Wiesenack, 2014; Flanagan et al., 2013), which show that a ratio 𝑇 ∕𝑇𝑜 of only 2–3 is sufficient to cause grain boundary rupture in unconstrained UO2 fuel. There are two possible explanations to this difference: either we underestimate the pre-accident gas pressure in the grain boundary bubbles, or we overestimate the grain boundary fracture energy. As regards the pre-accident gas pressure, we assume an equilibrium pressure in the bubbles before the simulated accident. This assumption is reasonable for the central part of the pellet, where the material is hot enough that equilibrium conditions can be established by diffusion processes, but grain boundary bubbles in the colder peripheral part of the pellet may be initially overpressurized. However, in light of experimental data (Cagna et al., 2016), it seems unlikely that the pre-accident gas pressure, resulting from steady-state operating conditions, would be 20 times higher than the equilibrium pressure for grain boundary bubbles with 𝑟𝑏 = 1 μm, i.e. around 400 MPa, according to Fig. 7. Bubble pressures of this magnitude have been reported only for ramp tested fuel, i.e. for transient conditions (Mogensen et al., 1993). It is more likely that we overestimate the grain boundary fracture energy. In the calculations, we use 𝒢𝑔𝑏 = 2 J⋅m−2 , which is a rough estimate based on results from mechanical testing of macro-scale specimens of unirradiated UO2 (Evans and Davidge, 1969; Kutty et al., 1987; Gatt et al., 2015). The results presented in Fig. 10 suggest that this value is about two orders of magnitude higher than the local fracture energy of the grain boundaries in irradiated fuel.

Fig. 9. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑃ℎ𝑜 = 20 MPa and 𝜙2 = 0.4, according to the criteria by Gruber et al. (1973), Olander (1997) and DiMelfi and Deitrich (1979).

Fig. 10. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑃ℎ𝑜 = 20 MPa and 𝜙2 = 0.4, according to the criteria by Likhanskii and Matveev (1999), Chakraborty-Tonks-Pastore (Chakraborty et al., 2014), Harwell staff (Finnis, 1976; Matthews and Wood, 1984; Matthews et al., 1990), original Worledge (1980) and the proposed improvement to Worledge’s criterion.

fractional coverage. This is illustrated in Figs. 11 and 12, which show the calculated trend of 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture versus 𝜙2 . All calculations were done for a fixed projected bubble radius 𝑟𝑏 of 0.2 μm and 𝑃ℎ𝑜 = 20 MPa. It is clear from the figures that three of the considered criteria do not account for the fractional coverage at all and that the criterion by Likhanskii and Matveev gives unphysical results in that it predicts that the propensity for grain boundary rupture decreases with increasing bubble coverage. The modified Worledge criterion proposed in the present work also exhibits this trend for 𝜙2 < 0.25. However, the behaviour of the rupture criteria is of little practical importance for such low values of 𝜙2 , since grain boundary rupture is expected to occur when the fractional coverage is fairly large. The remaining criteria give physically admissible results, although the differences are fairly large. In particular, we note that the stress-based rupture criterion by Olander stands out from the LEFM-based criteria, since it predicts that the grain boundary strength tends to infinity as the

3.3. Influence of bubble fractional coverage Figs. 9 and 10 show that there are differences among the rupture criteria with respect to their dependence on gas bubble radius. Likewise, the criteria differ with respect to their dependence on gas bubble 9

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Fig. 13. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑟𝑏 = 0.2 μm and 𝜙2 = 0.4, according to the criteria by Gruber et al. (1973), Olander (1997) and DiMelfi and Deitrich (1979).

