*Article* **Influence of the Additive of Ceramic and Intermetallic Powders on the Friction Properties and Temperature of the Wet Clutch Disc**

**Aleksander Yevtushenko <sup>1</sup> , Michal Kuciej 1,\* , Piotr Grzes <sup>1</sup> , Aleksander Ilyushchanka <sup>2</sup> and Andrey Liashok <sup>2</sup>**


**Abstract:** The basic function of friction clutches is to transfer the torque in the conditions of its smooth engagement without vibrations. Hard working conditions under high thermal and mechanical loads, leading to high temperature in the contact area, intense wear, and instability of the coefficient of friction impose restrictive criteria in the design of friction materials. In this paper, the results of experimental research of the effect of ceramic and intermetallic additives to the copper-based material of the friction disc of the clutch on the thermophysical and frictional properties were presented. Next, these properties were incorporated in the proposed contact 3D numerical model of the clutch to carry out computer simulations of the heating process and subsequent cooling. Based on the obtained experimental data and transient temperature changes of the friction and steel discs, the relations between the powder additives, thermophysical properties of the five friction materials, and coefficients of friction, wear, and temperature reached were discussed. Among these, it was found that when working with the lubrication, the largest values of the coefficient of friction 0.068 and wear 13.5 μm km−<sup>1</sup> were reached when using the 3 wt.% SiC additive.

**Keywords:** friction material; clutch; temperature; frictional heating; finite element analysis

### **1. Introduction**

Powder sintered friction materials (PSFM) have become widely used in the friction units of the automotive vehicles, motorcycles, tractors, airplanes, boats, machine tools, etc. Such friction units include, in particular, hydro mechanical gearboxes, oil-cooled brakes, clutches, etc. [1,2]. Powder metallurgy allows obtaining composite materials using powders of different types and at various chemical compositions.

The PSFM should comply with the following requirements: stable value of the coefficient of friction, high wear resistance, effective adaptation, and high thermal conductivity [3]. As a rule, PSFM on the basis of copper are used to operate under lubrication conditions, while materials based on iron are used at dry friction.

The achievement of the given level of tribotechnical and operating properties of the PSFM has been achieved by the use of additives of various kinds, and granulometric composition, the content of which is within the range 0.5–15 vol.%. The additives are able to interact with the metal base, and to localize in the form of individual inclusions. The main additive used in the composition of PSFM are graphites of various types, as well as carbon-containing additives with an amorphous structure, such as coke and anthracite [4,5].

It has been shown that graphite with a size of 80 μm provides a high value of the coefficient of friction, compared with graphite with a size of 8–10 μm. However, a much greater increase in the coefficient of friction was achieved with the use of coke powder [6]. An additive in the form of graphite allows increasing the operating properties of the friction

**Citation:** Yevtushenko, A.; Kuciej, M.; Grzes, P.; Ilyushchanka, A.; Liashok, A. Influence of the Additive of Ceramic and Intermetallic Powders on the Friction Properties and Temperature of the Wet Clutch Disc. *Materials* **2022**, *15*, 5384. https:// doi.org/10.3390/ma15155384

Academic Editor: Shengqiang Ma

Received: 28 June 2022 Accepted: 31 July 2022 Published: 4 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

unit material [7–9]. For a material containing 5 masses % of graphite, an increase in the sliding velocity leads to a sharp increase in the coefficient of friction, while the material itself is characterized by the increased antifriction and anti-pressure properties, and performance under increased load-velocity conditions. For the tin bronze containing 10% of graphite, the wear resistance is significantly higher than bronze with the graphite contents [10].

However, the use of graphite by itself does not provide specific tribological properties. Their further improvement is achieved through the use of additives in the PSFM structure of solid ceramic powders and their compounds.

The composition of friction material (FM) for clutches and brake units, which has a high coefficient of friction and a small difference between dynamic and static coefficients of friction, was presented in the patent [11]. It was noted that this effect is achieved using 8–15% of Al2O3. By means of a method of powder metallurgy, a metal-based FM class with high coefficient of friction and wear resistance, as well as reduced acoustic characteristics was created [12–17]. These materials contain in their composition 2–30% of the component of solid particles selected from metal oxides (composites), metal nitrides (carbonitrides), metal carbides, metal borides, intermetalides, and minerals with a Mohs hardness of 3.5 or more.

Titanium dioxide is widely used in industry as an addition of a small cost, characterized by stable properties and non-toxicity. This additive is characterized by a very high specific surface area, up to 600 m<sup>2</sup> g<sup>−</sup>1, and a low thermal conductivity [18].

The carbides of transition metals are characterized by high hardness and, at the same time, brittleness. The most commonly used is silicon carbide, which has high hardness and thermal conductivity at low density. Its main disadvantage is the low (2 − <sup>3</sup> MPa m1/2) viscosity of destruction [19].

Currently, the use of the intermetallic powder additives in tribotechnical materials is of great interest. The intermetalides Ni3Al and NiAl appear in difficult operating conditions due to the set of unique properties, such as increased value of impact toughness, resistance to oxidation at elevated temperatures and thermal resistance. The above-mentioned intermetalides have a density 7.3 and 5.9 g cm<sup>−</sup>3, respectively, are characterized by a high Young's elasticity, and may be used in products for constructional and tribotechnical purposes [20]. An effect of the addition of intermetallic powder Ti-46Al-8Cr, obtained by the method of mechanoactivable self-propagating high-temperature synthesis, on the tribological properties of the copper-based antifriction material was investigated [21]. It was shown that an increase in the content of aluminite from 0.5% to 1% leads to a decrease in the intensity of wear of the material by more than 3 times. The inclusion of the additive of NiAl/Al2O3 powder system in the frictional material based on copper in the range of 0.5–2.5% revealed an increase in the dynamic coefficient of friction from 0.040 to 0.051, while wear ranged from 4.2 to 5.7 μm km−<sup>1</sup> [22].

Newly designed PSFM materials used for friction elements of clutches, before implementation to production process, undergo a series of restrictive tests, both in full scale and laboratory tests. Even at the stage of their preselection and elimination of the least promising ones, this may be an expensive and time-consuming process. Numerical models are effective (time, costs) in this first phase of designing a new friction pair, allowing for a preliminary analysis of the level and temperature distribution of the friction pair. The novelty presented in this article is the proposed 3D numerical models to analyze the transient temperature fields of the new designed PSFM friction materials. The study presented an extensive numerical finite element (FE) model (from ref. [6]) of friction heating for the estimation of temperature distributions in a wet clutch. Unlike the numerical model from the article [6], this model takes into account the change of thermal properties of the steel disc under the influence of temperature and two different phases of the clutch operation, i.e., after its engagement (heat generation) and disconnection (cooling). The analysis of temperature distribution, based on the structural and tribological tests of newly developed PSFM materials, allowed selecting the most effective friction pair for the use in the wet clutch disc.

### **2. Materials and Experimental Methods**

As a basis for the friction material, a mixture of copper powders (81 wt.%), tin (11 wt.%), and elemental graphite GE-1 (8 wt.%) was used. The initial mixture was prepared in a blade mixer by mixing, within 45 min, the copper powders obtained by electrolysis with a mean particle size of 100 μm (Figure 1a), the tin obtained by spraying the melt, with a mean particle size of 20 μm (Figure 1b), the elemental graphite GE-1, the natural origin obtained by extraction, grinding, and processing in acid solution and having a scaly shape with a mean size of 100 μm (Figure 1c). As test additives, the powders of silicon carbide with a size 4–9 μm (Figure 1d) and the titanium dioxide, which is a conglomerate with a size of 100–150 μm, were used. The conglomerate consisted of the ultra-disperse powders of predominantly spherical shape up to 0.2 μm in size (Figure 1e) and the intermetallic powder Ti-46Al-8Cr. Particles of the powder Ti-46Al-8Cr in the size of 50–500 nm formed agglomerates in the size 5–20 μm with a high specific surface area (Figure 1f). The micro hardness of the powder particles was 4000–5140 MPa. The powders were supplied by manufacturers.

The TI-46AL-8CR system was obtained by the method of the mechanoactivated selfpropagating high-temperature synthesis (MASHS) [23]. The preliminary mechanical processing of the reaction mixture of the powders of the titanium, aluminum, and chromium was carried out in a mill A-4.5 with the following parameters: rotational speed of the impeller shaft 360 r min<sup>−</sup>1, the ratio of the mass of spheres and powder 10:1, and the duration of processing 3 h. The subsequent self-propagating high-temperature synthesis was carried out in the experimental reactor for MASHS in argon environment. The mixture

of powders was ignited with a tungsten spiral heated by the passage of an electric current. After cooling, the resulting sinter was milled in the planetary mill Pulverisette 6 (Fritsch, Germany) in an alcohol medium at the following parameters: the diameter of the spheres 5 mm, the mass ratio of the spheres and the powder 20:1, rotational speed of the impeller drive shaft 400 r min<sup>−</sup>1, and the grinding time 30 min.

The synthesized material Ti-46Al-8Cr, according to X-ray diffraction, consisted of the basis in the form of intermetalide *γ* − TiAl (the spatial group P4/mmm) (Figure 2a,d phase 1), doped by chromium, containing 64–68 at. % Ti, 30–34 at. % Al and up to 5 at. % Cr, the inclusions of double intermetalides Ti3Al and AlCr2, and the triple intermetalide Al0.67Cr0.08Ti0.25 (Figure 2a,c). Thin secondary *τ*-phase Al0.67Cr0.08Ti0.25 smaller than 0.5 μm (Figure 2b, phase 3), falling out in grains of titanium monoaluminide, contained about 68–71 at. % Al, 20–25 at. % Ti, 7–12 at. % Cr and had a cubic grate of Pm-3m type, which provided coherence of boundaries with the *γ*-phase.


**Figure 2.** Microstructure and phase composition of the synthesized SHS powder of the Ti-46Al-8Cr system before grinding: (**a**,**b**) structure; (**c**) radiograph; (**d**) results of the micro X-ray spectral analysis (MRSA).

