- Email: [email protected]
Adaptive finite element simulation of wear evolution in radial sliding bearings
Wear 296 (2012) 660–671
Contents lists available at SciVerse ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Adaptive finite element simulation of wear evolution in radial sliding bearings Ali Rezaei a,n, Wim Van Paepegem a, Patrick De Baets b, Wouter Ost b, Joris Degrieck a a b
Department of Materials Science and Engineering, Ghent University, Belgium Department of Mechanical construction and production, Ghent University, Belgium
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 March 2012 Received in revised form 2 August 2012 Accepted 8 August 2012 Available online 30 August 2012
This article employs an adaptive wear modeling method to study the wear progress in radial sliding bearings contacting with a rotary shaft. Mixed Lagrangian–Eulerian formulation has been used to simulate the contact condition between the bearing and the shaft, and the local wear evolution is modeled using the Archard equation. In the developed wear processor algorithm, not only remeshing is performed on the contact elements, but also is executed for their proximity elements. In this way the wear simulation becomes independent of the size of the contact elements. Validation was done for a laminated polymeric composite bearing. The composite has been modeled as a linear orthotropic material. The wear coefficients were obtained from flat-on-flat experiments and were applied as pressure and velocity dependent parameters in the wear processor. Finally, the effect of the clearance on the wear of the radial bearings has been studied numerically. The simulations also demonstrate how the contact pressure evolves during the wear process, and how the clearance influences this evolution. & 2012 Elsevier B.V. All rights reserved.
Keywords: Wear Finite element Sliding bearing
1. Introduction Wear is one of the most critical parameters that substantially affects the life span of bearings. In addition, existence of wear in the bearings of a mechanical system can deteriorate the performance of the entire system through the changing kinematics. Therefore, understanding the wear evolution and its effect on the deformation, stress fields and kinematics of bearings can be very helpful for design engineers. Very often pin-on-disc type standard tribometers are used to study the wear of materials. Since the tribosystem parameters such as contact conditions can be strongly different in practice, these tribotests are not adequate for accurate wear prediction in the design phase. Another method to incorporate wear into design is performance of full-scale tribotests that may mimic operating conditions. This technique, however, is time consuming and can be very expensive. Numerical simulations in addition to these experimental methods can improve the bearing’s design, and can overcome the limitations of the experimental methods. In one hand, simulations help to study the effect of the wear evolution on the contact stress distribution and deformation of a bearing, which can hardly be measured with the experimental techniques. On the other hand, once the simulations are verified with the basic standard or
n
Corresponding author. Tel.: þ32 9 331 0424. E-mail addresses: [email protected], [email protected] (A. Rezaei).
0043-1648/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.wear.2012.08.013
full-scale experiments, they can be used for parametric studies, or to predict the effect of the wear process on the performance of more complex mechanical systems. Recently, many efforts have been put into developing wear simulation techniques. In the most common simulation method the wear process is simulated based on evolving contact conditions. In this method (called ‘‘adaptive wear simulation’’) the contact geometry varies gradually and results into an iterative procedure in which the contact pressure and the sliding distance are altered at each iteration [1,2]. Yet by now, most of those numerical works have been performed on metallic and bulk polymers and fewer studies have been made for simulation of the wear in anisotropic materials. The few available studies simplify the material properties and geometry of the models. For example, in 2004, Hegadekatte et al. [2,3] have simulated the dry sliding wear of a brass–steel ring-on-ring test setup and wear in a silicon-nitride pin-on-disc tribometer using the adaptive FE wear modeling. In their model, the Archard coefficient is a fixed value independent of the pressure and sliding velocity, and all materials are linear isotropic. In 2010, Hegadekatte et al. [4] have used their algorithm to simulated the wear of a microplanetary silicon gear train. Similar to their first model the materials are linear isotropic. ¨ In 2009, Soderberg and Andersson [5] have utilized the adaptive FE method to simulate the wear of a pad-to-rotor mechanism. Using the ANSYS FEM package, a simplified 3D model is built-up which employs the Archard wear equation. The rotor is simulated as a rigid
A. Rezaei et al. / Wear 296 (2012) 660–671
body and its deformation is not taken into account. In addition, while the pad is a composite, it is simplified with a linear isotropic material model. In 2007, Rawlinson et al. [6] have studied the wear of the ProDiscL disc replacement with the adaptive finite element method. In their article, a simplified 3D model is built up which contains a polyethylene bearing and a rigid plate. The model includes a conforming contact and the material is linear isotropic. Gonzalez et al. [7] have simulated the pin on disc tests on Al–Li/SiC composites. The composite material is assumed to behave as isotropic thermo-elastoplastic. Simulation of the thermo-elastoplastic behavior makes the model very interesting. However, the model is a simplified plane stress model with a rigid shaft. To the best of our knowledge, by this time, adaptive finite element techniques have been employed only for isotropic materials. Moreover, in the available studies, a fixed Archard wear coefficient is used. While, in principle, for polymeric and polymeric composite materials the wear coefficient can be dependent to the pressure and sliding velocity. In this work, the wear of a laminated composite radial bearing is simulated using the adaptive finite element method. Contrary to the available models, the anisotropy of the composite material is considered and the bearing is simulated as a linear orthotropic material. Moreover, the wear coefficient depends on the pressure and sliding velocity. To this purpose, a wear-processing program has been developed using the UMESHMOTION feature of ABAQUS with FORTRAN programming language. This wear-processor calculates and implements the incremental wear evolution into the finite element model performed by ABAQUS. The developed wear-processor was first verified with the flat-onflat tribotesting of ORKOT [8] marine composite bearings. The tests were performed on a setup designed for tribotesting of mediumscale flat bearing specimens. Through these experiments the Archard wear coefficients corresponding to the normal pressure and sliding velocity of contacting surfaces were obtained. The obtained coefficients were implemented in a two-dimensional finite element model. Finally, the simulation results were compared to the wear depth measured in the experiments. The final work implements the developed wear-processor into the model of a full-scale radial sliding bearing test setup. In this model all clearances and degrees of freedom of the system are simulated, which makes the model more realistic.
661
After validation of the model, effects of the clearance between the bearing and the shaft on the wear of the composite bearings were studied numerically. It must be noted that in these simulations wear of the shaft was not simulated because it was negligible compared to the wear of the bearings.
2. Methodology of the adaptive FE wear modeling The adaptive finite element wear modeling has been developed to simulate the wear of two contacting surfaces through the movement of the contact nodes based on the following parameters:
The normal stresses at the contacting nodes. Relative sliding of the contacting nodes. Empirically determined wear coefficients. The model provides a time dependent geometry of the contacting surfaces, which gives a more realistic simulation of the contact stresses and kinematics of the worn components. The concept of the adaptive FE wear modeling is presented in Fig. 1. The simulation process starts through solving the initial finite element model based on the original mesh, material model, boundary conditions, and contact conditions. Therefore, in the first step the static infinitesimal contact between the deformable bodies is simulated. Afterwards, simulation is followed by the second step which is called the wear-processing. In the wearprocessing step, the relative motion between the contacting surfaces is added to the first model. The incremental solution procedure is considered for this step, and the step time is divided into very small intervals. Simulation of the boundary problem for the first increment of this step is accomplished by invoking the wear-processor algorithm. The wear-processor accesses the contacting nodes one-by-one, and calls and records the following properties for each node: nodal coordinates, contact stresses, and relative slip rate in the last increment. Based on the recorded nodal stress and slip rate, the algorithm calculates the adequate wear coefficient of the current node. Knowing the wear coefficient, contact pressure and sliding distance leads to the calculation of the wear depth. This is then followed by computing the inward surface normal at the current node. Finally, once the above process is accomplished for all contact nodes, the nodes are swept in the inward normal direction and the material quantities
Fig. 1. Conceptual diagram of the adaptive finite element wear simulation.
662
A. Rezaei et al. / Wear 296 (2012) 660–671
are re-mapped from the old positions to the new positions. This phase is performed via the adaptive meshing algorithm. After remeshing of the contact elements, the next increment of the FE model is computed for the remeshed geometry and once again at the end of each increment the wear-processing phase is executed. The total time of the step depends onto the required sliding distance and is defined by the user of the algorithm. 2.1. Wear rate calculation In the current work, a form of Archard wear model is employed to implement the local wear rate at each node (Eq. (1)). dw Lv m3 Nm=s ¼K ; ¼ : ð1Þ dt H s N=m2 In this equation dw/dt (m3/s) is defined to be the wear volume rate, L is the loading force (N), v denotes the sliding speed (m/s), H is the hardness of the surface region, and K is referred to as the dimensionless wear coefficient [9]. It is considered that the hardness of the worn body (composite bearing) is constant, hence the equation is altered to: dw ¼ kH Lv: dt
ð2Þ
where kH ¼K/H and is defined as the wear coefficient with the dimension1 of (m2/N). Dividing both sides of Eq. (2) by the real area of contact ‘‘s’’, the wear depth rate will become 1 dw L ¼ kH v n s dt s
ð3Þ
dh ¼ kH Pv dt
ð4Þ
In this equation dh/dt is the wear depth rate (mm/s) and P is the nodal pressure (N/mm2). This wear model is applicable to the global scale and in the current work it is supposed that it can also be applied to the local scale. Therefore, for a very small increment dt the wear depth become dh ¼ kH Pvdt
ð5Þ
In this equation dh is the wear depth (mm). By ablation of the contact surface, the real contact area changes and consequently the contact pressure also changes. Hence it is essential to consider enough increment numbers to obtain a more realistic wear model. The total wear depth of the ith increment will be hi ¼ hi1 þ dhi
ð6Þ
where hi is the total wear depth up to the increment number of i, hi 1 is the total wear depth up to the increment number ‘i-1’ and dhi is the wear depth of the current increment. The above equation calculates the wear of a node which is continuously in contact with its counterface. It is obvious that when a node is apart from its mating surface the wear depth of the relevant increment will become zero. Since wear simulation is mostly performed for a very long time resulting in a very large number of increments, a proper incrementation technique is essential to have a computationally efficient simulation. In this work, an automatic variable incremental technique has been employed. In the wear processor, a threshold wear depth has been considered for the maximum allowable local wear increment. At each increment before applying the local displacement to a contact node, the wear depth is calculated based on the current increment time, recommended by the finite element software. If the wear depth is bigger than the predefined threshold, the algorithm stops the current increment 1
kH is often called the specific wear rate and quoted in units of (mm3/Nm) [7].