Fig. 11. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑃ℎ𝑜 = 20 MPa and 𝑟𝑏 = 0.2 μm, according to the criteria by Gruber et al. (1973), Olander (1997) and DiMelfi and Deitrich (1979).

boundary rupture is reduced by a factor 48/8 to 41/1, depending on bubble shape, as 𝑃ℎ𝑜 increases from 0 to 40 MPa. The calculated results shown in Figs. 13 and 14 underline the importance of pre-accident stress state to the grain boundary rupture behaviour under transient heating. For example, a change of preaccident hydrostatic stress 𝑃ℎ𝑜 from 5 to 10 MPa implies a reduction in calculated 𝑇 ∕𝑇𝑜 at grain boundary rupture by 28 to 45% for the considered case with 𝑟𝑏 = 0.2 μm. The reduction depends on the assumed shape of the bubble. For larger bubbles, the reduction in 𝑇 ∕𝑇𝑜 at grain boundary rupture would tend to 50% as 𝑃ℎ𝑜 is changed from 5 to 10 MPa. Hence, to accurately calculate when fission gas induced grain boundary rupture occurs under transient heating, it is important to accurately model and calculate the pre-accident stress state in the fuel. The pre-accident hydrostatic stress 𝑃ℎ𝑜 may have a strong impact on the pre-accident gas content and pressure in grain face bubbles. This is particularly true for large bubbles, in which the steady-state, pre-accident equilibrium gas pressure 𝑃𝑜 = 𝑃ℎ𝑜 + 𝑃𝑠 ≈ 𝑃ℎ𝑜 . 4. Results and discussion Fig. 12. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑃ℎ𝑜 = 20 MPa and 𝑟𝑏 = 0.2 μm, according to the criteria by Likhanskii and Matveev (1999), Chakraborty-Tonks-Pastore (Chakraborty et al., 2014), Harwell staff (Finnis, 1976; Matthews and Wood, 1984; Matthews et al., 1990), original Worledge (1980) and the proposed improvement to Worledge’s criterion.

In Section 3, the criteria presented in Section 2 were compared by use of a simple test case, designed to simulate time histories of fuel temperature and hydrostatic pressure under a hypothetical loss-ofcoolant accident that initiates from steady-state equilibrium conditions. The comparison was made by combining Amontons’ law with the eight different rupture criteria for calculating the relative temperature increase, 𝑇 ∕𝑇𝑜 , required for breaking the grain boundaries. This is a convenient parameter to compare, and there are also some experimental data on typical values for 𝑇 ∕𝑇𝑜 at time of grain boundary rupture, obtained from recent LOCA simulation tests on high burnup UO2 fuel (Puranen et al., 2013; Oberländer and Wiesenack, 2014; Turnbull et al., 2015; Bianco et al., 2015). The calculations were done for a range of bubble sizes and for different grain face area fractions covered with bubbles. These parametric studies served to map out the dependence of each rupture criterion on the aforementioned parameters. In reality, however, the bubble size and area coverage are not independent parameters. It is well known that the area coverage increases by growth and coalescence of bubbles, meaning that the two parameters are correlated (White, 2004). From the comparison in Section 3, it is clear that rupture criteria based on LEFM predict that large grain face bubbles are more detrimental for grain face rupture than small bubbles, whereas the opposite

bubble fractional coverage turns to zero; this is also clear from Eq. (7). The rupture criterion by Olander shows the strongest dependence on fractional coverage. 3.4. Influence of pre-accident hydrostatic pressure Finally, we consider a case with 𝑟𝑏 = 0.2 μm and 𝜙2 = 0.4, for which we let the pre-accident hydrostatic pressure 𝑃ℎ𝑜 vary from 0 to 40 MPa. Depending on bubble shape, the calculated capillary pressure 𝑃𝑠 ranges from 1 to 8 MPa for bubbles with 𝑟𝑏 = 0.2 μm. Hence, the initial gas pressure 𝑃𝑜 = 𝑃ℎ𝑜 + 𝑃𝑠 ranges from about 1 to 48 MPa in these calculations. Figs. 13 and 14 show the calculated trend of 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture versus 𝑃ℎ𝑜 . Since 𝑃𝑔 in itself is independent of 𝑃ℎ𝑜 , all the curves follow the relation 𝑃𝑔 ∕𝑃𝑜 ∝ 1∕(𝑃ℎ𝑜 + 𝑃𝑠 ). For this reason, the ratio 𝑇 ∕𝑇𝑜 or 𝑃𝑔 ∕𝑃𝑜 at grain 10

