# A Stress-Criterion-Based Model for the Prediction of the Size of Wear Particles in Boundary Lubricated Contacts

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## Abstract

In this paper, the formulation and validation of a model for the prediction of the wear particles size in boundary lubrication is described. An efficient numerical model based on a well-established BEM formulation combined with a mechanical wear criterion was applied. The behavior of the model and particularly the influence of the initial surface roughness and load was explored. The model was validated using measurements of the wear particles formed in steel–steel and steel–brass contacts. In the case of steel–steel contact, a reasonable quantitative agreement was observed. In the case of steel–brass contact, formation of the brass transfer layer dominates the particles generation process. To include this effect, a layered material model was introduced.

## Keywords

Wear particles size Simulation Boundary element method## List of symbols

- \(x,y\), \(\tilde{z}\)
Spatial coordinates, \([{\text{m}}]\)

- \(p(x,y)\), \(s(x,y)\)
Normal and tangential surface stresses, \([{\text{Pa}}]\)

- \(u(x,y)\)
Deflection due to pressure \(p(x,y)\), \([{\text{m}}]\)

- \(E,E_{1} ,E_{2}\)
Young’s modulus of the materials in contact, \([{\text{Pa}}]\)

- \(\nu ,\nu_{1} ,\nu_{2}\)
Poisson’s ratio of the cylinder and the substrate, \([ - ]\)

- \(E^{'}\)
Composite Young’s modulus, \([{\text{Pa}}]\) \(\frac{2}{{E^{'} }} = \frac{{1 - \nu_{1}^{2} }}{{E_{1} }} + \frac{{1 - \nu_{2}^{2} }}{{E_{2} }}\)

- \(z(x,y)\)
Measured surface roughness, \([{\text{m}}]\)

- \(h_{s} (x,y)\)
Separation distance between the bodies, \([{\text{m}}]\)

- \(F_{C}\)
Load carried by surface contacts, \([N]\)

- \(A_{c}\)
Direct contact area, \([{\text{m}}^{2} ]\)

- \(N_{x} ,N_{y} ,N_{z}\)
Number of grid points in spatial directions \([ - ]\)

- \(R\)
Radius of the cylinder, \([{\text{m}}]\)

- \(f_{C}^{b}\)
Friction coefficient in boundary lubrication, \([ - ]\)

- \(B\)
Width of the cylinder, \([{\text{m}}]\)

- \(H\)
Hardness, \([{\text{Pa}}]\)

- \(\sigma_{y}\)
Yield stresses, \([{\text{Pa}}]\)

- \(\sigma_{ij}\)
Subsurface stresses, \([{\text{Pa}}]\)

- \(R_{q}\)
Surface roughness (standard deviation of surface heights), \([{\text{m}}]\)

## 1 Introduction

Failure of equipment due to wear has led to considerable effort in understanding wear and in the development of predictive models for wear. For example, Meng and Ludema [1] have identified 182 equations for different types of wear. Among them were empirical relations, contact mechanics-based approaches and equations based on material failure mechanisms.

One of the most famous and frequently used wear equations was developed by Holm and Archard in 1953 [2]. The model considers adhesive wear and assumes the sliding spherical asperities to deform fully plastic. The wear volume formed in a sliding distance \(s\) under the applied normal load \(F\) equals to \(V_{T} = k*F/H*s\). The coefficient \(k\) is known as a wear coefficient and is frequently used to compare the degree of the wear resistance of systems [3, 4]. It can be regarded as the probability that a wear particle will be formed during contact. In general, this wear coefficient is estimated experimentally. Although Archard originally developed this equation to model adhesive wear, it is now widely used for modeling abrasive, fretting and other types of wear [5] as well.

In general, wear models can be empirical or based on physical phenomena occurring in the system. Empirical models typically fit the experimental data with algebraic equations of arbitrary form. Although they require less investment in the model development, a good predictive capability can be achieved only with a great number of experimental data. Physically based models use the physical properties of the system and physical phenomena to predict wear. By doing so, it is possible to significantly reduce the experimental data needed for the model development. These models can be classified in analytical, semi-analytical and numerical methods. Most of them are used to predict the wear volume and resulting life-time, rather than focusing on the size of the resulting wear particles. On the other hand, the particles size is more relevant in some applications. For example, oxidation of grease thickener and base oil is accelerated by metal particles [6, 7], the severity of this phenomenon depends on the total surface area of the particles and therefore on their size, shape and number. Another example is that an autoimmune reaction of the body is highly dependent on the size of formed wear fragments in artificial joint replacements [8, 9]. Also, the environmental impact of small particles is of increasing concern [10].

*Analytical* wear models often link global parameters, such as load, speed, temperature with the wear volume or the wear track depth [11, 12, 13, 14, 15]. Although they are applicable only in specific cases, these models are very useful for understanding the relationship between governing parameters such as hardness, load or sliding speed.

