Lifetime prediction of linear slide rails based on surface abrasion and rolling contact fatigue-induced damage

Author’s Accepted Manuscript Lifetime prediction of linear slide rails based on surface abrasion and rolling contact fatigue-induced damage Dongwook Kim, Luca Quagliato, Donghwi Park, Naksoo Kim www.elsevier.com/locate/wear

PII: DOI: Reference:

S0043-1648(18)30603-3 https://doi.org/10.1016/j.wear.2018.10.015 WEA102525

To appear in: Wear Received date: 20 May 2018 Revised date: 29 August 2018 Accepted date: 1 October 2018 Cite this article as: Dongwook Kim, Luca Quagliato, Donghwi Park and Naksoo Kim, Lifetime prediction of linear slide rails based on surface abrasion and rolling contact fatigue-induced damage, Wear, https://doi.org/10.1016/j.wear.2018.10.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Lifetime prediction of linear slide rails based on surface abrasion and rolling contact fatigue-induced damage Dongwook Kim1, Luca Quagliato1, Donghwi Park1, and Naksoo Kim1 1

Sogang University, Department of Mechanical Engineering, 35 Baekbeom-ro, Daeheung-dong, Mapo-gu, Seoul, Republic of Korea

Corresponding author’s details: Prof. Naksoo Kim Adam Schall Hall, 601, Sogang University, 35, Baekbeom-ro, 04107, Seoul (South Korea) Tel.: +82 27038635, Fax: +82 27120799, Email address: [email protected]

Abstract The research presented in this paper deals with the development of an integrated numerical model for the estimation of the incremental surface wear and damage accumulation in linear slide rails. The target is the estimation of the progressive increment of the end-point deflection of the last member of the slide rail during the operational lifetime. The surface abrasion is accounted for by utilizing a modified Archard equation with the aim of estimating the amount of wear along the vertical direction of the slide rails members. In addition to that, the Lemaitre damage model is utilized for the estimation rolling contact fatigue (pitting), considering the total strain and not only the plastic strain. Experiments have been carried out on a small-scale slide rail testing machine in order to define the wear increment on the slide rail inner groove for increasing number of cycles and, accordingly, estimate the modified Archard model constants. In addition to that, the wear parameters for the Lemaitre damage model have been inversely calibrated from the results of tensile tests. A numerical model has been implemented in ABAQUS/Explicit and an external geometry-update subroutine has been employed to update the geometry of the slide rail groove for increasing number of cycles as a consequence of wear and roll contact fatigue. The comparison between numerical and experimental results on real rails have shown a maximum deviation equal to 12.9%, supporting the reliability of the proposed approach.

Keywords: Abrasive wear; Combined wear model; Modified Archard model; Rolling contact fatigue; Modified Lemaitre damage model; Efficient lifetime prediction.

1 Introduction Linear slide rails are largely utilized in several industrial and white-goods products and, during their operational lifetime, they are likely to be subjected to heavy and variable loading conditions. In slide rails products, the continuous rolling and sliding contacts between rail grooves and rotational elements (balls or rollers) result in abrasive wear and contact fatigue. These two issues have to be accounted for a reasonable estimation of the slide rail operational lifetime [1]. Due to the greater hardness of the rotational elements material (balls), in comparison to that of the rails, the inner surfaces of the rail grooves present an increasing wear that results in a reduction of the cross-section of the rail element, with a consequent reduction of its stiffness [2]. The loss of stiffness results in an increasing deflection of the last element of the slide rail, Fig.1, until the moment the deflection increases to such a point that either the rail breaks or the product becomes unusable.

Figure 1 – Slide rail (a) initial state and (b) progressive increment of the deflection due to wear. In the literature, various contributions deal with the estimation of wear and rolling contact fatigue (RCF) for increasing contact distance, or for increasing number of cycles. Most of the scholars focused their attention on the wear of wheels acting on linear rails, such as the case of railway systems, or on bearings applications. In both the above-mentioned cases, as well as for the case of slide rails, rolling elements are in contact with a locally-linear surface thus, although different contact areas are present, similar working conditions, wear and fatigue patterns can be identified. In all these cases, because of friction, lubrication, material properties and contact mechanisms, the surface of the rail shows an increasing loss of material due to wear and RCF [3]. For this reason, although the final applications differ, in principle, from the topic of the present research work, a literature summary concerning the contact problems for wheel-on-rail and ball- and roller-bearings is hereafter proposed, aiming to highlight the wear and fatigue patterns which can arise when rolling and sliding contacts act at the same time. In bearing applications, a lubricant layer is normally placed between the rolling elements and the groove to attenuate the wear. However, as demonstrated in the work of Olofsson et al. [4] and Olofsson [5, 6], during the operational lifetime, some specific contact zones are likely to reach a boundary lubricated condition which may deteriorate into a mild wear contact with a quasi-uniform abrasion of the race surface. The mild wear represents the first stage of the wear accumulation and has been largely