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(needle shape), in contrast to the fragments formed in more central parts of the fuel pellet (OECD/NEA, 2016). This difference in fragment shape suggests that the stress state is more hydrostatic (triaxial) in the pellet centre than in the periphery, and/or that the fragmentation in the central part of the pellet is dictated mainly by grain boundary orientation and not much affected by the stress state. Grain boundaries are probably less important for fragments formed in the HBS, since the HBS has a sub-micron grain structure with pores that are significantly larger than the typical grain size. Olander’s criterion suggests that small grain face bubbles are more limiting than large bubbles for causing grain boundary rupture when the fuel is overheated. This is opposite to the calculated trend from the other two criteria on the shortlist, which are the one by Chakraborty et al. (2014) and the improved Worledge criterion proposed in the present work. Both these criteria are based on linear elastic fracture mechanics and suggest that grain boundary rupture will occur predominantly at large bubbles. Due to lack of experimental data, it is impossible to settle whether this result is more correct than the contradicting outcome from stress based criteria. In practice, however, the difference in calculated trend with regard to bubble size between the two categories of criteria may be less important. The reason is that bubble size is closely correlated to the grain face fraction covered with bubbles, and that differences between the criteria with regard to this parameter partly offset the differences with regard to bubble size; compare Olander’s criterion in Fig. 11 with the C-T-P and improved Worledge criteria in Fig. 12. The criterion by Chakraborty et al. (2014) and the improved Worledge criterion proposed in the present work are similar with regard to their dependence on key parameters. Which one to use is a matter of taste: the criterion by Chakraborty–Tonks–Pastore is simple in its formulation, but the formulation is on the other hand restricted to a specific bubble geometry (lenticular bubbles with 𝜃 = 50◦ ). The improved Worledge criterion is more general, but since iterations are needed, it comes at a higher computational cost. Moreover, it is clear from our assessment that the lack of experimental data for the grain boundary strength and fracture energy poses a problem. In the presented calculations, the bulk strength and fracture energy of un-irradiated UO2 , determined through mechanical testing of large-scale specimens, were used as first order estimates for the local grain boundary properties. The stress-based rupture criteria gave reasonable results when using the bulk strength, which suggests that the tensile strength of the grain boundaries is not so different from the tensile strength of the bulk material. On the other hand, the criteria based on linear elastic fracture mechanics significantly overestimated the grain boundary strength, when using the bulk fracture energy of the material for the grain boundaries. More precisely, our calculations suggest that the local fracture energy of the grain boundaries in irradiated UO2 fuel is roughly two orders of magnitude lower than the bulk fracture energy of un-irradiated fuel. No measured data are available for the local fracture energy of grain boundaries, but molecular dynamics simulations indicate that it is comparable to that of bulk polycrystalline UO2 in unirradiated conditions (Nerikar et al., 2011; Zhang et al., 2014). These results suggest that the grain faces would be significantly weakened by irradiation effects, and a burnup dependence of the grain face fracture energy would thus be expected. Anyhow, due to the lack of data, the local fracture energy of the grain boundaries must be determined by inverse modelling, i.e. by calibrating the rupture criteria against results from annealing tests or integral type tests on irradiated fuel. This is attempted in Jernkvist (2019). Finally, it should be remarked that there is a difference between UO2 and (U,Pu)O2 mixed oxide (MOX) fuel in that the latter, at comparable burnup, generally has a higher concentration of fission gas retained in grain boundaries (Lemoine, 2006). The difference, which is most pronounced at low to moderate fuel burnup, is related to the heterogeneous microstructure of prevalent types of MOX fuel. At low to moderate fuel pellet average burnup, most of the fissioning in MOX fuel

