Abstract
To help overcome the problem of horizontalaxis windturbine (HAWT) gearbox rollerbearing prematurefailure, the root causes of this failure are currently being investigated using mainly laboratory and fieldtest experimental approaches. In the present work, an attempt is made to develop complementary computational methods and tools which can provide additional insight into the problem at hand (and do so with a substantially shorter turnaround time). Toward that end, a multiphysics computational framework has been developed which combines: (a) quantummechanical calculations of the grainboundary hydrogenembrittlement phenomenon and hydrogen bulk/grainboundary diffusion (the two phenomena currently believed to be the main contributors to the rollerbearing prematurefailure); (b) atomicscale kinetic Monte Carlobased calculations of the hydrogeninduced embrittling effect ahead of the advancing cracktip; and (c) a finiteelement analysis of the damage progression in, and the final failure of a prototypical HAWT gearbox rollerbearing inner raceway. Within this approach, the key quantities which must be calculated using each computational methodology are identified, as well as the quantities which must be exchanged between different computational analyses. The work demonstrates that the application of the present multiphysics computational framework enables prediction of the expected life of the most failureprone HAWT gearbox bearing elements.
Introduction
Windturbine gearbox failure remains one of the major problems to the windenergy industry (Ref 13). The combination of these high failure rates, long down times, and the high cost of gearbox repair has contributed to: (a) increased cost of wind energy; (b) increased sales price of windturbines due to higher warranty premiums; and (c) a higher cost of ownership due to the need for funds to cover repair after warranty expiration. To bring the windenergy cost down, durability and reliability of gearboxes have to be substantially improved. These goals are currently being pursued using mainly laboratory and fieldtest experimental approaches. While these empirical approaches are valuable in identifying shortcomings in the current design of the gearboxes and the main phenomena and processes responsible for the premature failure of windturbine gearboxes, advanced computational engineering methods, and tools can not only complement these approaches but also provide additional insight into the problem at hand (and do so in a relatively short time). The present work demonstrates the use of these methods and tools, and discusses the benefits offered by them within the context of horizontalaxis windturbine (HAWT) gearbox rollerbearing prematurefailure.
A photograph of an offshore wind turbine is provided in Fig. 1. All major components of the turbine are labeled for identification. Examination of this figure shows that the HAWT rotor (typically consisting of three blades) attached to a horizontal hub (which is connected to the rotor of the electrical generator, via a gearbox/drivetrain system, housed within the nacelle). The rotor/hub/nacelle assembly is placed on a tower, which is in turn anchored to the ocean floor (or ground, in the case of onshore HAWTs).
Turbine blades and the gearbox (or, in general, the windturbine drivetrain) are perhaps the most critical components/subsystems in the present designs of wind turbines. The present work deals only with the issues related to the performance, reliability, and modes of failure of gearbox components. In our recent work (Ref 4, 5), the problem of inadequate durability and premature failure of windturbine blades was investigated computationally.
The main function of the gearbox is to convert the low speed of the rotor shaft to the high speed of the electrical generator shaft. A labeled schematic of a prototypical wind turbine gearbox is shown in Fig. 2. The lowspeed stage of the gearbox is a planetary configuration with either spur (the present case, Fig. 2) or helical gears. In this configuration, the planetarygear carrier is driven by the windturbine rotor, the ring gear is stationary/reactionary, while the sun pinion shaft drives the intermediate gearbox stage, and, in turn, the highspeed stage (the one connected to the rotor of the electric generator). Predominantly, the gearbox failure initiation is observed in planet bearings, intermediateshaft bearings, and highspeedshaft bearings (each labeled in Fig. 2).
In principle, the predicted servicelife of properly maintained rollerbearings operating under elastohydrodynamic lubrication conditions, and foreseen loading/environmental conditions, is wellpredicted using the wellestablished bearingreliability assessment procedures (Ref 68). The basic assumptions used in these procedures are that the bearinglife is controlled by the material highcycle fatigue (typically within the bearing races/rings), commonly referred to as the rollerbearing contact fatigue (RCF) failure. The inservice cycling stresses arise from the repeated exposure of the ring material to ring/rollerelement nonconformal contact stresses. Under welllubricated/cleanlubricant conditions, RCF is typically initiated by the nucleation of subsurface cracks (in regions associated with critical combinations of the largest shear stress and the presence of highpotency microstructural defects). During subsequent repeated loading, cracks tend to advance toward the inner surfaces of the raceways, giving rise to the formation of spalls/fragments.
As mentioned earlier, rollerbearings in windturbine gearboxes tend to fail much earlier relative to their expected service life. In addition, the damage mechanism responsible for, and the appearance of rollerbearing prototypical prematurefailure seems different from the classic RCF failure. A detailed description of the visual appearance of the standard RCF failure is given in ISO 15243 (Ref 7), and hence will not be repeated here. Instead, a simple schematic of the subsurface region containing RCF dark and white bands as well as a single chevronshape crack is depicted in Fig. 3(a). In the case of the premature rollerbearing failure, the damaged region acquires a characteristic “White Etching Crack” appearance, and is initially localized at the contact surfaces or slightly beneath them. This appearance is related to the weak etching response of the associated local microstructure (mainly consisting of equiaxed submicronsize carbidefree ferrite grains) during the standard preparation of metallographic samples. In addition to the chevronshaped cracks like the one shown in Fig. 3(a), socalled butterfly whiteetching cracks are also often observed in the case of RCF failure. These cracks are formed at greater depths and are normally associated with excessive loading. In sharp contrast, whiteetch cracking in the case of premature rollerbearing failure, Fig. 3(b), is believed to be a surface or nearsurface phenomenon (Ref 9). Specifically, it is believed that a combination of disturbed bearing kinematics, unfavorable loading, and inadequate lubrication can lead to local surface or nearsurface tensilestress concentrations, at the root of surface asperities and/or at the nearsurface inclusion/matrix interfaces. If these stress concentrations and the number of loading cycles are sufficiently high, surface and/or subsurface cracks can nucleate. Due to proximity of the contact surfaces, subsurface cracks can readily extend to these surfaces (becoming surface cracks).
