Theory of contact interaction. Analysis of scientific publications in the framework of the mechanics of contact interaction. Approximate solution of the two-dimensional contact problem of linear creep for smooth cylindrical bodies

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

Mechanics of contact interaction deals with the calculation of elastic, viscoelastic and plastic bodies in static or dynamic contact. The mechanics of contact interaction is a fundamental engineering discipline, mandatory in the design of reliable and energy-saving equipment. It will be useful in solving many contact problems, for example, wheel-rail, in the calculation of clutches, brakes, tires, plain and rolling bearings, internal combustion engines, joints, seals; in stamping, metalworking, ultrasonic welding, 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 applications in micro- and nanosystems.

classical mechanics contact interactions 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 middle of the 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 the bodies in contact. Roughness leads to real area contact between rubbing bodies is much smaller than the apparent contact area. These ideas have 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 the contact interaction of rough surfaces.

Pioneer work in this area is the work of Archard (1957), who came to the conclusion that when elastic rough surfaces are in contact, the contact area is approximately proportional to the normal force. Further important contributions to the theory of rough surface contact 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 in a rough approximation is proportional to the normal force, while the characteristics of an individual microcontact (pressure, microcontact size) weakly depend on the load.

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

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 the contact radius are related by the following relationship:

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

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

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 the bodies touch, an adhesive neck appears as a result of the action of van der Waals forces. In order for the bodies to break again, it is necessary to apply a certain minimum force, called the adhesion force. Similar phenomena take place in the contact of two solid bodies separated by a very soft layer, such as in a sticker or plaster. Adhesion can be both of technological interest, for example, in adhesive bonding, and be an interfering factor, for example, preventing the rapid opening of elastomeric valves.

Adhesion force between parabolic solid and an elastic half-space was first found in 1971 by Johnson, Kendall and Roberts. She is equal

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

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

C. L. Johnson, C. Kendal, and A. D. Roberts (JKR - by the first letters of their surnames) took this theory as the basis for calculating the theoretical shear or depth of indentation in the presence of adhesion in their landmark paper "Surface energy and contact of elastic solid particles ”, 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 is known among researchers as the DMT theory, and from which Hertz’s formulation follows under zero adhesion.

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

Both theories, both DMT and JKR, are the basis of contact interaction mechanics, on which all contact transition models are based, and which are used in calculations of nanoshifts and electron microscopy. Thus, Hertz's research in his days 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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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 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 the equalization of pressure on the contact area and the increase in 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. -WITH. 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. - No. 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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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

ThesisHelp writingFind out the cost my work

However, the modern theory of elastic contact does not allow to sufficiently search for a rational geometric shape of the contacting 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 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. Systematic analysis of the current state and trends in the field of improving the elastic contact of bodies of complex shape
  - 1. 1. 1. The current state of the 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 rotation
- 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 performance 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 indicators of the process of shaping superfinishing depending on the processing mode
- 5. 4. General mathematical model of the process of shaping superfinishing
- 5. 5. The 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

The cost of a unique work

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.

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. Form-forming superfinishing methods developed by the Saratov scientific school are very effective, but their practical application is designed only for processing outer surfaces such as raceways of roller bearing inner rings, 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 most important directions for further improving 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 a mechanism 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 engineering.

The aim of the work is to develop an applied theory of local contact interaction of elastic bodies and 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 methods of experiment planning, experimental data processing, 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. The paper developed an applied theory of local contact interaction of elastic bodies and created on its basis the processes of shaping friction-rolling bearings with rational geometry, opening 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 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 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 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 rolling tracks 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 area 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 operational 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 shaping 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 constructed, 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 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, is proposed, 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.

The cost of a unique work

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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, Aleksandrov 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 machine mechanisms 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, 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 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 the 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 the radius of the contact zone 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 the modified fluoroplast, while for pure fluoroplast at significantly lower contact pressures, the radius of the contact area is larger than for 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 conditions of interaction 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 magnitude 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 parameters of the contact parameters of the contact. 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 deformation histories 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 control mode of the machine, 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.

480 rub. | 150 UAH | $7.5 ", MOUSEOFF, FGCOLOR, "#FFFFCC",BGCOLOR, "#393939");" onMouseOut="return nd();"> Thesis - 480 rubles, shipping 10 minutes 24 hours a day, 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. ... Dr. Phys.-Math. Sciences: 01.02.04: Cheboksary, 2004 275 p. RSL OD, 71:05-1/66

Introduction

1. Modern problems of contact interaction mechanics 17

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

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

1.3. Estimation of convergence of rough surfaces 20

1.4. Analysis of the contact interaction of multilayer structures 27

1.5. Relationship between mechanics and problems of friction and wear 30

1.6. Features of the use 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 a smooth isotropic disk and a plate with a cylindrical cavity 37

2.1.1. General Formulas 38

2.1.2. Derivation of the boundary condition for displacements in the contact area 39

2.1.3. Integral equation and its solution 42

2.1.3.1. Investigation of the resulting equation 4 5

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

2.1.3.1.2. Estimating 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 at internal contact of cylinders of close radii 68

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

2.5.1. Problem Statement 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 correcting the curvature of a deformed surface 80

3.1. Spatial non-local theory. Geometric assumptions 83

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

3.3. Method for Analytical Evaluation of the Influence of Roughness Deformation 88

3.4. Determination of displacements in the area of contact 89

3.5. Definition of auxiliary coefficients 91

3.6. Determination of 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 area of contact close to the line 102

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

3.10. Peculiarities of Averaging Pressures and Strains in Solving the Two-Dimensional Problem of Internal Contact of Rough Cylinders with 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. Definition of auxiliary coefficients

Conclusions and main results of the third chapter

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 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 case

uniformly aging isotropic plate 130

Conclusions and main results of the fourth chapter 135

5. Surface creep 136

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

5.2. Construction of a surface deformation model 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 deformations of the rough layer and average pressures 144

5.2.4. Definition of auxiliary coefficients 146

5.2.5. Determining the dimensions of the elliptical contact area 149

5.2.6. Determination of the dimensions of the circular contact area 152

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

5.3. Solution of the 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. Definition 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 modules in the theory of composites 169

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 173

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

6.3. 1 Statement of the problem and basic formulas 179

6.3.2. Derivation of the boundary condition for displacements 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. Determination of the effect of viscoelastic aging coating on the change in contact parameters 191

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

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

6.7.1. Contact of a plastic-coated ball 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 of problem 200

6.7.1.3. Determination of the maximum contact approach 204

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

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

Conclusions and main results of the sixth chapter 217

7. Solution of Mixed Boundary Value Problems Taking into Account the Wear of the Surfaces of Interacting Bodies 218

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

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

7.3. The method of theoretical wear assessment taking into account surface creep 229

7.4. Coating influence wear method 233

7.5. Concluding remarks on the formulation of plane problems with allowance for wear 237

Conclusions and main results of the seventh chapter 241

Conclusion 242

List of sources used

Introduction to work

The relevance of the dissertation topic. At present, 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 the calculation of wear of materials to problems of structural wear resistance.

It should be noted that the most extensive studies of the contact interaction were 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 individual sites that form the actual contact area. Therefore, when modeling the approach of surfaces, it is necessary to use parameters that characterize the real surface.

The cumbersomeness of the mathematical apparatus used in solving contact problems for rough bodies, the need to use powerful computing tools significantly hinder the use of existing theoretical developments in solving applied problems. And, despite the successes 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 solids are established is commensurate with the contact area.

All this requires the development of a unified approach to solving contact problems, which most fully takes into account both the geometry of interacting bodies, microgeometric and rheological characteristics of surfaces, their wear resistance characteristics, and the possibility of obtaining an approximate solution of 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, swivel joints, interference joints. Therefore, these tasks are usually chosen as model ones when conducting such studies.

Intensive work carried out in recent years in Belarusian National Technical University

to solve this problem and form the basis of nastdzddodood^y.

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); "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 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. 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 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:

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

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

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

Develop a method for the 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 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 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 spatial non-local model of the contact
interactions 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 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 the Research and Development Center “Cycloprivod”, NGOs Altech.

The main provisions of the dissertation submitted for defense:

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

Solution of non-local boundary value problems of the 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 the method for calculating the displacements of the boundary of cylindrical bodies due to the deformation of the superficial layers. The results obtained allow us to develop a theoretical approach,

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

Modeling of the viscoelastic interaction between the disk and the cavity in
plate of aging material, ease of implementation of the 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 to the loading of mates.

Construction of a non-local model and determination of the influence of the surface 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 anti-friction 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 focus on the study of phenomena occurring V contact areas.

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), 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).

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 full volume of the dissertation is 2-M "pages, including the volume occupied by illustrations - 14 pages, tables - 1 page. The number of sources used includes 310 titles.

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

Practical obtaining of analytical dependences 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 depend weakly on the configuration of the bodies far from the contact area, as well as on the method of their fixing. In this case, stresses with a fairly good degree of reliability can be calculated by considering each body as an infinite elastic medium bounded by a flat surface, i.e. as an elastic half-space.

The surface of each of the bodies is assumed to be topographically smooth at the micro- and macrolevels. At the micro level, this means the absence or neglect of microroughnesses of the contacting surfaces, which would cause an incomplete fit of the contact surfaces. Therefore, the real contact area, which is formed at the tops of the protrusions, is much smaller than the theoretical one. At the macro level, the surface profiles are considered continuous in the contact zone, together with the 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 over the contact surface, but are not applicable, in particular, to low-modulus materials. In addition, the conditions under which the Hertz 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 grow rapidly 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 in solving contact problems is the consideration of friction forces. At the same time, the latter on the interface between two bodies of a consistent shape, which 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 hypotheses listed above. It was carried out in the following main directions: the complication of the physical model of deformation of solids and (or) the rejection of the hypotheses of smoothness and uniformity of their surfaces.

Interest in creep has increased dramatically in connection with the development of technology. Among the first researchers who discovered the phenomenon of deformation of materials in time under constant load were Vika, 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 later by Volterra, is now a classical branch of the theory of integral equations.

Until the middle of the last century, elements of the theory of deformation of materials in time found little use in the practice of calculating engineering structures. However, with the development of power plants, chemical-technological apparatuses operating at higher temperatures and pressures, it became necessary to take into account the phenomenon of creep. The demands of mechanical engineering led to a huge scope of experimental and theoretical research in the field of creep. Due to the 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 in the contact interaction of solids is important for a number of applied and fundamental reasons. So, even under constant loads, the shape of the interacting bodies and their stress state, as a rule, change, 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 ideas of the theory of dislocations. 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 sliding in the metal. The result of dislocation motion is a shift by one interatomic distance. The stressed state of the body facilitates the movement of dislocations, reducing potential barriers.

The time laws of creep depend on the structure of the material, which changes with the course of creep. An exponential dependence of the steady-state creep rates on stresses at relatively high stresses (-10" and more on the elastic modulus) has been experimentally obtained. In a significant stress range, the experimental points on a logarithmic grid are usually grouped near a certain straight line. This means that in the considered stress interval (- 10 "-10" from the modulus of elasticity) there is a power-law dependence of strain rates on stress. It should be noted that at low stresses (10" or less on the modulus of elasticity), this dependence is linear. A number of works present 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=0 and this equation becomes an integral equation of the first kind. The features of the theory of analytic functions make it possible in this case, using additional conditions, to obtain a unique solution . These are the so-called inversion formulas for singular integral equations, which make it possible to obtain the solution of the problem in explicit form. 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 of the ends 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 the solution is constructed, which in the first case includes the general solution of the homogeneous equation. Given the behavior of the solution at infinity and the corner points of the contact area, based on physically justified assumptions, a unique solution is constructed that satisfies the indicated restrictions.

Thus, the uniqueness of the solution of this problem is understood in the sense of the accepted restrictions. It should be noted that when solving contact problems in the theory of elasticity, the most common restrictions are the requirement that the solution vanishes at the ends of the contact area and the assumption that stresses and rotations disappear at infinity. In the case when the integration area makes up 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 the representation of the Cauchy integral in the form of a series.

It should be noted that in the above general information from the theory of singular integral equations, the properties of the contours of the studied areas are not stipulated in any way, since in this case, it is known that the arc of the 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 domain boundary can be found in the AI monograph. Danilyuk.

Of greatest interest is the general case of the equation when 7i 0. The absence of methods for constructing an exact solution in this case leads to the need to apply the 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 of an equation by a functional of a certain type. The amount of accumulated results in this area makes it possible to identify the main criteria by which these methods are usually compared when they are used in applied problems. First of all, the simplicity of the physical analogy of the proposed approach (usually, in one form or another, this is the method of superposition of a system of certain solutions); the amount 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 as much as possible the features of its structure and, accordingly, allows obtaining a numerical result with the greatest speed. It should be noted that the last criterion plays an essential role only in the case of systems of high-order linear equations. All this determines the effectiveness of the approach used. At the same time, it should be stated that, to date, there are only a few 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 following form: V is an arc of a circle of unit radius enclosed between two points with angular coordinates -cc0 and a0, a0 є(0,l/2); y1 is a real coefficient determined by the elastic characteristics of the interacting bodies (2.6); f(t) is a known function determined by the applied loads (2.6). In addition, we recall that ar(m) vanishes at the ends of the integration interval.

Relative convergence 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 posed by him, it was assumed that the external load acting on the inner and outer cylinders along their surfaces is carried out in the form of a normal pressure diametrically opposite to the contact pressure. When deriving the equation of the problem, the decision on the compression of the cylinder by two opposite forces and the solution of a similar problem for the exterior of a circular hole in an elastic medium were used. He obtained an explicit expression for the displacement of the points of the contour of the cylinder and the hole through the integral operator of the stress function. This expression has been used by a number of authors to estimate the contact stiffness.

Using a heuristic approximation for the distribution of contact stresses for the I.Ya. Shtaerman, A.B. Milov obtained a simplified dependence for maximum contact displacements. However, he found that the obtained theoretical estimate differs 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 the coefficient of transition from a three-dimensional problem to a plane one is proposed.

A similar approach was used by M.I. Warm, asking for an approximate solution of a slightly different kind. It should be noted that in this work, in addition, a second-order linear differential equation was obtained to determine the contact displacements in the case of the circuit shown in Figure 2.1. This equation follows directly from the method of obtaining an integro-differential equation for determining normal radial stresses. In this case, the complexity of the right-hand side determines the awkwardness 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 constants, it is possible to determine the sum of radial displacements of diametrically opposite points of the contours of the hole and the shaft.

Thus, despite the relevance of the problem of determining the contact stiffness, the analysis of literary sources did not allow us to identify a method for solving it, which makes it possible to reasonably establish the magnitude of the largest normal contact displacements due to the deformation of the surface layers without taking into account the deformations of the 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, we will proceed from the following definitions: displacements under the action of the main vector of forces (without taking into account the features of the 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. 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 of concepts allows us to separate the displacements obtained from the solution of the problem of the theory of elasticity, and shows that the estimate of the contact stiffness of cylindrical bodies obtained by A.B. Milovsh from IL's solution. Shtaerman is true only for the given loading scheme.

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 the solution of the problem of a concentrated force in an infinite region. This explains the presence of a logarithmic singularity. The second terms (2.30) and (2.32) are determined by the absence of tangential stresses on the disk and hole contours, and also by the condition of the analytic 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 main vector of forces acting on the hole (or disk) contour. All this makes it possible to express in terms of 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 the combined base model, according to which, in an elastic body, in addition to displacements caused by the action of normal pressure and determined by the solution of the corresponding problems of the theory of elasticity, additional normal displacements arise due to purely local deformations that depend on the microstructure of the contacting surfaces. I.Ya.Shtaerman suggested that the additional displacement is proportional to the normal pressure, and the coefficient of proportionality 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.

In a number of works, it is assumed that additional normal displacements due to the deformation of the microprotrusions of the contacting bodies are proportional to the macrostress to some extent . This is based on equating the average displacements and stresses within the basic length of the surface roughness measurement. However, despite the rather 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 the microgeometry, is correct for small base lengths, i.e. high surface cleanliness, and, consequently, with the validity of the hypothesis of topographic smoothness at the micro and macro levels. It should also be noted that the equation becomes much more complicated when using such an approach and the impossibility of describing the effect of waviness with its help.

Despite the well-developed apparatus for solving contact problems, taking into account the layer of increased compliance, there are still a number of methodological issues that make it difficult to use in engineering practice of calculations. As already noted, the 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 the use by some authors of the direct relationship between macropressures and roughness deformations in the form: where s is the surface point.

It should also be noted that the problem is solved using the assumption about the transformation of 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 obtaining 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 the elastic radial displacements in the contact area, due to the deformation of micro-roughness, are constant and proportional to the average contact stress t to some extent k. However, despite its obvious simplicity, the disadvantage of this approach is that with this method of accounting for roughness, its influence gradually increases with increasing load, which is not observed in practice (Figure 3A).

AUTOMIR

Theory of contact interaction. Analysis of scientific publications in the framework of the mechanics of contact interaction. Approximate solution of the two-dimensional contact problem of linear creep for smooth cylindrical bodies

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Bibliography

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

Integral equation and its solution

Relative convergence of two parallel circles determined by roughness deformation

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