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A new efficient rolling element – Spall edge interaction model
Journal Pre-proofs A New Efficient Rolling Element – Spall Edge Interaction Model Dmitri Gazizulin, Lewis Rosado, Roni Schneck, Renata Klein, Jacob Bortman PII: DOI: Reference:
S0142-1123(19)30434-7 https://doi.org/10.1016/j.ijfatigue.2019.105330 JIJF 105330
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
1 August 2019 2 October 2019 9 October 2019
Please cite this article as: Gazizulin, D., Rosado, L., Schneck, R., Klein, R., Bortman, J., A New Efficient Rolling Element – Spall Edge Interaction Model, International Journal of Fatigue (2019), doi: https://doi.org/10.1016/ j.ijfatigue.2019.105330
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A New Efficient Rolling Element – Spall Edge Interaction Model Dmitri Gazizulin1, Dr. Lewis Rosado4, Prof. Roni Schneck2, Dr. Renata Klein3, and Prof. Jacob Bortman1 1PHM Laboratory, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel 2Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel 3R.K. Diagnostics, Gilon, P.O.B. 101, D.N. Misgav 20103, Israel 4Propulsion Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA Abstract This work presents a new method for a reliable modeling of a rolling element (RE)-spall edge interactions. The model aims to describe the material response within the spall edge as a result of repeated RE impacts. Two complementary RE-spall edge interaction models were developed and integrated, non-linear dynamic and finite element (FE). In order to demonstrate the advantages of the developed model, qualitative damage initiation simulations were conducted. The simulation results were validated and showed a good agreement with the published experimental results. To our best knowledge, this is the first physics-based attempt to simulate damage evolution within the spall edge. Keywords Rolling elements bearing, spall propagation, dynamic model, finite element
Corresponding author: [email protected] (Jacob Bortman)
1. Introduction Rolling element bearings (REB) are one of the basic components of rotating machines. They are considered critical components since a failure can lead to extended downtime and costly repairs and in aerospace applications, to safety issues and potential loss of life. One common failure mode of REBs is raceway spallation due to rolling contact fatigue. During service, the bearing raceways accumulate subsurface damage due to cyclic loading which eventually leads to crack initiation [1, 2]. These subsurface cracks propagate towards the surface resulting in spall formation. Conventional REB life models only address the lifetime of bearings to damage initiation, i.e., initial spall formation [3, 4, 5]. However, after the first defect formation, the bearing might still be fully operational for millions of cycles before the defect is detected or before catastrophic failure occurs. The spall growth mechanism of bearings was not thoroughly studied with only a few reports in the open literature [6, 7, 8]. Hence, the objective of the present study is to further investigate the spall propagation and to develop a suitable physics-based mathematical model that describes the spall growth or propagation characteristics of REBs. It has been postulated that the spall propagation in REBs is a three phase process [9, 10, 11]. After initial formation, the spall widens axially across the width of the raceway. Once the spall is wide enough, the rolling element (RE) descends into the spall. The descended RE is pushed by the cage and impacts the trailing edge of the spall which results in spall propagation in a circumferential direction of the raceway, Fig. Figure 1. There is a need to understand the damage mechanism during the propagation phase. Over the past two decades, studies attempting to understand the damage growth mechanism and to develop damage propagation models have been published [6, 8, 12, 13]. Nevertheless, the damage mechanism is only partially understood, the existing models are inefficient and the physical phenomena are not well represented.
2
Figure 1: Schematic bearing with spall, explaining the unloading of the ball in the spall and subsequent impact at the spall trailing edge [14].
Arakere et al. [7] and Branch et al. [15, 16] were among the first who tried to understand the damage mechanisms that cause spalls to propagate. Their work investigates the spall propagation process of thrust-loaded angular contact ball bearings thru contact and impact stress distribution analysis. Their main assumption is that the spall propagation process is driven by the impact forces and resulting stress fields generated in each RE i.e., ball impacts the trailing edge of the spall. The multiple impacts lead to crack formation within the trailing edge of the spall. The crack growth liberates material from the edge, which leads to spall propagation. Their assumption is supported by experimental observations from bearing endurance experiments that showed cracks near the spall trailing edge, Fig. Figure 2. It is well known that residual tensile stresses encourage crack formation and growth. Therefore, Arakere et al. [7] and Branch et al. [15, 16] used a dynamic 3D Finite Element (FE) elastic-plastic model to capture the critical stresses and strains states that develop in the vicinity of a spall edge as a result of RE-edge impact loading. Their simulation results showed that the location of the tensile residual hoop stresses at the spall edge surface agrees with the location of the cracks near the spall edge, Fig. Figure 2. Based on their experimental observations, Branch et al. [16] stated that the geometry of the spall edge remains roughly constant 3
during the spall propagation process, Fig. Figure 3(a). Also, the plastic strain contours from their simulation results appear consistent with the corresponding shape of representative material fragments as these detach from the raceway surface during spall growth, Fig. Figure 3. The plastic strain contour also closely matches the profilometer traces of the spall trailing edge as shown in Fig. Figure 3. This result can explain why the spall edge profile is unchanged during the spall propagation.
Figure 2: Cracks on the trailing edge of the spall [16].
4
(a)
(b)
(c) Figure 3: (a) Profilometer tracings of four different spall trailing edges [16], (b) schematic of the material fragment detachment from the raceway surface and (c) maximum principal strains contour within spall edge [15].
The main goal of the present work is to develop a simple, physics based simulation of RE-spall edge interaction. It is desired to be computationally efficient so it can be used in future research for more complex scenarios, e.g. damage evolution. Common dynamic models of the defected bearings assume spherical, cylindrical, or sharp spall edge shapes [1, 9, 12]. However, based on the experimental observations in [11, 16] an oblique surface appears to be more appropriate for the spall trailing edge representation. Based on experimental observations the range of the surface slopes, tan edge , is 0.8-2.5 (°°) [11, 16], Figs. Figure 3(a) and Figure 4. Hence we extend the dynamic model developed by Kogan et al. [17] to the case of an oblique spall edge. This extension should provide a more realistic representation of the RE-edge interaction.
5
Figure 4: Spall front edges with profile measurements [11].
In the present work a 2D FE model of the RE-spall edge interaction was developed to study the material behavior at the trailing edge of the spall during multiple RE impacts. It integrates the dynamic model results such as impact location, contact load, etc. as boundary and initial conditions. The integration of the two models, dynamic and FE, has the potential to significantly reduce the computational time and produce reliable results. These features of the model, efficiency and reliability, will be useful for more complex scenarios, e.g. damage or fatigue simulations. The damage driven mechanism of the spall propagation process is still not well understood. Thus, there is a need to develop a physics-based spall damage evolution model. Moreover, the damage evolution simulations can also be challenging in terms of the computational efficiency. The damage or fatigue simulations necessitate many of load cycles, which results in a very long run-time. Thus, the use of 3D non-linear dynamic, elasto-plastic, contact FE model is not feasible. In order to demonstrate the advantages of the proposed modeling approach, a damage evolution process was simulated and a qualitative analysis was conducted. For this purpose, a damage model based on Continuum Damage Mechanics (CDM) was utilized [18]. The simulation results showed a good agreement with the published experimental observations.
