Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces kravchuk alexander stepanovich. Analysis of scientific publications within the framework of the mechanics of contact interaction T

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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, for example, wheel-rail, in the calculation of clutches, brakes, tires, plain and rolling bearings, gears. gear wheels, hinges, seals; electrical contacts, etc. It covers a wide range of tasks, ranging from strength calculations of tribosystem interface elements, taking into account the lubricating medium and material structure, to application in micro- and nanosystems.

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

1. Classical problems of contact mechanics

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

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

The force required for this is

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

2. Contact between two balls

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

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

with maximum pressure in the center

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

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

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

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

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

The relationship between penetration depth and normal force is given by

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

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

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

The pressure distribution is determined by the formula

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

6. Contact between two cylinders with parallel axes

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

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

as in the case of contact between two balls.

The maximum pressure is

7. Contact between rough surfaces

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

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

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

then the microroughnesses are completely in a plastic state.

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

2. Accounting for roughness

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

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

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

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

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

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

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

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

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

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

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

Let's rewrite this expression as

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

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

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

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

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

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

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

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

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

Since the actual contact area is

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

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

Let us determine Smax from the known expression

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

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

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

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

Here, r is the radius of the roughness tip.

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

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

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

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

Then we find the total convergence from the following relation:

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

3. Experiment

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

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

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

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

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

Table 1

Roughness parameters

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

table 2

Approach of a spherical body to a rough surface

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

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

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

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

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

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

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

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

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

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

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

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

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

Bibliography

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

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

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

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

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

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

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

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

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

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

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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 areas: complication physical model deformation of solid bodies and (or) rejection of hypotheses of smoothness and homogeneity 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 steady-state creep rates on stresses at relatively high stresses (-10" and more on the modulus of elasticity) has been experimentally obtained. In a significant range of stresses, the experimental points on a logarithmic grid are usually grouped around a certain straight line. This means that in the stress range under consideration (- 10 "-10" from the 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 a solution to the problem in an 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 of the theory of elasticity, the most common restrictions are the requirements for the solution to vanish at the ends of the contact area and the assumption that stresses and rotations disappear at infinity. In the case when the integration area makes up the entire boundary of the area (body), then the uniqueness of the solution is guaranteed by the Cauchy formulas. Moreover, the simplest and most common method for solving applied problems in this case is the representation of the Cauchy integral in the form of a series.

It should be noted that in the above general information From the theory of singular integral equations, the properties of the contours of the studied areas are not stipulated in any way, since in this case it is known that the arc of the circle (the curve along which the integration is performed) satisfies the Lyapunov condition. A generalization of the theory of two-dimensional boundary value problems in the case of more general assumptions on the smoothness of the domain boundary can be found in the AI monograph. Danilyuk.

Of greatest interest is the general case of the equation when 7i 0. The absence of methods for constructing an exact solution in this case leads to the need to apply the methods of numerical analysis and approximation theory. In fact, as already noted, numerical methods for solving integral equations are usually based on approximating the solution of an equation by a functional of a certain type. The amount of accumulated results in this area makes it possible to 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 important role only in the case of systems of high-order linear equations. All this determines the effectiveness of the approach used. However, it should be noted that, to date, there are only a few studies on comparative analysis and possible simplifications in solving practical problems using various approximations.

Note that the integro-differential equation can be reduced to the following form: V is an arc of a circle of unit radius enclosed between two points with angular coordinates -cc0 and a0, a0 є(0,l/2); y1 is a real coefficient determined by the elastic characteristics of the interacting bodies (2.6); f(t) is a known function determined by the applied loads (2.6). In addition, we recall that ar(m) vanishes at the ends of the integration interval.

Relative convergence of two parallel circles determined by roughness deformation

The problem of internal compression of circular cylinders of close radii was first considered by I.Ya. Shtaerman. When solving the problem posed by him, it was assumed that the external load acting on the inner and outer cylinders along their surfaces is carried out in the form of 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 the constants, it is possible to determine the sum of the radial displacements of the diametrically opposite points of the contours of the hole and the shaft.

Thus, despite the relevance of the problem of determining the contact stiffness, the analysis of literary sources did not allow us to identify a method for solving it, which makes it possible to reasonably establish the magnitudes of the largest normal contact displacements due to deformation surface layers without taking into account the deformations of interacting bodies as a whole, which is explained by the lack of a formalized definition of the concept of "contact stiffness".

When solving the problem, we will proceed from the following definitions: displacements under the action of the main vector of forces (without taking into account the features of contact interaction) will be called the approach (removal) of the center of the disk (hole) and its surface, which does not lead to a change in the shape of its boundary. Those. 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), (2.32) are determined by the absence of tangential stresses on the contour of the disk and hole, 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 model of a combined base, according to which, in an elastic body, in addition to displacements caused by the action of normal pressure and determined by the solution of the corresponding problems of the theory of elasticity, additional normal displacements arise due to purely local deformations that depend on the microstructure of the contacting surfaces. I.Ya.Shtaerman suggested that the additional displacement is proportional to the normal pressure, and the coefficient of proportionality is a constant value for a given material. Within the framework of this approach, he was the first to obtain the equation of a plane contact problem for an elastic rough body, i.e. body having a layer of increased compliance.

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

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

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

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

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 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 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"

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.

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 rough surface contact were made by Greenwood and Williamson (1966) and Persson (2002). The main result of these works is the proof that the actual contact area of rough surfaces in a rough approximation is proportional to the normal force, while the characteristics of an individual microcontact (pressure, microcontact size) weakly depend on the load.

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

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

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

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

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

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

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

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

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

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

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

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

C. L. Johnson, C. Kendal, and A. D. Roberts (JKR - by the first letters of their surnames) took this theory as the basis for calculating the theoretical shear or depth of indentation in the presence of adhesion in their landmark paper "Surface energy and contact of elastic solid particles ”, published in 1971 in the proceedings of the Royal Society. Hertz's theory follows from their formulation, provided that the adhesion of materials is zero.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The main provisions of the dissertation submitted for defense:

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

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

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

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

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

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

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

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

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

334 Conclusions:

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

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

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

CONCLUSION

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces kravchuk alexander stepanovich. Analysis of scientific publications within the framework of the mechanics of contact interaction T

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

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

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

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