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Applied theory of contact interaction of elastic bodies and the creation on its basis of the processes of shaping friction-rolling bearings with rational geometry

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However modern theory elastic contact does not allow to sufficiently search for a rational geometric shape contact surfaces in a fairly wide range of operating conditions of rolling friction bearings. Experimental research in this area is limited by the complexity of the measuring technique and experimental equipment used, as well as by high labor intensity and duration...

• ACCEPTED SYMBOLS
• CHAPTER 1. CRITICAL ANALYSIS OF THE STATE OF THE ISSUE, GOALS AND OBJECTIVES OF THE WORK
  • 1. 1. System analysis of the current state and trends in the field of improving the elastic contact of bodies of complex shape
    • 1. 1. 1. Current state theory of local elastic contact of bodies of complex shape and optimization of the geometric parameters of the contact
    • 1. 1. 2. The main directions for improving the technology of grinding the working surfaces of rolling bearings of complex shape
    • 1. 1. 3. Modern technology of shaping superfinishing of surfaces of revolution
  • 1. 2. Research objectives
• CHAPTER 2 MECHANISM OF ELASTIC CONTACT OF BODIES
• COMPLEX GEOMETRIC SHAPE
  • 2. 1. The mechanism of the deformed state of elastic contact of bodies of complex shape
  • 2. 2. The mechanism of the stress state of the contact area of ​​elastic bodies of complex shape
  • 2. 3. Analysis of the Influence of the Geometric Shape of Contacting Bodies on the Parameters of Their Elastic Contact
• conclusions
• CHAPTER 3 FORM FORMATION OF RATIONAL GEOMETRIC SHAPE OF PARTS IN GRINDING OPERATIONS
  • 3. 1. Formation of the geometric shape of rotation parts by grinding with a circle inclined to the axis of the part
  • 3. 2. Algorithm and program for calculating the geometric shape of parts for grinding operations with an inclined wheel and the stress-strain state of the area of ​​​​its contact with an elastic body in the form of a ball
  • 3. 3. Analysis of the influence of the parameters of the grinding process with an inclined wheel on the bearing capacity of the ground surface
  • 3. 4. Investigation of the technological possibilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and the operational properties of bearings made with its use
• conclusions
• CHAPTER 4 BASIS FOR SHAPING THE PROFILE OF PARTS IN SUPERFINISHING OPERATIONS
  • 4. 1. Mathematical model of the mechanism of the process of shaping parts during superfinishing
  • 4. 2. Algorithm and program for calculating the geometric parameters of the machined surface
  • 4. 3. Analysis of the influence of technological factors on the parameters of the surface shaping process during superfinishing
• conclusions
• CHAPTER 5 RESULTS OF STUDYING THE EFFICIENCY OF THE PROCESS OF SHAPE-SHAPING SUPERFINISHING
  • 5. 1. Methodology of experimental research and processing of experimental data
  • 5. 2. Regression analysis of indicators of the process of shaping superfinishing depending on the characteristics of the tool
  • 5. 3. Regression analysis of the indicators of the process of shaping superfinishing depending on the processing mode
  • 5. 4. General mathematical model forming superfinishing process
  • 5. 5. Performance of roller bearings with a rational geometric shape of the working surfaces
• conclusions
• CHAPTER 6 PRACTICAL APPLICATION OF RESEARCH RESULTS
  • 6. 1. Improving the designs of friction-rolling bearings
  • 6. 2. Bearing ring grinding method
  • 6. 3. Method for monitoring the profile of the raceways of bearing rings
  • 6. 4. Methods for superfinishing details such as rings of a complex profile
  • 6. 5. The method of completing bearings with a rational geometric shape of the working surfaces
• conclusions

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Applied theory of contact interaction of elastic bodies and the creation on its basis of the processes of shaping friction-rolling bearings with rational geometry ( abstract , term paper , diploma , control )

It is known that the problem of economic development in our country largely depends on the rise of industry based on the use of progressive technology. This provision primarily applies to bearing production, since the activities of other sectors of the economy depend on the quality of bearings and the efficiency of their production. Improving the operational characteristics of rolling friction bearings will increase the reliability and service life of machines and mechanisms, the competitiveness of equipment in the world market, and therefore is a problem of paramount importance.

A very important direction in improving the quality of rolling friction bearings is the technological support of the rational geometric shape of their working surfaces: rolling bodies and raceways. In the works of V. M. Aleksandrov, O. Yu. Davidenko, A.V. Koroleva, A.I. Lurie, A.B. Orlova, I.Ya. Shtaerman et al. convincingly showed that giving the working surfaces of elastically contacting parts of mechanisms and machines of a rational geometric shape can significantly improve the parameters of elastic contact and significantly increase the operational properties of friction units.

However, the modern theory of elastic contact does not allow to sufficiently search for a rational geometric shape of the contact surfaces in a fairly wide range of operating conditions for rolling friction bearings. Experimental search in this area is limited by the complexity of the measuring technique and experimental equipment used, as well as by the high labor intensity and duration of research. Therefore, at present there is no universal method for choosing a rational geometric shape of the contact surfaces of machine parts and devices.

A serious problem on the way to the practical use of rolling friction units of machines with a rational contact geometry is the lack of effective methods for their manufacture. Modern methods of grinding and finishing the surfaces of machine parts are designed mainly for the manufacture of surfaces of parts of a relatively simple geometric shape, the profiles of which are outlined by circular or straight lines. Methods of shaping superfinishing developed by the Saratov scientific school, are very effective, but their practical application is designed only for the processing of external surfaces such as the raceways of the inner rings of roller bearings, which limits their technological capabilities. All this does not allow, for example, to effectively control the form of contact stress diagrams for a number of designs of rolling friction bearings, and, consequently, to significantly affect their performance properties.

Thus, providing a systematic approach to improving the geometric shape of the working surfaces of rolling friction units and its technological support should be considered as one of the major areas further improvement of the operational properties of mechanisms and machines. On the one hand, the study of the influence of the geometric shape of contacting elastic bodies of complex shape on the parameters of their elastic contact makes it possible to create a universal method for improving the design of rolling friction bearings. On the other hand, the development of the basics of technological support for a given shape of parts ensures the efficient production of rolling friction bearings for mechanisms and machines with improved performance properties.

Therefore, the development of theoretical and technological foundations for improving the parameters of elastic contact of parts of rolling friction bearings and the creation on this basis of highly efficient technologies and equipment for the production of parts of rolling bearings is scientific problem, which is important for the development of domestic engineering.

The aim of the work is to develop an applied theory of local contact interaction of elastic bodies and to create on its basis the processes of shaping friction-rolling bearings with rational geometry, aimed at improving the performance of bearing units of various mechanisms and machines.

Research methodology. The work is based on the fundamental provisions of the theory of elasticity, modern methods of mathematical modeling of the deformed and stressed state of locally contacting elastic bodies, modern provisions of mechanical engineering technology, the theory of abrasive processing, probability theory, mathematical statistics, mathematical methods of integral and differential calculus, numerical calculation methods.

Experimental studies were carried out using modern techniques and equipment, using the methods of planning an experiment, processing experimental data, and regression analysis, as well as using modern software packages.

Reliability. The theoretical provisions of the work are confirmed by the results of experimental studies carried out both in laboratory and in production conditions. The reliability of theoretical positions and experimental data is confirmed by the implementation of the results of the work in production.

Scientific novelty. In this paper, an applied theory of local contact interaction of elastic bodies has been developed and, on its basis, the processes of shaping friction-rolling bearings with rational geometry have been created, which open up the possibility of a significant increase in the operational properties of bearing supports and other mechanisms and machines.

The main provisions of the dissertation submitted for defense:

1. Applied theory of local contact of elastic bodies of complex geometric shape, taking into account the variability of the eccentricity of the contact ellipse and various shapes of the initial gap profiles in the main sections, described by power-law dependences with arbitrary exponents.

2. Results of studies of the stress state in the region of elastic local contact and analysis of the influence of the complex geometric shape of elastic bodies on the parameters of their local contact.

3. The mechanism of shaping the parts of rolling friction bearings with a rational geometric shape in the technological operations of grinding the surface with a grinding wheel inclined to the axis of the workpiece, the results of the analysis of the influence of grinding parameters with an inclined wheel on the bearing capacity of the ground surface, the results of studying the technological possibilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and operational properties of bearings made with its use.

Fig. 4. The mechanism of the process of shaping parts during superfinishing, taking into account the complex kinematics of the process, the uneven degree of clogging of the tool, its wear and shaping in the process of processing, the results of the influence analysis various factors for the metal removal process various points profile of the workpiece and the formation of its surface

5. Regression multifactorial analysis of the technological capabilities of the process of forming superfinishing of bearing parts on superfinishing machines of the latest modifications and operational properties of bearings manufactured using this process.

6. A technique for the purposeful design of a rational design of the working surfaces of parts of complex geometric shape such as parts of rolling bearings, an integrated technology for manufacturing parts of rolling bearings, including preliminary, final processing and control of the geometric parameters of working surfaces, the design of new technological equipment created on the basis of new technologies and intended for manufacturing parts of rolling bearings with a rational geometric shape of the working surfaces.

This work is based on the materials of numerous studies of domestic and foreign authors. Great help in the work was provided by the experience and support of a number of specialists from the Saratov Bearing Plant, the Saratov Research and Production Enterprise for Non-Standard Engineering Products, the Saratov State Technical University and other organizations who kindly agreed to take part in the discussion of this work.

The author considers it his duty to express special gratitude for the valuable advice and multilateral assistance provided in the course of this work to Honored Scientist of the Russian Federation, Doctor of Technical Sciences, Professor, Academician of the Russian Academy of Natural Sciences Yu.V. Chebotarevskii and Doctor of Technical Sciences, Professor A.M. Chistyakov.

The limited amount of work did not allow to give exhaustive answers to a number of questions raised. Some of these issues are more fully considered in the published works of the author, as well as in joint work with graduate students and applicants ("https: // site", 11).

334 Conclusions:

1. A method is proposed for the purposeful design of a rational design of the working surfaces of parts of a complex geometric shape, such as parts of rolling bearings, and as an example, a new design of a ball bearing with a rational geometric shape of the raceways is proposed.

2. A comprehensive technology has been developed for manufacturing parts of rolling bearings, including preliminary, final processing, control of the geometric parameters of working surfaces and the assembly of bearings.

