Theory of contact interaction. Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces kravchuk alexander stepanovich. Relationship of work with major scientific programs

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.

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 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 relation:

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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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 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 of 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 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. - No. 3. - S. 231-237.

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

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

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

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

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

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

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

The surface of each of the bodies is assumed to be topographically smooth at the micro- and macrolevels. At the micro level, this means the absence or neglect of microroughnesses of the contacting surfaces, which would cause an incomplete fit of the contact surfaces. Therefore, the real contact area, which is formed at the tops of the protrusions, is much smaller than the theoretical one. At the macro level, the surface profiles are considered continuous in the contact zone, together with the second derivatives.

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

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

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

Interest in creep has increased dramatically in connection with the development of technology. Among the first researchers who discovered the phenomenon of deformation of materials in time under constant load were Vika, Weber, Kohlrausch. Maxwell first presented the law of deformation in time in the form of a differential equation. Somewhat later, Bolygman created a general apparatus for describing the phenomena of linear creep. This apparatus, significantly developed later by Volterra, is now a classical branch of the theory of integral equations.

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

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

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

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

Integral equation and its solution

Note that if the elastic constants of the disk and plate are equal, then yx=0 and this equation becomes an integral equation of the first kind. The features of the theory of analytic functions allow in this case, using additional terms, get the only solution . These are the so-called inversion formulas for singular integral equations, which make it possible to obtain the solution of the problem in explicit form. The peculiarity is that in the theory of boundary value problems three cases are usually considered (when V is part of the boundary of the bodies): the solution has a singularity at both ends of the integration domain; the solution has a singularity at one of the ends of the integration domain, and vanishes at the other; the solution vanishes at both ends. Depending on the choice of one or another option, a general form of the solution is constructed, which in the first case includes the general solution of the homogeneous equation. Given the behavior of the solution at infinity and the corner points of the contact area, based on physically justified assumptions, a unique solution is constructed that satisfies the indicated restrictions.

Thus, the uniqueness of the solution of this problem is understood in the sense of the accepted restrictions. It should be noted that when solving contact problems 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 of linear equations; the required size of the system of linear equations to achieve the required accuracy of the solution; the use of a numerical method for solving a system of linear equations, which takes into account the features of its structure as much as possible and, accordingly, allows obtaining a numerical result with the greatest speed. It should be noted that the last criterion plays an essential role only in the case of systems of high-order linear equations. All this determines the effectiveness of the approach used. At the same time, it should be stated that, to date, there are only a few studies devoted to comparative analysis and possible simplifications in solving practical problems using various approximations.

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

Relative convergence of two parallel circles determined by roughness deformation

The problem of internal compression of circular cylinders of close radii was first considered by I.Ya. Shtaerman. When solving the problem posed by him, it was assumed that the external load acting on the inner and outer cylinders along their surfaces is carried out in the form of 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 scheme shown in Figure 2.1. This equation follows directly from the method of obtaining an integro-differential equation for determining normal radial stresses. In this case, the complexity of the right-hand side determines the awkwardness of the resulting expression for displacements. In addition, in this case, the values of the coefficients in the solution of the corresponding homogeneous equation remain unknown. At the same time, it is noted that, without setting the values of constants, it is possible to determine the sum of radial displacements of diametrically opposite points of the contours of the hole and the shaft.

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

When solving the problem, we will proceed from the following definitions: displacements under the action of the main vector of forces (without taking into account the features of 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 effect of roughness during internal contact of cylindrical bodies, based on the assumption that the elastic radial displacements in the contact area, due to the deformation of micro-roughness, are constant and proportional to the average contact stress t to some extent k. However, despite its obvious simplicity, the disadvantage of this approach is that with this method of accounting for roughness, its influence gradually increases with increasing load, which is not observed in practice (Figure 3A).

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

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

Beginning at 13:00, room 1624.

annotation

The main tactic of isogeometric analysis is the direct embedding of mechanics models in a complete description of a geometric object in order to formulate an efficient computational strategy. Such advantages of isogeometric analysis as a complete description of the geometry of an object 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"