The *α*<sup>2</sup> − Ti3Al phase (the spatial group P63/mmc) was localized mainly along the boundaries of grains and contained about 2 at. % Cr (Figure 2b, the phase 2). In addition, at the grain boundaries of titanium monoaluminide, there were also inclusions of excessive phases of chromium compounds with aluminum and titanium containing 46–53 at. % Al, 45–60 at. % Cr, and 3–5 at. % Ti (Figure 2b, phase 4), the formation of which was probably due to the problems of diffusion redistribution of components in the SHS process under conditions of predominantly solid-phase interaction.

The samples of the friction discs for testing were made as follows: obtained charge from the initial powders was applied by free filling to the surface of the steel base using special technological equipment, and then preliminary sintering was carried out in dissociated ammonia at a temperature of 840 ◦C within 50 min. For forming a system of oil-removing channels and grooves on the surface of the sintered material, as well as obtaining a porosity of 12–18%, the sintered workpiece of the friction disc was subjected to plastic deformation (embossing) with a punch having a profile in the form of a "grid" on the surface. Then, the final sintering was carried out at a pressure of 0.1 MPa in a medium of dissociated ammonia, which contains 75% H2 and 25% N2 at a temperature of 840 ◦C within 3 h. The friction and steel discs are shown in Figure 3.

The study of the tribotechnical properties of the friction material was carried out on a friction machine IM-58 according to the scheme friction disc-counter body at the following input parameters. The initial velocity of braking was 10 m s<sup>−</sup>1, the contact pressure was 4 MPa, the moment of inertia of rotating masses was 0.56 Nms2, and the work of friction was 27.5 kJ. As a counterpart, a disc made of 65H steel with a hardness of 260–320 HB and a surface roughness of Ra = 0.7–0.8 were used [6]. The bedding-in (burnishing) of the working surfaces by 300 engaging cycles was carried out. Then, 10 measurements of the values of the coefficients of friction and wear were made. From these data, mean values were determined.

The investigation of structure was carried out by means of the optical microscope MEF-3 (Austria). The morphology of the surface of the friction disc and its microstructure were studied on a high-resolution scanning electron microscope MIRA (Czech Republic) with a micro-X-ray spectral console INCA 350 of the Oxford Instruments (UK) company. The phase composition was examined on an X-ray diffractometer Ultima IV (Rigaky) in Cu Kα-radiation at an X-ray tube voltage of 40 kV and the anode current of 40 mA. The parameters of the crystal grate of the alloys were determined by diffraction lines located at the large scattering angles. For a phase analysis, a standard PDF card files was used. The thermophysical properties of investigated compositions of friction materials were carried out on the analyzer of thermal properties Hot Disk TPS2500S. As a sensor, a spiral, being a source of heat, was used. The sensor was located between the sample under the study and the sample, with the known thermophysical properties. Ten measurements were made after a given period of time, and the mean values of the thermophysical properties were established. The tested samples had a diameter of 50 mm, a thickness of 10 mm, and were obtained by compressing at the pressure of 2.5 t cm−<sup>2</sup> and sintering at 840 ◦C for 3 h.

### **3. Results of Experimental Investigations**

The results of the study of the physical and frictional properties of five compositions of friction materials are given in Table 1.


**Table 1.** Influence of the type of carbon-containing additive on the thermophysical properties and coefficient of friction.

The data obtained showed that use of additive SiC obtains the greatest value of the coefficient of friction. The solid inclusions of SiC in the process of friction are crumbled, displacing coarsely dispersed graphite from the surface of the friction material. The change in the morphology of the surface layer, the closure of pores, and increase in the area of the metal phase were fixed (Figure 4b).

**Figure 4.** Morphology of the friction surface of the: (**a**) basic material; with powder additives (**b**) SiC; (**c**) Ti-46Al-8Cr; (**d**) 2 wt.% TiO2; (**e**) 5 wt.% TiO2.

The introduction of the additive of the intermetallic powder of the Ti-46Al-8Cr in an amount of 2 wt.% showed an increase in the coefficient of friction to 0.055, whereas for the basic composition, without additives of powders, it was 0.036. An analysis of the morphology of the surface layer showed that the initial porosity of the friction material was preserved, and there is no replacement of graphite particles (Figure 4c), which is characteristic of the basic composition of the friction material (Figure 4a).

−− − −−

The use of TiO2 powder additive in an amount of 2 wt.% and 5 wt.% led to an increase in the coefficient of friction to 0.043 and 0.052, respectively. An increase in the addition of TiO2 powder from 2 to 5 wt.% showed a change in the morphology of the friction surface of the friction material with a slight increase in the area of the metal phase (Figure 4d,e).

### **4. Numerical Simulation of the Temperature Mode of the Clutch**

*Operating Parameters*

The aim of the numerical simulations was to investigate an effect of the abovementioned powder additives, namely one ceramic (SiC) denoted as variant 1 and three intermetalides (2–Ti-46Al-8Cr, 3–2 wt.% TiO2 and 4–5 wt.% TiO2) to 0—the friction base material, on the clutch temperature, presented in Figure 3. The analyzed friction pair consisted of two discs—a fixed one with a steel substrate (65H) and a friction material applied to it—and a steel (65H) disc rotating against the specimen. The thermophysical properties of the materials at the ambient (initial) temperature *T*<sup>0</sup> = 20 ◦C are presented in Table 1. The changes in the properties of 65H steel with temperature increasing from 20 ◦C to 800 ◦C are shown in Table 2.


**Table 2.** Temperature-dependent properties of the steel 65H [24].

The dimensions of the clutch components and the initial kinetic energy of the system were the same as in the article [6] (Table 2). The calculations were carried out for five variants of friction materials: 0—basic, 1—TiC, 2—Ti-46Al-8Cr, 3—2 wt.% TiO2, 4—5 wt.% TiO2 with the corresponding values of the coefficients of friction listed in Table 1.

### **5. Heating Taking into Account the Thermal Sensitivity of 65H Steel (First Calculation Model)**

Two 3D numerical models were developed using the finite element method (FEM) adapted in the Heat Transfer Module of the COMSOL Multiphysics® programme. The first model was a generalization of the linear (with material properties unchanged) model from the article [6] for the case of thermally sensitive materials (with temperature-varying properties of 65H steel). The finite element analysis was limited only to the friction heating stage during braking. The results of the calculations are presented in Figure 5 and in Table 3.

The evolutions of temperature of the friction surfaces at the equivalent radius *req* = 39.4 mm shown in Figure 5 for thermosensitive (dashed lines) and constant (solid lines) properties of materials revealed typical changes for braking at constant deceleration. Namely, temperature increased rapidly at the beginning, reached maximum value, and decreased until the stop. The obtained maximum temperature values did not exceed 165 ◦C (higher values appeared when taking into account thermosensitivity of the steel), hence the omission of the thermal sensitivity of the friction materials hardly influencing the simulation results. It should also be noted that in this temperature range (from 20 ◦C to 165 ◦C) the changes in the properties of steel 65H are negligible (Table 2).

Comparison of the results for 5 materials analyzed shows how braking time affects the maximum value of the temperature. Since frictional sliding lasts only a few seconds, generated heat cannot be absorbed by the components of the clutch and convection. Therefore, differences in braking times have a strong effect on the maximum temperature reached. The highest value (at thermosensitive material) is equal to 161.8 ◦C, whereas the lowest is equal to 123.1 ◦C.

**Figure 5.** Evolution of the temperature of the friction surfaces of the clutch on the equivalent radius *req* = 39.4 mm with constant (solid lines) and temperature-dependent (dashed lines) properties of 65H steel for five friction materials. Numbers 0–4 denote friction materials given in Table 1.


**Table 3.** Calculated parameters of the braking process for five friction materials.

### **6. Heating with Subsequent Cooling of the Clutch Elements (Second Calculation Model)**

The second computational model concerned both the clutch heating stage due to friction during operation as well as the next, after stopping, disengagement of the discs and their oil cooling. Due to the negligible influence of the thermal sensitivity of 65H steel on temperature (Figure 5), the calculations were performed with the constant, adapted to the initial temperature, material properties. This stemmed from the relatively short heating time of the clutch, less than 7 s, and thus limited ability to heat conduction to other neighboring parts of the assembly. On the other hand, the cooling step following the friction heating and lasting ≈90 s took place in the environment of the oil, which absorbed heat from the surface of the components intensely compared to air. The construction stages of the second model are presented below.

### *6.1. Boundary Conditions*

As mentioned above, the analyzed friction pair consisted of three geometric objects representing the basic elements of the clutch. The calculations were divided into two stages:

1. Heating of the friction surfaces during sliding contact with convection cooling of the side free surfaces;

2. Exclusive convection cooling of the lateral surfaces and working faces of the discs, where frictional heating occurred in the first stage. The second stage simulated the state when the components were disconnected (no friction).

It should be noted that in both stages the surfaces of the discs parallel to the friction surfaces were adiabatic.

In the first stage, during the frictional heating with duration time denoted *ts*,*i*, *i* = 0, 1, 2, 3, 4, the type of connection of geometric elements "create union" was used. This meant that the conditions of temperature continuity and heat flux intensity (perfect thermal contact) were required at the interface between the steel substrate and the clutch facing (friction material). On the other hand, on the contact surface of the friction material and the steel counterpart, there was a perfect thermal contact of friction, which consisted of meeting the following equality of:


On the lateral surfaces of both discs, heat exchange with the surrounding environment according to Newton's law of cooling at the constant heat transfer coefficient *h* = 600 W m−<sup>2</sup> K−<sup>1</sup> took place.

After stopping and disconnecting the clutch components, it was necessary to change the connection type of the parts in the "geometry" domain in COMSOL. Such a change affects almost all stages of the model creation (finite element mesh, selecting surfaces for heat transfer due to convection, etc.). Therefore, a new file was created, into which the temperature field from the last time step from the braking stage study was imported. Then, modifications were made to rebuild the geometry into an assembly. Creating an assembly, instead of the union formulation, allowed for the separation of the objects and the introduction of heat transfer due to convection also on the friction surfaces. The presence of such cooling better reflects the actual conditions in the clutch on the test bench. It was not possible when using the "create union" option in the computational model from the article [6].