characterized by utilizing the Archard wear model [7], as shown in the work of Ramalho et al. [8] for the case of rail/wheel contact in railway applications or in that of Tao et al. [9] for the prediction of the wear in roller liner guides. In addition to that, for materials with a high strain hardening effect, not only the material hardness, included in the original Archard formulation [7], but also the strain hardening of the material should be considered [10]. This can be done by modifying the original Archard model by including an exponent to the “L” term, in order to account for hardening of the material as the number of cycles increases. However, as proven in Lee at al. [11], both the initial hardness as well as the hardening effect arising during the progressive contact cannot describe alone all the wear phenomenon, thus cannot be regarded as the only influencing parameters for the lifetime prediction. As defined in the work of Feng et al. [12], the prediction of the lifetime for product subjected to sliding and rolling wear is of key importance to avoid any disastrous loss in terms of performances and maintenance. In their work, they successfully developed a lifetime prediction formulation for slewing bearing based on the results of accelerated life tests, allowing to predict the amount of wear based on the applied load. Same concerning slewing bearing application, Potočnik et al. [13] studied the balls contact pressure on inner and outer races, showing how it increases along with the increase of the deformation. In other words, for a constant external load on the bearing, the deformation field along the cross-section of the inner and outer races increase with the number of revolutions of the bearing. This fact is of key importance for the explanation of the progressive deterioration of the race surface in bearing applications, or of the rail surface for the case of slide rails. Once the surface layer of the rail material, directly in contact with the rolling elements, is deteriorated to such a point that fracture occurs, the layer immediately beneath it starts accumulating damage from an already advanced-damage state and will fail for a number of cycles/rotations lower than that which was required for the first layer. During the rolling contact, predominant also in the slide rail application, different rail surface topologies are likely to arise as a consequence of wear evolution. In the first stage, mild wear is likely to arise in those contact zone previously subjected to boundary lubrication. This initial stage is mainly characterized by abrasive wear and, except for the case of relatively high loads, not fatigue phenomena are visible [14]. Following this initial phase, because of the progressive contact, the RCF influence starts to increase and small cracks begin to nucleate and propagate on the less hard material surface. During the crack propagation, the wear remains almost unchanged for an increasing number of cycles and the length of the crack-propagation zone mainly depends on the applied load [15]. Finally, after a certain lifetime interval or, in other words, after a certain number of cycles, the cracks growth result in a large increase of the damage which, eventually, leads to the final failure. As defined in the work of Sadeghi et al. [16], the RCF is the unavoidable failure mode when rolling and sliding element interact with each other. In addition to that, Sadeghi et al. [16] also proposed a summary of the RCF estimations models, highlighting main assumptions and limitations. This progressive failure mechanism has been studied in detail by Sciammarella et al. [17] for the case of railway wheels, where the experimental procedure for the estimation of the RCF has been also detailed. In addition to that, the above-mentioned final failure is represented by the so-called spalling effect, largely studied in the literature [18-22] characterized by the detachment of a portion of the race or rail. The accumulation of more spalling all over the rail or race surface lead, unavoidably, to the ultimate failure of the component. Although several different contributions deal with the estimation of abrasive wear and RCF for bearing and railway applications, only sporadic contributions are focused on slide rail applications and no contributions seem to deal with the combined sliding and rolling contact effect on the wear evolution for these typologies of products. Therefore, the target of the present research work is the definition of an

investigation procedure through which to estimate the progressive surface wear and RCF due to rolling and sliding contact in slide rail application. The surface wear is accounted for by utilizing a modified Archard model including an exponent of the material hardness to account for the material hardening effect. In addition to that, the original Archard formulation [7] is also modified in order to estimate the vertical material loss and not the whole volume loss. This last modification allows correlating the wear to the progressive deflection of the last member of the slide rail for an increasing number of cycles. As concerns the estimation of the RCF, a modified Lemaitre damage model formulation [23, 24] has been employed considering the total equivalent strain and not only the equivalent plastic strain, as in the original formulation. The Lemaitre damage model has been successfully used for the estimation of the tool wear in Cheon et al. [25] and the same formulation is employed also in this research. Since the Lemaitre damage model allows coupling damage state and flow stress of the material, the progressive reduction of the material tenacity can be considered, allowing to estimate the failure of a portion of the rail surface those have reached a critical damage state. To determine the model constants for the modified Archard wear model, laboratory experiments on simple slide rail and under accelerated life-testing conditions have been carried out. In addition to that, the modified Lemaitre damage model constants have been estimated by inverse calibration from the results of the tensile test on specimens realized with the same materials of the rail guides. A numerical model has been implemented in ABAQUS/Explicit and, in order to limit the computational effort, an external geometry update algorithm (GUA) has been developed as well. After the initial estimation of the static deflection of the slide rail, stresses and strains are calculated in the ABAQUS numerical simulation and the results, in terms of stresses and strains, are inputted in the GUA. Considering a linear propagation with the number of cycles, the GUA algorithm estimated the stress state at the end of the first cycles interval, calculates wear and damage, and finally updates the rail groove and balls surfaces, generating the new input files for the following numerical simulation. This procedure is repeated until the final number of cycles, which is a user-defined parameter. The proposed approach has been validated by comparing the numerical results with those of industrial experiments carried out on real slide rail under simulated real-working conditions, for different weight loads and for an increasing number of cycles. In addition to that, two different typologies of slide rails, namely 2- and 4-balls versions, as shown in Fig. 3a and 3b respectively, have been taken into account and tested. The comparison between numerical and experimental results allowed concluding that, for a chosen combination of applied load and number of cycles, the developed approach can estimate the end-point deflection of the last member of the slide rail with an accuracy always higher than 91.4%.

