*2.7. Wear Mechanism Calculation*

Wear resistance is an important operational property of machines and mechanisms. Largely, wear resistance is caused by the contact interaction in tribo-conjugations, which is based on the properties of the surface layers. The processes implemented in triboconjugation depend on roughness. The rough surface model makes it possible to determine the type of tribo-tension stress state (elastic contact or plastic contact) and the wear mechanism and to evaluate the relationship of plasma electrolytic treatment conditions and the microgeometry parameters of the surface layer.

The model is based on experimental profilograms of friction tracks. The calculation is performed in relation to the contact of a rough surface with a smooth solid surface since the roughness of the sample surface significantly exceeds the roughness of the counterbody. This calculation is faster and easier to perform than the contact of two rough surfaces [32,33].

To describe the microgeometry of the surface of friction tracks, it is necessary to know the function of vertical material distribution throughout the rough layer and the function of the vertical material distribution throughout the single micro-roughness of the rough surface. The distribution of the material over the height of the rough layer is described by the curve of the support surface (Abbott curve). Curves are taken directly from the profiler by at least 15 pieces per 1 friction track on each ring.

To approximate the experimental reference curve, the Demkin function is selected [34].

$$\eta(\varepsilon) = l\_m \left(\frac{z}{R\_p}\right)^\nu = b \left(\frac{z}{R\_{\text{max}}}\right)^\nu = b \varepsilon\_{\text{max}}^\nu = \frac{A\_r}{A\_c} = \frac{P\_c}{P\_r} = \frac{n\_r}{n\_c},\tag{1}$$

where *z* is the profile cross-section level measured from the protrusions line; *Ar* is the actual contact area; *Ac* is the contour contact area; *Pr* is the average actual pressure on the friction contact; *Pc* is the contour pressure; *nr* is the number of contacting protrusions; *nc* is the number of all protrusions on the contour area.

It is more convenient to calculate the heights of the experimental profile in relative terms:

$$
\varepsilon = \frac{z}{R\_{\!\!\!P}}, \ \varepsilon\_{\text{max}} = \frac{z}{R\_{\text{max}}},\tag{2}
$$

where *Rp* is the smoothing height (the distance from the protrusions line to the midline in terms of the base length); *R*max is the maximum height of irregularities; *z* is the profile cross-section level measured from the protrusions line; *ε* and *ε*max are relative dimensionless profile heights relative to the midline.

The parameters of the reference curve *v* and *b* are determined experimentally from the results of measurements of the parameters of the rough body profile:

$$\nu = 2l\_m \left(\frac{R\_p}{R\_a}\right) - 1,\tag{3}$$

$$b = l\_{\rm{fl}} \left( \frac{R\_{\rm{max}}}{R\_p} \right)^{\nu},\tag{4}$$

where *Ra* is the arithmetic mean deviation of the profile. Additionally, *lm* is the relative reference length of the profile on the midline:

$$d\_m = \frac{\sum\_{i=1}^{n} \Delta l\_i}{l},\tag{5}$$

where *l* is the base length; Δ*li* is the length of the segments cut off by the middle line in the profile. The *lm* parameter is determined by direct measurements of the profilometer on the friction tracks.

To calculate the actual pressure *Pr* at the tops of the micro-steps, it is necessary to determine the type of deformations on the friction contact: elastic or plastic. The evaluation is made using the Greenwood–Williamson criterion:

$$K\_p = \frac{\Theta}{HB} \sqrt{\frac{R\_p}{r}} \,\,\,\,\tag{6}$$

Θ is the reduced modulus of elasticity:

$$
\Theta = \left(\frac{1-\mu\_1^2}{E\_1} + \frac{1-\mu\_2^2}{E\_2}\right)^{-1},
\tag{7}
$$

where *μ<sup>i</sup>* and *Ei* are Poisson's coefficients and the elastic modulus of interacting bodies, respectively.

The dimensionless parameter *Kp* takes into account the roughness and physical properties of the material at the same time. It describes the deformation properties of a rough surface. If the value of this parameter is lower than 3, then the deformations of the irregularities in contact with a flat surface will be completely elastic; if *Kp* exceeds 3, then the deformations will be predominantly plastic.