and recommends another increment time using the following procedure [10]. According to Eq. (5), the local wear depth of increment i equals dhi ¼ kH Pvdt i
ð7Þ
Consider dhmax as the maximum allowable local wear increment, therefore dhi must be less than dhmax. If dhi 4 dhmax then the wear processor modifies the incremental time in such a way that dhi equals dhmax. The new increment time then becomes dti_new ¼ dt i n
dhmax dhi
ð8Þ
It must be noted that the threshold wear depth is chosen based on the simulation conditions. If the difference between the nodal wear increments of a certain iteration becomes very large, the simulation results might be unreliable. Therefore it is very important to choose a proper dhmax. This approach is more suitable for complex contact configurations such as curved surfaces, where the wear is not uniform. 2.2. Calculation of the wear direction Once the incremental wear depth is calculated from the pressure and the slip rate is extracted from the finite element model, the next important step is determination of the wear direction [11]. Depending on the profile of the contacting surfaces, contact nodes are categorized into two groups:
interior nodes edge nodes Fig. 2 presents a schematic of two contacting bodies. In this arrangement part A is pressed against part B, and part B moves from left to right. The nodes on the contact surface of part B are labeled from ‘a’ to ‘f’. In this configuration, nodes b, c, d, and e are interior contact nodes and nodes a and f are edge nodes. The wear direction is determined through two different mechanisms. For interior contact nodes, the wear direction is determined by the inward surface normal. Fig. 3 and Eq. (9) represent how the inward surface normal ni is calculated for node i, in a twodimensional model. ni ¼
ni1 þ ni2 :ni1 þ ni2 :
ð9Þ
For an edge node, the wear direction is determined based on the vector from the edge node to the corresponding subsurface node. For instance in Fig. 4 node i is an edge node and l is the corresponding subsurface node for node i. A vector from node i to node l is computed
Fig. 2. Schematic of two contacting bodies including interior and edge contact nodes.
A. Rezaei et al. / Wear 296 (2012) 660–671
663
Fig. 3. Inward surface normal at an interior contacting node. ri1 is the edge vector corresponding to the element 1. ri2 is the edge vector corresponding to the element 2. si is the subsurface vector formed between node i and node l. ni1 is the normal vector to ri1. ni2 is the normal vector to ri2. ni is the inward surface normal vector of node i [11].
Fig. 5. Remeshing of the proximity of the contact zone. (a) Domain O contacts with the counterface. (b) Remeshing steps during the wear simulation: (b-1) the initial configuration of the elements 1, 2, 3 and 4, (b-2) movement of the contact node 1 based on the Archard wear model, (b-3) movement of the node 3 if the aspect ratio of element 1 is in a critical condition, (b-4) movement of the node 5 if the aspect ratio of element 2 is in a critical condition.
Fig. 4. Direction of the wear for edge nodes.
and is divided by its magnitude to get the unit vector ni. For the edge node i wear is applied in the direction of this vector [12]. Incremental wear simulation is accomplished by sweeping the contact nodes by an amount equal to the calculated wear depth, over the wear direction. In this article sweeping of contact nodes has been performed using the adaptive meshing feature of ABAQUS. Following the adjustment of nodes through the mesh sweeping process, material point quantities are advected for the revised meshing of the model in its current configuration [12]. 2.3. Global remeshing In the simulation of a wear process, the wear depth is limited by the size and aspect ratio of the contact elements if the adaptive remeshing is only performed on those elements. Taking into account that generally a very fine discretization is required for a proper contact simulation, this limitation will become more severe. To overcome this restriction, in the current work, not only the contact elements are remeshed, but also their proximity elements can be remeshed by the wear processor, if required. The developed remeshing technique is demonstrated using Fig. 5. In this figure, domain O contacts with a counterface. For instance, consider the contact element 1 and its proximity elements numbered 2, 3 and 4. During the wear process, node 1 is displaced according to the incremental wear procedure. After a certain number of iterations the wear depth of the node 1 equals h1. If H1 3 is the initial distance of the nodes 1 and 3, their current distance Hi13 becomes the following: Hi13 ¼ H13 h1
ð10Þ
Hcr 13
We define as the critical distance between node 1 and node i 3. As the wear progresses, Hi13 approaches Hcr 13 . Once H 13 equals Hc13 , the wear processor moves node 3 toward node 5. This movement is also incremental, and is about dhmax at each increment. By moving node 3, the current distance of nodes 3 and 5 becomes Hi35 ¼ H35 h3
ð11Þ
In this equation, H35 is the initial distance of nodes 3 and 5, and h3 is the total displacement of node 3. Similar to the element
Fig. 6. Comparison of the two remeshing techniques in the adaptive finite element wear simulation. (a) Initial mesh of the curved surface, (b) deformed elements after a wear simulation in which remeshing has been only performed on the contact elements, and (c) deformed elements after a wear simulation in which remeshing has been performed on three layers of elements in the vicinity of the contact zone.
1, a critical distance is defined for nodes 3 and 5. The moment that Hi35 equals Hcr 35 , the wear processor moves the node 5 toward the node 7. This incremental remeshing can be performed for any desired number of layers of elements in the proximity of the contact zone. In this regard, the wear depth of the simulation is not limited to the contact elements’ size. This remeshing technique provides a safe aspect ratio for the contact elements, while the total wear depth can even become larger than the contact element size. It must be noted that a structured discretization in the vicinity of contact zone helps to have a relatively inexpensive computational model in the explained configuration. Fig. 6 schematically compares the results of the proposed remeshing technique with the common remeshing method which is limited to the contact elements. Under the same simulation condition the contact elements have been entirely distorted when remeshing is only performed for the contact elements, while the
664
A. Rezaei et al. / Wear 296 (2012) 660–671
This section implements and validates the developed wear processor model to the experiments performed on a laminated composite bearing. First, the test setup and experiments are discussed and later on the finite element model is explained.
(1) placed in the holders (3). The normal load on the test specimens and the counterplates is applied through a hydraulic clamp (7) with a maximum capacity of 225 kN. In order to apply pure axial load on the bearings, the load of the load-clamp is transferred to the holders through two spherical plain bearings (6). The test material specimens (1) and the holders (3) are supported by the reaction-fork (5). The test bench is controlled by means of PC-controlled servo-hydraulics. All measuring signals are continuously and digitally registered with a data-acquisition card mounted in the PC. The major characteristics of the test setup are summarized in Table 1.
3.1. Test bench for tribotesting of flat bearings
3.2. Test material
The developed wear-processor was verified with the flat-on-flat tribotesting of ORKOT marine bearings [8,13]. The tests were performed on a medium-scale flat-on-flat tribotester, shown in Fig. 7. In this setup, two steel counterplates (2) are mounted (bolt connection) on a central vertical sliding block (4). This central sliding block is connected to the motion actuator (8) and is moved in the vertical direction such that it slides against two identical test specimens
All the tests were performed on the Orkot marine bearings which are composed of polyester woven fabrics impregnated with a polyester resin. The test materials were acquired from the manufacturer in cubic form with 30 mm long sides and manufacturing tolerance of 70.05 mm (Fig. 8). The friction surface of these bearings contained PTFE dispersed in the subsurface layers to provide self-lubricating properties. The Orkot bearing is an orthotropic material with the engineering constants presented in Table 2.
mesh is still structured very well when the remeshing has been performed for three layers of elements in the vicinity of the contact zone.
3. Implementation and validation of the wear processor
3.3. Experiments For each test two identical bearings were mounted in the holders and examined simultaneously. Since the wear volume was calculated through the weight loss, both specimens were placed in a dry oven for 24 h at a temperature of 60 1C before each test and after drying Table 1 Technical specifications of flat-on-flat test setup. Dimensions of test specimen Dimensions of counterplates Maximum stroke Maximum normal load Maximum vertical (friction) load Maximum frequency at maximum stroke
30*30*30 80*200*10 100 225 200 1
mm3 mm3 mm kN kN Hz
Fig. 8. Laminated composite bearing acquired from the manufacturer.
Table 2 Engineering constants of the ORKOT bearing.
Fig. 7. Schematic diagram of the flat-on-flat test setup. (1) Bearing, (2) counterplates, (3) holder, (4) central sliding block, (5) reaction fork, (6) spherical selfaligning bearing, (7) hydraulic loading clamp, (8) motion actuator, and (9) foundation plate.
E11 E22 E33 u12 u13 u23 G12 G13
2800 2560 2560 0.3 0.45 0.45 500 500
GPa GPa GPa
G23
1000
GPa
GPa GPa
A. Rezaei et al. / Wear 296 (2012) 660–671
the specimens were weighted. To this purpose, a balance with a range of 0–200 g and an accuracy of 70.001 g was used. In these experiments two counterplates made of 42Cr–Mo–4V steel alloy were employed. For real applications it is recommended by the manufacturer to use a very smooth counterface with an average roughness between Ra0.1 and Ra0.8. Such a small roughness leads to a very limited wear, which needs a very long test time [8]. Therefore, to perform the experiments in a reasonable time scope, the surfaces of the counterplates were roughened to attain an average roughness of about Ra3. The experiments were performed at room conditions, the sliding velocity of 10 mm/s, the normal pressures of 20, 30 and 40 MPa, and total sliding distance of 864,000 mm. Table 3 describes the details of the experiments. In Fig. 9 a tested bearing and its corresponding counterplate are displayed. After each test the bearings were cleaned and placed in a dry oven for 24 h at a temperature of 60 1C, and then the dried specimens were weighted. Knowing the weight of the specimens before and after the test, the wear depth can be calculated as below
Dh ¼
T DM M
ð12Þ
where Dh refers to the total wear depth in (mm), T is the thickness of the test specimen before the test in (mm), M is the weight of the Table 3 Test conditions for the flat bearing experiments. Test name
A
B
C
Sliding velocity (mm/s) Applied pressure (MPa)
10 20
30
40
Test environment Sliding Test duration (h)
Ambient Dry 24
Total sliding distance (mm)
864,000
665
test specimen before the test in (g), and DM is the weight loss in (g). In this relation it is supposed that the wear is distributed uniformly over the entire contact surface. Once the total wear depth of the test bearing is defined the wear coefficient is calculated from the Archard’s relation (Eq. (5)). Table 4 describes the results of the experiments. As it is seen, increasing the applied pressure leads to an increasing weight loss. Consequently, the total wear depth (Dh) and wear depth rate will increase as well. Moreover, since the surface of the counterface CF1 (Ra3.2) is rougher than the surface of the counterface C2 (Ra2.7), the wear depth in the bearings versus the counterface C1 is larger than that for the bearings versus the counterface CF2. The lowermost wear depth is 0.0408 mm that corresponds to the bearing subjected to the 10 mm/s slip rate, normal pressure of 19.8 MPa, and counterface CF2. The highest wear depth is 0.0837 mm that corresponds to the bearing subjected to the 10 mm/s slip rate, normal pressure of 38.82 MPa, and counterface CF1. 3.4. Verification of the wear-processor program The finite element model of the flat-on-flat test setup is shown in Fig. 10. Considering the geometry and loading conditions in the test setup, a two-dimensional plane strain FE model was built up. The bearing has been discretized using 900 quadrilateral elements, with the aspect ratio of about 1. Because the stiffness of the counterface and the holder (210 GPa) are almost 90 times bigger than that of the bearing (3 GPa), the holder and counterface have been simulated as rigid bodies. The holder is fixed and its inner surface is in contact with the side walls of the bearing. The counterface is pushed against and slides over the bearing’s surface. All the contacts were simulated using the surface-tosurface discretization and finite-sliding tracking approach. According to the experimental data (Tables 3 and 4), three simulations were performed which are described in Table 5. The wear simulations were carried out for 864,000 mm sliding distance. Each simulation consists of three major steps called
Fig. 9. A sample experiment; a tested bearing and its corresponding counterface.
