Wind-Turbine Gear-Box Roller-Bearing Premature-Failure ......Keywords gear-box premature failure,...

Wind-Turbine Gear-Box Roller-Bearing Premature-FailureCaused by Grain-Boundary Hydrogen Embrittlement:

A Multi-physics Computational InvestigationM. Grujicic, V. Chenna, R. Galgalikar, J.S. Snipes, S. Ramaswami, and R. Yavari

(Submitted April 16, 2014; in revised form July 14, 2014; published online August 9, 2014)

To help overcome the problem of horizontal-axis wind-turbine (HAWT) gear-box roller-bearing premature-failure, the root causes of this failure are currently being investigated using mainly laboratory and field-testexperimental approaches. In the present work, an attempt is made to develop complementary computa-tional methods and tools which can provide additional insight into the problem at hand (and do so with asubstantially shorter turn-around time). Toward that end, a multi-physics computational framework hasbeen developed which combines: (a) quantum-mechanical calculations of the grain-boundary hydrogen-embrittlement phenomenon and hydrogen bulk/grain-boundary diffusion (the two phenomena currentlybelieved to be the main contributors to the roller-bearing premature-failure); (b) atomic-scale kinetic MonteCarlo-based calculations of the hydrogen-induced embrittling effect ahead of the advancing crack-tip; and(c) a finite-element analysis of the damage progression in, and the final failure of a prototypical HAWTgear-box roller-bearing inner raceway. Within this approach, the key quantities which must be calculatedusing each computational methodology are identified, as well as the quantities which must be exchangedbetween different computational analyses. The work demonstrates that the application of the present multi-physics computational framework enables prediction of the expected life of the most failure-prone HAWTgear-box bearing elements.

Keywords gear-box premature failure, horizontal-axis wind-turbine, multi-physics modeling

1. Introduction

Wind-turbine gear-box failure remains one of the majorproblems to the wind-energy industry (Ref 1-3). The combi-nation of these high failure rates, long down times, and the highcost of gear-box repair has contributed to: (a) increased cost ofwind energy; (b) increased sales price of wind-turbines due tohigher warranty premiums; and (c) a higher cost of ownershipdue to the need for funds to cover repair after warrantyexpiration. To bring the wind-energy cost down, durability andreliability of gear-boxes have to be substantially improved.These goals are currently being pursued using mainly labora-tory and field-test experimental approaches. While theseempirical approaches are valuable in identifying shortcomingsin the current design of the gear-boxes and the main phenomenaand processes responsible for the premature failure of wind-turbine gear-boxes, advanced computational engineering meth-ods, and tools can not only complement these approaches butalso provide additional insight into the problem at hand (and do

so in a relatively short time). The present work demonstratesthe use of these methods and tools, and discusses the benefitsoffered by them within the context of horizontal-axis wind-turbine (HAWT) gear-box roller-bearing premature-failure.

A photograph of an offshore wind turbine is provided inFig. 1. All major components of the turbine are labeled foridentification. Examination of this figure shows that the HAWTrotor (typically consisting of three blades) attached to ahorizontal hub (which is connected to the rotor of the electricalgenerator, via a gear-box/drive-train system, housed within thenacelle). The rotor/hub/nacelle assembly is placed on a tower,which is in turn anchored to the ocean floor (or ground, in thecase of onshore HAWTs).

Turbine blades and the gear-box (or, in general, the wind-turbine drive-train) are perhaps the most critical components/subsystems in the present designs of wind turbines. The presentwork deals only with the issues related to the performance,reliability, and modes of failure of gear-box components. In ourrecent work (Ref 4, 5), the problem of inadequate durability andpremature failure of wind-turbine blades was investigatedcomputationally.

The main function of the gear-box is to convert the lowspeed of the rotor shaft to the high speed of the electricalgenerator shaft. A labeled schematic of a prototypical windturbine gear-box is shown in Fig. 2. The low-speed stage of thegear-box is a planetary configuration with either spur (thepresent case, Fig. 2) or helical gears. In this configuration,the planetary-gear carrier is driven by the wind-turbine rotor,the ring gear is stationary/reactionary, while the sun pinionshaft drives the intermediate gear-box stage, and, in turn, thehigh-speed stage (the one connected to the rotor of the electricgenerator). Predominantly, the gear-box failure initiation is

M. Grujicic, V. Chenna, R. Galgalikar, J.S. Snipes, S. Ramaswamiand R. Yavari, Department of Mechanical Engineering, 241Engineering Innovation Building, Clemson University, Clemson,SC 29634-0921. Contact e-mail: [email protected].

JMEPEG (2014) 23:3984–4001 �ASM InternationalDOI: 10.1007/s11665-014-1188-0 1059-9495/$19.00

3984—Volume 23(11) November 2014 Journal of Materials Engineering and Performance

observed in planet bearings, intermediate-shaft bearings, andhigh-speed-shaft bearings (each labeled in Fig. 2).

In principle, the predicted service-life of properly main-tained roller-bearings operating under elasto-hydrodynamiclubrication conditions, and foreseen loading/environmentalconditions, is well-predicted using the well-established bear-ing-reliability assessment procedures (Ref 6-8). The basicassumptions used in these procedures are that the bearing-life iscontrolled by the material high-cycle fatigue (typically withinthe bearing races/rings), commonly referred to as the roller-bearing contact fatigue (RCF) failure. The in-service cyclingstresses arise from the repeated exposure of the ring material toring/roller-element non-conformal contact stresses. Under well-lubricated/clean-lubricant conditions, RCF is typically initiatedby the nucleation of subsurface cracks (in regions associated

with critical combinations of the largest shear stress and thepresence of high-potency microstructural defects). Duringsubsequent repeated loading, cracks tend to advance towardthe inner surfaces of the raceways, giving rise to the formationof spalls/fragments.

As mentioned earlier, roller-bearings in wind-turbine gear-boxes tend to fail much earlier relative to their expected servicelife. In addition, the damage mechanism responsible for, andthe appearance of roller-bearing prototypical premature-failureseems different from the classic RCF failure. A detaileddescription of the visual appearance of the standard RCF failureis given in ISO 15243 (Ref 7), and hence will not be repeatedhere. Instead, a simple schematic of the sub-surface regioncontaining RCF dark and white bands as well as a singlechevron-shape crack is depicted in Fig. 3(a). In the case of thepremature roller-bearing failure, the damaged region acquires acharacteristic ‘‘White Etching Crack’’ appearance, and isinitially localized at the contact surfaces or slightly beneaththem. This appearance is related to the weak etching responseof the associated local microstructure (mainly consisting ofequiaxed sub-micron-size carbide-free ferrite grains) during thestandard preparation of metallographic samples. In addition tothe chevron-shaped cracks like the one shown in Fig. 3(a), so-called butterfly white-etching cracks are also often observed inthe case of RCF failure. These cracks are formed at greaterdepths and are normally associated with excessive loading. Insharp contrast, white-etch cracking in the case of prematureroller-bearing failure, Fig. 3(b), is believed to be a surface ornear-surface phenomenon (Ref 9). Specifically, it is believedthat a combination of disturbed bearing kinematics, unfavorableloading, and inadequate lubrication can lead to local surface ornear-surface tensile-stress concentrations, at the root of surfaceasperities and/or at the near-surface inclusion/matrix interfaces.If these stress concentrations and the number of loading cyclesare sufficiently high, surface and/or subsurface cracks cannucleate. Due to proximity of the contact surfaces, subsurface

Fig. 2 Schematic of a prototypical wind-turbine gear-box. The major components and sub-systems are identified. Failure typically occurswithin the (planet, intermediate shaft, and high-speed shaft) roller-bearings

Fig. 1 A photograph of a typical off-shore wind farm, with themajor wind turbine sub-systems identified

Journal of Materials Engineering and Performance Volume 23(11) November 2014—3985

cracks can readily extend to these surfaces (becoming surfacecracks).