### *6.2. Modeling Rotational Motion*

As on the test stand, in the developed numerical models, it was assumed that the discs with the clutch facing are stationary, and the steel counterpart rotates at the angular velocity *ω*. The rotation of the counterpart in relation to the stationary disc was carried out using the well-known and verified approach of changing the velocity field at each point of the rotating part. The components of the linear velocity *V* vector were determined respectively from the dependence *Vx* = −*yω* and *Vy* = *xω* using a special tool available as the Translational Motion option of the Heat Transfer module of the COMSOL Multiphysics® software (Heat Transfer in Solids-Solid-Translational Motion).

#### *6.3. Construction of a Finite Element Mesh of the Clutch*

Apart from the counterpart (steel disc) characterized by geometrical axial symmetry (mounting elements were omitted) (Figure 3b), there were differences in the shape in the circumferential direction of the other parts (steel plate with the clutch facing) (Figure 3a). The spatial (3D) model of the clutch was selected for the thermal finite element analysis.

When dividing the 3D geometric objects of the clutch into finite elements, an automatic mesh generator with an option of tetrahedral elements (free tetrahedral) and the general default size appearing under the name "normal" was used. This method takes into account the type of the problem as well as the curvature and geometric details that change mesh (divide into smaller elements) only in critical areas. Initial attempts to manually create mapped or free quad elements and then building regular hexagonal finite elements on the basis of the sweep method showed a number of warnings and errors at the edges of the objects. This was due to the large difference in the size of the contacting edges of the two parts—the smallest edges in the case of a friction material with many cuts on the working surface, steel plate, and the counterpart.

The final mesh created from of tetrahedral elements is shown in Figure 6, and before the actual calculations, it was additionally verified in terms of distributions and maximum values of temperature in the braking process.

**Figure 6.** Finite element mesh used in finite element analysis of: (**a**) heating; (**b**–**d**) cooling.

### *6.4. Results of Computer Simulations*

The second order shape function (quadratic Lagrange) of elements was used to calculate the temperature fields at both stages (heating and convection cooling). Such finite elements generated the most accurate results without the need to use an extremely fine mesh in the area of high temperature gradients. An experience in the construction of a finite element grid was obtained from previously conducted simulations of heat generation in disc brakes [25] and tread brakes (wheel-rail) of railway vehicles [26]. It was found that the linear finite elements significantly falsify the calculations (over 20%) even at many times higher than the default mesh density. The results of the calculations of the working surfaces temperature of the clutch are shown in Figure 7 and in Table 4.

**Figure 7.** Evolutions of the temperature of the friction surfaces of the clutch on the equivalent radius *req* = 39.4 mm obtained by means of the computational model: from the article [6]—connected clutch; developed in this paper—disconnected clutch. Numbers 0–4 denote friction materials given in Table 1.


**Table 4.** Calculated parameters for the cooling stage for five friction materials.

In order to investigate the effect of oil cooling in the contact area, the temperature distributions of the clutch in the cross-section (*r*, *z*) were compared under the condition of perfect thermal contact and with the disconnected parts after stopping time moments *ts* + 0 s, *ts* + 15 s, and *ts* + 55 s (Figures 8–10). It should be noted that because of different braking durations for each of the five friction materials, the presented distributions occur at slightly different points in time from the beginning at *t* = 0 s.

The temperature distributions in Figure 8 show the stopping times *ts* + 0 s. Slight differences in the distributions for variants a and b result from the fact that for connected clutch components (variant a) these are the values calculated and displayed from the model in which the perfect thermal contact condition was maintained all the time, while for disconnected components (variant b) the field is imported to the model with the separate cooling. The highest temperature is accumulated in the central part of the friction path near the friction radius.

Significant differences in temperature distributions resulting from the cooling method appeared after time *ts* + 15 s (Figure 9). Due to the smaller total cooled area of the clutch, at this time moment, significantly higher temperature values were achieved for the model in which the friction pair remained connected (Figure 9a). Only for materials 1 and 2, for which the shortest cooling times take place, was the maximum temperature of the friction disc for variant b equally high.

The temperature evolutions are confirmed by the temperature distributions shown in Figure 10. It can be clearly seen that the temperature field for each of the tested materials was similar. However, while at the time moment *ts* + 15 s, a higher temperature was

obtained for the friction disc, and at *ts* + 55 s a higher average temperature occurred for the steel disc.

**Figure 8.** Temperature distribution at time *ts* + 0 s obtained using: (**a**) connected [6]; and (**b**) disconnected parts of the clutch for materials no. 0, 1, 2, 3, 4.

**Figure 9.** Temperature distribution at time *ts* + 15 s obtained using: (**a**) connected [6]; and (**b**) disconnected parts of the clutch for materials no. 0, 1, 2, 3, 4.

**Figure 10.** Temperature distribution at time *ts* + 55 s obtained using: (**a**) connected [6]; and (**b**) disconnected parts of the clutch for materials no. 0, 1, 2, 3, 4.

### **7. Results and Discussion**

The article presented an experimental analysis of material properties and thermal finite element analysis of friction heating for new PSFM materials used for clutch facing under lubricated conditions. Experimental tests were carried out on the IM-58 friction machine for four different PSFM materials with different additives (SiC, Ti-46Al-8Cr, and TiO2) and one base material. The materials produced were formed into friction discs and combined with a steel 65H disc, determining the values of the friction coefficients for each pair. The thermophysical properties for the new materials were investigated using the Hot Disk TPS2500S analyzer of thermal properties. These properties and values of the friction coefficient, as well as the input parameters (initial velocity, contact pressure, moment of inertia of rotating masses) of the experiment were adapted to 3D numerical models of friction heating. Based on the computer simulations carried out for the heating stage only with five friction materials, the temperature distributions (its maximum value on the contact surface and the time to reach this value), taking into account the temperature changes of the material properties of the steel disc, were analyzed. In the second part of the numerical tests, both the friction heating stage and the cooling stage after the clutch was disengaged were taken into account.

One of the main results of the material, tribological, and numerical tests carried out is the selection of such additives that had the greatest impact on the operation of the friction pair, and thus also on the temperature level during clutch engagement. It was shown that the greatest change of the tribological properties was obtained using addition of 3 wt.% SiC in the composition of the friction material based on copper with 12% tin and 30 vol.% graphite GE-1, namely, the coefficient of friction increased from 0.036 to 0.068. At the same time, wear increased from 3.1 to 13.5 μm km−<sup>1</sup> . The least influence on the tribological properties of the base material has 2 wt.% TiO2 powder, i.e. the coefficient of friction was the smallest (0.043) at the greatest wear resistance.

The basic factors influencing the changes in temperature distribution in the friction pair components and the evolution of the maximum temperature in the contact zone include (1) the amount of mechanical energy converted into heat, and thus the initial angular velocity and the moment of inertia of rotating masses; (2) the velocity at which this energy is dissipated, i.e. the braking torque dependent on the clamping force, coefficient of friction, and the friction radius; (3) type and dimensions of the given friction pair (thickness, number of neighboring elements absorbing heat), (4) thermophysical properties, and (5) cooling conditions due to convection and thermal radiation.

Assuming that in the analyzed friction pairs, braking takes place at the same input parameters (initial angular velocity, moment of inertia of rotating masses and clamping force), and assuming that the process time is short enough to ignore the influence of cooling, the key factors that affect the maximum temperature are the thermophysical properties and the coefficient of friction. As shown in Table 1, the thermophysical properties were very similar, while the greatest difference in the values of the friction coefficients was 89% (SiC in relation to the base material). Therefore, it is the coefficient of friction and the resulting braking time that in this case play a key role in reaching the maximum temperature value. For the higher coefficient of friction, the braking time is shorter, and the maximum temperature higher since the time for heat dissipation from the contact area due to conduction being limited.

The shortest braking time *ts*,1 = 3.61 s and the highest temperature value equal to 160.2 ◦C among the five numerically tested materials was for the material with the addition of ceramic powder (SiC)—the greatest value of the coefficient of friction (Table 1). The longest braking time *ts*,0 = 6.82 s and lowest temperature on the working surfaces, equal to 122.1 ◦C, was reached for the base material—the least value of the coefficient of friction.

Taking into account the disconnection of the clutch elements after stopping and convection cooling of the working surface with oil at the heat transfer coefficient *h* influences the value of the maximum contact surface temperature. The difference in the average temperature value for the five materials with the clutch components disconnected and the average temperature value obtained while maintaining the condition of perfect thermal contact was about 13 ◦C (30%) in the middle of the cooling stage (*t* ≈ 40 s) and 5 ◦C (17%) at the end (*t* ≈ 90 s).

Based on the presented research, we can conclude that the most promising from the point of view of achieving the shortest braking time with the same total friction work is the friction material with a 3% addition of SiC ceramics.

As a part of the future research, it is planned to determine the mechanical properties of the considered friction materials and to carry out numerical calculations of thermal stresses. In addition, attempts will be made to take into account the thermal contact resistance instead of using the perfect thermal contact condition.

**Author Contributions:** Conceptualization, A.Y., M.K. and A.I.; methodology, A.Y., M.K. and A.I.; software, P.G. and A.L.; validation, M.K. and A.L.; formal analysis, A.Y., M.K. and A.I.; investigation, P.G. and A.L.; resources, A.Y., A.I., P.G. and A.L.; data curation, P.G. and A.L.; writing—original draft preparation, M.K., P.G. and A.L.; writing—review and editing, A.Y., M.K. and A.I.; visualization, P.G. and A.L.; supervision, A.Y., M.K. and A.I.; project administration, M.K.; funding acquisition, A.Y., M.K. and P.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** Project financing through the program of the Minister of Science and Higher Education of Poland named "Regional Initiative of Excellence" in 2019–2022, project number 011/RID/2018/19, amount of financing 12,000,000 PLN.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Nomenclature**


### **References**


**Roman Kushnir 1,2, Anatoliy Yasinskyy 1, Yuriy Tokovyy 1,2,\* and Eteri Hart <sup>3</sup>**


**Abstract:** Within the framework of the one-dimensional model for a tribo-couple consisting of two elastic cylinders accounting for the frictional heat generation on the interface due to the roughness of the contacting dissimilar materials, a problem on the identification of the unknown temperature on one of the limiting surfaces of either inner or outer cylindrical layers is formulated and reduced to an inverse thermoelasticity problem via the use of the circumferential strain given on the other surface. To solve the latter problem, a semi-analytical algorithm is suggested, and its stability with respect to the small errors in the input data is analyzed. The efficiency of the proposed solution algorithm is validated numerically by comparing its results with the solution of a corresponding direct problem. The temperature and thermal stresses in the tribo-couple are analyzed.