The average radius of a single micrometer, which characterizes the shape of the protrusion, is calculated directly from the profilometer data:

$$r = \frac{S\_m^2}{8R\_a} \cdot \frac{\gamma\_1}{\gamma\_2^{2'}}\tag{8}$$

where *Sm* is the average step of the irregularities; *γ*<sup>1</sup> is the vertical increase in the profiler; *γ*<sup>2</sup> is the horizontal increase in the profiler.

With elastic contact, the deformation of individual protrusions can be calculated according to the classical Hertz contact problem. Then, the average actual pressure at the contact is determined by the following expression:

$$P\_r = (0.43 \Theta)^{\frac{2v}{2v+1}} \left(\frac{2N}{\eta A\_c}\right)^{\frac{1}{2v+1}} \left(\frac{R\_p}{r}\right)^{\frac{2v}{2v+1}},\tag{9}$$

where *N* is the normal load. With plastic contact, the average voltage at the contact is numerically equal to the microhardness *Pr* ≈ HB.

The actual contact area is determined by the ratio of the normal load to the actual pressure:

$$A\_r = \frac{N}{P\_r}.\tag{10}$$

Substituting the reference curve (1) *z* = *h* into the equation leads to the following expression of the absolute convergence of the surfaces:

$$h = R\_{\text{max}} \left( \frac{P\_{\mathcal{C}}}{bP\_r} \right)^{\frac{1}{v}}. \tag{11}$$

Then, the ratio of the number of contacting protrusions *nr* to the number of all protrusions on the contour area *nc* can be determined using the Demkin function (1):

$$\frac{m\_r}{m\_c} = \left(\frac{N}{P\_r \cdot l\_m}\right)^{\frac{\nu - 1}{\nu}},\tag{12}$$

where *lm* is the relative reference length of the profile at the midline level.

The relative embedding of the sample and the counterbody is the ratio of the absolute embedding to the average radius of the micronerosity:

$$\frac{h}{r} = \frac{R\_{\text{max}}}{r} \cdot \left(\frac{N}{b \cdot P\_r}\right)^{\frac{1}{v}} = \frac{8R\_aR\_{\text{max}}}{S\_m^2} \cdot \left(\frac{N}{b \cdot P\_r}\right)^{\frac{1}{v}} \cdot \frac{\gamma\_2^2}{\gamma\_1}.\tag{13}$$

Relative convergence (13) characterizes the type of violation of frictional bonds in tribocontact. For steel surfaces, in the case when *h*/*r* < 0.01, destruction occurs because of friction fatigue, and friction bonds are broken due to elastic displacement of the sample material. At a value of *h*/*r* < 0.1, low-cycle friction fatigue develops, and the friction surfaces are destroyed due to plastic displacement of the sample material with residual deformation of the friction track after the passage of micro-steps along it.

To assess the bearing capacity of roughness, the dimensionless Kragelsky–Kombalov criterion is calculated [35]:

$$
\Delta = \left(\frac{100}{l\_m}\right)^{\frac{1}{v}} \cdot \left(\frac{R\_p}{r}\right). \tag{14}
$$

The complex (14) represents the most complete roughness assessment, including not only geometric but also statistical characteristics of the height distribution of the protrusions as well as the average radius of the rounding of the micro-protrusions. On a friction-worn surface, Δ shows how much its bearing capacity has been preserved. The lower the calculated value of Δ on the friction track, the higher the bearing capacity of the rough profile and the more favorable conditions for friction with the minimum possible wear.

The mathematical expression for the approximation of the Abbott curve (1) allows us to give a theoretical estimate of the wear value of the sample. If only the volume of embedded irregularities *dV* with density *ρ* is involved in the deformation, then

$$dm = \rho \cdot dV = \rho A\_r dz,\tag{15}$$

where the contour area is defined by expression (10), and *dz* is the depth of the embedding of irregularities, which can vary from zero to the full value of the relative embedding of *h*, as defined by expression (11). Thus, weight wear is determined by volume *V* and is directly proportional to the area of the actual contact:

$$
\Delta m = \rho V = \rho \int\_0^z A\_r dz = \rho \int\_0^\varepsilon b \varepsilon^\nu d\varepsilon = \frac{\rho A\_r h}{\nu + 1}.\tag{16}
$$

## **3. Results**