Cracks nucleated at the contact surfaces are subsequentlyinfiltrated by the lubricant. The lubricant contains various (e.g.,extreme pressure, EP, and anti-wear, AW) additives and may becontaminated with water. Passage of the rolling elements overthe damaged area can result in hydrodynamic effects, leading tocrack spreading, and branching. Newly formed ‘‘clean-metal’’crack faces tend to readily chemically interact with thelubricant, causing the formation of a chemically alteredfracture-toughness-inferior region at the crack tip. Thesechanges, in turn, lead to a transition from a purely mechan-ical-fatigue-cracking regime to a corrosion-assisted fatigue-cracking regime. The same chemical reactions tend to producehydrogen, which diffuses into the surrounding crack-tip region,primarily along the grain boundaries. This (embrittling) processreduces grain-boundary cohesion and promotes inter-granularcracking (Ref 9).

In our recent work (Ref 10, 11), the problem of gear-boxfailure caused by gear-tooth bending-induced fatigue wasinvestigated computationally. However, even when the ultimatefailure of the gear-box is localized within the gears, the initialdamage which led to this failure could be traced back to one ormore gear-box roller-bearings.

In the present work, a multi-physics computational approachis proposed to address the problem of HAWT gear-box roller-bearing premature-failure. While such a computational ap-proach is not a substitute for the re-engineer-and-field-testapproach, it can provide complementary insight into theproblem of wind-turbine gear-box failure and help gain insightinto the nature of the main cause of this failure. In addition,computational engineering analyses enable investigation of thegear-box failure in a relatively short time, under a variety ofwind-loading conditions comprising both the expected design-load spectrum as well as the unexpected extreme loadingconditions.

Considering the aforementioned potential benefits of thecomputational analysis, the main objective of the present workis to develop a multi-physics computational framework com-prising: (a) quantum-mechanical analysis of grain-boundaryhydrogen embrittlement phenomenon and hydrogen-atomjumps along the grain-boundary and through the bulk; (b)atomistic-level kinetic Monte Carlo analysis of hydrogen inter-granular and bulk diffusion and determination of the effectivegrain-boundary diffusivity; and (c) continuum-type analysis ofsurface-crack inter-granular spreading and branching assistedby hydrogen-embrittlement and corrosion effects under normal-wind loading conditions which may result in the formation ofspalls/fragments within the inner race and, in turn, lead to thebearing-element failure. A flowchart of the multi-physicscomputational approach used in the present work is shown inFig. 4.

A concise summary of the computational approach used inthe investigation of wind-turbine gear-box roller-bearingpremature-failure and the associated hydrogen-induced grain-boundary embrittlement is presented in section 2. The keyresults yielded by the present investigation are presented and

Fig. 3 Schematics of: (a) the classic rolling-contact fatigue (RCF)fracture/failure; and (b) white-etch cracking (WEC) premature-failure

Fig. 4 A flow-chart showing the multi-length-scale computationalanalysis, of HAWT-gearbox roller-bearing premature-failure, utilizedin the present work. Nomenclature: A—hydrogen diffusion parame-ters; B—hydrogen embrittlement parameters; C—effective grain-boundary hydrogen diffusivity


discussed in section 3, while the main conclusions resultingfrom the present work are summarized in section 4.

2. Quantum-Mechanical Analysis of Hydrogen-Induced Grain-Boundary Embrittlement

Premature failure of wind-turbine gear-box roller-bearings istaken to be a manifestation of a hydrogen-induced grain-boundary embrittlement phenomenon. The local extent ofhydrogen-induced grain-boundary embrittlement is controlledby two phenomena/processes: (a) the extent of local hydrogensegregation to the grain-boundary; and (b) the extent of grain-boundary normal and tangential strength sensitivity to the localgrain-boundary hydrogen concentration. To quantify thesephenomena/processes, a series of quantum-mechanical andatomistic-level kinetic Monte Carlo analyses is conducted. Thequantum-mechanical calculations, presented in this section, areused to: (a) assess the sensitivity of grain-boundary strength tothe local concentration of grain-boundary hydrogen; and (b) toquantify key parameters such as the activation energy and thevibrational frequency associated with the hydrogen diffusionand required in the kinetic Monte Carlo hydrogen-diffusionanalysis. Kinetic Monte Carlo simulations, presented in thenext section, are used to assess the rate at which hydrogensegregation from the crack-tip/-faces to the adjacent (connectedto the crack tip) grain-boundaries takes place.

2.1 Problem Formulation

The main problem analyzed in this portion of the work is theextent of grain-boundary hydrogen embrittlement as a functionof the degree of hydrogen segregation to the grain-boundaryand the structure of the grain-boundary itself. All the calcula-tions carried out in this portion of the work involve the use ofthe semi-empirical quantum method AM1, as implemented inVAMP computer program (Ref 12).

2.2 Computational Modeling and Analysis

2.2.1 Grain-Boundary (GB) Modeling. To constructGBs investigated in the present work, the standard procedure isused which involves: (a) rotation of the two crystals either aboutan axis residing in the GB plane (for tilt GBs) or an axis normal tothe GB plane (for twist GBs); (b) sectioning of the two crystalsparallel to the GB plane; and (c) joining the two crystals over theGB plane to form a bicrystal (the computational cell).

2.2.2 Semi-empirical AM1 Method. For the reasonsexplained in our prior work (Ref 13), the rigorous ab initioquantum mechanical calculations were not used in the presentwork. Rather, the AM1 method (Ref 14), a computationallyefficient semi-empirical quantum mechanical method, is uti-lized. Within the AM1 method, diatomic differential overlapeffects are neglected (i.e., the overlap matrix is replaced withthe unit matrix) which enables simplification of the Hartree-Fock secular equation governing the quantum-mechanicalbehavior of the material system. The AM1 method was chosenbecause it appears to be the most suitable semi-empiricalquantum-mechanical method for the analysis of the hydrogendissolution in BCC iron (used to represent the conventionalbearing steel, within the quantum mechanical framework),since it is parameterized to reproduce experimental resultspertaining to hydrogen dissolution energy.

A typical semi-empirical quantum-mechanical calculationinvolves the following three steps: (a) construction of thematerial model; (b) application of the quantum-mechanicalmethod; and (c) post-processing analysis and evaluation of therelevant material quantity/quantities.

2.3 Results and Discussion

The results presented in Fig. 5 reveal the effect of hydrogen,under the condition of full GB saturation, on the GB normalstrength. In this figure, the GB structure varied along the x-axisis represented by the volume of the corresponding interstitial-site Voronoi cells. The results displayed in Fig. 5 reveal that,under the condition of full saturation, the percent decrease inthe GB normal strength due to the presence of hydrogen is not asensitive function of the grain-boundary type. On the otherhand, as evidenced by the results also displayed in Fig. 5, thehydrogen-free GB normal strength is a fairly sensitive functionof the GB structure. Specifically, as the GB structure becomesmore open, as represented by larger Voronoi cells, thehydrogen-free GB normal strength decreases.

