The Construction of Models of Rough Surfaces’ Interaction: Markov’s Approach

Abstract

This article shows that the change of rough surfaces in the way of their contact interaction can be analyzed using the theory of Markov’s processes. In the framework of existing models, this problem cannot be solved. In this article, the idea of reducing to Markov’s model is shown on a simple discrete scheme, which is then generalized. The approach was applied to the analysis of the friction process, to the fatigue failure mode, in which the surface element changes after multiple contacts, possibly many millions. The Kolmogorov-Feller’s equations for the model of this regime were presented and the model of the influence of lubrication is offered. A calculated example of estimating the evolution of surfaces separated by a lubricant layer is given. Additionally, the technical characteristics as functions of the friction path and the load were evaluated.

1. Introduction

The interaction of rough surfaces determines many processes in electrical and heat engineering in the transmission of electrical and thermal energy, through mobile and fixed contacts, friction, lubrication and wear processes in mechanical engineering, the automobile industry, engine building and other industries [1,2,3,4]. These processes determine the performance and service life of the main tribo-assemblies and units. The aim of this article is to show that the change of rough surfaces in the way of their contact interaction can be analyzed by using the theory of Markov’s processes.

The fundamental idea of Markov’s process is as follows: the future depends only on the present and does not depend on the past.

Interaction characteristics depend on many factors, first of all, on surface microgeometry, the physical characteristics of materials, as well as regime and force factors [5,6]. The relief of a rough surface represents a function random in space. The problems of interaction contacting surfaces are most actively considered in mechanics [5,7,8] and are caused by the processes of friction and wear. It is known that the interaction process is a complex process of physico-chemical and molecular mechanics, and it is not possible to describe it without reasonable simplifications, therefore, various models of the friction process have become widespread [7,9,10,11,12]. It is widely recognized that existing methods are not perfect, and work in this direction continues.

Most analysis methods are based on functions that are random in their spatial argument. Gauss functions are used starting with work [13], as well as functions composed of projections of a certain shape in such a way as to have a given law of distribution of heights; this approach is first proposed in work [14]. Considering [15,16], the shapes of protrusions are in the form of columns, hemispheres, wedges, etc. An important engineering task is to calculate the change of the surfaces during the process of friction and the corresponding friction characteristics. Existing models of friction do not solve this problem.

This article describes an approach in which each of the two surfaces is represented by a set of protrusions. Each ledge is described by its random state (a set of selected characteristics, for example, the height and radius of curvature of the vertex). Each surface is described by the probability distribution on the set of states. When the surfaces move mutually, the interaction of the elementary protrusions has a place, and the result for the two projections describes a two-dimensional function of interaction, which is considered known. The state of the fixed ledge at the next moment is determined by the previous state, and the accidental impact of the ledge of another surface. If the effects are independent, the process of state change can be considered Markov’s, and its change is described by difference, differential or integro-differential equations (depending on the assumptions). By counting distribution recurrently, the results of time-varying distribution are received, which are determined by the desired characteristics of the interaction. Choosing different variants of what is the state of the projection, and what is the function of interaction, models of varying complexity and accuracy are obtained. This approach to the analysis of the interaction of surfaces (in particular, for friction) has not been described in the literature; the only exceptions are the work of the authors, [17,18,19,20]; this work generalizes on previous ones.

In the second section, the idea of reducing to the Markov model is shown in a simple discrete scheme. Each of the surfaces consist of projections arranged in “orderly rows” at the same distance from each other, and these distances are the same on two surfaces; mutual movement occurs discretely: shift, interaction, shift, interaction, etc.; the state of the projection is its random height, which changes under the influence of the projections of another surface. The process of changing the state of the ledge is a Markov’s chain, if the disturbances are random and independent. The entire surface is characterized by the distribution of probabilities on a set of the heights of projections, for the probability distribution recurrent recalculation in time is fair. The distribution of heights determines such characteristics as surface contact area, the average height of the protrusions, the frictional force, and wear, etc., so that we can assess the change in the surface over time and the characteristics of the interaction can be assessed.

In the third section, the model is generalized: it is assumed that the location of the projections is random, the characteristics of the location for the two surfaces are different, and the mutual movement is continuous. The protrusion state is a multidimensional random variable value (for example, three-dimensional: height, radius of curvature of the protrusion vertex, temperature). The state changes under the influence of the protrusions of another surface, and the height plays a special role since the interaction of the protrusions occurs only under the condition of contact. Under some assumptions, the state change process is considered to be Markov’s, for the distribution of the process is still a fair recurrent recalculation in time. Technical characteristics can be estimated by distribution.