Table 4 Results of the experiments with flat-on-flat tribotester. CF1: Counterface with the overall surface roughness of Ra3.2; CF2: Counterface with the overall surface roughness of Ra2.7. Test no.
Part no.
Sliding velocity (mm/s)
Corresponding counterface
Average pressure (MPa)
Average friction coefficient
Weight before the test (g)
Weight after the test (g)
Total weight loss (g)
Total wear depth (mm)
Wear depth rate 10 7 (mm/s)
Specific wear rate10 6 (mm3/Nm)
A
P1 P2 P3 P4 P5 P6
10
CF1 CF2 CF1 CF2 CF1 CF2
19.8
0.103
29.6
0.090
38.8
0.083
34.615 34.174 34.221 34.998 34.748 34.362
34.548 34.128 34.129 34.935 34.650 34.294
0.067 0.046 0.092 0.063 0.097 0.068
0.0585 0.0408 0.0810 0.0547 0.0837 0.0594
6.79 4.72 9.39 6.34 9.78 6.89
3.43 2.38 3.17 2.14 2.52 1.78
B C
666
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 10. Finite element model of flat-on-flat tribotester.
Table 5 Parameters for flat-on-flat simulations.
Table 6 Comparison of the simulation and experimental results for wear depth of flat laminated composite bearings.
Name Sliding velocity (mm/s)
Normal pressure (MPa)
Friction Wear coefficient coefficient (mm/N2)
Wear simulation time (h)
A-P1 B-P3 C-P5
20 30 40
0.10 0.09 0.08
24 24 24
10 10 10
3.43E-09 3.17E-09 2.52E-09
Name
Dh(A-P1) Dh(B-P3) Dh(C-P5)
Simulation (mm)
Experiment
Agreement (%)
0.05919 0.08199 0.08483
0.0585 0.0810 0.0837
98.82 98.77 98.64
Table 6 compares the average wear depth of the bearing’s surface nodes and the experiments. As seen the simulation results are in a very good agreement with the experiments. The overall agreement is more than 98%. Small deviations among the agreements are due to the rounding of the implemented wear coefficients and loads in the simulations.
4. Wear in a radial sliding bearing The main objective of the wear simulation is to predict the wear evolution of more complex contact geometries, using the wear coefficients obtained from standard small scale tribotests. In this section, the developed wear-processor is used to predict the wear in radial composite sliding bearings. The simulations are performed on the ORKOT material, and the wear coefficients obtained from the previous section are applied. At first, the model is verified with the experiments performed on a radial bearing tribotester. Afterwards, the effects of the clearance (between the bearing and shaft) on the wear evolution and contact stress in the bearing are investigated numerically. 4.1. Small scale radial bearing tribotester (SSRB)
Fig. 11. Displacement of the interior contact nodes of the bearing surface. In these graphs Dh(A-P1), Dh(B-P3) and Dh(C-P5) denote the wear depth of the corresponding simulation in Table 5.
loading-step, wearing-step and unloading-step. In the loadingstep the bearing deforms elastically, and then the wear starts and evolves during the wearing-step. Finally in the unloading-step the wear process stops, the load is removed, and elastic springback occurs in the bearing. The distance between the initial position of the contact nodes and their position after the unloading-step is the wear depth. Fig. 11 shows the displacement of the middle nodes of the bearing’s surface.
Fig. 12 shows the schematic of the Radial Bearing tribotester designed for bearings with an internal diameter of about 40 mm. In this configuration, the radial composite bearing (1) is fixed inside the backing (2) and then these two are mounted inside the housing (3). The composite bearing is in contact with the counterface (4) which is mounted on the shaft axis (5). The shaft axis, made of 42Cr–Mo–4V alloy steel, is supported with two ballbearings (SKF 23044 [14]) (6, 7). The lever arm (8) transfers the friction torque between the bearing and the counterface to the friction-torque load-cell (9). The setup is equipped with a calibrated load-cell in the range of 0–300 N (SENCY 5932) [15]. However, depending on the friction force, the load-cell can be changed. The pneumatic muscle (12)
A. Rezaei et al. / Wear 296 (2012) 660–671
667
Fig. 12. Schematic diagram of the small-scale radial-bearing tribotester. (1) Composite bearing, (2) backing, (3) housing, (4) counterface, (5) shaft axis, (6,7) shaft supports, (8) lever arm, (9) friction-torque load-cell, (10) load-lever, (11) actuator’s load-cell, (12) pneumatic muscle, and (13) transmission trolley.
provides a compressive load with the range of 0–6 kN [16]. The load of the pneumatic muscle is transferred by the load-lever (10) to the housing, through the transmission trolley (13). The transmission trolley includes two rollers that allow small rotations of the housing without disturbing the load transmission. An electric motor with the maximum power of 2.2 kW drives the shaft, and the shaft rotation is measured by the LITEON optical encoder [17]. All measuring signals are continuously and digitally registered with the NI-9215 USB data acquisition card [18]. 4.2. Finite element model of SSRB In this work, wear of radial composite bearings in contact with a rotary shaft was simulated using the configuration of the explained radial bearing tribotester. Mixed Lagrangian–Eulerian [12,19] formulation was utilized to simulate the behavior of the setup. In the model, it is supposed that the load was uniformly distributed over the bearing and the axial strain in the bearing is negligible. Hence, a two-dimensional plane-strain model was built up which is shown in Fig. 13. The characteristics of the model are as below:
The loading-subassembly was simplified with a mechanism
composed of four springs and two rigid rollers. In order to provide an accurate radial pressure on the housing, the rotational degrees of freedom of the rollers are independent of the springs. The model was discretized with 26,640 linear elements. To reduce the calculation time, the lever-arm was simulated with a rigid body. The friction-torque load-cell was simulated as a solid beam in the interaction with two rigid pins. The reference point of the upper pin was tied to the lever-arm, and the reference point of the lower pin was fixed. The clearance between the pins and the loadcell was 0.1 mm, and the friction coefficient was 0.1. The surface-to-surface discretization method and finite-sliding contact tracking approach was employed for the contact boundaries. Since the friction between the bearing and the shaft during the wear process is dynamic friction, only the dynamic friction coefficient was implemented and the static friction was ignored. According to the experimental measurements, the dynamic friction coefficient between the bearing and the shaft was about 0.1. The rotation of the shaft was simulated by the Eulerian formulation, and its deformation was simulated by the Lagrangian formulation. The elastic modulus of the load-cell, the housing, and the shaft was considered 210 GPa with the Poisson’s ratio of 0.33 (mechanical properties of steel alloys).
Fig. 13. Finite element model of the SSRB test setup. (a) Entire model, (b) meshing of the shaft and the bearing in detail, and (c) friction-torque load-cell and its connection pin.
The bearing was simulated as an orthotropic material. The wear was simulated using the developed wear-processor program.
The wear coefficients in the simulation were implemented from the corresponding flat-on-flat experiments.
4.3. Verification The finite element model was verified with experiments performed on radial ORKOT bearings. The bearing material was the same as presented in Section 3.2. The bearings were acquired from the manufacturer with a diameter of 40 mm, width of 10 mm, and thickness of 3 mm (Fig. 14). They were mounted with the freeze fitting method inside the backing. In this experiment the counterface was made of 42Cr–Mo–4V steel alloy, similar to the flat-on-flat experiments. It was tried to roughen the surface of the counterface similar to the counterface
668
A. Rezaei et al. / Wear 296 (2012) 660–671
of the flat-on-flat setup. However, in practice it is very difficult to obtain the same surface topography. The average surface roughness of the counterface was Ra ¼2.6 mm (Fig. 15). Table 7 depicts the details of the performed experiment, in which the applied load by the pneumatic muscle was about 1200 N and the shaft rotation was 0.5 rad/s. The test was performed for 24 h. The wear volume in the experiment was measured through the calculation of the weight loss in the bearings (the same method as for flat-on-flat experiments). The weight loss after 24 h was 0.019 g. The corresponding wear volume for the test was 15.2 mm3. Table 7 summarizes the experimental conditions and the results. As noticed before, the developed finite element wear-processor was used to simulate the experiment. Based on the experimental measurements the inputs of the finite element model become as Table 8. The wear coefficients were implemented from the flat-on-flat experiments. Considering the surface roughness of the counterface (Ra 2.6 mm) and the sliding velocity (10 mm/s), the relevant flat-on-flat experiments are A-P2, B-P4 and C-P6 (Table 4). In Fig. 16, KH versus Pv (pressure velocity) is plotted for these experiments. Fitting a line to the experimental results provides a Pv dependent wear coefficient (Eq. (13)). This equation was used as the wear coefficient function in the simulations. kH ¼ ð2:09 1012 ÞPv þ ð2:68 109 Þ
Fig. 17 compares the simulation and experimental results for the friction and vertical forces. The friction coefficient calculated from the experiment is about 0.093. The normal force is about Table 8 The input of the simulation. Simulation inputs Applied force (kN) Shaft rotation (rad/s) Simulation duration (h) Shaft diameter (mm) Bearing diameter (mm) Bearing thickness (mm) Clearance of the load-cell connections (mm) Friction coefficient between the shaft and the bearing Friction coefficient in the load cell connections Simulation
6 0.5 24 39.62 40.10 3 0.1 0.1 0.1 A
ð13Þ
Fig. 16. Wear coefficient (kH) versus Pv (pressure velocity).
