CASL Structural Mechanics Modeling of Grid-to-Rod Fretting(GTRF)

WEI LU,1,4 M.D. THOULESS,1,2 ZUPAN HU,1 HAI WANG,1

RAMIN GHELICHI,3 CHEN-HUNG WU,3 KEN KAMRIN,3

and DAVID PARKS3

1.—Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA.2.—Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109,USA. 3.—Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,MA 02139, USA. 4.—e-mail: weilu@umich.edu

Fluid-induced grid-to-rod fretting (GTRF) wear is responsible for over 70% offuel leaks in pressurized water reactors (PWRs) in the US. The Consortium forAdvanced Simulation of Light Water Reactors (CASL) has identified GTRF asa challenge problem that is very important to nuclear plants. GTRF is acomplex problem that involves multiple physical phenomena. This papersummarizes several GTRF-related problems being addressed by the Univer-sity of Michigan and the Massachusetts Institute of Technology, CASL part-ners. These include analyses of cladding creep, wear, structural mechanics,and the effects of the rod-to-grid gap. Also outlined are additional aspects ofmaterial science and computational modeling that will be needed to realize theultimate goal of high-fidelity predictive modeling and design tools to addressGTRF.

INTRODUCTION

Grid-to-rod fretting (GTRF) wear is responsiblefor over 70% of the fuel leaks in pressurized-waterreactors (PWRs) in the US.1 One of the main causesof failure has been identified as the relaxation of thegrid-to-rod support in the assembly and flow-in-duced vibration.

The occurrence and progression of GTRF isclosely related to the structure of the fuel assembly.In a fuel assembly, the fuel rods are supported bythe springs and dimples of the spacer grid (Fig. 1).The initial preload between the grid and the fuel rodcan prevent gross sliding between the two compo-nents in response to the turbulent flow of thecoolant. However, this flow can cause partial slipat the edges of each contact, resulting in localfretting wear. Furthermore, the initial preload willrelax over time because of deformation processesthat take place as the reactor operates. As a resultof this, the contact force between the grid and thefuel rod can relax to such an extent that grosssliding occurs between them, which enhances thefretting wear. The wear scar that forms during thisprocess further accelerates the relaxation of the

contact force. Eventually, the grid can no longermaintain continuous contact with the fuel rod, anda gap will open up between them. This gap caninduce larger amplitudes of vibration and dynamicimpact between the contacts, and some havesuggested this may result in a significant increasein the wear.2,3

Reliable simulations of GTRF require integrationof quasi-static (in the early stage of wear) anddynamic (after a gap opens) structural analyses,turbulent-flow calculations, the modeling of mate-rial behavior at high temperatures and in anirradiation environment, and the simulation of localcontact and wear. Here, we review recent advancesin modeling techniques for GTRF-related problems,with a summary of some results of completed andin-progress research related to creep, wear, turbu-lent flow, structural analysis and the coupling ofwear and creep.

CREEP

To model creep correctly one needs to be able tocapture the changes in the dominant mechanismthat occur with changes in both temperature and

JOM, Vol. 68, No. 11, 2016

DOI: 10.1007/s11837-016-2095-7� 2016 The Minerals, Metals & Materials Society

2922 (Published online September 7, 2016)

stress state. This can be done within a finite-element calculation by incorporating mechanism-based models that include the appropriate activa-tion energies, stress dependencies, and other mate-rial parameters. There is no need to make any apriori guesses about what creep law to include.Instead, the dominant law evolves naturally, andcan change appropriately if the conditions change.

To develop such a model for zircaloy-45 thepublished thermal creep data for the alloy over thelast 35 years were divided into groups of resultsfrom similar experimental conditions that each hadremarkable internal consistency. These groups were

used to identify the creep mechanisms, based on theforms of the relationships between stress, temper-ature and strain rate, from which the appropriateactivation energies and other associated creepparameters were deduced. Figure 2 shows theresultant deformation-mechanism map constructedfor unirradiated zircaloy-4. It portrays how differentcreep mechanisms evolve and dominate under var-ious stress and temperature conditions. By incorpo-rating the concepts of such a creep-mechanism mapinto an FEM code, the controlling creep rateemerges naturally during any numerical analysis.

