General properties of directed ordering near line defects in elastic crystals undergoing phase transition are studied using the two-component time-dependent Ginzburg-Landau equation. Upon quenching the system below its transition point, the temporal evolution of the order parameter in the vicinity of the defect is evaluated. The development of vortices is explored and their interaction with the structural defect is examined. Finally, phase transitions in improper ferroelectrics in the context of the model are discussed.

The problem of heterogeneous nucleation of second-phase in alloys in the vicinity of elastic defects is considered. The defect can be a dislocation line or a crack tip residing in a crystalline solid. We use a compressible Ising-like model, represented by the Ginzburg-Landau equation, to describe the spatiotemporal evolution of the order parameter in the environs of the defect. The model accounts for the elasticity of the solid and the interaction of order parameter field with the elastic field of the defect. A finite volume numerical method is used to solve the governing partial differential equation for the order parameter. We examine the nature of the phase transition and discuss the phase diagram topology near and away from the defect. We discuss our calculations in light of observations of formation of hydrides in zirconium and titanium alloys.

General properties of directed ordering near line defects, in particular an edge dislocation, in elastic crystals undergoing phase transition are studied using the two-component time-dependent Ginzburg-Landau equation in two dimensions or 2D-XY model. The associated Landau potential comprises a sixth-order term, cubic anisotropy terms and the field of the dislocation. In thermodynamic equilibrium, the phase diagram for the model is delineated. Upon quenching the system below its transition point, the temporal evolution of the order parameter components in the vicinity of the defect is numerically evaluated. The development of vortices, emanated from the model, is explored and their interaction with the dislocation is examined. The dislocation produced a vortex free circular region whose diameter grew almost linearly with time. The time-dependence of vortex density for various settings of the Landau potential coefficients are evaluated. The vortex density ( in 2D) decreased inversely with time, albeit faster in the absence of dislocation. By computing the two-point correlation function, we established that the dynamic scaling law is satisfied for the considered model if the distance is scaled by L = t(1/2) or by its half-width L-1/2 for a dislocation free crystal. Finally, phase transitions in improper ferroelectrics in the context of the model are discussed.

The time-dependent Ginzburg-Landau (TDGL) equation for a single component non-conservative structural order parameter is used to study the spatio-temporal evolution of a second phase in the vicinity of an edge dislocation in an elastic crystalline solid. A symmetric Landau potential of sixth-order is employed. Dislocation field and elasticity modify the second-order and fourth-order coefficients of the Landau polynomial, respectively, where the former makes the coefficient singular at the origin. The TDGL equation is solved numerically using a finite volume method, where a wide range of parameter sets is explored. Computations are made for temperatures both above and below the transition temperature of a defect-free crystal Tc0. In both cases, the effects of the elastic properties of the solid and the strength of interaction between the order parameter and the displacement field are examined. If the system is quenched below Tc0, a steady state is first reached on the compressive side of the dislocation. On the tensile side, the growth is held back. The effect of thermal noise term in the TDGL equation is studied. We find that if the dislocation is introduced above Tc0, thermal noise supports the nucleation of the second phase, and a steady state will be attained earlier than if the thermal noise was absent. For a dislocation-free solid, we have compared our numerical computations for a mean-field (spatially averaged) order parameter versus time with the late time growth of the ensemble-averaged order parameter, calculated analytically, and find that both results follow upper asymptotes of sigmoid curves.

Hydrogen induced embrittlement and associated cracking effects in Zirconium alloys impose a serious problem, for example, within the nuclear industry where these materials are used in a range of applications, including fuel claddings. To get a fuller understanding of the hydrogen diffusion, hydride formation and phase transformations taking place at conditions of external stress and at elevated temperature it is crucial to augment experimental measurements with multi-level modelling. It is, for example, possible to simulate hydride phase-transformation trough phase-field calculations[1]. In order to get accurate results from these calculations, however, it is important to have basic physical information, such as interfacial energies and elastic constants. The latter were obtained completely from first principle calculations of density functional theory. In pure zirconium, at low hydrogen concentrations and at low temperatures, hydrogen atoms are dissolved into the matrix and preferentially occupy tetrahedral sites within the thermodynamically stable aZr hexagonal closed packed unit cell. At higher concentrations the various face centred cubic and tetragonal hydride phases start to form. Using the plane-wave code DACAPO, within the generalized gradient approximation, the lattice parameters and the ionic positions of bulk aZr, dZrHx and gZrHx were relaxed with respect to total energies. The formation energies as well as bulk moduli were then obtained by fitting total energies as a function of lattice parameters with a 3rd or 4th order polynomial. By comparing the present results with previous calculations and experiments it was possible to validate the methods for further use. The results, thus, comprise a foundation and serves to verify methods and codes. The next step is to compute more complex properties such as surface free energies, or interfacial energies, of bi-phase Zr-H systems, so that the full multiscale model can be implemented. 1. V. Vaithyanathan et al., Acta Materialia 52 (2004) p. 2973