In the fourth section, the approach is applied to the analysis of the fatigue failure mode resulting from repeated contact of the projections; the load is assumed to be constant. Fatigue failure is modeled by the fact that there is a low probability of separation of the particle after the contact of the projections, and the height of the projection is reduced by, generally speaking, a random variable.

The fifth section provides a computational example of the evolution of surfaces (distributions) separated by a lubricant layer. The technical characteristics as functions of the friction and load path are evaluated. The wear of the main bearing of the crankshaft of the diesel engine is estimated.

2. Idea: Simple Discrete Model

In the work [17], the model of change of a surface during the time is offered. The surface is represented by a set of projections with random heights (and with very flat tops), Figure 1, located at a distance along the line of motion, which reflects the interaction of two friction surfaces.

Movement appears discrete: in one step with an interval $\mathsf{\tau }=\Delta s/V$ there is a progress with a speed $V$ on one element $\Delta s$ and interaction of opposite ledges. The height $\mathsf{\xi }$ of the projection of the lower surface is exposed to moving projections with random independent heights of the upper surface. The result ${\mathsf{\xi }}_k\left(n+1\right)$ of interaction at the element step $n+1$ with the number $k$ depends on ${\mathsf{\xi }}_k\left(n\right)$ and the level ${\mathsf{\eta }}_{k-1}\left(n\right)$ of the neighboring element:

$${\mathsf{\xi }}_k\left(n+1\right)=\mathsf{\Phi }\left({\mathsf{\xi }}_k\left(n\right),{\mathsf{\eta }}_{k-1}\left(n\right)\right),$$

(1)

Let us fix any element with a number and simplify the recording:

$${\mathsf{\xi }}_{n+1}=\mathsf{\Phi }\left({\mathsf{\xi }}_n,{\mathsf{\eta }}_n\right),$$

(2)

If we consider the sequence ${\mathsf{\eta }}_n,n=0,1,2,\dots$ to be independent random variables, then the sequence ${\mathsf{\xi }}_n,n=0,1,2,\dots$ is Markov’s. It is given that not only the upper sequence ${\mathsf{\eta }}_n$ acts on the lower protrusion with height ${\mathsf{\xi }}_n$, but also on any fixed protrusion of the upper surface with a level ${\mathsf{\eta }}_n$, the sequence ${\mathsf{\xi }}_n$ acts, and it is possible for any element of the upper surface to write similarly:

$${\mathsf{\eta }}_{n+1}=\mathsf{\Psi }\left({\mathsf{\eta }}_n,{\mathsf{\xi }}_n\right),$$

(3)

Functions $\mathsf{\Phi }$ and $\mathsf{\Psi }$ set the mechanism of interaction between the two projections. The choice of functions $\mathsf{\Phi }$ and $\mathsf{\Psi }$ can implement various schemes of interaction. These two general relations define Markov’s sequences $\left\{{\mathsf{\eta }}_n\right\}$ and $\left\{{\mathsf{\xi }}_n\right\}$ changes in heights and the evolution of the corresponding distributions $p_n\left(x\right)$ and $q_n\left(y\right)$. By introducing a discretization on the level of the vertices of the projections, we obtain a simple Markov’s chain with the following recalculation of the distributions:

$$\begin{array}{c}{p}_{n+1}=p_n{\mathsf{\Pi }}_{\mathsf{\xi }}\left(q_n,\mathsf{\Phi }\right),\\ q_{n+1}=q_n{\mathsf{\Pi }}_{\mathsf{\eta }}\left(p_n,\mathsf{\Psi }\right),\end{array},$$

(4)

where ${\mathsf{\Pi }}_{\mathsf{\xi }},{\mathsf{\Pi }}_{\mathsf{\eta }}$—matrix of transition probabilities that depend on the distributions $p_n\left(x\right)$, $q_n\left(y\right)$ and transformation functions $\mathsf{\Phi }$ and $\mathsf{\Psi }$; $p_n$ and $q_n$—vectors of the row distributions. The matrix elements (transition probabilities from $x$ to $y$) in general for ${\mathsf{\xi }}_n$ in (2) are defined as follows:

$$\begin{array}{l}{\mathsf{\pi }}_{{\mathsf{\xi }}_n}\left(x,y\right)\equiv P\left\{{\mathsf{\xi }}_{n+1}=y|{\mathsf{\xi }}_n=x\right\}=\\ =P\left\{\mathsf{\Phi }\left(x,{\mathsf{\eta }}_n\right)=y\right\}={\displaystyle \sum _{z:\mathsf{\Phi }\left(x,z\right)=y}{q}_n\left(z\right)}\end{array},$$