This framework and the implemented finite-ele-ment program provide an important tool for assess-ing the effects of stress and temperature in the earlydesign stage, especially when multiple mechanismregimes are involved at the same time, or when thedominant mechanism changes over time. We haveused it to simulate the creep-induced contact-forcerelaxation and structural deformation in the fuelassembly. Because the current deformation-mecha-nism map is for unirradiated zircaloy-4, additionalmodels need to be developed to link the effects ofradiation to creep, including irradiation growth.

WEAR

Archard’s relationship6 is popularly used todescribe the wear between two surfaces. The weardepth, w(s, t), at any location along a contact, s, andtime, t, can be calculated by integrating the gov-erning equation:

@w

@Dðs; tÞ ¼ lKpðs; tÞ ð1Þ

where K is the wear coefficient, p(s, t) is the localcontact pressure, D(s, t) is the local relative slipbetween the two surfaces at the contact, and l is thefriction coefficient. The local wear rate, W ¼ @w=@t,depends on the contact pressure, and even relativelysmall amounts of wear can change the profiles of thecontacting bodies sufficiently to have a significantinfluence on the distribution of contact pressure.

The contact and wear problems are coupled,requiring an incremental algorithm for finite-ele-ment simulations. At each time step, the contactpressure is calculated based on the instantaneousmorphology of the contact surfaces, and the localwear rate is obtained from the wear model. Themorphology of the contact surface is then updatedby adjusting the position of the surface nodes tomatch the wear. The process is then repeated forsuccessive cycles. A well-known challenge withwear calculations is the need for constant remesh-ing to track the wear profile: it is not only compu-tationally expensive but can also lead to numericalinstability and non-convergence. We have developedan approach in which the morphology update isachieved by imposing fictitious eigenstrains in theelements at the contact surface. The magnitudes ofthese eigenstrains are calculated to produce the

Fig. 2. A deformation-mechanism map for unirradiated zircaloy-4with a grain size of 150 lm. Reprinted from Wang et al.,5 with per-mission from Elsevier.

Fig. 1. A cell of the fuel assembly showing a fuel rod and the spacergrid. This model is based on a standard design for the 17 9 17Vantage5H fuel.4

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change of surface morphology consistent with thatcaused by wear.7 This method avoids severe numer-ical problems associated with remeshing, particu-larly if there are residual stresses induced by theslip process, and enables highly efficient modeling ofwear evolution in GTRF that can be used withessentially all FEM packages.

STRUCTURAL MECHANICS MODELING

Multi-rod Grid Coupling

Traditional approaches8 modeling turbulence-dri-ven rod vibration adopt statistical characterizationsof fluctuating fluid-pressure distributions on therods, based on random vibration theory.9 Each rod,modeled by elastic-beam theory, is coupled togrounded springs representing discrete points ofmechanical interaction (contact, friction, sliding)with spacer-grid springs and dimples. Dynamichistories of point-contact force and relative slidingat each spring and dimple site are generated andanalyzed for sliding wear work-rate. These historiescan also be used to drive more detailed continuummodeling of the wear processes. Detailed continuumsimulations incorporating structural creep andwear-scar evolution can be used to update grid/rodcontact forces and/or gap openings, enabling simu-lations of continued turbulence-driven rodvibration.

