30.10.202127.05.2017

Stresses in the contact area under simultaneous loading by normal and tangential forces. Stresses determined by photoelasticity method

Mechanics of contact interaction deals with the calculation of elastic, viscoelastic and plastic bodies under static or dynamic contact. Contact interaction mechanics is a fundamental engineering discipline that is mandatory when designing reliable and energy-saving equipment. It will be useful in solving many contact problems, for example, wheel-rail, when calculating couplings, brakes, tires, plain and rolling bearings, internal combustion engines, hinges, seals; in stamping, metalworking, ultrasonic welding, electrical contacts, etc. It covers wide range tasks, ranging from calculating the strength 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. Only a century later, Johnson, Kendal and Roberts found a similar solution for adhesive contact (JKR - theory).

Further progress in the mechanics of contact interaction in the mid-20th century is associated with the names of Bowden and Tabor. They were the first to point out the importance of taking into account the surface roughness of contacting bodies. Roughness leads to the fact that the actual contact area between the rubbing bodies is much smaller than the apparent contact area. These ideas significantly changed the direction of many tribological studies. The work of Bowden and Tabor gave rise to a number of theories of the mechanics of contact interaction of rough surfaces.

Pioneering work in this area is the work of Archard (1957), who concluded that when elastic rough surfaces come into contact, the contact area is approximately proportional to the normal force. Further important contributions to the theory of contact of rough surfaces were made by Greenwood and Williamson (1966) and Persson (2002). The main result of these works is the proof that the actual contact area of rough surfaces is, in a rough approximation, proportional to the normal force, while the characteristics of an individual microcontact (pressure, microcontact size) weakly depend on the load.

Contact between a solid 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

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 related by the following relationship:

The voltage at the apex of the cone (in the center of the contact area) varies logarithmically. The total force is calculated as

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

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

as in the case of contact between two balls. The maximum pressure is

The phenomenon of adhesion is most easily observed in the contact of a solid body with a very soft elastic body, for example, with jelly. When bodies touch, an adhesive neck appears as a result of the action of van der Waals forces. In order to break the bodies again, it is necessary to apply a certain minimum force, called the adhesion force. Similar phenomena occur in the contact of two solids, separated by a very soft layer, such as in a sticker or adhesive. Adhesion can either be of technological interest, for example, in adhesive joints, or be an interfering factor, for example, preventing the rapid opening of elastomeric valves.

The adhesion force between a parabolic rigid body and an elastic half-space was first found in 1971 by Johnson, Kendall and Roberts. It is equal

More complex forms begin to come off “from the edges” of the form, after which the separation front spreads toward the center until a certain critical state is reached. The process of adhesive contact separation can be observed in the study.

Many problems in the mechanics of contact interaction can be easily solved by the dimension reduction method. In this method, the original three-dimensional system is replaced by a one-dimensional elastic or viscoelastic base (figure). If the parameters of the base and the shape of the body are chosen based on the simple rules of the reduction method, then the macroscopic properties of the contact coincide exactly with the properties of the original.

C. L. Johnson, K. Kendal, and A. D. Roberts (JKR) used this theory as the basis for calculating the theoretical shear or indentation depth in the presence of adhesion in their landmark paper “Surface Energy and Contact of Elastic Solids.” ", published in 1971 in the proceedings of the Royal Society. Hertz's theory follows from their formulation, provided that the adhesion of materials is zero.

Similar to this theory, but based on other assumptions, in 1975 B.V. Deryagin, V.M. Muller and Yu.P. Toporov developed another theory, which among researchers is known as the DMT theory, and from which Hertz’s formulation also follows: zero adhesion.

The DMT theory was subsequently revised several times before it was accepted as another contact theory in addition to the JKR theory.

Both DMT and JKR theories are the basis of contact interaction mechanics, on which all contact transition models are based, and which are used in nanoshear calculations and electron microscopy. Thus, Hertz’s research in the days of his work as a lecturer, which he himself, with his sober self-esteem, considered trivial, even before his great works on electromagnetism, fell into the age of nanotechnology.

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

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However, the modern theory of elastic contact does not allow a sufficient search for the rational geometric shape of the contacting 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 technology and experimental equipment used, as well as the high complexity and duration...

ACCEPTED CONVENTIONS
CHAPTER 1. CRITICAL ANALYSIS OF THE STATE OF THE ISSUE, GOALS AND OBJECTIVES OF THE WORK
- 1. 1. Systematic analysis of the current state and trends in the field of improving elastic contact of bodies of complex shape
  - 1. 1. 1. Current state of the theory of local elastic contact of bodies of complex shape and optimization geometric parameters contact
  - 1. 1. 2. The main directions for improving the technology for grinding working surfaces of rolling bearings of complex shape
  - 1. 1. 3. Modern technology of shape-forming superfinishing of surfaces of rotation
- 1. 2. Research objectives
CHAPTER 2. MECHANISM OF ELASTIC CONTACT OF BODIES
COMPLEX GEOMETRIC SHAPE
- 2. 1. Mechanism of the deformed state of elastic contact of bodies of complex shape
- 2. 2. Mechanism of the stressed 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. SHAPE FORMATION OF RATIONAL GEOMETRICAL SHAPE OF PARTS DURING GRINDING OPERATIONS
- 3. 1. Shaping the geometric shape of rotating parts by grinding with a wheel inclined to the axis of the part
- 3. 2. Algorithm and program for calculating the geometric shape of parts during grinding with an inclined wheel and the stress-strain state of the region of its contact with an elastic body in the form of a ball
- 3. 3. Analysis of the influence of inclined wheel grinding process parameters on the supporting ability of the ground surface
- 3. 4. Research into the technological capabilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and the operational properties of bearings manufactured using it
conclusions
CHAPTER 4. BASICS OF PART PROFILE FORMATION IN SUPERFINISHING OPERATIONS
- 4. 1. Mathematical model of the mechanism of the process of forming 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. RESEARCH RESULTS ON THE EFFECTIVENESS OF THE PROCESS OF FORM-BUILDING SUPERFINISHING
- 5. 1. Methodology of experimental research and processing of experimental data
- 5. 2. Regression analysis of parameters of the shaping superfinishing process depending on the characteristics of the tool
- 5. 3. Regression analysis of indicators of the form-building superfinishing process depending on the processing mode
- 5. 4. General mathematical model of the process of shaping superfinishing
- 5. 5. Performance of roller bearings with rational geometric shape of working surfaces
conclusions
CHAPTER 6. PRACTICAL APPLICATION OF RESEARCH RESULTS
- 6. 1. Improving the designs of rolling friction bearings
- 6. 2. Bearing ring grinding method
- 6. 3. Method for monitoring the profile of bearing ring raceways
- 6. 4. Methods for superfinishing parts such as rings with complex profiles
- 6. 5. A method for assembling bearings with a rational geometric shape of working surfaces
conclusions