(5)

Similarly, for ${\mathsf{\eta }}_n$ in (3):

$$\begin{array}{l}{\mathsf{\pi }}_{{\mathsf{\eta }}_n}\left(x,y\right)\equiv P\left\{{\mathsf{\eta }}_{n+1}=y|{\mathsf{\eta }}_n=x\right\}=\\ =P\left\{\mathsf{\Psi }\left(x,{\mathsf{\xi }}_n\right)=y\right\}={\displaystyle \sum _{z:\mathsf{\Psi }\left(x,z\right)=y}{p}_n\left(z\right)}\end{array},$$

(6)

$$\begin{array}{c}{p}^{\ast }=p^{\ast }{\mathsf{\Pi }}_{\mathsf{\xi }}\left(q^{\ast },\mathsf{\Phi }\right),\\ q^{\ast }=q^{\ast }{\mathsf{\Pi }}_{\mathsf{\eta }}\left(p^{\ast },\mathsf{\Psi }\right).\end{array},$$

(7)

In the idea to trace the evolution of the process according to the scheme (1), (7) was detailed and implemented as a modeling program of the protrusions’ interaction; the results are described in [17,18]. A fatigue cycle distribution based on Markov chains has been presented in the following work [21]. The transformation functions $\mathsf{\Phi }$ and $\mathsf{\Psi }$, adopted there, reflect the process of elastic and plastic deformations under constant load, as well as fatigue failure. The implementation of the model on a computer allows, through the physical properties of materials, the characteristics of the initial roughness and the applied normal pressure to evaluate the characteristics of friction in time. The results of the program are shown in Figure A1 and Figure A2 of Appendix A. The application and efficiency of this approach are shown in [19,20,22].

With the help of the described model, the evolution of distributions is observed not by simulation modeling, but by the usual for Markov’s processes recalculation of distributions, similar to (4).

3. Generalization of the Model

It is assumed that the mutual movement is continuous, the location of the projections is random, and the characteristics of the location for the two surfaces are different. The state of the surface protrusion (conditionally lower) will be considered as a two-dimensional random variable $\mathsf{\xi }=\left({\mathsf{\xi }}_1,{\mathsf{\xi }}_2\right),$ (dimension, as it is clear from the following, is easy to increase) where ${\mathsf{\xi }}_1$—is the height, ${\mathsf{\xi }}_2$—is another parameter, for example, the radius of curvature of the vertex. Component ${\mathsf{\xi }}_1$—height plays a special role, because the fact of interaction between the protrusions is determined by the intersection of the protrusions. The state of the protrusion on the other (conditionally, upper) surface will be considered similarly $\mathsf{\eta }=\left({\mathsf{\eta }}_1,{\mathsf{\eta }}_2\right)$.

To fix any protrusion of the lower surface, it is exposed to the flow of moving protrusions of the upper surface with random locations and heights. Note that the interaction of protrusions (as a result of mechanical contact) occurs only between the highest protrusions, the share of which, according to the assumption, is one percent. Additionally, take into account that the flow of outputs of a random process for a certain high level $x$ (when the probability of exceeding the level is small) is approximately Poisson’s. These circumstances allow us to assume that the flow of contacts (interactions) for a protrusion with a level ${\mathsf{\xi }}_1\left(t\right)=x_1$ is Poisson, so the flow parameter ${\mathsf{\lambda }}_{\mathsf{\eta }}\left(x_1,t\right)$ is equal to the average number of events per unit time:

$${\lambda }_{\mathsf{\eta }}\left(x_1,t\right)=\left(V/\Delta s_{\mathsf{\eta }}\right)F_{\mathsf{\eta }1}\left(x_1,t\right),$$