Cracks nucleated at the contact surfaces are subsequently infiltrated by the lubricant. The lubricant contains various (e.g., extreme pressure, EP, and antiwear, AW) additives and may be contaminated with water. Passage of the rolling elements over the damaged area can result in hydrodynamic effects, leading to crack spreading, and branching. Newly formed “cleanmetal” crack faces tend to readily chemically interact with the lubricant, causing the formation of a chemically altered fracturetoughnessinferior region at the crack tip. These changes, in turn, lead to a transition from a purely mechanicalfatiguecracking regime to a corrosionassisted fatiguecracking regime. The same chemical reactions tend to produce hydrogen, which diffuses into the surrounding cracktip region, primarily along the grain boundaries. This (embrittling) process reduces grainboundary cohesion and promotes intergranular cracking (Ref 9).
In our recent work (Ref 10, 11), the problem of gearbox failure caused by geartooth bendinginduced fatigue was investigated computationally. However, even when the ultimate failure of the gearbox is localized within the gears, the initial damage which led to this failure could be traced back to one or more gearbox rollerbearings.
In the present work, a multiphysics computational approach is proposed to address the problem of HAWT gearbox rollerbearing prematurefailure. While such a computational approach is not a substitute for the reengineerandfieldtest approach, it can provide complementary insight into the problem of windturbine gearbox failure and help gain insight into the nature of the main cause of this failure. In addition, computational engineering analyses enable investigation of the gearbox failure in a relatively short time, under a variety of windloading conditions comprising both the expected designload spectrum as well as the unexpected extreme loading conditions.
Considering the aforementioned potential benefits of the computational analysis, the main objective of the present work is to develop a multiphysics computational framework comprising: (a) quantummechanical analysis of grainboundary hydrogen embrittlement phenomenon and hydrogenatom jumps along the grainboundary and through the bulk; (b) atomisticlevel kinetic Monte Carlo analysis of hydrogen intergranular and bulk diffusion and determination of the effective grainboundary diffusivity; and (c) continuumtype analysis of surfacecrack intergranular spreading and branching assisted by hydrogenembrittlement and corrosion effects under normalwind loading conditions which may result in the formation of spalls/fragments within the inner race and, in turn, lead to the bearingelement failure. A flowchart of the multiphysics computational approach used in the present work is shown in Fig. 4.
A concise summary of the computational approach used in the investigation of windturbine gearbox rollerbearing prematurefailure and the associated hydrogeninduced grainboundary embrittlement is presented in section 2. The key results yielded by the present investigation are presented and discussed in section 3, while the main conclusions resulting from the present work are summarized in section 4.
QuantumMechanical Analysis of HydrogenInduced GrainBoundary Embrittlement
Premature failure of windturbine gearbox rollerbearings is taken to be a manifestation of a hydrogeninduced grainboundary embrittlement phenomenon. The local extent of hydrogeninduced grainboundary embrittlement is controlled by two phenomena/processes: (a) the extent of local hydrogen segregation to the grainboundary; and (b) the extent of grainboundary normal and tangential strength sensitivity to the local grainboundary hydrogen concentration. To quantify these phenomena/processes, a series of quantummechanical and atomisticlevel kinetic Monte Carlo analyses is conducted. The quantummechanical calculations, presented in this section, are used to: (a) assess the sensitivity of grainboundary strength to the local concentration of grainboundary hydrogen; and (b) to quantify key parameters such as the activation energy and the vibrational frequency associated with the hydrogen diffusion and required in the kinetic Monte Carlo hydrogendiffusion analysis. Kinetic Monte Carlo simulations, presented in the next section, are used to assess the rate at which hydrogen segregation from the cracktip/faces to the adjacent (connected to the crack tip) grainboundaries takes place.
Problem Formulation
The main problem analyzed in this portion of the work is the extent of grainboundary hydrogen embrittlement as a function of the degree of hydrogen segregation to the grainboundary and the structure of the grainboundary itself. All the calculations carried out in this portion of the work involve the use of the semiempirical quantum method AM1, as implemented in VAMP computer program (Ref 12).
Computational Modeling and Analysis
GrainBoundary (GB) Modeling
To construct GBs investigated in the present work, the standard procedure is used which involves: (a) rotation of the two crystals either about an axis residing in the GB plane (for tilt GBs) or an axis normal to the GB plane (for twist GBs); (b) sectioning of the two crystals parallel to the GB plane; and (c) joining the two crystals over the GB plane to form a bicrystal (the computational cell).
Semiempirical AM1 Method
For the reasons explained in our prior work (Ref 13), the rigorous ab initio quantum mechanical calculations were not used in the present work. Rather, the AM1 method (Ref 14), a computationally efficient semiempirical quantum mechanical method, is utilized. Within the AM1 method, diatomic differential overlap effects are neglected (i.e., the overlap matrix is replaced with the unit matrix) which enables simplification of the HartreeFock secular equation governing the quantummechanical behavior of the material system. The AM1 method was chosen because it appears to be the most suitable semiempirical quantummechanical method for the analysis of the hydrogen dissolution in BCC iron (used to represent the conventional bearing steel, within the quantum mechanical framework), since it is parameterized to reproduce experimental results pertaining to hydrogen dissolution energy.
A typical semiempirical quantummechanical calculation involves the following three steps: (a) construction of the material model; (b) application of the quantummechanical method; and (c) postprocessing analysis and evaluation of the relevant material quantity/quantities.
Results and Discussion