2 Wear and damage models theoretical background In the present research work the original Archard formulation [7], Eq. (1), has been modified to account for the wear along the vertical direction of the inner groove (Fig. 1) of the slide rail as well as to account for possible hardening, or softening, of the material for increasing number of cycles.

V

KWL H

(1)

In Eq. (1), V stands for the amount of volume loss due to abrasive wear, K is a material constant, W is the normal load applied, L is the sliding distance and H is the material hardness. In the original Archard model, being L the only variable, the amount of material volume loss is proportional to the linear distance. Being both K and H material constants, in the modified Archard model proposed in this paper, they have been combined together in the term k . In Addition to that, to account for possible hardening or softening of the material for increasing sliding distance, an exponent has been added to the term N relevant for the number of cycles. Finally, the effects of the load on the abrasive wear are not only to increase the amount of material loss but also to change the shape of the deflection vs cycles curves. This effect is accounted for by adding an exponent to the load term, defined as P . Therefore, the proposed modified Archard model is reported in Eq. (2).

  kPa N b

(2)

Moreover, to account for the rolling contact fatigue on the rail surfaces the Lemaitre damage model has been employed. However, differently compared with the original formulation, since rolling contact fatigue phenomena are likely to arise already at elastic strains, the incremental damage is calculated according to the equivalent (total) strain and not only based on the equivalent plastic strain, as shown in Eq. (3). In Eq. (3), S0 and b are material constants derived from the inverse calibration of the stressstrain curve resulting from the tensile test. On the other hand, Y is defined as the strain energy released rate caused by the damage, as shown in Eq. (4).

1  Y  D   tot   1  D  S0 

Y 

b

 s  eq,vM  2  1    3 1  2   dev  2E (1  D)  3  eq ,vM

(3)

  

(4)

The strain energy released rate depends on the von Mises equivalent stress,  eq ,vM , and on the deviatoric stress, sdev . Both the von Mises and the deviatoric stress are estimated considering the total strain and not only the plastic strain only.

3 Material characterization In order to estimate the constants for the modified Archard model and for the Lemaitre damage model, laboratory experiments have been carried and both experimental procedures and results are reported in this section of the paper. For the case of the Archard model, the pin-on-disc test is normally utilized to determine the model constants those allow to estimate the amount of material loss for a certain number of cycles, or rotations [26]. However, the research presented in this paper focuses on the increase of the vertical displacement of the rail for increasing number of cycles thus, the test through which to determine

the model constants should account for a linear contact with an inversion of the motion after a certain distance, to resemble the real working condition of slide rails (Fig. 1). To this aim, authors designed and realized a testing machine which replicates the movements of the slide rails but slide rails with a limited amount of balls (1/5 of the original model) have been utilized. The utilization of the same rails and balls materials allows to achieve the same ball-groove contact conditions, but the reduced number of balls allows achieving accelerated life testing conditions, reducing the time required to perform the experiments. The accelerated life testing experiments have been carried out on two different typologies of rails, as detailed in section 3.1, where different materials are also employed. Simple tension tests have been carried out on the two materials composing the tested rails in order to calibrate the Lemaitre damage model. Numerical inverse calibration and best fitting of the results was used to acquire damage model constants. In the following two sub-sections of the paper the methodology, the test settings and the results of the material characterization campaign are reported in detail, especially for the case of the linear wear machine for accelerated life tests.

3.1 Accelerated life tests As previously mentioned, a testing machine has been specifically designed and realized to determine the material constants for the modified Archard model, Eq. (2), as presented in Fig. 2. The rationale behind the choice of realizing a specific machine for the test is given by the fact that, in slide rail application, the work hardening plays an important role in changing the mechanical properties of both rails and balls, thus a standard pin on disc test would not be appropriate to account for these phenomena.

Figure 2 – Abrasive wear accelerated life-testing machine. As previously mentioned, two different typologies of rails have been considered, as shown in Fig. 3 (concept) and Fig. 4 (real products) and are normally utilized for different loading conditions. The 2-balls rails (Fig. 3a and 4a) are realized with the ASTM A109 steel whereas the 4-balls rails (Fig. 3b and 4b) with the SAE 1010 steel. The higher number of balls and the better mechanical properties of the latter one makes it to be more suitable for higher load applications. In the experiments, five loading conditions have been applied to both typologies of rail by staking and bolting various steel bars (with different weights) on the top of the rail, as detailed in Table 1.