We adopt certain features used in the VITRANrod-vibration modeling software,10 including theidealization of turbulent loading using a constant-amplitude, band-limited power spectral density(PSD) at discrete transverse point loads down-stream of each spacer grid. The PSD magnitudescales that of observables such as mid-span velocityand acceleration, guiding calibration. An FFT algo-rithm spans the assumed frequency window, gen-erating a set of temporal sine waves of randomphase shift, constraining half of them to produce areal signal. Using explicit integration of projectedmodal dynamics, and accounting for solution-de-pendent grid/rod contact interactions, rod vibrationis simulated over a sufficiently large time interval.Multiple realizations of random phase shifts provideconverged estimates of the mean and the standarddeviation of frictional work (rate) at each contact.

The support stiffnesses of the springs and dimplescomprising a given grid cell facet’s interaction withits adjacent fuel rod(s) are intrinsically coupled,owing to local bending flexibility of the orthogonalgrid strips. Figure 3a shows a representative gridstrip and illustrates the support coupling of twoadjacent rods with the two dimples and one springcomprising a single panel of the grid cell. Non-negative contact forces FI at the spring and dimplesof a panel segment induce corresponding normaldeflections dJ. A 3 9 3 elastic stiffness matrix kIJconnects these quantities according to FI ¼ kIJdJ.

Vibration simulations incorporating 2-sided cou-pled contact stiffness within structural mechanicsmodels of multi-rod/multi-grid assemblies demon-strate spatial patterning of peak sliding wear work-rate. Figure 3b shows the spatial patterning ofcomputed wear work-rate in a model 5 9 5 assem-bly. Each rod was supported by 7 spacer grids andsubjected to identical initial support conditions ofzero gap and zero normal force, along with span-by-span turbulent loading of constant PSD. The peakmean wear work-rate at any contact point alongeach rod was normalized by that for a single rodidentically supported by coupled grid panels com-prising each spacer grid cell. No rod in the assemblyhad a peak wear work-rate exceeding that of thesingle-rod model, suggesting that the latter providesa conservative estimate. The peak wear work for therod in the corner of the assembly that was supportedexternally by (stiff) dimples and internally by(compliant) springs matched that of the single rod,while that of the rod in the diagonally opposedassembly corner, supported externally by springsand internally by dimples, was only half as large,with intermediate values along external assemblyedges having a dimple support. The reduced wearwork-rates at interior rod locations are due tochanges in average contact force because adjacent-cell rod contact elastically deflects contact points onshared grid cell panels. The spatial wearing pat-terns in this simple model bear similarities topatterns of leaking rods within a representativeassembly, which are often concentrated alongassembly external corners and edges.11

Multi-rod Fluid–Structure Coupling

Typically, flow-induced vibrations in GTRF aremodeled by studying the force spectrum on anisolated rigid rod interacting with an inflow at highReynolds number. An important consideration iswhether flow through a rod assembly causes geo-metric coupling with the fluid phase, resulting incertain rods receiving different forcing signaturesthan others.

This question can be addressed with a fluid–structure interaction study. The simulation methodused is the Reference Map Technique,12,13 an Eule-rian numerical framework that permits the simula-tion of a dynamic fluid phase interacting withpossibly nonlinear continuum solid phases on onenumerical grid.

Five parallel rods are simulated, idealized as 2-Dsolid elastic bodies anchored firmly at their endsand weakly (109 smaller) at three internal points tomodel grid supports. Anchoring is modeled withspring forces spread over the pictured interactionarea (Fig. 3c). An incoming transverse fluid flow isintroduced. To bring out possible coupling of rodmotions, the rods are given an artificially high

Lu, Thouless, Hu, Wang, Ghelichi, Wu, Kamrin, and Parks2924

elastic compliance. Non-laminar fluid motion gen-erates flow-induced vibrations of the rods (Reynoldsnumber = 12,000; see Fig. 3e).