Fig. 14. Radial composite bearing acquired from the manufacturer.
Fig. 15. Counterface used in the SSRB experiments.
Fig. 17. Comparison of the simulation and experimental results for the normal and friction forces. FF: friction force, FN: normal force, and COF: coefficient of friction.
Table 7 Performed experiments on the SSRB setup. Experiment characteristics Applied vertical load (kN) Shaft rotation (rad/s) Test duration (h) Shaft diameter (mm) Average roughness of the shaft (mm) Bearing diameter after fitting process (mm)
Experiment results 6 0.5 24 39.62 2.6 40.10
Weight before the test (g) Weight after the test (g) Weight loss (g) Density (g/mm3) Wear volume (mm3)
5.626 5.607 0.019 1.25 15.2
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 18. Wear depth in the bearing’s surface simulated by finite element simulation.
669
Fig. 19. Contact pressure distribution on the surface of the bearing for various steps of the wear process.
6150 N for the experiment and 6000 N for the simulation. The friction force is about 580 N for the experiment and 604 N for the simulation. The small deviations are due to the fluctuation of the applied force by the pneumatic muscle and the errors of the friction measurements. In general, the errors are very small and the simulation results correspond closely to the experiments. Fig. 18 depicts the graph of the wear depth of the bearing’s surface nodes. The maximum wear depth is 0.089 mm. The area under the graph is equal to the worn area. Multiplication of the area with the bearing width results into the wear volume, which is 18.4 mm3. The wear volume measured in the experiment is 15.2 mm3. The deviation of the simulation results is 18% for the test. This can be explained by the following causes:
The first and simplest error source is the calculation method of
the wear volume in the experiments. In reality the wear rate of the fibers and matrix is not equal. The volume fraction of the matrix and fibers in the worn particles are different from that of the composite bearing. Therefore, the density of the worn particles will not be equal to the density of the bearing. Because the wear volume in the experiments is calculated based on the density of the bearing’s material, this can cause inaccuracy in the calculation of the experimental results. The second reason is that the surface topography of the counterface differs from that of the flat-on-flat tribotests. Surface topography may have a severe effect on the wear of the material. Therefore, the wear coefficients obtained from the flat-on-flat experiments are not perfectly adaptable to the SSRB experiments. The third possible error source can be the different tribosystem of radial and flat bearings. In radial bearings, wear debris might be trapped on the surface of bearing and play the role of solid lubricant. While in the flat bearings wear debris are swiped away from the bearing’s surface [20]. Therefore the wear rate of the tested radial bearing can become slightly lower than that of the flat bearing.
The evolution of the contact stresses on the bearing is illustrated in Fig. 19. This figure plots the contact stress on the bearing surface, for various steps of the wear process. As seen, the contact pressure progressively decreases during the wear process. The maximum contact pressure decreases about 22 MPa after 24 h. By progressing the time of wear, the stress level decreases with a lower rate. However, Fig. 20 shows that the integration of the contact pressure over the contact area grows linearly by progress of the wear.
Fig. 20. Integral of the contact pressure over the contact area during the wear process.
5. Parametric study: clearance effect In this section, the developed finite element model is employed to investigate the effects of the clearance, between the composite bearing and shaft, on the wear of the bearing. The simulations were performed for the clearance range between 0.1 mm and 1.2 mm (0.1, 0.3, 0.6, 0.9, and 1.2 mm). In these simulations, the sliding velocity was 0.5 rad/s (10 mm/s), the vertical load was 6 kN, the shaft diameter was 40 mm, the wear coefficient was as Eq. (13) and the simulation time was 24 h. The simulations were performed based on the configuration of the SSRB setup. In the models, 232,324 elements were utilized, and the friction coefficient was 0.1. Fig. 21 shows the effect of the clearance on the contact stresses at the beginning and end of the wear process. At the first hour of the process the maximum contact pressure for the clearances of 0.1, 0.3, 0.6, 0.9 and 1.2 mm is respectively 29, 38, 46, 52, and 58 MPa. After 24 h, the maximum pressure for those clearances is respectively 20, 22, 24, 26, and 28 MPa. As seen, after evolution of the wear, the stress graphs get closer to each other. In the beginning, the difference between the maximum pressures for the clearances 0.1 and 1.2 mm is about 30 MPa, while this quantity is 8 MPa after 24 h. Fig. 22 depicts the graphs of the wear depth after 24 h of the process. The maximum wear depth for the clearances 0.1, 0.3, 0.6, 0.9, and 1.2 mm is respectively 0.115, 0.105, 0.092, 0.078 and 0.062 mm. The wear volume of the bearing for each graph was
670
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 21. Effect of the clearance on the contact pressure of the bearing at the beginning (left) and end of the wear process (right).
Fig. 23. Effect of the clearance on the contact area during the wear process.
Fig. 22. Effect of the clearance on the wear depth at the end of the wear process.
calculated by computing the area under the graphs. The calculations indicate that the wear volume for those clearances is respectively 18.2, 17.5, 17.4, 17.5, and 17.8 mm3. These quantities show that the clearance does not have a significant effect on the wear volume. Fig. 23 shows the effect of the clearance on the contact area. It is seen that the contact area grows with a similar rate for all the graphs. In the beginning of the process the contact area for the clearances 0.1, 0.3, 0.6, 0.9 and 1.2 mm is respectively 310, 230, 180, 150, and 140 mm2. After 24 h, those quantities are 460, 360, 300, 265, and 245 mm2.
6. Conclusion An adaptive finite element wear processor was developed to simulate the wear of composite sliding bearings through the movement of the contacting nodes. The developed wear processor performs an iterative remeshing not only on the contact elements but also on the proximity of them. With this method of remeshing, a large wear depth independent of the size of the contact elements can be simulated.
The wear phenomena were modeled via the interaction of normal stresses at the contacting elements, relative sliding of the contacting nodes, and empirically determined wear coefficients. Wear coefficients were obtained using the flat-on-flat experimental setup designed especially for tribotesting of flat bearings. This is essential to obtain the wear coefficients for different contact pressures and sliding velocities. Applying the pressure and sliding velocity dependent wear coefficient provides a better wear model for more complex contact conditions, such as radial bearings, where the contact pressure is not uniformly distributed. This wear simulation also helps to study the evolution of the contact stresses during the wear process, which is not easily measurable with experimental techniques. In this study, particularly, the simulations show that the contact pressure in the radial bearing decreases during the wear process. In the beginning, the pressure decreases with a high ratio and by progression of the wear it decreases with a lower ratio. On the contrary, the contact area increases during the wear process. In the beginning, it increases with a high ratio and by progression of the wear it increases with a lower ratio. However, the integral of the contact pressure over the contact area increases with a fixed ratio over the time. The results also show that while the wear depth is directly dependent to clearance size, the wear volume is not influenced considerably by the initial clearance size. In summary, comparing the simulations and the experiments proves that the developed model can be a useful tool in the prediction of the wear of radial composite bearings. In a similar
A. Rezaei et al. / Wear 296 (2012) 660–671
way, effects of the other parameters such as e.g. material properties and friction coefficient can be predicted using this method. Considering the costs of the tribological experiments, specifically for full-scale experiments, these simulations can help to reduce the number of the required experiments. These simulations will be more important when the geometry of worn surface is more complex, and the wear progress cannot be calculated easily using the analytical solutions.
References [1] S. Mukras, N.H. Kim, W.G. Sawyer, D.B. Jackson, L.W. Bergquist, Numerical integration schemes and parallel computation for wear prediction using finite element method, Wear 266 (2009) 822–831. [2] V. Hegadekatte, N. Huber, O. Kraft, Finite element based simulation of dry sliding wear, Modelling and Simulation in Materials Science and Engineering 13 (2004) 57–75. [3] V. Hegadekatte, N. Huber, O. Kraft, Modeling and simulation of wear in a pin on disc tribometer, Tribology Letters 24 (2006) 51–60. [4] V. Hegadekatte, J. Hilgert, O. Kraft, N. Huber, Multi time scale simulations for wear prediction in micro-gears, Wear 268 (2010) 316–324. ¨ [5] A. Soderberg, S. Andersson, Simulation of wear and contact pressure distribution at the pad-to-rotor interface in a disc brake using general purpose finite element analysis software, Wear 267 (2009) 2243–2251.
671
[6] J.J. Rawlinson, K.P. Punga, K.L. Gunsallus, D.L. Bartel, T.M. Wright, Wear simulation of the prodisc-l disc replacement using adaptive finite element analysis, Journal of Neurosurgery-Spine 7 (2007) 165–173. [7] C. Gonza´lez, A. Martı´n, J. Llorca, M.A. Garrido, M.T. Go´mez, A. Rico, J. Rodrı´guez, Numerical analysis of pin on disc tests on Al–Li/SiC composites, Wear 259 609–612. [8] ORKOT, Marine bearings Engineering Manual, December 2010. /http://www. orkot.comS. [9] K.C. Ludema, Friction Wear Lubrication: A Textbook in Tribology, CRC Press, 1996. ~ [10] P. Podra, S. Andersson, Simulating sliding wear with finite element method, Tribology International 32 (1999) 71–81. [11] V. Hegadekatte, Modeling and simulation of dry sliding wear for micromachine applications, University of Karlsruhe, Karlsruhe, 2006 (pp. 111). [12] ABAQUS analysis user’s manual, in: Analysis Procedures. [13] ORKOT, Tlmm & txmm Brochure, December 2010. /http://www.orkot.comS. [14] SKF, Split Plummer Block Housings, Saf Series With Spherical Roller Bearings on an Adapter Sleeve, January 2011. /http://www.skf.comS. [15] SENCY, Miniature Tension and Compression Load Cells, December 2010. /http://www.sensy.comS. [16] FESTO, Fluidic Muscle Dmsp/Mas, January 2011. /http://www.festo.comS. [17] LITEON, Lth-301-07 Data Sheet, January 2011. /http://html.alldatasheet.comS. [18] N. Instruments, Ni 9215; 4-channel, 100 ks/s/ch, 16-bit, 7 10 v Analog Input Module, January 2011. /http://sine.ni.comS. [19] A. Rezaei, W. Ost, W.V. Paepegem, J. Degrieck, P. Debaets, Experimental study and numerical simulation of the large-scale testing of polymeric composite journal bearings: two-dimensional modeling and validation, Tribology Letters 37 (2009) 12. [20] D. An, J.H. Choi, T.L. Schmitz, N.H. Kim, In situ monitoring and prediction of progressive joint wear using bayesian statistics, Wear 270 (2011) 828–838.