The accuracy of a numerical fission gas release algorithm developed by Forsberg and Massih for solving the problem of diffusive flow to a spherical grain boundary is analysed. Estimates of numerical errors are derived for both steady-state and time varying conditions. We also present a method through which the accuracy of the algorithm can be improved or optimised for most applications.

A computational model for hydrogen transport, hydrogen induced deformation and fracture in metals that form binary hydrides, such as Zr and Ti alloys, is presented. The model uses a continuum description of the two-phase (metal + hydride) material, and solves the multi-field partial differential equations for temperature and stress-directed hydrogen diffusion together with mechanical equilibrium in a three-dimensional finite element setting. Point-kinetics models are used for metal-hydride phase transformation and stress-directed orientation of hydride precipitates, while a cohesive zone fracture model caters for initiation and propagation of cracks. The local fracture properties of the hydrided material are correlated to the calculated local concentration and orientation of the hydride precipitates, which have a strong embrittling effect on the material. In Part I of this two-part paper, we present sub-models applied for the aforementioned phenomena together with a detailed description of their numerical implementation. The applicability of the model is then demonstrated by simulating five independent experiments on hydrogen transport, metal-hydride phase transformation and stress-directed hydride orientation in zirconium alloys. Based on the results, we conclude that the model captures these phenomena over a wide range of thermo-mechanical loading conditions, including thermal cycling. Part II of the paper is focussed on fracture, and includes details on the fracture model and its validation against tests and experiments on initiation and propagation of hydride induced cracks.

An integrated numerical model for hydride-induced failure of zirconium alloys is presented. The model solves the time-dependent and interconnected problems of temperature- and stress-directed diffusion of hydrogen, metal-hydride phase transformation, stress-directed hydride orientation, hydrogen-induced expansion and fracture within a two- dimensional finite element framework. The finite element method allows representation of arbitrary two-dimensional geometries and thermo-mechanical loading conditions. The model has a wide range of application, but is intended primarily for analyses of delayed hydride cracking in cladding tubes of light water reactor nuclear fuel rods.

We study initiation of hydride induced cracks in Zr-2.5%Nb CANDU reactor pressure tubes, containing blunt notches and exposed to temperature cycles by use of a novel computational model. The model uses a continuum description of the two-phase (alpha-phase metal plus delta-phase hydride) material, and solves the multi-field partial differential equations for temperature- and stress-directed hydrogen diffusion together with mechanical equilibrium in a finite element setting. Point-kinetics models are used for metal-hydride phase transformation and stress-directed orientation of hydrides, while a cohesive zone fracture model caters for initiation and propagation of cracks. Our calculations show that the propensity for crack initiation through hydride fracture at the notch during an overload event depends on the preceding histories of stress and temperature, and in particular, on the number of ratcheting thermal cycles experienced by the hydrided material. The calculated results agree favorably with experiments reported in open literature.

This work deals with a computational model for hydrogen transport, hydrogen induced deformation, embrittlement and fracture in hydride forming metals, notably Ti and Zr. The model uses a continuum description of the two-phase (alpha-phase metal plus delta-phase hydride) material, and solves the multi-field partial differential equations for temperature- and stress-directed hydrogen diffusion together with mechanical equilibrium in a three-dimensional finite element setting. Point-kinetics models are used for metal-hydride phase transfor¬mation and stress-directed orientation of hydrides, while a cohesive zone fracture model caters for initiation and propagation of cracks. The model as a whole is versatile and can be used to study a wide range of problems and conditions involving transport of hydrogen by directed diffusion in combination with hydride precipitation and fracture. The applicability of the model is demon-strated by simulations of fracture tests on a hydrogen-charged Zr-Nb alloy.