The driving force for hydrogen segregation to the GBs is thehydrogen solution-energy. The lower is this energy, the higheris the tendency of hydrogen to segregate to the GBs. Thequantum-mechanical results obtained in the present workpertaining to the effect of grain-boundary structure on thehydrogen solution-energy are depicted in Fig. 6(a). It is seenthat as the GB structure becomes more open, the expectedextent of hydrogen segregation to the GBs increases. Inaddition to depending on the GB structure, hydrogen solution-energy is also a function of the distance from the GB. This isexemplified by the results displayed in Fig. 6(b) for the case ofthe 3 112ð Þ 1�10½ � tilt GB. It is seen that as the distance from theGB increases, the hydrogen solution-energy increases (i.e., theexpected extent of hydrogen segregation decreases). However,it should be noted that the expected hydrogen segregation to theregion adjacent to the grain-boundary is still quite highconsidering the associated lower solution-energy.

Fig. 5 The effect of grain-boundary (GB) structure (as representedby the corresponding interstitial-site Voronoi cell volume) on the GBcohesion strength, and the change in the GB cohesive strength at fullsaturation with hydrogen


The quantum-mechanical calculations carried out in thissection provide the critical data for the kinetic Monte Carlohydrogen diffusion analysis. An example of such data is shownin Fig. 7. In this figure, variation in the computational-cellenergy as a function of the hydrogen-diffusion distance in the001½ � direction is shown. Two cases are considered: (a)diffusion through the bulk; and (b) diffusion along the3 112ð Þ 1�10½ � tilt GB. It should be noted that the system-energyrange in each case is a measure of the corresponding hydrogen-diffusion activation energy. Examination of the results dis-played in Fig. 7 reveals that the bulk-diffusion activationenergy is only about 60% of its grain-boundary counterpart. Atroom temperature, this difference in the diffusion activationenergy translates into an approximately 1100-fold increase inthe diffusion rate through the bulk relative to that along thegrain-boundary. This finding reveals that, while hydrogen has ahigh tendency to diffuse to the grain-boundaries, hydrogenresiding on the grain-boundary has quite low diffusivity alongthe grain-boundary.

3. Atomic-Scale Hydrogen-Diffusion Analysis

The analysis carried out here utilizes the following resultsyielded by the quantum-mechanical calculations described inthe previous section: (i) the number of interstitial-site types; (ii)the coordination of different interstitial-site types; (iii) variationin the system energy along different diffusion paths of thehydrogen atom from the given (occupied) interstitial-site; and(iv) activation energy associated with hydrogen-atom jumpfrom a given (occupied) interstitial site to a neighboring(unoccupied) interstitial-site. It should be noted that only a fewselected results yielded by the quantum-mechanical calcula-tions were shown in the previous section, Fig. 6(a), (b), and 7.Many more such results, which were generated (but not shown)in the previous section, are used in this section.


The problem analyzed here involves: (a) modeling ofhydrogen diffusion from the crack-tip wake region into theadjacent (connected to the crack tip) grain-boundaries; and (b)determination of the effective hydrogen diffusion coefficient.All the calculations carried out utilized atomic-scale kineticMonte Carlo simulations. To monitor the progress of thehydrogen-diffusion process, hydrogen tracer atoms are intro-duced (with identical properties as the remaining hydrogenatoms).

3.2 Computational Modeling and Analysis

3.2.1 Grain-Boundary (GB) Modeling. Essentially thesame geometrical models for the grain boundaries are used inthe kinetic Monte Carlo analysis as in the previous quantum-mechanical analysis. Fe-Fe, Fe-H, and H-H interactions arerepresented using the embedded atom method interatomicpotentials (Ref 15). An example of the computational domainused is depicted in Fig. 8(a). In this figure, iron atoms are

Fig. 6 Dependence of the hydrogen solution-energy on: (a) struc-ture of the grain-boundary (GB) (as represented by the correspond-ing interstitial-site Voronoi cell volume); and (b) distance from theGB for the case of the 3 112ð Þ 1�10½ � tilt GB

Fig. 7 Variation in the computational-cell energy as a function ofthe hydrogen-diffusion distance in the 001½ � direction for the caseof bulk-diffusion and diffusion along the 3 112ð Þ 1�10½ � tilt grain-boundary


colored cyan and hydrogen atoms residing on the grain-boundary are colored white, while a single hydrogen traceratom is colored yellow.

3.2.2 Kinetic Monte Carlo Method. The temporal evo-lution of the hydrogen tracer-atom position during the diffusionanalysis is simulated using the version of the kinetic MonteCarlo method originally developed by Grujicic and Lai(Ref 16). Within this method, one atomic jump is allowed totake place from one (occupied) interstitial site to the neighbor-ing (unoccupied) interstitial site during each time step. Theoccurrence of one such jump at one of the (occupied) sites istermed an event. At each time step, a list of all possible eventsis first constructed, and then the probability for each event is setproportional to the rate at which the associated hydrogen-atomjumps take place. Next, the event to take place is selectedrandomly. This is accomplished by generating, at each timestep, a random number aw from a uniform distribution functionin the (0, 1) range. The value of a is next used to select event mfrom M possible events in accordance with the proceduredescribed in Ref 9. After an event has occurred, the totalnumber of possible eventsM is updated and the aforementionedprocedure is repeated. The kinetic Monte Carlo methodemployed in the present work uses a variable time-step toaccount for the fact that different events require differentamounts of time to occur. At each simulation step the timeincrement is computed using the procedure described in Ref 9.According to this procedure, the maximum time incrementallowed is controlled (dynamically and stochastically) by the

fastest event(s). Consequently, significant reductions in thecomputational time are achieved relative to the fixed time-stepkinetic Monte Carlo methods.


An example of the kinetic Monte Carlo results for theadvancement of the hydrogen front from the crack-tip along theadjoining grain-boundary, showing the (yellow) tracer hydro-gen atom: (a) originally in the wake of the crack-tip; (b) and (c)diffusing through the bulk iron adjacent to the grain-boundary;and (d) ultimately arriving at the hydrogen-concentration frontalong the grain-boundary, causing this front to grow, is depictedin Fig. 8(a)-(d). For brevity, other results obtained in thisportion of the work will not be presented. Rather, the keyfindings can be summarized as:

(a) Diffusion rate of hydrogen along the grain-boundaries isquite small in comparison to its counterpart through thebulk, which is in complete agreement with the quantum-mechanical predictions;

(b) Once a hydrogen atom has arrived at a grain-boundary,it becomes effectively trapped with very little mobilityboth in the directions along and normal to the grain-boundary;

(c) Mass transport of hydrogen from the crack-tip wake re-gion along the adjoining grain-boundaries is effectivelycontrolled not by the grain-boundary diffusion but rather

Fig. 8 Kinetic Monte Carlo results for the advancement of the hydrogen front from the crack-tip along the adjoining grain-boundary, showingthe (yellow) tracer hydrogen atom: (a) originally in the wake of the crack-tip; (b) and (c) diffusing through the bulk iron adjacent to the grain-boundary; and (d) ultimately arriving at the hydrogen-concentration front along the grain-boundary, causing this front to grow


by the diffusion of hydrogen through the adjoiningbulk material and its deposition onto the advancingGB-hydrogen front (Ref 17-20);

(d) While the hydrogen diffusion from the crack-tip throughthe bulk to the advancing hydrogen-front is associatedwith a longer diffusion path, the effective hydrogengrain-boundary diffusivity is still controlled by its bulkcomponent. This is the result of two effects: (i) substan-tially higher bulk diffusivity of hydrogen compared toits inter-granular counterpart; and (ii) the fact that the in-ter-granular and trans-granular diffusion paths are con-nected in parallel; and

(e) Using the procedure described in Ref. 21, the effective(averaged) grain-boundary diffusivity for hydrogen atroom temperature has been found to be(1.7± 0.1) 10�11 m2/s. This value will be used in sec-tion 4 to model the progress of hydrogen segregation tothe grain-boundaries ahead of the advancing crack-tip.