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Applied theory of contact interaction of elastic bodies and the creation on its basis of processes of shaping friction-rolling bearings with rational geometry ( essay, coursework, diploma, test)

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 national economy depend on the quality of bearings and the efficiency of their production. Improving the performance 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 technological support for the rational geometric shape of their working surfaces: bodies and raceways. In the works of V. M. Alexandrov, O. Yu. Davidenko, A. B. Koroleva, A.I. Lurie, A.B. Orlova, I.Ya. Shtaerman et al. have convincingly shown that giving the working surfaces of elastically contacting parts of mechanisms and machines a rational geometric shape can significantly improve the parameters of elastic contact and significantly increase the operational properties of friction units.

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

Thus, ensuring 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 most important directions for further improving the operational properties of mechanisms and machines. On the one hand, studying 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 fundamentals of technological support for a given shape of parts ensures the effective production of rolling friction bearings 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 a scientific problem that is important for the development of domestic mechanical engineering.

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

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

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

Credibility. The theoretical provisions of the work are confirmed by the results of experimental studies performed both in laboratory and production conditions. The reliability of the theoretical principles and experimental data is confirmed by the implementation of the results of the work in production.

Scientific novelty. In this work, an applied theory of local contact interaction of elastic bodies has been developed and, on its basis, processes for the formation of friction-rolling bearings with rational geometry have been created, which open up the possibility of significantly increasing 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 relations 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 forming parts of rolling friction bearings with a rational geometric shape during technological operations of grinding a 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 supporting ability of the ground surface, the results of a study of the technological capabilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and operational properties of bearings manufactured using it.

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

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

6. Methodology for the targeted design of a rational design of the working surfaces of parts of complex geometric shapes such as parts of rolling bearings, an integrated technology for the manufacture of parts of rolling bearings, including preliminary, final processing and control of the geometric parameters of the 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 working surfaces.

This work is based on materials from numerous studies by domestic and foreign authors. The work was greatly assisted by the experience and support of a number of specialists from the Saratov Bearing Plant, the Saratov Research and Production Enterprise for Non-Standard Mechanical 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 implementation of this work to the Honored Scientist of the Russian Federation, Doctor of Technical Sciences, Professor, Academician of the Russian Academy of Natural Sciences Yu. V. Chebotarevsky and Doctor of Technical Sciences, Professor A.M. Chistyakov.

The limited volume of work did not allow us to provide comprehensive answers to a number of questions raised. Some of these issues are more fully discussed in the author's published works, as well as in joint work with graduate students and applicants ("https://site", 11).

334 Conclusions:

1. A method for the targeted design of a rational design of the working surfaces of parts of complex geometric shapes such as rolling bearing parts is proposed, 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 for manufacturing parts of rolling bearings has been developed, including preliminary and final processing, control of geometric parameters of working surfaces and assembly of bearings.

3. 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 for searching for a rational geometric shape of locally contacting elastic bodies and the technological basis for their formation have been developed, which opens up prospects for increasing 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-law relationships with arbitrary exponents. The proposed model generalizes the previously obtained solutions and significantly expands the scope of practical application of the exact solution of contact problems.

3. A mathematical model of the stressed state of the region of elastic local contact of bodies of complex shape has been developed, showing that the proposed solution to the contact problem gives a fundamentally new result, opening 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 performance of friction units of mechanisms and cars

4. A numerical solution for 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 allow purposefully designing rational designs of the working surfaces of parts.

5. An analysis of the influence of the geometric shape of elastic bodies on the parameters of their local contact has been carried out, showing that by changing the shape of the bodies it is possible to simultaneously control the shape of the diagram of contact stresses, their magnitude and the dimensions of the contact area, which makes it possible to ensure high support capacity of the contacting surfaces, and therefore, significantly increase the performance properties of contact surfaces.

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

7. A technology for grinding ball bearings with a grinding wheel inclined to the axis of the workpiece and a mathematical model for the formation of the grinded surface have been developed. It is shown that the resulting shape of the polished surface, in contrast to the traditional shape of a circular arc, has four geometric parameters, which significantly expands the ability to control the supporting ability of the surface being processed.

8. A set of programs has been proposed that provide calculation of the geometric parameters of the surfaces of parts obtained by grinding with an inclined wheel, the stress and deformation state of the elastic body in the rolling bearings at various grinding parameters. An analysis of the influence of grinding parameters with an inclined wheel on the supporting ability 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 contact stresses and at the same time vary the size of the contact area, which significantly increases the bearing capacity of the contact surface and helps reduce friction on the contact. Checking the adequacy of the proposed mathematical model gave positive results.

9. Research has been carried out on the technological capabilities of the grinding process with a grinding wheel inclined to the axis of the workpiece and the operational properties of bearings manufactured using it. It has been shown that the grinding process with an inclined wheel helps to increase processing productivity compared to conventional grinding, as well as improve the quality of the machined surface. Compared to standard bearings, the durability of bearings made using inclined wheel grinding increases by 2-2.5 times, waviness decreases by 11 dB, friction torque decreases by 36%, and speed increases more than twice.