**Keywords:** tribo-couple; cylindrical layers; frictional heating; unknown thermal loading; inverse thermoelasticity problem; Volterra integral equation; stable algorithm

### **1. Introduction**

Construction and improvement of the elements of present-day techniques, along with the development and implementation of new materials with advanced properties, necessitate the comprehensive analysis of the heat transfer and the stress–strain state in composite materials under the simultaneous action of force and thermal fields while accounting for a wide range of the operational and constructional features, as well as the interaction of the structural elements of different geometry [1]. The importance of such analysis for both mechanical engineering and material science is also motivated by the prioritized implementation of non-destructive testing, which is important for ensuring the safety and durability of the operational performance of the heat and power equipment [2–4].

The comprehensive thermoelastic analysis is extremely important for the structural elements, some surface parts of which appear to be inaccessible (due to specific structural, technological, operational, or environmental reasons) for the direct reading of the thermal and mechanical signatures that are to be in use as the boundary conditions for the corresponding direct heat-transfer and thermoelasticity problems. As a result, the corresponding heat-transfer and thermoelasticity problems for such structural elements become ill-posed and require some supplementary information about the thermal or mechanical process, collected, preferably, on the accessible segments of the surface. It is worth noting that the type of additional information can be regarded as a critical point of the methodologies for solving the ill-posed problems of this kind.

If, for example, the original problem is supplemented with the information about some parameters of the thermal process (e.g., temperature or heat flux) at some points of a solid or its surface, the problem of the identification of the unknown thermal loading can be reduced to solving an inverse heat conduction problem [5,6]. The inverse problems

**Citation:** Kushnir, R.; Yasinskyy, A.; Tokovyy, Y.; Hart, E. Inverse Thermoelastic Analysis of a Cylindrical Tribo-Couple. *Materials* **2021**, *14*, 2657. https://doi.org/ 10.3390/ma14102657

Academic Editor: Aleksander Yevtushenko

Received: 8 April 2021 Accepted: 3 May 2021 Published: 19 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

obtained in this case are substantially ill-posed and their solution is concerned with the application of the corresponding regularizing algorithms.

There are numerous practical cases, however, when the reproduction of all the components of thermal loading appears to be impossible within the framework of an inverse heat-transfer problem alone. While accounting for coupling between the temperature and strain fields, the original mathematical models in these cases can be extended to non-classical thermoelastic problems by implementing the additional information on the thermomechanical parameters (displacements, strains, or stresses) on the accessible segment of the surface. The problem of determining the temperature and thermostressed fields in a solid under the above conditions can thereby be reduced to an inverse thermoelasticity problem [7]. The inverse problems of this kind are conditionally well-posed; i.e., they may become well-posed under certain supplementary conditions. This can be explained by the fact that the components of the stress–strain state have the form of integral dependences on the temperature at all points of a solid, including its boundary [1,8,9]. For non-stationary processes, these conditions usually express the fitting between the input data at the initial moment of time or interrelation of the mechanical components on the surfaces of the solid [1,10]. Some methods for solving one- and two-dimensional inverse thermoelasticity problems have been addressed in [11–16].

The inverse analysis of the temperature on inaccessible surfaces is an important issue in the analysis of tribo-systems. Such analysis is a key point in evaluating the characteristics of frictional interaction and material properties and is vital for both mechanical engineering and material science. Therefore, many practical cases of thermoelasticity problems, those that focus on the coupling between the thermal and mechanical fields, are concerned with frictional heating induced by the roughness of the contacting surfaces of interacting solids (see, e.g., [17–19]).

In [20–22], a technique for solving the inverse thermoelasticity problems was presented based on the reduction to integral equations. Particularly in [21,22], one-dimensional thermoelasticity problems were considered for interacting layers with friction. In engineering practice and material science experiments, numerous tribo-systems involve elements of cylindrical shape. In this paper, we extend the technique for solving inverse one-dimensional thermoelasticity problems, which are obtained for the identification of the time-dependence of a temperature of one of the circumferences of a cylindrical tribo-couple by making use of the additionally known circumferential strain on the surface where the thermal loading is known.

### **2. Formulation of the Problem**

Consider a one-dimensional model of a cylindrical tribo-couple consisting of two cylindrical elements made of dissimilar materials generating heat due to mutual interfacial friction [17]. The model is schematized in Figure 1 and presented by a circular hollow cylinder "1" of the inner and outer radii *r* = *R*<sup>1</sup> and *r* = *R*<sup>0</sup> encapsulated without tension and gap into another cylinder "2" of the same shape with inner and outer radii *r* = *R*<sup>0</sup> and *r* = *R*2, respectively. Assume the inner, *r* = *R*1, and outer, *r* = *R*2, circumferences of the tribo-couple to be kept under the given transient temperatures *t* ∗ <sup>1</sup> (*τ*∗) and *t* ∗ <sup>2</sup> (*τ*∗) while being subjected to the compressive forces *P*1(*τ*∗) and *P*2(*τ*∗). Here, *r* is the radial coordinate and *τ*<sup>∗</sup> is time. The mechanical and thermal contact of the cylindrical layers occurring on the interface *r* = *R*<sup>0</sup> is assumed to be imperfect in view of the roughness of the material on contacting surfaces so that the linear relationship

$$
\widehat{u}^{(i)}(\tau\_\*) = (-1)^i n\_i P(\tau\_\*)\_\prime \; i = 1, 2 \tag{1}
$$

obtains between the radial displacements on the interface *<sup>u</sup>*(1)(*τ*∗) and *<sup>u</sup>*(2)(*τ*∗) induced by the deformation of micro-roughness and the contact pressure *P*(*τ*∗). Here, *ni* are the coefficients characterizing the deformative features of the contacting surfaces.

**Figure 1.** The scheme of the considered tribo-couple, where the inner and outer cylindrical layers are denoted by "1" and "2", respectively; the thermal and force loadings are imposed on the inner and outer surfaces *R*<sup>1</sup> and *R*<sup>2</sup> and the frictional heating occurs on the interface *R*0.

Assume that one of the cylinders (let it be the outer one) rotates against the other cylinder. Let us restrict our attention to the case when the rotation reaches a steady-state condition at a constant angular velocity *ω* = *const*. Due to the frictional forces according to Amonton's law, the interface *r* = *R*<sup>0</sup> is subjected to the non-stationary heat generation, and the specific power of the frictional heating sources equals the specific work of the friction forces. The mechanical and thermo-physical properties of cylinders 1 and 2 are constant and indicated with upper indices accordingly. Within the framework of the formulated problem, the transient temperature field in the considered tribo-couple varies along the radial coordinate *r* only, and in view of the plane strain condition, *u*(1) *<sup>z</sup>* <sup>=</sup> *<sup>u</sup>*(2) *<sup>z</sup>* = 0, where *u*(*i*) *<sup>z</sup>* is the axial displacement of the *i*th layer of the cylinder.

In view of the foregoing model, the one-dimensional thermoelasticity problem for the tribo-couple is governed by the following system of equations, including:

(*i*) the heat-transfer equation

$$\frac{\partial^2 T\_i^\*(\rho, \tau)}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial T\_i^\*(\rho, \tau)}{\partial \rho} = \frac{1}{b\_i} \frac{\partial T\_i^\*(\rho, \tau)}{\partial \tau} \tag{2}$$

(*ii*) and the Lamé equations

$$\frac{\partial^2 u\_r^{(i)}(\rho,\tau)}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u\_r^{(i)}(\rho,\tau)}{\partial \rho} - \frac{u\_r^{(i)}(\rho,\tau)}{\rho^2} = \beta\_i \mathbb{R}\_0 \frac{\partial \ T\_i^\*(\rho,\tau)}{\partial \rho} \tag{3}$$

$$\frac{\partial^2 u\_{\rho}^{(i)}(\rho,\tau)}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u\_{\rho}^{(i)}(\rho,\tau)}{\partial \rho} - \frac{u\_{\rho}^{(i)}(\rho,\tau)}{\rho^2} = 0 \tag{4}$$

under the set of complementary conditions consisting of:

(*i*) the mechanical boundary conditions

$$
\sigma\_{rr}^{(i)}(k\_{i\prime}\tau) = -P\_{i}(\tau),\ \mu\_{\varphi}^{(i)}(k\_{i\prime}\tau) = 0\tag{5}
$$

(*ii*) the mechanical interface conditions

$$\begin{array}{ll} \sigma\_{rr}^{(1)}(1,\tau) = \sigma\_{rr}^{(2)}(1,\tau) = -P(\tau), \ \sigma\_{r\varphi}^{(1)}(1,\tau) = \sigma\_{r\varphi}^{(2)}(1,\tau) = -fP(\tau) \\\ u\_{r}^{(1)}(1,\tau) + \hat{u}^{(1)}(\tau) = u\_{r}^{(2)}(1,\tau) + \hat{u}^{(2)}(\tau), \ u\_{z}^{(i)}(\rho,\tau) = 0 \end{array} \tag{6}$$

(*iii*) the thermal boundary conditions

$$T\_i^\*\left(k\_{i\prime}\tau\right) = t\_i^\*\left(\tau\right) \tag{7}$$

(*iv*) the thermal interface conditions

$$\begin{split} \lambda\_1 \frac{\frac{\partial T\_1^\*(1,\tau)}{\partial \rho} - \lambda\_2 \frac{\partial \, T\_2^\*(1,\tau)}{\partial \, \rho} = \omega R\_0^2 f P(\tau) \\ \lambda\_1 \frac{\partial \, T\_1^\*(1,\tau)}{\partial \rho} + \lambda\_2 \frac{\partial \, T\_2^\*(1,\tau)}{\partial \rho} = \frac{R\_0}{R} \left( T\_2^\*(1,\tau) - T\_1^\*(1,\tau) \right) \end{split} \tag{8}$$