A fluctuating force of the grids on the rods isapparent, consistent with turbulent flow inducedvibration of the rods. Interestingly, there is also a

Fig. 3. (a) Schematic illustration of rod contacts at the 2 dimples and 1 opposing spring comprising a panel of a spacer grid cell. Contact force atany site on the panel elastically deflects all grid contact points on the panel segment. (b) Summary of normalized mean peak wear work-rate, inper cent, for a model 5 9 5 rod assembly. Normalization is with respect to the corresponding mean peak wear work in a single-rod model,supported at each spacer grid by four stiffness-coupled panels. (c)–(e) include fluid. (c) Simulation setup of five rods anchored at their endssubject to a rightward incoming uniform flow. Weak supports provided at three interior locations. (d) Transverse forces at the interior supports oneach rod. (e) Snapshot of relative pressure and vorticity (in SI) during simulation.

CASL Structural Mechanics Modeling of Grid-to-Rod Fretting (GTRF) 2925

non-negligible mean normal force, which vanisheson the central rod and pushes outward on theextreme rods (Fig. 3d). A mean force is rarely if everconsidered in typical FIV studies used as inputs toGTRF. As previously noted, rods at the extremallocations of an assembly tend to be more susceptibleto wear.11 Likewise, fluid coupling of the rod systemappears to provide another explanation for thiseffect, on top of the effect of grid-wise couplingdescribed previously.

EFFECTS OF GAP SIZE AND EXCITATIONFREQUENCY ON GTRF WEAR

The wear rate is sensitive to the gap size betweenthe rod and grid. A close fit between them results ina larger contact pressure and frictional force, lead-ing to a higher wear rate if sliding happens. A tightfit can make it difficult for sliding to happen, leadingto a lower wear rate. Conversely, a loose fit mayresult in dynamic impacts during oscillation, lead-ing to a higher wear rate. However, if the gap size istoo large, the amplitude of oscillation may beinsufficient to cause regular contact, leading to avery low wear rate. Obviously, this would not bedesirable, despite the low wear rate, because thegrid would not sufficiently support the fuel rod.

The wear rate also depends on the excitationfrequency from the coolant flow. The frequency andamplitude of the responding vibration of the rod areintercorrelated. As we show here, even if oneassumes a periodic excitation frequency from theflow, the rod responds in a complicated fashion,

exhibiting subharmonic, period-doubling and chao-tic modes of vibration, depending on the excitationfrequency and gap size.

We have developed 3-D finite-element modeling toinvestigate (1) the dynamic response of a fuel rodunder the driving forces of various frequenciesinduced by the coolant, (2) the effects of the gapsize and initial interference on the wear rate of therod, and (3) the effect of the excitation frequency onthe wear rate. Figure 4 shows the dependence of thewear rate on the gap size and excitation fre-quency.14 There is a critical size of gap thatcorresponds to a subharmonic response, and whichcauses a maximum rate of wear for a given excita-tion frequency. The gap size that results in amaximum wear rate for a given excitation frequencycorresponds to the gap size that results in theexcitation frequency matching the natural fre-quency of the system, where the natural frequencyis defined as being the frequency of the first mode offree vibration of the system consisting of the rod andsupport plates. This finding suggests that themaximum wear rate is directly related to thenatural resonance of the system. When the gap islarger than the critical size for a given excitationfrequency, the response of the rod becomes chaotic,and the wear rate reduces greatly.

COUPLING WEAR AND CREEP

The characteristic time scales of creep and fret-ting wear are quite different. A single cycle of wearcan occur within a second. In contrast, significantcreep can take place at the scale of days or months.Using the small time increments suitable for mod-eling each wear cycle would be too computationallyexpensive and impractical. On the other hand, usingthe relatively large time increments suitable forcreep may miss significant changes in contact stressinduced by wear.

To resolve this large difference in time scales, wehave developed an ‘‘effective-cycle’’ approach. Up tosome limit, several successive vibration cycles canbe combined into a single effective cycle with alarger period without loss of accuracy. In calculatingan optimal effective period, it is noted that bothwear and creep cause stress redistribution, and bothaffect each other because of this stress change. Theeffective-cycle technique takes care of this interac-tion systematically. For instance, creep causesstress relaxation within each effective cycle, andthis change in stress inherently interacts with thewear calculation, since both wear and creep arecalculated simultaneously at each time step withinthe effective cycle. The period of the effective cycle isadjusted dynamically based on the current rate ofstress relaxation from both creep and wear inaccordance with any required accuracy. This auto-matic adjustment is achieved by continuously mon-itoring the change of contact pressure.