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 metal removal at any point in the profile, reflects the process of forming the tool profile during processing, and the complex mechanism of its clogging and wear.

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

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

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

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

Cost of unique work

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480 rub. | 150 UAH | $7.5 ", MOUSEOFF, FGCOLOR, "#FFFFCC",BGCOLOR, "#393939");" onMouseOut="return nd();"> Dissertation - 480 RUR, delivery 10 minutes, around the clock, seven days a week and holidays

Kravchuk Alexander Stepanovich. Theory of contact interaction of deformable solids with circular boundaries taking into account the mechanical and microgeometric characteristics of surfaces: Dis. ... Doctor of Physics and Mathematics Sciences: 02/01/04: Cheboksary, 2004 275 p. RSL OD, 71:05-1/66

Introduction

1. Contemporary issues mechanics of contact interaction 17

1.1. Classical hypotheses used in solving contact problems for smooth bodies 17

1.2. The influence of creep of solids on their shape change in the contact area 18

1.3. Assessment of the convergence of rough surfaces 20

1.4. Analysis of contact interaction of multilayer structures 27

1.5. The relationship between mechanics and problems of friction and wear 30

1.6. Features of the application of modeling in tribology 31

Conclusions on the first chapter 35

2. Contact interaction of smooth cylindrical bodies 37

2.1. Solution of the contact problem for smooth isotropic disk and plate with a cylindrical cavity 37

2.1.1. General formulas 38

2.1.2. Derivation of boundary conditions for movements in the contact area 39

2.1.3. Integral equation and its solution 42

2.1.3.1. Study of the resulting equation 4 5

2.1.3.1.1. Reducing a singular integro-differential equation to an integral equation with a kernel having a logarithmic singularity 46

2.1.3.1.2. Estimation of the norm of a linear operator 49

2.1.3.2. Approximate Solution of Equation 51

2.2. Calculation of a fixed connection of smooth cylindrical bodies 58

2.3. Determination of displacement in a movable connection of cylindrical bodies 59

2.3.1. Solution of an auxiliary problem for an elastic plane 62

2.3.2. Solution of an auxiliary problem for an elastic disk 63

2.3.3. Determination of maximum normal radial displacement 64

2.4. Comparison of theoretical and experimental data on the study of contact stresses during internal contact of cylinders of close radii 68

2.5. Modeling of spatial contact interaction of a system of coaxial cylinders of finite dimensions 72

2.5.1. Statement of problem 73

2.5.2. Solving auxiliary two-dimensional problems 74

2.5.3. Solution of the original problem 75

Conclusions and main results of the second chapter 7 8

3. Contact problems for rough bodies and their solution by adjusting the curvature of the deformed surface 80

3.1. Spatial nonlocal theory. Geometric assumptions 83

3.2. Relative approach of two parallel circles determined by roughness deformation 86

3.3. Method for analytical assessment of the influence of roughness deformation 88

3.4. Determination of movements in the contact area 89

3.5. Determination of auxiliary coefficients 91

3.6. Determining the dimensions of the elliptical contact area 96

3.7. Equations for determining the contact area close to circular 100

3.8. Equations for determining the contact area close to line 102

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

3.10. Features of averaging pressures and deformations when solving the two-dimensional problem of internal contact of rough cylinders of close radii 1 and 5

3.10.1. Derivation of the integro-differential equation and its solution in the case of internal contact of rough cylinders 10"

3.10.2. Determination of auxiliary coefficients

Conclusions and main results of the third chapter

4. Solving contact problems of viscoelasticity for smooth bodies

4.1. Basic provisions

4.2. Compliance Principles Analysis

4.2.1. Volterra's principle

4.2.2. Constant coefficient of transverse expansion under creep deformation 123

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 128

4.3.3. Fixed connection solution 129

4.3.4. Modeling of contact interaction in the case

uniformly aging isotropic plate 130

Conclusions and main results of the fourth chapter 135

5. Surface creep 136

5.1. Features of contact interaction of bodies with low yield strength 137

5.2. Construction of a model of surface deformation taking into account creep in the case of an elliptical contact area 139

5.2.1. Geometric assumptions 140

5.2.2. Surface Creep Model 141

5.2.3. Determination of average strains of the rough layer and average pressures 144

5.2.4. Determination of auxiliary coefficients 146

5.2.5. Determining the dimensions of the elliptical contact area 149

5.2.6. Determining the dimensions of the circular contact area 152

5.2.7. Determining the width of the contact area in the form of a strip 154

5.3. Solution of a two-dimensional contact problem for internal touch

rough cylinders taking into account surface creep 154

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

differential equation 156

5.3.2. Determination of auxiliary coefficients 160

Conclusions and main results of the fifth chapter 167

6. Mechanics of interaction of cylindrical bodies taking into account the presence of coatings 168

6.1. Calculation of effective moduli in the theory of composites 169

6.2. Construction of a self-consistent method for calculating effective coefficients of inhomogeneous media taking into account the spread of physical and mechanical properties 173

6.3. Solution of the contact problem for a disk and a plane with an elastic composite coating on the contour of a hole 178

6.3. 1 Statement of the problem and basic formulas 179

6.3.2. Derivation of boundary conditions for movements in the contact area 183

6.3.3. Integral equation and its solution 184

6.4. Solution of the problem in the case of an orthotropic elastic coating with cylindrical anisotropy 190

6.5. Determining the influence of a viscoelastic aging coating on changes in contact parameters 191

6.6. Analysis of the features of contact interaction between a multicomponent coating and disk roughness 194

6.7. Modeling of contact interaction taking into account thin metal coatings 196

6.7.1. Contact between a plastic-coated sphere and a rough half-space 197

6.7.1.1. Basic hypotheses and model of interaction of solids 197

6.7.1.2. Approximate solution to problem 200

6.7.1.3. Determination of maximum contact approach 204

6.7.2. Solution of the contact problem for a rough cylinder and a thin metal coating on the contour of a hole 206

6.7.3. Determination of contact stiffness for internal contact of cylinders 214

Conclusions and main results of the sixth chapter 217

7. Solution of mixed boundary value problems taking into account wear of surfaces of interacting bodies 218

7.1. Features of solving the contact problem taking into account wear of surfaces 219

7.2. Statement and solution of the problem in the case of elastic deformation of roughness 223

7.3. Method for theoretical assessment of wear taking into account surface creep 229

7.4. Method for assessing wear taking into account the influence of coating 233

7.5. Concluding remarks on the formulation of plane problems taking into account wear 237

Conclusions and main results of the seventh chapter 241

Conclusion 242

List of sources used

Introduction to the work

Relevance of the dissertation topic. Currently, significant efforts of engineers in our country and abroad are aimed at finding ways to determine the contact stresses of interacting bodies, since contact problems of the mechanics of a deformable solid play a decisive role in the transition from calculating the wear of materials to the problems of structural wear resistance.

It should be noted that the most extensive studies of contact interaction have been carried out using analytical methods. At the same time, the use of numerical methods significantly expands the possibilities of analyzing the stress state in the contact area, taking into account the properties of the surfaces of rough bodies.

The need to take into account the surface structure is explained by the fact that the protrusions formed during technological processing have a different distribution of heights and the contact of microroughnesses occurs only on separate areas that form the actual contact area. Therefore, when modeling the convergence of surfaces, it is necessary to use parameters that characterize the real surface.

The cumbersomeness of the mathematical apparatus used to solve contact problems for rough bodies and the need to use powerful computing tools significantly hinder the use of existing theoretical developments in solving applied problems. And despite achievements achieved, it is still difficult to obtain satisfactory results taking into account the features of the macro- and microgeometry of the surfaces of interacting bodies, when the surface element on which the roughness characteristics of solid bodies are established is comparable to the contact area.

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

Contact problems for bodies with circular boundaries form the theoretical basis for the calculation of such machine elements as bearings, hinge joints, and tension joints. Therefore, these problems are usually chosen as model ones when conducting such studies.

Intensive work carried out in last years V Belarusian National Technical University

to solve this problem form the basis of our national strategy.

Connection of work with major scientific programs and topics.

The research was carried out in accordance with the following topics: “Develop a method for calculating contact stresses during elastic contact interaction of cylindrical bodies, not described by Hertz’s theory” (Ministry of Education of the Republic of Belarus, 1997, No. GR 19981103); “The influence of micro-irregularities of contacting surfaces on the distribution of contact stresses during the interaction of cylindrical bodies with similar radii” (Belarusian Republican Foundation for Basic Research, 1996, No. GR 19981496); “To 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 of 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. GR2000G251)

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 solid bodies and the presence of coatings on the stress state in the contact area, as well as the establishment on this basis of patterns of changes in contact stiffness and wear resistance of joints using the example of the interaction of bodies with circular boundaries.

To achieve this goal, the following problems need to be solved:

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

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

Justify an approach to correct curvature
interacting surfaces due to roughness deformation.

Develop a method for approximate solution of contact problems for a disk and isotropic, orthotropic With cylindrical anisotropy and viscoelastic aging coatings on the hole in the 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 contact interaction With plastic coating on the counter body.

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 antifriction 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, using the example of which in this work a comprehensive method for analyzing changes in the stress state in the contact area depending on quality indicators is developed their surfaces.

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

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

Methodology and methods of the study. When conducting research, the basic equations of mechanics of a deformable solid, tribology, and functional analysis were used. A method has been developed and justified that makes it possible to correct the curvature of loaded surfaces due to deformations of microroughnesses, which significantly simplifies the analytical transformations carried out and makes it possible to obtain analytical dependencies for the size of the contact area and contact stresses taking into account the specified parameters without using the assumption that the base length of the measurement of roughness characteristics relative to the dimensions is small contact areas.

When developing a method for theoretically predicting 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 comparison with the results of some solutions found by other methods.

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

The complex of studies carried out made it possible to present in the dissertation a theoretically based method for solving problems of solid mechanics, based on the consistent consideration of macroscopically observable 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 posed:

A spatial nonlocal model of contact
interaction of solids with isotropic surface roughness.

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 was studied, 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 results obtained. The results of the theoretical study have been brought to acceptable methods for practical use and can be directly applied when carrying out engineering calculations of bearings, sliding supports, and gears. The use of the proposed solutions will reduce the time for creating new machine-building structures, as well as predict their service characteristics with great accuracy.

Some results of the research carried out were implemented at NPP "Cyclodrive", NGO"Altech".

The main provisions of the dissertation submitted for defense:

Approximately solve problems in the mechanics of deformed
solid body about the contact interaction of smooth cylinders and
cylindrical cavity in the plate, with sufficient accuracy
describing the phenomenon under study using the minimum
number of independent parameters.

Solution of nonlocal boundary value problems in the mechanics of a deformable solid, taking into account the geometric and rheological characteristics of their surfaces, based on a method that allows one to correct the curvature of interacting surfaces due to roughness deformation. The absence of the assumption that the geometric dimensions of the basic roughness measurement lengths are small compared to the dimensions of the contact area allows us to proceed to the development of multi-level models of deformation of the surface of solid bodies.

determining the contact rigidity of the mates With taking into account the joint influence of all features of the state of the surfaces of real bodies.

Modeling of viscoelastic interaction between a disc and a cavity in
plate made of aging material, ease of implementation of results
which allows them to be used for a wide range of applications
tasks.

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

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

Development of a method for solving boundary value problems With taking into account the wear of cylindrical bodies, the quality of their surfaces, as well as the presence of antifriction coatings. On this basis, a methodology has been proposed that focuses mathematical and physical methods in the study of wear resistance, which makes it possible, instead of studying real friction units, to place the main emphasis on studying the phenomena that occur V contact areas.

Personal contribution of the applicant. All results submitted for defense were obtained by the author personally.

Approbation of the dissertation results. 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), P 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).

Structure and scope of the dissertation. The dissertation consists of an introduction, seven chapters, a conclusion, a list of sources used and an appendix. The full volume of the dissertation is 2" pages, including the volume occupied by illustrations - 14 pages, tables - 1 page. The number of sources used includes 310 titles.

The influence of creep of solids on their shape change in the contact area

Practical obtaining of analytical dependencies for stresses and displacements in a closed form for real objects, even in the simplest cases, is associated with significant difficulties. As a result, when considering contact problems, it is customary to resort to idealization. Thus, it is believed that if the dimensions of the bodies themselves are large enough compared to the dimensions of the contact area, then the stresses in this zone weakly depend on the configuration of the bodies far from the contact area, as well as the method of their fastening. In this case, stresses can be calculated with a fairly good degree of reliability, considering each body as an infinite elastic medium limited by a flat surface, i.e. like an elastic half-space.

The surface of each of the bodies is assumed to be topographically smooth at the micro- and macrolevel. At the micro level, this means the absence or failure to take into account micro-irregularities of the contacting surfaces, which would cause an incomplete fit of the contact surfaces. Therefore, the actual contact area that forms at the tops of the protrusions is significantly smaller than the theoretical one. At the macro level, surface profiles are considered continuous in the contact zone along with second derivatives.

These assumptions were first used by Hertz in solving the contact problem. The results obtained on the basis of his theory satisfactorily describe the deformed state of ideally elastic bodies in the absence of friction along the contact surface, but are not applicable, in particular, to low-modulus materials. In addition, the conditions under which Hertz's theory is used are violated when considering the contact of matched surfaces. This is explained by the fact that, due to the application of a load, the dimensions of the contact area quickly grow and can reach values comparable to the characteristic dimensions of the contacting bodies, so that the bodies cannot be considered as elastic half-spaces.

Of particular interest when solving contact problems is taking into account friction forces. At the same time, the latter, on the interface between two bodies of consistent shape that are in normal contact, plays a role only at relatively high values of the friction coefficient.

The development of the theory of contact interaction of solids is associated with the rejection of the above hypotheses. It was carried out in the following main directions: complication of the physical model of deformation of solids and (or) rejection of the hypotheses of smoothness and homogeneity of their surfaces.

Interest in creep has increased sharply due to the development of technology. Among the first researchers to discover the phenomenon of deformation of materials over time under constant load were Wick, Weber, Kohlrausch. Maxwell first presented the law of deformation in time in the form of a differential equation. Somewhat later, Bolygman created a general apparatus for describing the phenomena of linear creep. This apparatus, significantly developed subsequently by Volterra, is currently a classical branch of the theory of integral equations.

Until the middle of the last century, elements of the theory of deformation of materials over time found little application in the practice of calculating engineering structures. However, with the development of power plants and chemical technological devices operating at higher temperatures and pressures, it became necessary to take into account the phenomenon of creep. Requests from mechanical engineering have led to a huge scope of experimental and theoretical research in the field of creep. Due to the emerging need for accurate calculations, the phenomenon of creep began to be taken into account even in materials such as wood and soils,

The study of creep during contact interaction of solids is important for a number of applied and fundamental reasons. Thus, even under constant loads, the shape of interacting bodies and their stress state, as a rule, changes, which must be taken into account when designing machines.

A qualitative explanation of the processes occurring during creep can be given based on the basic concepts of dislocation theory. Thus, various local defects can occur in the structure of the crystal lattice. These defects are called dislocations. They move, interact with each other and cause various types of slip in the metal. The result of dislocation motion is a shift of one interatomic distance. The stressed state of the body facilitates the movement of dislocations, reducing potential barriers.

The temporal laws of creep depend on the structure of the material, which changes with creep. An exponential dependence of the rates of steady-state creep on stresses at relatively high stresses (-10" or more from the elastic modulus) was experimentally obtained. In a significant stress range, experimental points on a logarithmic grid are usually grouped around a certain straight line. This means that in the stress range under consideration (- 10" -10" from the elastic modulus) there is a power-law dependence of strain rates on stress. It should be noted that at low stresses (10" or less from the elastic modulus) this dependence is linear. A number of works provide various experimental data on the mechanical properties of various materials in a wide range of temperatures and strain rates.

Integral equation and its solution

Note that if the elastic constants of the disk and plate are equal, then yx = O and this equation becomes an integral equation of the first kind. Features of the theory of analytic functions allow in this case, using additional conditions, to obtain a unique solution. These are the so-called inversion formulas for singular integral equations, which allow one to obtain an explicit solution to the problem posed. The peculiarity is that in the theory of boundary value problems three cases are usually considered (when V is part of the boundary of the bodies): the solution has a singularity at both ends of the integration domain; the solution has a singularity at one end of the integration domain and vanishes at the other; the solution vanishes at both ends. Depending on the choice of one or another option, a general form of solution is constructed, which in the first case includes the general solution of a homogeneous equation. Specifying the behavior of the solution at infinity and corner points of the contact area, based on physically based assumptions, a unique solution is constructed that satisfies the specified restrictions.

Thus, the uniqueness of the solution to this problem is understood in the sense of the accepted restrictions. It should be noted that when solving contact problems of the theory of elasticity, the most common restrictions are the requirements for the solution to vanish at the ends of the contact area and the assumption that stresses and rotations disappear at infinity. In the case when the area of integration constitutes the entire boundary of the area (body), then the uniqueness of the solution is guaranteed by the Cauchy formulas. Moreover, the simplest and most common method for solving applied problems in this case is to represent the Cauchy integral in the form of a series.

It should be noted that the above general information from the theory of singular integral equations does not in any way specify the properties of the contours of the regions under study, since in this case, it is known that the arc of a circle (the curve along which the integration is performed) satisfies the Lyapunov condition. A generalization of the theory of two-dimensional boundary value problems in the case of more general assumptions on the smoothness of the boundaries of domains can be found in the monograph of AI. Danilyuk.

Of greatest interest is the general case of the equation, when 7i 0. The lack of methods for constructing an exact solution in this case leads to the need to use methods of numerical analysis and approximation theory. In fact, as already noted, numerical methods for solving integral equations are usually based on approximating the solution to the equation by a functional of a certain type. The volume of accumulated results in this area allows us to identify the main criteria by which these methods are usually compared when used in applied problems. First of all, the simplicity of the physical analogy of the proposed approach (usually this is, in one form or another, a method of superposition of a system of certain solutions); the volume of necessary preparatory analytical calculations used to obtain the corresponding system of linear equations; the required size of the system of linear equations to achieve the required accuracy of the solution; the use of a numerical method for solving a system of linear equations, which takes into account the features of its structure as much as possible and, accordingly, allows one to obtain a numerical result with the greatest speed. It should be noted that the last criterion plays a significant role only in the case of systems of linear equations of large order. All this determines the effectiveness of the approach used. At the same time, it should be noted that to date there are only isolated studies devoted to comparative analysis and possible simplifications in solving practical problems using various approximations.

Note that the integro-differential equation can be reduced to the form: V is an arc of a circle of unit radius, enclosed between two points with angular coordinates -сс0 and а0, а0 є(0,л/2); y1 is a real coefficient determined by the elastic characteristics of interacting bodies (2.6); f(t) is a known function determined by the applied loads (2.6). In addition, recall that cm(t) vanishes at the ends of the integration segment.

Relative approach of two parallel circles determined by roughness deformation

The problem of internal compression of circular cylinders of close radii was first considered by I.Ya. Shtaerman. When solving the problem he posed, it was accepted that the external load acting on the inner and outer cylinders along their surfaces is carried out in the form of normal pressure, diametrically opposite to the contact pressure. When deriving the equation of the problem, we used the solution of compression of a cylinder by two opposite forces and the solution of a similar problem for the outside of a circular hole in an elastic medium. He obtained an explicit expression for the displacements of the contour points of the cylinder and hole through the integral operator of the stress function. This expression has been used by a number of authors to estimate contact stiffness.

Using a heuristic approximation for the distribution of contact stresses for the I.Ya. Shtaerman, A.B. Milov obtained a simplified relationship for maximum contact displacements. However, he found that the resulting theoretical estimate differed significantly from the experimental data. Thus, the displacement determined from the experiment turned out to be 3 times less than the theoretical one. This fact is explained by the author by the significant influence of the features of the spatial loading scheme and a coefficient of transition from a three-dimensional problem to a flat one is proposed.

A similar approach was used by M.I. Warm, having asked for an approximate solution of a slightly different type. It should be noted that in this work, in addition, a second-order linear differential equation was obtained to determine contact displacements in the case of the circuit shown in Figure 2.1. This equation follows directly from the method of obtaining the integro-differential equation for determining normal radial stresses. In this case, the complexity of the right side determines the cumbersomeness of the resulting expression for displacements. In addition, in this case, the values of the coefficients in the solution of the corresponding homogeneous equation remain unknown. At the same time, it is noted that, without setting the values of the constants, it is possible to determine the sum of the radial movements of diametrically opposite points of the contours of the hole and shaft.

Thus, despite the relevance of the problem of determining contact stiffness, the analysis of literature sources did not allow us to identify a method for solving it that would allow us to reasonably establish the values of the largest normal contact movements caused by the deformation of surface layers without taking into account the deformations of interacting bodies as a whole, which is explained by the lack of a formalized definition of the concept of “contact stiffness” ".

When solving the problem posed, we will proceed from the following definitions: movements under the influence of the main vector of forces (without taking into account the features of contact interaction) will be called the approach (removal) of the center of the disk (hole) and its surface, which does not lead to a change in the shape of its boundary. Those. This is the rigidity of the body as a whole. Then the contact stiffness is the maximum displacement of the center of the disk (hole) without taking into account the displacement of the elastic body under the action of the main vector of forces. This system concepts allows us to separate the displacements obtained from solving the problem of the theory of elasticity, and shows that the estimate of the contact stiffness of cylindrical bodies obtained by A.B. Milovs from the decision of IL. Shtaerman, is true only for this loading scheme.

Let us consider the problem posed in section 2.1. (Figure 2.1) with boundary condition (2.3). Taking into account the properties of analytic functions, from (2.2) we have that:

It is important to emphasize that the first terms (2.30) and (2.32) are determined by solving the problem of a concentrated force in an infinite region. This explains the presence of a logarithmic singularity. The second terms (2.30), (2.32) are determined by the absence of tangential stresses on the contour of the disk and hole, as well as by the condition of the analytical behavior of the corresponding terms of the complex potential at zero and at infinity. On the other hand, the superposition of (2.26) and (2.29) ((2.27) and (2.31)) gives a zero principal vector of forces acting on the contour of the hole (or disk). All this allows us to express through the third term the magnitude of radial displacements in an arbitrary fixed direction C, in the plate and in the disk. To do this, we find the difference between Фпд(г), (z) and Фп 2(2), 4V2(z):

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

The idea of the need to take into account the microstructure of the surface of compressible bodies belongs to I.Ya. Shtaerman. He introduced a model of a combined foundation, according to which in an elastic body, in addition to displacements caused by the action of normal pressure and determined by solving the corresponding problems of the theory of elasticity, additional normal displacements arise due to purely local deformations, depending on the microstructure of the contacting surfaces. I.Ya. Shtaerman suggested that additional movement is proportional to normal pressure, and the proportionality coefficient is a constant value for a given material. Within the framework of this approach, he was the first to obtain the equation of a plane contact problem for an elastic rough body, i.e. body having a layer of increased compliance.

A number of works suggest that additional normal displacements due to the deformation of microprotrusions of contacting bodies are proportional to the macrostress to some extent. This is based on equating the average displacements and stresses within the reference length of the surface roughness measurement. However, despite a fairly well-developed apparatus for solving problems of this class, a number of methodological difficulties have not been overcome. Thus, the hypothesis used about the power-law relationship between stresses and displacements of the surface layer, taking into account the real characteristics of microgeometry, is correct at small base lengths, i.e. high surface cleanliness, and, therefore, the validity of the hypothesis of topographic smoothness at the micro and macro level. It should also be noted that the equation becomes significantly more complicated when using this approach and the impossibility of describing the influence of waviness using it.

Despite a fairly well-developed apparatus for solving contact problems taking into account a layer of increased compliance, a number of methodological issues remain that complicate its use in engineering calculation practice. As already noted, surface roughness has a probabilistic distribution of heights. The commensurability of the dimensions of the surface element on which the roughness characteristics are determined with the dimensions of the contact area is the main difficulty in solving the problem and determines the incorrectness of some authors in using the direct connection between macropressures and roughness deformations in the form: where s is a surface point.

It should also be noted that the solution to the problem posed using the assumption of transforming the type of pressure distribution into parabolic, if the deformations of the elastic half-space in comparison with the deformations of the rough layer can be neglected. This approach leads to a significant complication of the integral equation and allows one to obtain only numerical results. In addition, the authors used the already mentioned hypothesis (3.1).

It is necessary to mention an attempt to develop an engineering method for taking into account the influence of roughness during internal contact of cylindrical bodies, based on the assumption that elastic radial movements in the contact area, caused by the deformation of micro-roughness, are constant and proportional to the average contact stress m to some extent k. However, Despite its obvious simplicity, the disadvantage of this approach is that with this method of taking roughness into account, its influence gradually increases with increasing load, which is not observed in practice (Figure 3 L).

1. MODERN PROBLEMS OF CONTACT MECHANICS

INTERACTIONS

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

1.2. The influence of creep of solids on their shape change in the contact area

1.3. Estimation of the convergence of rough surfaces

1.4. Analysis of contact interaction of multilayer structures

1.5. The relationship between mechanics and friction and wear problems

1.6. Features of the application 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 smooth isotropic disk and plate with a cylindrical cavity

2.1.1. General formulas

2.1.2. Derivation of boundary conditions 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. Reducing a singular integrodifferential equation to an integral equation with a kernel having a logarithmic singularity

2.1.3.1.2. Estimation of 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 movement

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

2.5. Modeling of spatial contact interaction of a system of coaxial cylinders of finite dimensions

2.5.1. Formulation of the problem

2.5.2. Solving auxiliary two-dimensional problems

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

3. CONTACT PROBLEMS FOR ROUGH BODIES AND THEIR SOLUTION USING CORRECTION OF THE CURVATURE OF THE DEFORMED SURFACE

3.1. Spatial nonlocal theory. Geometric assumptions

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

3.3. Method for analytical assessment of the influence of roughness deformation

3.4. Determination of movements in the contact area

3.5. Determination of auxiliary coefficients

3.6. Determining the dimensions of the elliptical contact area

3.7. Equations for determining the contact area close to circular

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

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

3.10. Peculiarities of averaging pressures and deformations when solving the two-dimensional problem of internal contact of rough cylinders of close radii 10

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

3.10.2. Determination of auxiliary coefficients ^ ^

3.10.3. Stressed fit of rough cylinders ^ ^ CONCLUSIONS AND MAIN RESULTS OF THE THIRD CHAPTER

4. SOLUTION OF CONTACT PROBLEMS OF VISCOELASTICITY FOR SMOOTH BODIES

4.1. Basic provisions

4.2. Compliance Principles Analysis

4.2.1. Volterra's 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 a uniformly aging isotropic plate

CONCLUSIONS AND MAIN RESULTS OF CHAPTER FOUR

5. SURFACE CREEP

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

5.2. Construction of a model of surface deformation 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 the rough layer and average pressures

5.2.4. Determination of auxiliary coefficients

5.2.5. Determining 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 in the form of a strip

5.3. Solution of a two-dimensional contact problem for internal contact of rough cylinders taking into account 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 TAKEN INTO ACCOUNT OF THE PRESENCE OF COATINGS

6.1. Calculation of effective moduli in the theory of composites

6.2. Construction of a self-consistent method for calculating 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 contour of a hole

6.3.1. Statement of the problem and basic formulas

6.3.2. Derivation of boundary conditions 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 influence of a viscoelastic aging coating on changes in contact parameters

6.6. Analysis of the features of contact interaction of a multicomponent coating and disk roughness

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

6.7.1. Contact between a plastic-coated sphere and a rough half-space

6.7.1.1. Basic hypotheses and model of interaction of solids

6.7.1.2. Approximate solution to the problem

6.7.1.3. Determination of maximum contact approach

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

6.7.3. Determination of contact stiffness for internal contact of cylinders

CONCLUSIONS AND MAIN RESULTS OF CHAPTER SIX

7. SOLUTION OF MIXED BOUNARY-VALUE PROBLEMS TAKEN INTO ACCOUNT FOR SURFACE WEAR

INTERACTING BODIES

7.1. Features of solving the contact problem taking into account surface wear

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

7.3. Method for theoretical assessment of wear taking into account surface creep

7.4. Wear assessment method taking into account the influence of coating

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

CONCLUSIONS AND MAIN RESULTS OF CHAPTER SEVEN

Recommended list of dissertations

On the contact interaction between thin-walled elements and viscoelastic bodies during torsion and axisymmetric deformation, taking into account the aging factor 1984, Candidate of Physical and Mathematical Sciences Davtyan, Zaven Azibekovich
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Introduction of the dissertation (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 field of research into the performance of machines and their elements. Increasing their reliability and durability is the most important factor determining the growth of competitiveness. In addition, extending the service life of machinery and equipment, even to a small extent with a high saturation of technology, is tantamount to introducing significant new production capacities.

The current state of the theory of machine operating processes, combined with extensive experimental technology for determining work loads and a high level of development of the applied theory of elasticity, with existing 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 breakdowns under normal conditions services. At the same time, the tendency to reduce the weight and size parameters of the latter with a simultaneous increase in their energy saturation forces us to reconsider the known approaches and assumptions when determining the stressed state of parts and requires the development of new calculation models, as well as the improvement of experimental research methods. Analysis and classification of failures of mechanical engineering products have shown that the main cause of failure under operating conditions is not breakdown, but wear and damage to their working surfaces.

Increased wear of parts in joints in some cases violates the tightness of the working space of the machine, in others it disrupts the normal lubrication regime, and in others it leads to a loss of kinematic accuracy of the mechanism. Wear and damage to surfaces reduce 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 emergency destruction.

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 on the basis of achievements in physics, chemistry, and metallurgy to lay the foundations of the doctrine of friction, wear and lubrication in joints.

Currently, 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, because For the transition from calculating the wear of materials to problems of structural wear resistance, contact problems of the mechanics of a deformable solid play a decisive role. Solutions of contact problems in the theory of elasticity for bodies with circular boundaries are of significant importance for engineering practice. They form the theoretical basis for the calculation of such machine elements as bearings, hinge joints, some types of gears, and interference joints.

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 cumbersomeness of the mathematical apparatus and the need to use powerful computing tools significantly hinder the use of existing theoretical developments in solving applied problems. Thus, one of the current 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 progress achieved, it is still difficult to obtain satisfactory results taking into account the local design features and microgeometry of interacting bodies.

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

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, characteristics of their wear resistance, and the possibility of obtaining an approximate solution with the smallest number of independent parameters.

Connection of work with major scientific programs and topics. The research was carried out in accordance with the following topics: “Develop a method for calculating contact stresses during elastic contact interaction of cylindrical bodies, not described by Hertz’s theory” (Ministry of Education of the Republic of Belarus, 1997, No. GR 19981103); “The influence of micro-irregularities of contacting surfaces on the distribution of contact stresses during the interaction of cylindrical bodies with similar radii” (Belarusian Republican Foundation for Basic Research, 1996, No. GR 19981496); “To 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 of 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 solid bodies and the presence of coatings on the stress state in the contact area, as well as the establishment on this basis of patterns of changes in contact stiffness and wear resistance of joints using the example of the interaction of bodies with circular boundaries.

To achieve this goal, the following problems need to be solved:

To develop a method for approximate solution of problems in the theory of elasticity and viscoelasticity about 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 contact interaction of bodies, taking into account the microgeometric, rheological characteristics of surfaces, as well as the presence of plastic coatings.

Justify an approach that allows you to correct the curvature of interacting surfaces due to roughness deformation.

To develop a method for 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 antifriction coatings.

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

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

As part of solving the problem posed:

A spatial nonlocal model of contact interaction of solids 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.

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

The main provisions of the dissertation submitted for defense:

An approximate solution to the problem of mechanics of a deformed solid about the contact interaction of a smooth cylinder and a cylindrical cavity in a plate, which describes the phenomenon under study with sufficient accuracy using a minimum number of independent parameters.

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

Modeling of the viscoelastic interaction between a disk and a cavity in a plate made of an aging material, the ease of 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 elastic modulus on the loading of joints.

Construction of a nonlocal model and determination of the influence of the roughness characteristics 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 antifriction coatings. On this basis, a methodology has been proposed that focuses mathematical and physical methods in the study of wear resistance, which makes it possible, instead of studying real friction units, to place the main emphasis on studying the phenomena occurring in the contact area.

Personal contribution of the applicant. All results submitted for defense were obtained by the author personally.

Approbation of the dissertation results. 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 have been published, including: 1 monograph, 19 articles in magazines and collections, including 15 articles under personal authorship. The total number of pages of published materials is 370.

Structure and scope of the dissertation. The dissertation consists of an introduction, seven chapters, a conclusion, a list of sources used and an appendix. The full 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 titles.

Similar dissertations in the specialty "Mechanics of deformable solids", 01.02.04 code VAK

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Numerical modeling of dynamic contact interaction of elastoplastic bodies 2001, Candidate of Physical and Mathematical Sciences Sadovskaya, Oksana Viktorovna
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Conclusion of the dissertation on the topic “Mechanics of deformable solids”, Kravchuk, Alexander Stepanovich

CONCLUSION

In the course of the research, a number of static and quasi-static problems in the mechanics of a deformable solid 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 parameters of machines, the use of new technological and design solutions, leads to the need to revise and clarify the approaches and assumptions used in determining the stress state , movements and wear in the mates. On the other hand, the cumbersomeness of the mathematical apparatus and the need to use powerful computing tools significantly hinder the use of existing theoretical developments in solving applied problems and define as one of the main directions of development of mechanics the obtaining of explicit approximate solutions to the problems posed, ensuring the simplicity of their numerical implementation.

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

3. For the first time, nonlocal boundary value problems of the theory of elasticity have been solved, taking into account the geometric and rheological characteristics of roughness, based on a method that makes it possible to correct the curvature of interacting surfaces. The absence of an assumption about the smallness of the geometric dimensions of the basic roughness measurement lengths compared to the dimensions of the contact area makes it possible to correctly pose and solve problems about the interaction of solids, taking into account the microgeometry of their surfaces at relatively small contact sizes, and also to proceed to the creation of multi-level models of roughness deformation.

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

5. A simulation of the viscoelastic interaction between a disk and a cavity in a plate made of an aging material was carried out, the ease of implementation of the results of which allows them to be used for a wide range of applied problems.

6. Solved 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. This makes it possible to evaluate the effect of composite antifriction coatings with a low elastic modulus.

7. A model was constructed 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 was 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 and the quality of their antifriction coatings. boundary value problems taking into account wear of surfaces, as well as the presence

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Bibliography

The influence of creep of solids on their shape change in the contact area

Integral equation and its solution

Relative approach of two parallel circles determined by roughness deformation

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

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