4. Roller-Bearing Inner-Race Damage/FailureAnalysis

In this section, the problem of surface-crack spreading andbranching in the (inner) raceway is analyzed under rolling-element/raceway contact-stresses.


The main problem addressed here is the determination of theaverage time it will take a surface-crack to penetrate deepenough into the raceway, turn around and return to the surface,forming a (potentially failure-causing) spall. It should be notedthat the process of nucleation of the surface-crack is notanalyzed. Rather, it is assumed that a combination of theunfavorable kinematic conditions within the rolling-elementand excessive loading can cause such a crack to nucleate. Thetime for the surface-crack to propagate as described, whencombined with the average time for surface-crack nucleation(not dealt with in the present work) could then serve as anestimate of the roller-bearing predicted life. All simulations inthis portion of the work were conducted using ABAQUS/Explicit, a general-purpose finite-element program (Ref 22).

4.2 Computational Model and Analysis

4.2.1 Contact-Region Finite-Element Model. To ac-count for the fact that: (a) the rolling-element/raceway contactpatch is quite small in comparison to the characteristicdimensions of these components; and (b) the premature failureis frequently found to occur in inner races, the geometricalmodel analyzed consists of a two-dimensional rectangularregion residing fully within the inner race, Fig. 9(a). The roller-element is not modeled explicitly but is rather replaced by the(moving) surface-distribution of the contact-stresses (markedschematically in Fig. 9b) of the type determined in our previouswork (Ref 13). To recognize the fact that: (a) surface-crackspreading and branching will be localized near the surface; and(b) crack growth is primarily of the inter-granular character, theupper portion of the computational domain surrounding thepreviously nucleated crack was assigned a granular structure

[using two-dimensional Poisson-type Voronoi cells (Ref 13)].In this way, common edges of the adjacent Voronoi cells act as(two-dimensional) grain boundaries. The remainder of thecomputational domain is treated as a featureless, two-dimen-sional continuum.

Both the granular and the featureless portions of thecomputational domain are meshed using a combination oftriangular and quadrilateral plane-strain continuum finite ele-ments. In addition, grain boundaries are meshed using four-node traction/separation cohesive-zone elements. These ele-ments enable modeling of the normal- and/or shear-tractioninduced grain-boundary decohesion (i.e., inter-granular crack-ing). Furthermore, in order to model hydraulic loading of thepartially or fully debonded grain-boundaries, the cohesive-zoneelements are assigned additional degrees of freedom (i.e., thepore-pressure). A close-up of the meshed model used in thisportion of the work is depicted in Fig. 9(c). It should be notedthat, since each cohesive-zone element initially involves twopairs of coincident nodes, they are not (visually) resolved inFig. 9(c). The mesh typically contains 15,000-20,000 contin-uum and 4000-5000 cohesive-zone elements.

4.2.2 Analysis of Crack Spreading/Branching UsingABAQUS/Explicit. The finite-element analysis (FEA)employed in this portion of the work requires specification ofthe following: (a) geometrical model; (b) meshed model; (c)computational algorithm; (d) initial conditions; (e) boundaryconditions; (f) contact interactions; (g) material model(s); and(h) computational tool. Details regarding the geometrical andmeshed models are presented above. Since the remaining itemshave been described in our prior work (Ref 13), they will not berepeated here. However, unique features of the FEA used in thisportion of the work are presented in the remainder of thissection. These include:

(a) To represent traction-separation constitutive relations forthe cohesive-zone elements under hydrogen-/corrosion-assisted inter-granular cracking conditions, a user sub-routine had to be created and linked with the ABAQUSsolver. The essential feature of these relations is that asthe grain-boundary separation increases, the correspond-ing traction first increases, then reaches a peak and sub-sequently continues to decrease toward zero (indicatinga complete loss of grain-boundary cohesion). Detailsregarding the functional forms for these constitutive rela-tions are presented in the Appendix. Since the parame-ters of these functional relationships (e.g., the peaktraction, i.e., the cohesive strength) are affected by theextent of hydrogen segregation to the grain-boundariesahead of the advancing cracks, they were formulatedusing the results of: (i) a quantum-mechanical analysisrevealing the sensitivity of the grain-boundary cohesionstrength to the presence of hydrogen (section 2); and (ii)an atomic-level analysis of hydrogen diffusion from thecrack-tip wake into the surrounding grain-boundaries(section 3);

(b) A moving distributed normal-loading (representing roll-ing-element/inner-race interactions) is applied over thetop edge of the computational domain, in a periodicsense. That is, when a portion of the distributed loadingleaves the top edge of the computational domain on oneside (the right side, in Fig. 9b), an equivalent ‘‘missingportion’’ of the profile contacts the top edge at the otherside (the left side, in Fig. 9b). Furthermore, to represent


Fig. 9 Schematic of diffusion of hydrogen from crack tip through grain boundary and bulk due to lubricant being hydraulically loaded bypassage of roller-element


hydrodynamic loading exerted on the crack facesextending to the surface by the infiltrating and pressur-ized lubricant, a periodic pore-pressure loading functionis applied to the ‘‘cracked’’ grain boundaries;

(c) Inter-granular fracture is modeled as the process of deg-radation and ultimate failure of the grain-boundary cohe-sive-zone elements. Once a cohesive-zone finite elementis completely degraded/failed, it is removed from themodel. Thereafter, the faces of the corresponding ele-ments of the adjacent grains are prevented from inter-penetration by activating a penalty-type contactalgorithm between them (Ref 23-27); and

(d) To account for the fact that the evolution portion of theroller-bearing premature-failure process is stress-controlled, the continuum-material (AISI 4340 steel)model is assumed to be of a (isotropic) linearly elasticcharacter.

4.2.3 Prediction of the Bearing-Life Controlled byInter-granular Cracking. Surface-crack spreading andbranching, and thus the associated bearing-life are modeledexplicitly in this section. As mentioned above, the rate at whichsurface-cracks spread/branch and, thus, the bearing-life arecontrolled by the rate of hydrogen-segregation to the grain-boundaries ahead of the advancing cracks and the extent ofgrain-boundary cohesive-strength sensitivity to the presence ofhydrogen (grain-boundary hydrogen-embrittlement).

To include the effect of grain-boundary hydrogen-embrit-tlement into the present FEA, the following procedure shouldbe implemented at each computational step:

(a) since the hydrogen ingress is believed to be diffusion-and not chemical-reaction-controlled (i.e., chemical reac-tions between the freshly formed crack surfaces and lu-bricant additives are quite fast), a fixed hydrogenconcentration is assumed to exist at the tip of eachcrack. This concentration corresponds to the conditionof hydrogen chemical-potential equality in the lubricantlocated in the wake of the crack-tip and at the adjacentcrack-faces. Then, using the information about the effec-tive grain-boundary diffusivity of hydrogen (as providedby the atomic-scale analysis presented in section 3), theconcentration profiles and the associated average con-centrations of hydrogen along the (still bonded) grain-boundaries connected to the crack-tip are calculated. Tohelp clarify the procedure implemented in this portion ofthe work, a simple schematic of a lubricant-filled andhydraulically loaded crack impinging on the two grain-boundaries is depicted in Fig. 10;

(b) the information obtained in (a), in conjunction with thesensitivity of the grain-boundary strength to the presenceof hydrogen (as provided by the quantum-mechanicalanalysis, section 2), is used to impart the hydrogen-induced embrittling effect to the grain-boundaries inquestion. This was done by properly reducing the grain-boundary normal/tangential cohesive-strengths.