Contents lists available at SciVerse ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Adaptive finite element simulation of wear evolution in radial sliding bearings Ali Rezaei a,n, Wim Van Paepegem a, Patrick De Baets b, Wouter Ost b, Joris Degrieck a a b
Department of Materials Science and Engineering, Ghent University, Belgium Department of Mechanical construction and production, Ghent University, Belgium
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 March 2012 Received in revised form 2 August 2012 Accepted 8 August 2012 Available online 30 August 2012
This article employs an adaptive wear modeling method to study the wear progress in radial sliding bearings contacting with a rotary shaft. Mixed Lagrangian–Eulerian formulation has been used to simulate the contact condition between the bearing and the shaft, and the local wear evolution is modeled using the Archard equation. In the developed wear processor algorithm, not only remeshing is performed on the contact elements, but also is executed for their proximity elements. In this way the wear simulation becomes independent of the size of the contact elements. Validation was done for a laminated polymeric composite bearing. The composite has been modeled as a linear orthotropic material. The wear coefficients were obtained from flat-on-flat experiments and were applied as pressure and velocity dependent parameters in the wear processor. Finally, the effect of the clearance on the wear of the radial bearings has been studied numerically. The simulations also demonstrate how the contact pressure evolves during the wear process, and how the clearance influences this evolution. & 2012 Elsevier B.V. All rights reserved.
Keywords: Wear Finite element Sliding bearing
1. Introduction Wear is one of the most critical parameters that substantially affects the life span of bearings. In addition, existence of wear in the bearings of a mechanical system can deteriorate the performance of the entire system through the changing kinematics. Therefore, understanding the wear evolution and its effect on the deformation, stress fields and kinematics of bearings can be very helpful for design engineers. Very often pin-on-disc type standard tribometers are used to study the wear of materials. Since the tribosystem parameters such as contact conditions can be strongly different in practice, these tribotests are not adequate for accurate wear prediction in the design phase. Another method to incorporate wear into design is performance of full-scale tribotests that may mimic operating conditions. This technique, however, is time consuming and can be very expensive. Numerical simulations in addition to these experimental methods can improve the bearing’s design, and can overcome the limitations of the experimental methods. In one hand, simulations help to study the effect of the wear evolution on the contact stress distribution and deformation of a bearing, which can hardly be measured with the experimental techniques. On the other hand, once the simulations are verified with the basic standard or
n
Corresponding author. Tel.: þ32 9 331 0424. E-mail addresses: [email protected], [email protected] (A. Rezaei).
0043-1648/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.wear.2012.08.013
full-scale experiments, they can be used for parametric studies, or to predict the effect of the wear process on the performance of more complex mechanical systems. Recently, many efforts have been put into developing wear simulation techniques. In the most common simulation method the wear process is simulated based on evolving contact conditions. In this method (called ‘‘adaptive wear simulation’’) the contact geometry varies gradually and results into an iterative procedure in which the contact pressure and the sliding distance are altered at each iteration [1,2]. Yet by now, most of those numerical works have been performed on metallic and bulk polymers and fewer studies have been made for simulation of the wear in anisotropic materials. The few available studies simplify the material properties and geometry of the models. For example, in 2004, Hegadekatte et al. [2,3] have simulated the dry sliding wear of a brass–steel ring-on-ring test setup and wear in a silicon-nitride pin-on-disc tribometer using the adaptive FE wear modeling. In their model, the Archard coefficient is a fixed value independent of the pressure and sliding velocity, and all materials are linear isotropic. In 2010, Hegadekatte et al. [4] have used their algorithm to simulated the wear of a microplanetary silicon gear train. Similar to their first model the materials are linear isotropic. ¨ In 2009, Soderberg and Andersson [5] have utilized the adaptive FE method to simulate the wear of a pad-to-rotor mechanism. Using the ANSYS FEM package, a simplified 3D model is built-up which employs the Archard wear equation. The rotor is simulated as a rigid
A. Rezaei et al. / Wear 296 (2012) 660–671
body and its deformation is not taken into account. In addition, while the pad is a composite, it is simplified with a linear isotropic material model. In 2007, Rawlinson et al. [6] have studied the wear of the ProDiscL disc replacement with the adaptive finite element method. In their article, a simplified 3D model is built up which contains a polyethylene bearing and a rigid plate. The model includes a conforming contact and the material is linear isotropic. Gonzalez et al. [7] have simulated the pin on disc tests on Al–Li/SiC composites. The composite material is assumed to behave as isotropic thermo-elastoplastic. Simulation of the thermo-elastoplastic behavior makes the model very interesting. However, the model is a simplified plane stress model with a rigid shaft. To the best of our knowledge, by this time, adaptive finite element techniques have been employed only for isotropic materials. Moreover, in the available studies, a fixed Archard wear coefficient is used. While, in principle, for polymeric and polymeric composite materials the wear coefficient can be dependent to the pressure and sliding velocity. In this work, the wear of a laminated composite radial bearing is simulated using the adaptive finite element method. Contrary to the available models, the anisotropy of the composite material is considered and the bearing is simulated as a linear orthotropic material. Moreover, the wear coefficient depends on the pressure and sliding velocity. To this purpose, a wear-processing program has been developed using the UMESHMOTION feature of ABAQUS with FORTRAN programming language. This wear-processor calculates and implements the incremental wear evolution into the finite element model performed by ABAQUS. The developed wear-processor was first verified with the flat-onflat tribotesting of ORKOT [8] marine composite bearings. The tests were performed on a setup designed for tribotesting of mediumscale flat bearing specimens. Through these experiments the Archard wear coefficients corresponding to the normal pressure and sliding velocity of contacting surfaces were obtained. The obtained coefficients were implemented in a two-dimensional finite element model. Finally, the simulation results were compared to the wear depth measured in the experiments. The final work implements the developed wear-processor into the model of a full-scale radial sliding bearing test setup. In this model all clearances and degrees of freedom of the system are simulated, which makes the model more realistic.
661
After validation of the model, effects of the clearance between the bearing and the shaft on the wear of the composite bearings were studied numerically. It must be noted that in these simulations wear of the shaft was not simulated because it was negligible compared to the wear of the bearings.
2. Methodology of the adaptive FE wear modeling The adaptive finite element wear modeling has been developed to simulate the wear of two contacting surfaces through the movement of the contact nodes based on the following parameters:
The normal stresses at the contacting nodes. Relative sliding of the contacting nodes. Empirically determined wear coefficients. The model provides a time dependent geometry of the contacting surfaces, which gives a more realistic simulation of the contact stresses and kinematics of the worn components. The concept of the adaptive FE wear modeling is presented in Fig. 1. The simulation process starts through solving the initial finite element model based on the original mesh, material model, boundary conditions, and contact conditions. Therefore, in the first step the static infinitesimal contact between the deformable bodies is simulated. Afterwards, simulation is followed by the second step which is called the wear-processing. In the wearprocessing step, the relative motion between the contacting surfaces is added to the first model. The incremental solution procedure is considered for this step, and the step time is divided into very small intervals. Simulation of the boundary problem for the first increment of this step is accomplished by invoking the wear-processor algorithm. The wear-processor accesses the contacting nodes one-by-one, and calls and records the following properties for each node: nodal coordinates, contact stresses, and relative slip rate in the last increment. Based on the recorded nodal stress and slip rate, the algorithm calculates the adequate wear coefficient of the current node. Knowing the wear coefficient, contact pressure and sliding distance leads to the calculation of the wear depth. This is then followed by computing the inward surface normal at the current node. Finally, once the above process is accomplished for all contact nodes, the nodes are swept in the inward normal direction and the material quantities
Fig. 1. Conceptual diagram of the adaptive finite element wear simulation.
662
A. Rezaei et al. / Wear 296 (2012) 660–671
are re-mapped from the old positions to the new positions. This phase is performed via the adaptive meshing algorithm. After remeshing of the contact elements, the next increment of the FE model is computed for the remeshed geometry and once again at the end of each increment the wear-processing phase is executed. The total time of the step depends onto the required sliding distance and is defined by the user of the algorithm. 2.1. Wear rate calculation In the current work, a form of Archard wear model is employed to implement the local wear rate at each node (Eq. (1)). dw Lv m3 Nm=s ¼K ; ¼ : ð1Þ dt H s N=m2 In this equation dw/dt (m3/s) is defined to be the wear volume rate, L is the loading force (N), v denotes the sliding speed (m/s), H is the hardness of the surface region, and K is referred to as the dimensionless wear coefficient [9]. It is considered that the hardness of the worn body (composite bearing) is constant, hence the equation is altered to: dw ¼ kH Lv: dt
ð2Þ
where kH ¼K/H and is defined as the wear coefficient with the dimension1 of (m2/N). Dividing both sides of Eq. (2) by the real area of contact ‘‘s’’, the wear depth rate will become 1 dw L ¼ kH v n s dt s
ð3Þ
dh ¼ kH Pv dt
ð4Þ
In this equation dh/dt is the wear depth rate (mm/s) and P is the nodal pressure (N/mm2). This wear model is applicable to the global scale and in the current work it is supposed that it can also be applied to the local scale. Therefore, for a very small increment dt the wear depth become dh ¼ kH Pvdt
ð5Þ
In this equation dh is the wear depth (mm). By ablation of the contact surface, the real contact area changes and consequently the contact pressure also changes. Hence it is essential to consider enough increment numbers to obtain a more realistic wear model. The total wear depth of the ith increment will be hi ¼ hi1 þ dhi
ð6Þ
where hi is the total wear depth up to the increment number of i, hi 1 is the total wear depth up to the increment number ‘i-1’ and dhi is the wear depth of the current increment. The above equation calculates the wear of a node which is continuously in contact with its counterface. It is obvious that when a node is apart from its mating surface the wear depth of the relevant increment will become zero. Since wear simulation is mostly performed for a very long time resulting in a very large number of increments, a proper incrementation technique is essential to have a computationally efficient simulation. In this work, an automatic variable incremental technique has been employed. In the wear processor, a threshold wear depth has been considered for the maximum allowable local wear increment. At each increment before applying the local displacement to a contact node, the wear depth is calculated based on the current increment time, recommended by the finite element software. If the wear depth is bigger than the predefined threshold, the algorithm stops the current increment 1
kH is often called the specific wear rate and quoted in units of (mm3/Nm) [7].