2. RE-spall edge interaction model A 2D model was built for estimating the stresses and strains imposed within the spall edge during its interaction with the RE. Figure Figure 5 illustrates the modeling approach. First, the operational condition of the 6
analyzed bearing must be determined, e.g. rotating speed, bearing dimensions, spall width, etc. Next, this information is used as an input to the non-linear dynamic model developed by Kogan et al. [17] with modifications for realistic case. The model yields the kinematics and the dynamic responses due to RE-spall edge interaction. Finally, the results of the dynamic model are used as an input of the spalled bearing FE model. This section includes the formulation of the dynamic model and its integration with the FE spalled bearing model.
Figure 5: A modeling approach for the evaluation of the stress and strain fields within the spall edge.
2.1. Non-linear dynamic model The goal of the dynamic model is to estimate the contact load between the RE and spall trailing edge during their interaction. Kogan et al. [17] assumed an infinitesimal small radius of the spall edge, i.e. sharp edge. However, this geometry is not representative of real spall edge and thus, not practical when building the FE model. The model was extended to the case of oblique spall edge. It is assumed that the highest level of the stresses/strains will be reached at the first RE-spall interaction. Thus, not only the contact load but also the first impact location is of interest and will be used later to define the contact point between the RE and spall edge in the FE model. The formulation of the first impact location is based on different geometrical parameters of the problem, which are schematically presented in Figs. Figure 6 and Figure 7.
7
Figure 6: Schematic of a spall trailing edge.
(a)
(b) Figure 7: Geometric parameters at the moment of the RE (a) disconnection from the raceways and (b) impact on the trailing edge.
X , Y which is parallel to the leading edge, the center location Rdis Using the Cartesian coordinates system of the RE and its velocity R dis at the moment of disconnection from the raceways, are expressed by
R dis RRE , 2 RRE . c Dp 2 ,1 R dis 2 RRE
8
(1)
where, is the initial deflection of the RE into the inner raceway, RRE is the RE radius, DP is the pitch diameter, and c is the constant cage speed. Details about the initial condition formulation, can be found in [17]. The impact location is given by
x , y R cos h, R R sin RE imp y RE imp imp h R imp x cos h d / RRE d
(2)
where d is the spall depth. R imp is the RE center at the moment of impact, Fig. Figure 7(b): 2 Dp 2 timp c Dc 2 timp g c R imp x RRE 2 2 RRE 2 c Dp timp R imp 2 RRE y 2
(3)
where timp is the time to impact (TTI). TTI or timp is the time between two events, the entrance of the RE into the spall and when it exits from the spall, Fig. Figure 8. Based on Eqs. (1)-(3) the spall width s can be expressed by s R imp RRE d sin y
(4)
Knowing the spall width, s , the TTI can be estimated, and vice versa. It is noteworthy to mention that the expression of the first impact location ximp , yimp is valid only when edge .
9
Figure 8: Definition of the of the time interval timp
A simplified estimation of the impact response as a result of the RE-spall edge interaction is presented next. The impact direction n (Fig. Figure 7(b)) is approximated by
nˆ n1, n2 cos ,sin
(5)
The normal contact force, Fn , between the trailing edge and the RE at the moment of impact is given by the elastic impact model [17, 19]
Fn k RRE no nRE
n
3 1 e 2 nRE no 1 4
mREnRE Fn mRE gn1 mono Fn mo gn1
(6)
(7)
where k is the contact stiffness between the RE and the outer raceway, e is the coefficient of restitution, no and
nRE are the ring and the RE displacements in the direction of n , respectively. mo and mRE are the massof the outer ring and the RE, respectively, and is the collision speed that is given by
R imp nˆ .
The values of those parameters, at the moment of impact, are given by
10
(8)
nRE RRE nRE no 0 n 0 o
(9)
.
Assuming a constant RE acceleration, the velocity of the RE at impact R imp is given by, . . R imp R dis Rtimp c Dp 2 . ,1 R dis 2 RRE g Dp 2 , 0 R c 2
where
R
is
the
acceleration
of
the
RE
center
(10)
at
the
moment
of
impact.
A more
detailed description of the model formulation can be found in [17]. 2.2. FE model The RE-spall interaction was simulated using a two-dimensional, plane strain, quasi-static model within ABAQUS/Standard. The ABAQUS/Standard implicit solver was chosen due to the following reasons [20]. The implicit solver must iterate to satisfy the stability of the model. We developed a relatively simple FE model, where only a few iterations are required to achieve a stable solution. Moreover, as long as the stability is satisfied there is no size limit on the step time icrement. Thus, large time increments can be used to reduce the total simulation time. When using an implicit solver a stable time increment must be satisfied. The stable time increment, tstable , is given by
tstable
Le min cd
(11)
where Le and cd are the characteristic element length and wave speed of the material, respectively. The stable time increment is dictated by the smallest element in the mesh. In the case of contact simulations with high local 11
stresses, a fine mesh is needed. Moreover, in case of damage simulations, e.g. crack formation, a fine mesh along the crack path must be applied. Using a fine mesh in the ABAQUS/Explicit could result in a large number of time increments, i.e. high computational cost and time. The ABAQUS/Explicit is particularly well-suited to simulate the dynamic event. However, the inertial effects and parameters, e.g. mass, velocity, acceleration, were taken into account in the non-dynamic model. The details above suggest that for the developed FE model, the ABAQUS/Standard is more efficient solver and can produce reliable results. The assembly of the FE model includes two instances: RE and spall. To improve computational efficiency, the spall part is only represented by the trailing edge. The schematic of the edge geometry and the FE mesh are shown in Fig. Figure 10. The edge material was assumed to be an elastic-plastic material with anisotropic strain hardening. The RE was modeled as a rigid surface with a radius RRE . The contact between the RE and spall edge was assumed to be frictionless. Since the stresses within the spall edge were expected to vary with high gradients, a very fine mesh, with CPE4R elements type, was used in the vicinity of the contact zone. The minimum element size at the vicinity of the contact region was determined based on a mesh convergence study. This study aimed to investigate the relationship between the element size, discretization error, analysis accuracy and the computation time. The convergence analysis based on error relative to the finest mesh, in total strain energy and the reaction force at the RE center, was carried out. Different meshes with varying element sizes, 1 10 m , in the vicinity of the contact zone were examined. From these convergence studies, it was found, that for all examined cases the discretization error was sufficiently small. Thus the minimum element size was determined based on the computation time, Fig. Figure 9. The mesh becomes gradually coarser as the distance from the contact zone increases, Fig. Figure 10(b). Symmetric boundary constraints (XSYM) were applied to the right and left vertical edges of the spall part. The bottom edge was fixed. The repeated impacts, i.e., cyclic load, was defined as an indentation of the RE into the spall edge. First, the indentation or the displacement of the RE center, nRE , is calculated using Eqs. (1)-(10). Then, it is applied and released gradually to the center of the RE part as a boundary condition. 12
Figure 9: Normalized computational time versus element size within the vicinity of the contact region.