(*v*) and the initial condition

$$T\_i^\*(\rho, 0) = T\_0 = \text{const} \neq 0 \tag{9}$$

where, *i* = 1, 2 (*i* = 1 corresponds to the range *ρ* ∈ [*k*1, 1) and *i* = 2 corresponds to the range *ρ* ∈ (1, *k*2]), *ρ* = *r*/*R*<sup>0</sup> is the dimensionless radial coordinate *ρ* ∈ [*k*1, *k*2], *ki* = *Ri* / *R*0, *<sup>τ</sup>* <sup>=</sup> *<sup>a</sup>*2*τ*∗/*R*<sup>2</sup> <sup>0</sup> is the Fourier criterion, *τ* ∈ (0, *τm*], *τ<sup>m</sup>* is a constant parameter, *b* <sup>1</sup> = *a*<sup>1</sup> /*a*2, *<sup>b</sup>*<sup>2</sup> <sup>=</sup> 1, *<sup>a</sup> <sup>i</sup>* is the coefficient of thermal diffusivity, *<sup>β</sup> <sup>i</sup>* <sup>=</sup> *<sup>α</sup>*(*i*) *<sup>T</sup>* (<sup>1</sup> <sup>+</sup> *<sup>ν</sup>i*)/(<sup>1</sup> <sup>−</sup> *<sup>ν</sup>i*), *<sup>α</sup>*(*i*) *<sup>T</sup>* is the coefficient of linear thermal expansion, *ν<sup>i</sup>* is the Poisson ratio, *λ<sup>i</sup>* denotes the heatconduction coefficient, *T*∗ *<sup>i</sup>* is the temperature, *<sup>u</sup>*(*i*) *<sup>r</sup>* and *<sup>u</sup>*(*i*) *ϕ* are the radial and circumferential displacements, *σ*(*i*) *rr* and *<sup>σ</sup>*(*i*) *<sup>r</sup><sup>ϕ</sup>* are the radial and tangential stress-tensor components, *f* is the coefficient of friction, and *R* is the coefficient of contact thermal resistance.

It is well known that in the case of plane strain, the thermoelasticity problem (3) and (4) can be represented by two independent problems [23], when (i) *u*(*i*) *<sup>r</sup>* = 0, *ε* (*i*) *rr* = 0, *ε* (*i*) *ϕϕ* <sup>=</sup> 0, *<sup>u</sup>*(*i*) *<sup>ϕ</sup>* = *ε* (*i*) *<sup>r</sup><sup>ϕ</sup>* <sup>=</sup> 0 and (ii) *<sup>u</sup>*(*i*) *<sup>ϕ</sup>* = 0, *ε* (*i*) *<sup>r</sup><sup>ϕ</sup>* <sup>=</sup> 0, *<sup>u</sup>*(*i*) *<sup>r</sup>* =*ε* (*i*) *rr* =*ε* (*i*) *ϕϕ* = 0, *i* = 1, 2. Here, *ε* (*i*) *rr* , *ε* (*i*) *ϕ ϕ*, and *ε* (*i*) *<sup>r</sup><sup>ϕ</sup>* are, respectively, the radial, circumferential, and tangential strains of the *i*th cylindrical layer.

If all the input functions and coefficients in Equations (2)–(9) are properly imposed, then the formulated problem appears to be a well-posed direct thermoelasticity problem. Assuming, however, the transient temperature *t* ∗ <sup>1</sup> (*τ*), *τ* ∈ [0, *τm*], on the inner surface *ρ* = *k*<sup>1</sup> to be unknown (a typical situation due to the inaccessibility of the inner surface for the direct measurement) necessitates the determination of this function prior to solving the direct problem.

In order to identify this function appearing in the boundary condition (7), we use the supplementary information about the thermo-mechanical state of the compound cylinder, i.e., the condition

$$
\varepsilon^{(2)}\_{\varphi\varphi}(k\_2, \tau) = \varphi\_\*(\tau), \ \tau \in [0, \ \tau\_m] \tag{10}
$$

imposing the circumferential strain measured on the accessible outer surface *ρ* = *k*2. Here, *ϕ*∗(*τ*) is a given function of time.

Let us determine the temperature field and thermal stresses in the considered tribocouple by making use of condition (10) in order to identify the unknown temperature distribution *t* ∗ <sup>1</sup> (*τ*) on the inner circumference of the cylinder.

#### **3. Solution Technique**

By implementing the technique suggested in [17], a solution to the formulated thermoelastic problem (1), (3)–(6) can be given in the form expressing the circumferential

strain in the cylindrical tribo-couple explicitly through the force loadings and thermal field as follows:

$$\begin{split} \boldsymbol{\varepsilon}\_{\boldsymbol{\varrho}\boldsymbol{\varrho}}^{(i)}(\boldsymbol{\rho},\boldsymbol{\tau}) &= \boldsymbol{a}\_{T}^{(i)}(1+\nu\_{i})T\_{0} + \left((1-\overline{\nu}\_{i}) + \frac{1+\overline{\nu}\_{i}}{\rho^{2}}\right) \frac{c\_{i}p\_{i}(\boldsymbol{\tau})}{2} \\ &- \left((1-\overline{\nu}\_{i}) + \boldsymbol{k}\_{i}^{2}\frac{1+\overline{\nu}\_{i}}{\rho^{2}}\right) \frac{c\_{i}p(\boldsymbol{\tau})}{2\boldsymbol{k}\_{i}^{2}} + (-1)^{i+1}\frac{1-\overline{\nu}\_{i}}{1+\overline{\nu}\_{i}}\frac{\beta\_{i}T\_{0}}{1-\boldsymbol{k}\_{i}^{2}} \int\_{\boldsymbol{v}\_{1}^{(i)}}^{\boldsymbol{\xi}} \boldsymbol{\xi}T\_{i}(\boldsymbol{\xi},\boldsymbol{\tau}) \mathrm{d}\boldsymbol{\xi} \\ &+ \frac{\beta\_{i}T\_{0}}{2\rho^{2}} \int\_{0}^{\boldsymbol{v}\_{2}^{(i)}} \boldsymbol{\xi} \left((-1)^{i+1}\frac{1+\underline{\boldsymbol{k}}\_{i}^{2}}{1-\boldsymbol{k}\_{i}^{2}} + \text{sgn}(\rho-\boldsymbol{\xi})\right) T\_{i}(\boldsymbol{\xi},\boldsymbol{\tau}) \mathrm{d}\boldsymbol{\xi} \end{split} \tag{11}$$

and

$$c\_3 p(\tau) = c\_1 p\_1(\tau) - c\_2 p\_2(\tau) + \ell\_1 \int\_{k\_1}^1 \xi T\_1(\xi, \tau) d\xi + \ell\_2 \int\_1^{k\_2} \xi T\_2(\xi, \tau) d\xi + (\overline{a}\_1 - \overline{a}\_2) T\_0 \tag{12}$$

where *i* = 1, 2, *p*(*τ*) = *P*(*τ*)/*σ*<sup>∗</sup> and *pi*(*τ*) = *Pi*(*τ*)/*σ*<sup>∗</sup> are the dimensionless contact pressure and compressive pressures on the inner and outer surfaces, *σ*<sup>∗</sup> is a constant in the dimension of stresses, *<sup>ν</sup><sup>i</sup>* = *<sup>ν</sup>i*/(<sup>1</sup> − *<sup>ν</sup>i*), *Ei* = *Ei*/(<sup>1</sup> − *<sup>ν</sup>*<sup>2</sup> *<sup>i</sup>* ), *Ti* = (*T*<sup>∗</sup> *<sup>i</sup>* − *T*0)/*T*0, *v* (*i*) *<sup>i</sup>* = *ki*, *v* (2) <sup>1</sup> = *v* (1) <sup>2</sup> = 1, *<sup>i</sup>* <sup>=</sup> <sup>2</sup>*α*(*i*) *<sup>T</sup>* (<sup>1</sup> + *<sup>ν</sup>i*)*T*0/(<sup>1</sup> − *<sup>k</sup>*<sup>2</sup> *<sup>i</sup>* ), *<sup>α</sup><sup>i</sup>* <sup>=</sup> *<sup>α</sup>*(*i*) *<sup>T</sup>* (<sup>1</sup> + *<sup>ν</sup>i*), *ci* = <sup>2</sup>*k*<sup>2</sup> *<sup>i</sup> <sup>σ</sup>*∗/((<sup>1</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *<sup>i</sup>* )*Ei*), *Ei* denotes the Young modulus of the *i*th cylindrical layer, and

$$c\_3 = \sum\_{i=1}^{2} \left(-1\right)^{i+1} \frac{1 - \overline{v}\_i + (1 + \overline{v}\_i)k\_i^2}{1 - k\_i^2} \frac{\sigma\_\*}{\overline{E}\_i} + \frac{(n\_1 + n\_2)\sigma\_\*}{R\_0}$$

A general solution to Equation (4) for the circumferential strain *u*(*i*) *ϕ* can be given [23] as

$$
\mu\_{\phi}^{(i)}(\rho,\tau) = \frac{A\_{\ i}(\tau)R\_{0}\rho}{2} + \frac{B\_{i}(\tau)}{R\_{0}\rho}
$$

where *A <sup>i</sup>* (*τ*) and *B <sup>i</sup>*(*τ*) are arbitrary and yet unknown functions of time, *i* = 1, 2. By making use of conditions (5) and (6) for the displacement *u*(*i*) *<sup>ϕ</sup>* and stress *<sup>σ</sup>*(*i*) *<sup>r</sup><sup>ϕ</sup>* , we can finally derive

$$\begin{aligned} u\_{\boldsymbol{\rho}}^{(i)}(\boldsymbol{\rho},\boldsymbol{\tau}) &= \frac{f(1+\overline{\nu}\_{i})\mathbb{R}\_{0}\mathbb{P}(\boldsymbol{\tau})}{\mathbb{E}\_{i}\boldsymbol{\rho}} \left(1 - \frac{\boldsymbol{\rho}^{2}}{k\_{i}^{2}}\right) \\ \varepsilon\_{r\boldsymbol{\rho}}^{(i)}(\boldsymbol{\rho},\boldsymbol{\tau}) &= -\frac{f(1+\overline{\nu}\_{i})\mathbb{P}(\boldsymbol{\tau})}{\mathbb{E}\_{i}\boldsymbol{\rho}^{2}}, \; \sigma\_{r}^{(i)}(\boldsymbol{\rho},\boldsymbol{\tau}) = -\frac{f\mathbb{P}(\boldsymbol{\tau})}{\rho^{2}} \end{aligned} \tag{13}$$

Equation (13) allow for expressing the thermal stresses and displacements in the two-layer cylindrical tribo-couple through the contact pressure found by formula (12).