and recommends another increment time using the following procedure [10]. According to Eq. (5), the local wear depth of increment i equals dhi ¼ kH Pvdt i
ð7Þ
Consider dhmax as the maximum allowable local wear increment, therefore dhi must be less than dhmax. If dhi 4 dhmax then the wear processor modifies the incremental time in such a way that dhi equals dhmax. The new increment time then becomes dti_new ¼ dt i n
dhmax dhi
ð8Þ
It must be noted that the threshold wear depth is chosen based on the simulation conditions. If the difference between the nodal wear increments of a certain iteration becomes very large, the simulation results might be unreliable. Therefore it is very important to choose a proper dhmax. This approach is more suitable for complex contact configurations such as curved surfaces, where the wear is not uniform. 2.2. Calculation of the wear direction Once the incremental wear depth is calculated from the pressure and the slip rate is extracted from the finite element model, the next important step is determination of the wear direction [11]. Depending on the profile of the contacting surfaces, contact nodes are categorized into two groups:
interior nodes edge nodes Fig. 2 presents a schematic of two contacting bodies. In this arrangement part A is pressed against part B, and part B moves from left to right. The nodes on the contact surface of part B are labeled from ‘a’ to ‘f’. In this configuration, nodes b, c, d, and e are interior contact nodes and nodes a and f are edge nodes. The wear direction is determined through two different mechanisms. For interior contact nodes, the wear direction is determined by the inward surface normal. Fig. 3 and Eq. (9) represent how the inward surface normal ni is calculated for node i, in a twodimensional model. ni ¼
ni1 þ ni2 :ni1 þ ni2 :
ð9Þ
For an edge node, the wear direction is determined based on the vector from the edge node to the corresponding subsurface node. For instance in Fig. 4 node i is an edge node and l is the corresponding subsurface node for node i. A vector from node i to node l is computed
Fig. 2. Schematic of two contacting bodies including interior and edge contact nodes.
A. Rezaei et al. / Wear 296 (2012) 660–671
663
Fig. 3. Inward surface normal at an interior contacting node. ri1 is the edge vector corresponding to the element 1. ri2 is the edge vector corresponding to the element 2. si is the subsurface vector formed between node i and node l. ni1 is the normal vector to ri1. ni2 is the normal vector to ri2. ni is the inward surface normal vector of node i [11].
Fig. 5. Remeshing of the proximity of the contact zone. (a) Domain O contacts with the counterface. (b) Remeshing steps during the wear simulation: (b-1) the initial configuration of the elements 1, 2, 3 and 4, (b-2) movement of the contact node 1 based on the Archard wear model, (b-3) movement of the node 3 if the aspect ratio of element 1 is in a critical condition, (b-4) movement of the node 5 if the aspect ratio of element 2 is in a critical condition.
Fig. 4. Direction of the wear for edge nodes.
and is divided by its magnitude to get the unit vector ni. For the edge node i wear is applied in the direction of this vector [12]. Incremental wear simulation is accomplished by sweeping the contact nodes by an amount equal to the calculated wear depth, over the wear direction. In this article sweeping of contact nodes has been performed using the adaptive meshing feature of ABAQUS. Following the adjustment of nodes through the mesh sweeping process, material point quantities are advected for the revised meshing of the model in its current configuration [12]. 2.3. Global remeshing In the simulation of a wear process, the wear depth is limited by the size and aspect ratio of the contact elements if the adaptive remeshing is only performed on those elements. Taking into account that generally a very fine discretization is required for a proper contact simulation, this limitation will become more severe. To overcome this restriction, in the current work, not only the contact elements are remeshed, but also their proximity elements can be remeshed by the wear processor, if required. The developed remeshing technique is demonstrated using Fig. 5. In this figure, domain O contacts with a counterface. For instance, consider the contact element 1 and its proximity elements numbered 2, 3 and 4. During the wear process, node 1 is displaced according to the incremental wear procedure. After a certain number of iterations the wear depth of the node 1 equals h1. If H1 3 is the initial distance of the nodes 1 and 3, their current distance Hi13 becomes the following: Hi13 ¼ H13 h1
ð10Þ
Hcr 13
We define as the critical distance between node 1 and node i 3. As the wear progresses, Hi13 approaches Hcr 13 . Once H 13 equals Hc13 , the wear processor moves node 3 toward node 5. This movement is also incremental, and is about dhmax at each increment. By moving node 3, the current distance of nodes 3 and 5 becomes Hi35 ¼ H35 h3
ð11Þ
In this equation, H35 is the initial distance of nodes 3 and 5, and h3 is the total displacement of node 3. Similar to the element
Fig. 6. Comparison of the two remeshing techniques in the adaptive finite element wear simulation. (a) Initial mesh of the curved surface, (b) deformed elements after a wear simulation in which remeshing has been only performed on the contact elements, and (c) deformed elements after a wear simulation in which remeshing has been performed on three layers of elements in the vicinity of the contact zone.
1, a critical distance is defined for nodes 3 and 5. The moment that Hi35 equals Hcr 35 , the wear processor moves the node 5 toward the node 7. This incremental remeshing can be performed for any desired number of layers of elements in the proximity of the contact zone. In this regard, the wear depth of the simulation is not limited to the contact elements’ size. This remeshing technique provides a safe aspect ratio for the contact elements, while the total wear depth can even become larger than the contact element size. It must be noted that a structured discretization in the vicinity of contact zone helps to have a relatively inexpensive computational model in the explained configuration. Fig. 6 schematically compares the results of the proposed remeshing technique with the common remeshing method which is limited to the contact elements. Under the same simulation condition the contact elements have been entirely distorted when remeshing is only performed for the contact elements, while the
664
A. Rezaei et al. / Wear 296 (2012) 660–671
This section implements and validates the developed wear processor model to the experiments performed on a laminated composite bearing. First, the test setup and experiments are discussed and later on the finite element model is explained.
(1) placed in the holders (3). The normal load on the test specimens and the counterplates is applied through a hydraulic clamp (7) with a maximum capacity of 225 kN. In order to apply pure axial load on the bearings, the load of the load-clamp is transferred to the holders through two spherical plain bearings (6). The test material specimens (1) and the holders (3) are supported by the reaction-fork (5). The test bench is controlled by means of PC-controlled servo-hydraulics. All measuring signals are continuously and digitally registered with a data-acquisition card mounted in the PC. The major characteristics of the test setup are summarized in Table 1.
3.1. Test bench for tribotesting of flat bearings
3.2. Test material
The developed wear-processor was verified with the flat-on-flat tribotesting of ORKOT marine bearings [8,13]. The tests were performed on a medium-scale flat-on-flat tribotester, shown in Fig. 7. In this setup, two steel counterplates (2) are mounted (bolt connection) on a central vertical sliding block (4). This central sliding block is connected to the motion actuator (8) and is moved in the vertical direction such that it slides against two identical test specimens
All the tests were performed on the Orkot marine bearings which are composed of polyester woven fabrics impregnated with a polyester resin. The test materials were acquired from the manufacturer in cubic form with 30 mm long sides and manufacturing tolerance of 70.05 mm (Fig. 8). The friction surface of these bearings contained PTFE dispersed in the subsurface layers to provide self-lubricating properties. The Orkot bearing is an orthotropic material with the engineering constants presented in Table 2.
mesh is still structured very well when the remeshing has been performed for three layers of elements in the vicinity of the contact zone.
3. Implementation and validation of the wear processor
3.3. Experiments For each test two identical bearings were mounted in the holders and examined simultaneously. Since the wear volume was calculated through the weight loss, both specimens were placed in a dry oven for 24 h at a temperature of 60 1C before each test and after drying Table 1 Technical specifications of flat-on-flat test setup. Dimensions of test specimen Dimensions of counterplates Maximum stroke Maximum normal load Maximum vertical (friction) load Maximum frequency at maximum stroke
30*30*30 80*200*10 100 225 200 1
mm3 mm3 mm kN kN Hz
Fig. 8. Laminated composite bearing acquired from the manufacturer.
Table 2 Engineering constants of the ORKOT bearing.
Fig. 7. Schematic diagram of the flat-on-flat test setup. (1) Bearing, (2) counterplates, (3) holder, (4) central sliding block, (5) reaction fork, (6) spherical selfaligning bearing, (7) hydraulic loading clamp, (8) motion actuator, and (9) foundation plate.
E11 E22 E33 u12 u13 u23 G12 G13
2800 2560 2560 0.3 0.45 0.45 500 500
GPa GPa GPa
G23
1000
GPa
GPa GPa
A. Rezaei et al. / Wear 296 (2012) 660–671
the specimens were weighted. To this purpose, a balance with a range of 0–200 g and an accuracy of 70.001 g was used. In these experiments two counterplates made of 42Cr–Mo–4V steel alloy were employed. For real applications it is recommended by the manufacturer to use a very smooth counterface with an average roughness between Ra0.1 and Ra0.8. Such a small roughness leads to a very limited wear, which needs a very long test time [8]. Therefore, to perform the experiments in a reasonable time scope, the surfaces of the counterplates were roughened to attain an average roughness of about Ra3. The experiments were performed at room conditions, the sliding velocity of 10 mm/s, the normal pressures of 20, 30 and 40 MPa, and total sliding distance of 864,000 mm. Table 3 describes the details of the experiments. In Fig. 9 a tested bearing and its corresponding counterplate are displayed. After each test the bearings were cleaned and placed in a dry oven for 24 h at a temperature of 60 1C, and then the dried specimens were weighted. Knowing the weight of the specimens before and after the test, the wear depth can be calculated as below
Dh ¼
T DM M
ð12Þ
where Dh refers to the total wear depth in (mm), T is the thickness of the test specimen before the test in (mm), M is the weight of the Table 3 Test conditions for the flat bearing experiments. Test name
A
B
C
Sliding velocity (mm/s) Applied pressure (MPa)
10 20
30
40
Test environment Sliding Test duration (h)
Ambient Dry 24
Total sliding distance (mm)
864,000
665
test specimen before the test in (g), and DM is the weight loss in (g). In this relation it is supposed that the wear is distributed uniformly over the entire contact surface. Once the total wear depth of the test bearing is defined the wear coefficient is calculated from the Archard’s relation (Eq. (5)). Table 4 describes the results of the experiments. As it is seen, increasing the applied pressure leads to an increasing weight loss. Consequently, the total wear depth (Dh) and wear depth rate will increase as well. Moreover, since the surface of the counterface CF1 (Ra3.2) is rougher than the surface of the counterface C2 (Ra2.7), the wear depth in the bearings versus the counterface C1 is larger than that for the bearings versus the counterface CF2. The lowermost wear depth is 0.0408 mm that corresponds to the bearing subjected to the 10 mm/s slip rate, normal pressure of 19.8 MPa, and counterface CF2. The highest wear depth is 0.0837 mm that corresponds to the bearing subjected to the 10 mm/s slip rate, normal pressure of 38.82 MPa, and counterface CF1. 3.4. Verification of the wear-processor program The finite element model of the flat-on-flat test setup is shown in Fig. 10. Considering the geometry and loading conditions in the test setup, a two-dimensional plane strain FE model was built up. The bearing has been discretized using 900 quadrilateral elements, with the aspect ratio of about 1. Because the stiffness of the counterface and the holder (210 GPa) are almost 90 times bigger than that of the bearing (3 GPa), the holder and counterface have been simulated as rigid bodies. The holder is fixed and its inner surface is in contact with the side walls of the bearing. The counterface is pushed against and slides over the bearing’s surface. All the contacts were simulated using the surface-tosurface discretization and finite-sliding tracking approach. According to the experimental data (Tables 3 and 4), three simulations were performed which are described in Table 5. The wear simulations were carried out for 864,000 mm sliding distance. Each simulation consists of three major steps called
Fig. 9. A sample experiment; a tested bearing and its corresponding counterface.