Table 1 – Accelerated life tests loading conditions. ASTM A109 (2-balls rail) SAE 1010 (4-balls rail) 22.5 N (2.3 kg) 46.9 N (4.80 kg) 30.0 N (3.0 kg) 62.5 N (6.4 kg) 39.2 N (4.0 kg) 81.6 N (8.3 kg) 48.3 N (4.9 kg) 100.7 N (10.3 kg) 57.5 N (5.8 kg) 119.8 N (12.2 kg) Due to the higher bearable load of the 4-balls rails, in comparison to the 2-balls ones, heavier loading conditions have been applied to the latter one.

Figure 3 – (a) 2-balls rail and (b) 4-balls rail concepts.

Figure 4 – (a) 2-balls and (b) 4-balls slide rail real products. The test loads have been chosen according to the specific working conditions of the slide rails, normally installed in domestic white goods, such as refrigerator drawers. All the tests have been run utilizing a rotational speed for the motor equal to 120 rpm. As concerns to lubrication conditions, during the manufacturing process, a thin lubricant grease layer of about 0.1 mm is automatically sprayed on the

rails inner race to reduce the friction. However, as it will be shown from the results of the accelerated tests, the progressive friction between balls and rails creates some debris which, eventually, end in mixing with the grease. As the test proceeds, more debris are generated by the wear and the RCF, reducing the effectiveness of the lubrication grease, To measure the increasing deflection of the slide rail, a linear scale element has been added to the test machine, as shown in Fig. 2 and 5. As the test proceeds, the inner groove of the rail element (top, middle, and bottom) will be subjected to increase wear and, after a certain number of cycles, RCF arises, leading to huge increase in the top rail deflection for a small increase of the number of cycles. The linear scale element has an accuracy equal to 0.001 mm thus a precise measurement of the increasing deflection of the slide rail is granted.

Figure 5 – Slide rail test machine section detail and elements. The linear scale element is set to zero before the connection to the top rail element and, once it is connected and the loads are stacked on its top, the initial static deflection of the top member is measured. During the test, only the middle member translates whereas top and bottom ones are fixed. Due to the fact that, the bottom surface of the top member, the top and the bottom surface of the middle member and the top surface of the bottom member will all be subjected to wear, measured by the linear scale, the recorded values from the linear scale sensor must be divided by 4 in order to calculate the vertical wear  , as shown in Eq. (2). Considering the rotational speed of the motor and the accelerated life testing conditions, one test setting, relevant for a rail model (2 or 4 balls) and a loading condition, requires 24 hours to achieve 100,000 cycles, set as target values for the test. Since both the materials utilized for the manufacturing of the top-, middle- and bottom-members, namely the ASTM A109 and the SAE 1010, have a hardness equal to 50HRB and 56HRB, lower than that of the balls, equal to 65HRB, in the present research the deflection caused by increasing wear has been considered to be related only to the sliding elements and not the balls. As it will be shown in the result section, although this assumption neglects the wear on the balls, it well represents the reality of the wear conditions on the slide rail elements and good agreements between numerically estimated and experimental results, for the case of real slide rails, can be achieved. According to the above-mentioned loading conditions, a total of five experiments have been carried out for each of the two slide rails models and the graphical results, in terms of increasing vertical wear on the top element, for increasing number of cycles, are reported in following Fig. 6a, for the ASTM A109 material (2-balls concept rail), and in Fig. 6b, for the SAE 1010 material (4-balls concept rail), respectively.

Figure 6 – Slide rail linear wear test results. Thanks to the higher material hardness as well as to the greater number of balls, the 4-balls concept slide rail presents a lower wear for increasing number of cycles. In addition to that, as shown for the 100.7 N and 119.8 N curves in Fig. 6a, around 60,000 and 80,000 cycles an abrupt increase of the wear is seen, and it has to be related to the initiation of the crack propagation in the rail surfaces, as a consequence of the RCF. Although surface wear and RCF are modelled as two different phenomena, from the results of the experiments shown in Fig. 6 it is not possible to clearly define for which number of cycles the RCF becomes predominant on the abrasive wear thus, for the calculation of the modified Archard model constants, the whole curve, until 100,000 have been considered. By utilizing the Matlab® curve fitting tool, the model constants for Eq. (2) have been calculated for both rails models and are reported in Table 2. These parameters will be subsequently utilized in the numerical model for the estimation of the abrasive wear on the rail inner groove surfaces.

Parameter

k a b

Table 2 – Modified Archard model constants. ASTM A109 (2-balls rail) SAE 1010 (4-balls rail) 5.01E-06 3.77E-06 1.026 0.4666

1.256 0.4265

According to the results presented in Fig. 6, also summarized in Table 2, the maximum deviation between the fitting curves and the experimental values is calculated in 13.3%, for the ASTM A109 material and in 17.8% for the SAE 1010 material whereas the correlation factors (R2) are equal to 0.956 and 0.942, respectively.