Fig. 14. Calculated ratios 𝑃𝑔 ∕𝑃𝑜 or 𝑇 ∕𝑇𝑜 at time of grain boundary rupture for a case with 𝑟𝑏 = 0.2 μm and 𝜙2 = 0.4, according to the criteria by Likhanskii and Matveev (1999), Chakraborty-Tonks-Pastore (Chakraborty et al., 2014), Harwell staff (Finnis, 1976; Matthews and Wood, 1984; Matthews et al., 1990), original Worledge (1980) and the proposed improvement to Worledge’s criterion.

is true for stress based criteria. The trend observed for the LEFM-based rupture criteria is caused by their inherent defect size dependence, while the trend for the stress based criteria is mainly a result of the influence of capillary pressure on the initial bubble pressure; see Fig. 7. It is also clear that the assumptions made about the shape of the grain face bubbles are important for the performance of a specific criterion. Lenticular bubbles are assumed in most criteria, but spherical or vermicular bubbles are considered in a few exceptional cases. Two of the rupture criteria (DiMelfi and Deitrich, 1979; Likhanskii and Matveev, 1999) can be discarded, since they give results that contradict observed fuel behaviour or exhibit unphysical trends with regard to key parameters. Three other criteria (Matthews et al., 1990; Gruber et al., 1973; Worledge, 1980) are deemed unfit, since they do not satisfactorily account for the grain face area fraction covered with bubbles. This deficiency can possibly be remedied by extending two of the criteria (Matthews et al., 1990; Gruber et al., 1973) such that bubble interaction effects on stress or stress intensity are considered. Useful studies of these effects for co-planar arrays of internally pressure loaded defects are currently unavailable in literature, but in principle, the interaction effects can be quantified by finite element analyses or analytical methods (Lekesiz et al., 2013). In fact, the C-T-P rupture criterion (Chakraborty et al., 2014) can be seen as an extension of the Harwell criterion (Matthews et al., 1990), in which the influence of 𝜙2 in Eq. (11) has been determined by finite element analysis: confer Eqs. (9) and (10). Yet, in their original form, only three of the assessed rupture criteria seem useful. The shortlist includes the stress-based rupture criterion by Olander (1997), which gives reasonable results in spite of its simplicity. Several computer programs that are specifically designed for analyses of LWR fuel rods in RIA conditions (Moal et al., 2014; Suzuki et al., 2012; Khvostov, 2018) use this criterion as part of their models for transient fission gas release. It is clear from Eq. (7) that no particular assumptions are made for the bubble size or shape in Olander’s criterion, which means that it is applicable also to micron-sized pores in the fuel pellet high burnup structure. However, in contrast to grain face bubbles, these pores are not concentrated to specific planes in the material. Based on the Delesse principle, the grain face area fraction covered with bubbles, i.e. 𝜙2 in Eq. (7), should be replaced with the volume fraction of pores in case the criterion is applied to the HBS at the pellet rim. It has been reported that fine fuel fragments formed in the HBS are elongated 11

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occurs in Pu-rich agglomerates, at which gaseous fission products are concentrated and the HBS starts to form (Boulore et al., 2015). Whether the fracture properties of the grain boundaries differ between UO2 and (U,Pu)O2 fuel is unclear, but the difference in fission gas distribution leads to slightly higher transient fission gas release by grain boundary rupture in MOX fuel (Cazalis and Georgenthum, 2012). Nonetheless, the applicability of the grain boundary rupture criteria considered in this paper is not expected to be affected by the differences in fission gas distribution between UO2 and (U,Pu)O2 fuel.