(c) loading induced by the passage of the roller-elementsover the computational domain of the raceway is thenused within the FEA to further (mechanically) degradethe grain-boundary elements. Two components of thisloading are included: (i) surface-type loading associatedwith the passage of the contact-pressure profile over thetop edge of the computational domain; and (ii) hydraulic

loading associated with the pressurization of the lubri-cant, residing within the surface-originating crack(s), dueto the passage and the piston-like action of the roller-element over the crack mouth; and

(d) the progress of a spread-out and branched crack-front ismonitored in order to identify the instant when one ofthe fronts arrives at the raceway free surface. Thisinstant is then identified as the moment of failure of theroller-bearing.

The procedure described above should be applied at eachcomputational step, for the most accurate determination of theinitiation and evolution of damage within the computationaldomain. However, due to a very large number of loading cyclesrequired to produce a spall, a ‘‘jump-in-cycles’’ approach wasimplemented. This procedure involves the following steps: (i)first, an integer value is assigned to the number of cycles to beskipped, i.e., to the cycle-jump; (ii) the analysis is thenperformed over a number of consecutive time-steps required tocomplete one loading cycle, i.e., one passage of the Hertziancontact-pressure over the top edge of the computationaldomain; (iii) then, the clock is advanced by the time equal tothe product of the cycle-jump and the total duration of the just-completed loading cycle; (iv) the time obtained in (iii) is usedto update the grain-boundary hydrogen concentration, and thenormal and shear cohesive strengths; and (v) the next loadingcycle is applied, and the sequence of steps (i)-(v) repeated.


In this section, the main results of the finite-elementstructural analysis, as related to the wind-turbine gear-boxroller-bearing premature-failure, and of the associated hydro-gen-induced grain-boundary embrittlement, are presented anddiscussed. While the present computational framework enablesthe generation of results under various mechanical loadingconditions and hydrogen-source conditions in the crack-front

Fig. 10 (a) Close-up view of the transverse section of a roller-bearing; (b) geometrical model; and (c) meshed model of the two-dimensional computational domain analyzed


wake, only a few prototypical results will be presented anddiscussed due to space limitations. In our future communica-tion, a more detailed parametric study of the effect of variousloading and environmental conditions on the roller-bearingservice-life will be presented.

As suggested earlier, premature-failure of wind-turbine gear-box roller-bearings is generally assumed to be initiated by oneof the surface-distress processes. Typically, these processesresult in the formation of surface cracks. In the present work, itis assumed that the grain subdomain contains, from the onset,two such surface cracks. Using the FEA, it is then investigatedhow the presence of these cracks and the absence/presence ofhydrogen-induced grain-boundary embrittlement affect thetemporal evolution and spatial distribution of the contact-region stresses and the inter-granular damage/failure.

All the results presented in the remainder of this sectionwere obtained under the following loading and environmentalconditions: (a) Hertzian peak-pressure of 3 GPa; (b) contact-patch half-width b = 100 lm; (c) the intra-crack lubricanthydraulic-pressure of 375 MPa; and (d) crack-tip hydrogenconcentration of 49 10�3 at.%. For the loading conditions (a)and (b), the values chosen are consistent with the normaloperating conditions of an intermediate-speed-shaft (ISS) typeof bearing in a prototypical 5 MW HAWT gear-box. The valuefor the loading condition (c) was obtained in a separate analysisin which the piston-like action of the roller-element onto theintra-crack lubricant, with a pressure-dependent bulk modulus,was investigated. In the absence of any experimental data, theenvironmental condition (d) is set equal to the nominal surfacesolubility limit of hydrogen in BCC iron.

4.3.1 Temporal Evolution/Spatial Distribution ofContact-Region Stresses. For clarity, the results correspond-ing to the absence and the presence of the grain-boundaryembrittling effects are presented and discussed separately.

In the Absence of Grain-Boundary Hydrogen-Embrittle-ment. Spatial distribution of the von Mises stress at fourdifferent times during the 25-millionth passage of the Hertziancontact-pressure profile over the top edge of the computationaldomain, in the absence of grain-boundary hydrogen-embrittle-ment effects, is shown in Fig. 11(a)-(d). The four figurescorrespond, respectively, to the cases when the Hertziancontact-pressure profile: (a) has been fully applied to theleftmost top portion of the computational domain; (b) hasadvanced to the right and its leading half acts on the leftmosttop portion of the granular subdomain; (c) has advanced furtherto the right and its trailing half acts on the rightmost top portionof the granular subdomain; and (d) has advanced still further sothat it has been fully applied to the rightmost top portion of thecomputational domain. The corresponding spatial distributionof the von Mises stress during the 100-millionth passage of thesame Hertzian contact-pressure profile is shown in Fig. 12(a)-(d). In both sets of figures, the position of the Hertzian contact-pressure profile on the top edge of the computational domain isindicated using a solid thick line.

Examination of the results displayed in Fig. 11(a), (d) and12(a)-(d) reveals that: (a) passage of the Hertzian contact-pressure profile over the top side of the computational domaincauses an increase in the von Mises stress in the region adjacentto the (moving) contact-patch; (b) while the von Mises stressfield within the non-granular portion of the computationaldomain is quite smooth, the same field shows discontinuities inthe granular subdomain. This finding is consistent with thepresence of the grain-boundary cohesive-zone elements sepa-rating the adjacent grains; and (c) spatial distribution of thevon Mises stress is not substantially different between the25-millionth and the 100-millionth passage of the Hertziancontact-pressure profile over the top edge of the computationaldomain. As will be shown in section 4.3.2, this finding is the

Fig. 11 Spatial distribution of the von Mises stress during the 25-millionth passage of the Hertzian contact-pressure profile over the top edge ofthe computational domain, in the absence of grain-boundary hydrogen-embrittlement effects. The position of the Hertzian contact-pressure profileon the top edge of the computational domain is indicated in (a), (b), (c), and (d) using a solid thick line. Please also see the text for details


result of the fact that the repeated passage of the Hertziancontact-pressure profile causes only relatively minor degrada-tion of the grain-boundaries connected to the initial cracks. Itshould be noted that in the present work, no attempt was madeto model RCF failure. This failure-mode is of a trans-granularcharacter and involves local-plasticity-based shear localization.The analysis carried out in the present work is of a purelyelastic character, although the normal-tensile and shear stiff-nesses of the grain-boundary cohesive-zone elements areallowed to degrade as a result of the combined effects ofhydrogen-induced grain-boundary embrittlement and excessiveloading. Consequently, if one includes the effects of the localplasticity and shear localization in the FEA, more significantdifferences may be expected between the corresponding vonMises stress spatial-distribution results corresponding to the25-millionth and the 100-millionth passage of the Hertziancontact-pressure profile.

In the Presence of Grain-Boundary Hydrogen-Embrittle-ment. Spatial distribution of the von Mises stress at fourdifferent times during the 25-millionth passage of the Hertziancontact-pressure profile over the top edge of the computationaldomain, in the presence of grain-boundary hydrogen-embrit-tlement effects, is shown in Fig. 13(a)-(d). The four figures areassociated with the same four positions of the Hertzian contact-pressure profile as in Fig. 11(a)-(d) and 12(a)-(d). The corre-sponding spatial distribution of the von Mises stress during the100-millionth passage of the same Hertzian contact-pressureprofile is shown in Fig. 14(a)-(d).