(8)

where $\Delta s_{\mathsf{\eta }}$—is the average distance between the projections of the upper surface; $V$—the rate of slippage; $V/\Delta s_{\mathsf{\eta }}$—the number of passing projections per unit time; $F_{\mathsf{\eta }1}\left(x_1,t\right)$—the conditional probability of contact, equal to the distribution function of a random variable, which defines as:

$$\begin{array}{l}P\left\{\mathsf{\delta }\left(t\right)={\mathsf{\xi }}_1\left(t\right)-{\mathsf{\eta }}_1\left(t\right)>0|{\mathsf{\xi }}_1\left(t\right)=x_1\right\}=\\ =P\left\{{\mathsf{\eta }}_1\left(t\right)>x_1\right\}={\displaystyle \underset{-\infty }{\overset{x_1}{\int }}{q}_1\left(z,t\right)dz}=F_{\mathsf{\eta }1}\left(x_1,t\right)\end{array},$$

(9)

If the contact moments $t_1,t_2,\dots ,t_k,\dots$—are Poisson’s, and the corresponding states $\mathsf{\eta }\left(t_1\right),\mathsf{\eta }\left(t_2\right),\dots ,\mathsf{\eta }\left(t_k\right),\dots$ are independent, then the process $\mathsf{\xi }\left(t\right)$ is two—dimensional Markov’s with jumps. It is very important that the moments of contact are determined only by heights, and therefore—distributions:

$$\begin{array}{l}{p}_1\left(x_1,t\right)={\displaystyle \int p_{\mathsf{\xi }}\left(x_1,x_2;t\right)d{x}_2}\\ q_1\left(x_1,t\right)={\displaystyle \int q_{\mathsf{\eta }}\left(x_1,x_2;t\right)d{x}_2}\end{array},$$

(10)

where $p_{\mathsf{\xi }}\left(x_1,x_2;t\right)$, $q_{\mathsf{\eta }}\left(x_1,x_2;t\right)$—are distributions of two-dimensional random variables, respectively, $\mathsf{\xi }\left(t\right)=\left({\mathsf{\xi }}_1\left(t\right),{\mathsf{\xi }}_2\left(t\right)\right)$ and $\mathsf{\eta }\left(t\right)=\left({\mathsf{\eta }}_1\left(t\right),{\mathsf{\eta }}_2\left(t\right)\right)$.

Due to the proposed Poisson’s belonging, the conditional probability for a protrusion with a height level ${\mathsf{\xi }}_1\left(t\right)=x_1$ to get a contact in a short time $\Delta t$ is $\widehat{p}={\lambda }_{\mathsf{\eta }}\left(x_1,t\right)\Delta t+\mathsf{o}\left(\Delta t\right)$.

It can be recorded for ${\mathsf{\xi }}_1(t+\Delta t)$ provided to ${\mathsf{\xi }}_1\left(t\right)=x_1$:

$$\begin{array}{l}\mathsf{\xi }\left(t+\Delta t\right)=\mathsf{\xi }\left(t\right)+\\ \{\begin{array}{c}\Delta W_{\mathsf{\xi }}\left({\mathsf{\xi }}_t,{\mathsf{\eta }}_t\right)withprobability\widehat{p},\\ 0withprobability1-\widehat{p},\end{array}\end{array},$$

(11)

where ${\mathsf{\xi }}_t,{\mathsf{\eta }}_t$—is a simplified designation for $\mathsf{\xi }\left(t\right)$, $\mathsf{\eta }\left(t\right)$; $\Delta W_{\mathsf{\xi }}\left({\mathsf{\xi }}_t,{\mathsf{\eta }}_t\right)$—abrupt change of state, a random variable (in our case, two-dimensional, and depending on the result of the interaction of the projections with the states ${\mathsf{\xi }}_t$ and ${\mathsf{\eta }}_t$) with distribution $h\left(y|x\right)\equiv h\left(y_1,y_2|x_1,x_2\right)$ of the transition from $x=\left(x_1,x_2\right)$ into $y=\left(y_1,y_2\right)$.

For Markov’s processes with jumps, at $\Delta t\to 0$ for density $p_{\mathsf{\xi }}\left(y,t\right)$ the Kolmogorov-Feller’s equation is fair:

$$\begin{array}{l}\frac{\partial p_{\mathsf{\xi }}\left(y,t\right)}{\partial t}=-{\lambda }_{\mathsf{\eta }}\left(y,t\right)p_{\mathsf{\xi }}\left(y,t\right)+\\ {\displaystyle {\int }_{-\infty }^{\infty }{\lambda }_{\mathsf{\eta }}\left(x,t\right)p_{\mathsf{\xi }}\left(x,t\right)h_{\mathsf{\xi }}\left(y/x\right)dx}\end{array},$$

(12)

Which in the two-dimensional case is written as:

$$\begin{array}{l}\frac{\partial p_{\mathsf{\xi }}(y_1,y_2,t)}{\partial t}=-{\lambda }_n(y_1,t)\\ \cdot p_{\mathsf{\xi }}(y_1,y_2,t)+{\displaystyle \int {\lambda }_n(x_1,t)}{p}_1(x_1,t)\\ \cdot \left[{\displaystyle \int h_{\mathsf{\xi }}}\left(y_1,y_2|x_1,x_2\right)d{x}_2\right]d{x}_1\end{array},$$

(13)

where $p_1\left(x_1,t\right)$ determines by (10).