3. The designs of new technological equipment, created on the basis of new technologies, and intended for the manufacture of parts of rolling bearings with a rational geometric shape of working surfaces, are proposed.

CONCLUSION

1. As a result of research, a system has been developed for searching for a rational geometric shape of locally contacting elastic bodies and the technological foundations for their shaping, which opens up prospects for improving the performance of a wide class of other mechanisms and machines.

2. A mathematical model has been developed that reveals the mechanism of local contact of elastic bodies of complex geometric shape and takes into account the variability of the eccentricity of the contact ellipse and various shapes of the initial gap profiles in the main sections, described by power dependences with arbitrary exponents. The proposed model generalizes the solutions obtained earlier and significantly expands the field of practical application of the exact solution of contact problems.

3. A mathematical model of the stress state of the region of elastic local contact of bodies of complex shape has been developed, showing that the proposed solution of the contact problem gives a fundamentally new result, opening up a new direction for optimizing the contact parameters of elastic bodies, the nature of the distribution of contact stresses and providing an effective increase in the efficiency of friction units of mechanisms and machines.

4. A numerical solution of the local contact of bodies of complex shape, an algorithm and a program for calculating the deformed and stressed state of the contact area are proposed, which make it possible to purposefully design rational designs of the working surfaces of parts.

5. An analysis was made of the influence of the geometric shape of elastic bodies on the parameters of their local contact, showing that by changing the shape of the bodies, it is possible to simultaneously control the shape of the contact stress diagram, their magnitude and the size of the contact area, which makes it possible to provide a high support capacity of the contacting surfaces, and therefore, significantly improve the operational properties of contact surfaces.

6. Technological foundations for the manufacture of parts of rolling friction bearings with a rational geometric shape in the technological operations of grinding and shaping superfinishing have been developed. These are the most frequently used technological operations in precision engineering and instrumentation, which ensures a wide practical implementation of the proposed technologies.

7. A technology has been developed for grinding ball bearings with a grinding wheel inclined to the axis of the workpiece and a mathematical model for shaping the surface to be ground. It is shown that the formed shape of the ground surface, in contrast to the traditional form - the arc of a circle, has four geometric parameters, which significantly expands the possibility of controlling the bearing capacity of the machined surface.

8. A set of programs is proposed that provides the calculation of the geometric parameters of the surfaces of parts obtained by grinding with an inclined wheel, the stress and deformation state of an elastic body in rolling bearings for various grinding parameters. The analysis of the influence of grinding parameters with an inclined wheel on the bearing capacity of the ground surface was carried out. It is shown that by changing the geometric parameters of the grinding process with an inclined wheel, especially the angle of inclination, it is possible to significantly redistribute the contact stresses and simultaneously vary the size of the contact area, which significantly increases the bearing capacity of the contact surface and helps to reduce friction on the contact. The verification of the adequacy of the proposed mathematical model gave positive results.

9. Investigations of the technological possibilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and the performance properties of bearings made with its use were carried out. It is shown that the process of grinding with an inclined wheel contributes to an increase in processing productivity compared to conventional grinding, as well as to an increase in the quality of the machined surface. Compared to standard bearings, the durability of bearings made by grinding with an inclined circle is increased by 2–2.5 times, the waviness is reduced by 11 dB, the friction moment is reduced by 36%, and the speed is more than doubled.

10. A mathematical model of the mechanism of the process of forming parts during superfinishing has been developed. Unlike previous studies in this area, the proposed model provides the ability to determine the metal removal at any point of the profile, reflects the process of forming the tool profile during processing, the complex mechanism of its clogging and wear.

11. A set of programs has been developed that provides the calculation of the geometric parameters of the surface processed during superfinishing, depending on the main technological factors. The influence of various factors on the process of metal removal at various points of the workpiece profile and the formation of its surface is analyzed. As a result of the analysis, it was found that the clogging of the working surface of the tool has a decisive influence on the formation of the workpiece profile in the process of superfinishing. The adequacy of the proposed model was checked, which gave positive results.

12. A regression multifactorial analysis of the technological capabilities of the process of forming superfinishing of bearing parts on superfinishing machines of the latest modifications and the operational properties of bearings manufactured using this process was carried out. A mathematical model of the superfinishing process has been built, which determines the relationship between the main indicators of efficiency and quality of the processing process and technological factors and which can be used to optimize the process.

13. A method is proposed for the purposeful design of a rational design of the working surfaces of parts of a complex geometric shape, such as parts of rolling bearings, and as an example, a new design of a ball bearing with a rational geometric shape of the raceways is proposed. A complex technology has been developed for manufacturing parts of rolling bearings, including preliminary, final processing, control of the geometric parameters of working surfaces and the assembly of bearings.

14. Designs of new technological equipment, created on the basis of new technologies and intended for the manufacture of parts of rolling bearings with a rational geometric shape of working surfaces, are proposed.

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Mechanics of contact interaction

Introduction

mechanics pin roughness elastic

Contact mechanics is a fundamental engineering discipline that is extremely useful in designing reliable and energy efficient equipment. It will be useful in solving many contact problems, such as wheel-rail, in the calculation of clutches, brakes, tires, plain and rolling bearings, gears, joints, seals; electrical contacts, etc. It covers a wide range of tasks, ranging from strength calculations of tribosystem interface elements, taking into account the lubricating medium and material structure, to application in micro- and nanosystems.

The classical mechanics of contact interactions is associated primarily with the name of Heinrich Hertz. In 1882, Hertz solved the problem of the contact of two elastic bodies with curved surfaces. This classical result still underlies the mechanics of contact interaction today.

1. Classical problems of contact mechanics

1. Contact between a ball and an elastic half-space

A solid ball of radius R is pressed into an elastic half-space to a depth d (penetration depth), forming a contact area of ​​radius

The force required for this is

Here E1, E2 are elastic moduli; h1, h2 - Poisson's ratios of both bodies.

2. Contact between two balls

When two balls with radii R1 and R2 come into contact, these equations are valid for the radius R, respectively

The pressure distribution in the contact area is determined by the formula

with maximum pressure in the center

The maximum shear stress is reached under the surface, for h = 0.33 at.

3. Contact between two crossed cylinders with the same radii R

The contact between two crossed cylinders with the same radii is equivalent to the contact between a ball of radius R and a plane (see above).

4. Contact between a rigid cylindrical indenter and an elastic half-space

If a solid cylinder of radius a is pressed into an elastic half-space, then the pressure is distributed as follows:

The relationship between penetration depth and normal force is given by

5. Contact between a solid conical indenter and an elastic half-space

When indenting an elastic half-space with a solid cone-shaped indenter, the penetration depth and contact radius are determined by the following relationship:

Here and? the angle between the horizontal and the lateral plane of the cone.

The pressure distribution is determined by the formula

The stress at the top of the cone (in the center of the contact area) changes according to the logarithmic law. The total force is calculated as

6. Contact between two cylinders with parallel axes

In the case of contact between two elastic cylinders with parallel axes, the force is directly proportional to the penetration depth

The radius of curvature in this ratio is not present at all. The contact half-width is determined by the following relation

as in the case of contact between two balls.

The maximum pressure is

7. Contact between rough surfaces

When two bodies with rough surfaces interact with each other, the real contact area A is much smaller than the geometric area A0. At contact between a plane with a randomly distributed roughness and an elastic half-space, the real contact area is proportional to the normal force F and is determined by the following approximate equation:

At the same time, Rq? r.m.s. value of the roughness of a rough surface and. Average pressure in real contact area

is calculated to a good approximation as half the modulus of elasticity E* times the r.m.s. value of the surface profile roughness Rq. If this pressure is greater than the hardness HB of the material and thus

then the microroughnesses are completely in a plastic state.

For sh<2/3 поверхность при контакте деформируется только упруго. Величина ш была введена Гринвудом и Вильямсоном и носит название индекса пластичности.

2. Accounting for roughness

Based on the analysis of experimental data and analytical methods for calculating the parameters of contact between a sphere and a half-space, taking into account the presence of a rough layer, it was concluded that the calculated parameters depend not so much on the deformation of the rough layer, but on the deformation of individual irregularities.

When developing a model for the contact of a spherical body with a rough surface, the results obtained earlier were taken into account:

- at low loads, the pressure for a rough surface is less than that calculated according to the theory of G. Hertz and is distributed over a larger area (J. Greenwood, J. Williamson);

- the use of a widely used model of a rough surface in the form of an ensemble of bodies of a regular geometric shape, the height peaks of which obey a certain distribution law, leads to significant errors in estimating the contact parameters, especially at low loads (N.B. Demkin);

– there are no simple expressions suitable for calculating contacting parameters and the experimental base is not sufficiently developed.

This paper proposes an approach based on fractal concepts of a rough surface as a geometric object with a fractional dimension.

We use the following relations, which reflect the physical and geometric features of the rough layer.

The modulus of elasticity of the rough layer (and not the material that makes up the part and, accordingly, the rough layer) Eeff, being a variable, is determined by the dependence:

where E0 is the modulus of elasticity of the material; e is the relative deformation of the irregularities of the rough layer; w is a constant (w = 1); D is the fractal dimension of the rough surface profile.

Indeed, the relative approach characterizes in a certain sense the distribution of the material along the height of the rough layer and, thus, the effective modulus characterizes the features of the porous layer. At e = 1, this porous layer degenerates into a continuous material with its own modulus of elasticity.

We assume that the number of touch spots is proportional to the size of the contour area with radius ac:

Let's rewrite this expression as

Let us find the coefficient of proportionality C. Let N = 1, then ac=(Smax / p)1/2, where Smax is the area of ​​one contact spot. Where

Substituting the obtained value of C into equation (2), we obtain:

We believe that the cumulative distribution of contact patches with an area greater than s obeys the following law

The differential (modulo) distribution of the number of spots is determined by the expression

Expression (5) allows you to find the actual contact area

The result obtained shows that the actual contact area depends on the structure of the surface layer, determined by the fractal dimension and the maximum area of ​​an individual touch spot located in the center of the contour area. Thus, in order to estimate the contact parameters, it is necessary to know the deformation of an individual asperity, and not of the entire rough layer. The cumulative distribution (4) does not depend on the state of the contact patches. It is valid when contact spots can be in elastic, elastic-plastic and plastic states. The presence of plastic deformations determines the effect of adaptability of the rough layer to external influences. This effect is partially manifested in equalizing the pressure on the contact area and increasing the contour area. In addition, plastic deformation of multi-vertex protrusions leads to the elastic state of these protrusions with a small number of repeated loadings, if the load does not exceed the initial value.