Examination of the results displayed in Fig. 13(a)-(d) and14(a)-(d) and their comparison with the corresponding resultsshown in Fig. 11(a)-(d) and 12(a)-(d) reveals that:

(a) as in the case where grain-boundary hydrogen-embrittle-ment effects were absent, passage of the Hertzian con-tact-pressure profile over the top side of thecomputational domain causes an increase in the vonMises stress in the region adjacent to the (moving) con-tact-patch. However, the size of this region is somewhatincreased in the case of hydrogen-induced grain-bound-ary embrittlement (for example, please compare Fig. 11cand 13c);

(b) spatial distribution of the von Mises stress is substan-tially different between the 25-millionth and the 100-millionth passage of the Hertzian contact-pressure profileover the top edge of the computational domain. Specifi-cally, the region associated with the highest levels of thevon Mises stress is significantly increased (for example,please compare the results displayed in Fig. 11d and14d). As will be shown in section 4.3.2, this finding isthe result of the fact that the repeated passage of theHertzian contact-pressure profile, in the presence of thegrain-boundary hydrogen-embrittlement effects, causessignificant degradation of the grain-boundary structure,extending quite deep from the top edge of the granularportion of the computational domain; and

(c) differences in the effect of the repeated passage of theHertzian contact-pressure profile on the spatial distribu-tion of the von Mises stress, observed by comparing theresults displayed in Fig. 11(a)-(d) and 13(a)-(d), with theresults displayed in Fig. 12(a)-(d) and 14(a)-(d), respec-tively, are the manifestation of the more pronounceddegradation of the material structural integrity due to theoperation of hydrogen-induced grain-boundary embrit-tling processes.

Fig. 12 Spatial distribution of the von Mises stress during the 100-millionth passage of the Hertzian contact-pressure profile over the top edgeof the computational domain, in the absence of grain-boundary hydrogen-embrittlement effects. The position of the Hertzian contact-pressureprofile on the top edge of the computational domain is indicated in (a), (b), (c) and (d) using a solid thick line. Please also see the text fordetails


4.3.2 Temporal Evolution/Spatial Distribution of Contact-Region Damage. Again, for clarity, the results correspondingto the absence and the presence of the grain-boundary embrittlingeffects are presented and discussed separately.

In the Absence of Grain-Boundary Hydrogen-Embrittle-ment. Spatial distribution of the grain-boundary cohesive-zoneelements in which both the normal and the shear cohesive-strengths are degraded by at least 90% relative to their initial

Fig. 13 Spatial distribution of the von Mises stress during the 25-millionth passage of the Hertzian contact-pressure profile over the top edge ofthe computational domain, in the presence of grain-boundary hydrogen-embrittlement effects. The position of the Hertzian contact-pressureprofile on the top edge of the computational domain is indicated in (a), (b), (c), and (d) using a solid thick line. Please also see the text for details

Fig. 14 Spatial distribution of the von Mises stress during the 100-millionth passage of the Hertzian contact-pressure profile over the top edge ofthe computational domain, in the presence of grain-boundary hydrogen-embrittlement effects. The position of the Hertzian contact-pressure profileon the top edge of the computational domain is indicated in (a), (b), (c), and (d) using a solid thick line. Please also see the text for details


values, after the 25-, 50-, 75-, and 100-millionth passage of theHertzian contact-pressure profile over the top edge of thecomputational domain, in the absence of grain-boundaryhydrogen-embrittlement effects, is shown in Fig. 15(a)-(d).Examination of these results reveals that:

(a) the repeated passage of the Hertzian contact-pressurefirst causes some branching of the two initially inducedsurface cracks;

(b) after a sufficient number of Hertzian contact-pressurepassages, the newly formed cracks arrive to the top edgeof the granular subdomain, yielding the formation of rel-atively thin spalls/flakes; and

(c) subsequent loading of the computational domain by thepassing Hertzian contact-pressure profile does not pro-duce any additional damage. It should again be recalledthat in the present work, no attempt was made to modelRCF failure. Consequently, under the present loadingand environmental conditions, one cannot exclude thepossibility that the material would suffer additional dam-age due to the interplay of the local plasticity and shearlocalization phenomena (Ref 13). However, modeling ofthese phenomena is beyond the scope of the presentwork.

In the Presence of Grain-Boundary Hydrogen-Embrittle-ment. Spatial distribution of the grain-boundary cohesive-zoneelements in which both the normal and the shear cohesive-strengths are degraded by at least 90% relative to their initialvalues, due to a combined effect of the grain-boundaryhydrogen-embrittlement and excessive loading after the 25-,50-, 75-, and 100-millionth passage of the Hertzian

contact-pressure profile over the top edge of the computationaldomain, is shown in Fig. 16(a)-(d). Examination of theseresults and their comparison with the corresponding resultsdisplayed in Fig. 15(a)-(d) reveals that:

(a) the repeated passage of the Hertzian contact-pressureand the operation of the attendant grain-boundary hydro-gen-embrittlement processes cause extensive and deepspreading and branching of the two initially induced sur-face cracks;

(b) ultimately, some of the newly created cracks mutuallyconnect, forming a large spall/fragment; and

(c) the size of this spall, Fig. 16(d), is at least an order ofmagnitude larger than the one observed in the no-hydro-gen-embrittlement case, Fig. 15(d). This finding suggeststhat while in the absence of hydrogen-embrittlementeffects, surface-distress can induce only very shallowcraters into the raceways, affecting somewhat the roller-bearing performance, in the presence of the grain-boundaryembrittling effects, the raceways may experience a majordamage, resulting in the formation of large and deepcraters, and large-sized fragments. In this case, the func-tionality of the roller-bearing element may be compro-mised, as well as the functionality and structuralintegrity of the adjacent gears (should any of the largefragments propagate to the nearby gear-box stage andget caught between the teeth of meshing gears).

4.3.3 Prediction of the Roller-Bearing Service-Life. Inthis section, an attempt is made to estimate the service-life of aHAWT gear-box planetary-bearing element. Toward that end,one must determine not only the portion of the service-lifeassociated with the crack-spreading/-branching until the

Fig. 15 Spatial distribution of the grain-boundary cohesive-zone elements in which both the normal and the shear cohesive-strengths aredegraded by at least 90% relative to their initial values, after the: (a) 25-; (b) 50-; (c) 75-; and (d) 100-millionth passage of the Hertzian contact-pressure profile over the top edge of the computational domain, in the absence of grain-boundary hydrogen-embrittlement effects


formation of a spall, but also the portion of the service-lifeassociated with the crack-nucleation. Following the standardpractice, it is assumed in the present work that events such asemergency-shutdown control the onset of cracking whilenormal-wind loading controls the kinetics of crack spreading/branching.

The procedure for determination of the bearing-life in thecrack-spreading/-branching regime described in section 4.2yielded, for the aforementioned normal wind-loading condi-tions, the following results: number of cycles = 99 million;duration = 21 months.

To estimate the number of cycles until the nucleation of thesurface-cracks, the procedure described in our prior work(Ref 10, 11) is utilized. This procedure recognizes the strain-controlled character of the fatigue-crack initiation process, andmodels this process by combining:

(a) the conventional Coffin-Manson equation, De¢p/2 =e¢f(2Ni)

c, where Dep¢/2 is the equivalent plastic strainamplitude, e¢f is the fatigue ductility coefficient, c is thefatigue ductility exponent, Ni is the number of cycles re-quired to reach ath, and 2Ni is the corresponding numberof stress reversals; with

(b) the additive decomposition of the total equivalent strainamplitude De¢/2 into its elastic, De¢e/2, and plastic com-ponents;

(c) the fatigue micro-yielding constitutive law, De¢p/2 =e¢f(Dr¢/2r¢f)1/n¢, where Dr¢/2 is the equivalent-stressamplitude, n¢ is the cyclic strain-hardening exponent,and r¢f is the fatigue strength coefficient;

(d) Hooke�s law, Dr¢ = E ÆDe¢e, where E is the Young�smodulus; and

(e) stress-based fatigue-life relation, Dr¢/2 = r¢FL +r¢f(2Ni)

b, where r¢FL is the material fatigue/endurancelimit and b is a material parameter.