Table 4 Results of the experiments with flat-on-flat tribotester. CF1: Counterface with the overall surface roughness of Ra3.2; CF2: Counterface with the overall surface roughness of Ra2.7. Test no.
Part no.
Sliding velocity (mm/s)
Corresponding counterface
Average pressure (MPa)
Average friction coefficient
Weight before the test (g)
Weight after the test (g)
Total weight loss (g)
Total wear depth (mm)
Wear depth rate 10 7 (mm/s)
Specific wear rate10 6 (mm3/Nm)
A
P1 P2 P3 P4 P5 P6
10
CF1 CF2 CF1 CF2 CF1 CF2
19.8
0.103
29.6
0.090
38.8
0.083
34.615 34.174 34.221 34.998 34.748 34.362
34.548 34.128 34.129 34.935 34.650 34.294
0.067 0.046 0.092 0.063 0.097 0.068
0.0585 0.0408 0.0810 0.0547 0.0837 0.0594
6.79 4.72 9.39 6.34 9.78 6.89
3.43 2.38 3.17 2.14 2.52 1.78
B C
666
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 10. Finite element model of flat-on-flat tribotester.
Table 5 Parameters for flat-on-flat simulations.
Table 6 Comparison of the simulation and experimental results for wear depth of flat laminated composite bearings.
Name Sliding velocity (mm/s)
Normal pressure (MPa)
Friction Wear coefficient coefficient (mm/N2)
Wear simulation time (h)
A-P1 B-P3 C-P5
20 30 40
0.10 0.09 0.08
24 24 24
10 10 10
3.43E-09 3.17E-09 2.52E-09
Name
Dh(A-P1) Dh(B-P3) Dh(C-P5)
Simulation (mm)
Experiment
Agreement (%)
0.05919 0.08199 0.08483
0.0585 0.0810 0.0837
98.82 98.77 98.64
Table 6 compares the average wear depth of the bearing’s surface nodes and the experiments. As seen the simulation results are in a very good agreement with the experiments. The overall agreement is more than 98%. Small deviations among the agreements are due to the rounding of the implemented wear coefficients and loads in the simulations.
4. Wear in a radial sliding bearing The main objective of the wear simulation is to predict the wear evolution of more complex contact geometries, using the wear coefficients obtained from standard small scale tribotests. In this section, the developed wear-processor is used to predict the wear in radial composite sliding bearings. The simulations are performed on the ORKOT material, and the wear coefficients obtained from the previous section are applied. At first, the model is verified with the experiments performed on a radial bearing tribotester. Afterwards, the effects of the clearance (between the bearing and shaft) on the wear evolution and contact stress in the bearing are investigated numerically. 4.1. Small scale radial bearing tribotester (SSRB)
Fig. 11. Displacement of the interior contact nodes of the bearing surface. In these graphs Dh(A-P1), Dh(B-P3) and Dh(C-P5) denote the wear depth of the corresponding simulation in Table 5.
loading-step, wearing-step and unloading-step. In the loadingstep the bearing deforms elastically, and then the wear starts and evolves during the wearing-step. Finally in the unloading-step the wear process stops, the load is removed, and elastic springback occurs in the bearing. The distance between the initial position of the contact nodes and their position after the unloading-step is the wear depth. Fig. 11 shows the displacement of the middle nodes of the bearing’s surface.
Fig. 12 shows the schematic of the Radial Bearing tribotester designed for bearings with an internal diameter of about 40 mm. In this configuration, the radial composite bearing (1) is fixed inside the backing (2) and then these two are mounted inside the housing (3). The composite bearing is in contact with the counterface (4) which is mounted on the shaft axis (5). The shaft axis, made of 42Cr–Mo–4V alloy steel, is supported with two ballbearings (SKF 23044 [14]) (6, 7). The lever arm (8) transfers the friction torque between the bearing and the counterface to the friction-torque load-cell (9). The setup is equipped with a calibrated load-cell in the range of 0–300 N (SENCY 5932) [15]. However, depending on the friction force, the load-cell can be changed. The pneumatic muscle (12)
A. Rezaei et al. / Wear 296 (2012) 660–671
667
Fig. 12. Schematic diagram of the small-scale radial-bearing tribotester. (1) Composite bearing, (2) backing, (3) housing, (4) counterface, (5) shaft axis, (6,7) shaft supports, (8) lever arm, (9) friction-torque load-cell, (10) load-lever, (11) actuator’s load-cell, (12) pneumatic muscle, and (13) transmission trolley.
provides a compressive load with the range of 0–6 kN [16]. The load of the pneumatic muscle is transferred by the load-lever (10) to the housing, through the transmission trolley (13). The transmission trolley includes two rollers that allow small rotations of the housing without disturbing the load transmission. An electric motor with the maximum power of 2.2 kW drives the shaft, and the shaft rotation is measured by the LITEON optical encoder [17]. All measuring signals are continuously and digitally registered with the NI-9215 USB data acquisition card [18]. 4.2. Finite element model of SSRB In this work, wear of radial composite bearings in contact with a rotary shaft was simulated using the configuration of the explained radial bearing tribotester. Mixed Lagrangian–Eulerian [12,19] formulation was utilized to simulate the behavior of the setup. In the model, it is supposed that the load was uniformly distributed over the bearing and the axial strain in the bearing is negligible. Hence, a two-dimensional plane-strain model was built up which is shown in Fig. 13. The characteristics of the model are as below:
The loading-subassembly was simplified with a mechanism
composed of four springs and two rigid rollers. In order to provide an accurate radial pressure on the housing, the rotational degrees of freedom of the rollers are independent of the springs. The model was discretized with 26,640 linear elements. To reduce the calculation time, the lever-arm was simulated with a rigid body. The friction-torque load-cell was simulated as a solid beam in the interaction with two rigid pins. The reference point of the upper pin was tied to the lever-arm, and the reference point of the lower pin was fixed. The clearance between the pins and the loadcell was 0.1 mm, and the friction coefficient was 0.1. The surface-to-surface discretization method and finite-sliding contact tracking approach was employed for the contact boundaries. Since the friction between the bearing and the shaft during the wear process is dynamic friction, only the dynamic friction coefficient was implemented and the static friction was ignored. According to the experimental measurements, the dynamic friction coefficient between the bearing and the shaft was about 0.1. The rotation of the shaft was simulated by the Eulerian formulation, and its deformation was simulated by the Lagrangian formulation. The elastic modulus of the load-cell, the housing, and the shaft was considered 210 GPa with the Poisson’s ratio of 0.33 (mechanical properties of steel alloys).
Fig. 13. Finite element model of the SSRB test setup. (a) Entire model, (b) meshing of the shaft and the bearing in detail, and (c) friction-torque load-cell and its connection pin.
The bearing was simulated as an orthotropic material. The wear was simulated using the developed wear-processor program.
The wear coefficients in the simulation were implemented from the corresponding flat-on-flat experiments.
4.3. Verification The finite element model was verified with experiments performed on radial ORKOT bearings. The bearing material was the same as presented in Section 3.2. The bearings were acquired from the manufacturer with a diameter of 40 mm, width of 10 mm, and thickness of 3 mm (Fig. 14). They were mounted with the freeze fitting method inside the backing. In this experiment the counterface was made of 42Cr–Mo–4V steel alloy, similar to the flat-on-flat experiments. It was tried to roughen the surface of the counterface similar to the counterface
668
A. Rezaei et al. / Wear 296 (2012) 660–671
of the flat-on-flat setup. However, in practice it is very difficult to obtain the same surface topography. The average surface roughness of the counterface was Ra ¼2.6 mm (Fig. 15). Table 7 depicts the details of the performed experiment, in which the applied load by the pneumatic muscle was about 1200 N and the shaft rotation was 0.5 rad/s. The test was performed for 24 h. The wear volume in the experiment was measured through the calculation of the weight loss in the bearings (the same method as for flat-on-flat experiments). The weight loss after 24 h was 0.019 g. The corresponding wear volume for the test was 15.2 mm3. Table 7 summarizes the experimental conditions and the results. As noticed before, the developed finite element wear-processor was used to simulate the experiment. Based on the experimental measurements the inputs of the finite element model become as Table 8. The wear coefficients were implemented from the flat-on-flat experiments. Considering the surface roughness of the counterface (Ra 2.6 mm) and the sliding velocity (10 mm/s), the relevant flat-on-flat experiments are A-P2, B-P4 and C-P6 (Table 4). In Fig. 16, KH versus Pv (pressure velocity) is plotted for these experiments. Fitting a line to the experimental results provides a Pv dependent wear coefficient (Eq. (13)). This equation was used as the wear coefficient function in the simulations. kH ¼ ð2:09 1012 ÞPv þ ð2:68 109 Þ
Fig. 17 compares the simulation and experimental results for the friction and vertical forces. The friction coefficient calculated from the experiment is about 0.093. The normal force is about Table 8 The input of the simulation. Simulation inputs Applied force (kN) Shaft rotation (rad/s) Simulation duration (h) Shaft diameter (mm) Bearing diameter (mm) Bearing thickness (mm) Clearance of the load-cell connections (mm) Friction coefficient between the shaft and the bearing Friction coefficient in the load cell connections Simulation
6 0.5 24 39.62 40.10 3 0.1 0.1 0.1 A
ð13Þ
Fig. 16. Wear coefficient (kH) versus Pv (pressure velocity).
Fig. 14. Radial composite bearing acquired from the manufacturer.
