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. Theory of contact interaction of deformable solid bodies with circular boundaries, taking into account mechanical

1. MODERN PROBLEMS OF CONTACT MECHANICS

INTERACTIONS

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

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

1.3. Estimation of convergence of rough surfaces

1.4. Analysis of the contact interaction of multilayer structures

1.5. Relationship between mechanics and problems of friction and wear

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

2. CONTACT INTERACTION OF SMOOTH CYLINDRICAL BODIES

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

2.1.1. General formulas

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

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

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

2.1.3.1.2. Estimating the Norm of a Linear Operator

2.1.3.2. Approximate solution of the equation

2.2. Calculation of a fixed connection of smooth cylindrical bodies

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

2.3.1. Solution of an auxiliary problem for an elastic plane

2.3.2. Solution of an auxiliary problem for an elastic disk

2.3.3. Determination of maximum normal radial displacement

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

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

2.5.1. Formulation of the problem

2.5.2. Solution of auxiliary two-dimensional problems

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

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

3.1. Spatial non-local theory. geometric assumptions

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

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

3.4. Definition of displacements in the area of contact

3.5. Definition of auxiliary coefficients

3.6. Determination of the dimensions of the elliptical contact area

3.7. Equations for determining the contact area close to circular

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

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

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

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

3.10.2. Definition of auxiliary coefficients ^ ^

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

4. SOLUTION OF CONTACT PROBLEMS OF VISCOELASTICITY FOR SMOOTH BODIES

4.1. Basic provisions

4.2. Compliance principles analysis

4.2.1. Volterra principle

4.2.2. Constant coefficient of transverse expansion under creep deformation

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

4.3.1. General case of viscoelasticity operators

4.3.2. Solution for a monotonically increasing contact area

4.3.3. Fixed connection solution

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

CONCLUSIONS AND MAIN RESULTS OF THE FOURTH CHAPTER

5. SURFACE CREEP

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

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

5.2.1. geometric assumptions

5.2.2. Surface Creep Model

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

5.2.4. Definition of auxiliary coefficients

5.2.5. Determination of the dimensions of the elliptical contact area

5.2.6. Determining the dimensions of the circular contact area

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

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

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

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

6. MECHANICS OF INTERACTION OF CYLINDRICAL BODIES WITH COVERINGS

6.1. Calculation of effective modules in the theory of composites

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

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

6.3.1. Statement of the problem and basic formulas

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

6.3.3. Integral equation and its solution

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

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

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

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

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

6.7.1.1. Main hypotheses and model of interaction of rigid bodies

6.7.1.2. Approximate solution of the problem

6.7.1.3. Determination of the maximum contact approach

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

6.7.3. Determination of contact stiffness at internal contact of cylinders

CONCLUSIONS AND MAIN RESULTS OF CHAPTER SIX

7. SOLUTION OF MIXED BOUNDARY PROBLEM WITH SURFACE WEAR INCLUDED

OF INTERACTING BODIES

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

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

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

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

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

CONCLUSIONS AND MAIN RESULTS OF THE SEVENTH CHAPTER

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

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

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

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

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

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

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

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

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

Connection of work with major scientific programs, topics. The studies were carried out in accordance with the following topics: "To 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. GR 20001251)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

As part of solving the problem:

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

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

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

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

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

The main provisions of the dissertation submitted for defense:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

CONCLUSION

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

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

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

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

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

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

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

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

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

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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.

Story

The classical mechanics of contact interactions is associated primarily with the name of Heinrich Hertz. In 1882, Hertz solved the problem of the contact of two elastic bodies with curved surfaces. This classical result still underlies the mechanics of contact interaction today. Only a century later, Johnson, Kendal and Roberts found a similar solution for adhesive contact (JKR - theory).

Further progress in the mechanics of contact interaction in the 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 the fact that the actual area of contact between rubbing bodies is much less than the apparent area of contact. 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 contact between rough surfaces were made by Greenwood and Williamson (1966) and Persson (2002). The main result of these works is the proof that the actual contact area of rough surfaces 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.

Classical problems of contact interaction mechanics

Contact between a ball and an elastic half-space

A solid ball of radius is pressed into the elastic half-space to a depth (penetration depth), forming a contact area of radius .

The force required for this is

And here the moduli of elasticity, and and - Poisson's ratios of both bodies.