(a)
(b)
Figure 10: (a) Schematic (not to scale) and (b) FE mesh of the spalled section. The dimensions were selected to diminish the effect of the RE-spall contact at the boundaries of the spall edge model.
2.3. Model validation To validate the proposed modeling approach, we compared it with the work published by Branch et al. [15, 16], and Arakere et al. [7]. In these works the RE-spall edge impact was studied using a simplified 3D FE model. The model was developed for the experiments carried out by Rosado et al. [10]. It simulates repeated impacts of a rigid ball on the elastic-plastic spall edge, Fig. Figure 11. Branch et al. [15] simulated three impacts of a rigid ball on the spall edge. After the third impact, no significant changes occurred in the stresses and strains fields within the spall edge. It is noteworthy that the 3D FE model does not take into account parameters such as defect size, and impact location. These parameters can lead to a different solution. For example, in the case of relatively large spall, the contact of the RE is not only with the spall edge but also with the spall bottom. Thus, the parameters 13
for the validation case were chosen based on the following criteria. During the impact, the RE in contact only with the pall edge and the maximum contact pressure is 7.8 GPa. These criteria meet impact conditions of the 3D FE model [15, 16].
Figure 11: 3D FE impact model introduced by Branch et al. [16].
Model parameters for the simulated case are summarized in Tables Table 1 and Table 2. Table Table 1 lists values of the various physical parameters which describe the experiments conducted by Rosado et al. [10]. These parameters were used as an input to the non-linear dynamic model described in subsection 2.1. The results of the dynamic model were used as input parameters for the FE model, Table Table 2. The stress-strain curve defining the elastic-plastic material response of the FE model is depicted in Fig. Figure 12. The load cycle of the spall edge was represented by the indentation of the RE into the spall edge. This process is repeated until a steadystate solution is achieved. The steady response was measured thru equivalent plastic deformation (PEEQ). If the change in the PEEQ was less than 5% for the next load cycle, then it is considered to have reached a steady-state solution. This condition was achieved after four load cycles, Fig. Figure 13.
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Table 1: Model parameters of the non-linear dynamic model.
15
Parameter
Value
Units
Pitch diameter
60
mm
Ball radius
6.4
mm
Inner raceway normalized curvature
0.52
-
Outer raceway normalized curvature
0.53
-
Contact stiffness exponent
1.5
-
Clearance diameter
1.5 10 2
mm
Number of balls
11
-
Modulus of elasticity for balls
310
GPa
Modulus of elasticity for rings
200
GPa
Poisson's ratio for balls and rings
0.3
-
Ball mass
2.8
g
Mass attached to the inner ring
166
kg
Mass attached to the outer ring
9.5
kg
Radial gravity
9.8
m/sec2
Inner ring rotation speed
166.67
rps
Stiffness of the contact between the ball and the inner ring
3.91 1010
N/m1.5
Stiffness of the contact between the ball and the outer ring
3.41 1010
N/m1.5
Spall edge slope
45
°
Time to impact
1 10 4
sec
Table 2: FE Model parameters for the simulated case.
Parameter
Value
Units
Impact location
1.3, 1.8 10
mm
Impact direction 90
80
°
Maximum deflection of the ball center
1.6 10 2
mm
Spall depth
0.125
mm
Spall width/size
2.64
mm
3
Figure 12 Stress-strain curve [15].
(a)
(b)
Figure 13: PEEQ contour plots after (a) first and (b) fourth load cycle.
The 3D model developed by Branch et al. [15] was meshed with a linear hexahedral mesh. The element size near the edge location was 10 m . The total CPU time of one impact was 3.5 105 sec [7]. In the proposed 2D 16
FE model, an element size near the spall edge was 4 m , and CPU time of one impact was 200 sec. During the present simulation, the contact pressure reaches a maximum value of 7.8 GPa. This result is similar to that of the 3D FE model introduced by Arakere et al. [7]. The results of the proposed model are shown in Fig. Figure 14. The results are in a good agreement with those obtained by Branch et al. [15], Fig. Figure 15. Table 3 summarizes the main differences between the two modeling approaches. Branch et al. [15] determined the impact velocity of the RE into the spall edge based on the kinematics of the inner raceway rotational speed and bearing geometry. The RE impact mass is assumed to be that of the ball alone [7]. The proposed model utilizes a non-linear dynamic model to estimate the dynamics and kinematics during the RE-spall edge interaction. Parameters such as initial load of the RE before disconnection from the raceways, spall length, impact location, etc. were taken into account. During the RE-spall edge interaction, a small contact area is formed. In the presented case the width of the contact between the RE and spall edge is 110 m . Thus, in order to achieve more reliable predictions of the stress distribution in the contact vicinity, a fine mesh is required. The proposed model allows increasing mesh refinement without a significant computational cost. Thus, the main difference between the two models, 3D FE model developed by Branch et al. [15] and the proposed 2D FE model, is that the latter has the potential to produce more reliableresults and to significantly reduce the computation time, Table 3.
(a)
(b)
(c)
Figure 14: The results of 2D FE model: (a) maximum principal plastic strain, (b) residual hoop (MPa) and (c) maximum principal stresses (MPa).
17
(a)
(b)
(c)
Figure 15: The results of 3D FE model obtained by Branch et al. [15]: (a) maximum principal plastic strain, (b) residual hoop (MPa) and (c) maximum principal stresses (MPa). Table 3: Main differences between the proposed model and the model developed by Branch et al. [15].
3D dynamic FE model [15]
2D quasi-static FE model
Dynamics and Kinematics
Simplified approximation
Non-linear dynamic model [17]
Element size - contact location
10 m
4 m
CPU time – One impact
3.5 105 sec
200 sec
3. Results and discussion 3.1. Material response analysis Branch et al. [15] chose to represent the spall edge in the FE model as a 45 degree inclined surface. They examined the contribution of different bearing properties, e.g ball density, raceway surface hardness, strength, etc., to the magnitude and extent of damage due to RE impact. The accumulated plastic strain and critical stresses were used as indicators of the damage severity. In the current work, we have extended the damage severity analysis by examining the effect of the spall edge geometry on the damage indicators. As mentioned above, based on published experimental observations the spall edge surface slope varies, Figs. Figure 3(a) and Figure 4. The spall edge surface slopes of 38o, 45o and 55o were studied. It is noteworthy to mention that the damage mechanism 18
governing the spall propagation is still not fully understood. Thus, we examined three damage indicators that can lead to the spall propagation, maximum principal and radial residual stresses, and plastic strain. During the loading stage, i.e. RE impact, the trailing edge of the spall is under compressive stress. In this stage, the likelihood of crack formation and propagation within the spall edge is low. Only upon the unloading stage, a portion of the spall edge turns into tension due to the plastic deformation underneath, Fig. Figure 14(c). The location of the tensile stress on the surface corresponds to the crack formation in front of the spall edge, in similarity with the experimental observation in Fig. Figure 2. Figure 16 shows the contour plots of the maximum principal residual stress of varying edge slopes. The observed change in the location and magnitude of the residual stress is insignificant. However, the surface area under the tensile stress increases as the slope becomes sharper. Thus, assuming that crack formation on the raceway surface is governed by the maximum principal residual stress, the sharp edges generate a more favorable condition for the evolution of cracks.