By analogy with expression (4), we write the integral distribution function of the areas of contact spots in the form

The differential form of expression (7) is represented by the following expression:

Then the mathematical expectation of the contact area is determined by the following expression:

Since the actual contact area is

and, taking into account expressions (3), (6), (9), we write:

Assuming that the fractal dimension of the rough surface profile (1< D < 2) является величиной постоянной, можно сделать вывод о том, что радиус контурной площади контакта зависит только от площади отдельной максимально деформированной неровности.

Let us determine Smax from the known expression

where b is a coefficient equal to 1 for the plastic state of the contact of a spherical body with a smooth half-space, and b = 0.5 for an elastic one; r -- radius of curvature of the top of the roughness; dmax - roughness deformation.

Let us assume that the radius of the circular (contour) area ac is determined by the modified formula of G. Hertz

Then, substituting expression (1) into formula (11), we obtain:

Equating the right parts of expressions (10) and (12) and solving the resulting equality with respect to the deformation of the maximum loaded unevenness, we write:

Here, r is the radius of the roughness tip.

When deriving equation (13), it was taken into account that the relative deformation of the most loaded unevenness is equal to

where dmax is the greatest deformation of the roughness; Rmax -- the highest profile height.

For a Gaussian surface, the fractal dimension of the profile is D = 1.5 and at m = 1, expression (13) has the form:

Considering the deformation of irregularities and the settlement of their base as additive quantities, we write:

Then we find the total convergence from the following relation:

Thus, the expressions obtained allow us to find the main parameters of the contact of a spherical body with a half-space, taking into account the roughness: the radius of the contour area was determined by expressions (12) and (13), convergence? according to formula (15).

3. Experiment

The tests were carried out on an installation for studying the contact stiffness of fixed joints. The accuracy of measuring contact strains was 0.1–0.5 µm.

The test scheme is shown in fig. 1. The experimental procedure provided for smooth loading and unloading of samples with a certain roughness. Three balls with a diameter of 2R=2.3 mm were placed between the samples.

Samples with the following roughness parameters were studied (Table 1).

In this case, the upper and lower samples had the same roughness parameters. Sample material - steel 45, heat treatment - improvement (HB 240). The test results are given in table. 2.

It also presents a comparison of the experimental data with the calculated values ​​obtained on the basis of the proposed approach.

Table 1

Roughness parameters

Sample number	Surface roughness parameters of steel specimens
					Reference Curve Fitting Parameters

table 2

Approach of a spherical body to a rough surface

	Sample No. 1	Sample #2
		dosn, µm		Experiment		dosn, µm		Experiment

A comparison of the experimental and calculated data showed their satisfactory agreement, which indicates the applicability of the considered approach to estimating the contact parameters of spherical bodies, taking into account roughness.

On fig. Figure 2 shows the dependence of the ratio ac/ac (H) of the contour area, taking into account the roughness, to the area calculated according to the theory of G. Hertz, on the fractal dimension.

As seen in fig. 2, with an increase in the fractal dimension, which reflects the complexity of the profile structure of a rough surface, the value of the ratio of the contour contact area to the area calculated for smooth surfaces according to the theory of G. Hertz increases.

Rice. 1. Test scheme: a - loading; b - the location of the balls between the test samples

The given dependence (Fig. 2) confirms the fact of an increase in the area of ​​contact of a spherical body with a rough surface in comparison with the area calculated according to the theory of G. Hertz.

When evaluating the actual area of ​​contact, it is necessary to take into account the upper limit equal to the ratio of load to Brinell hardness of the softer element.

The area of ​​the contour area, taking into account the roughness, is found using formula (10):

Rice. Fig. 2. Dependence of the ratio of the radius of the contour area, taking into account the roughness, to the radius of the Hertzian area on the fractal dimension D

To estimate the ratio of the actual contact area to the contour area, we divide expression (7.6) into the right side of equation (16)

On fig. Figure 3 shows the dependence of the ratio of the actual contact area Ar to the contour area Ac on the fractal dimension D. As the fractal dimension increases (roughness increases), the Ar/Ac ratio decreases.

Rice. Fig. 3. Dependence of the ratio of the actual contact area Ar to the contour area Ac on the fractal dimension

Thus, the plasticity of a material is considered not only as a property (physico-mechanical factor) of the material, but also as a carrier of the effect of adaptability of a discrete multiple contact to external influences. This effect manifests itself in some equalization of pressures on the contour area of ​​contact.

Bibliography

1. Mandelbrot B. Fractal geometry of nature / B. Mandelbrot. - M.: Institute of Computer Research, 2002. - 656 p.

2. Voronin N.A. Patterns of contact interaction of solid topocomposite materials with a rigid spherical stamp / N.A. Voronin // Friction and lubrication in machines and mechanisms. - 2007. - No. 5. - S. 3-8.

3. Ivanov A.S. Normal, angular and tangential contact stiffness of a flat joint / A.S. Ivanov // Vestnik mashinostroeniya. - 2007. - No. 1. pp. 34-37.

4. Tikhomirov V.P. Contact interaction of a ball with a rough surface / Friction and lubrication in machines and mechanisms. - 2008. - No. 9. -FROM. 3-

5. Demkin N.B. Contact of rough wavy surfaces taking into account the mutual influence of irregularities / N.B. Demkin, S.V. Udalov, V.A. Alekseev [et al.] // Friction and wear. - 2008. - T.29. - Number 3. - S. 231-237.

6. Bulanov E.A. Contact problem for rough surfaces / E.A. Bulanov // Mechanical Engineering. - 2009. - No. 1 (69). - S. 36-41.

7. Lankov, A.A. Probability of elastic and plastic deformations during compression of rough metal surfaces / A.A. Lakkov // Friction and lubrication in machines and mechanisms. - 2009. - No. 3. - S. 3-5.

8. Greenwood J.A. Contact of nominally flat surfaces / J.A. Greenwood, J.B.P. Williamson // Proc. R. Soc., Series A. - 196 - V. 295. - No. 1422. - P. 300-319.

9. Majumdar M. Fractal model of elastic-plastic contact of rough surfaces / M. Majumdar, B. Bhushan // Modern mechanical engineering. ? 1991.? No. ? pp. 11-23.

10. Varadi K. Evaluation of the real contact areas, pressure distributions and contact temperatures during sliding contact between real metal surfaces / K. Varodi, Z. Neder, K. Friedrich // Wear. - 199 - 200. - P. 55-62.

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1. Analysis of scientific publications within the framework of the mechanics of contact interaction 6

2. Analysis of the influence of the physical and mechanical properties of materials of contact pairs on the contact zone in the framework of the theory of elasticity in the implementation of the test problem of contact interaction with a known analytical solution. 13

3. Investigation of the contact stress state of elements of a spherical bearing part in an axisymmetric formulation. 34

3.1. Numerical analysis of the bearing assembly design. 35

3.2. Investigation of the influence of grooves with lubricant on a spherical sliding surface on the stress state of the contact assembly. 43

3.3. Numerical study of the stress state of the contact node for different materials of the antifriction layer. 49

Conclusions.. 54

References.. 57

Analysis of scientific publications in the framework of the mechanics of contact interaction

Many components and structures used in mechanical engineering, construction, medicine and other fields operate in the conditions of contact interaction. These are, as a rule, expensive, hard-to-repair critical elements, which are subject to increased requirements regarding strength, reliability and durability. In connection with the wide application of the theory of contact interaction in mechanical engineering, construction and other areas of human activity, it became necessary to consider the contact interaction of bodies of complex configuration (structures with anti-friction coatings and interlayers, layered bodies, nonlinear contact, etc.), with complex boundary conditions in the contact zone, in static and dynamic conditions. The foundations of the mechanics of contact interaction were laid by G. Hertz, V.M. Aleksandrov, L.A. Galin, K. Johnson, I.Ya. Shtaerman, L. Goodman, A.I. Lurie and other domestic and foreign scientists. Considering the history of the development of the theory of contact interaction, the work of Heinrich Hertz "On the contact of elastic bodies" can be singled out as a foundation. At the same time, this theory is based on the classical theory of elasticity and continuum mechanics, and was presented to the scientific community in the Berlin Physical Society at the end of 1881. Scientists noted the practical importance of the development of the theory of contact interaction, and Hertz's research was continued, although the theory did not receive due development. The theory did not initially become widespread, since it determined its time and gained popularity only at the beginning of the last century, during the development of mechanical engineering. At the same time, it can be noted that the main drawback of the Hertz theory is its applicability only to ideally elastic bodies on contact surfaces, without taking into account friction on mating surfaces.

At the moment, the mechanics of contact interaction has not lost its relevance, but is one of the most rapidly fluttering topics in the mechanics of a deformable solid body. At the same time, each task of the mechanics of contact interaction carries a huge amount of theoretical or applied research. The development and improvement of the contact theory, when proposed by Hertz, was continued by a large number of foreign and domestic scientists. For example, Alexandrov V.M. Chebakov M.I. considers problems for an elastic half-plane without taking into account and taking into account friction and cohesion, also in their formulations, the authors take into account lubrication, heat released from friction and wear. Numerical-analytical methods for solving non-classical spatial problems of the mechanics of contact interactions are described in the framework of the linear theory of elasticity. A large number of authors have worked on the book, which reflects the work up to 1975, covering a large amount of knowledge about contact interaction. This book contains the results of solving contact static, dynamic and temperature problems for elastic, viscoelastic and plastic bodies. A similar edition was published in 2001 containing updated methods and results for solving problems in contact interaction mechanics. It contains works of not only domestic, but also foreign authors. N.Kh. Harutyunyan and A.V. Manzhirov in his monograph investigated the theory of contact interaction of growing bodies. A problem was posed for non-stationary contact problems with a time-dependent contact area and methods for solving were presented in .Seimov V.N. studied dynamic contact interaction, and Sarkisyan V.S. considered problems for half-planes and strips. In his monograph, Johnson K. considered applied contact problems, taking into account friction, dynamics and heat transfer. Effects such as inelasticity, viscosity, damage accumulation, slip, and adhesion have also been described. Their studies are fundamental for the mechanics of contact interaction in terms of creating analytical and semi-analytical methods for solving contact problems of a strip, half-space, space and canonical bodies, they also touch upon contact issues for bodies with interlayers and coatings.