Downward axial relocation of fuel pellet fragments may occur when overheated and internally overpressurized cladding tubes of light water reactor fuel rods distend due to creep during a loss-of-coolant accident (LOCA). The relocation is of safety concern, since it changes the axial distribution of heat load along the rod and also has the potential to increase the amount of fuel material dispersed into the reactor coolant, should the cladding fail. Here, we present a computational model that calculates the fuel relocation on the basis of estimated fuel fragment size distributions and the calculated cladding distension along the fuel rod. The model has been implemented and fully integrated with the FRAPTRAN-1.5 computer program, such that thermal feedback effects of fuel relocation on the axial redistribution of fuel mass, stored heat and power are accounted for in FRAPTRAN’s calculations of the fuel rod thermo-mechanical behaviour. The model has been validated against the IFA-650.4 integral LOCA test in the Halden reactor, Norway, which was done on a very high burnup UO2 fuel rodlet and resulted in extensive fuel pellet pulverization, axial relocation and dispersal into the coolant. Our simulations of this test suggest that thermal feedback effects from axial fuel relocation are strong enough to significantly affect the dynamics of cladding ballooning and rupture, in spite of the short duration of these processes. Moreover, for the considered LOCA test, the axial relocation has a strong effect on the calculated peak cladding temperature and oxidation after rupture.

Haverifall med kylmedelsförlust (LOCA) i lättvattenreaktorer kan leda till överhettning av bränslestavar, utvidgning av stavarnas kapslingsrör, samt axiell omflyttning av brän- slekutsfragment i de delar av stavarna som expanderat. Bränsleomflyttningen kan kon- centrera värmebelastningen till en begränsad del av stavarna och därigenom öka risken för kapslingsskador och förvärra kapslingsrörens oxidation lokalt. Den kan också medföra att mängden bränsle som sprids ut i kylvattnet ökar i händelse av kapslingsskada. Nyligen genomförda LOCA-experiment har återuppväckt tillsynsmyndigheternas intresse för dessa fenomen, då provresultaten antyder att högutbränt (> 65 MWd/kgU) UO2- bränsle kan pulveriseras till mycket små (< 0.2 mm) fragment, vilket leder till en större risk för omflyttning och efterföljande utspridning i kylvattnet än vad som tidigare observerats för bränsle med låg eller medelhög utbränning. För att analysera dessa frågor utvecklas här en beräkningsmodell för axiell omflyttning av bränslefragment under LOCA och dess effekter på bränslestavens värmebelastning och skadeprocesser. Modellen införs i SSM:s version av FRAPTRAN-1.5, ett beräknings- program avsett för termomekanisk analys av bränslestavar under transienter och olyckor. Bränsleomflyttningen beräknas på grundval av uppskattad storleksfördelning för bränsle- fragmenten samt kapslingsrörets beräknade deformationsprofil längs bränslestaven, och bränsleomflyttningens inverkan på axiell omfördelning av effekt och lagrad värme beaktas i termomekaniska analyser av bränslestaven. Vår modell tar således full hänsyn till termiska återkopplingseffekter av bränsleomflyttningen, i motsats till existerande beräkningsmodeller. Den tillhandahåller även uppskattningar av den bränslemängd som potentiellt kan spridas ut i kylvattnet vid kapslingsbrott i någon del av staven. Modellen valideras genom jämförelser mot mätdata och diskuteras mot bakgrund av experimentella resultat. I synnerhet studerar vi LOCA-experiment IFA-650.4, som genomförts i Haldenreaktorn, Norge. Provet gjordes på en högutbränd (92.3 MWd/kgU) UO2-provstav, och resulterade i omfattande bränslepulverisering, axiell omflyttning och bränsleutspridning i kylvattnet. Våra simuleringar av detta prov antyder att termiska återkopplingseffekter från den axiella bränsleomflyttningen är tillräckligt starka för att märkbart påverka dynamiken för kapslingens deformation och brott, trots att den beräknade varaktigheten hos dessa processer är högst 7–8 sekunder. Dessutom har den axiella bränsleomflyttningen stor inverkan på kapslingens beräknade maximala temperatur och oxidation efter kapslingsbrottet för det beaktade LOCA-experimentet. Avslutningsvis skall nämnas att vårt arbete pekar mot att ovan nämnda pulveriserings- mekanism för högutbränt bränsle är betydelsefull för axiell bränsleomflyttning, då den kan öka bränslefragmentens packningstäthet. Pulveriseringen underlättar därmed axiella rörelser hos bränslekutspelaren och ökar den lokala värmebelastningen i områden där kutsfragment ansamlas. Våra beräkningar antyder att bränsle med en genomsnittlig kutsutbränning runt 70–75 MWd/kgU torde vara särskilt benäget för axiell omflyttning, på grund av dess förväntade fragmentstorleksfördelning.