This formula (13) can be used for Markov processes with jumps at 1 $\Delta t\to 0$.

Such reasoning can lead to any ledge of upper surface and records the ratio analogous to (13), [23]. There are also other related details about Equation (13).

Equation (13) has to be solved numerically, moving to the discretization in time and states. As it is so small, the probability for any protrusion to achieve more than one impact can be neglected. Let’s denote: $t_n=n\Delta t$, ${\mathsf{\eta }}_n=\mathsf{\eta }\left(t_n\right)$, $n=0,1,2,\dots$, For two-dimensional processes ${\mathsf{\xi }}_n$ and ${\mathsf{\eta }}_n$ can be written the ratio (1) with the only difference that the values ${\mathsf{\xi }}_n,{\mathsf{\eta }}_n$ and functions $\mathsf{\Phi },\mathsf{\Psi }$—are two-dimensional [24].

4. The Process of Friction: A Model of Fatigue Fracture, Taking into Account the Lubricant

4.1. Mode without Lubrication

The model for this very important regime of fatigue failure at constant load is considered in [19]. It is assumed that the process of fatigue failure leads to the fact that with multiple contacts of the projections, the separation of the material particle occurs, and the height of the protrusion decreases by a certain amount, generally speaking, it is random. The condition of the projection is considered to be a one-dimensional parameter—the height (or rather, the level of a node). The ratio (2) takes the form (with precision $o\left(\Delta t\right)$):

$${\mathsf{\xi }}_{n+1}=\mathsf{\Phi }\left({\mathsf{\xi }}_n,{\mathsf{\eta }}_n\right)={\mathsf{\xi }}_n-W_{\mathsf{\xi }}\cdot {\epsilon }_{\mathsf{\xi }}\left({\mathsf{\delta }}_n\right)+h_n/2,$$

(14)

if ${\mathsf{\xi }}_n=x$, so

$$\begin{array}{l}{\epsilon }_{\mathsf{\xi }n}\left({\mathsf{\delta }}_n\right)=\\ \{\begin{array}{l}1,withprobability{P}_{W_{\mathsf{\xi }}}{\lambda }_{\mathsf{\eta }}\left(x,t\right)\Delta t,\\ 0,withprobability1-P_{W_{\mathsf{\xi }}}{\lambda }_{\mathsf{\eta }}\left(x,t\right)\Delta t.\end{array}\end{array},$$

(15)

This entry means that if the contact occurs, i.e., the value of the contact intersection is ${\mathsf{\delta }}_n={\mathsf{\xi }}_n-{\mathsf{\eta }}_n>0$ (the probability of this event is ${\lambda }_{\mathsf{\eta }}\left(x,t_n\right)\Delta t$), then the probability of destruction is $P_{W_{\mathsf{\xi }}}$. The decrease in height occurs by the value $W_{\mathsf{\xi }}$; ${\mathsf{\epsilon }}_{\mathsf{\xi }n}\left({\mathsf{\delta }}_n\right)$ is a random variable; the flow parameter:

$${\mathsf{\lambda }}_{\mathsf{\eta }}\left(x,t_n\right)=\left(V/\Delta s_{\mathsf{\eta }}\right)F_{\mathsf{\eta }}\left(x,t_n\right),$$

(16)

according to (8) at $x_1=x$. The value $h_n$ of the approximation of surfaces is introduced to ensure the condition of the constant force of resistance to a constant external load (this non-random value, defined below, is a functional of the distributions $p_n$ and $q_n$; explanation: with a decrease of the height the resistance force falls, and under the action of a constant normal load the surfaces converge; for the purposes of symmetry, the approximation is provided by a half value for each surface). Adding a constant $h_n/2$ shifts the distribution of ($n+1$) moment, so it is convenient to make Markov’s recalculation without taking it into account, and then shift the distribution. The ratio (3) takes the form similar to (16) (with obvious difference in increment signs). In [10] this scheme is analyzed, transition probabilities for Markov’s chains ${\mathsf{\xi }}_n$ and ${\mathsf{\eta }}_n$ are derived, and the conversion ratio $p_n$ in $p_{n+1}$ and $q_n$ in $q_{n+1}$ are obtained. There is also derived a formula for convergence $h_n$. In addition, in [10] the question of determining the incremental way of friction ΔL in terms of distributions is solved.