Among the *numerical* models, a large group is represented by finite element analysis (FEM) [16, 17]. The application of these models is restricted to the analysis of simplified problems, such as smooth surfaces, single asperities, due to the associated computational costs. The numerical models frequently incorporate Archard’s law or its modifications using local contact stress distributions obtained by FEM [18, 19, 20, 21, 22, 23, 24, 25]. A number of researchers also considered multi-scale FEM-MD (molecular dynamic approach) coupling in order to perform nanometer-scale wear simulations [26, 27, 28].

*Semi*-*Analytical* models combine numerical methods with analytical solutions to speed up the calculations. This approach is also applicable to the calculation of the subsurface stresses [29, 30, 31] and temperatures in the contact [32]. Boundary element methods (BEM) are widely spread in this group. These models are extensively used in simulations of the contact of rough surfaces [29, 33, 34, 35] and rely on the assumption that the contacting bodies are semi-infinite.

Application of BEM to model wear may be performed in the same way as is done in FEM, for example, by using Archard’s wear law locally [36, 37]. Alternatively, a particle-by-particle removal model can be built. This type of wear model requires the formulation of a material rupture criterion, which is used to indicate the regions of the wear volume generation. A number of criteria can be found in the literature, including critical accumulated dissipated energy [13], critical accumulated plastic strain [17], critical accumulated damage [38], critical von Mises stress [39] and their variants [14, 15, 40, 41].

Critical damage-based models are typically used to simulate fatigue type of wear, in which the failure criterion is based on a critical number of cycles and stress intensity. A combination of a local Archard wear equation and cumulative damage model (Palmgren–Miner) was used by Morales-Espejel et al. [42, 43] to simulate mild wear and to predict the generation of micro-pits. The model showed a good qualitative agreement with the experimental data in the prediction of the size of the pits [42], as well as in the topography evolution [44]. At the same time, this approach requires the assessment of the fatigue limits at various loads and it introduces additional parameters that need to be determined. In practice, these values are rather hard to determine using representative conditions. This approach is definitely required to simulate the evolution of the surfaces in time. However, in this paper, this time scale is not addressed since the objective here is to predict the particle generation until a steady-state situation has occurred. The objective is not to predict when the particles are generated.

In the current paper, the focus is rather on the running-in process of sliding contacts. The time scale of the problem is not relevant. The wear process is considered not to be time but only strain/stress driven [39]. Oila [45] studied the mechanisms of the material degradation due to wear in high carbon steels and concluded that the wear particle formation is associated with the so called plastically deformed regions beneath the surface. Based on this work, Nelias [46] proposed a running-in wear model, using the accumulated plastic strain criterion to distinguish these regions. Bosman et al. [39] employed the Von Mises yield criterion to identify the plastically deformed regions and simulated the wear particles generation. In addition, they combined a thermal and wear model [39, 47, 48]. They successfully applied the combined model to predict the transition from mild to severe wear, as well as the level of severe wear and running-in. It should be noted that the application of the stress-based criterion is computationally more efficient than the plastic strain-based criterion. A dense mesh is required which results in much longer calculation times for the plastic strains.

In the present paper, the attention will be given to steel–steel and steel–brass materials, frequently encountered in rolling element—cage contacts in bearings. A simplified version of the model developed by Bosman et al. [39] will be used here. An engineering semi-analytical model (BEM) for the simulation of wear particle formation in boundary lubrication will be formulated based on a critical Von Mises stress and validated using experimental measurements of the wear particles’ size. The model can be combined with the Archard’s wear law to estimate the number of generated particles, as discussed in the current paper.

## 2 Materials and Methods

### 2.1 Experimental Procedures

Experimental data on the wear particles size formed in boundary lubrication sliding contacts of steel–steel and steel–brass were taken from an accompanying publication by the authors [49].

In the current paper, several additional sliding tests of steel cylinder–brass disk contacts were performed to study the formation of a brass transfer layer and obtain its thickness (it was not measured in Ref. [49]). The tests were performed at the same conditions as in Ref. [49] and are described below. Experiments were conducted under boundary lubrication conditions with a polyalphaolefin oil (PAO) as a lubricant. The viscosity of the oil was 68 cSt at 40 °C. AISI 52100 steel cylinders with surface roughness (root mean square, \(R_{q}\)) of around 50 nm were worn against brass disks with approximately 760 nm roughness. The cylinder was stationary, while the disk was rotating. The radius of the wear track was 17 mm. The radius of the cylinder was 2 mm and the width was 6 mm. The normal load was taken to be 2.5, 5 or 10 N (corresponding to 88, 130 and 175 MPa) under a constant temperature of 80 °C and a sliding velocity of 0.01 m/s.