Figure 16: Contour plots of maximum principal residual stress (MPa) with varying edge slopes, from left to right: 38, 45, and 55 degrees.
Based on the published experimental observations, Figs. Figure 2 and Figure 4, it can be assumed that cracks initiate at the rolling surface. However, it is also possible that these cracks represent the final stage of the damage, which was initiated below the surface. Figure 17 the shows radial residual stress contour plots from the simulations of varying edge slopes. It shows the presence of tensile stresses beneath the spall edge surface. This stress can lead to generation of subsurface cracks parallel to the surface. Moreover, in the case of moderate edges 19
slope, the subsurface tensile area increases, generating a more favorable condition for the formation of subsurface cracks.
Figure 17: Contour plots of radial residual stress (MPa) with varying edge slopes, from left to right: 38, 45, and 55 degrees.
Branch et al. [15] suggested that the plastic strain could represent damage severity due to multiple ball-trailing edge impacts. Figure 18 shows the contour plots of maximum plastic strain from the simulations of varying edge slopes. The 55 degrees slope edge accumulated the highest plasticity (red area). The results suggest that more material deforms plastically (red area in the PEEQ contour plot) when the edge is sharp, i.e. large slopes. Assuming that the accumulated plasticity provides a major contribution to the damage magnitude and rate,, the result suggests that the spall will propagate more rapidly when encountering sharp spall edges. Moreover, according to Branch et al. [15], the plastic strain contour was proposed to correspond to the shape of a representative material fragment. Figure 18 implies that the spall edge profile will remain constant in similarity with Figs. 3(a) and 4.
20
Figure 18: Contour plots of the maximum principal plastic strain with varying edge slopes, from left to right: 38, 45, and 55 degrees. The potential material fragment can be represented by a plastic strain zone.
3.2. Damage evolution – Qualitative analysis In order to demonstrate the advantages of the proposed modeling approach, a qualitative damage initiation analysis was carried out. In the current analysis, we used a model that has been implemented in the author's previous work on the subject of damage initiation in a rotating element bearing [2]. The damage evolution was represented by CDM based model. The damage model provides a representation of failure mechanism by the definition of a nondimensional damage variable, D . It is assumed that the damage variable D affects the elastic modulus E of the material as
E 1 D E
(12)
where E is the damaged elastic modulus. The presence of the damage reduces the material stiffness [18]. The damage variable D has values ranging from 0, which represents undamaged material, to Dmax , which represents a completely damaged material:
0 D Dmax
(13)
When damage at some point reaches its critical value, Dmax , it corresponds to the initiation of a macro-crack. The damage rate evolution model is given by 21
dD eff dN r 1 D
m
(14)
where N is the number of stress cycles and eff is the effective stress that causing the damage, r and m are material-dependent parameters. More detailed description of the damage model algorithm and its application can be found in Gazizulin et al. [2]. The existent experimental observations of the spall edge are very limited. Only cracks on the surface of the spall edge are visible. Thus, qualitative damage simulations were conducted under the assumption that cracks initiate at the surface of the edge. The effective stress was chosen to be the maximum principal residual stress,
eff Max.Princ. . In this analysis, we have assumed that the maximum principal residual stress is the only damage driven mechanism. In future research, the contribution of the plasticity and radial stresses to the damage mechanisms should be also examined. It is noteworthy to mention that the material properties within the spall edge are unknown [15]. These properties might change during the bearing operation and RE impacts. Thus, the values of material-dependent parameters were taken arbitrarily to be similar to those used in Gazizulin et al., [2]. Three different cases with hundreds of load cycles were simulated within a reasonable time-frame. The simulation results show that the proposed damage model and criterion, predict well the location and direction of the initial damage evolution stage. The damage simulation results are presented in Figs. Figure 19 and Figure 20. The cracks started to evolve on the surface. Their location corresponded to the surface area under tensile stress, Fig. Figure 16. After the surface cracks evolved, they propagated downward into the edge. Upon reaching the subsurface area under compression, their propagation ceased, Fig. Figure 20. Figure 19 shows cracks, which evolved after hundreds of load cycles. In all simulated cases, the location of the cracks corresponds to those observed in the published experimental observations [16], Fig. Figure 2.
22
Figure 19: Cracks generated within the spall edge, from top to bottom: 38, 45, and 55 degrees.
Figure 20: Contour plots of maximum principal residual stress at the end of damage simulations, from left to right: 38, 45, and 55 degrees.
4. Conclusions We proposes a new RE-spall edge interaction model. It combines non-linear dynamic and FE models which were developed based on experimental observations and physical insights. The dynamic and kinematic responses of the interaction were estimated using non-linear dynamic model [17]. The results of the dynamic model were used as load parameters and boundary conditions for the FE model. Next, the material response, stress/strain 23
fields, due to the interactions were estimated using the FE model. This modeling approach provides, both computational efficiency as well as reliable results. The results of the proposed model were validated with the 3D dynamic FE model developed by Arakere et al. [7]. Several cases of the RE-spall edge interactions were simulated and potential damage driven mechanisms were analyzed. Since the damage mechanism is not fully understood, several potential damage indicators were examined. The simulation results showed that the location of the maximum principal residual stress at the spall edge surface agrees with the location of the cracks near the spall edge. The maximum principal residual stress spreads over a larger area as the edge becomes sharper, i.e increased probability of crack generation. In a case where cracks are generated beneath the surface, the simulation results showed that the radial residual stress might be responsible for sub-surface crack generation. Unlike the maximum principal residual stress, the radial residual stress field decreases as the edge becomes sharper. In the future, for a better understanding of the damage mechanism, an extensive bearing endurance tests and metallurgical analysis of the spalled bearings should be conducted. Based on the conclusions from this analysis qualitative damage simulations were carried out. One goal of the damage analysis was to demonstrate the advantages of a fast and reliable model. The preliminary results of the damage simulations showed a good agreement with published experimental observations. This work provides new insights regarding the damage evolution within the spall edge as a result of impacts with RE. 5. Acknowledgments The authors would like to express their deepest appreciations to the Pearlstone Center for their support andfunding of this work.