It is assumed, that the effect of lubrication, [24], consists mainly in the following:

−

the presence of grease in the gap of the two contacting bodies, as a result of the load, leads to the appearance of the field of hydrodynamic pressures and, accordingly, the appearance of the reaction of the lubricant layer, which counteracts the external load, and therefore the load in the lubricated contact of the counterbody decreases;

−

as a result, the lubricant “pushes” the surfaces, and the number of contacts (the proportion of the contact surface) decreases;

−

the molecular (adhesion) component of the friction force of solids disappears;

−

the convergence of the surfaces at some moment ends, as well as wear, because there is a hydrodynamic mode of friction, due to a decrease of roughness. The onset of the hydrodynamic regime is estimated by the criterion $\mathsf{\lambda }$ [5] (p. 275):

$$\mathsf{\lambda }=h_{\min }/\left(R_{z1}+R_{z2}\right),$$

(17)

where $R_{z1},R_{z2}$—are the parameters of roughness of working surfaces, determined experimentally using a profilometer. When $\mathsf{\lambda }>3$, the hydrodynamic regime of friction comes; the wear ends.

4.2. Definition of Initial Convergence D₀

Before starting the calculations of the mechanical contact, $d$ is great, $P_k\left(d\right)=0$. At the convergence (reduction of $d$) increase the proportion of $P_k\left(d\right)$ contacts, and elastic counter force $F_{elast}\left(d\right)$; if $F_{elast}\left(d\right)+F_{oil}\left(d\right)<F_N$, so the convergence continues until equality

$$F_{elast}\left(d\right)+F_{oil}\left(d\right)=F_N,$$

(18)

moreover

$$\begin{gathered} F_{elast}\left(d\right)=P_k\left(d\right)ScM\mathsf{\delta }E, \\ {F}_{oil}\left(d\right)=\left(1-P_k\left(d\right)\right)S{\sigma }_{oil}, \end{gathered}$$

(19)

where $d$—the distance between the surfaces (the value of convergence); $P_k\left(d\right)$—the proportion of the surface of the mechanical contact, depending on the $d$ convergence; $S$—nominal area of interface surfaces; $P_k\left(d\right)S$—surface area of mechanical contact; $\left(1-P_k\left(d\right)\right)S$—the surface area of the lubricant; ${\sigma }_{oil}$—the pressure under which the lubricant is supplied (this value is an order of magnitude less than the pressure due to the load); $E$—elastic modulus, $c$—constant (in Hooke’s law, $c=1/L,L$—is the height of the rod, i.e., protrusion), M—number of contact projections.

The external load $F_N$ is balanced by the elastic force of the contact lugs $F_{elast}\left(d\right)$ and the pressure force of the lubricant $F_{oil}\left(d\right)$. When wear $P_k\left(d\right)$ increases, and due to the preservation of the balance of forces (14), the d convergence decreases, and so it will be until the moment of hydrodynamic regime.

4.3. Scheme of Plastic Deformation

The described approach was also used in [16] to estimate the change of surfaces in the run-in mode when plastic deformations act. The following scheme is adopted: if the contact intersection is $\mathsf{\delta }\le \Delta ,\Delta >0$ where $\Delta$—is the threshold of elasticity (i.e., the maximum value $\mathsf{\delta }$ of the elastic (recoverable) deformation), then the change of the element height does not occur; if $\mathsf{\delta }>\Delta$ then the total decrease in elevation $D_{\mathsf{\xi }}+D\mathsf{\eta }$ takes place on the value $\mathsf{\delta }-\Delta$. This value is distributed between $D_{\mathsf{\xi }}$ and $D\mathsf{\eta }$ depending on the ratios of physical properties of materials. The dependence $D_{\mathsf{\xi }}\left(\mathsf{\delta }\right)$ is assumed to be random; it is believed that the change in height $\left(\mathsf{\delta }-\Delta \right)$ occurs only for one element: with probability $P_m$ for ${\mathsf{\xi }}_n$ and with probability $1-P_m$—for ${\mathsf{\eta }}_n$. For this scheme, transition probability matrices for ${\mathsf{\xi }}_n$. and ${\mathsf{\eta }}_n$ are defined in [16]. There are also calculations with elastic and plastic deformations and fatigue failure under the action of a constant load; the evolution of distributions over time, the influence of the load, the threshold of elasticity, and the initial distributions on the limit distributions and roughness are analyzed there. In work [19] the described approach is used for the analysis of wear process of the piston of the diesel engine, and the estimation of the parameters of friction defining its resource is made. There is also the choice of the sampling interval for the recalculation of the distributions was substantiated, which had greatly accelerated the numerical evaluation of the model by a change of the matrix of transition probabilities.