Assuming the function *t* ∗ <sup>1</sup> (*τ*) to be known for *τ* ≥ 0 and making use of the integral Laplace transform [24] by the time-variable *τ* yields a solution to the heat-conduction problem (2), (7), (8) and (9) in the form as follows

$$T\_i(\rho, \tau) = \sum\_{j=1}^2 \int\_0^\tau G\_j^{(i)}(\rho, \tau - \tilde{\xi}) t\_j(\tilde{\xi}) \, \mathrm{d}\tilde{\xi} + \Omega \int\_0^\tau G\_3^{(i)}(\rho, \tau - \tilde{\xi}) p(\tilde{\xi}) \, \mathrm{d}\tilde{\xi} \tag{14}$$

where *i* = 1, 2,

*G*(1) <sup>1</sup> (*ρ*, *<sup>τ</sup>*) = <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) (2*λZ*(1) <sup>10</sup> (1, *<sup>ρ</sup>*,*sn*) *<sup>Z</sup>*(2) <sup>10</sup> (1, *k*2,*sn*) +*ϑ*(*λZ*(2) <sup>00</sup> (*k*2, 1,*sn*)*Z*(1) <sup>10</sup> (1, *<sup>ρ</sup>*,*sn*) + *<sup>Z</sup>*(2) <sup>10</sup> (1, *<sup>k</sup>*2,*sn*)*Z*(1) <sup>00</sup> (1, *ρ*,*sn*))) *G*(1) <sup>2</sup> (*ρ*, *<sup>τ</sup>*) = *<sup>ϑ</sup>* <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) *<sup>Z</sup>*(1) <sup>00</sup> (*ρ*, *k*1,*sn*) *G*(1) <sup>3</sup> (*ρ*, *<sup>τ</sup>*) = <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) *Z*(2) <sup>10</sup> (1, *<sup>k</sup>*2,*sn*) + *<sup>ϑ</sup>Z*(2) <sup>00</sup> (*k*2, 1,*sn*) *Z*(1) <sup>00</sup> (*ρ*, *k*1,*sn*) *G*(2) <sup>1</sup> (*ρ*, *<sup>τ</sup>*) = <sup>−</sup>*λϑ* <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) *<sup>Z</sup>*(2) <sup>00</sup> (*ρ*, *k*2,*sn*) *G*(2) <sup>2</sup> (*ρ*, *<sup>τ</sup>*) = <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) (2*λZ*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*sn*) *<sup>Z</sup>*(2) <sup>10</sup> (1, *ρ*,*sn*) +*ϑ*(*λZ*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*sn*)*Z*(2) <sup>00</sup> (*ρ*, 1,*sn*) + *<sup>Z</sup>*(1) <sup>00</sup> (1, *<sup>k</sup>*1,*sn*)*Z*(2) <sup>10</sup> (1, *ρ*,*sn*))) *G*(2) <sup>3</sup> (*ρ*, *<sup>τ</sup>*) = <sup>−</sup> <sup>∞</sup> ∑ *n*=1 exp(−*μ*<sup>2</sup> *<sup>n</sup>τ*) *<sup>∂</sup>s*(Δ(*sn*)) (*λZ*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*sn*) + *<sup>ϑ</sup>Z*(1) <sup>00</sup> (1, *<sup>k</sup>*1,*sn*)) *<sup>Z</sup>*(2) <sup>00</sup> (*ρ*, *k*2,*sn*) Δ(*s*) = 2*λZ*(2) <sup>10</sup> (1, *<sup>k</sup>*2,*s*)*Z*(1) <sup>10</sup> (1, *k*1,*s*) +*ϑ λZ*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*s*)*Z*(2) <sup>00</sup> (*k*2, 1,*s*) + *<sup>Z</sup>*(2) <sup>10</sup> (1, *<sup>k</sup>*2,*s*)*Z*(1) <sup>00</sup> (1, *k*1,*s*) *<sup>Z</sup>*(*j*) <sup>10</sup> (*x*, *y*,*s*) = *qjx*(*I*1(*qjx*)*K*0(*qjy*) + *I*0(*qjy*)*K*1(*qjx*)) *<sup>Z</sup>*(*j*) *kk* (*x*, *y*,*s*) = *Ik*(*qjx*)*Kk*(*qjy*) + *Ik*(*qjy*)*Kk*(*qjx*)

*j* = 1, 2; *k* = 0, 1, Ω = *ωR*<sup>2</sup> <sup>0</sup> *f σ*∗/(*λ*2*T*0) is the dimensionless angular velocity, *<sup>λ</sup>* <sup>=</sup> *<sup>λ</sup>*1/*λ*2, *<sup>ϑ</sup>* <sup>=</sup> *Rs*/*R*, *Rs* <sup>=</sup> *<sup>R</sup>*0/(*R*∗*λ*2), *<sup>q</sup>*<sup>2</sup> <sup>1</sup> = *<sup>s</sup>*/*b*1, *<sup>q</sup>*<sup>2</sup> <sup>2</sup> = *s*, *R* = *R*/*R*<sup>∗</sup> is the dimensionless interfacial thermal resistance, *R*<sup>∗</sup> is a constant in the dimension of thermal resistance, *Ik*(*s*) and *Kk*(*s*) are the modified Bessel functions of the first and second kind, *k* = 0, 1, *s* stands for the parameter of the Laplace transform, *∂<sup>s</sup>* denotes the partial derivative by *s*, and *sn* = −*μ*<sup>2</sup> *<sup>n</sup>* are the roots of the characteristic equation Δ(*s*) = 0, *μ<sup>n</sup>* > 0, *n* = 1, 2, . . .

Formula (14) expresses the dependence of the temperature field within the tribocouple on the contact pressure, while formula (12) shows the dependence of the contact pressure on the temperature. By making use of these two formulas along with expression (11) for the circumferential strain, the condition for the radial displacement in (5) and (6) yields the following formula for the contact pressure on the interface:

$$\begin{split} p(\boldsymbol{\tau}) &= \int\_{0}^{\tau} M(\boldsymbol{\tau} - \boldsymbol{\eta}) (c\_{1} p\_{1}(\boldsymbol{\eta}) - c\_{2} p\_{2}(\boldsymbol{\eta})) \mathrm{d}\boldsymbol{\eta} + \sum\_{i=1}^{2} \int\_{0}^{\tau} N\_{i}(\boldsymbol{\tau} - \boldsymbol{\eta}) t\_{i}(\boldsymbol{\eta}) \mathrm{d}\boldsymbol{\eta} \\ &+ \Big( (1 + \nu\_{1}) a\_{T}^{(1)} - (1 + \nu\_{2}) a\_{T}^{(2)} \Big) T\_{0} \int\_{0}^{\tau} M(\boldsymbol{\eta}) \mathrm{d}\boldsymbol{\eta} \end{split} \tag{15}$$

where

*<sup>M</sup>*(*τ*) = <sup>∞</sup> ∑ *n*=1 Δ(*s*∗ *<sup>n</sup>*) exp(*s*<sup>∗</sup> *<sup>n</sup>τ*) *∂s*(Δ∗(*s*<sup>∗</sup> *<sup>n</sup>*)) , *Ni*(*τ*) = <sup>∞</sup> ∑ *n*=1 *Vi*(*s*<sup>∗</sup> *<sup>n</sup>*) exp(*s*<sup>∗</sup> *<sup>n</sup>τ*) *∂s*(Δ∗(*s*<sup>∗</sup> *<sup>n</sup>*)) *V*1(*s*) = -1 <sup>2</sup>*λk*1*Z*(1) <sup>11</sup> (1, *<sup>k</sup>*1,*s*)*Z*(2) <sup>10</sup> (1, *k*2,*s*) + *ϑ <sup>λ</sup>k*1*Z*(1) <sup>11</sup> (1, *<sup>k</sup>*1,*s*)*Z*(2) <sup>00</sup> (*k*2, 1,*s*) + *Z*(2) <sup>10</sup> (1, *k*2,*s*) *<sup>Z</sup>*(1) <sup>10</sup> (*k*1,1,*s*)−1 *q*2 1 − -<sup>2</sup>*λϑ* <sup>1</sup>−*Z*(2) <sup>10</sup> (1,*k*2,*s*) *q*2 2 *V*2(*s*) = -<sup>1</sup>*<sup>ϑ</sup> <sup>Z</sup>*(1) <sup>10</sup> (1,*k*1,*s*)−1 *q*2 1 + -2 <sup>2</sup>*λk*2*Z*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*s*)*Z*(2) <sup>11</sup> (*k*2, 1,*s*) +*ϑ λZ*(1) <sup>10</sup> (1, *k*1,*s*) *<sup>Z</sup>*(2) <sup>10</sup> (*k*2,1,*s*)−1 *q*2 2 <sup>+</sup> *<sup>k</sup>*2*Z*(1) <sup>00</sup> (1, *<sup>k</sup>*1,*s*)*Z*(2) <sup>11</sup> (*k*2, 1,*s*) Δ∗(*s*) = *c*3Δ(*s*) − Ω -1 *Z*(2) <sup>10</sup> (1, *<sup>k</sup>*2,*s*) + *<sup>ϑ</sup>Z*(2) <sup>00</sup> (*k*2, 1,*s*) *<sup>Z</sup>*(1) <sup>10</sup> (1,*k*1,*s*)−1 *q*2 1 −-2 *λZ*(1) <sup>10</sup> (1, *<sup>k</sup>*1,*s*) + *<sup>ϑ</sup>Z*(1) <sup>00</sup> (1, *k*1,*s*) 1−*Z*(2) <sup>10</sup> (1,*k*2,*s*) *q*2 2 

*s*∗ *<sup>n</sup>* are the roots of the characteristic equation Δ∗(*s*) = 0, *n* = 1, 2, . . .