To obtain the thickness of the transfer layer formed during the steel–brass wear tests, indentation marks were placed across the contact line of a cylinder before the test. Subsequently, the roughness profile, including the dents, was measured. After the wear test, the profile measurement was taken again and the initial and worn profiles were matched using the indent marks. After the test, an increase in the initial height profile of the cylinder was observed due to the presence of the transfer layer. The thickness of the transfer layer was obtained by measuring the difference in the height between the worn and initial profiles.

The roughness measurement was taken using a Keyence VK9700 laser scanning microscope. Indentations and hardness measurements were taken using a LECO micro-hardness tester LM100AT.

### 2.2 Simulation Parameters

During the simulations, a cylinder-on-disk configuration was used, corresponding to the experimental conditions [49], as discussed in the previous section.

Properties of the materials

Property | Cylinder | Disk, steel | Disk, brass |
---|---|---|---|

\(E\), GPa | 210 | 210 | 88 |

\(\nu\) | 0.3 | 0.3 | 0.3 |

\(H\), GPa | 6 | 1.8 | 2.64 |

\(\sigma_{y}\), GPa | – | 0.64 | 0.94 |

\(R\), mm | 2 | ∞ | ∞ |

\(R_{q}\), nm | 50 | 610 | 760 |

Skewness | −0.27 | −0.84 | −1 |

Kurtosis | 8.6 | 4.46 | 4.94 |

Slope | 0.22 | 0.28 | 0.15 |

### 2.3 Contact Model

In the present model, the contacting bodies were assumed to be semi-infinite solids and both surfaces were assumed to be rough. In tribological systems running in boundary lubrication, the elastic properties near the surface may be significantly different from the bulk due to the presence of chemically/physically absorbed layers. However, as it was shown in [48], the thickness of the altered layers is relatively small and the presence of the layer can be neglected in the subsurface stress state calculation. Therefore, in this work, a homogeneous material model was assumed, unless explicitly mentioned.

*y*direction) was considered. Periodic boundary conditions were applied in this direction to mimic the full cylinder geometry.

*f*

_{ c }

^{ b }·

*p*). Equation (2) can be used for the calculation of the displacements in homogeneous [48], but also layered systems. For a layered body, expressions for \(K + S\) are given in the frequency domain [51, 52].

Details are found in references [30, 53]. In case of layered surfaces, these functions are given in the frequency domain [51].

### 2.4 Wear Particle Formation Criterion and Wear Model

### 2.5 Surface Roughness Simulation

The influence of the initial surface roughness on the wear particle’s size was explored on artificially generated surfaces with different values for \(R_{q}\), following the approach from Hu [55].

For each surface type, three surfaces were generated: R_{q} = 500, R_{q} = 1000 and R_{q} = 1500 nm for the softer surface and R_{q} = 50, R_{q} = 500 and R_{q} = 1500 nm for the harder surface.

## 3 Results and Discussion

The simulations were done using homogenous material properties unless explicitly mentioned. The computational area was a square with a size of 4 times the Hertz contact radius.

### 3.1 Numerical Convergence

In this section, the convergence of the wear algorithm with respect to the particle’s size is evaluated. Simplified simulations were performed by assuming that the steel disk is worn against a rigid flat surface using a normal load of 10 N.

*d*

_{ x }=

*d*

_{ y }= 305,

*d*

_{ z }= 95 nm) to \(N_{x} \times N_{y} \times N_{z} = 320 \times 320 \times 64\) (

*d*

_{ x }=

*d*

_{ y }= 76 nm, d

_{z}= 95 nm). Variation of the average wear particle size is given in Fig. 4a (normalized to the value at the lowest grid). It can be seen that with the increase in the mesh density, the mean wear particle’s size decreases and converges to a certain value. It does not change significantly with a further increase in the grid points.

In addition, the influence of the number of grid points in vertical direction on the results was studied. The number was gradually increased from \(N_{z} = 8\) (\(d_{z} = 1500 \,{\text{nm}}\)) to \(N_{z} = 64\) (\(d_{z} = 95 \,{\text{nm}}\)). The results are presented in Fig. 4b. As it was mentioned earlier, during the wear simulation, an interpolation is used in the z direction. Due to the interpolation, the influence of the initial depth resolution is decreased and the size of the particle does not depend on it, as shown in Fig. 4. More details in the description of the numerical algorithm are found in Ref. [54].

### 3.2 General Simulation Results

In this section, some typical simulation results are shown for a steel–steel contact, at a load of 10 N and a grid of \(N_{x} \times N_{y} \times N_{z} = 240 \times 240 \times 64\) points with \(d_{x} = d_{y} = 102 \,{\text{nm }}\) and \(d_{z} = 95 \,{\text{nm}}\).