Downward axial relocation of fuel fragments within distending fuel rods may occur during loss-of-coolant accidents (LOCAs) in light water reactors. The fuel relocation may localize the heat load to “ballooned” parts of the rod, thereby increasing the risk for cladding failure and aggravating local oxidation. It may also increase the amount of fuel dispersed into the coolant, should the cladding fail. Recent LOCA tests have revived interest in the relocation and dispersion phenomena, since the test results suggest that high burnup UO2 fuel pellets may pulverize into very fine fragments, with a higher potential for axial relocation and subsequent dispersal than observed earlier for low to medium burnup fuel. To improve our understanding of these phenomena, a computational model for axial relocation of fuel fragments during LOCA and its effects on the fuel rod heat load and failure processes has been developed and introduced in the FRAPTRAN-1.5 computer program. The axial fuel relocation is calculated on the basis of estimated fuel fragment size distributions and the calculated cladding distension along the fuel rod, and its effects on the axial redistribution of fuel mass, stored heat and power are accounted for in FRAPTRAN’s calculations of the fuel rod thermo-mechanical behaviour. The model has been validated against the IFA-650.4 integral LOCA test in the Halden reactor, Norway, which was done on a very high burnup UO2 fuel rodlet and resulted in extensive fuel pulverization, axial relocation and fuel dispersal into the coolant. Our simulations of this test suggest that thermal feedback effects from axial fuel relocation are strong enough to significantly affect the dynamics of cladding ballooning and rupture, even though the calculated duration of these processes is no more than 7–8 seconds. Moreover, for the considered LOCA test, the axial relocation has a strong effect on the calculated peak cladding temperature and oxidation after rupture.

The nonlinear dynamic behavior of damped beam oscillator with elastic two-sided amplitude constraints is analyzed. The structure is modeled by a Bernoulli-Euler beam supported by elastic springs. Finite element method is used for discretization in space and time integration is performed by Newmark’s method. Rayleigh damping is assumed for the structure. Symmetric and elastic double-impact motions, both harmonic and subharmonic, are studied by way of a Poincare´ mapping that relates the states at subsequent impacts. We have found that by increasing the forcing frequency (ω) for the beam at a certain frequency a stable period one motion (solution) turns into a stable period two motion and subsequently without bifurcation it transits to an infinite number of solutions characteristic of chaotic behavior. By further increasing ω a series of windows in the bifurcation diagram (impact velocity vs. ω) comprising periodic solutions within the chaotic domain appear.

We model an interface layer connecting two parts of a solid body by 𝑁 parallel elastic springs connecting two rigid blocks. We load the system by a shear force acting on the top side. The springs have equal stiffness but are ruptured randomly when the load reaches a critical value. For the considered system, we calculate the shear modulus 𝐺 as a function of the order parameter 𝜙 describing the state of damage, and also the “spalled” material (burst) size distribution. In particular, we evaluate the relation between the damage parameter and the applied force and explore the behavior in the vicinity of material breakdown. Using this simple model for material breakdown, we show that damage, caused by applied shear forces, is analogous to a first-order phase transition. The scaling behavior of 𝐺 with 𝜙 is explored analytically and numerically, close to 𝜙=0 and 𝜙=1 and in the vicinity of 𝜙𝑐, when the shear load is close to but below the threshold force that causes material breakdown. Our model calculation represents a first approximation of a system subject to wear induced loads.

The dynamics of non-linear oscillators comprising of a single-degree-of- freedom system and beams with elastic two-sided amplitude constraints subject to harmonic loads is analyzed. The beams are clamped at one end, and constrained against unilateral contact sites near the other end. The structures are modelled by a Bernoulli-type beam supported by springs using the finite element method. Rayleigh damping is assumed. Symmetric and elastic double-impact motions, both harmonic and sub-harmonic, are studied by way of a Poincaré mapping that relates the states at subsequent impacts. Stability and bifurcation analyses are performed for these motions, and domains of instability are delineated. Impact work rate, which is the rate of energy dissipation to the impacting surfaces, is evaluated and discussed. In addition, an experiment conducted by Moon and Shaw on the vibration of a cantilevered beam with one-sided amplitude constraining stop is modelled. Bifurcation observed in the experiment could be captured.