5. Results of Modeling

The described approach differs from the known ones by the possibility of obtaining estimates of the changes in characteristics over time. In this section, a specific example demonstrates how the method works to obtain the required time dependencies. It is considered a mate “crankshaft—main bearing” of the diesel engine. Changes in time of friction characteristics, such as roughness, average contact area, average contact intersection, friction coefficient and wear, are estimated. The dependence of these characteristics on the applied load is also estimated. The initial data are given in

Table 1. Initial parameters of mating surfaces.

Parameters	Crankshaft Journal (Surface $\mathsf{\eta }$)	Half-Bearing (Surface $\mathsf{\xi }$)
$R_a$—arithmetic mean deviation of the profile, μm	0.083	1.099
Standard deviation ($1.25R_a$), μm	0.103	1.373
$\Delta s$—the average distance between the projections, μm	18	30
$E$—the Young’s modulus, GPa	200	75
$\nu$—Poisson ratio	0.28	0.31
${\mathsf{\sigma }}_{\mathrm{T}}$–yield point, MPa	4250	1200

and below in the text. The data of Table 1 were obtained experimentally and from reference books.

The nominal area of the contact area S (this is a small part of the cylindrical surface along the generatrix with an angle of about 10 degrees), taking into account the pulsating load, is determined by the engineers as the value $S=22.62$ cm² the maximum permissible minimum thickness of the lubricant layer is $h_{\min }=1$ μm. The probability of failure $P_W$ for the half-bearing is taken to be 4·10⁻⁷, and 10⁻⁹ for the crankshaft journal, i.e., a hundred times less, because it is made of a much harder material, and it is known that its surface is practically unchanged. The initial distribution densities $p_0\left(x\right)\mathrm{and}{q}_0\left(x\right)$ for the heights of the surface protrusions $\mathsf{\xi }$ (half-bearing) and $\mathsf{\eta }$ (crankshaft journal) were, respectively, approximated by shifted and stretched beta-distributions (density $C(a,b)x^{a-1}{(1-x)}^{b-1}$, $x\in \left[0,1\right]$). Parameters $a$ and $b$ were selected on the base of profilograms so that after the shift and stretching in the desired height range to ensure the equality of the first two moments. For $p_0\left(x\right)$ (half-bearing) it is received: $a=1.82$; $b=1.86$. For $q_0\left(x\right)$ (crankshaft journal): $a=1.71$; $b=4.60$.

Analyzed coupling works in the hydrodynamic regime of friction most of the time, but in some points in time, characterized by increased load, it goes to mixed and boundary friction regimes (with regards to these regimes, see for example [5], p. 274), which determine resource coupling. In the considered computational model, first, for a short time, the coupling operates in the run-in mode, then—in the fatigue failure mode (boundary mode), after which the hydrodynamic friction mode occurs. The run-in time is less than 1% of the considered friction path. Boundary friction ranged from 30 to 40% of the friction path depending on the load.

Figure 2a shows the initial densities of the distribution $p_0\left(x\right),q_0\left(x\right)$ of the peak levels of the lower and upper surfaces; the level scale of the peaks of the treads is rigidly connected with the half-bearing to observe the change in the distribution of the half-bearing $p\left(x\right)$. The levels of the protrusion peaks of the half-bearing (curve $p_0\left(x\right)$) are in the range from 2.5 µm to 8.5 µm; the levels of the crankshaft journal protrusion peaks are in the range from 8.2 µm to 8.7 µm. The normal force is taken to be 90 kN.