Further development of the mechanics of contact interaction is reflected in the works of Goryacheva I.G., Voronin N.A., Torskaya E.V., Chebakov M.I., M.I. Porter and other scientists. A large number of works consider the contact of a plane, half-space or space with an indenter, contact through an interlayer or thin coating, as well as contact with layered half-spaces and spaces. Basically, the solutions of such contact problems are obtained using analytical and semi-analytical methods, and mathematical contact models are quite simple and, if they take into account friction between mating parts, they do not take into account the nature of the contact interaction. In real mechanisms, parts of a structure interact with each other and with surrounding objects. Contact can occur both directly between the bodies and through various layers and coatings. Due to the fact that the mechanisms of machines and their elements are often geometrically complex structures operating within the framework of contact interaction mechanics, the study of their behavior and deformation characteristics is an urgent problem in the mechanics of a deformable solid body. Examples of such systems include plain bearings with a composite material interlayer, a hip endoprosthesis with an antifriction interlayer, a bone-articular cartilage junction, road pavement, pistons, bearing parts of bridge superstructures and bridge structures, etc. Mechanisms are complex mechanical systems with a complex spatial configuration, having more than one sliding surface, and often contact coatings and interlayers. In this regard, the development of contact problems, including contact interaction through coatings and interlayers, is of interest. Goryacheva I.G. In her monograph, she studied the influence of surface microgeometry, the inhomogeneity of the mechanical properties of surface layers, as well as the properties of the surface and films covering it on the characteristics of contact interaction, friction force, and stress distribution in near-surface layers under different contact conditions. In her study, Torskaya E.V. considers the problem of sliding a rigid rough indenter along the boundary of a two-layer elastic half-space. It is assumed that friction forces do not affect the distribution of contact pressure. For the problem of frictional contact of an indenter with a rough surface, the influence of the friction coefficient on the stress distribution is analyzed. The studies of the contact interaction of rigid stamps and viscoelastic bases with thin coatings for cases where the surfaces of stamps and coatings are mutually repeating are presented in. The mechanical interaction of elastic layered bodies is studied in the works, they consider the contact of a cylindrical, spherical indenter, a system of stamps with an elastic layered half-space. A large number of studies have been published on the indentation of multilayer media. Aleksandrov V.M. and Mkhitaryan S.M. outlined the methods and results of research on the impact of stamps on bodies with coatings and interlayers, the problems are considered in the formulation of the theory of elasticity and viscoelasticity. It is possible to single out a number of problems on contact interaction, in which friction is taken into account. In the plane contact problem on the interaction of a moving rigid stamp with a viscoelastic layer is considered. The die moves at a constant speed and is pressed in with a constant normal force, assuming that there is no friction in the contact area. This problem is solved for two types of stamps: rectangular and parabolic. The authors experimentally studied the effect of interlayers of various materials on the heat transfer process in the contact zone. About six samples were considered and it was experimentally determined that stainless steel filler is an effective heat insulator. In another scientific publication, an axisymmetric contact problem of thermoelasticity was considered on the pressure of a hot cylindrical circular isotropic stamp on an elastic isotropic layer, there was a non-ideal thermal contact between the stamp and the layer. The works discussed above consider the study of more complex mechanical behavior on the site of contact interaction, but the geometry remains in most cases of the canonical form. Since there are often more than 2 contact surfaces in contacting structures, complex spatial geometry, materials and loading conditions that are complex in their mechanical behavior, it is almost impossible to obtain an analytical solution for many practically important contact problems, therefore, effective solution methods are required, including numerical. At the same time, one of the most important tasks of modeling the mechanics of contact interaction in modern applied software packages is to consider the influence of the materials of the contact pair, as well as the correspondence of the results of numerical studies to existing analytical solutions.

The gap between theory and practice in solving problems of contact interaction, as well as their complex mathematical formulation and description, served as an impetus for the formation of numerical approaches to solving these problems. The most common method for numerically solving problems of contact interaction mechanics is the finite element method (FEM). An iterative solution algorithm using the FEM for the one-sided contact problem is considered in. The solution of contact problems is considered using the extended FEM, which makes it possible to take into account friction on the contact surface of contacting bodies and their inhomogeneity. The considered publications on the FEM for problems of contact interaction are not tied to specific structural elements and often have a canonical geometry. An example of considering a contact within the framework of the FEM for a real design is , where the contact between the blade and disk of a gas turbine engine is considered. Numerical solutions to the problems of contact interaction of multilayer structures and bodies with antifriction coatings and interlayers are considered in. The publications mainly consider the contact interaction of layered half-spaces and spaces with indenters, as well as the conjugation of canonical bodies with interlayers and coatings. Mathematical models of contact are of little content, and the conditions of contact interaction are described poorly. Contact models rarely consider the possibility of simultaneous sticking, sliding with different types of friction and detachment on the contact surface. In most publications, the mathematical models of the problems of deformation of structures and nodes are described little, especially the boundary conditions on the contact surfaces.

At the same time, the study of the problems of contact interaction of bodies of real complex systems and structures assumes the presence of a base of physical-mechanical, frictional and operational properties of materials of contacting bodies, as well as anti-friction coatings and interlayers. Often one of the materials of contact pairs are various polymers, including antifriction polymers. Insufficiency of information about the properties of fluoroplastics, compositions based on it and ultra-high molecular weight polyethylenes of various grades is noted, which hinders their effectiveness in use in many industries. On the basis of the National Material Testing Institute of the Stuttgart University of Technology, a number of full-scale experiments were carried out aimed at determining the physical and mechanical properties of materials used in Europe in contact nodes: ultra-high molecular weight polyethylenes PTFE and MSM with carbon black and plasticizer additives. But large-scale studies aimed at determining the physical, mechanical and operational properties of viscoelastic media and a comparative analysis of materials suitable for use as a material for sliding surfaces of critical industrial structures operating in difficult conditions of deformation in the world and Russia have not been carried out. In this regard, there is a need to study the physical-mechanical, frictional and operational properties of viscoelastic media, build models of their behavior and select constitutive relationships.

Thus, the problems of studying the contact interaction of complex systems and structures with one or more sliding surfaces are an actual problem in the mechanics of a deformable solid body. Topical tasks also include: determination of physical-mechanical, frictional and operational properties of materials of contact surfaces of real structures and numerical analysis of their deformation and contact characteristics; carrying out numerical studies aimed at identifying patterns of influence of physical-mechanical and antifriction properties of materials and geometry of contacting bodies on the contact stress-strain state and, on their basis, developing a methodology for predicting the behavior of structural elements under design and non-design loads. And also relevant is the study of the influence of physical-mechanical, frictional and operational properties of materials entering into contact interaction. The practical implementation of such problems is possible only by numerical methods oriented towards parallel computing technologies, with the involvement of modern multiprocessor computer technology.

Analysis of the influence of the physical and mechanical properties of materials of contact pairs on the contact zone in the framework of the theory of elasticity in the implementation of the test problem of contact interaction with a known analytical solution

Let us consider the influence of the properties of materials of a contact pair on the parameters of the contact interaction area using the example of solving the classical contact problem on the contact interaction of two contacting spheres pressed against each other by forces P (Fig. 2.1.). We will consider the problem of the interaction of spheres within the framework of the theory of elasticity; the analytical solution of this problem was considered by A.M. Katz in .

Rice. 2.1. Contact diagram

As part of the solution of the problem, it is explained that, according to the Hertz theory, the contact pressure is found according to the formula (1):

, (2.1)

where is the radius of the contact area, is the coordinate of the contact area, is the maximum contact pressure on the area.

As a result of mathematical calculations in the framework of the mechanics of contact interaction, formulas were found for determining and presented in (2.2) and (2.3), respectively:

, (2.2)

, (2.3)

where and are the radii of the contacting spheres, , and , are the Poisson's ratios and the moduli of elasticity of the contacting spheres, respectively.

It can be seen that in formulas (2-3) the coefficient responsible for the mechanical properties of the contact pair of materials has the same form, so let's denote it , in this case formulas (2.2-2.3) have the form (2.4-2.5):

, (2.4)

. (2.5)

Let us consider the influence of the properties of materials in contact in the structure on the contact parameters. Consider, within the framework of the problem of contacting two contacting spheres, the following contact pairs of material: Steel - Fluoroplastic; Steel - Composite antifriction material with spherical bronze inclusions (MAK); Steel - Modified PTFE. Such a choice of contact pairs of materials is due to further studies of their work with spherical bearings. The mechanical properties of contact pair materials are presented in Table 2.1.

Table 2.1.

Material properties of contacting spheres

No. p / p	Material 1 sphere	Material 2 spheres
	Steel	Fluoroplast
, N/m2		, N/m2
2E+11	0,3	5.45E+08	0,466
	Steel	POPPY
, N/m2		, N/m2
2E+11	0,3		0,4388
	Steel	Modified fluoroplast
, N/m2		, N/m2
2E+11	0,3		0,46

Thus, for these three contact pairs, one can find the coefficient of the contact pair, the maximum radius of the contact area and the maximum contact pressure, which are presented in Table 2.2. Table 2.2. the contact parameters are calculated under the condition of action on spheres with unit radii ( , m and , m) of compressive forces , N.

Table 2.2.

Contact area options

Rice. 2.2. Contact pad parameters:

a), m 2 /N; b) , m; c) , N / m 2

On fig. 2.2. a comparison of the contact zone parameters for three contact pairs of sphere materials is presented. It can be seen that pure fluoroplastic has a lower value of maximum contact pressure compared to the other 2 materials, while its contact zone radius is the largest. The parameters of the contact zone for the modified fluoroplast and MAK differ insignificantly.

Let us consider the influence of the radii of the contacting spheres on the parameters of the contact zone. At the same time, it should be noted that the dependence of the contact parameters on the radii of the spheres is the same in formulas (4)-(5), i.e. they enter the formulas in the same way, therefore, to study the influence of the radii of the contacting spheres, it is enough to change the radius of one sphere. Thus, we will consider an increase in the radius of the 2nd sphere at a constant value of the radius of 1 sphere (see Table 2.3).