Fig. 4. A wear-rate map for a large spectrum of gap sizes andexcitation frequencies. The excitation frequency is f. The first naturalfrequency for the vibration of a free rod is fn. The inset shows the 3-Dmodel: a fuel rod with a cladding thickness of Dc, four support plateseach connected to springs with a stiffness of ks, and the gap betweenthe rod and the plate is g. The amplitude of force exerted on the rodby the turbulent flow is Fa. The critical gap size, which is associatedwith the maximum wear rate, lies within the subharmonic regime. Inthe no-wear region, the amplitude of the rod vibration is smaller thanthe gap size so that no impact between the rod and plate can hap-pen. The curve of the natural frequency of the system appears tooverlap with the peaks in the contour.14

Lu, Thouless, Hu, Wang, Ghelichi, Wu, Kamrin, and Parks2926

The effective-cycle approach, the fictitious eigen-strain approach for wear, and the mechanism-basedcreep model have been coupled to evaluate gap

formation in GTRF. These three tools were used tomodel gap formation in a 2-D model of a unitthickness in the axial direction of the rod. Figure 5

Fig. 5. The contact force, or integrated normal pressure over the contact surface, relaxes over time. The amplitude of force per rod lengthexerted on the rod by the turbulent flow is Fa with a period of 0.1 s. Two stages exist: partial slip and full slip. When partial slip occurs, creepdominates the stress relaxation. When full slip occurs, wear becomes the dominant stress relaxation mechanism. The wear scar shows differentgeometry in the partial slip and full slip stages.15

CASL Structural Mechanics Modeling of Grid-to-Rod Fretting (GTRF) 2927

shows the contact force relaxation process.15 Forcomparison, stress relaxation if only creep is active,and no wear, is shown on the figure. The comparisonshows that when partial slip occurs, creep domi-nates the stress relaxation. When full slip occurs,wear becomes the dominant stress relaxation mech-anism. The result not only gives a prediction of theservice time before loss of contact but also showsthat the process comprises two stages: partial slipwhere wear happens at the edge of the contactsurface, and full slip where wear happens at theentire surface. Though the amount of wear issmaller in the partial slip stage, this stage occupiesmost of the time and therefore partial slip isimportant in GTRF. The wear scar shows differentgeometries in the two stages.

At present we have not coupled the chaotic effectsof turbulent flow, and the excitation force wasmodeled as a sinusoidal wave. In future studies,the excitation force will be replaced by the fullspectrum obtained from computational fluiddynamics.

INFLUENCE OF FLUID IN OPENGRID-ROD GAPS

One potential factor yet to be discussed is the rolethe coolant fluid plays in the wearing process. In earlystages when gaps remain closed or separated onlymarginally, there is evidence the presence of station-ary fluid does not significantly change rates of wear;16

however, when fluid is flowing over the contactregion, wear rates can increase notably due toparticles being carried off by the flow. As gaps openfurther, the effect of the fluid can become moresevere, as separated surfaces produce a squeeze-filmof fluid that assists in removing third-body particlesand builds up hydrodynamic stresses as contactsform. It has been shown that wear rates increasedirectly with gap size beyond a critical gap opening influid submerged tests.17 Among various wearingconditions, it was shown18 that grid–rod interactionsin a fluid with separation sufficient to permit impactsproduce an effective wear coefficient �100 timeslarger than that of dry rods in close sliding conditions.

The effect of fluid can be understood considering asimplified version of the system. Figure 6 shows amodel system of a submerged roughened rod mak-ing contact with a roughened dimple surface,bounded by a bath pressure.