3.2 Simple tension test experiments For the determination of the material properties and the model constants for the Lemaitre damage model, tensile tests have been carried out on the two materials utilized for the manufacturing of the 2balls and 4-balls slide rails, ASTM A109 and the SAE 1010 steel alloys respectively. Since these two materials are made available to the market into two different shapes, the first one in plates whereas the latter in rods, two different typologies of specimens have been considered, as shown in Fig. 7 and Fig. 8. The simple tensions tests have been conducted utilizing a Universal Testing Machine (INSTRON 5882), under room temperature conditions and utilizing a tensile speed of 2 mm/min. The tests have been repeated three times for each specimen to assure the repeatability of the results and the average curves are reported in Fig. 9a, for the ASTM A109 material, and in Fig. 9b, for the SAE 1010 material, respectively.

Figure 7 – ASTM A109 tensile test specimen (ASTM E8 standard, dimensions in mm).

Figure 8 – SAE 1010 tensile test specimen (ASTM E8 standard, dimensions in mm).

Figure 9 – Tensile test and numerical inverse calibration curve for the (a) ASTM A109 material and (b) for the SAE 1010 material.

The results of the experiments have been inputted in a numerical model implemented in ABAQUS/Standard for the inverse calibration of the mechanical properties and for the calculation of the Lemaitre damage model parameters. Thanks to the implementation of a user-defined subroutine, the damage is calculated based on the total strain and not on the basis of the only plastic strain, as in the original Lemaitre damage model. This approach allows accounting for the damage accumulation already in the elastic part region and, as it will be shown in the result section, it allows obtaining a very good agreement between experimental and numerical results. In the tensile test numerical simulation, only one side of the specimen, for a length equal to 50 mm, has been fixed whereas to the other part the same displacement conditions of 2 mm/min has been set. Both tensile test specimen models have been meshed utilizing the C3D8R element type for a total count of 10,500 elements for the ASTM A109 specimen and 24,600 elements for the SAE 1010 specimen, respectively. The numerical models’ implementation and the fracture points are reported in following Fig. 10.

Figure 10 – Numerical model and fracture of the (a) ASTM A109 and SAE 1010 (b) specimens. Finally, from the results of the inverse calibrations, the parameters for the Lemaitre damage models have been estimated as reported in Table 3. In Table 2, Dcr stands for the critical damage value.

Parameter

b S0 Dcr

Table 3 – Lemaitre damage model constants. ASTM A109 (2-balls rail) SAE 1010 (4-balls rail) 32.7 45.6 0.56

0.78

0.68

0.72

If an element of the mesh reached the critical damage value it is considered to have reached the limit of its loading capability and it is therefore removed from the mesh.

4 Numerical model implementation In order to estimate the end-point deflection of the last member of slide rail products for increasing number of cycles, a numerical model has been implemented in ABAQUS/Explicit. Moreover, an external geometry update algorithm (GUA) has been developed to allow the update of the rail inner groove surface as a consequence of both wear and RCF. The GUA has been implemented in a Fortran95 based program which receives, as input, the results of the ABAQUS numerical simulation, estimates wear and damage for each node/element of the mesh and automatically generates the new input file for the following numerical simulation. In order to reduce the computational time, wear and RCF are estimated on a userdefined cycles-interval which means that, between two consecutive intervals, the variation of the rails’ surface is assumed to vary according to a linear law. Although the nature of the wear is, in principle, not linear, the utilization of a 10,000 cycles-interval grants a good accuracy while limiting the computational time. As concerns the geometry update algorithm (GUA), its working principle is shown in Fig. 11.

Figure 11 – ABAQUS-GUA interaction procedure and wear and RCF geometry update. For the update of the rail groove geometry in the GUA, the nodal results coming from the ABAQUS numerical simulation are utilized and the following rationale is applied. As concerns the abrasive wear, its depth is estimated for each one of the nodes in contact with the balls and the nodes are moved inward, along the rail thickness, of the relevant wear-caused-distance, as shown in Fig. 12a. In relation to the RCF, the total equivalent strain (elastic and plastic) resulting from the numerical simulation is inputted in the geometry calculation algorithm and, if one or more elements of the mesh have reached the critical damage value, the nodes relevant for those elements are moved inward along the normal direction, Fig. 12b. Although the abrasive wear is estimated only on the surface nodes, the RCF is estimated for all the elements of the mesh thus the damage can start its accumulation also on elements not directly in contact with the balls. Following this rationale, wear and RCF are modeled as different phenomena, according to the formulations presented in previous section 3, but both act at the same time as a sort of superpositioning effect. The wear effect is calculated according to the material properties and has an exponential effect based on the number of cycles “N” (variable) and the exponent “b” (constant, materialbased). For this reason, after a certain number of cycles, the wear of all the elements those have been in contact with the balls will be the same. No local-wear effect on the mesh is considered in the developed numerical model. On the other hand, the RCF is considered as a local phenomenon and is related to the specific stress and strain field on the nodes of the mesh. This competitiveness between wear and RCF

makes some element of the mesh to be completely erased due to wear and some other to fail due to RCF although the wear has had only a faint effect on them. The combination of both these effects allows estimation of the increasing gap between balls and rails inner grove which results in the increase of the endpoint deflection of the slide rail.