24
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26
V. ABAQUS, “6.14 documentation,” Dassault Systemes Simulia Corporation, 2014.
Highlights
Non-linear dynamic model with a realistic representation of the spall edge. Rolling element-spall edge interaction Finite Element model: simple and efficient. Physics-based damage driven mechanism of the spall propagation process. Damage evolution simulations within the spall edge.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0142-1123(19)30434-7 https://doi.org/10.1016/j.ijfatigue.2019.105330 JIJF 105330
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
1 August 2019 2 October 2019 9 October 2019
Please cite this article as: Gazizulin, D., Rosado, L., Schneck, R., Klein, R., Bortman, J., A New Efficient Rolling Element – Spall Edge Interaction Model, International Journal of Fatigue (2019), doi: https://doi.org/10.1016/ j.ijfatigue.2019.105330
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A New Efficient Rolling Element – Spall Edge Interaction Model Dmitri Gazizulin1, Dr. Lewis Rosado4, Prof. Roni Schneck2, Dr. Renata Klein3, and Prof. Jacob Bortman1 1PHM Laboratory, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel 2Department of Materials Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel 3R.K. Diagnostics, Gilon, P.O.B. 101, D.N. Misgav 20103, Israel 4Propulsion Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA Abstract This work presents a new method for a reliable modeling of a rolling element (RE)-spall edge interactions. The model aims to describe the material response within the spall edge as a result of repeated RE impacts. Two complementary RE-spall edge interaction models were developed and integrated, non-linear dynamic and finite element (FE). In order to demonstrate the advantages of the developed model, qualitative damage initiation simulations were conducted. The simulation results were validated and showed a good agreement with the published experimental results. To our best knowledge, this is the first physics-based attempt to simulate damage evolution within the spall edge. Keywords Rolling elements bearing, spall propagation, dynamic model, finite element
Corresponding author: [email protected] (Jacob Bortman)
1. Introduction Rolling element bearings (REB) are one of the basic components of rotating machines. They are considered critical components since a failure can lead to extended downtime and costly repairs and in aerospace applications, to safety issues and potential loss of life. One common failure mode of REBs is raceway spallation due to rolling contact fatigue. During service, the bearing raceways accumulate subsurface damage due to cyclic loading which eventually leads to crack initiation [1, 2]. These subsurface cracks propagate towards the surface resulting in spall formation. Conventional REB life models only address the lifetime of bearings to damage initiation, i.e., initial spall formation [3, 4, 5]. However, after the first defect formation, the bearing might still be fully operational for millions of cycles before the defect is detected or before catastrophic failure occurs. The spall growth mechanism of bearings was not thoroughly studied with only a few reports in the open literature [6, 7, 8]. Hence, the objective of the present study is to further investigate the spall propagation and to develop a suitable physics-based mathematical model that describes the spall growth or propagation characteristics of REBs. It has been postulated that the spall propagation in REBs is a three phase process [9, 10, 11]. After initial formation, the spall widens axially across the width of the raceway. Once the spall is wide enough, the rolling element (RE) descends into the spall. The descended RE is pushed by the cage and impacts the trailing edge of the spall which results in spall propagation in a circumferential direction of the raceway, Fig. Figure 1. There is a need to understand the damage mechanism during the propagation phase. Over the past two decades, studies attempting to understand the damage growth mechanism and to develop damage propagation models have been published [6, 8, 12, 13]. Nevertheless, the damage mechanism is only partially understood, the existing models are inefficient and the physical phenomena are not well represented.
2
Figure 1: Schematic bearing with spall, explaining the unloading of the ball in the spall and subsequent impact at the spall trailing edge [14].
Arakere et al. [7] and Branch et al. [15, 16] were among the first who tried to understand the damage mechanisms that cause spalls to propagate. Their work investigates the spall propagation process of thrust-loaded angular contact ball bearings thru contact and impact stress distribution analysis. Their main assumption is that the spall propagation process is driven by the impact forces and resulting stress fields generated in each RE i.e., ball impacts the trailing edge of the spall. The multiple impacts lead to crack formation within the trailing edge of the spall. The crack growth liberates material from the edge, which leads to spall propagation. Their assumption is supported by experimental observations from bearing endurance experiments that showed cracks near the spall trailing edge, Fig. Figure 2. It is well known that residual tensile stresses encourage crack formation and growth. Therefore, Arakere et al. [7] and Branch et al. [15, 16] used a dynamic 3D Finite Element (FE) elastic-plastic model to capture the critical stresses and strains states that develop in the vicinity of a spall edge as a result of RE-edge impact loading. Their simulation results showed that the location of the tensile residual hoop stresses at the spall edge surface agrees with the location of the cracks near the spall edge, Fig. Figure 2. Based on their experimental observations, Branch et al. [16] stated that the geometry of the spall edge remains roughly constant 3
during the spall propagation process, Fig. Figure 3(a). Also, the plastic strain contours from their simulation results appear consistent with the corresponding shape of representative material fragments as these detach from the raceway surface during spall growth, Fig. Figure 3. The plastic strain contour also closely matches the profilometer traces of the spall trailing edge as shown in Fig. Figure 3. This result can explain why the spall edge profile is unchanged during the spall propagation.
Figure 2: Cracks on the trailing edge of the spall [16].
4
(a)
(b)
(c) Figure 3: (a) Profilometer tracings of four different spall trailing edges [16], (b) schematic of the material fragment detachment from the raceway surface and (c) maximum principal strains contour within spall edge [15].
The main goal of the present work is to develop a simple, physics based simulation of RE-spall edge interaction. It is desired to be computationally efficient so it can be used in future research for more complex scenarios, e.g. damage evolution. Common dynamic models of the defected bearings assume spherical, cylindrical, or sharp spall edge shapes [1, 9, 12]. However, based on the experimental observations in [11, 16] an oblique surface appears to be more appropriate for the spall trailing edge representation. Based on experimental observations the range of the surface slopes, tan edge , is 0.8-2.5 (°°) [11, 16], Figs. Figure 3(a) and Figure 4. Hence we extend the dynamic model developed by Kogan et al. [17] to the case of an oblique spall edge. This extension should provide a more realistic representation of the RE-edge interaction.
5
Figure 4: Spall front edges with profile measurements [11].
In the present work a 2D FE model of the RE-spall edge interaction was developed to study the material behavior at the trailing edge of the spall during multiple RE impacts. It integrates the dynamic model results such as impact location, contact load, etc. as boundary and initial conditions. The integration of the two models, dynamic and FE, has the potential to significantly reduce the computational time and produce reliable results. These features of the model, efficiency and reliability, will be useful for more complex scenarios, e.g. damage or fatigue simulations. The damage driven mechanism of the spall propagation process is still not well understood. Thus, there is a need to develop a physics-based spall damage evolution model. Moreover, the damage evolution simulations can also be challenging in terms of the computational efficiency. The damage or fatigue simulations necessitate many of load cycles, which results in a very long run-time. Thus, the use of 3D non-linear dynamic, elasto-plastic, contact FE model is not feasible. In order to demonstrate the advantages of the proposed modeling approach, a damage evolution process was simulated and a qualitative analysis was conducted. For this purpose, a damage model based on Continuum Damage Mechanics (CDM) was utilized [18]. The simulation results showed a good agreement with the published experimental observations.