Table 2.3.

Radii of contacting spheres

No. p / p	, m	, m

Table 2.4

Contact zone parameters for different radii of contacting spheres

No. p / p	Steel-Photoplast	Steel-MAK	Steel-Mod PTFE
, m	, N/m2	, m	, N/m2	, m	, N/m2
	0,000815	719701,5	0,000707	954879,5	0,000701	972788,7477
	0,000896	594100,5	0,000778	788235,7	0,000771	803019,4184
	0,000953		0,000827	698021,2	0,000819	711112,8885
	0,000975	502454,7	0,000846	666642,7	0,000838	679145,8759
	0,000987	490419,1	0,000857	650674,2	0,000849	662877,9247
	0,000994	483126,5	0,000863	640998,5	0,000855	653020,7752
	0,000999		0,000867	634507,3	0,000859	646407,8356
	0,001003		0,000871	629850,4	0,000863	641663,5312
	0,001006		0,000873	626346,3	0,000865	638093,7642
	0,001008	470023,7	0,000875	623614,2	0,000867	635310,3617

Dependences on the parameters of the contact zone (the maximum radius of the contact zone and the maximum contact pressure) are shown in fig. 2.3.

Based on the data presented in fig. 2.3. it can be concluded that as the radius of one of the contacting spheres increases, both the maximum radius of the contact zone and the maximum contact pressure become asymptotic. In this case, as expected, the law of distribution of the maximum radius of the contact zone and the maximum contact pressure for the three considered pairs of contacting materials are the same: as the maximum radius of the contact zone increases, and the maximum contact pressure decreases.

For a more visual comparison of the influence of the properties of the contacting materials on the contact parameters, we plot on one graph the maximum radius for the three contact pairs under study and, similarly, the maximum contact pressure (Fig. 2.4.).

Based on the data shown in Figure 4, there is a noticeably small difference in the contact parameters between MAC and modified PTFE, while pure PTFE at significantly lower contact pressures has a larger contact area radius than the other two materials.

Consider the distribution of contact pressure for three contact pairs of materials with increasing . The distribution of contact pressure is shown along the radius of the contact area (Fig. 2.5.).

Rice. 2.5. Distribution of contact pressure along the contact radius:

a) Steel-Ftoroplast; b) Steel-MAK;

c) Steel-modified PTFE

Next, we consider the dependence of the maximum radius of the contact area and the maximum contact pressure on the forces bringing the spheres together. Consider the action on spheres with unit radii ( , m and , m) of forces: 1 N, 10 N, 100 N, 1000 N, 10000 N, 100000 N, 1000000 N. The contact interaction parameters obtained as a result of the study are presented in Table 2.5.

Table 2.5.

Contact options when zoomed in

P, N	Steel-Photoplast	Steel-MAK	Steel-Mod PTFE
, m	, N/m2	, m	, N/m2	, m	, N/m2
	0,0008145	719701,5	0,000707	954879,5287	0,000700586	972788,7477
	0,0017548		0,001523	2057225,581	0,001509367	2095809,824
	0,0037806		0,003282	4432158,158	0,003251832	4515285,389
	0,0081450		0,007071	9548795,287	0,00700586	9727887,477
	0,0175480		0,015235	20572255,81	0,015093667	20958098,24
	0,0378060		0,032822	44321581,58	0,032518319	45152853,89
	0,0814506		0,070713	95487952,87	0,070058595	97278874,77

The dependences of the contact parameters are shown in fig. 2.6.

Rice. 2.6. Dependencies of contact parameters on

for three contact pairs of materials: a), m; b), N / m 2

For three contact pairs of materials, with an increase in squeezing forces, both the maximum radius of the contact area and the maximum contact pressure increase (Fig. 2.6. At the same time, similarly to the previously obtained result for pure fluoroplast at a lower contact pressure, the contact area of ​​a larger radius.

Consider the distribution of contact pressure for three contact pairs of materials with increasing . The distribution of contact pressure is shown along the radius of the contact area (Fig. 2.7.).

Similar to the previously obtained results, with an increase in the approaching forces, both the radius of the contact area and the contact pressure increase, while the nature of the distribution of the contact pressure is the same for all calculation options.

Let's implement the task in the ANSYS software package. When creating a finite element mesh, the element type PLANE182 was used. This type is a four-nodal element and has a second order of approximation. The element is used for 2D modeling of bodies. Each element node has two degrees of freedom UX and UY. Also, this element is used to calculate problems: axisymmetric, with a flat deformed state and with a flat stressed state.

In the studied classical problems, the type of contact pair was used: "surface - surface". One of the surfaces is assigned as the target ( TARGET), and another contact ( CONTA). Since a two-dimensional problem is considered, the finite elements TARGET169 and CONTA171 are used.

The problem is implemented in an axisymmetric formulation using contact elements without taking into account friction on mating surfaces. The calculation scheme of the problem is shown in fig. 2.8.

Rice. 2.8. Design scheme of spheres contact

The mathematical formulation of the problems of squeezing two contiguous spheres (Fig. 2.8.) is implemented within the framework of the theory of elasticity and includes:

equilibrium equations

geometric relationships

, (2.7)

physical ratios

, (2.8)

where and are the Lame parameters, is the stress tensor, is the strain tensor, is the displacement vector, is the radius vector of an arbitrary point, is the first invariant of the strain tensor, is the unit tensor, is the area occupied by sphere 1, is the area occupied by sphere 2, .

The mathematical statement (2.6)-(2.8) is supplemented by boundary conditions and symmetry conditions on the surfaces and . Sphere 1 is subjected to a force

force acts on sphere 2

. (2.10)

The system of equations (2.6) - (2.10) is also supplemented by the interaction conditions on the contact surface , while two bodies are in contact, the conditional numbers of which are 1 and 2. The following types of contact interaction are considered:

– sliding with friction: for static friction

, , , , (2.8)

wherein , ,

– for sliding friction

, , , , , , (2.9)

wherein , ,

– detachment

, , (2.10)

- full grip

, , , , (2.11)

where is the coefficient of friction; is the value of the vector of tangential contact stresses.

The numerical implementation of the solution of the problem of contacting spheres will be implemented using the example of a contact pair of materials Steel-Ftoroplast, with compressive forces H. This choice of load is due to the fact that for a smaller load, a finer breakdown of the model and finite elements is required, which is problematic due to limited computing resources.

In the numerical implementation of the contact problem, one of the primary tasks is to estimate the convergence of the finite element solution of the problem from the contact parameters. Below is table 2.6. which presents the characteristics of finite element models involved in the assessment of the convergence of the numerical solution of the partitioning option.

Table 2.6.

Number of Nodal Unknowns for Different Sizes of Elements in the Problem of Contacting Spheres

On fig. 2.9. the convergence of the numerical solution of the problem of contacting spheres is presented.

Rice. 2.9. Convergence of the numerical solution

One can notice the convergence of the numerical solution, while the distribution of the contact pressure of the model with 144 thousand nodal unknowns has insignificant quantitative and qualitative differences from the model with 540 thousand nodal unknowns. At the same time, the program computation time differs by several times, which is a significant factor in the numerical study.

On fig. 2.10. a comparison of the numerical and analytical solutions of the problem of contacting spheres is shown. The analytical solution of the problem is compared with the numerical solution of the model with 540 thousand nodal unknowns.

Rice. 2.10. Comparison of analytical and numerical solutions

It can be noted that the numerical solution of the problem has small quantitative and qualitative differences from the analytical solution.

Similar results on the convergence of the numerical solution were also obtained for the remaining two contact pairs of materials.

At the same time, at the Institute of Continuum Mechanics, Ural Branch of the Russian Academy of Sciences, Ph.D. A.Adamov carried out a series of experimental studies of the deformation characteristics of antifriction polymeric materials of contact pairs under complex multi-stage history of deformation with unloading. The cycle of experimental studies included (Fig. 2.11.): tests to determine the hardness of materials according to Brinell; research under conditions of free compression, as well as constrained compression by pressing in a special device with a rigid steel holder of cylindrical samples with a diameter and a length of 20 mm. All tests were carried out on a Zwick Z100SN5A testing machine at strain levels not exceeding 10%.

Tests to determine the hardness of materials according to Brinell were carried out by pressing a ball with a diameter of 5 mm (Fig. 2.11., a). In the experiment, after placing the sample on the substrate, a preload of 9.8 N is applied to the ball, which is maintained for 30 sec. Then, at a machine traverse speed of 5 mm/min, the ball is introduced into the sample until a load of 132 N is reached, which is maintained constant for 30 seconds. Then there is unloading to 9.8 N. The results of the experiment to determine the hardness of the previously mentioned materials are presented in table 2.7.

Table 2.7.

Material hardness

Cylindrical specimens with a diameter and height of 20 mm were studied under free compression. To implement a uniform stress state in a short cylindrical sample, three-layer gaskets made of a fluoroplastic film 0.05 mm thick, lubricated with a low-viscosity grease, were used at each end of the sample. Under these conditions, the specimen is compressed without noticeable “barrel formation” at strains up to 10%. The results of free compression experiments are shown in Table 2.8.

Results of free compression experiments

Studies under conditions of constrained compression (Fig. 2.11., c) were carried out by pressing cylindrical samples with a diameter of 20 mm, a height of about 20 mm in a special device with a rigid steel cage at permissible limiting pressures of 100-160 MPa. In the manual mode of machine control, the sample is loaded with a preliminary small load (~ 300 N, axial compressive stress ~ 1 MPa) to select all gaps and squeeze out excess lubricant. After that, the sample is kept for 5 min to dampen the relaxation processes, and then the specified loading program for the sample begins to be worked out.

The obtained experimental data on the nonlinear behavior of composite polymer materials are difficult to compare quantitatively. Table 2.9. the values ​​of the tangential modulus M = σ/ε, which reflects the rigidity of the sample under conditions of a uniaxial deformed state, are given.