By putting (14) and (15) into formula (11) at *i* = 2 and *ρ* = *k*<sup>2</sup> within the context of the supplementary condition (10) for the circumferential strain, we arrive at the convolutiontype Volterra integral equation of the first kind [25] for the determination of function *t*1(*τ*) in the following form:

$$\begin{aligned} \stackrel{\tau}{\underset{0}{\mathbb{T}}} K\_{1}(\tau-\eta) \; t\_{1}(\eta) \mathbf{d}\eta &= \varrho\_{\ast}(\tau) - \stackrel{\tau}{\underset{0}{\mathbb{T}}} K\_{2}(\tau-\eta) \; t\_{2}(\eta) \mathbf{d}\eta \\ - \stackrel{\tau}{\underset{0}{\mathbb{T}}} L(\tau-\eta) \; (c\_{1}p\_{1}(\eta) - c\_{2}p\_{2}(\eta)) \mathbf{d}\eta &- \stackrel{\tau\_{2}}{2k\_{2}} (1+k\_{2}^{2}+(1-k\_{2}^{2})\overline{\nu}\_{2}) p\_{2}(\tau) \\ - \left((1+\nu\_{1})a\_{T}^{(1)} - (1+\nu\_{2})a\_{T}^{(2)}\right) T\_{0} \stackrel{\tau}{\underset{0}{\mathbb{T}}} L(\eta) \mathbf{d}\eta - (1+\nu\_{2})k\_{2}a\_{T}^{(2)}T\_{0} \end{aligned} \tag{16}$$

where *τ* ∈ [0, *τm*] and

$$\begin{split} K\_{i}(\tau) &= \sum\_{k=1}^{\infty} \frac{\mathcal{U}\_{i}(s\_{k}) \exp(s\_{k}\tau)}{\mathcal{U}\_{i}(\Delta(s\_{k})\Delta\_{\*}(s\_{k}))},\ L(\tau) = \sum\_{n=1}^{\infty} \frac{\mathcal{V}\_{3}(s\_{n}^{\*}) \exp(s\_{n}^{\*}\tau)}{\mathcal{S}\_{i}(\Delta\_{\*}(s\_{n}^{\*}))} \\ \mathcal{U}\_{1}(s) &= \mathcal{V}\_{1}(s)\mathcal{V}\_{3}(s) + k\_{2}\ell\_{2}\lambda\vartheta \frac{1-Z\_{10}^{(1)}(1,k\_{2}s)}{q\_{2}^{2}}\Delta\_{\*}(s) \\ \mathcal{U}\_{2}(s) &= \mathcal{V}\_{2}(s)\mathcal{V}\_{3}(s) - k\_{2}\ell\_{2}(2\lambda k\_{2}Z\_{10}^{(1)}(1,k\_{1},s)Z\_{11}^{(2)}(k\_{2},1,s) \\ &+ \theta\left(\lambda Z\_{10}^{(1)}(1,k\_{1},s)\frac{Z\_{10}^{(2)}(k\_{2}1,s)-1}{q\_{2}^{2}} + k\_{2}Z\_{00}^{(1)}(1,k\_{1},s)Z\_{11}^{(2)}(k\_{2},1,s)\right)\right)\Delta\_{\*}(s) \\ \mathcal{V}\_{3}(s) &= -\frac{c\_{2}}{k\_{2}}\Delta(s) + \Omega\,k\_{2}\ell\_{2}\left(\lambda Z\_{10}^{(1)}(1,k\_{1},s) + \theta\,Z\_{00}^{(1)}(1,k\_{1},s)\right)\frac{1-Z\_{10}^{(2)}(1,k\_{2},s)}{q\_{2}^{2}} \end{split}$$

*sk* are roots of equations Δ(*s*) = 0 and Δ∗(*s*) = 0 combined, which are negative real numbers *sk* = −*γ*<sup>2</sup> *<sup>k</sup>* , *γ<sup>k</sup>* > 0, *k* = 1, 2, ..., when the angular velocity does not exceed a critical value [17].

By setting *τ* = 0 in (16) and allowing *ti*(0) = 0, *i* = 1, 2, we derive the fitting condition for the initial temperature, the circumferential strain imposed on the outer surface *ρ* = *k*2, and the dimensionless pressures on the inner and outer circumferences of the tribo-couple at the initial moment of time in the form as follows:

$$\begin{cases} \frac{c\_2}{2k\_2}(1+k\_2^2+(1-k\_2^2)\overline{\nu}\_2)p\_2(0) - \frac{c\_2}{k\_2c\_3}(c\_1p\_1(0)-c\_2p\_2(0))\\ -\frac{c\_2}{k\_2c\_3}((1+\nu\_1)a\_T^{(1)}-(1+\nu\_2)a\_T^{(2)})T\_0+(1+\nu\_2)k\_2a\_T^{(2)}T\_0 = \rho\_\*(0) \end{cases}$$

The latter condition ensures the continuity of the solution of integral Equation (16).

In such a manner, the original heat-conduction problem for the considered cylindrical tribo-couple with frictional hating is reduced to an inverse thermoelasticity problem, which is verbalized by the integral Equation (16) and implies the determination of the temperature on the inner surface via the temperature and circumferential strain given on the outer surface.

It can be shown that the kernel *K*1(*τ* − *η*) of Equation (16) is always positive for *η* ∈ [0, *τ*], increases monotonically and suffers the root singularity at *η* = *τ*. This means that Equation (16) is the Abel integral equation [25]. The fact that the kernel *K*1(*τ* − *η*) has the integrable singularity at *η* = *τ* implies the absence of the time delay in the maximum response of the thermal constituent of the circumferential strain *ϕ*∗(*τ*) to the variation of temperature *t*1(*τ*).

Assume the unknown temperature *t*1(*η*) to be a continuous function on the interval [0, *τ*], i.e., *t*1(*η*) ∈ C[0,*τ*], to construct a solution to Equation (16). Let us represent the time interval [0, *τm*] by the mesh consisting of *m* intervals of the length *h* = *τm*/*m* and represent the sought-out function on each of these intervals by a linear spline

*S*(1) *<sup>j</sup>* (*τ*) = ((*τ<sup>j</sup>* − *τ*)*t* (*j*−1) <sup>1</sup> + (*τ* − *τj*−1)*t* (*j*) <sup>1</sup> )*h*<sup>−</sup>1, *<sup>τ</sup>* <sup>∈</sup> [*τj*−1, *<sup>τ</sup>j*], *<sup>τ</sup><sup>j</sup>* <sup>=</sup> *hj*, *<sup>t</sup>* (*j*) <sup>1</sup> = *t*1(*τj*), *j* = 1, . . . , *m*. As a result, Equation (16) yields the following system of linear algebraic equations:

$$t\_1^{(1)} = \frac{\Phi\_1}{\mathfrak{c}\_0}, \sum\_{j=1}^{l-1} \Theta\_{l'j} t\_1^{(j)} + t\_1^{(l)} = \frac{\Phi\_l}{\mathfrak{c}\_0}, l = 2, \dots, m \tag{17}$$

Here, Φ*<sup>l</sup>* = Φ(*τl*) is the values of the right-hand side of Equation (16) at the knots of the mesh *τ* = *τ<sup>l</sup>* and

$$\begin{split} \Theta\_{lj} \equiv q\_{l-j} &= \frac{1}{\varepsilon\_0} \sum\_{k=1}^{\infty} \frac{\mathcal{U}\_1(s\_k)}{\frac{\mathcal{S}\_1(\Delta(s\_k)\Delta\_\*(s\_k))}{\mathcal{S}\_1\hbar^2}} \frac{(1 - \exp(-\gamma\_k^2 h))^2}{\gamma\_k^4 \hbar^2} \exp\left(-\gamma\_k^2 h (l - j - 1)\right), j < l \\ &\mathcal{C}\_0 = \sum\_{k=1}^{\infty} \frac{\mathcal{U}\_1(s\_k)}{\mathcal{S}\_1(\Delta(s\_k)\Delta\_\*(s\_k))} \frac{1}{\gamma\_k^2 h} \left(1 - \frac{1 - \exp(-\gamma\_k^2 h)}{\gamma\_k^2 h}\right) \end{split}$$

The matrix of system (17) is the lower diagonal matrix with equal elements on each diagonal below the main one:

$$\mathbf{Q}\_1 = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \\ q\_1 & 1 & 0 & \cdots & 0 & 0 \\ q\_2 & q\_1 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ q\_{m-1} & q\_{m-2} & q\_{m-3} & \cdots & q\_1 & 1 \end{pmatrix}, \\ 0 < q\_1 < 1, \ q\_{i+1} < q\_i, \ i = 1, \dots, m-1$$

It can be shown that for *h* > 0, the norm **Q**1 = max *j* ∑ *i* Θ*i j* < ∞. System (17) can be represented in the following form:

$$\mathbf{T} = \mathbf{Q}\_2 \mathbf{T} + \mathbf{F} \tag{18}$$

where

$$\mathbf{Q}\_2 = \begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 \\ q\_1^\* & 0 & 0 & \cdots & 0 & 0 \\ q\_2^\* & q\_1^\* & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ q\_{m-1}^\* & q\_{m-2}^\* & q\_{m-3}^\* & \cdots & q\_1^\* & 0 \end{pmatrix}, \mathbf{T} = \begin{pmatrix} t\_1^{(1)} \\ t\_1^{(2)} \\ t\_1^{(3)} \\ \vdots \\ t\_1^{(m)} \end{pmatrix}, \mathbf{F} = \frac{1}{c\_0} \begin{pmatrix} \Phi\_1 \\ \Phi\_2 - \Phi\_1 \\ \Phi\_3 - \Phi\_2 \\ \vdots \\ \Phi\_m - \Phi\_{m-1} \end{pmatrix}$$

*q*∗ <sup>1</sup> = 1 − *q*1, *q*<sup>∗</sup> *<sup>i</sup>* = *qi*−<sup>1</sup> − *qi*, *q*<sup>∗</sup> *<sup>i</sup>* > 0, *i* = 2, ... , *m* − 1. Due to the fact that **Q**2 <sup>=</sup> *<sup>m</sup>*−<sup>1</sup> ∑ *i*=1 *q*∗ *<sup>i</sup>* =1 − *qm* < 1 for *h* > 0, the simple iteration routine [26,27] implies that the problem on solving the system of Equation (18), and, consequently, (17) is well-posed. Based on this fact, system (17) allows for deriving a recursive formula for determination of *t* (*l*) <sup>1</sup> , *l* = 1, . . . , *m*.