2. RE-spall edge interaction model A 2D model was built for estimating the stresses and strains imposed within the spall edge during its interaction with the RE. Figure Figure 5 illustrates the modeling approach. First, the operational condition of the 6
analyzed bearing must be determined, e.g. rotating speed, bearing dimensions, spall width, etc. Next, this information is used as an input to the non-linear dynamic model developed by Kogan et al. [17] with modifications for realistic case. The model yields the kinematics and the dynamic responses due to RE-spall edge interaction. Finally, the results of the dynamic model are used as an input of the spalled bearing FE model. This section includes the formulation of the dynamic model and its integration with the FE spalled bearing model.
Figure 5: A modeling approach for the evaluation of the stress and strain fields within the spall edge.
2.1. Non-linear dynamic model The goal of the dynamic model is to estimate the contact load between the RE and spall trailing edge during their interaction. Kogan et al. [17] assumed an infinitesimal small radius of the spall edge, i.e. sharp edge. However, this geometry is not representative of real spall edge and thus, not practical when building the FE model. The model was extended to the case of oblique spall edge. It is assumed that the highest level of the stresses/strains will be reached at the first RE-spall interaction. Thus, not only the contact load but also the first impact location is of interest and will be used later to define the contact point between the RE and spall edge in the FE model. The formulation of the first impact location is based on different geometrical parameters of the problem, which are schematically presented in Figs. Figure 6 and Figure 7.
7
Figure 6: Schematic of a spall trailing edge.
(a)
(b) Figure 7: Geometric parameters at the moment of the RE (a) disconnection from the raceways and (b) impact on the trailing edge.
X , Y which is parallel to the leading edge, the center location Rdis Using the Cartesian coordinates system of the RE and its velocity R dis at the moment of disconnection from the raceways, are expressed by
R dis RRE , 2 RRE . c Dp 2 ,1 R dis 2 RRE
8
(1)
where, is the initial deflection of the RE into the inner raceway, RRE is the RE radius, DP is the pitch diameter, and c is the constant cage speed. Details about the initial condition formulation, can be found in [17]. The impact location is given by
x , y R cos h, R R sin RE imp y RE imp imp h R imp x cos h d / RRE d
(2)
where d is the spall depth. R imp is the RE center at the moment of impact, Fig. Figure 7(b): 2 Dp 2 timp c Dc 2 timp g c R imp x RRE 2 2 RRE 2 c Dp timp R imp 2 RRE y 2
(3)
where timp is the time to impact (TTI). TTI or timp is the time between two events, the entrance of the RE into the spall and when it exits from the spall, Fig. Figure 8. Based on Eqs. (1)-(3) the spall width s can be expressed by s R imp RRE d sin y
(4)
Knowing the spall width, s , the TTI can be estimated, and vice versa. It is noteworthy to mention that the expression of the first impact location ximp , yimp is valid only when edge .
9
Figure 8: Definition of the of the time interval timp
A simplified estimation of the impact response as a result of the RE-spall edge interaction is presented next. The impact direction n (Fig. Figure 7(b)) is approximated by
nˆ n1, n2 cos ,sin
(5)
The normal contact force, Fn , between the trailing edge and the RE at the moment of impact is given by the elastic impact model [17, 19]
Fn k RRE no nRE
n
3 1 e 2 nRE no 1 4
mREnRE Fn mRE gn1 mono Fn mo gn1
(6)
(7)
where k is the contact stiffness between the RE and the outer raceway, e is the coefficient of restitution, no and
nRE are the ring and the RE displacements in the direction of n , respectively. mo and mRE are the massof the outer ring and the RE, respectively, and is the collision speed that is given by
R imp nˆ .
The values of those parameters, at the moment of impact, are given by
10
(8)
nRE RRE nRE no 0 n 0 o
(9)
.
Assuming a constant RE acceleration, the velocity of the RE at impact R imp is given by, . . R imp R dis Rtimp c Dp 2 . ,1 R dis 2 RRE g Dp 2 , 0 R c 2
where
R
is
the
acceleration
of
the
RE
center
(10)
at
the
moment
of
impact.
A more
detailed description of the model formulation can be found in [17]. 2.2. FE model The RE-spall interaction was simulated using a two-dimensional, plane strain, quasi-static model within ABAQUS/Standard. The ABAQUS/Standard implicit solver was chosen due to the following reasons [20]. The implicit solver must iterate to satisfy the stability of the model. We developed a relatively simple FE model, where only a few iterations are required to achieve a stable solution. Moreover, as long as the stability is satisfied there is no size limit on the step time icrement. Thus, large time increments can be used to reduce the total simulation time. When using an implicit solver a stable time increment must be satisfied. The stable time increment, tstable , is given by
tstable
Le min cd
(11)
where Le and cd are the characteristic element length and wave speed of the material, respectively. The stable time increment is dictated by the smallest element in the mesh. In the case of contact simulations with high local 11
stresses, a fine mesh is needed. Moreover, in case of damage simulations, e.g. crack formation, a fine mesh along the crack path must be applied. Using a fine mesh in the ABAQUS/Explicit could result in a large number of time increments, i.e. high computational cost and time. The ABAQUS/Explicit is particularly well-suited to simulate the dynamic event. However, the inertial effects and parameters, e.g. mass, velocity, acceleration, were taken into account in the non-dynamic model. The details above suggest that for the developed FE model, the ABAQUS/Standard is more efficient solver and can produce reliable results. The assembly of the FE model includes two instances: RE and spall. To improve computational efficiency, the spall part is only represented by the trailing edge. The schematic of the edge geometry and the FE mesh are shown in Fig. Figure 10. The edge material was assumed to be an elastic-plastic material with anisotropic strain hardening. The RE was modeled as a rigid surface with a radius RRE . The contact between the RE and spall edge was assumed to be frictionless. Since the stresses within the spall edge were expected to vary with high gradients, a very fine mesh, with CPE4R elements type, was used in the vicinity of the contact zone. The minimum element size at the vicinity of the contact region was determined based on a mesh convergence study. This study aimed to investigate the relationship between the element size, discretization error, analysis accuracy and the computation time. The convergence analysis based on error relative to the finest mesh, in total strain energy and the reaction force at the RE center, was carried out. Different meshes with varying element sizes, 1 10 m , in the vicinity of the contact zone were examined. From these convergence studies, it was found, that for all examined cases the discretization error was sufficiently small. Thus the minimum element size was determined based on the computation time, Fig. Figure 9. The mesh becomes gradually coarser as the distance from the contact zone increases, Fig. Figure 10(b). Symmetric boundary constraints (XSYM) were applied to the right and left vertical edges of the spall part. The bottom edge was fixed. The repeated impacts, i.e., cyclic load, was defined as an indentation of the RE into the spall edge. First, the indentation or the displacement of the RE center, nRE , is calculated using Eqs. (1)-(10). Then, it is applied and released gradually to the center of the RE part as a boundary condition. 12
Figure 9: Normalized computational time versus element size within the vicinity of the contact region.