The results presented in Fig. 5 reveal the effect of hydrogen, under the condition of full GB saturation, on the GB normal strength. In this figure, the GB structure varied along the xaxis is represented by the volume of the corresponding interstitialsite Voronoi cells. The results displayed in Fig. 5 reveal that, under the condition of full saturation, the percent decrease in the GB normal strength due to the presence of hydrogen is not a sensitive function of the grainboundary type. On the other hand, as evidenced by the results also displayed in Fig. 5, the hydrogenfree GB normal strength is a fairly sensitive function of the GB structure. Specifically, as the GB structure becomes more open, as represented by larger Voronoi cells, the hydrogenfree GB normal strength decreases.
The driving force for hydrogen segregation to the GBs is the hydrogen solutionenergy. The lower is this energy, the higher is the tendency of hydrogen to segregate to the GBs. The quantummechanical results obtained in the present work pertaining to the effect of grainboundary structure on the hydrogen solutionenergy are depicted in Fig. 6(a). It is seen that as the GB structure becomes more open, the expected extent of hydrogen segregation to the GBs increases. In addition to depending on the GB structure, hydrogen solutionenergy is also a function of the distance from the GB. This is exemplified by the results displayed in Fig. 6(b) for the case of the \(\varSigma 3\left( {112} \right)\left[ {1\bar{1}0} \right]\) tilt GB. It is seen that as the distance from the GB increases, the hydrogen solutionenergy increases (i.e., the expected extent of hydrogen segregation decreases). However, it should be noted that the expected hydrogen segregation to the region adjacent to the grainboundary is still quite high considering the associated lower solutionenergy.
The quantummechanical calculations carried out in this section provide the critical data for the kinetic Monte Carlo hydrogen diffusion analysis. An example of such data is shown in Fig. 7. In this figure, variation in the computationalcell energy as a function of the hydrogendiffusion distance in the \(\left[ {001} \right]\) direction is shown. Two cases are considered: (a) diffusion through the bulk; and (b) diffusion along the \(\varSigma 3\left( {112} \right)\left[ {1\bar{1}0} \right]\) tilt GB. It should be noted that the systemenergy range in each case is a measure of the corresponding hydrogendiffusion activation energy. Examination of the results displayed in Fig. 7 reveals that the bulkdiffusion activation energy is only about 60% of its grainboundary counterpart. At room temperature, this difference in the diffusion activation energy translates into an approximately 1100fold increase in the diffusion rate through the bulk relative to that along the grainboundary. This finding reveals that, while hydrogen has a high tendency to diffuse to the grainboundaries, hydrogen residing on the grainboundary has quite low diffusivity along the grainboundary.
AtomicScale HydrogenDiffusion Analysis
The analysis carried out here utilizes the following results yielded by the quantummechanical calculations described in the previous section: (i) the number of interstitialsite types; (ii) the coordination of different interstitialsite types; (iii) variation in the system energy along different diffusion paths of the hydrogen atom from the given (occupied) interstitialsite; and (iv) activation energy associated with hydrogenatom jump from a given (occupied) interstitial site to a neighboring (unoccupied) interstitialsite. It should be noted that only a few selected results yielded by the quantummechanical calculations 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.
Problem Formulation
The problem analyzed here involves: (a) modeling of hydrogen diffusion from the cracktip wake region into the adjacent (connected to the crack tip) grainboundaries; and (b) determination of the effective hydrogen diffusion coefficient. All the calculations carried out utilized atomicscale kinetic Monte Carlo simulations. To monitor the progress of the hydrogendiffusion process, hydrogen tracer atoms are introduced (with identical properties as the remaining hydrogen atoms).
Computational Modeling and Analysis
GrainBoundary (GB) Modeling
Essentially the same geometrical models for the grain boundaries are used in the kinetic Monte Carlo analysis as in the previous quantummechanical analysis. FeFe, FeH, and HH interactions are represented using the embedded atom method interatomic potentials (Ref 15). An example of the computational domain used is depicted in Fig. 8(a). In this figure, iron atoms are colored cyan and hydrogen atoms residing on the grainboundary are colored white, while a single hydrogen tracer atom is colored yellow.
Kinetic Monte Carlo Method
The temporal evolution of the hydrogen traceratom position during the diffusion analysis is simulated using the version of the kinetic Monte Carlo method originally developed by Grujicic and Lai (Ref 16). Within this method, one atomic jump is allowed to take place from one (occupied) interstitial site to the neighboring (unoccupied) interstitial site during each time step. The occurrence of one such jump at one of the (occupied) sites is termed an event. At each time step, a list of all possible events is first constructed, and then the probability for each event is set proportional to the rate at which the associated hydrogenatom jumps take place. Next, the event to take place is selected randomly. This is accomplished by generating, at each time step, a random number αψ from a uniform distribution function in the (0, 1) range. The value of α is next used to select event m from M possible events in accordance with the procedure described in Ref 9. After an event has occurred, the total number of possible events M is updated and the aforementioned procedure is repeated. The kinetic Monte Carlo method employed in the present work uses a variable timestep to account for the fact that different events require different amounts of time to occur. At each simulation step the time increment is computed using the procedure described in Ref 9. According to this procedure, the maximum time increment allowed is controlled (dynamically and stochastically) by the fastest event(s). Consequently, significant reductions in the computational time are achieved relative to the fixed timestep kinetic Monte Carlo methods.
Results and Discussion
An example of the kinetic Monte Carlo results for the advancement of the hydrogen front from the cracktip along the adjoining grainboundary, showing the (yellow) tracer hydrogen atom: (a) originally in the wake of the cracktip; (b) and (c) diffusing through the bulk iron adjacent to the grainboundary; and (d) ultimately arriving at the hydrogenconcentration front along the grainboundary, causing this front to grow, is depicted in Fig. 8(a)(d). For brevity, other results obtained in this portion of the work will not be presented. Rather, the key findings can be summarized as:

(a)
Diffusion rate of hydrogen along the grainboundaries is quite small in comparison to its counterpart through the bulk, which is in complete agreement with the quantummechanical predictions;

(b)
Once a hydrogen atom has arrived at a grainboundary, it becomes effectively trapped with very little mobility both in the directions along and normal to the grainboundary;

(c)
Mass transport of hydrogen from the cracktip wake region along the adjoining grainboundaries is effectively controlled not by the grainboundary diffusion but rather by the diffusion of hydrogen through the adjoining bulk material and its deposition onto the advancing GBhydrogen front (Ref 1720);

(d)
While the hydrogen diffusion from the cracktip through the bulk to the advancing hydrogenfront is associated with a longer diffusion path, the effective hydrogen grainboundary diffusivity is still controlled by its bulk component. This is the result of two effects: (i) substantially higher bulk diffusivity of hydrogen compared to its intergranular counterpart; and (ii) the fact that the intergranular and transgranular diffusion paths are connected in parallel; and

(e)
Using the procedure described in Ref. 21, the effective (averaged) grainboundary diffusivity for hydrogen at room temperature has been found to be (1.7 ± 0.1) 10^{−11} m^{2}/s. This value will be used in section 4 to model the progress of hydrogen segregation to the grainboundaries ahead of the advancing cracktip.
RollerBearing InnerRace Damage/Failure Analysis
In this section, the problem of surfacecrack spreading and branching in the (inner) raceway is analyzed under rollingelement/raceway contactstresses.
Problem Formulation
The main problem addressed here is the determination of the average time it will take a surfacecrack to penetrate deep enough into the raceway, turn around and return to the surface, forming a (potentially failurecausing) spall. It should be noted that the process of nucleation of the surfacecrack is not analyzed. Rather, it is assumed that a combination of the unfavorable kinematic conditions within the rollingelement and excessive loading can cause such a crack to nucleate. The time for the surfacecrack to propagate as described, when combined with the average time for surfacecrack nucleation (not dealt with in the present work) could then serve as an estimate of the rollerbearing predicted life. All simulations in this portion of the work were conducted using ABAQUS/Explicit, a generalpurpose finiteelement program (Ref 22).
Computational Model and Analysis
ContactRegion FiniteElement Model
To account for the fact that: (a) the rollingelement/raceway contact patch is quite small in comparison to the characteristic dimensions of these components; and (b) the premature failure is frequently found to occur in inner races, the geometrical model analyzed consists of a twodimensional rectangular region residing fully within the inner race, Fig. 9(a). The rollerelement is not modeled explicitly but is rather replaced by the (moving) surfacedistribution of the contactstresses (marked schematically in Fig. 9b) of the type determined in our previous work (Ref 13). To recognize the fact that: (a) surfacecrack spreading and branching will be localized near the surface; and (b) crack growth is primarily of the intergranular character, the upper portion of the computational domain surrounding the previously nucleated crack was assigned a granular structure [using twodimensional Poissontype Voronoi cells (Ref 13)]. In this way, common edges of the adjacent Voronoi cells act as (twodimensional) grain boundaries. The remainder of the computational domain is treated as a featureless, twodimensional continuum.
Both the granular and the featureless portions of the computational domain are meshed using a combination of triangular and quadrilateral planestrain continuum finite elements. In addition, grain boundaries are meshed using fournode traction/separation cohesivezone elements. These elements enable modeling of the normal and/or sheartraction induced grainboundary decohesion (i.e., intergranular cracking). Furthermore, in order to model hydraulic loading of the partially or fully debonded grainboundaries, the cohesivezone elements are assigned additional degrees of freedom (i.e., the porepressure). A closeup of the meshed model used in this portion of the work is depicted in Fig. 9(c). It should be noted that, since each cohesivezone element initially involves two pairs of coincident nodes, they are not (visually) resolved in Fig. 9(c). The mesh typically contains 15,00020,000 continuum and 40005000 cohesivezone elements.
Analysis of Crack Spreading/Branching Using ABAQUS/Explicit
The finiteelement analysis (FEA) employed in this portion of the work requires specification of the following: (a) geometrical model; (b) meshed model; (c) computational algorithm; (d) initial conditions; (e) boundary conditions; (f) contact interactions; (g) material model(s); and (h) computational tool. Details regarding the geometrical and meshed models are presented above. Since the remaining items have been described in our prior work (Ref 13), they will not be repeated here. However, unique features of the FEA used in this portion of the work are presented in the remainder of this section. These include:

(a)
To represent tractionseparation constitutive relations for the cohesivezone elements under hydrogen/corrosionassisted intergranular cracking conditions, a user subroutine had to be created and linked with the ABAQUS solver. The essential feature of these relations is that as the grainboundary separation increases, the corresponding traction first increases, then reaches a peak and subsequently continues to decrease toward zero (indicating a complete loss of grainboundary cohesion). Details regarding the functional forms for these constitutive relations are presented in the Appendix. Since the parameters of these functional relationships (e.g., the peak traction, i.e., the cohesive strength) are affected by the extent of hydrogen segregation to the grainboundaries ahead of the advancing cracks, they were formulated using the results of: (i) a quantummechanical analysis revealing the sensitivity of the grainboundary cohesion strength to the presence of hydrogen (section 2); and (ii) an atomiclevel analysis of hydrogen diffusion from the cracktip wake into the surrounding grainboundaries (section 3);

(b)
A moving distributed normalloading (representing rollingelement/innerrace interactions) is applied over the top edge of the computational domain, in a periodic sense. That is, when a portion of the distributed loading leaves the top edge of the computational domain on one side (the right side, in Fig. 9b), an equivalent “missing portion” of the profile contacts the top edge at the other side (the left side, in Fig. 9b). Furthermore, to represent hydrodynamic loading exerted on the crack faces extending to the surface by the infiltrating and pressurized lubricant, a periodic porepressure loading function is applied to the “cracked” grain boundaries;

(c)
Intergranular fracture is modeled as the process of degradation and ultimate failure of the grainboundary cohesivezone elements. Once a cohesivezone finite element is completely degraded/failed, it is removed from the model. Thereafter, the faces of the corresponding elements of the adjacent grains are prevented from interpenetration by activating a penaltytype contact algorithm between them (Ref 2327); and

(d)
To account for the fact that the evolution portion of the rollerbearing prematurefailure process is stresscontrolled, the continuummaterial (AISI 4340 steel) model is assumed to be of a (isotropic) linearly elastic character.
Prediction of the BearingLife Controlled by Intergranular Cracking
Surfacecrack spreading and branching, and thus the associated bearinglife are modeled explicitly in this section. As mentioned above, the rate at which surfacecracks spread/branch and, thus, the bearinglife are controlled by the rate of hydrogensegregation to the grainboundaries ahead of the advancing cracks and the extent of grainboundary cohesivestrength sensitivity to the presence of hydrogen (grainboundary hydrogenembrittlement).
To include the effect of grainboundary hydrogenembrittlement into the present FEA, the following procedure should be implemented at each computational step:

(a)
since the hydrogen ingress is believed to be diffusion and not chemicalreactioncontrolled (i.e., chemical reactions between the freshly formed crack surfaces and lubricant additives are quite fast), a fixed hydrogen concentration is assumed to exist at the tip of each crack. This concentration corresponds to the condition of hydrogen chemicalpotential equality in the lubricant located in the wake of the cracktip and at the adjacent crackfaces. Then, using the information about the effective grainboundary diffusivity of hydrogen (as provided by the atomicscale analysis presented in section 3), the concentration profiles and the associated average concentrations of hydrogen along the (still bonded) grainboundaries connected to the cracktip are calculated. To help clarify the procedure implemented in this portion of the work, a simple schematic of a lubricantfilled and hydraulically loaded crack impinging on the two grainboundaries is depicted in Fig. 10;

(b)
the information obtained in (a), in conjunction with the sensitivity of the grainboundary strength to the presence of hydrogen (as provided by the quantummechanical analysis, section 2), is used to impart the hydrogeninduced embrittling effect to the grainboundaries in question. This was done by properly reducing the grainboundary normal/tangential cohesivestrengths.

(c)
loading induced by the passage of the rollerelements over the computational domain of the raceway is then used within the FEA to further (mechanically) degrade the grainboundary elements. Two components of this loading are included: (i) surfacetype loading associated with the passage of the contactpressure profile over the top edge of the computational domain; and (ii) hydraulic loading associated with the pressurization of the lubricant, residing within the surfaceoriginating crack(s), due to the passage and the pistonlike action of the rollerelement over the crack mouth; and

(d)
the progress of a spreadout and branched crackfront is monitored in order to identify the instant when one of the fronts arrives at the raceway free surface. This instant is then identified as the moment of failure of the rollerbearing.
The procedure described above should be applied at each computational step, for the most accurate determination of the initiation and evolution of damage within the computational domain. However, due to a very large number of loading cycles required to produce a spall, a “jumpincycles” approach was implemented. This procedure involves the following steps: (i) first, an integer value is assigned to the number of cycles to be skipped, i.e., to the cyclejump; (ii) the analysis is then performed over a number of consecutive timesteps required to complete one loading cycle, i.e., one passage of the Hertzian contactpressure over the top edge of the computational domain; (iii) then, the clock is advanced by the time equal to the product of the cyclejump and the total duration of the justcompleted loading cycle; (iv) the time obtained in (iii) is used to update the grainboundary hydrogen concentration, and the normal and shear cohesive strengths; and (v) the next loading cycle is applied, and the sequence of steps (i)(v) repeated.
Results and Discussion
In this section, the main results of the finiteelement structural analysis, as related to the windturbine gearbox rollerbearing prematurefailure, and of the associated hydrogeninduced grainboundary embrittlement, are presented and discussed. While the present computational framework enables the generation of results under various mechanical loading conditions and hydrogensource conditions in the crackfront wake, only a few prototypical results will be presented and discussed due to space limitations. In our future communication, a more detailed parametric study of the effect of various loading and environmental conditions on the rollerbearing servicelife will be presented.
As suggested earlier, prematurefailure of windturbine gearbox rollerbearings is generally assumed to be initiated by one of the surfacedistress processes. Typically, these processes result in the formation of surface cracks. In the present work, it is assumed that the grain subdomain contains, from the onset, two such surface cracks. Using the FEA, it is then investigated how the presence of these cracks and the absence/presence of hydrogeninduced grainboundary embrittlement affect the temporal evolution and spatial distribution of the contactregion stresses and the intergranular damage/failure.
All the results presented in the remainder of this section were obtained under the following loading and environmental conditions: (a) Hertzian peakpressure of 3 GPa; (b) contactpatch halfwidth b = 100 μm; (c) the intracrack lubricant hydraulicpressure of 375 MPa; and (d) cracktip hydrogen concentration of 4 × 10^{−3} at.%. For the loading conditions (a) and (b), the values chosen are consistent with the normal operating conditions of an intermediatespeedshaft (ISS) type of bearing in a prototypical 5 MW HAWT gearbox. The value for the loading condition (c) was obtained in a separate analysis in which the pistonlike action of the rollerelement onto the intracrack lubricant, with a pressuredependent bulk modulus, was investigated. In the absence of any experimental data, the environmental condition (d) is set equal to the nominal surface solubility limit of hydrogen in BCC iron.
Temporal Evolution/Spatial Distribution of ContactRegion Stresses
For clarity, the results corresponding to the absence and the presence of the grainboundary embrittling effects are presented and discussed separately.
In the Absence of GrainBoundary HydrogenEmbrittlement
Spatial distribution of the von Mises stress at four different times during the 25millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain, in the absence of grainboundary hydrogenembrittlement effects, is shown in Fig. 11(a)(d). The four figures correspond, respectively, to the cases when the Hertzian contactpressure profile: (a) has been fully applied to the leftmost top portion of the computational domain; (b) has advanced to the right and its leading half acts on the leftmost top portion of the granular subdomain; (c) has advanced further to the right and its trailing half acts on the rightmost top portion of the granular subdomain; and (d) has advanced still further so that it has been fully applied to the rightmost top portion of the computational domain. The corresponding spatial distribution of the von Mises stress during the 100millionth passage of the same Hertzian contactpressure profile is shown in Fig. 12(a)(d). In both sets of figures, the position of the Hertzian contactpressure profile on the top edge of the computational domain is indicated using a solid thick line.
Examination of the results displayed in Fig. 11(a), (d) and 12(a)(d) reveals that: (a) passage of the Hertzian contactpressure profile over the top side of the computational domain causes an increase in the von Mises stress in the region adjacent to the (moving) contactpatch; (b) while the von Mises stress field within the nongranular portion of the computational domain is quite smooth, the same field shows discontinuities in the granular subdomain. This finding is consistent with the presence of the grainboundary cohesivezone elements separating the adjacent grains; and (c) spatial distribution of the von Mises stress is not substantially different between the 25millionth and the 100millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain. As will be shown in section 4.3.2, this finding is the result of the fact that the repeated passage of the Hertzian contactpressure profile causes only relatively minor degradation of the grainboundaries connected to the initial cracks. It should be noted that in the present work, no attempt was made to model RCF failure. This failuremode is of a transgranular character and involves localplasticitybased shear localization. The analysis carried out in the present work is of a purely elastic character, although the normaltensile and shear stiffnesses of the grainboundary cohesivezone elements are allowed to degrade as a result of the combined effects of hydrogeninduced grainboundary embrittlement and excessive loading. Consequently, if one includes the effects of the local plasticity and shear localization in the FEA, more significant differences may be expected between the corresponding von Mises stress spatialdistribution results corresponding to the 25millionth and the 100millionth passage of the Hertzian contactpressure profile.
In the Presence of GrainBoundary HydrogenEmbrittlement
Spatial distribution of the von Mises stress at four different times during the 25millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain, in the presence of grainboundary hydrogenembrittlement effects, is shown in Fig. 13(a)(d). The four figures are associated with the same four positions of the Hertzian contactpressure profile as in Fig. 11(a)(d) and 12(a)(d). The corresponding spatial distribution of the von Mises stress during the 100millionth passage of the same Hertzian contactpressure profile is shown in Fig. 14(a)(d).
Examination of the results displayed in Fig. 13(a)(d) and 14(a)(d) and their comparison with the corresponding results shown in Fig. 11(a)(d) and 12(a)(d) reveals that:

(a)
as in the case where grainboundary hydrogenembrittlement effects were absent, passage of the Hertzian contactpressure profile over the top side of the computational domain causes an increase in the von Mises stress in the region adjacent to the (moving) contactpatch. However, the size of this region is somewhat increased in the case of hydrogeninduced grainboundary embrittlement (for example, please compare Fig. 11c and 13c);

(b)
spatial distribution of the von Mises stress is substantially different between the 25millionth and the 100millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain. Specifically, the region associated with the highest levels of the von Mises stress is significantly increased (for example, please compare the results displayed in Fig. 11d and 14d). As will be shown in section 4.3.2, this finding is the result of the fact that the repeated passage of the Hertzian contactpressure profile, in the presence of the grainboundary hydrogenembrittlement effects, causes significant degradation of the grainboundary structure, extending quite deep from the top edge of the granular portion of the computational domain; and

(c)
differences in the effect of the repeated passage of the Hertzian contactpressure profile on the spatial distribution of the von Mises stress, observed by comparing the results displayed in Fig. 11(a)(d) and 13(a)(d), with the results displayed in Fig. 12(a)(d) and 14(a)(d), respectively, are the manifestation of the more pronounced degradation of the material structural integrity due to the operation of hydrogeninduced grainboundary embrittling processes.
Temporal Evolution/Spatial Distribution of ContactRegion Damage
Again, for clarity, the results corresponding to the absence and the presence of the grainboundary embrittling effects are presented and discussed separately.
In the Absence of GrainBoundary HydrogenEmbrittlement
Spatial distribution of the grainboundary cohesivezone elements in which both the normal and the shear cohesivestrengths are degraded by at least 90% relative to their initial values, after the 25, 50, 75, and 100millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain, in the absence of grainboundary hydrogenembrittlement effects, is shown in Fig. 15(a)(d). Examination of these results reveals that:

(a)
the repeated passage of the Hertzian contactpressure first causes some branching of the two initially induced surface cracks;

(b)
after a sufficient number of Hertzian contactpressure passages, the newly formed cracks arrive to the top edge of the granular subdomain, yielding the formation of relatively thin spalls/flakes; and

(c)
subsequent loading of the computational domain by the passing Hertzian contactpressure profile does not produce any additional damage. It should again be recalled that in the present work, no attempt was made to model RCF failure. Consequently, under the present loading and environmental conditions, one cannot exclude the possibility that the material would suffer additional damage due to the interplay of the local plasticity and shear localization phenomena (Ref 13). However, modeling of these phenomena is beyond the scope of the present work.
In the Presence of GrainBoundary HydrogenEmbrittlement
Spatial distribution of the grainboundary cohesivezone elements in which both the normal and the shear cohesivestrengths are degraded by at least 90% relative to their initial values, due to a combined effect of the grainboundary hydrogenembrittlement and excessive loading after the 25, 50, 75, and 100millionth passage of the Hertzian contactpressure profile over the top edge of the computational domain, is shown in Fig. 16(a)(d). Examination of these results and their comparison with the corresponding results displayed in Fig. 15(a)(d) reveals that:

(a)
the repeated passage of the Hertzian contactpressure and the operation of the attendant grainboundary hydrogenembrittlement processes cause extensive and deep spreading and branching of the two initially induced surface cracks;

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

(c)
the size of this spall, Fig. 16(d), is at least an order of magnitude larger than the one observed in the nohydrogenembrittlement case, Fig. 15(d). This finding suggests that while in the absence of hydrogenembrittlement effects, surfacedistress can induce only very shallow craters into the raceways, affecting somewhat the rollerbearing performance, in the presence of the grainboundary embrittling effects, the raceways may experience a major damage, resulting in the formation of large and deep craters, and largesized fragments. In this case, the functionality of the rollerbearing element may be compromised, as well as the functionality and structural integrity of the adjacent gears (should any of the large fragments propagate to the nearby gearbox stage and get caught between the teeth of meshing gears).
Prediction of the RollerBearing ServiceLife
In this section, an attempt is made to estimate the servicelife of a HAWT gearbox planetarybearing element. Toward that end, one must determine not only the portion of the servicelife associated with the crackspreading/branching until the formation of a spall, but also the portion of the servicelife associated with the cracknucleation. Following the standard practice, it is assumed in the present work that events such as emergencyshutdown control the onset of cracking while normalwind loading controls the kinetics of crack spreading/branching.
The procedure for determination of the bearinglife in the crackspreading/branching regime described in section 4.2 yielded, for the aforementioned normal windloading conditions, the following results: number of cycles = 99 million; duration = 21 months.
To estimate the number of cycles until the nucleation of the surfacecracks, the procedure described in our prior work (Ref 10, 11) is utilized. This procedure recognizes the straincontrolled character of the fatiguecrack initiation process, and models this process by combining:

(a)
the conventional CoffinManson equation, Δɛ′_{p}/2 = ɛ′_{f}(2N _{ i })^{c}, where Δɛ _{p}′/2 is the equivalent plastic strain amplitude, ɛ′_{f} is the fatigue ductility coefficient, c is the fatigue ductility exponent, N _{ i } is the number of cycles required to reach a _{th}, and 2N _{ i } is the corresponding number of stress reversals; with

(b)
the additive decomposition of the total equivalent strain amplitude Δɛ′/2 into its elastic, Δɛ′_{e}/2, and plastic components;

(c)
the fatigue microyielding constitutive law, Δɛ′_{p}/2 = ɛ′_{f}(Δσ′/2σ′_{f})^{1/n′}, where Δσ′/2 is the equivalentstress amplitude, n′ is the cyclic strainhardening exponent, and σ′_{f} is the fatigue strength coefficient;

(d)
Hooke’s law, Δσ′ = E · Δɛ′_{e}, where E is the Young’s modulus; and

(e)
stressbased fatiguelife relation, Δσ′/2 = σ′_{FL} + σ′_{f}(2N _{ i })^{b}, where σ′_{FL} is the material fatigue/endurance limit and b is a material parameter.
This procedure yields the following equation:
Equation 1 can be solved iteratively to get the number of cycles to fatiguecrack initiation N _{ i } for a given combination of bearingmaterial and cyclic loading (as represented by Δσ′/2).
Equation 1 enables determination of N _{ i } under constantamplitude cyclicloading conditions. In the present investigation, surfacecrack initiation was assumed to be controlled by unfavorable kinematic and excessive loading conditions accompanying emergency shutdown. Under such circumstances, cyclic loading is not of a constant amplitude (and is also intermittent). To account for these effects, a procedure proposed in our prior work (Ref 4, 5) which combined the socalled rainflow cyclecounting algorithm, Goodman diagram and Miner’s Rule, was used. Furthermore, to account for the intermittency of the cyclic loading, a frequency of the emergencyshutdown events had to be assumed. A combination of Eq 1 and these procedures then yields N _{ i }, which can be readily converted into τ_{ i }, the bearinglife within the cracknucleation regime.
Application of the aforementioned procedure for the prediction of the bearingelement life before surfacecrack initiation yielded the following results: (a) normalwind loading—3.1 billion cycles, time = ca. 54 years; and (b) emergency shutdown (assuming a prototypical number of shutdown/restart cycles of 3000 per year and 10 torqueoscillation cycles accompanying each shutdown/restart cycle)—19 million cycles, time = ca. 4 months.
Making the assumption that events such as emergencyshutdown control the onset of cracking while normalwind 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 components, is a sensitive function of the normalwind loading and emergency shutdown conditions simulated, bearingmaterials used, as well as the chemistry and the additive content of the lubricant. The extent of this sensitivity will be investigated in our future work.
Summary and Conclusions
Based on the results obtained in the present work, the following main summary remarks and conclusions can be drawn:

1.
A new multiphysics computational framework has been developed to investigate windturbine gearbox rollerbearing prematurefailure. Within this framework, three distinct computational analyses are carried out and their results combined. The three analyses involve: (a) quantummechanical study; (b) atomicscale simulations; and (c) a finiteelement calculation.

2.
Within the quantummechanical analysis, the phenomenon of hydrogeninduced grainboundary embrittlement and hydrogenatom jumps along the grainboundary and through the bulk are investigated.

3.
Within a kinetic Monte Carlo atomicscale analysis, the phenomenon of hydrogen intergranular and bulk diffusion is investigated and the associated effective grainboundary diffusivity determined.

4.
Within the finiteelement continuumtype analysis, the processes associated with surfacecrack intergranular spreading and branching assisted by hydrogenembrittlement and corrosion effects under normalwind loading conditions which may result in the formation of spalls/fragments within the inner race and, in turn, lead to the bearingelement failure, are investigated.