1. MODERN PROBLEMS OF CONTACT MECHANICS

INTERACTIONS

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

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

1.3. Estimation of convergence of rough surfaces

1.4. Analysis of the contact interaction of multilayer structures

1.5. Relationship between mechanics and problems of friction and wear

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

2. CONTACT INTERACTION OF SMOOTH CYLINDRICAL BODIES

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

2.1.1. General formulas

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

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

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

2.1.3.1.2. Estimating the Norm of a Linear Operator

2.1.3.2. Approximate solution of the equation

2.2. Calculation of a fixed connection of smooth cylindrical bodies

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

2.3.1. Solution of an auxiliary problem for an elastic plane

2.3.2. Solution of an auxiliary problem for an elastic disk

2.3.3. Determination of maximum normal radial displacement

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

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

2.5.1. Formulation of the problem

2.5.2. Solution of auxiliary two-dimensional problems

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

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

3.1. Spatial non-local theory. geometric assumptions

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

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

3.4. Definition of displacements in the area of contact

3.5. Definition of auxiliary coefficients

3.6. Determination of the dimensions of the elliptical contact area

3.7. Equations for determining the contact area close to circular

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

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

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

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

3.10.2. Definition of auxiliary coefficients ^ ^

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

4. SOLUTION OF CONTACT PROBLEMS OF VISCOELASTICITY FOR SMOOTH BODIES

4.1. Key points

4.2. Compliance principles analysis

4.2.1. Volterra principle

4.2.2. Constant coefficient of transverse expansion under creep deformation

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

4.3.1. General case of viscoelasticity operators

4.3.2. Solution for a monotonically increasing contact area

4.3.3. Fixed connection solution

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

CONCLUSIONS AND MAIN RESULTS OF THE FOURTH CHAPTER

5. SURFACE CREEP

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

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

5.2.1. geometric assumptions

5.2.2. Surface Creep Model

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

5.2.4. Definition of auxiliary coefficients

5.2.5. Determination of the dimensions of the elliptical contact area

5.2.6. Determining the dimensions of the circular contact area

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

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

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

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

6. MECHANICS OF INTERACTION OF CYLINDRICAL BODIES WITH COVERINGS

6.1. Calculation of effective modules in the theory of composites

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

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

6.3.1. Statement of the problem and basic formulas

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

6.3.3. Integral equation and its solution

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

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

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

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

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

6.7.1.1. Main hypotheses and model of interaction of rigid bodies

6.7.1.2. Approximate solution of the problem

6.7.1.3. Determination of the maximum contact approach

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

6.7.3. Determination of contact stiffness at internal contact of cylinders

CONCLUSIONS AND MAIN RESULTS OF CHAPTER SIX

7. SOLUTION OF MIXED BOUNDARY PROBLEM WITH SURFACE WEAR INCLUDED

OF INTERACTING BODIES

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

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

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

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

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

CONCLUSIONS AND MAIN RESULTS OF THE SEVENTH CHAPTER

Recommended list of dissertations

On the contact interaction between thin-walled elements and viscoelastic bodies under torsion and axisymmetric deformation, taking into account the aging factor 1984, candidate of physical and mathematical sciences Davtyan, Zaven Azibekovich
Static and dynamic contact interaction of plates and cylindrical shells with rigid bodies 1983, candidate of physical and mathematical sciences Kuznetsov, Sergey Arkadievich
Technological support of the durability of machine parts based on hardening treatment with simultaneous application of anti-friction coatings 2007, doctor of technical sciences Bersudsky, Anatoly Leonidovich
Thermoelastic contact problems for bodies with coatings 2007, candidate of physical and mathematical sciences Gubareva, Elena Aleksandrovna
A technique for solving contact problems for bodies of arbitrary shape, taking into account surface roughness by the finite element method 2003, Candidate of Technical Sciences Olshevsky, Alexander Alekseevich

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

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

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

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

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

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

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

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

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

Connection of work with major scientific programs, topics. The studies were carried out in accordance with the following topics: "Develop a method for calculating contact stresses with elastic contact interaction of cylindrical bodies, not described by the Hertz theory" (Ministry of Education of the Republic of Belarus, 1997, No. GR 19981103); "Influence of microroughnesses of contacting surfaces on the distribution of contact stresses in the interaction of cylindrical bodies with similar radii" (Belarusian Republican Foundation for Fundamental Research, 1996, No. GR 19981496); "Develop a method for predicting the wear of sliding bearings, taking into account the topographic and rheological characteristics of the surfaces of interacting parts, as well as the presence of anti-friction coatings" (Ministry of Education of the Republic of Belarus, 1998, No. GR 1999929); "Modeling the contact interaction of machine parts, taking into account the randomness of the rheological and geometric properties of the surface layer" (Ministry of Education of the Republic of Belarus, 1999 No. GR 20001251)

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

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

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

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

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

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

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

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

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

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

As part of solving the problem:

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

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

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

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

The main provisions of the dissertation submitted for defense:

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

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

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

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

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

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

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

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

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

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

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

Development and research of the process of smoothing the surface of gas-thermal coatings of textile machine parts in order to increase their performance 1999, candidate of technical sciences Mnatsakanyan, Victoria Umedovna
Numerical simulation of dynamic contact interaction of elastoplastic bodies 2001, candidate of physical and mathematical sciences Sadovskaya, Oksana Viktorovna
Solution of contact problems in the theory of plates and plane non-Hertzian contact problems by the boundary element method 2004, candidate of physical and mathematical sciences Malkin, Sergey Aleksandrovich
Discrete Simulation of the Rigidity of Jointed Surfaces in the Automated Estimation of the Accuracy of Process Equipment 2004, Candidate of Technical Sciences Korzakov, Alexander Anatolyevich
Optimal design of contact pair parts 2001, doctor of technical sciences Hajiyev Vahid Jalal oglu

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

CONCLUSION

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

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

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

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

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

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

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

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

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

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Theory of contact interaction. Theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces kravchuk alexander stepanovich. Relationship of work with major scientific programs

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