(a)
(b)
Figure 10: (a) Schematic (not to scale) and (b) FE mesh of the spalled section. The dimensions were selected to diminish the effect of the RE-spall contact at the boundaries of the spall edge model.
2.3. Model validation To validate the proposed modeling approach, we compared it with the work published by Branch et al. [15, 16], and Arakere et al. [7]. In these works the RE-spall edge impact was studied using a simplified 3D FE model. The model was developed for the experiments carried out by Rosado et al. [10]. It simulates repeated impacts of a rigid ball on the elastic-plastic spall edge, Fig. Figure 11. Branch et al. [15] simulated three impacts of a rigid ball on the spall edge. After the third impact, no significant changes occurred in the stresses and strains fields within the spall edge. It is noteworthy that the 3D FE model does not take into account parameters such as defect size, and impact location. These parameters can lead to a different solution. For example, in the case of relatively large spall, the contact of the RE is not only with the spall edge but also with the spall bottom. Thus, the parameters 13
for the validation case were chosen based on the following criteria. During the impact, the RE in contact only with the pall edge and the maximum contact pressure is 7.8 GPa. These criteria meet impact conditions of the 3D FE model [15, 16].
Figure 11: 3D FE impact model introduced by Branch et al. [16].
Model parameters for the simulated case are summarized in Tables Table 1 and Table 2. Table Table 1 lists values of the various physical parameters which describe the experiments conducted by Rosado et al. [10]. These parameters were used as an input to the non-linear dynamic model described in subsection 2.1. The results of the dynamic model were used as input parameters for the FE model, Table Table 2. The stress-strain curve defining the elastic-plastic material response of the FE model is depicted in Fig. Figure 12. The load cycle of the spall edge was represented by the indentation of the RE into the spall edge. This process is repeated until a steadystate solution is achieved. The steady response was measured thru equivalent plastic deformation (PEEQ). If the change in the PEEQ was less than 5% for the next load cycle, then it is considered to have reached a steady-state solution. This condition was achieved after four load cycles, Fig. Figure 13.
14
Table 1: Model parameters of the non-linear dynamic model.
15
Parameter
Value
Units
Pitch diameter
60
mm
Ball radius
6.4
mm
Inner raceway normalized curvature
0.52
-
Outer raceway normalized curvature
0.53
-
Contact stiffness exponent
1.5
-
Clearance diameter
1.5 10 2
mm
Number of balls
11
-
Modulus of elasticity for balls
310
GPa
Modulus of elasticity for rings
200
GPa
Poisson's ratio for balls and rings
0.3
-
Ball mass
2.8
g
Mass attached to the inner ring
166
kg
Mass attached to the outer ring
9.5
kg
Radial gravity
9.8
m/sec2
Inner ring rotation speed
166.67
rps
Stiffness of the contact between the ball and the inner ring
3.91 1010
N/m1.5
Stiffness of the contact between the ball and the outer ring
3.41 1010
N/m1.5
Spall edge slope
45
°
Time to impact
1 10 4
sec
Table 2: FE Model parameters for the simulated case.
Parameter
Value
Units
Impact location
1.3, 1.8 10
mm
Impact direction 90
80
°
Maximum deflection of the ball center
1.6 10 2
mm
Spall depth
0.125
mm
Spall width/size
2.64
mm
3
Figure 12 Stress-strain curve [15].
(a)
(b)
Figure 13: PEEQ contour plots after (a) first and (b) fourth load cycle.
The 3D model developed by Branch et al. [15] was meshed with a linear hexahedral mesh. The element size near the edge location was 10 m . The total CPU time of one impact was 3.5 105 sec [7]. In the proposed 2D 16
FE model, an element size near the spall edge was 4 m , and CPU time of one impact was 200 sec. During the present simulation, the contact pressure reaches a maximum value of 7.8 GPa. This result is similar to that of the 3D FE model introduced by Arakere et al. [7]. The results of the proposed model are shown in Fig. Figure 14. The results are in a good agreement with those obtained by Branch et al. [15], Fig. Figure 15. Table 3 summarizes the main differences between the two modeling approaches. Branch et al. [15] determined the impact velocity of the RE into the spall edge based on the kinematics of the inner raceway rotational speed and bearing geometry. The RE impact mass is assumed to be that of the ball alone [7]. The proposed model utilizes a non-linear dynamic model to estimate the dynamics and kinematics during the RE-spall edge interaction. Parameters such as initial load of the RE before disconnection from the raceways, spall length, impact location, etc. were taken into account. During the RE-spall edge interaction, a small contact area is formed. In the presented case the width of the contact between the RE and spall edge is 110 m . Thus, in order to achieve more reliable predictions of the stress distribution in the contact vicinity, a fine mesh is required. The proposed model allows increasing mesh refinement without a significant computational cost. Thus, the main difference between the two models, 3D FE model developed by Branch et al. [15] and the proposed 2D FE model, is that the latter has the potential to produce more reliableresults and to significantly reduce the computation time, Table 3.
(a)
(b)
(c)
Figure 14: The results of 2D FE model: (a) maximum principal plastic strain, (b) residual hoop (MPa) and (c) maximum principal stresses (MPa).
17
(a)
(b)
(c)
Figure 15: The results of 3D FE model obtained by Branch et al. [15]: (a) maximum principal plastic strain, (b) residual hoop (MPa) and (c) maximum principal stresses (MPa). Table 3: Main differences between the proposed model and the model developed by Branch et al. [15].
3D dynamic FE model [15]
2D quasi-static FE model
Dynamics and Kinematics
Simplified approximation
Non-linear dynamic model [17]
Element size - contact location
10 m
4 m
CPU time – One impact
3.5 105 sec
200 sec
3. Results and discussion 3.1. Material response analysis Branch et al. [15] chose to represent the spall edge in the FE model as a 45 degree inclined surface. They examined the contribution of different bearing properties, e.g ball density, raceway surface hardness, strength, etc., to the magnitude and extent of damage due to RE impact. The accumulated plastic strain and critical stresses were used as indicators of the damage severity. In the current work, we have extended the damage severity analysis by examining the effect of the spall edge geometry on the damage indicators. As mentioned above, based on published experimental observations the spall edge surface slope varies, Figs. Figure 3(a) and Figure 4. The spall edge surface slopes of 38o, 45o and 55o were studied. It is noteworthy to mention that the damage mechanism 18
governing the spall propagation is still not fully understood. Thus, we examined three damage indicators that can lead to the spall propagation, maximum principal and radial residual stresses, and plastic strain. During the loading stage, i.e. RE impact, the trailing edge of the spall is under compressive stress. In this stage, the likelihood of crack formation and propagation within the spall edge is low. Only upon the unloading stage, a portion of the spall edge turns into tension due to the plastic deformation underneath, Fig. Figure 14(c). The location of the tensile stress on the surface corresponds to the crack formation in front of the spall edge, in similarity with the experimental observation in Fig. Figure 2. Figure 16 shows the contour plots of the maximum principal residual stress of varying edge slopes. The observed change in the location and magnitude of the residual stress is insignificant. However, the surface area under the tensile stress increases as the slope becomes sharper. Thus, assuming that crack formation on the raceway surface is governed by the maximum principal residual stress, the sharp edges generate a more favorable condition for the evolution of cracks.