Having derived the temperature *t*1(*τ*), *τ* ∈ [0, *τm*], by means of the foregoing routine, we can use Formula (14) to determine the temperature field within the tribo-couple. The thermal stresses and displacements can be computed accordingly by making use of the Formulae (11)–(13) and (15), along with the basic thermoelasticity equations [23].

### **4. Numerical Example and Discussion**

In order to verify the proposed solution technique, consider a solution to the formulated identification problem for the tribo-couple, whose inner layer 1 is made of steel (*λ*<sup>1</sup> <sup>=</sup> 21 [Wt/(m <sup>×</sup> K)], *<sup>a</sup>* <sup>1</sup> <sup>=</sup> 5.9 <sup>×</sup> <sup>10</sup>−<sup>6</sup> [m2/s], *<sup>α</sup>* (1) *<sup>T</sup>* = <sup>14</sup> × <sup>10</sup>−<sup>6</sup> [1/K], *<sup>E</sup>*<sup>1</sup> = 190 [GPa], and *ν*<sup>1</sup> = 0.3) and the outer one 2 is made of copper (*λ*<sup>2</sup> = 381 [Wt/(m × K)], *<sup>a</sup>* <sup>2</sup> <sup>=</sup> 101.9 <sup>×</sup> <sup>10</sup>−<sup>6</sup> [m2/s], *<sup>α</sup>* (2) *<sup>T</sup>* = <sup>17</sup> × <sup>10</sup>−<sup>6</sup> [1/K], *<sup>E</sup>*<sup>2</sup> = 121 [GPa], and *<sup>ν</sup>*<sup>2</sup> = 0.33).

Herein, we employ the following commonly used verification strategy [5] with two stages. In the first stage, we formulate a direct problem by imposing the temperature *t* ∗ <sup>1</sup> (*τ*) on the inner circumference of the tribo-couple. Together with the given temperature on the outer surface and the interface thermal conditions (8), this would allow us to compute the thermal field in the tribo-couple. Making use of the determined temperature, a solution of the thermoelasticity problem (3)–(6) is constructed analytically. The latter solution can then be used to derive an expression for the circumferential strain on the outer surface of the tribo-couple. In the second stage, we formulate the inverse problem, where condition (10) is used together with the circumferential strain computed on the previous stage in order to restore the temperature on the inner surface by making use of the proposed algorithm. By comparing the solution of the inverse problem with the temperature *t* ∗ <sup>1</sup> (*τ*) imposed when formulating the direct problem on stage 1, we can draw a conclusion about the efficiency of the algorithm. When solving the inverse problem in this stage, we also introduce some random small errors in the distribution of the circumferential strain in order to verify the stability of the algorithm.

By following this strategy, let us first consider the direct heat-conduction and thermoelasticity problems by imposing the following boundary temperatures

$$t\,^\*\_1(\tau) = T\_0 + B(1 - \cos 2\tau), \, t\,^\*\_2(\tau) = T\_0 \tag{19}$$

and pressures *p*1(*τ*) = *C*H(*τ*) and *p*2(*τ*) = 0, where *B*, *C* = const, H(*τ*) is the Heaviside step function, to determine the circumferential strain distribution on the outer surface *ρ* = *k*2. Then, we can approximate the constructed strain within certain accuracy and use it as the input data for the inverse problem to determine the temperature *t* (*l*) <sup>1</sup> , *l* = 1, ... , *m*, on the inner surface *<sup>ρ</sup>* <sup>=</sup> *<sup>k</sup>*1. By comparing the computed values *<sup>t</sup>* (*l*) <sup>1</sup> , *l* = 1, ... , *m*, with the actual *t*1(*τ*), *τ* ∈ [0, *τm*], imposed in (19), we can evaluate the accuracy of the proposed solution algorithm for the considered inverse problem of thermoelasticity.

The distribution of the dimensionless circumferential strain *ε*(*τ*) = *ε* (2) *ϕϕ*(*k*2, *<sup>τ</sup>*) × <sup>10</sup><sup>4</sup> on the outer surface *ρ* = *k*<sup>2</sup> is shown in Figure 2a. The strain was computed from the direct problem under the thermal loading (19) for the following parameters *<sup>R</sup>* <sup>0</sup> = 5.0 × <sup>10</sup>−<sup>2</sup> [m], *<sup>R</sup>* <sup>1</sup> = 3.5 × <sup>10</sup>−<sup>2</sup> [m], *<sup>R</sup>* <sup>2</sup> = 6.0 × <sup>10</sup>−<sup>2</sup> [m], *<sup>n</sup>* <sup>1</sup> = <sup>10</sup>−<sup>3</sup> [m/GPa], *<sup>n</sup>*<sup>2</sup> = <sup>10</sup>−<sup>4</sup> [m/GPa], *<sup>R</sup>* <sup>=</sup> 5.0 <sup>×</sup> <sup>10</sup>−<sup>3</sup> [m2 <sup>×</sup> K/Wt], *<sup>R</sup>* <sup>∗</sup> <sup>=</sup> 1.1 <sup>×</sup> <sup>10</sup>−<sup>3</sup> [m<sup>2</sup> <sup>×</sup> K/Wt]; *<sup>σ</sup>*<sup>∗</sup> <sup>=</sup> <sup>10</sup><sup>2</sup> [MPa], *<sup>T</sup>*<sup>0</sup> <sup>=</sup> 20 [K], *B* = 200 [K], *C* = 102, *f* = 0.25, *ω* = 1.22 [rad/s], and *τ <sup>m</sup>* = 2.5.

Now we can use the computed strain as the input data for solving the inverse problem in order to reconstruct the thermal loading on the inner circumference of cylinder 1. It is also important to analyze the effect of small errors in the input data (which can be induced by the errors in the stain measurement, etc.). For modeling of such errors, let us substitute the strain distribution at the discrete time moments *<sup>τ</sup><sup>i</sup>* with the values *<sup>ε</sup>*(*τi*) computed by the formula *<sup>ε</sup>*(*τi*) = *<sup>ε</sup>*(*τi*)(<sup>1</sup> <sup>+</sup> *<sup>θ</sup><sup>i</sup>* <sup>×</sup> <sup>10</sup>−2), where *<sup>θ</sup><sup>i</sup>* are arbitrary numbers from the interval [−1, 1] with the uniform distribution law and represent *<sup>ε</sup>*(*τ*) by a linear spline. This means that the input data are encountered with an arbitrary error falling within 1%.

In Figure 2b, the open circles denote the time distribution of the temperature *t* (*i*) 1 , *i* = 1, 250 on the inner surface of the cylinder 1, found by solving the inverse thermoelasiticy problem with the computational step *h* = 0.01. It is shown that the maximum relative error of the computed values in comparison to the corresponding values imposed in the direct problem (19) falls within 1.8%, which confirms the stability of the proposed solution algorithm with respect to the small errors in the input data. Due to the fact that the solutions to well-posed direct problems are stable with respect to small errors in the input data, the error in computing the thermal stresses, strains, and displacements by using the thermal loading (19) of the one computed by solving the inverse problem can be dismissed.

**Figure 2.** The circumferential elastic strain *ε*(*τ*) = *ε* (2) *ϕϕ*(*k*2, *<sup>τ</sup>*) <sup>×</sup> <sup>10</sup><sup>4</sup> (**a**) computed by the temperature *t*1(*τ*) given in (19) by solving the direct problem versus the dimensionless time *τ*; the dimensionless temperature *<sup>t</sup>* (*i*) <sup>1</sup> on the inner circumference (**b**) as given by formula (19) (solid lines) and computed by solving the inverse problem (open circles).

### **5. Conclusions**

A problem on the determination of temperature and thermal-stress fields in a cylindrical tribo-couple with frictional heating on the interface is formulated for the case when the thermal loading on one of its circumferences is unknown. The additional information about the transient variation of the circumferential strain on the surface where the thermal loading is known was used as a supplementary condition for the formulated inverse thermoelasticity problem governed by a Volterra integral equation of the first kind. Due to the fact that the kernel of this integral equation *K*1(*τ* − *η*) takes only positive values on the interval *η* ∈ [0, *τ*], monotonically increases for the entire range of variables, and suffers a root singularity at the point *η* = *τ*, this integral equation can be regarded as one of Abel kind. The presence of the integrable singularity in this kernel at *η* = *τ* implies that there is no delay in the maximum thermal response of the circumferential strain *ϕ*∗(*τ*) to a change in the variation profile of temperature *t*1(*τ*) at *η* = *τ* in view of the integral dependence of this strain on the temperature within the cylindrical layers of the tribo-couple. This feature of kernel *K*1(*τ* − *η*) ensures the conditional correctness of the inverse problem. The correctness condition in this case was derived in the form of the fitting condition for the, circumferential strain on the periphery of the tribo-couple, and the pressures applied to its surfaces at the initial moment of time.

It is worth noting that the analogous kernels within the framework of inverse heatconduction problems solely exhibit quite different features, which, in the final count, makes these problems ill-posed [5,6].

Another advantage of the proposed technique is that the system of algebraic Equation (17), which is the discrete analog for Equation (16), was represented in the form (18). This, in view of the appearance of its matrix **Q**<sup>2</sup> ensures the stability of its solution with respect to small errors in the input data. An algorithm for solving the formulated inverse problem is suggested on the basis of the linear spline approximation technique. The efficiency of the algorithm was verified by solving the direct problem under the given thermal loading in order to determine the circumferential strain, which was then used as the input data for the inverse problem on the reconstruction of thermal loading.

These key features of the proposed algorithm may serve for benefit of setting up technological and experimental cylindrical tribo-systems expecting incomplete information about thermal loading for engineering applications and the wear analysis [28,29].

**Author Contributions:** Conceptualization, R.K.; methodology, A.Y.; validation and formal analysis, Y.T.; computation, E.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by the budget program of Ukraine "Support for the Development of Priority Research Areas" (CPCEC 6451230) under the grant No. 0120U100499.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data Sharing is not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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