Contact between two balls

When two balls with radii and are in contact, these equations are valid, respectively, for the radius

The pressure distribution in the contact area is calculated as

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

Contact between two crossed cylinders with the same radii

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

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

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

Contact between a cone 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:

There is an 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

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

Contact between rough surfaces

When two bodies with rough surfaces interact with each other, then the real contact area is much smaller than the apparent area. 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 and is determined by the following equation:

In this case - the root mean square value of the roughness of the plane and . Average pressure in real contact area

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

then the microroughnesses are completely in a plastic state. For the surface upon contact is deformed only elastically. The value was introduced by Greenwood and Williamson and is called the index of plasticity. The fact of deformation of a body, elastic or plastic, does not depend on the applied normal force.

Literature

K. L. Johnson: contact mechanics. Cambridge University Press, 6. Nachdruck der 1. Auflage, 2001.
Popov, Valentin L.: Kontaktmechanik und Reibung. Ein Lehr- und Anwendungsbuch von der Nanotribologie bis zur numerischen Simulation, Springer-Verlag, 2009, 328 S., ISBN 978-3-540-88836-9 .
Popov, Valentin L.: Contact Mechanics and Friction. Physical Principles and Applications, Springer-Verlag, 2010, 362 p., ISBN 978-3-642-10802-0 .
I. N. Sneddon: The Relationship between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile. Int. J. Eng. Sc., 1965, v. 3, pp. 47–57.
S. Hyun, M. O. Robbins: Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths. Trobology International, 2007, v.40, pp. 1413–1422.

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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. Basic provisions

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 progress made, 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 last 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: "To 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 about 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.

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

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.

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.

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.

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.

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 in 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. Yes, in the building crystal lattice various local defects may occur. 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) was experimentally obtained. In a significant stress range, experimental points on a logarithmic grid are usually grouped near a certain straight line. This means that in the considered stress range (- 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. In a number of works, 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 given 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 view of the solution is constructed, which in the first case includes common decision 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. At the same time, 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 single out 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 linear equations; the required size of the system of linear equations to achieve the required accuracy of the solution; the use of a numerical method for solving a system of linear equations, which takes into account the features of its structure as much as possible and, accordingly, allows 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 separate 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 σi(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 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 allows one 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 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 effect 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).

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

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

Beginning at 13:00, room 1624.

annotation

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

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

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

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

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. Study 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 their 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 models contacts 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 bodies and through various layers and coatings. Due to the fact that the mechanisms of machines and their elements are often geometrically complex structures operating within the framework of the mechanics of contact interaction, the study of their behavior and deformation characteristics is topical issue 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 contact pressure distribution. 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 methods solutions, including numerical ones. 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, slipping with different type friction and slip. 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 deformation conditions 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 computing 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 contact pair coefficient, 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 modified PTFE, while pure PTFE at significantly lower contact pressures has a larger contact area radius than the other two materials.

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

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

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

c) Steel-Modified PTFE

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

Table 2.5.

Contact options when zoomed in

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

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

Rice. 2.6. Dependencies of contact parameters on

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

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

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

Similarly 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 PLANE182 element type was used. This type is a four-nodal element and has a second order of approximation. The element is used for 2D modeling of bodies. Each element node has two degrees of freedom UX and UY. Also, this element is used to calculate problems: axisymmetric, with a flat deformed state and with a flat stressed state.

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

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

Rice. 2.8. Design scheme of spheres contact

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

equilibrium equations

geometric relationships

, (2.7)

physical ratios

, (2.8)

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

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

force acts on sphere 2

. (2.10)

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

– sliding with friction: for static friction

, , , , (2.8)

wherein , ,

– for sliding friction

, , , , , , (2.9)

wherein , ,

– detachment

, , (2.10)

- full grip

, , , , (2.11)

where is the coefficient of friction; is the 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 contact parameters. Below is table 2.6. which presents the characteristics of finite element models involved in the assessment of the convergence of the numerical solution of the partitioning option.

Table 2.6.

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

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

Rice. 2.9. Convergence of the numerical solution

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

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

Rice. 2.10. Comparison of analytical and numerical solutions

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

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

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

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

Table 2.7.

Material hardness

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

Results of free compression experiments

Studies under conditions of constrained compression (Fig. 2.11., c) were carried out by pressing cylindrical samples with a diameter of 20 mm, a height of about 20 mm in a special device with a rigid steel cage at permissible limiting pressures of 100-160 MPa. In the manual 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 held 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 also obtained mechanical characteristics materials: 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 a real design of a spherical bearing.

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. Theory of contact interaction of deformable solid bodies with circular boundaries, taking into account mechanical

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