Rigidity of specimens under conditions of uniaxial deformed state

From the test results, the mechanical characteristics of materials are also obtained: modulus of elasticity, Poisson's ratio, strain diagrams

0,000 0,000 -0,000 1154,29 -0,353 -1,923 1226,43 -0,381 -2,039 1298,58 -0,410 -2,156 1370,72 -0,442 -2,268 2405,21 -0,889 -3,713 3439,70 -1,353 -4,856 4474,19 -1,844 -5,540 5508,67 -2,343 -6,044 6543,16 -2,839 -6,579 7577,65 -3,342 -7,026 8612,14 -3,854 -7,335 9646,63 -4,366 -7,643 10681,10 -4,873 -8,002 11715,60 -5,382 -8,330 12750,10 -5,893 -8,612 13784,60 -6,403 -8,909 14819,10 -6,914 -9,230 15853,60 -7,428 -9,550 16888,00 -7,944 -9,865 17922,50 -8,457 -10,184 18957,00 -8,968 -10,508 19991,50 -9,480 -10,838 21026,00 -10,000 -11,202

Table 2.11

Deformation and Stresses in Samples of an Antifriction Composite Material Based on Fluoroplast with Spherical Bronze Inclusions and Molybdenum Disulfide

Number	Time, sec	Elongation, %	Stress, MPa
		0,00000	-0,00000
	1635,11	-0,31227	-2,16253
	1827,48	-0,38662	-2,58184
	2196,16	-0,52085	-3,36773
	2933,53	-0,82795	-4,76765
	3302,22	-0,99382	-5,33360
	3670,9	-1,15454	-5,81052
	5145,64	-1,81404	-7,30133
	6251,69	-2,34198	-8,14546
	7357,74	-2,85602	-8,83885
	8463,8	-3,40079	-9,48010
	9534,46	-3,90639	-9,97794
	10236,4	-4,24407	-10,30620
	11640,4	-4,92714	-10,90800
	12342,4	-5,25837	-11,18910
	13746,3	-5,93792	-11,72070
	14448,3	-6,27978	-11,98170
	15852,2	-6,95428	-12,48420
	16554,2	-7,29775	-12,71790
	17958,2	-7,98342	-13,21760
	18660,1	-8,32579	-13,45170
	20064,1	-9,01111	-13,90540
	20766,1	-9,35328	-14,15230
		-9,69558	-14,39620
		-10,03990	-14,57500

Deformation and Stresses in Samples of Modified Fluoroplastic

Number	Time, sec	Axial deformation, %	Conditional stress, MPa
	0,0	0,000	-0,000
	1093,58	-0,32197	-2,78125
	1157,91	-0,34521	-2,97914
	1222,24	-0,36933	-3,17885
	2306,41	-0,77311	-6,54110
	3390,58	-1,20638	-9,49141
	4474,75	-1,68384	-11,76510
	5558,93	-2,17636	-13,53510
	6643,10	-2,66344	-14,99470
	7727,27	-3,16181	-16,20210
	8811,44	-3,67859	-17,20450
	9895,61	-4,19627	-18,06060
	10979,80	-4,70854	-18,81330
	12064,00	-5,22640	-19,48280
	13148,10	-5,75156	-20,08840
	14232,30	-6,27556	-20,64990
	15316,50	-6,79834	-21,18110
	16400,60	-7,32620	-21,69070
	17484,80	-7,85857	-22,18240
	18569,00	-8,39097	-22,65720
	19653,20	-8,92244	-23,12190
	20737,30	-9,45557	-23,58330
	21821,50	-10,00390	-24,03330

According to the data presented in tables 2.10.-2.12. deformation diagrams are constructed (Fig. 2.2).

Based on the results of the experiment, it can be assumed that the description of the behavior of materials is possible within the framework of the deformation theory of plasticity. On test problems, the influence of the elastoplastic properties of materials was not tested due to the lack of an analytical solution.

The study of the influence of the physical and mechanical properties of materials when working as a contact pair material is considered in Chapter 3 on the real design of a spherical bearing part.

1. MODERN PROBLEMS OF CONTACT MECHANICS

INTERACTIONS

1.1. Classical hypotheses used in solving contact problems for smooth bodies

1.2. Influence of Creep of Solids on Their Shape Change in the Contact Area

1.3. Estimation of convergence of rough surfaces

1.4. Analysis of the contact interaction of multilayer structures

1.5. Relationship between mechanics and problems of friction and wear

1.6. Features of the use of modeling in tribology 31 CONCLUSIONS ON THE FIRST CHAPTER

2. CONTACT INTERACTION OF SMOOTH CYLINDRICAL BODIES

2.1. Solution of the contact problem for a smooth isotropic disk and a plate with a cylindrical cavity

2.1.1. General formulas

2.1.2. Derivation of the boundary condition for displacements in the contact area

2.1.3. Integral equation and its solution 42 2.1.3.1. Study of the resulting equation

2.1.3.1.1. Reduction of a singular integro-differential equation to an integral equation with a kernel having a logarithmic singularity

2.1.3.1.2. Estimating the Norm of a Linear Operator

2.1.3.2. Approximate solution of the equation

2.2. Calculation of a fixed connection of smooth cylindrical bodies

2.3. Determination of displacement in a movable connection of cylindrical bodies

2.3.1. Solution of an auxiliary problem for an elastic plane

2.3.2. Solution of an auxiliary problem for an elastic disk

2.3.3. Determination of maximum normal radial displacement

2.4. Comparison of theoretical and experimental data on the study of contact stresses at internal contact of cylinders of close radii

2.5. Modeling of Spatial Contact Interaction of a System of Coaxial Cylinders of Finite Sizes

2.5.1. Formulation of the problem

2.5.2. Solution of auxiliary two-dimensional problems

2.5.3. Solution of the original problem 75 CONCLUSIONS AND MAIN RESULTS OF THE SECOND CHAPTER

3. CONTACT PROBLEMS FOR ROUGH BODIES AND THEIR SOLUTION BY CORRECTING THE CURVATURE OF A DEFORMED SURFACE

3.1. Spatial non-local theory. geometric assumptions

3.2. Relative convergence of two parallel circles determined by roughness deformation

3.3. Method for Analytical Evaluation of the Influence of Roughness Deformation

3.4. Definition of displacements in the area of ​​contact

3.5. Definition of auxiliary coefficients

3.6. Determination of the dimensions of the elliptical contact area

3.7. Equations for determining the contact area close to circular

3.8. Equations for determining the area of ​​contact close to the line

3.9. Approximate determination of the coefficient a in the case of a contact area in the form of a circle or a SW strip

3.10. Peculiarities of averaging pressures and strains in solving the two-dimensional problem of internal contact of rough cylinders with close radii Yu

3.10.1. Derivation of the integro-differential equation and its solution in the case of internal contact of rough cylinders Yu

3.10.2. Definition of auxiliary coefficients ^ ^

3.10.3. Stress fit of rough cylinders ^ ^ CONCLUSIONS AND MAIN RESULTS OF CHAPTER THREE

4. SOLUTION OF CONTACT PROBLEMS OF VISCOELASTICITY FOR SMOOTH BODIES

4.1. Key points

4.2. Compliance principles analysis

4.2.1. Volterra principle

4.2.2. Constant coefficient of transverse expansion under creep deformation

4.3. Approximate solution of the two-dimensional contact problem of linear creep for smooth cylindrical bodies ^^

4.3.1. General case of viscoelasticity operators

4.3.2. Solution for a monotonically increasing contact area

4.3.3. Fixed connection solution

4.3.4. Modeling of contact interaction in the case of uniformly aging isotropic plate

CONCLUSIONS AND MAIN RESULTS OF THE FOURTH CHAPTER

5. SURFACE CREEP

5.1. Features of the contact interaction of bodies with low yield strength

5.2. Construction of a Surface Deformation Model Taking into Account Creep in the Case of an Elliptical Contact Area

5.2.1. geometric assumptions

5.2.2. Surface Creep Model

5.2.3. Determination of average deformations of a rough layer and average pressures

5.2.4. Definition of auxiliary coefficients

5.2.5. Determination of the dimensions of the elliptical contact area

5.2.6. Determining the dimensions of the circular contact area

5.2.7. Determining the width of the contact area as a strip

5.3. Solution of a 2D Contact Problem for Internal Touch of Rough Cylinders with Allowance for Surface Creep

5.3.1. Statement of the problem for cylindrical bodies. Integro-differential equation

5.3.2. Determination of auxiliary coefficients 160 CONCLUSIONS AND MAIN RESULTS OF THE FIFTH CHAPTER

6. MECHANICS OF INTERACTION OF CYLINDRICAL BODIES WITH COVERINGS

6.1. Calculation of effective modules in the theory of composites

6.2. Construction of a self-consistent method for calculating the effective coefficients of inhomogeneous media, taking into account the spread of physical and mechanical properties

6.3. Solution of the contact problem for a disk and a plane with an elastic composite coating on the hole contour

6.3.1. Statement of the problem and basic formulas

6.3.2. Derivation of the boundary condition for displacements in the contact area

6.3.3. Integral equation and its solution

6.4. Solution of the Problem in the Case of an Orthotropic Elastic Coating with Cylindrical Anisotropy

6.5. Determination of the effect of viscoelastic aging coating on the change in contact parameters

6.6. Analysis of the Features of the Contact Interaction of a Multicomponent Coating and the Roughness of a Disc

6.7. Modeling of contact interaction taking into account thin metal coatings

6.7.1. Contact of a plastic-coated ball and a rough half-space

6.7.1.1. Main hypotheses and model of interaction of rigid bodies

6.7.1.2. Approximate solution of the problem

6.7.1.3. Determination of the maximum contact approach

6.7.2. Solution of the contact problem for a rough cylinder and a thin metal coating on the hole contour

6.7.3. Determination of contact stiffness at internal contact of cylinders

CONCLUSIONS AND MAIN RESULTS OF CHAPTER SIX

7. SOLUTION OF MIXED BOUNDARY PROBLEM WITH SURFACE WEAR INCLUDED

OF INTERACTING BODIES

7.1. Features of the solution of the contact problem, taking into account the wear of surfaces

7.2. Statement and solution of the problem in the case of elastic deformation of roughness

7.3. The method of theoretical wear assessment taking into account surface creep

7.4. Method for assessing wear taking into account the influence of the coating

7.5. Concluding remarks on the formulation of plane problems with wear taken into account

CONCLUSIONS AND MAIN RESULTS OF THE SEVENTH CHAPTER

Recommended list of dissertations

• On the contact interaction between thin-walled elements and viscoelastic bodies under torsion and axisymmetric deformation, taking into account the aging factor 1984, candidate of physical and mathematical sciences Davtyan, Zaven Azibekovich

• Static and dynamic contact interaction of plates and cylindrical shells with rigid bodies 1983, candidate of physical and mathematical sciences Kuznetsov, Sergey Arkadyevich

• Technological support of the durability of machine parts based on hardening treatment with simultaneous application of anti-friction coatings 2007, doctor of technical sciences Bersudsky, Anatoly Leonidovich

• Thermoelastic contact problems for bodies with coatings 2007, candidate of physical and mathematical sciences Gubareva, Elena Alexandrovna

• A technique for solving contact problems for bodies of arbitrary shape, taking into account surface roughness by the finite element method 2003, Candidate of Technical Sciences Olshevsky, Alexander Alekseevich

Introduction to the thesis (part of the abstract) on the topic "Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces"

The development of technology poses new challenges in the study of the performance of machines and their elements. Increasing their reliability and durability is the most important factor determining the growth of competitiveness. In addition, the lengthening of the service life of machinery and equipment, even to a small extent with a high saturation of technology, is tantamount to the commissioning of significant new production capacities.