Impact work-rate of a weakly damped beam with elastic two-sided amplitude constraints subject to harmonic excitation is calculated. Impact work-rate is the rate of energy dissipation to the impacting surfaces. The beam is clamped at one end and constrained by unilateral contact sites near the other end. This system was an object of a vibro-impact experiment which was analyzed in our earlier paper (Knudsen and Massih 2000). Detailed nonlinear dynamic behavior of this system is evaluated in our companion paper (Knudsen and Massih 2002b). Computations show that the work-rate for asymmetric orbits is signifi-cantly higher than for symmetric orbits at or near the same frequency. For the vibro-impacting beam, under conditions that exhibit a stable attractor, calculation of work-rate allows us to predict the “lifetime” of the contacting beam due to fretting-wear damage by extending the stable branch and using the local gap between contacting surfaces as a control parameter. That is, upon computation of the impact work-rate, the fretting-wear process time is calculated through back-substitution of the work-rate and gap-width in a given wear law. 

The vibration and impact dynamics of a periodically forced loosely supported beam are analyzed. The wear work rates at impact points are evaluated. The considered beam is clamped at one end, and constrained against unilateral contact at contact sites, with or without friction, near the other end. In this work, the structure is modeled by a Bernoulli-type beam supported by springs using finite element method. Our model calculations are compared with measurements of contact forces and displacements made on a loosely supported rod that was subjected to harmonic loading. Furthermore, the dynamics of vibro-impacts are characterized by evaluating the impact velocity as a function of harmonic excitation frequency for two idealizations of the aforementioned structure. [S0094-9930(00)01002-7] 

An integrated computer model for reactor fuel cladding rupture under loss-of-coolant accident (LOCA) conditions has been implemented in the transient fuel rod performance code FRAPTRAN. The model treats the zirconium alloy solid-to-solid phase transformation kinetics, cladding oxidation, cladding deformation, and eventually cladding rupture concurrently. It is for use together with the recently developed finite element based solution module in FRAPTRAN-1.4. The model has been employed to calculate ex-reactor single-rod transient burst tests in which the rod internal pressure and the heating rate were kept constant during the tests. The calculations are compared with experimental data on cladding rupture strain, temperature and pressure. Furthermore this model in FRAPTRAN-1.4 has been used to evaluate a LOCA simulation test within the IFA-650 series performed in the Halden boiling heavy- water reactor. The Halden test was made on a pressurized water reactor fuel rod with Zircaloy-4 cladding. It simulated the blowdown and refill (heatup) phase of a LOCA. The results of the FRAPTRAN-1.4 calculations are compared with experimental data on (i) maximum hoop strain at rupture, (ii) cladding diameter increase versus axial position, (iii) internal rod pressure versus time, (iv) peak cladding temperature at rupture, and (v) post-test cladding outer surface layer thickness. The results are discussed in terms of various uncertainties in the calculations. Nevertheless, the new implementation of the cladding rupture models indicates an overall improvement of the code for fuel rod LOCA analysis.

We present a unified model for calculation of zirconium alloy fuel cladding rupture during a postulated loss-of-coolant accident in light water reactors. The model treats the Zr alloy solid-to-solid phase transformation kinetics, cladding creep deformation, oxidation, and rupture as functions of temperature and time in an integrated fashion during the transient. The fuel cladding material considered here is Zircaloy-4, for which material property data (model parameters) are taken from the literature. We have modelled and simulated single-rod transient burst tests in which the rod internal pressure and the heating rate were kept constant during each test. The results are compared with experimental data on cladding rupture strain, temperature, and pressure. The agreement between computations and measurements in general is satisfactory. The effects of heating rate and rod internal pressure on the rupture strain are evaluated on the basis of systematic parameter variations of these quantities. In the α-phase of Zr, the burst strain decreases with increasing heating rate, whereas in the two-phase coexistence (α+β) domain and β-phase, the situation is more complex. Also, the mechanism for creep deformation in the (α+β) domain is not well understood; hence, its mechanistic constitutive relation is presently unknown.

Bränslestavprover under LOCA förhållanden i Halden reaktorn, IFA-650 seriens prov 2, 3, och 4, utvärderas med två olika versioner av datorprogrammet FRAPTRAN-1.4. Provobjektet vid IFA-650.2 var en färsk bränslestav, dvs. i obestrålat tillstånd, med stavegenskaper karakteristiska för tryckvattenreaktor (PWR) bränsle. Experimentstavarna för proven IFA- 650.3 och IFA-650.4 tillverkades från bränslestavar, förbestrålade i en PWR, till en stavutbränning av 82 MWd/kgU för det första provet och 92 MWd/kgU för det andra. Alla tre bränslestavar havererade under LOCA proven vid temperaturer omkring eller under 800◦C, genom kapslingsbrott. Resultaten från våra datorberäkningar jämförs med mätdata för följande parametrar: (i) kapslingstemperatur som funktion av tid; (ii) bränslestavtryck som funktion av tid; (iii) kapslingsdiameter vid brott längs staven; (iv) maximal kapslingstemperatur vid brott; och (v) maximal oxidtjocklek efter LOCA transienten (prov 2). Överensstämmelsen mellan beräkningar och mätningar samt mellan de två olika versionerna av beräkningsprogrammet är tillfredsställande. I rapporten ges beskrivningar av de olika proven, datorprogrammen, beräkningarna och en sammanfattning av resultaten.