5.
The present multiphysics computational framework enabled determination of the portion of the rollerbearing servicelife associated with the spreading and branching of surfacecracks until the formation of a spall. These results are combined with a strainbased lowcycle fatigue analysis, to determine the portion of the bearinglife before the nucleation of surfacecracks. The total predicted rollerbearing servicelife was found to be considerably shorter than its design servicelife of ca. 20 years.
Appendix: GrainBoundary CohesiveZone Potential
As mentioned earlier, the constitutive response of grain boundaries is modeled, in the present work, using the “cohesive zone framework” originally proposed by Needleman (Ref 28). The cohesivezone/grainboundary is assumed to have a negligible thickness when compared with other characteristic lengths of the problem, such as the grain size, contactpatch width, etc. The constitutive response of the cohesive zone is characterized by a tractiondisplacement relation, which is introduced through the definition of a grainboundary potential, ψ. The perfectly bonded grain boundary is assumed to be in a stable equilibrium, in which case the potential ψ has a minimum and all tractions vanish. For any other configuration, the value of the potential is taken to depend only on the displacement discontinuities (jumps) across the grain boundary.
For a twodimensional problem, as in the present case, the grainboundary displacement jump (i.e., the grainboundary separation) is expressed in terms of its normal component, U _{ n }, and a tangential component, U _{ t }, where both components lie in the xy plane of the Cartesian coordinate system. Differentiating the interface potential function \(\Psi = \hat{\Psi }\left( {U_{n} } \right.,\left. {U_{t} } \right)\) with respect to U _{ n } and U _{ t } yields, respectively, the normal and tangential components of F, the force per unit grainboundary area in the deformed configuration, as:
The grainboundary traction/separation constitutive relations are thus fully defined by specifying the form for the grainboundary potential function \(\hat{\Psi }(U_{n} ,U_{t} )\). The grainboundary potential of the following form initially proposed by Xu and Needleman (Ref 29) is used in the present study:
where the parameters ϕ_{ n } and ϕ_{ t } are the work of (pure) normal and (pure) shear separation, respectively, q = ϕ_{ t }/ϕ_{ n }, δ_{ n }, and δ_{ t } are the normal and tangential interface characteristiclengths, r = δ ^{*}_{ n } /δ_{ n }, δ ^{*}_{ n } (set to zero, in the present work) is the value of δ_{ n } after complete shear separation takes place under the condition of normal tension being zero. Differentiation of Eq A3 with respect to U _{ n } and U _{ t } yields the following expressions for the normal and tangential grainboundary tractions:
The works of normal and shear separations, ϕ_{ n } and ϕ_{ t }, are related to σ_{max} = max (F _{ n }) and τ_{max} = max (F _{ t }), respectively, as:
Graphical representations of the two functions defined by Eqs. A4 and A5 are given in Fig. A1(a) and (b), respectively. In these plots, the following normalization of the independent and dependent variables was used: U _{ n }/δ_{ n }, U _{ t }/δ_{ t }, F _{ n }/σ_{max}, F _{ t }/τ_{max}. Examination of Fig. A1(a) and (b) shows that: (a) at a given value of U _{ t }, F _{ n } peaks at U _{ n } = δ_{ n }; (b) at a given value of U _{ n }, F _{ t } peaks at \(U_{t} = \updelta_{t} /\sqrt 2\); (c) a nonzero value of U _{ t } reduces the normaltraction maximum value, max (F _{ n }); and (d) a nonzero value of U _{ n } reduces the tangentialtraction maximum value, max (F _{ t }).
An inspection of Eqs. (A3)(A7) shows that the grainboundary behavior is characterized by five independent parameters: σ_{max}, τ_{max}, δ_{ n }, δ_{ t }, and r.
The grainboundary decohesion potential presented above is next incorporated into a User Element Library (VUEL) subroutine of ABAQUS/Explicit. The VUEL subroutine allows the user to define the contribution of the grainboundary elements to the global finite element model. In other words, for the given nodal displacements of the cohesivezone grainboundary elements provided to the VUEL by ABAQUS, the contribution of the elements to the global vector of residual forces and to the global Jacobian (element stiffness matrix) is computed in the VUEL subroutine and passed back to ABAQUS/Explicit. The implementation of the grainboundary decohesion potential in the VUEL subroutine is discussed below.
For the twodimensional case analyzed here, each grainboundary element is defined as a fournode isoparametric element on the grainboundary, as shown schematically in Fig. A2. In the undeformed configuration (not shown for brevity), nodes 1 and 4 in Fig. A2, and nodes 2 and 3 coincide, respectively. A local coordinate system, consistent with the directions that are tangent, t, and normal, n, to the interface, is next assigned to each element. This is done by introducing two internal nodes, A and B, located at the midpoints of the lines 14 and 23, connecting the corresponding grainboundary nodes. The grainboundary displacements at the internal nodes A and B are expressed in terms of the displacements of the element nodes 14 as in the global coordinate system zr, as:
An isoparametric coordinate η is next introduced along the tangent direction with η(A) = −1 and η(B) = 1 and two linear Lagrangian interpolation functions are defined as N _{ A }(η) = (1 − η)/2 and N _{ B }(η) = (1 + η)/2. These interpolation functions allow the normal and the tangential components of the grainboundary displacements to be expressed in the form of their values at the internal nodes A and B as:
The tangential and normal components of the forces at nodes A and B, i.e., \(F_{t}^{A}\),\(F_{t}^{B}\), \(F_{n}^{A}\), and \(F_{n}^{B}\), which are work conjugates of the corresponding nodal displacements \(U_{t}^{A}\), \(U_{t}^{B}\), \(U_{n}^{A}\), and \(U_{n}^{B}\) are next determined through the application of the virtual work to the grainboundary element as:
where L is the AB element length. To determine the grainboundary (normal and tangential) tractions at the internal nodes A and B, the grainboundary potential is perturbed and the result expressed in terms of the perturbations of the grainboundary displacements at the internal nodes, \(U_{t}^{A}\), \(U_{t}^{B}\), \(U_{n}^{A}\), and \(U_{n}^{B}\) as:
By substituting Eq A15 into A14 and by choosing one of the \(\updelta U_{I}^{N} \left( {N = A,B;I = t,n} \right)\) perturbations at a time to be unity and the remaining perturbations to be zero, the corresponding \(F_{I}^{N}\) component of the nodal force can be expressed as:
Using a straightforward geometrical procedure and imposing the equilibrium condition, the corresponding residual nodal forces \(R_{r}^{i}\) and \(R_{z}^{i} (i = 1  4)\) in the global rz coordinate system, are defined as:
The components of the grainboundaryelement Jacobian are next defined as:
where the components of the internal Jacobian \(\partial F_{i}^{N} /\partial U_{j}^{M} (i,j = n,t;N,M = A,B)\) are calculated by differentiation of Eq A16.
To summarize, the residual nodal forces given by Eq A17 and the element Jacobian given by Eq A18 are computed in the VUEL subroutine, and passed to ABAQUS/Explicit for use in its global Newton scheme for accurate assessment of the kinematics in the problem at hand.
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Grujicic, M., Chenna, V., Galgalikar, R. et al. WindTurbine GearBox RollerBearing PrematureFailure Caused by GrainBoundary Hydrogen Embrittlement: A Multiphysics Computational Investigation. J. of Materi Eng and Perform 23, 3984–4001 (2014). https://doi.org/10.1007/s1166501411880
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DOI: https://doi.org/10.1007/s1166501411880
Keywords
 gearbox premature failure
 horizontalaxis windturbine
 multiphysics modeling