material properties. In addition to integral type LOCA and RIA tests, out-of-reactor fuel annealing tests under well controlled temperature and pressure conditions are needed on irradiated fuel material that is carefully characterized with regard to the pre-test state. In particular, characterization of the pre-test population of grain boundary bubbles with respect to bubble size, shape, area fraction and fission gas content is essential for providing the data needed for validation and calibration of grain boundary rupture criteria. Declaration of competing interest

5. Conclusions The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Seven criteria for fission gas induced grain boundary rupture of oxide nuclear fuel were reviewed and assessed with the aim to identify suitable criteria for implementation in computer programs for fuel rod thermal–mechanical analyses. The criteria were originally formulated for application to RIA conditions, but here, we also consider their applicability to conditions typical for design basis LOCA in LWRs. The assessed criteria are based either on the notion of a critical grain boundary stress or on concepts from linear elastic fracture mechanics. Irrespective of the approach, all the criteria provide an estimate for the gas pressure required in intergranular bubbles for the grain boundary to break, and another common feature is that all criteria make use of idealized representations of the grain boundary and gas bubble geometry to calculate this critical gas pressure. The simplest rupture criteria are derived by considering isolated bubbles on a planar grain boundary (grain face), whereas more advanced criteria account for bubble-to-bubble interaction. This is without exception done by making the simplifying assumption that the grain face is covered with a periodic array of bubbles with equal size and shape. A simple test case, simulating the fuel conditions under a hypothetical LWR LOCA, were used for comparing the rupture criteria. Parametric studies, based on this test case, were conducted to investigate the impact of bubble size, bubble fractional coverage of the grain face and pre-accident hydrostatic pressure on the rupture predictions of each criterion. The outcome of these studies does not depend on the considered test case, which means that the results of the assessment apply also to RIA conditions. The LOCA test case was selected, since it allows fairly straightforward comparisons of calculated results with data from LOCA simulation tests and experiments. The parametric studies showed that the calculated trends with regard to bubble size and fractional coverage differ substantially among the criteria. They also suggest that the pre-accident hydrostatic pressure in the fuel is important for the rupture behaviour under transient heating, in particular when the grain boundaries contain large bubbles and pores. To accurately calculate when fission gas induced grain boundary rupture occurs under transient heating, it is therefore important that the computer program hosting the criteria is able to accurately model the pre-accident stress state in the fuel, especially under PCMI conditions. Five of the criteria were deemed unfit, since they exhibit unphysical trends with regard to the aforementioned parameters and/or give results that contradict the fuel behaviour observed in LOCA simulation tests. However, as shown in this work, one of the deficient criteria can be improved by a slight re-formulation. In conclusion, we identify three rupture criteria as being suitable for modelling fuel fine fragmentation under RIA and LOCA conditions. One of these criteria is believed to be applicable not only to grain boundary bubbles, but also to micron-sized pores in the fuel pellet high burnup structure. Finally, it should be recognized that the considered rupture criteria contain poorly known material properties in terms of the local strength or fracture energy for the grain boundaries. Our assessment indicates that, for irradiated UO2 fuel, the local tensile strength of the grain boundary is comparable to the macroscopic tensile strength of the material, whereas the local fracture energy of the grain boundary is significantly lower than the fracture energy of the polycrystalline bulk material. Additional experiments are needed to determine these local