This procedure yields the following equation:

Dε0

2¼ Dr0=2ð Þ � r0FL

Eþ ε0f

Dr0=2ð Þ � r0FL2r0f

� �1=n0

¼ r0fE

2Nið Þbþε0f 2Nið Þc:ðEq 1Þ

Equation 1 can be solved iteratively to get the number ofcycles to fatigue-crack initiation Ni for a given combination ofbearing-material and cyclic loading (as represented by Dr¢/2).

Equation 1 enables determination of Ni under constant-amplitude cyclic-loading conditions. In the present investiga-tion, surface-crack initiation was assumed to be controlled byunfavorable kinematic and excessive loading conditionsaccompanying emergency shutdown. Under such circum-stances, cyclic loading is not of a constant amplitude (and isalso intermittent). To account for these effects, a procedureproposed in our prior work (Ref 4, 5) which combined the so-called rainflow cycle-counting algorithm, Goodman diagramand Miner�s Rule, was used. Furthermore, to account for theintermittency of the cyclic loading, a frequency of theemergency-shutdown events had to be assumed. A combinationof Eq 1 and these procedures then yields Ni, which can bereadily converted into si, the bearing-life within the crack-nucleation regime.

Application of the aforementioned procedure for the pre-diction of the bearing-element life before surface-crack initia-tion yielded the following results: (a) normal-windloading—3.1 billion cycles, time = ca. 54 years; and (b)emergency shutdown (assuming a prototypical number ofshutdown/restart cycles of 3000 per year and 10 torque-oscillation cycles accompanying each shutdown/restartcycle)—19 million cycles, time = ca. 4 months.

Making the assumption that events such as emergency-shutdown control the onset of cracking while normal-wind

Fig. 16 Spatial distribution of the grain-boundary cohesive-zone elements in which both the normal and the shear cohesive-strengths are degradedby at least 90% relative to their initial values, due to a combined effect of the grain-boundary hydrogen-embrittlement and excessive loading afterthe: (a) 25-; (b) 50-; (c) 75-; and (d) 100-millionth passage of the Hertzian contact-pressure profile over the top edge of the computational domain


loading controls the kinetics of crack spreading/branching,the total life of the subject bearing is estimated as 4 months(for crack initiation) + 21 months (for spall formation) =25 months. Clearly, this estimate, as well as its two compo-nents, is a sensitive function of the normal-wind loading andemergency shutdown conditions simulated, bearing-materialsused, as well as the chemistry and the additive content of thelubricant. The extent of this sensitivity will be investigated inour future work.

5. Summary and Conclusions

Based on the results obtained in the present work, thefollowing main summary remarks and conclusions can bedrawn:

1. A new multi-physics computational framework has beendeveloped to investigate wind-turbine gear-box roller-bearing premature-failure. Within this framework, threedistinct computational analyses are carried out and theirresults combined. The three analyses involve: (a) quan-tum-mechanical study; (b) atomic-scale simulations; and(c) a finite-element calculation.

2. Within the quantum-mechanical analysis, the phenome-non of hydrogen-induced grain-boundary embrittlementand hydrogen-atom jumps along the grain-boundary andthrough the bulk are investigated.

3. Within a kinetic Monte Carlo atomic-scale analysis, thephenomenon of hydrogen inter-granular and bulk diffu-sion is investigated and the associated effective grain-boundary diffusivity determined.

4. Within the finite-element continuum-type analysis, theprocesses associated with surface-crack inter-granularspreading and branching assisted by hydrogen-embrittle-ment and corrosion effects under normal-wind loadingconditions which may result in the formation of spalls/fragments within the inner race and, in turn, lead to thebearing-element failure, are investigated.

5. The present multi-physics computational framework en-abled determination of the portion of the roller-bearingservice-life associated with the spreading and branchingof surface-cracks until the formation of a spall. These re-sults are combined with a strain-based low-cycle fatigueanalysis, to determine the portion of the bearing-life be-fore the nucleation of surface-cracks. The total predictedroller-bearing service-life was found to be considerablyshorter than its design service-life of ca. 20 years.

6. Appendix: Grain-Boundary Cohesive-ZonePotential

As mentioned earlier, the constitutive response of grainboundaries is modeled, in the present work, using the‘‘cohesive zone framework’’ originally proposed by Needleman(Ref 28). The cohesive-zone/grain-boundary is assumed to havea negligible thickness when compared with other characteristiclengths of the problem, such as the grain size, contact-patchwidth, etc. The constitutive response of the cohesive zone is

characterized by a traction-displacement relation, which isintroduced through the definition of a grain-boundary poten-tial, w. The perfectly bonded grain boundary is assumed to bein a stable equilibrium, in which case the potential w has aminimum and all tractions vanish. For any other configuration,the value of the potential is taken to depend only on thedisplacement discontinuities (jumps) across the grain boundary.

For a two-dimensional problem, as in the present case, thegrain-boundary displacement jump (i.e., the grain-boundaryseparation) is expressed in terms of its normal component, Un,and a tangential component, Ut, where both components lie inthe x-y plane of the Cartesian coordinate system. Differentiatingthe interface potential function W ¼ W Unð ;UtÞ with respect toUn and Ut yields, respectively, the normal and tangentialcomponents of F, the force per unit grain-boundary area in thedeformed configuration, as:

Fn Un;Utð Þ ¼ @WðUn;UtÞ@Un

; ðEq A1Þ

Ft Un;Utð Þ ¼ @WðUn;UtÞ@Ut

: ðEq A2Þ

The grain-boundary traction/separation constitutive relationsare thus fully defined by specifying the form for the grain-boundary potential function WðUn;UtÞ. The grain-boundarypotential of the following form initially proposed by Xu andNeedleman (Ref 29) is used in the present study:

WðUn;UtÞ ¼/nþ/n exp �Un

dn

� �

1� rþUn

dn

� �1� q

r� 1

� �� qþ r� q

r� 1

� �Un

dn

� �exp �U2

t

d2t

!( );

ðEq A3Þ

where the parameters /n and /t are the work of (pure) nor-mal and (pure) shear separation, respectively, q = /t//n, dn,and dt are the normal and tangential interface characteristic-lengths, r = dn

*/dn, dn* (set to zero, in the present work) is the

value of dn after complete shear separation takes place underthe condition of normal tension being zero. Differentiation ofEq A3 with respect to Un and Ut yields the following expres-sions for the normal and tangential grain-boundary tractions:

FnðUn;UtÞ ¼ /n

Un

d2n

!exp �Un

dn

� �q exp �U 2

t

d2t

!þ 1� qð Þ

( );

ðEq A4Þ

FtðUn;UtÞ ¼ /n 1þ Un

dn

� �exp �Un

dn

� �2qUt

d2t

!exp �U 2

t

d2t

!( ):

ðEq A5Þ

The works of normal and shear separations, /n and /t, arerelated to rmax = max (Fn) and smax = max (Ft), respectively,as:

/n ¼ exp 1:0ð Þrmaxdn; ðEq A6Þ

/t ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexp 1:0ð Þ=2

psmaxdt: ðEq A7Þ

Graphical representations of the two functions defined byEqs. A4 and A5 are given in Fig. A1(a) and (b), respectively. In


these plots, the following normalization of the independent anddependent variables was used: Un/dn, Ut/dt, Fn/rmax, Ft/smax.Examination of Fig. A1(a) and (b) shows that: (a) at a givenvalue of Ut, Fn peaks at Un = dn; (b) at a given value of Un, Ft

peaks at Ut ¼ dt=ffiffiffi2p

; (c) a nonzero value of Ut reduces thenormal-traction maximum value, max (Fn); and (d) a nonzerovalue of Un reduces the tangential-traction maximum value,max (Ft).