De främsta skälen till att tillföra små mängder av metalloxider, som exempelvis Cr2O3, till kärnbränsle av UO2 är att öka materialets kornstorlek och densitet, samt om möjligt göra bränslekutsarna mjukare. En ökning av bränslets kornstorlek ( > 30 µm) ökar diffusionslängden för gasformiga fissionsprodukter till materialets korngränser, genom vilka den största delen gas avges från bränslekutsen. Resultatet torde vara en fördröjning av termiskt aktiverad gasavgivning från bränslet vid en given temperatur. En ökning av bränslets densitet ger större mängd U-235 per bränsleknippe och leder till mindre bränsleförtätning under bestrålning. Mjukare kutsar, det vill säga bränsle med en högre kryptöjningshastighet och/eller lägre sträckgräns, kan mildra mekanisk växelverkan mellan kuts och kapsling under effekthöjningar (ramper) vid reaktordrift, vilket skulle minska risken för kapslingsbrott. Tillsatser kan också påverka UO2-bränslets termofysikaliska egenskaper, såsom värmekapacitet, termisk längdutvidgning och värmeledningsförmåga. Experimentella data och teoretisk analys antyder emellertid att om koncentrationen av tillsatserna är låg (t.ex. < 0.2 viktprocent av tillsatsämnet Cr2O3), så påverkas dessa egenskaper endast marginellt. Målet med denna rapport är att utvärdera data och modeller för viktiga egenskaper hos UO2-baserat bränsle innehållande tillsatser. Tillsatserna som beaktas är de för vilka studier finns rapporterade i öppen litteratur. Huvudakligen diskuteras Cr2O3, men vi inkluderar även Al2O3,MgO och Nb2O5. Lämpliga modeller för termofysikaliska egenskaper utvärderas och rekommenderas för UO2-bränsle med tillsatser av typen M2O3 (M: metall), men även för MgO-dopad UO2. Data och korrelationer för diffusivitet hos fissionsgas i UO2 med nämnda tillsatsämnen analyseras och används i en standardmodell för fissionsgasavgivning och gassvällning, i syfte att utvärdera dessa egenskaper och hur de påverkas av tillsatsämnena. Även inverkan av bränslets kornstorlek på gasavgivning och svällning utvärderas. Tillgängliga data och korrelationer för termiskt kryp i Nb2O5- och Cr2O3-dopat bränsle utvärderas kritiskt, och möjliga krypmekanismer beskrivs. Resultat från utvalda reaktorbestrålningsprogram, rampprov och transienter på bränsle med tillsatser granskas översiktligt. Rapporten avses tjäna som underlag för implementering av modeller i beräkningsprogram för bränsle-stavanalys.

High-temperature (≈ 900−1400 K) steady-state creep test data on as-received zirconium alloys, Zr-1wt%Nb and Zircaloy-4 used as fuel cladding materials in light water reactors are evaluated by employing two sets of models. In particular, the focus of the paper is on the former alloy and in the two-phase coexistence region, i.e. the (α+β)-domain of the alloy. In one modeling approach, the constitutive relations for the two single phase regions (α and β) are combined through a phase transition kinetic model and a phase mixing rule; in another, a superplasticity model is used directly to calculate the creep deformation rate as a function of stress and temperature in the (α+β)-domain. The results show that the former approach is inadequate in retrodicting the experimental data, while the latter one gives a fair overall agreement. The paper describes the details of the models, the data, and derivations of the constitutive laws.

A model for nucleation of second phase at or around dislocation in a crystalline solid is considered. The model employs the Ginzburg-Landau theory of phase transition comprising the sextic term in order parameter in the Landau free energy. The ground state solution of the linearized time-independent Ginzburg-Landau equation has been derived, through which the spatial variation of the order parameter has been delineated. Moreover, a generic phase diagram indicating a tricritical behavior near and away from the dislocation is depicted. The relation between the classical nucleation theory and the Ginzburg-Landau approach has been discussed, for which the critical formation energy of nucleus is related to the maximal of the Landau potential energy. A numerical example illustrating the application of the model to the case of nucleation of hydrides in zirconium alloys is provided.