The current state of the theory of working processes of machines, combined with extensive experimental techniques for determining the working loads and a high level of development of the applied theory of elasticity, with the available knowledge of the physical and mechanical properties of materials, make it possible to ensure the overall strength of machine parts and apparatus with a fairly large guarantee against breakage under normal conditions services. At the same time, the trend towards a decrease in the weight and size indicators of the latter with a simultaneous increase in their energy saturation makes it necessary to revise the known approaches and assumptions in determining the stress state of parts and require the development of new calculation models, as well as the improvement of experimental research methods. Analysis and classification of failures of mechanical engineering products showed that the main cause of failure under operating conditions is not breakage, but wear and damage to their working surfaces.

Increased wear of parts in the joints in some cases violates the tightness of the working space of the machine, in others - the normal lubrication regime, in the third - leads to a loss of the kinematic accuracy of the mechanism. Wear and damage to surfaces reduces the fatigue strength of parts and can cause their destruction after a certain service life with minor structural and technological concentrators and low rated stresses. Thus, increased wear disrupts the normal interaction of parts in assemblies, can cause significant additional loads and cause accidental damage.

All this attracted a wide range of scientists of various specialties, designers and technologists to the problem of increasing the durability and reliability of machines, which made it possible not only to develop a number of measures to increase the service life of machines and create rational methods for caring for them, but also based on the achievements of physics, chemistry, and metal science to lay the foundations for the doctrine of friction, wear and lubrication in mates.

At present, significant efforts of engineers in our country and abroad are aimed at finding ways to solve the problem of determining the contact stresses of interacting parts, since for the transition from the calculation of the wear of materials to the problems of structural wear resistance, the contact problems of the mechanics of a deformable solid have a decisive role. Solutions of contact problems of elasticity theory for bodies with circular boundaries are of essential importance for engineering practice. They form the theoretical basis for the calculation of such machine elements as bearings, swivel joints, some types of gears, interference connections.

The most extensive studies have been carried out using analytical methods. It is the presence of fundamental connections between modern complex analysis and potential theory with such a dynamic field as mechanics that determined their rapid development and use in applied research. The use of numerical methods significantly expands the possibilities of analyzing the stress state in the contact area. At the same time, the bulkiness of the mathematical apparatus, the need to use powerful computing tools significantly hinders the use of existing theoretical developments in solving applied problems. Thus, one of the topical directions in the development of mechanics is to obtain explicit approximate solutions to the problems posed, ensuring the simplicity of their numerical implementation and describing the phenomenon under study with sufficient accuracy for practice. However, despite the successes achieved, it is still difficult to obtain satisfactory results taking into account the local design features and microgeometry of the interacting bodies.

It should be noted that the properties of the contact have a significant impact on the wear processes, since, due to the discreteness of the contact, the contact of microroughnesses occurs only on separate areas that form the actual area. In addition, the protrusions formed during processing are diverse in shape and have a different distribution of heights. Therefore, when modeling the topography of surfaces, it is necessary to introduce parameters characterizing the real surface into the statistical distribution laws.

All this requires the development of a unified approach to solving contact problems taking into account wear, which most fully takes into account both the geometry of interacting parts, microgeometric and rheological characteristics of surfaces, their wear resistance characteristics, and the possibility of obtaining an approximate solution with the least number of independent parameters.

Connection of work with major scientific programs, topics. The studies were carried out in accordance with the following topics: "Develop a method for calculating contact stresses with elastic contact interaction of cylindrical bodies, not described by the Hertz theory" (Ministry of Education of the Republic of Belarus, 1997, No. GR 19981103); "Influence of microroughnesses of contacting surfaces on the distribution of contact stresses in the interaction of cylindrical bodies with similar radii" (Belarusian Republican Foundation for Fundamental Research, 1996, No. GR 19981496); "Develop a method for predicting the wear of sliding bearings, taking into account the topographic and rheological characteristics of the surfaces of interacting parts, as well as the presence of anti-friction coatings" (Ministry of Education of the Republic of Belarus, 1998, No. GR 1999929); "Modeling the contact interaction of machine parts, taking into account the randomness of the rheological and geometric properties of the surface layer" (Ministry of Education of the Republic of Belarus, 1999 No. GR 20001251)

Purpose and objectives of the study. Development of a unified method for theoretical prediction of the influence of geometric, rheological characteristics of the surface roughness of solids and the presence of coatings on the stress state in the contact area, as well as the establishment on this basis of the patterns of change in contact stiffness and wear resistance of mates using the example of the interaction of bodies with circular boundaries.

To achieve this goal, it is necessary to solve the following problems:

To develop a method for the approximate solution of problems in the theory of elasticity and viscoelasticity on the contact interaction of a cylinder and a cylindrical cavity in a plate using a minimum number of independent parameters.

Develop a non-local model of the contact interaction of bodies, taking into account the microgeometric, rheological characteristics of surfaces, as well as the presence of plastic coatings.

Substantiate an approach that allows correcting the curvature of interacting surfaces due to roughness deformation.

To develop a method for the approximate solution of contact problems for a disk and isotropic, orthotropic with cylindrical anisotropy and viscoelastic aging coatings on a hole in a plate, taking into account their transverse deformability.

Build a model and determine the influence of microgeometric features of the surface of a solid body on the contact interaction with a plastic coating on the counterbody.

To develop a method for solving problems taking into account the wear of cylindrical bodies, the quality of their surfaces, as well as the presence of anti-friction coatings.

The object and subject of the study are non-classical mixed problems of the theory of elasticity and viscoelasticity for bodies with circular boundaries, taking into account the non-locality of the topographic and rheological characteristics of their surfaces and coatings, on the example of which a complex method for analyzing the change in the stress state in the contact area depending on the quality indicators is developed in this paper. their surfaces.

Hypothesis. When solving the set boundary problems, taking into account the quality of the surface of the bodies, a phenomenological approach is used, according to which the deformation of the roughness is considered as the deformation of the intermediate layer.

Problems with time-varying boundary conditions are considered as quasi-static.

Methodology and methods of the research. When conducting research, the basic equations of mechanics of a deformable solid body, tribology, and functional analysis were used. A method has been developed and substantiated that makes it possible to correct the curvature of loaded surfaces due to deformations of microroughnesses, which greatly simplifies the ongoing analytical transformations and makes it possible to obtain analytical dependences for the size of the contact area and contact stresses, taking into account the indicated parameters without using the assumption of the smallness of the value of the base length for measuring the roughness characteristics relative to the dimensions. contact areas.

When developing a method for theoretical prediction of surface wear, the observed macroscopic phenomena were considered as the result of the manifestation of statistically averaged relationships.

The reliability of the results obtained in the work is confirmed by comparisons of the obtained theoretical solutions and the results of experimental studies, as well as by comparison with the results of some solutions found by other methods.

Scientific novelty and significance of the obtained results. For the first time, using the example of the contact interaction of bodies with circular boundaries, a generalization of studies was carried out and a unified method for complex theoretical prediction of the influence of non-local geometric, rheological characteristics of rough surfaces of interacting bodies and the presence of coatings on the stress state, contact stiffness and wear resistance of interfaces was developed.

The complex of researches carried out made it possible to present in the dissertation a theoretically substantiated method for solving problems of solid mechanics, based on the consistent consideration of macroscopically observed phenomena, as a result of the manifestation of microscopic bonds statistically averaged over a significant area of ​​the contact surface.

As part of solving the problem:

A three-dimensional non-local model of the contact interaction of solid bodies with isotropic surface roughness is proposed.

A method has been developed for determining the influence of the surface characteristics of solids on the stress distribution.

The integro-differential equation obtained in contact problems for cylindrical bodies is investigated, which made it possible to determine the conditions for the existence and uniqueness of its solution, as well as the accuracy of the constructed approximations.

Practical (economic, social) significance of the obtained results. The results of the theoretical study have been brought to methods acceptable for practical use and can be directly applied in the engineering calculations of bearings, sliding bearings, and gears. The use of the proposed solutions will reduce the time of creating new machine-building structures, as well as predict their service characteristics with great accuracy.

Some of the results of the research carried out were implemented at NLP "Cycloprivod", NPO "Altech".

The main provisions of the dissertation submitted for defense:

Approximate solution of the problem of mechanics of a deformed solid on the contact interaction of a smooth cylinder and a cylindrical cavity in a plate, describing the phenomenon under study with sufficient accuracy using a minimum number of independent parameters.

Solution of non-local boundary value problems of mechanics of a deformable solid body, taking into account the geometric and rheological characteristics of their surfaces, based on a method that makes it possible to correct the curvature of interacting surfaces due to roughness deformation. The absence of an assumption about the smallness of the geometric dimensions of the base lengths of the roughness measurement in comparison with the dimensions of the contact area allows us to proceed to the development of multilevel models of deformation of the surface of solids.

Construction and substantiation of a method for calculating the displacements of the boundary of cylindrical bodies due to the deformation of surface layers. The results obtained make it possible to develop a theoretical approach that determines the contact stiffness of mates, taking into account the joint influence of all the features of the state of the surfaces of real bodies.