Figure 12 – (a) Abrasive wear and (b) RCF mesh update concepts. The numerical models implemented in ABAQUS/Explicit have been defined considering all the elements present in the real slide rail products, as reported in Fig. 3a, for the 2-balls type model, and in Fig. 3b, for the 4-balls type model. Both models are shown in Fig. 13 (lateral view) and Fig. 14 (crosssection view).

Figure 13 – Numerical model implementation showing load, constraints and boundary conditions. After the model settings, the first step is represented by the calculation of the static deflection of the rail due to the load applied to its top member. This initial step allows estimating the starting point of the deflection vs cycles curve. After this initial step, the results contained in the ABAQUS output file are exported to the GUA algorithm which reads the nodes coordinates and the results of the total strain, the deviatoric stress, the equivalent von Mises stress and the contact pressure and calculates the amount of abrasive wear and the damage, for each element of the mesh. Accordingly, wear and damage are linearly propagated for the chosen cycles interval and, once the calculation is completed, the new input file with the new nodal coordinates is generated and the following numerical simulation is started.

Figure 14 – Numerical model implementation, cross-section detail (zoomed views). Aiming to simulate the fix elements those are connected, for instance, with the main frame of the white good product, displacements and rotations of the vertical element have been blocked whereas the load has been distributed on the top element of the top member, as also done in the accelerated life testing experiments. In addition to that, in the contact regions between balls and sliding elements, a finer mesh has been implemented to better estimate the geometry variation for increasing wear and RCF. The different mesh sizes are also highlighted in Table 4 where the values between brackets represent the fine mesh sizes). Based on the original lubrication conditions in the real slide rails, friction has been modeled utilizing the Coulomb friction setting the friction factor to 0.2. Table 4 – Slide rail numerical model mesh settings (dimensions in mm). Element Top Middle Bottom Ball

Length 485 365 338 D=6.34

Element Top Middle Bottom Bracket Ball-D1 Ball-D2

Length 512.5 465.7 510.7 100 D=4.76 D=3.97

2-balls slide rail model Withdraw Mesh size 453.2 20 (0.3) 161 20 (0.3) 0 20 (0.3) 0.3 4-balls slide rail model Withdraw Mesh size 560.2 20 (0.3) 256.7 20 (0.3) 0 20 (0.3) 0 20 0.3 0.3

Mesh no. 20500 75900 28180 2050 (x40)

Mesh type C3D8R C3D8R C3D8R C3D8R

Mesh no. 13120 60370 13120 160 (x2) 2536 (x24) 3400 (x24)

Mesh type C3D8R C3D8R C3D8R C3D8R C3D8R C3D8R

Although friction plays an important role in the onset and development of wear and RCF, the elements of the slide rails assembly in the developed numerical model are not in real motion and only stress and strain fields are estimated based on the contact conditions. For this reason, although generally important in the real working conditions, in the numerical model, friction does not play a key role in the estimation of the vertical deflection of the top member of the slide rail. Nevertheless, as it will be shown in the results section, this simplification has a slight influence on the estimated deflection, which is only slightly lower than the experimental values. In the same way, the mesh sizes reported in Table 4 are the result of a mesh sensitivity analysis aimed at the optimization of the computational time, whose results are reported in section 5.

5 Results 5.1 Mesh sensitivity analysis In order to optimize the mesh sizes for the elements of the slide rail, a mesh sensitivity analysis have been run considering four different mesh levels. The target of the mesh sensitivity analysis is twofold: firstly, the influence of the mesh size on the results can be inferred, secondly, the results variation vs computational time can be. This allows defining the best compromise between the accuracy of the calculation and computational time. In Table 5 the results of the deflection calculated for the five cycles intervals, each one of them equal to 10,000 cycles, relevant for four different fine mesh sizes are reported. The values are related to the 2-balls rail simulation model with a load equal to 25 kgf and the maximum error, among the five cycle intervals, are reported in Table 6 along with the computational time. The validation experiments, utilized as a reference for the mesh sensitivity analysis, are presented in section 5.3. Table 5 – Mesh sensitivity analysis, comparison between experimental and numerical results. Cycles Experiment Mesh 0.1mm Error %_0.1mm Mesh 0.3mm Error %_0.3mm Mesh 0.5mm Error %_0.5mm Mesh 1.0mm Error %_1.0mm

0 6.05 5.84 3.52 5.79 4.31 5.65 6.68 5.42 10.40

10,000 6.75 6.29 7.64 6.24 8.41 6.10 10.79 5.87 14.51

20,000 7.17 6.50 11.12 6.45 11.88 6.31 14.28 6.08 17.98

30,000 7.31 6.70 10.07 6.65 10.86 6.51 13.24 6.28 16.96

40,000 7.6 6.81 12.99 6.77 13.77 6.62 16.15 6.40 19.87

50,000 7.55 6.96 9.82 6.91 10.60 6.77 12.98 6.54 16.68

Table 6 – Mesh sensitivity analysis, maximum errors and computational times. Error % Number of mesh [-] Computational time [min]