Fig. A1 (a) Normalized normal, Fn; (b) and normalized tangential components, Ft, of the traction per unit interface area, as a function of thenormalized normal, Un, and normalized tangential, Ut, components of the interface displacements


An inspection of Eqs. (A3)-(A7) shows that the grain-boundary behavior is characterized by five independent param-eters: rmax, smax, dn, dt, and r.

The grain-boundary decohesion potential presented above isnext incorporated into a User Element Library (VUEL)subroutine of ABAQUS/Explicit. The VUEL subroutine allowsthe user to define the contribution of the grain-boundaryelements to the global finite element model. In other words, forthe given nodal displacements of the cohesive-zone grain-boundary elements provided to the VUEL by ABAQUS, thecontribution of the elements to the global vector of residualforces and to the global Jacobian (element stiffness matrix) iscomputed in the VUEL subroutine and passed back toABAQUS/Explicit. The implementation of the grain-boundarydecohesion potential in the VUEL subroutine is discussedbelow.

For the two-dimensional case analyzed here, each grain-boundary element is defined as a four-node iso-parametricelement on the grain-boundary, as shown schematically inFig. A2. In the un-deformed configuration (not shown forbrevity), nodes 1 and 4 in Fig. A2, and nodes 2 and 3 coincide,respectively. A local co-ordinate system, consistent with thedirections that are tangent, t, and normal, n, to the interface, isnext assigned to each element. This is done by introducing twointernal nodes, A and B, located at the midpoints of the lines 1-4and 2-3, connecting the corresponding grain-boundary nodes.The grain-boundary displacements at the internal nodes A and Bare expressed in terms of the displacements of the elementnodes 1-4 as in the global coordinate system z-r, as:

UAn ¼ U4

z � U1z

cos h� U4

r � U1r

sin h; ðEq A8Þ

UAt ¼ U4

z � U1z

sin h� U4

r � U1r

cos h; ðEq A9Þ

UBn ¼ U3

z � U2z

cos h� U3

r � U2r

sin h; ðEq A10Þ

UBt ¼ U3

z � U2z

sin h� U3

r � U2r

cos h: ðEq A11Þ

An iso-parametric coordinate g is next introduced along thetangent direction with g(A) = �1 and g(B) = 1 and two linearLagrangian interpolation functions are defined as NA(g) =(1�g)/2 and NB(g) = (1 + g)/2. These interpolation functionsallow the normal and the tangential components of the grain-boundary displacements to be expressed in the form of theirvalues at the internal nodes A and B as:

Ut gð Þ ¼ NA gð ÞUAt þ NB gð ÞUB

t ; ðEq A12Þ

Un gð Þ ¼ NA gð ÞUAn þ NB gð ÞUB

n : ðEq A13Þ

The tangential and normal components of the forces atnodes A and B, i.e., FA

t , FBt , F

An , and FB

n , which are workconjugates of the corresponding nodal displacements UA

t , UBt ,

UAn , and UB

n are next determined through the application of thevirtual work to the grain-boundary element as:

L

Z1

�1

dU gð Þdg ¼XI¼n;t

XN¼A;B

FNI dUN

I ; ðEq A14Þ

where L is the A-B element length. To determine the grain-boundary (normal and tangential) tractions at the internal nodesA and B, the grain-boundary potential is perturbed and the resultexpressed in terms of the perturbations of the grain-boundarydisplacements at the internal nodes, UA

t , UBt , U

An , and U

Bn as:

dU ¼ @U UtðgÞ;UnðgÞ½ �@Un

NAðgÞdUAn þ NBðgÞdUB

n

� �

þ @U UtðgÞ;UnðgÞ½ �@Ut

NAðgÞdUAt þ NBðgÞdUB

t

� �:

ðEq A15Þ

By substituting Eq A15 into A14 and by choosing one ofthe dUN

I N ¼ A;B; I ¼ t; nð Þ perturbations at a time to be unityand the remaining perturbations to be zero, the correspondingFNI component of the nodal force can be expressed as:

FNI ¼ L

Z1

�1

@U UtðgÞ;UnðgÞ½ �@UI

NN ðgÞdg: ðEq A16Þ

Using a straightforward geometrical procedure and impos-ing the equilibrium condition, the corresponding residual nodalforces Ri

r and Rizði ¼ 1� 4Þ in the global r-z co-ordinate

system, are defined as:

R1r ¼ �R4

r ¼ FAt cos h� FA

n sin h;

R1z ¼ �R4

z ¼ FAt sin h� FA

n cos h; ðEq A17Þ

R2r ¼ �R3

r ¼ FBt cos h� FB

n sin h;

R2z ¼ �R3

z ¼ FBt sin h� FB

n cos h:

The components of the grain-boundary-element Jacobian arenext defined as:

Fig. A2 Definition of the linear, four-node axisymmetric interfaceelement. Nodes 1 and 4 and nodes 2 and 3 coincide in the equilib-rium (reference) configuration. Internal nodes A and B located at themidpoints of segments connected corresponding nodes in the metaland plastics sides of the interface; two integration points markedas + and a local t-n co-ordinate system are also indicated


@Rij

@Ukl

¼XI¼n;t

XN¼A;B

XJ¼n;t

XM¼A;B

@Rij

@FNI

@FNI

@UMJ

@UMJ

@Ukl

; ðEq A18Þ

where the components of the internal Jacobian@FN

i =@UMj ði; j ¼ n; t;N ;M ¼ A;BÞ are calculated by differen-

tiation of Eq A16.To summarize, the residual nodal forces given by Eq A17

and the element Jacobian given by Eq A18 are computed in theVUEL subroutine, and passed to ABAQUS/Explicit for use inits global Newton scheme for accurate assessment of thekinematics in the problem at hand.

References

1. US Department of Energy, 20% Wind Energy by 2030, Increasing WindEnergy�s Contribution to U.S. Electricity Supply, http://www.nrel.gov/docs/fy08osti/41869.pdf, accessed April 10, 2014

2. B. McNiff, W.D. Musial, and R. Errichello, Variations in Gear FatigueLife for Different Wind Turbine Braking Strategies, Solar EnergyResearch Institute, Golden, CO, 1990

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http://www.nrel.gov/docs/fy08osti/41869.pdf

http://www.nrel.gov/docs/fy08osti/41869.pdf

http://www.skf.com/group/knowledge-centre/aptitude-exchange/articles/index.html

http://www.skf.com/group/knowledge-centre/aptitude-exchange/articles/index.html

http://www.nrel.gov/wind/grc/pdfs/12_premature_bearing_failures.pdf

http://www.nrel.gov/wind/grc/pdfs/12_premature_bearing_failures.pdf

http://accelrys.com/products/datasheets/vamp.pdf

http://dx.doi.org/10.1177/1464420714544803

http://dx.doi.org/10.1177/1464420714544803

Wind-Turbine Gear-Box Roller-Bearing Premature-Failure ......Keywords gear-box premature failure,...

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