Metal oxides added to UO2 to improve material performance during irradiation, or as burnable absorbers to control reactor energy output, affect point defect processes in UO2. The oxygen and uranium Frenkel pairs and the uranium-oxygen Schottky defects regulate the O/U ratio, which in turn influence diffusion processes in UO2. The dopants considered here include Cr2O3 and Gd2O3. Using the law of mass action to the Frenkel and Schottky defects in doped UO2, we relate O/U to the dopant concentration. Also, we find relationships between the oxygen and uranium vacancies and dopant concentration. The uranium self-diffusion coefficient is proportional to concentration of uranium vacancies in the hyper-stoichiometric region, whereas to that of uranium interstitials in hypo-stoichiometric region. We relate this to the creep rate and diffusion coefficient of fission gases in UO2. We use the model to evaluate creep rate and gas diffusivity in doped UO2 in light of experimental data.

The creep of UO2 doped with Nb2O5 and Cr2O3 has been assessed using a point defect model based on the law of mass action, and the diffusional creep according to the Nabarro-Herring mechanism, which relates the creep rate to the lattice self-diffusivity, the inverse of grain area and the applied stress. The self-diffusion coefficients of cation (U) and anion (O) are directly proportional to the concentrations of ions, which in turn are functions of dopant concentrations. The model has been used to evaluate past creep experiments on UO2 doped with Nb2O5 and Cr2O3 in concentrations up to about 1 mol%, with a varying grain size at different temperatures and applied stresses. The creep rate increases significantly with the dopant concentration and the putative model, after a modification of the creep rate coefficient, retrodict the measured data satisfactorily. A number of factors affecting creep rate and thereby our model computations are discussed.

Vi söker i rapporten systematisera data avseende zirkoniumbaserade bränslekapslingsrörs brottbeteende under haverifall med kylmedelsförlust (LOCA), som rapporterats från ex- perimentella studier sedan slutet av 1970-talet. Vårt mål är att fastställa användbara data och utvärdera dessa gentemot de brottkriterier som är tillgängliga i QT/SSM:s version av beräkningsprogrammet FRAPTRAN. Detta program beräknar transientbeteendet hos kärn- bränslestavar i lättvattenreaktorer under reaktortransienter och hypotetiska olyckor, såsom LOCA. Databasen omfattar kapslingsmaterialen Zircaloy-4, ZIRLO och legeringar med sammansättningen Zr-1wt%Nb. Rapporten sammanfattar databasen, beräkningsmetodiken och uttrycken för de olika brottkriterierna, samt presenterar resultaten av vår utvärdering genom att jämföra beräkningsresultat med mätdata i diagram över tid till kapslingsbrott, brottemperatur, och kapslingens brottspänning och brottöjning. Dessutom ges en kort över- sikt av osäkerheterna i beräkningarna. Vi har funnit att Rosingers spänningsbaserade brottkriterium, vilket ursprungligen utvecklades för “best-estimate”-prediktering av kapslings- brott i Zircaloy-4, är tillämpbart för såväl Zircaloy-4 som ZIRLO-kapsling, om en bästa skattning av kapslingsbrott erfordras. Vad gäller ZIRLO-kapsling, kan nämnda brottkriterium förbättras ytterligare, under förutsättning att en tillräcklig mängd mätdata avseende brott- och materialegenskaper är tillgänglig.

The creep of UO2 doped with Nb2O5 or Cr2O3 has been assessed using a point defect model based on the law of mass action, and the diffusional creep according to the Nabarro-Herring mechanism, which relates the creep rate to the lattice self-diffusivity, the inverse of grain area and the applied stress. The self-diffusion coefficients of cation (U) and anion (O) are directly proportional to the ion concentrations, which in turn are functions of dopant concentrations. The model has been used to evaluate past creep experiments on doped UO2 that were made as a function of dopant concentration (up to about 1 mol%) with a varying grain size at different temperatures and applied stresses. The creep rate increases significantly with the dopant concentration and the model, after a modification of the creep rate coefficient, retrodict the measured data satisfactorily.

Stationära krypdata för obestrålad Zircaloy-4 och Zr1%Nb, som används i kapslingsrör till kärnbränslestavar i lättvattenreaktorer, rapporterade i litteraturen, har prövats på nytt. De beaktade kryptesterna utfördes i temperaturintervallet från 923 till 1873 K i inert miljö täckande samtliga stabila fasområden hos zirkoniumlegeringar: α, (α + β) och β som funktion av pålagd spänning. Resultat från modellberäkningar och mätdata jämförs, varvid den relativa skillnaden mellan uppmätta och beräknade värden på kryphastighet kvantifieras för en serie av tester. Modeller som beaktas är sådana som används eller skulle kunna användas i datorprogram för uppskatting av bränslestavbeteende under förhållanden rådande vid en olycka orsakad av kylmedelsförlust i lättvattenreaktorer.