Modeling of the viscoelastic interaction of a disk and a cavity in a plate made of aging material, the simplicity of the implementation of the results of which allows them to be used for a wide range of applied problems.

Approximate solution of contact problems for a disk and isotropic, orthotropic with cylindrical anisotropy, as well as viscoelastic aging coatings on a hole in a plate, taking into account their transverse deformability. This makes it possible to evaluate the effect of composite coatings with a low modulus of elasticity on the loading of interfaces.

Construction of a non-local model and determination of the influence of the characteristics of the roughness of the surface of a solid body on the contact interaction with a plastic coating on the counterbody.

Development of a method for solving boundary value problems, taking into account the wear of cylindrical bodies, the quality of their surfaces, as well as the presence of anti-friction coatings. On this basis, a methodology is proposed that focuses mathematical and physical methods in the study of wear resistance, which makes it possible, instead of studying real friction units, to focus on the study of phenomena occurring in the contact area.

Applicant's personal contribution. All results submitted for defense were obtained by the author personally.

Approbation of the results of the dissertation. The results of the research presented in the dissertation were presented at 22 international conferences and congresses, as well as conferences of the CIS and republican countries, among them: "Pontryagin readings - 5" (Voronezh, 1994, Russia), "Mathematical models of physical processes and their properties" ( Taganrog, 1997, Russia), Nordtrib"98 (Ebeltoft, 1998, Denmark), Numerical mathematics and computational mechanics - "NMCM"98" (Miskolc, 1998, Hungary), "Modelling"98" (Praha, 1998, Czech Republic), 6th International Symposium on Creep and Coupled Processes (Bialowieza, 1998, Poland), "Computational methods and production: reality, problems, prospects" (Gomel, 1998, Belarus), "Polymer composites 98" (Gomel, 1998, Belarus), " Mechanika"99" (Kaunas, 1999, Lithuania), II Belarusian Congress on Theoretical and Applied Mechanics

Minsk, 1999, Belarus), Internat. Conf. On Engineering Rheology, ICER"99 (Zielona Gora, 1999, Poland), "Problems of strength of materials and structures in transport" (St. Petersburg, 1999, Russia), International Conference on Multifield Problems (Stuttgart, 1999, Germany).

Publication of results. Based on the dissertation materials, 40 printed works were published, among them: 1 monograph, 19 articles in journals and collections, including 15 articles under personal authorship. The total number of pages of published materials is 370.

The structure and scope of the dissertation. The dissertation consists of an introduction, seven chapters, a conclusion, a list of references and an appendix. The total volume of the dissertation is 275 pages, including the volume occupied by illustrations - 14 pages, tables - 1 page. The number of sources used includes 310 items.

Similar theses in the specialty "Mechanics of a deformable solid body", 01.02.04 VAK code

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• Solution of contact problems in the theory of plates and plane non-Hertzian contact problems by the boundary element method 2004, candidate of physical and mathematical sciences Malkin, Sergey Aleksandrovich

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Dissertation conclusion on the topic "Mechanics of a deformable solid body", Kravchuk, Alexander Stepanovich

CONCLUSION

In the course of the research carried out, a number of static and quasi-static problems of the mechanics of a deformable solid body were posed and solved. This allows us to formulate the following conclusions and indicate the results:

1. Contact stresses and surface quality are one of the main factors determining the durability of machine-building structures, which, combined with the tendency to reduce the weight and size indicators of machines, the use of new technological and structural solutions, leads to the need to revise and refine the approaches and assumptions used in determining the stress state , displacements and wear in mates. On the other hand, the cumbersomeness of the mathematical apparatus, the need to use powerful computing tools significantly hinder the use of existing theoretical developments in solving applied problems and define one of the main directions in the development of mechanics to obtain explicit approximate solutions of the problems posed, ensuring the simplicity of their numerical implementation.

2. An approximate solution of the problem of mechanics of a deformable solid on the contact interaction of a cylinder and a cylindrical cavity in a plate with a minimum number of independent parameters is constructed, which describes the phenomenon under study with sufficient accuracy.

3. For the first time non-local boundary value problems of the theory of elasticity are solved taking into account the geometric and rheological characteristics of roughness on the basis of a method that allows correcting the curvature of interacting surfaces. The absence of an assumption about the smallness of the geometric dimensions of the base lengths of the roughness measurement in comparison with the dimensions of the contact area makes it possible to correctly formulate and solve problems of the interaction of solid bodies, taking into account the microgeometry of their surfaces at relatively small contact sizes, and also to proceed to the creation of multilevel models of roughness deformation.

4. A method for calculating the largest contact displacements in the interaction of cylindrical bodies is proposed. The results obtained made it possible to construct a theoretical approach that determines the contact stiffness of mates, taking into account the microgeometric and mechanical features of the surfaces of real bodies.

5. Modeling of the viscoelastic interaction between a disk and a cavity in a plate made of aging material was carried out, the simplicity of the implementation of the results of which makes it possible to use them for a wide range of applied problems.

6. Contact problems are solved for a disk and isotropic, orthotropic with cylindrical anisotropy, and viscoelastic aging coatings on a hole in a plate, taking into account their transverse deformability. This makes it possible to evaluate the effect of composite antifriction coatings with a low modulus of elasticity.

7. A model is built and the influence of the microgeometry of the surface of one of the interacting bodies and the presence of plastic coatings on the surface of the counterbody is determined. This makes it possible to emphasize the leading influence of the surface characteristics of real composite bodies in the formation of the contact area and contact stresses.

8. A general method has been developed for solving cylindrical bodies, the quality of their anti-friction coatings. boundary value problems, taking into account the wear of surfaces, as well as the presence

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At the meeting of the scientific seminar "Modern problems of mathematics and mechanics" November 24, 2017 a presentation by Alexander Veniaminovich Konyukhov (Dr. habil. PD KIT, Prof. KNRTU, Karlsruhe Institute of Technology, Institute of Mechanics, Germany)

Geometrically exact theory of contact interaction as a fundamental basis of computational contact mechanics

Beginning at 13:00, room 1624.

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The main tactic of isogeometric analysis is the direct embedding of mechanics models in a complete description of a geometric object in order to formulate an efficient computational strategy. Such advantages of isogeometric analysis as a complete description of the geometry of an object in the formulation of algorithms for computational contact mechanics can be fully expressed only if the kinematics of contact interaction is fully described for all geometrically possible contact pairs. The contact of bodies from a geometric point of view can be considered as the interaction of deformable surfaces of arbitrary geometry and smoothness. In this case, various conditions for the smoothness of the surface lead to the consideration of mutual contact between the faces, edges and vertices of the surface. Therefore, all contact pairs can be hierarchically classified as follows: surface-to-surface, curve-to-surface, point-to-surface, curve-to-curve, point-to-curve, point-to-point. The shortest distance between these objects is a natural measure of contact and leads to the Closest Point Projection (CPP) problem.

The first main task in constructing a geometrically exact theory of contact interaction is to consider the conditions for the existence and uniqueness of a solution to the PBT problem. This leads to a number of theorems that allow us to construct both three-dimensional geometric domains of existence and uniqueness of the projection for each object (surface, curve, point) in the corresponding contact pair, and the transition mechanism between contact pairs. These areas are constructed by considering the differential geometry of the object, in the metric of the curvilinear coordinate system corresponding to it: in Gaussian (Gauß) coordinate system for the surface, in the Frenet-Serret coordinate system for curves, in the Darboux coordinate system for curves on the surface, and using Euler coordinates (Euler) as well as quaternions to describe the final rotations around the object - the point.

The second main task is to consider the kinematics of the contact interaction from the point of view of the observer in the corresponding coordinate system. This allows us to define not only the standard measure of normal contact as "penetration" (penetration), but also geometrically precise measures of relative contact interaction: tangential sliding on the surface, sliding along individual curves, relative rotation of the curve (torsion), sliding of the curve along its own tangent, and along the tangential normal (“dragging”) as the curve moves along the surface. On the this stage, using the apparatus of covariant differentiation in the corresponding curvilinear coordinate system,
preparations are being made for the variational formulation of the problem, as well as for the linearization necessary for the subsequent global numerical solution, for example, for the Newton iterative method (Newton nonlinear solver). Linearization is understood here as Gateaux differentiation in covariant form in a curvilinear coordinate system. In a number of complex cases based on multiple solutions to the PBT problem, such as in the case of "parallel curves", it is necessary to build additional mechanical models (3D continuum model of the curved rope "Solid Beam Finite Element"), compatible with the corresponding contact algorithm "Curve To Solid Beam contact algorithm. An important step in describing the contact interaction is the formulation in covariant form of the most general arbitrary law of interaction between geometric objects, which goes far beyond the standard Coulomb friction law (Coulomb). In this case, the fundamental physical principle of “dissipation maximum” is used, which is a consequence of the second law of thermodynamics. This requires the formulation of an optimization problem with a constraint in the form of inequalities in covariant form. In this case, all the necessary operations for the chosen method of numerical solution of the optimization problem, including, for example, the "return-mapping algorithm" and the necessary derivatives, are also formulated in a curvilinear coordinate system. Here, an indicative result of a geometrically exact theory is both the ability to obtain new analytical solutions in a closed form (a generalization of the Euler problem of 1769 on the friction of a rope along a cylinder to the case of anisotropic friction on a surface of arbitrary geometry), and the ability to obtain in a compact form generalizations of the Coulomb friction law, which takes into account anisotropic geometric surface structure together with anisotropic micro-friction.

The choice of methods for solving the problem of statics or dynamics, provided that the laws of contact interaction are satisfied, remains extensive. These are various modifications of Newton's iterative method for a global problem and methods for satisfying constraints at the local and global levels: penalty (penalty), Lagrange (Lagrange), Nitsche (Nitsche), Mortar (Mortar), as well as an arbitrary choice of a finite difference scheme for a dynamic problem . The main principle is only the formulation of the method in covariant form without
consideration of any approximations. Careful passage of all stages of the construction of the theory makes it possible to obtain a computational algorithm in a covariant "closed" form for all types of contact pairs, including an arbitrarily chosen law of contact interaction. The choice of the type of approximations is carried out only at the final stage of the solution. At the same time, the choice of the final implementation of the computational algorithm remains very extensive: the standard Finite Element Method, High Order Finite Element, Isogeoemtric Analysis, Finite Cell Method, "submerged"