Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces kravchuk alexander stepanovich. Theory of contact interaction of deformable solid bodies with circular

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 application 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 relation:

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

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

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

3.1. Numerical analysis of the bearing assembly design. 35

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

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

Conclusions.. 54

References.. 57

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

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

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

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

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

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

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

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

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

Rice. 2.1. Contact diagram

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

, (2.1)

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

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

, (2.2)

, (2.3)

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

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

, (2.4)

. (2.5)

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

Table 2.1.

Material properties of contacting spheres

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

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

Table 2.2.

Contact area options

Rice. 2.2. Contact pad parameters:

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

On fig. 2.2. a comparison of the contact zone parameters for three contact pairs of sphere materials is presented. It can be seen that pure fluoroplastic has a lower value of maximum contact pressure compared to the other 2 materials, while 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 element type PLANE182 was used. This type is a four-nodal element and has a second order of approximation. The element is used for 2D modeling of bodies. Each element node has two degrees of freedom UX and UY. Also, this element is used to calculate problems: axisymmetric, with a flat deformed state and with a flat stressed state.

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

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

Rice. 2.8. Design scheme of spheres contact

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

equilibrium equations

geometric relationships

, (2.7)

physical ratios

, (2.8)

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

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

force acts on sphere 2

. (2.10)

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

– sliding with friction: for static friction

, , , , (2.8)

wherein , ,

– for sliding friction

, , , , , , (2.9)

wherein , ,

– detachment

, , (2.10)

- full grip

, , , , (2.11)

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

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

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

Table 2.6.

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

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

Rice. 2.9. Convergence of the numerical solution

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

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

Rice. 2.10. Comparison of analytical and numerical solutions

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

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

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

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

Table 2.7.

Material hardness

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

Results of free compression experiments

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

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

Rigidity of specimens under conditions of uniaxial deformed state

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

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

Table 2.11

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

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

Deformation and Stresses in Samples of Modified Fluoroplastic

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

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

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

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

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

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

Beginning at 13:00, room 1624.

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

The first main task in constructing a geometrically exact theory of contact interaction is to consider the conditions for the existence and uniqueness of a solution to the PBT problem. This leads to a number of theorems that allow us to construct both three-dimensional geometric domains of existence and uniqueness of the projection for each object (surface, curve, point) in the corresponding contact pair, and the transition mechanism between contact pairs. These areas are constructed by considering the differential geometry of the object, in the metric of the curvilinear coordinate system corresponding to it: in 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 of the optimization problem, including, for example, the "return-mapping algorithm" and the necessary derivatives, are also formulated in a curvilinear coordinate system. Here, an indicative result of a geometrically exact theory is both the ability to obtain new analytical solutions in a closed form (a generalization of the Euler problem of 1769 on the friction of a rope along a cylinder to the case of anisotropic friction on a surface of arbitrary geometry), and the ability to obtain in a compact form generalizations of the Coulomb friction law, which takes into account anisotropic geometric surface structure together with anisotropic micro-friction.

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

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Mechanics of contact interaction

Introduction

mechanics pin roughness elastic

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

1. Classical problems of contact mechanics

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

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

The force required for this is

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

2. Contact between two balls

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

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

with maximum pressure in the center

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

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

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

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

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

The relationship between penetration depth and normal force is given by

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

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

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

The pressure distribution is determined by the formula

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

6. Contact between two cylinders with parallel axes

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

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

as in the case of contact between two balls.

The maximum pressure is

7. Contact between rough surfaces

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

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

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

then the microroughnesses are completely in a plastic state.

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

2. Accounting for roughness

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

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

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

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

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

In this paper, we propose an approach based on fractal concepts of a rough surface as a geometric object with a fractional dimension.

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

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

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

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

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

Let's rewrite this expression as

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

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

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

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

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

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

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

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

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

Since the actual contact area is

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

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

Let us determine Smax from the known expression

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

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

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

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

Here, r is the radius of the roughness tip.

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

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

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

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

Then we find the total convergence from the following relation:

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

3. Experiment

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

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

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

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

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

Table 1

Roughness parameters

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

table 2

Approach of a spherical body to a rough surface

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

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

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

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

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

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

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

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

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

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

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

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

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

Bibliography

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

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

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

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

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

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

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

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

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

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

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