The wear volume per iteration follows the same trend. In the beginning, the wear volume per iteration step is high. It results in a large wear volume in the first 50 cycles. Later, the wear volume per iteration stabilizes and the accumulation of the wear volume becomes slower.

Based on the simulations of these cases, it can be concluded that some of the major characteristics of the wear process were captured. The running-in process is represented by the evolution of the surface profiles, wear volume, wear particles size and average contact pressure. In the following sections, the influence of the surface roughness and experimental validation of the model will be addressed.

### 3.3 Wear Particles Size

In case of brass, both AFM and DLS do not show noticeable change in the particles size with load, in contrast with the model predictions. According to both DLS and AFM measurement, the brass particles have to be smaller than the steel particles, but the simulation shows an opposite behavior. In addition, the brass transfer layer was found on the surface of a cylinder after testing. It was therefore concluded that a different primary wear mechanism was encountered in the steel–brass contact, which will be discussed later.

Specification of the particle’s sizes, steel–steel contact

Property | AFM, [49] | Theory | Load, N |
---|---|---|---|

Length, nm | 747 | 442 | 10 |

Width, nm | 541 | 245 | 10 |

Thickness, nm | 145 | 197 | 10 |

Length, nm | 600 | 267 | 2.5 |

Width, nm | 476 | 161 | 2.5 |

Thickness, nm | 129 | 151 | 2.5 |

### 3.4 Influence of Surface Roughness on Wear Particles Size

### 3.5 Brass Transfer Film and Wear of the Layered Surface

Hardness of the brass

Hardness of the surface, GPa | 2.64 |

Hardness of the transfer film, GPa | 2.96 |

- 1.
The particles detached from the brass disk only form a transfer film and do not create loose wear particles.

- 2.
Loose wear particles are only formed from the transferred layer with a given (uniform) thickness and based on the critical Von Mises stress criterion.

- 3.
There is no return of the brass wear particles.

Specification of the particle’s sizes, steel–brass contact

Property | AFM, [49] | Theory | Load, N |
---|---|---|---|

Length, nm | 751 | 635 | 10 |

Width, nm | 521 | 262 | 10 |

Thickness, nm | 126 | 95 | 10 |

Length, nm | 543 | 300 | 2.5 |

Width, nm | 397 | 195 | 2.5 |

Thickness, nm | 115 | 89 | 2.5 |

### 3.6 Discussion

The degradation of surfaces during the wear process is complex. It involves the influence of contact pressure, microstructural changes, material transformation, chemical reactions and material properties variation [45, 61].

The model described in this paper is a relatively simple method to predict the wear particles size distribution. If the removed volume is known, the size may also be used to calculate the number of wear particles resulting from the wear process. Unfortunately, the considered approach does not allow for the calculation of wear volume directly, due to the assumption of the immediate wear particle detachment if the critical stress is exceeded. In reality, the wear is a gradual, fatigue governed process and sometimes, even if the mentioned criterion is met, the particle will not detach, or if detached, stick to the surface again, and self-heal the surfaces. This leads to a several orders of magnitude overestimation of the wear volume. Therefore, to model the wear volume, at least an Archard’s type of approach will be necessary to introduce the probability of wear particle detachment.

As it was shown in the case of a steel–steel contact, reasonable agreement is found between wear particles size prediction and measurements. In the steel–brass contacts, the dominant particle formation mechanism is different. The brass from the disk is first transferred to the counter body, forming a new sliding pair with different properties, and the loose wear particles are detached mainly from the transfer layer. Therefore, the particles formation is driven by the growth and rupture of this layer [62]. A better agreement with the experiment was found by considering the layered structure of the contacting bodies.

Employment of the BEM methods makes it possible to run a considerable number of iterations and to address the dynamics of the wear process, by including the running-in and steady-state regimes.

## 4 Conclusions

A simple wear algorithm based on subsurface stress calculation was introduced and validated using experimental data on the wear particles size. The particle generation criterion was based on a critical stress and geometrical boundary conditions. It was shown that the model works well for steel–steel contacts. In case of steel–brass contact, a transfer film was observed and it was concluded that the main wear particle formation mechanism is linked to the rupture of this layer. The model was therefore extended by including the influence of the transfer layer after which a better agreement was observed.

It was found that the resultant wear particles size is independent of the softer surface profile. In contrast, an increase in roughness of the harder material resulted in an increase in the wear particles detached from the softer material.

Overall, the performance of the model can be considered as satisfactory to predict the formation of wear particles both by direct removal as well as through formation and rupture of a transfer layer.

## Notes

### Acknowledgments

The authors would like to thank SKF Engineering & Research Center, Nieuwegein, The Netherlands, for providing technical and financial support. This research was carried out under Project Number M21.1.11450 in the framework of the Research Program of the Materials innovation institute M2i.

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