Mesh 0.1mm 12.99 515160 460

Mesh 0.3mm 13.77 239050 170

Mesh 0.5mm 16.15 184930 130

Mesh 1.0mm 19.87 142890 120

From the results presented in Table 6, it is clear that the maximum error variation between the 0.1mm fine mesh case and the 0.3mm is lower than those between the previous two mesh approaches as well as the utilization of the 0.3mm fine mesh approach allows also limiting the computational time. For these reasons, a fine mesh size equal to 0.3 mm has been utilized to mesh the regions of the rail grooves in contact with the balls as well as the ball themselves. 5.2 Friction influence analysis In real slide rail products, friction is not constant during the operational lifetime due to the deterioration of the grease lubricant as well as due to the presence of debris in the gap between balls and rails. Although friction plays an important role, the concept of the developed numerical model makes it not to be an important parameter in the calculation, as hereafter explained. Differently compared with the real slide rail, in the developed numerical model the elements of the slide rail are not put into motions and wear and RCF are calculated only on the basis of the stress and strain field arising from the contact, as a consequence of the applied load. This concept has been largely explained in section 4 of the paper and is not here repeated. According to this modeling concept, where elements are not put into motion, friction is not a driving parameter in the estimation of the vertical deflection of the last member of the slide rail. To prove this statement, analysis have been run setting the friction coefficient to 0.05, 0.2 and 0.5. Since the calculation is carried out considering a steady state concept, only the initial static deflection for the four validation cases presented in following section 5.3 have been considered and the results are reported in Table 7. Table 7 – Influence of the friction coefficient of the slide rail deflection. Loading conditions Experiment μ = 0.05 Error%_μ = 0.05 μ = 0.2 Error%_μ = 0.2 μ = 0.5 Error%_μ = 0.5

25kgf 6.05 5.86 3.19 5.79 4.31 5.78 4.48

40kgf 9.15 9.53 -6.33 9.43 -4.68 9.42 -4.48

60kgf 7.8525 7.65 3.43 7.53 5.28 7.53 5.30

80kgf 10.47 10.06 6.78 9.97 8.23 9.97 8.25

Based on the results presented in Table 7, no substantial variation of the static deflection of the last member of the slide rail is seen even if the friction coefficient has been varied in a wide range where the minimum and maximum values are 10 times each other. According to these results, the previous statement concerning the negligible influence of the friction in the numerical calculation of the end-point deflection of the slide rail is verified. 5.3 Validation experiments on real slide rails In order to validate the proposed numerical model, experiments have been carried out on real slide rails installed on drawers aiming to simulate the real working conditions. The rails utilized for these validations experiments are realized with all the balls thus heavy loading conditions has to be applied in order to achieve realistic wear and RCF conditions in a reasonable amount of time. Based on the specifications of the slide rail manufacturing company, two loading conditions for the 2-balls rail, equal

to 25kgf and 40kgf, and two for the 4-balls rail, equal to 60kgf and 80kgf, have been tested. The experimental setting is shown in Fig. 15.

Figure 15 – Test settings on real slide rails (drawer test). The experimental data utilized for the model validation have been obtained from real product testing under real working conditions and, due to the heavy load and the resulting strong dynamic effects, the testing speed was considerably reduced in comparison to that of the accelerated life tests presented in previous section 3.1. Moreover, since real working conditions are applied, two slide rails are tested at the same time thus the applied load is divided by half, accordingly, the time required to achieve the same amount of wear or, in other words, deflection is doubled. 5.4 Slide rail end-member deflection prediction Utilizing the numerical model and the geometry update algorithm (GUA) described in the previous chapter of the paper, numerical simulations have been carried out to verify the reliability and accuracy of the developed numerical model. As previously mentioned, two different numerical models have been implemented, one for the 2-balls and one for the 4-balls slide rail models. Accordingly, as also done for the case of the validation experiments, two different loading conditions for each one of the models have been applied on the top of the last member of the slide rails, as shown in Fig. 13, and equal to 25 kgf and 40 kgf for the 2-balls model and to 60 kgf and 80 kgf for the 4-balls model, respectively. Considering the case of the 4-balls slide rail loaded with 80 kgf, the endpoint member deflection, at the beginning of the simulation (static loading) and after 100,000 cycles, considered as the final target for the test, is shown in Fig. 16. In addition to that, by utilizing the developed numerical model and the GUA, as described in the previous chapter 4 of the paper, the end-point deflection for the above-mentioned four validation cases have been calculated and the results are presented in Fig. 17 and Fig. 18. For the four analyzed study cases, the maximum deviation between experimental and numerical results are calculated in 11.8% and 12.9%, for the 25 kgf and 40 kgf cases (2-ball slide rail), and 12.3% and 10.0%, for the 60 kgf and 80 kgf cases (4-ball slide rail), respectively.

Figure 16 – Full simulation model results for (a) and (b) after 100,000 for the 4-balls slide rail with 80 kgf applied loads (front views are zoomed).

Figure 17 – End-point deflection for increasing number of cycles for the 2-balls slide rail with 25kgf and 40kgf applied loads, simulation vs experiments results comparison.