Fig. 15. Counterface used in the SSRB experiments.
Fig. 17. Comparison of the simulation and experimental results for the normal and friction forces. FF: friction force, FN: normal force, and COF: coefficient of friction.
Table 7 Performed experiments on the SSRB setup. Experiment characteristics Applied vertical load (kN) Shaft rotation (rad/s) Test duration (h) Shaft diameter (mm) Average roughness of the shaft (mm) Bearing diameter after fitting process (mm)
Experiment results 6 0.5 24 39.62 2.6 40.10
Weight before the test (g) Weight after the test (g) Weight loss (g) Density (g/mm3) Wear volume (mm3)
5.626 5.607 0.019 1.25 15.2
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 18. Wear depth in the bearing’s surface simulated by finite element simulation.
669
Fig. 19. Contact pressure distribution on the surface of the bearing for various steps of the wear process.
6150 N for the experiment and 6000 N for the simulation. The friction force is about 580 N for the experiment and 604 N for the simulation. The small deviations are due to the fluctuation of the applied force by the pneumatic muscle and the errors of the friction measurements. In general, the errors are very small and the simulation results correspond closely to the experiments. Fig. 18 depicts the graph of the wear depth of the bearing’s surface nodes. The maximum wear depth is 0.089 mm. The area under the graph is equal to the worn area. Multiplication of the area with the bearing width results into the wear volume, which is 18.4 mm3. The wear volume measured in the experiment is 15.2 mm3. The deviation of the simulation results is 18% for the test. This can be explained by the following causes:
The first and simplest error source is the calculation method of
the wear volume in the experiments. In reality the wear rate of the fibers and matrix is not equal. The volume fraction of the matrix and fibers in the worn particles are different from that of the composite bearing. Therefore, the density of the worn particles will not be equal to the density of the bearing. Because the wear volume in the experiments is calculated based on the density of the bearing’s material, this can cause inaccuracy in the calculation of the experimental results. The second reason is that the surface topography of the counterface differs from that of the flat-on-flat tribotests. Surface topography may have a severe effect on the wear of the material. Therefore, the wear coefficients obtained from the flat-on-flat experiments are not perfectly adaptable to the SSRB experiments. The third possible error source can be the different tribosystem of radial and flat bearings. In radial bearings, wear debris might be trapped on the surface of bearing and play the role of solid lubricant. While in the flat bearings wear debris are swiped away from the bearing’s surface [20]. Therefore the wear rate of the tested radial bearing can become slightly lower than that of the flat bearing.
The evolution of the contact stresses on the bearing is illustrated in Fig. 19. This figure plots the contact stress on the bearing surface, for various steps of the wear process. As seen, the contact pressure progressively decreases during the wear process. The maximum contact pressure decreases about 22 MPa after 24 h. By progressing the time of wear, the stress level decreases with a lower rate. However, Fig. 20 shows that the integration of the contact pressure over the contact area grows linearly by progress of the wear.
Fig. 20. Integral of the contact pressure over the contact area during the wear process.
5. Parametric study: clearance effect In this section, the developed finite element model is employed to investigate the effects of the clearance, between the composite bearing and shaft, on the wear of the bearing. The simulations were performed for the clearance range between 0.1 mm and 1.2 mm (0.1, 0.3, 0.6, 0.9, and 1.2 mm). In these simulations, the sliding velocity was 0.5 rad/s (10 mm/s), the vertical load was 6 kN, the shaft diameter was 40 mm, the wear coefficient was as Eq. (13) and the simulation time was 24 h. The simulations were performed based on the configuration of the SSRB setup. In the models, 232,324 elements were utilized, and the friction coefficient was 0.1. Fig. 21 shows the effect of the clearance on the contact stresses at the beginning and end of the wear process. At the first hour of the process the maximum contact pressure for the clearances of 0.1, 0.3, 0.6, 0.9 and 1.2 mm is respectively 29, 38, 46, 52, and 58 MPa. After 24 h, the maximum pressure for those clearances is respectively 20, 22, 24, 26, and 28 MPa. As seen, after evolution of the wear, the stress graphs get closer to each other. In the beginning, the difference between the maximum pressures for the clearances 0.1 and 1.2 mm is about 30 MPa, while this quantity is 8 MPa after 24 h. Fig. 22 depicts the graphs of the wear depth after 24 h of the process. The maximum wear depth for the clearances 0.1, 0.3, 0.6, 0.9, and 1.2 mm is respectively 0.115, 0.105, 0.092, 0.078 and 0.062 mm. The wear volume of the bearing for each graph was
670
A. Rezaei et al. / Wear 296 (2012) 660–671
Fig. 21. Effect of the clearance on the contact pressure of the bearing at the beginning (left) and end of the wear process (right).
Fig. 23. Effect of the clearance on the contact area during the wear process.
Fig. 22. Effect of the clearance on the wear depth at the end of the wear process.
calculated by computing the area under the graphs. The calculations indicate that the wear volume for those clearances is respectively 18.2, 17.5, 17.4, 17.5, and 17.8 mm3. These quantities show that the clearance does not have a significant effect on the wear volume. Fig. 23 shows the effect of the clearance on the contact area. It is seen that the contact area grows with a similar rate for all the graphs. In the beginning of the process the contact area for the clearances 0.1, 0.3, 0.6, 0.9 and 1.2 mm is respectively 310, 230, 180, 150, and 140 mm2. After 24 h, those quantities are 460, 360, 300, 265, and 245 mm2.
6. Conclusion An adaptive finite element wear processor was developed to simulate the wear of composite sliding bearings through the movement of the contacting nodes. The developed wear processor performs an iterative remeshing not only on the contact elements but also on the proximity of them. With this method of remeshing, a large wear depth independent of the size of the contact elements can be simulated.
The wear phenomena were modeled via the interaction of normal stresses at the contacting elements, relative sliding of the contacting nodes, and empirically determined wear coefficients. Wear coefficients were obtained using the flat-on-flat experimental setup designed especially for tribotesting of flat bearings. This is essential to obtain the wear coefficients for different contact pressures and sliding velocities. Applying the pressure and sliding velocity dependent wear coefficient provides a better wear model for more complex contact conditions, such as radial bearings, where the contact pressure is not uniformly distributed. This wear simulation also helps to study the evolution of the contact stresses during the wear process, which is not easily measurable with experimental techniques. In this study, particularly, the simulations show that the contact pressure in the radial bearing decreases during the wear process. In the beginning, the pressure decreases with a high ratio and by progression of the wear it decreases with a lower ratio. On the contrary, the contact area increases during the wear process. In the beginning, it increases with a high ratio and by progression of the wear it increases with a lower ratio. However, the integral of the contact pressure over the contact area increases with a fixed ratio over the time. The results also show that while the wear depth is directly dependent to clearance size, the wear volume is not influenced considerably by the initial clearance size. In summary, comparing the simulations and the experiments proves that the developed model can be a useful tool in the prediction of the wear of radial composite bearings. In a similar
A. Rezaei et al. / Wear 296 (2012) 660–671
way, effects of the other parameters such as e.g. material properties and friction coefficient can be predicted using this method. Considering the costs of the tribological experiments, specifically for full-scale experiments, these simulations can help to reduce the number of the required experiments. These simulations will be more important when the geometry of worn surface is more complex, and the wear progress cannot be calculated easily using the analytical solutions.
References [1] S. Mukras, N.H. Kim, W.G. Sawyer, D.B. Jackson, L.W. Bergquist, Numerical integration schemes and parallel computation for wear prediction using finite element method, Wear 266 (2009) 822–831. [2] V. Hegadekatte, N. Huber, O. Kraft, Finite element based simulation of dry sliding wear, Modelling and Simulation in Materials Science and Engineering 13 (2004) 57–75. [3] V. Hegadekatte, N. Huber, O. Kraft, Modeling and simulation of wear in a pin on disc tribometer, Tribology Letters 24 (2006) 51–60. [4] V. Hegadekatte, J. Hilgert, O. Kraft, N. Huber, Multi time scale simulations for wear prediction in micro-gears, Wear 268 (2010) 316–324. ¨ [5] A. Soderberg, S. Andersson, Simulation of wear and contact pressure distribution at the pad-to-rotor interface in a disc brake using general purpose finite element analysis software, Wear 267 (2009) 2243–2251.
671
[6] J.J. Rawlinson, K.P. Punga, K.L. Gunsallus, D.L. Bartel, T.M. Wright, Wear simulation of the prodisc-l disc replacement using adaptive finite element analysis, Journal of Neurosurgery-Spine 7 (2007) 165–173. [7] C. Gonza´lez, A. Martı´n, J. Llorca, M.A. Garrido, M.T. Go´mez, A. Rico, J. Rodrı´guez, Numerical analysis of pin on disc tests on Al–Li/SiC composites, Wear 259 609–612. [8] ORKOT, Marine bearings Engineering Manual, December 2010. /http://www. orkot.comS. [9] K.C. Ludema, Friction Wear Lubrication: A Textbook in Tribology, CRC Press, 1996. ~ [10] P. Podra, S. Andersson, Simulating sliding wear with finite element method, Tribology International 32 (1999) 71–81. [11] V. Hegadekatte, Modeling and simulation of dry sliding wear for micromachine applications, University of Karlsruhe, Karlsruhe, 2006 (pp. 111). [12] ABAQUS analysis user’s manual, in: Analysis Procedures. [13] ORKOT, Tlmm & txmm Brochure, December 2010. /http://www.orkot.comS. [14] SKF, Split Plummer Block Housings, Saf Series With Spherical Roller Bearings on an Adapter Sleeve, January 2011. /http://www.skf.comS. [15] SENCY, Miniature Tension and Compression Load Cells, December 2010. /http://www.sensy.comS. [16] FESTO, Fluidic Muscle Dmsp/Mas, January 2011. /http://www.festo.comS. [17] LITEON, Lth-301-07 Data Sheet, January 2011. /http://html.alldatasheet.comS. [18] N. Instruments, Ni 9215; 4-channel, 100 ks/s/ch, 16-bit, 7 10 v Analog Input Module, January 2011. /http://sine.ni.comS. [19] A. Rezaei, W. Ost, W.V. Paepegem, J. Degrieck, P. Debaets, Experimental study and numerical simulation of the large-scale testing of polymeric composite journal bearings: two-dimensional modeling and validation, Tribology Letters 37 (2009) 12. [20] D. An, J.H. Choi, T.L. Schmitz, N.H. Kim, In situ monitoring and prediction of progressive joint wear using bayesian statistics, Wear 270 (2011) 828–838.