Acknowledgements This work was funded by the Swedish Radiation Safety Authority (SSM) under research contracts SSM 2015-4050 and SSM 2016-3944. The author is grateful to Ali R. Massih for helpful comments on the manuscript. References Anderson, T.L., 2005. Fracture Mechanics: Fundamentals and Applications, third ed. CRC Press, Boca Raton, FL, USA. Bianco, A., 2015. Experimental iNvestigation on the Causes for Pellet Fragmentation Under LOCA Conditions (Ph.D. thesis). Technischen Universität München, Germany. Bianco, A., Vitanza, C., Seidl, M., Wensauer, A., Faber, W., Marcian-Juan, R., 2015. Experimental investigation on the causes for pellet fragmentation under LOCA conditions. J. Nucl. Mater. 465, 260–267. Boulore, A., Aufore, L., Federici, E., Blanpain, P., Blachier, R., 2015. Advanced characterization of MIMAS MOX fuel microstructure to quantify the HBS formation. Nucl. Engng. Des. 281, 79–87. Cagna, C., Zacharie-Aubrun, I., Bienvenu, P., Barrallier, L., Michel, B., Noirot, J., 2016. A complementary approach to estimate the internal pressure of fission gas bubbles by SEM-SIMS-EPMA in irradiated nuclear fuels. Mat. Sci. Engng. 109, 012002. Canon, R.F., Roberts, J.T.A., Beals, R.J., 1971. Deformation of UO2 at high temperature. J. Amer. Ceram. Soc. 54 (2), 105–112. Cazalis, B., Georgenthum, V., 2012. MOX fuel behaviour under reactivity initiated accident. In: Proceedings of TopFuel 2012. European Nuclear Society, Manchester, UK, pp. 176–182. Chakraborty, P., Tonks, M.R., Pastore, G., 2014. Modeling the influence of bubble pressure on grain boundary separation and fission gas release. J. Nucl. Mater. 452, 95–101. Coindreau, O., Fichot, F., Fleurot, J., 2013. Nuclear fuel rod fragmentation under accidental conditions. Nucl. Engng. Design 255, 68–76. Deitrich, L.W., Jackson, J.F., 1977. The role of fission products in whole core accidents - research in the USA. In: IAEA-IWGFR Specialist’S Meeting on Role of Fission Products in Whole Core Accidents. IWGFR–19, International Atomic Energy Agency, AERE, Harwell, UK, pp. 66–87. DiMelfi, R.J., Deitrich, L.W., 1979. The effects of grain boundary fission gas on transient fuel behavior. Nucl. Technol. 43, 328–337. DiMelfi, R.J., Kramer, J.M., 1983. Modeling fragmentation and spallation of oxide reactor fuel during transient heating including the effects of burnup and fission product distribution. Nucl. Technol. 62, 51–61. Eslami, M.R., Hetnarski, R.B., Ignaczak, J., Noda, N., Sumi, N., Tanigawa, Y., 2013. Theory of elasticity and thermal stresses: explanations, problems and solutions. In: Solid Mechanics and Its Applications, vol. 197, Springer, Dordrecht, Netherlands. Evans, A.G., Davidge, R.W., 1969. The strength and fracture of stoichiometric polycrystalline UO2 . J. Nucl. Mater. 33, 249–260. Finnis, M.W., 1976. (Report Unavailable). Harwell Research Report AERE-R8537, Atomic Energy Research Establishment, Harwell, Oxon, UK. Flanagan, M., Askeljung, P., Puranen, A., 2013. Post-Test Examination Results from Integral, High-Burnup, Fueled LOCA Tests at Studsvik Nuclear Laboratory. Report NUREG-2160, U.S. Nuclear Regulatory Commission, Washington, DC, USA. Fuketa, T., Nakmura, T., Sasajima, H., Nagase, F., Uetsuka, H., Kikuchi, K., Abe, T., 2000. Behavior of PWR and BWR fuels during reactivity-initiated accident conditions. In: 2000 International Topical Meeting on Light Water Reactor Fuel Performance. American Nuclear Society, Park City, UT, USA, pp. 359–374. Fuketa, T., Sasajima, H., Mori, Y., Ishijima, K., 1997. Fuel failure and fission gas release in high burnup PWR fuels under RIA conditions. J. Nucl. Mater. 248, 249–256. Gatt, J.M., Sercombe, J., Aubrun, I., Menard, J.C., 2015. Experimental and numerical study of fracture mechanisms in UO2 nuclear fuel. Engng. Failure Analysis 47, 299–311. 12

Progress in Nuclear Energy 119 (2020) 103188

L.O. Jernkvist

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