The operating mode is characterized by elastic and plastic deformations. Let us omit the description of this mode, because in our case this mode takes a negligible fraction of time; it is described in [15,16]. The convergence of the two surfaces and, as a consequence, an increase in the contact area (i.e., the number of contacting protrusions) and a decrease in stress, occurs almost due to plastic deformations. When they stop, this mode ends. The corresponding moment is seen in Figure 2b: a small “jump” in the distribution shows that a part of the protrusions has undergone plastic deformation, and changes in height; this distribution is the initial to analyze the mode of fatigue failure; accordingly, the height distribution $q\left(x\right)$ of the shaft protrusions is shifted by the amount ~0.3 µm in the surface of the half-bearing. On all the charts of Figure 2, vertices of peak density distributions are not shown, they are above the boundary of the figure; significant are not the peaks of peak density, but the location of the peaks on the horizontal axis of the levels of projection peaks.

The fatigue failure mode described in Section 3, and caused by the separation of the particle under multiple interactions, begins. Figure 2b shows the distribution after 240 km of friction; it can be seen that the distribution has changed significantly. The separation of the particles reduces the height of the projections, part of the height of the half-bearing from the upper levels goes to lower, forming a large proportion of projections with almost equal heights (this is the peak on the curve 4 $p\left(x\right)$ in the vicinity of the 6 microns level); distribution $q\left(x\right)$, respectively, almost unchanged shifted towards the half-bearing at 2.5 microns. The distribution shift $q\left(x\right)$ is determined by maintaining a constant load. The shift also determines the amount of wear. This slow mode ends when the roughness becomes so small that the condition (“lambda criterion”) of transition to the hydrodynamic friction regime is satisfied [5], (p 275):

$$\mathsf{\lambda }=h_{\min }/\left(R_{z1}+R_{z2}\right)>3,$$

(20)

where h_min—the minimum thickness of the lubricant layer, $R_{z1},R_{z2}$—surface roughness.

Fatigue failure mode stops. The distributions corresponding to this point are shown in the Figure 2d. The hydrodynamic regime starts, and the wear intensity decreases to zero, i.e., the convergence of the surfaces is terminated. Modeling of transient, without a doubt, a smooth process, requires additional analysis; in the given calculations the transition is described in conditional (spasmodic) form.

Friction characteristics were calculated at three values of normal force: 10, 30, 90 kN. Figure 3 and Figure 4 show the dependence of friction way. They show the possibility for engineers to quantify and compare the characteristics changing in the process of friction. As you can see, they agree with the known (qualitatively) dependences: the greater is the applied load, the faster the surface of the half-bearing smooths (Figure 3a); the proportion of contacting protrusions $P\left\{\mathsf{\delta }>0\right\}$ increases faster (Figure 3b), i.e., the proportion of contact area; the transition to the hydrodynamic regime comes faster (“lambda criterion” $\mathsf{\lambda }$, Figure 3c), intensive wear occurs (Figure 4b,c). Let us note that the parameter that determines the rate of change of the surface can be determined once by comparing the estimated time of the resource with empirically known.

6. Conclusions

An approach to modeling changes in rough surfaces during their interaction is described in a general form. Using the developed method, a computational analysis of the “crankshaft journal-liner” interface of a diesel engine was performed. Taking into account the necessary initial data obtained experimentally and taken from reference materials, an assessment was made of the change in the distribution of the heights of the liner irregularities. The evaluation was carried out depending on the friction path corresponding to the mode of operation of the diesel engine. Based on the calculation results, the following conclusions can be drawn. The proportion of high protrusions of the liner surface is small. They are subjected to pressure in excess of the yield strength. The height range for the shaft 0.5 microns is an order of magnitude narrower than for the liner. It can be seen that after running-in, the surface of the liner has undergone plastic deformation. The fatigue failure mode is caused by the separation of particles during multiple interactions, part of the liner heights from the upper levels moves to lower ones. The height shift of the liner determines the amount of wear. The described processes made it possible to calculate the coupling friction characteristics under three characteristic load conditions of a diesel engine: 10, 30, and 90 kN.

With an increase in load, the surface of the liner is smoothed out faster, and the proportion of the contact area increases faster; the transition to hydrodynamic friction occurs faster, and wear occurs more intensively. Thus, using the developed model in the future, it is possible to estimate the resource of a hydrodynamic interface operating in hydrodynamic and boundary friction modes. Such tribological couplings, in particular, include bearings of crankshafts of internal combustion engines, parts of the cylinder-piston group, as well as the couplings of high-pressure fuel pumps.