We treat the problem of diffusion of solute atoms around screw dislocations. In particular, we express and solve the diffusion equation, in radial symmetry, in an elastic field of a screw dislocation subject to the flux conservation boundary condition at the interface of a new phase. We consider an incoherent second-phase precipitate growing under the action of the stress field of a screw dislocation. The second-phase growth rate as a function of the supersaturation and a strain energy parameter is evaluated in spatial dimensions d=2 and d=3. Our calculations show that an increase in the amplitude of dislocation force, e.g. the magnitude of the Burgers vector, enhances the second-phase growth in an alloy. Moreover, a relationship linking the supersaturation to the precipitate size in the presence of the elastic field of dislocation is calculated.

The thermoelectric properties of pure and doped UO2, namely the thermal and electrical conductivities and the thermopower, are assessed. We adopt the small polaron theory of the Mott type insulators, wherein the charge carriers, the electron and hole on the U3+ and U5+ ions, are treated as small polarons. For the thermal conductivity, the small polaron theory is applicable at temperatures above 1500 K. A review of the experimental data on the temperature dependence of the aforementioned transport properties is made. The data include UO2 with dopants such as Cr2O3, Gd2O3, Y2O3 and Nb2O5. We compare the applications of the theory with the data. Two limiting regimes, adiabatic and nonadiabatic, with the ensuing expressions for the conductivities and the thermoelectric power are considered. We discuss both the merits and shortcomings of the putative small polaron model and the simplification thereof as applied to pure and doped uranium dioxide. (C) 2017 Elsevier B.V. All rights reserved.

A generic model for nucleation of oriented second-phase in alloys in the vicinity of cracks and dislocations, is considered. The model employs the Ginsburg-Landau approach, which accounts for the elasticity of crystalline solid and the interaction of structure/composition with the elastic field in the vicinity of the defect and in the crystalline bulk. We examine the nature of the structural phase transition and construct its phase diagram.

En mängd olika experiment har utförts tidigare för att kvantifiera och förstå bränslestavars beteende under LOCA (olycka orsakad av kylmedelsförlust) förhållanden i lättvattenreaktorer (LWR). Det experimentellt framtagna underlaget utgör basen för nuvarande acceptanskriterier för härdnödkylsystemet under LOCA förhållanden i LWR. Resultat från experiment under senare tid indikerar att kapslingsrörets legeringssammansättning och förändringar till följd av hög utbränning (högutbränningseffekter) påverkar marginalerna till acceptanskriterierna för LOCA. I denna rapport granskas några äldre viktiga och nyare experimentella resultat. Först diskuteras bakgrunden till acceptanskriterierna för LOCA, nämligen, fenomen som orsakar kapslingsförsprödning, kriterier för kapslingsförsprödning (begränsningar på maximal kapslingsoxidering och maximal kapslingstemperatur) och den experimentella basen för kriterierna. Två typer av omfattande tester har utförts under LOCA förhållanden: (i) Separateffekt tester för undersökning av kapslingsoxidation, kapslingsdeformation och brott, samt zirkoniumlegeringars allotropiska fasövergång under LOCA. (ii) Integrala LOCA tester, i vilka hela LOCA förloppet simuleras (i laboratorium eller i testreaktor) med en enskild stav eller arrangemang med flera stavar i ett bränsleknippe, för att studera det övergripande bränslestavbeteendet under LOCA. Separateffekt tester och resultat diskuteras, samt empiriska korrelationer härledda från dessa tester och kvantitativa modeller visas. Speciellt diskuteras inverkan av niob i zirkonium-bas kapsling och kapslingens väteinnehåll på dess allotropiska fasomvandling under LOCA samt även kapslingens brottspänning. Vidare granskas resultat från några nyare integrala LOCA tester med betoning på termiska chocktester. Avslutningsvis ges förslag på modellering och utvärdering av vissa experimentella resultat.

The overall solid-to-solid phase transformation kinetics under non-isothermal conditions in Zr alloys has been evaluated using a model presented on our preceding note. It uses as input an applied thermal history and calculates the time/temperature variation of the volume fraction of the new phase in α ↔ β transition in Zr alloys under heating/cooling, in concordance with experiments.