Figure 18 – End-point deflection for increasing number of cycles for the 4-balls slide rail with 60kgf and 80kgf applied loads, simulation vs experiments results comparison.

6 Discussion Considering the positive results comparison highlighted in chapter 5 of the paper, with a maximum deviation between experimental and numerical results equal to 12.9%, the proposed approach seems to be valid for the estimation of the end-point deflection of slide rails. Moreover, the previously mentioned assumption concerning the negligible effect of the friction in the estimations made by means of the developed numerical model is verified as well. In addition to that, other remarks concerning the developed model and the results should be addressed, as hereafter summarized. The time required for an industrial testing of a couple of slide rails, as shown in Fig. 15, requires between 8 to 13 days, according to the loading conditions and the typology of rail, especially in terms of material and geometrical design. On the other hand, by means of the proposed approach, the material characterization for one specific case last for approximately 24 hours and, once enough study cases in the min-max load range have been tested, the model constants for the modified Archard model can be estimated. Moreover, the Lemaitre damage model constants are also estimated in a relatively small amount of time by means of a room temperature tensile test. In addition to that, once the ball-rail material combinations have been tested, they can be applied also to different product those share the same combination of materials, even though their geometry is considerably different. Thus, the proposed approach allows to strongly reduce the time required to estimate the progressive increase of the slide rail deflection and can be utilized for an effective product life estimation. In addition to that, considering the results presented in Fig. 17 and Fig. 18, a pattern in the progressive deflection can be inferred and, for both typologies of rails, three main wear behavior regions are identified. In the first part, the lubricant layer sprayed in the inner rail surfaces progressively reduces its effect till the point of critical lubrication, which coincides with the beginning of mild wear. Although the transition zone is clearer for the 4-ball slide rail, due to higher applied load, Fig. 18, it is also visible for the case of the 2-ball slide rail, Fig. 17. After this initial phase, the inner surface of the rail grooves is progressively eroded by the rolling contact with the balls, a fact which brings a steadily increase of the end-point deflection. Finally, after a certain number of cycles, the end-point deflection abruptly increases due to

arising of diffused RCF, which brings to the detachment of an ever-growing portion of the slide element groove surface, with the consequent increase of the deflection. This third phase is simulated in the numerical model by the deletion of several surface elements of the mesh those make the end-point deflection to abruptly increase with a largely higher slope in comparison to the previous phase. According to the positive comparison between experimental and numerical results, the proposed approach can be utilized to analyze the effect of i) loading conditions, ii) rails grooves design and iii) balls and rails material properties on the velocity at which the wear progresses. In addition to that, if a failure criterion is defined for the end-point deflection, the proposed approach can also be utilized for the definition of the number of cycles the slide rail can operate before it has to be replaced. Ultimately, thanks to the smart implementation and the presence of an external program for the estimation of the geometry update for specific cycles-intervals, computational time can be also saved, making the proposed approach to be a reasonable compromise between accuracy and experimentcomputational time.

7 Conclusions In the research presented in this paper, an innovative approach through which to calculate the progressive deflection of linear slide rail, as a consequence of abrasive wear and rolling contact fatigue, is proposed and validated. The model constants, to be utilized in the modified Archard model, have been derived from the results of accelerated life tests carried out on simplified rails where only 1/5 of the original number of balls was installed. In addition to that, the constants for the Lemaitre damage model have been estimated from the inverse calibration of the results of tensile tests carried out on the materials composing the slide rail members. Thanks to the utilization of the total strain, and not only the plastic strain, the damage accumulation is considered to start already from the elastic range allowing to have a good approximation of the real RCF behavior of the rails during their operational lifetime. Finally, although the ball wear is ignored in the computations, the slide rail end-point deflection vs cycles is estimated with a maximum error equal to 12.9%, proving the low influence of this simplification on the results. In fact, the overall time required for the estimation of the results for a single case, by means of the proposed approach, is of 5 days for the accelerated life tests, one day for the tensile tests and one day for the computational time required by the numerical model. On the other hand, in the case of a fullexperimental approach, approximately 10 days are required to reach 150,000 cycles, thus a strong timereduction can be achieved. According to these considerations, the proposed approach can be utilized for a successful estimation of the slide rail life-time according to a defined failure criterion and can allow predicting after how many cycles the products should be replaced in order to avoid excessive deflection. In addition to that, it can also be extended to different typologies of products where both abrasive and fatigue wear are present and where the progressive variation of the geometry is aimed to be estimated.

Acknowledgment This work was supported by the Technological Innovation R&D Program (S2296061) funded by the Small and Medium Business Administration (SMBA, South Korea).

Data availability The raw and processed data required to reproduce these findings are available on request to the corresponding author.

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Highlights:

    

Modified Archard wear model to estimate the vertical wear Modified Lemeitre damage model to include the elastic strain contribution Numerical model to estimate the rail members deflection due to wear and RCF Reduction of the computational time due to external program Reliable experimental-numerical approach for the slide rail lifetime prediction