In the limit of a roughness-free surface, as the gapcloses fluid strain-rates grow to hypotheticallyunbounded values. This can aid in particle removalprocesses discussed previously, and, notably, causea hydrodynamic stress blow-up. In the rough case,these stresses can assist wearing and lubricationanalysis19 reveals peak pressure in the center of theroughened gap region, exceeding the bath pressureby �6gVL2(W2 + h2)/W5 and the fluid shear stressalong the surfaces grows from the center out,reaching a peak value of �1.5gVL/(W/2 � h)2,where g is the viscosity. The substantial rate thesestresses grow as gaps close (i.e. W/2 approaches h)suggests fluid stress on surfaces can contribute inthe surface removal process.

To study this effect more completely, a two-waysubmodeling capability is being built into thefluid-solid interaction method from the ‘‘StructuralMechanics Modeling’’ section to permit coupledsimulation of the rod-scale turbulent fluid motion,and the contact-scale lubrication behavior. Theroutine models the system with two separatenumerical domains: a fine-grid region encompass-ing the fluid–solid contact zones, and a coarse-griddomain for the remainder of the problem. The twodomains are coupled through their mutual bound-ary using interpolation. The key benefit is thesmaller time-step of the refined domain does notlimit the time-step of the larger, coarse domain.Initial verifications of the submodeling approachwere applied to a nonuniformly deforming solidwith a localized stress-concentration region. Whenthe stress-concentrated zone was refined two timesusing submodeling, the simulated motion on thewhole gives the same L2 accuracy as a simulationwith a uniform refined grid, but the submodeledsimulation completes 5 times faster. This ability tomulti-scale the numerics will be key to simulatingthe widely separated scales of fluid-solid interac-tion occurring in GTRF within a reasonable time.

Fig. 6. (a) Not-to-scale illustration of roughened rod and dimple surfaces making contact in a fluid. (b) Simplified lubrication model.

Lu, Thouless, Hu, Wang, Ghelichi, Wu, Kamrin, and Parks2928

CONCLUSION

The wear caused by GTRF is a complex andchallenging problem, which is significant to theoperation of PWRs. Solving the GTRF-relatedproblems requires integrating a number of differ-ent areas, including structural mechanics, defor-mation mechanisms under high-temperature andirradiation conditions such as creep and irradia-tion growth, fretting wear, computational fluiddynamics, and vibration. Some of the integration,such as of vibration, creep and wear, has beenperformed. More work needs to be done tointegrate other areas, such as turbulent excitationforces from fluid dynamic simulations, and pro-cesses such as corrosion on the surface of the fuelrod, which can alter the friction coefficient andwear coefficient.20 Experimental validation is anon-going challenge for GTRF models. An idealGTRF wear tester would operate under conditionsthat simulate the fuel-rod contact stresses,motions, coolant flow, frequency, amplitude, tem-perature, and state of irradiation. CASL is cur-rently designing a unique GTRF simulative wearbench tester: autoclave fretting-impact wear(AFIW) tester. This new equipment is designedto handle actual cladding and grid samples inheated pressurized water at temperatures up to220�C and pressures up to 2.4 MPa. Once thetester is available, systematic wear tests will beconducted at CASL at the modes of fretting only,impact only, and fretting plus impact. The creepand wear calculation developed at the Universityof Michigan, and the power spectrum analysis ofthe turbulent excitation forces developed at theMassachusetts Institute of Technology will beintegrated. This paper highlights some completedand on-going research related to the GTRF prob-lem, which we hope to provide an understandingof the current status and to stimulate the interestto continue exploring GRTF-related problems.

ACKNOWLEDGEMENTS

This research was supported by the Consortiumfor Advanced Simulation of Light Water Reactors(http://www.casl.gov), an Energy Innovation Hub(http://www.energy.gov/hubs) for Modeling andSimulation of Nuclear Reactors under US Depart-ment of Energy Contract No. DE-AC05-00OR22725.

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