Figure 16: Contour plots of maximum principal residual stress (MPa) with varying edge slopes, from left to right: 38, 45, and 55 degrees.
Based on the published experimental observations, Figs. Figure 2 and Figure 4, it can be assumed that cracks initiate at the rolling surface. However, it is also possible that these cracks represent the final stage of the damage, which was initiated below the surface. Figure 17 the shows radial residual stress contour plots from the simulations of varying edge slopes. It shows the presence of tensile stresses beneath the spall edge surface. This stress can lead to generation of subsurface cracks parallel to the surface. Moreover, in the case of moderate edges 19
slope, the subsurface tensile area increases, generating a more favorable condition for the formation of subsurface cracks.
Figure 17: Contour plots of radial residual stress (MPa) with varying edge slopes, from left to right: 38, 45, and 55 degrees.
Branch et al. [15] suggested that the plastic strain could represent damage severity due to multiple ball-trailing edge impacts. Figure 18 shows the contour plots of maximum plastic strain from the simulations of varying edge slopes. The 55 degrees slope edge accumulated the highest plasticity (red area). The results suggest that more material deforms plastically (red area in the PEEQ contour plot) when the edge is sharp, i.e. large slopes. Assuming that the accumulated plasticity provides a major contribution to the damage magnitude and rate,, the result suggests that the spall will propagate more rapidly when encountering sharp spall edges. Moreover, according to Branch et al. [15], the plastic strain contour was proposed to correspond to the shape of a representative material fragment. Figure 18 implies that the spall edge profile will remain constant in similarity with Figs. 3(a) and 4.
20
Figure 18: Contour plots of the maximum principal plastic strain with varying edge slopes, from left to right: 38, 45, and 55 degrees. The potential material fragment can be represented by a plastic strain zone.
3.2. Damage evolution – Qualitative analysis In order to demonstrate the advantages of the proposed modeling approach, a qualitative damage initiation analysis was carried out. In the current analysis, we used a model that has been implemented in the author's previous work on the subject of damage initiation in a rotating element bearing [2]. The damage evolution was represented by CDM based model. The damage model provides a representation of failure mechanism by the definition of a nondimensional damage variable, D . It is assumed that the damage variable D affects the elastic modulus E of the material as
E 1 D E
(12)
where E is the damaged elastic modulus. The presence of the damage reduces the material stiffness [18]. The damage variable D has values ranging from 0, which represents undamaged material, to Dmax , which represents a completely damaged material:
0 D Dmax
(13)
When damage at some point reaches its critical value, Dmax , it corresponds to the initiation of a macro-crack. The damage rate evolution model is given by 21
dD eff dN r 1 D
m
(14)
where N is the number of stress cycles and eff is the effective stress that causing the damage, r and m are material-dependent parameters. More detailed description of the damage model algorithm and its application can be found in Gazizulin et al. [2]. The existent experimental observations of the spall edge are very limited. Only cracks on the surface of the spall edge are visible. Thus, qualitative damage simulations were conducted under the assumption that cracks initiate at the surface of the edge. The effective stress was chosen to be the maximum principal residual stress,
eff Max.Princ. . In this analysis, we have assumed that the maximum principal residual stress is the only damage driven mechanism. In future research, the contribution of the plasticity and radial stresses to the damage mechanisms should be also examined. It is noteworthy to mention that the material properties within the spall edge are unknown [15]. These properties might change during the bearing operation and RE impacts. Thus, the values of material-dependent parameters were taken arbitrarily to be similar to those used in Gazizulin et al., [2]. Three different cases with hundreds of load cycles were simulated within a reasonable time-frame. The simulation results show that the proposed damage model and criterion, predict well the location and direction of the initial damage evolution stage. The damage simulation results are presented in Figs. Figure 19 and Figure 20. The cracks started to evolve on the surface. Their location corresponded to the surface area under tensile stress, Fig. Figure 16. After the surface cracks evolved, they propagated downward into the edge. Upon reaching the subsurface area under compression, their propagation ceased, Fig. Figure 20. Figure 19 shows cracks, which evolved after hundreds of load cycles. In all simulated cases, the location of the cracks corresponds to those observed in the published experimental observations [16], Fig. Figure 2.
22
Figure 19: Cracks generated within the spall edge, from top to bottom: 38, 45, and 55 degrees.
Figure 20: Contour plots of maximum principal residual stress at the end of damage simulations, from left to right: 38, 45, and 55 degrees.
4. Conclusions We proposes a new RE-spall edge interaction model. It combines non-linear dynamic and FE models which were developed based on experimental observations and physical insights. The dynamic and kinematic responses of the interaction were estimated using non-linear dynamic model [17]. The results of the dynamic model were used as load parameters and boundary conditions for the FE model. Next, the material response, stress/strain 23
fields, due to the interactions were estimated using the FE model. This modeling approach provides, both computational efficiency as well as reliable results. The results of the proposed model were validated with the 3D dynamic FE model developed by Arakere et al. [7]. Several cases of the RE-spall edge interactions were simulated and potential damage driven mechanisms were analyzed. Since the damage mechanism is not fully understood, several potential damage indicators were examined. The simulation results showed that the location of the maximum principal residual stress at the spall edge surface agrees with the location of the cracks near the spall edge. The maximum principal residual stress spreads over a larger area as the edge becomes sharper, i.e increased probability of crack generation. In a case where cracks are generated beneath the surface, the simulation results showed that the radial residual stress might be responsible for sub-surface crack generation. Unlike the maximum principal residual stress, the radial residual stress field decreases as the edge becomes sharper. In the future, for a better understanding of the damage mechanism, an extensive bearing endurance tests and metallurgical analysis of the spalled bearings should be conducted. Based on the conclusions from this analysis qualitative damage simulations were carried out. One goal of the damage analysis was to demonstrate the advantages of a fast and reliable model. The preliminary results of the damage simulations showed a good agreement with published experimental observations. This work provides new insights regarding the damage evolution within the spall edge as a result of impacts with RE. 5. Acknowledgments The authors would like to express their deepest appreciations to the Pearlstone Center for their support andfunding of this work.
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Highlights
Non-linear dynamic model with a realistic representation of the spall edge. Rolling element-spall edge interaction Finite Element model: simple and efficient. Physics-based damage driven mechanism of the spall propagation process. Damage evolution simulations within the spall edge.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: