At the meeting of the scientific seminar "Modern problems of mathematics and mechanics" November 24, 2017 Alexander Veniaminovich Konyukhova (Dr. Habil. PD Kit, Prof. Knrtu, Karlsruhe Institute Of Technology, Institute of Mechanics, Germany)
Geometrically accurate theory of contact interaction as a fundamental basis for computing contact mechanics
Start at 13:00, audience 1624.
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The main tactics of isogeometric analysis is a direct attachment of models of mechanics with the full description of the geometric object in order to formulate an effective computing strategy. Such advantages of isogeometric analysis as a complete description of the geometry of the object When formulating algorithms for computing contact mechanics can be fully expressed only if the kinematics of the contact interaction is fully described for all geometrically possible contact pairs. Contact bodies from a geometrical point of view can be considered as the interaction of deformable surfaces of arbitrary geometry and smoothness. In this case, different conditions of surface smoothness lead to the consideration of mutual contact between the edges, ribs and the vertices of the surface. Consequently, all contact pairs can be hierarchically classified as follows: the surface-in-surface, curve-in-surface, point-to-surface, curve curve, point-B-curve, point-to-point. The shortest distance between these objects is the natural measure of contact and leads to the task of the projection of the nearest point (PBT, English. Closest Point Projection, CPP).
The first main task in constructing the geometrically accurate theory of contact interaction is the consideration of the conditions for the existence and uniqueness of the solution of the PBT problem. This leads to a number of theorems that allow us to construct both three-dimensional geometric areas of the 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 built when considering the differential geometry of the object, in the metric of the curvilinear coordinate system of the coordinate to it: in Gaussian (Gauß) coordinate system for the surface, in the FREN-Serre coordinate system for curves, in the Darboux coordinate system for curves On the surface, and using the coordinates of Euler (Euler), as well as quaternions to describe the final turns around the object - points.
The second main task is to consider the kinematics of contact interaction from the point of view of the observer in the corresponding coordinate system. This allows you to determine not only the standard measure of normal contact as "penetration" (penetration), but also geometrically accurate measures of relative contact interaction: tangent slip on the surface, slip by separate curves, relative rotation of the curve (twisting), sliding the curve on its own tangential and By tangential normal ("Patch") when moving the curve over the surface. At this stage, using the covariant differentiation apparatus in the corresponding curvilinear coordinate system,
Preparation for the variational formulation of the problem, as well as to the linearization necessary for the subsequent global numerical solution, for example, for the Newton Nonlinear Solver iterative method (NEWTON NONLINEAR SOLVER). Linearization is understood as GATO (Gateaux) Differentiation in covariant form in a curvilinear coordinate system. In a number of complex cases, outgoing from a variety of solutions of the PBT problem, such as, for example, in the case of "parallel curves", it is necessary to build additional mechanical models (3D Continual model of the "Solid Beam Finite Element" curvilinear rope) compatible with the corresponding contact algorithm "Curve to Solid Beam Contact Algorithm. An important step to describe contact interaction is the wording in the covariant form of the most common arbitrary law of interaction between the geometric objects, which are far beyond the framework of the standard Cruton friction law (Coulomb). At the same time, the fundamental physical principle of the "dissipation maximum" is used, which is a consequence of the second law of thermodynamics. This requires the wording of the optimization problem with restriction in the form of inequalities in covariant form. In this case, all the necessary operations for the selected method of numerical solution of the optimization problem, including, for example, "Return-Mapping Algorithm" and the necessary derivatives are also formulated in the curvature coordinate system. Here, an indicative result of a geometrically accurate theory is as an opportunity to receive new analytical solutions in a closed form (generalization of the Euler problem 1769g. On the friction of the cylinder's rope in the case of anisotropic friction on the surface of arbitrary geometry) and the ability to receive in the compact form of a compact form of the coil friction law, taking into account Anisotropic geometric structure of the surface together with anisotropic micro-friction.
The choice of methods for solving the problem of statics or dynamics, subject to the satisfaction of the laws of contact interaction remains extensive. These are various modifications of the Newton iterative method for the global task and methods of satisfying restrictions on the local and global levels: fine (Penalty), Lagrange (Lagrange), Nitsche (Nitsche), Mortar (Mortar), and arbitrary selection of a finite difference scheme for a dynamic task . The basic principle is only the wording of the method in covariant form without
consideration of any approximations. Careful passage of all stages of building theory allows you to get a computational algorithm in a covariant "closed" form for all types of contact pairs, including an arbitrarily selected 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: standard method Finite Element Method, high-order final elements (High Order Finite Element), isogeometric analysis (IsogeeMtric Analysis), "Finite Cell Method)," Immersed "
1. Analysis of scientific publications within the mechanics of contact interaction 6
2. Analysis of the influence of the physical and mechanical properties of materials of contact pairs on the contact area within the framework of the theory of elasticity when the test task of contact interaction with a known analytical solution is implemented. 13
3. The study of the contact intense state of the elements of the spherical reference part in the axisymmetric formulation. 34.
3.1. Numerical analysis of the design of the reference part assembly. 35.
3.2. Study of the influence of grooves with a lubricant material of the spherical slide surface on the stress state of the contact node. 43.
3.3. Numerical study of the intense state of the contact node under different materials of the antifriction layer. 49.
Conclusions .. 54.
References .. 57
Analysis of scientific publications within the framework of mechanics of contact interaction
Many nodes and structures used in mechanical engineering, construction, medicine and other areas work under contact interaction. This is usually expensive, difficult to repairable responsible elements to which increased requirements for strength, reliability and durability are presented. Due to the wide use of the theory of contact interaction in mechanical engineering, construction and other areas of human activity, there was a need to consider the contact interaction of the bodies of a complex configuration (design with antifriction coatings and layers, layered bodies, nonlinear contact, etc.), with complex boundary conditions In the contact zone, in conditions of statics and speakers. The basis of the mechanics of contact interaction was laid down by G. Herz, V.M. Alexandrov, L.A. Galin, K. Johnson, I.Ya. Staperman, L. Gudman, A.I. Lurie and other domestic and foreign scientists. Considering the history of the development of the theory of contact interaction as a foundation, the work of Heinrich Hertz "On the contact of the elastic bodies" can be distinguished. In this case, this theory is based on the classical theory of elasticity and mechanics of continuous media, and was represented by the scientific community in the Berlin Physical Society At the end of 1881, scientists have noted the practical importance of the development of the theory of contact interaction, and the research of Hertz was continued, although the theory did not receive due development. The theory initially did not get spread, as it fastened its time and gained popularity only at the beginning of the last century, during the development of mechanical engineering. In this case, it can be noted that the main disadvantage of the theory of Hertz is its applicability only to ideally elastic bodies on the surfaces of the contact, without taking into account friction on the mating surfaces.
At the moment, the mechanics of contact interaction did not lose its relevance, but is one of the most rapidly fluttering the mechanics of the deformable solid. At the same time, each task of the mechanics of contact interaction carries a huge number of theoretical or applied research. Development and improvement of contact theory, when a large number of foreign and domestic scientists continued to develop a large number of foreign and domestic scientists. For example, Alexandrov V.M. Chebakov M.I. Considers the tasks for an elastic half-plane without taking into account and taking into account friction and clutch, also in their productions, the authors take into account lubricant, the heat allocated from friction and wear. The numerical analytical methods of solving the non-classical spatial problems of the mechanics of contact interactions within the linear theory of elasticity are described. A large number of authors worked on a book, which reflects the work until 1975, covering a large number of knowledge about contact interaction. This book contains the results of solutions of contact static, dynamic and temperature tasks for elastic, viscoelastic and plastic tel. A similar publication was published in 2001 containing updated methods and results of solving problems of contact interaction mechanics. It has works not only in domestic, but also foreign authors. N.Kh.Arutyunyan and A.V. Mandry in their monograph investigated the theory of contact interaction of growing tel. A task was set for non-stationary contact problems depending on the time of the contact area and sets out the methods of solving in .Sames V.N. I studied the dynamic contact interaction, and Sargsyan V.S. Considered tasks for semi-positions and strips. In his monograph, Johnson K. reviewed applied contact tasks with taking into account friction, dynamics and heat exchange. Effects were also described as inelasticity, viscosity, damage accumulation, sliding, clutch. Their studies are fundamental to the mechanics of contact interaction in terms of creating analytical and semi-analytical methods to solve the problems of contact of the band, half-space, space and bodies of the canonical form, they also affect contact for bodies with layers and coatings.
Further development of the mechanics of contact interaction is reflected in the works of Goryacheva I.G., Voronina N.A., Torka E.V., Chebakova M.I., M.I. Porter and other scientists. A large number of works considers contact of the plane, half-space or space with an indenter, contact through a layer or a thin coating, as well as contact with layered half-spaces and spaces. Basically, the solution of such contact problems were obtained using analytical and semi-analytical methods, and mathematical contact models are quite simple and, if they take into account the friction between the mating details, the nature of the contact interaction does not take into account. In real mechanisms, part of the design interact with each other and with the surrounding objects. Contact can occur both directly between the bodies and through various layers and coatings. Due to the fact that machine mechanisms and their elements are often geometrically complex structures that work within the framework of the mechanics of contact interaction, the study of their behavior and deformation characteristics is the actual problem of mechanics of the deformable solid body. As examples of such systems, the sliding bearings with a layer of a composite material, endoprosthesis of the thigh with an antifriction layer, a compound of bone and articular cartilage, road coating, pistons, supporting parts of the span structures of bridges and bridge structures, etc. Mechanisms are complex mechanical systems with a complex spatial configuration, which has more than one slide surface, and often contact coatings and layers. In this regard, it is interesting to develop contact problems, including contact interaction through coatings and layers. Goryacheva I.G. In its monograph, the effect of surface microgeometry, the heterogeneity of the mechanical properties of surface layers, as well as the properties of the surface and the films covering it on the characteristics of the contact interaction, the force of friction and the distribution of stresses in the surface layers under different contact conditions are investigated. In his research Torka E.V. Considers the task of sliding a rigid rough indenter along the border of a two-layer elastic half-space. It is assumed that friction forces do not affect the distribution of contact pressure. For the problem of frictional contact of the indenter with a rough surface, the effect of friction coefficient on the voltage distribution is analyzed. The studies of the contact interaction of rigid stamps and viscoelastic bases with subtle coatings are presented for cases when the surfaces of the stamps and coatings are mutual resourcing, are given in. The mechanical interaction of elastic layered bodies is studied in the works, they consider contact of cylindrical, spherical indesters, system of stamps with a elastic layered half-space. A large number of studies have been published about the insertion of multilayer media. Alexandrov V.M. and Mkhitaryan S.M. The methods and results of research on the effects of stamps on bodies with coatings and layers, tasks are considered in the formulation of the theory of elasticity and viscoelasticity. You can select a number of tasks about contact interaction in which friction is taken into account. The presence of a flat contact problem on the interaction of a moving rigid stamp with a viscoelastic layer is considered. The stamp moves with a constant speed and presses with a constant normal force, while it is assumed that friction in the contact area is absent. This task is solved for two types of stamps: rectangular and parabolic. The authors experimentally investigated the effect of sucks from various materials on the heat transfer process in the contact zone. About six samples were considered and experimentally determined that the effective heat insulator is stainless steel aggregate. In another scientific publication, an axisymmetric contact problem of thermoplasticity was considered about the pressure of a hot cylindrical circular isotropic stamp on the elastic isotropic layer, there was a nonideal thermal contact between the stamp and layer. The above works consider the study of a more complex mechanical behavior on the contact interaction site, but the geometry remains in most cases canonical form. Since often in contact structures there are more than 2 surfaces of contact, complex spatial geometry, complex in its mechanical behavior materials and loading conditions, analytical solution to obtain almost impossible for many practically important contact tasks, therefore effective decision methods are required, including numerical. In this case, one of the most important tasks of modeling the mechanics of contact interaction in modern application software packages is the consideration of the influence of the contact pair materials, as well as the compliance of the results of numerical research by existing analytical solutions.
The rupture of the theory and practice to solve the problems of contact interaction, as well as their complex mathematical formulation and descriptions served to the formation of numerical approaches to solving these problems. The most common methods of numerical solution of problems of contact interaction mechanics is the finite element method (MCE). The iterative algorithm of solutions using the MCE for the task of one-way contact is considered in. The solution of contact problems using an extended MCE, which allows to take into account friction on the surface of contacting contacting bodies and their inhomogeneity. The considered publications on the ICE for the problems of contact interaction are not tied to specific elements of the design and often have canonical geometry. An example of the consideration of contact within the framework of the ICE for a real design is where the contact between the blade and the gas turbine engine shovel is considered. Numerical solutions to the problems of contact interaction of multilayer structures and bodies with antifriction coatings and layers are considered in. Publications mainly considers the contact interaction of layered half-spaces and spaces with interements, as well as the pair of canonical-shaped bodies with layers and coatings. Mathematical models of contact are not enough, and the conditions of contact interaction are described poorly. The contact models rarely consider the possibility of preparing, slippages on the contact surface at the same time, slipping with different types of friction and digs. In most publications, mathematical models of problems of deformation of structures and nodes are described, especially boundary conditions on contact surfaces.
At the same time, the study of the tasks of contact interaction of the bodies of real complex systems and structures involves the presence of a base of physical and mechanical, frictional and operational properties of materials of contacting bodies, as well as antifriction coatings and grounds. Often one of the materials of contact pairs are various polymers, including antifriction polymers. The inadequacy of information on the properties of fluoroplastics, compositions on its basis and ultra-high-molecular weight polyethylene of various brands, which restrains their effectiveness to use in many industries. On the basis of National Material Testing Institute of The Stuttgart University Of The Stuttgart University of Technology, a number of inventive experiments were carried out on the definition of the physicomechanical properties of materials used in Europe in contact nodes: PTFE and MSM ultra-molecular weight polyethylene with supplements of soot and plasticizer. But large-scale studies aimed at determining the physico-mechanical and operational properties of viscoelastic media and a comparative analysis of materials suitable for use as material for the sliding surfaces of responsible industrial structures in difficult conditions for deformation in the world and Russia were not conducted. In this regard, there is a need to study the physico-mechanical, friction and operational properties of viscoelastic environments, the construction of models of their behavior and the choice of determining ratios.
Thus, the tasks of studying the contact interaction of complex systems and structures with one and more sliding surfaces are the actual problem of mechanics of the deformable solid body. To current tasks also include: determining the physicomechanical, frictional and operational properties of materials of contact surfaces of real structures and the numerical analysis of their deformation and contact characteristics; Conducting numerical studies aimed at identifying patterns of influence of the physicomechanical and antifriction properties of materials and geometry of contacting bodies to the contact stress-strain state and based on their basis, the development of the methods for predicting the behavior of structural elements during design and non-project loads. And also relevant research on the influence of the physicomechanical, friction and operational properties of materials entering into contact interaction. The practical implementation of such tasks is possible only with numerical methods focused on parallel computing technologies, with the involvement of modern multiprocessor computing.
Analysis of the influence of the physico-mechanical properties of materials of contact pairs on the contact area within the framework of the theory of elasticity when implementing the test task of contact interaction with a known analytical solution
The influence of the properties of the contact pair materials on the parameters of the contact interaction site will be considered on the example of solving the classical contact problem on the contact interaction of the two in contacts of the splashes (Fig. 2.1.). We will consider the task of the interaction of spheres within the framework of the theory of elasticity, the analytical solution of this problem is considered A.M. Katz in.
Fig. 2.1. Contact scheme
As part of the solution of the problem, it is explained that the theory of Hertz Contribute Pressure is to be in the formula (1):
, (2.1)
where - the radius of the contact site, the coordinate of the contact site, is the maximum contact pressure on the site.
As a result of mathematical calculations, as part of the mechanics of contact interaction, formulas were found for the definition and, presented in (2.2) and (2.3), respectively:
, (2.2)
, (2.3)
where and are the radii of contacting spheres, and, - the coefficients of Poisson and the elastic moduli of contacting spheres, respectively.
It can be noted that in formulas (2-3) the coefficient responsible for the mechanical properties of the contact pair of materials has the same look in this way, we denote it 
In this case, formula (2.2-2.3) have the form (2.4-2.5):
, (2.4)
. (2.5)
Consider the influence of the properties of the materials in contact in the design on the contact parameters. Consider within the framework of the task of contacting the two contacting areas of the following contact pairs of material: steel - fluoroplastic; Steel - composite antifriction material with spherical bronze inclusions (MAC); Steel - modified fluoroplastic. Such a choice of contact pairs of materials is due to further studies of their work with spherical reference parts. The mechanical properties of materials of contact pairs are presented in Table 2.1.
Table 2.1.
Properties of in contact areas
| No. p / p | Material 1 of the sphere | Material 2 spheth | |
| Steel | Fluoroplast | ||
| , N / m 2 | , N / m 2 | ||
| 2E + 11. | 0,3 | 5,45E + 08. | 0,466 |
| Steel | POPPY | ||
| , N / m 2 | , N / m 2 | ||
| 2E + 11. | 0,3 | 0,4388 | |
| Steel | Modified fluoroplastic | ||
| , N / m 2 | , N / m 2 | ||
| 2E + 11. | 0,3 | 0,46 |
Thus, for these three contact steam, you can find the contact pair coefficient, the maximum radius of the contact site and the maximum contact pressure, which are presented in Table 2.2. Table 2.2. The contact parameters are calculated under the condition of validity on the spheres with single radii (, m and, m) of comprehensive forces, N.
Table 2.2.
Parameters of contact area
Fig. 2.2. Contact site parameters:
a), m 2 / n; b), m; c), n / m 2
In fig. 2.2. A comparison of the parameters of the contact zone for three contact pairs of materials of the spheres is presented. It can be noted that pure fluoroplast has less, compared to the 2nd other materials, the value of the maximum contact pressure, while the radius of the contact zone has the greatest. The parameters of the contact zone in the modified fluoroplast and poppy are not significantly significant.
Consider the impact of the radii of contacting spheres on the parameters of the contact zone. It should be noted that the dependence of the contact parameters from the radii of the spheres is the same in formulas (4) - (5), i.e. They are part of the formula of the same type, therefore, in order to investigate the effect of contacting radii, it suffices to change the radius of one sphere. Thus, we consider an increase in the radius of the 2nd sphere with a constant value of the radius of the sphere (see Table 2.3).
Table 2.3.
Radius of contacting spheres
| No. p / p | , M. | , M. |
Table 2.4.
Contact zone parameters for different radii of contacting spheres
| No. p / p | Steel-photoplast | Steel-Mak. | Steel-Fluoroplast | |||
| , M. | , N / m 2 | , M. | , N / m 2 | , M. | , N / m 2 | |
| 0,000815 | 719701,5 | 0,000707 | 954879,5 | 0,000701 | 972788,7477 | |
| 0,000896 | 594100,5 | 0,000778 | 788235,7 | 0,000771 | 803019,4184 | |
| 0,000953 | 0,000827 | 698021,2 | 0,000819 | 711112,8885 | ||
| 0,000975 | 502454,7 | 0,000846 | 666642,7 | 0,000838 | 679145,8759 | |
| 0,000987 | 490419,1 | 0,000857 | 650674,2 | 0,000849 | 662877,9247 | |
| 0,000994 | 483126,5 | 0,000863 | 640998,5 | 0,000855 | 653020,7752 | |
| 0,000999 | 0,000867 | 634507,3 | 0,000859 | 646407,8356 | ||
| 0,001003 | 0,000871 | 629850,4 | 0,000863 | 641663,5312 | ||
| 0,001006 | 0,000873 | 626346,3 | 0,000865 | 638093,7642 | ||
| 0,001008 | 470023,7 | 0,000875 | 623614,2 | 0,000867 | 635310,3617 |
Depending on the parameters of the contact zone (maximum contact zone radius and maximum contact pressure) are presented in Fig. 2.3.
Based on the data presented in Fig. 2.3. It can be concluded that with an increase in the radius of one of the contacting areas as the maximum radius of the contact zone and the maximum contact pressure goes to the asymptot. At the same time, as expected, the permissions of the maximum radius of the contact zone and the maximum contact pressure for the three pairs under consideration of contacting materials are the same: as the maximum radius of the contact zone increases, and the maximum contact pressure decreases.
For a more visible comparing the effect of the properties of contacting materials on the contact parameters, we rebuild the maximum radius for the three-studied contact pairs and is similar to the maximum contact pressure (Fig. 2.4.).
Based on the data shown in Figure 4, a similarly small difference of contact parameters in poppy and a modified fluoroplastic, with a clean fluoroplastic with a significant smaller magnitudes of the contact pressure of the contact site radius more than two other materials.
Consider the distribution of contact pressure for three contact pairs of materials while increasing. The distribution of contact pressure is shown along the radius of the contact site (Fig. 2.5.).
 |
 |
 |
Fig. 2.5. Distribution of contact pressure on the contact radius:
a) steel fluoroplastic; b) steel-poppy;
c) steel-modified fluoroplastic
Next, consider the dependence of the maximum radius of the contact site and the maximum contact pressure from bringing closer areas. Consider the action on spheres with single radii (, m and, m) of forces: 1 H, 10 H, 100 H, 1000 H, 10000 H, 100000 H, 1000000 N. The resulting research parameters of contact interaction are presented in Table 2.5.
Table 2.5.
Contact options for magnification
| P, N. | Steel-photoplast | Steel-Mak. | Steel-Fluoroplast | |||
| , M. | , N / m 2 | , M. | , N / m 2 | , M. | , N / m 2 | |
| 0,0008145 | 719701,5 | 0,000707 | 954879,5287 | 0,000700586 | 972788,7477 | |
| 0,0017548 | 0,001523 | 2057225,581 | 0,001509367 | 2095809,824 | ||
| 0,0037806 | 0,003282 | 4432158,158 | 0,003251832 | 4515285,389 | ||
| 0,0081450 | 0,007071 | 9548795,287 | 0,00700586 | 9727887,477 | ||
| 0,0175480 | 0,015235 | 20572255,81 | 0,015093667 | 20958098,24 | ||
| 0,0378060 | 0,032822 | 44321581,58 | 0,032518319 | 45152853,89 | ||
| 0,0814506 | 0,070713 | 95487952,87 | 0,070058595 | 97278874,77 |
The dependences of the contact parameters are presented in Fig. 2.6.
 |
Fig. 2.6. The dependences of the contact parameters from
for three contact pairs of materials: a), m; b), n / m 2
For three contact pairs of materials, with the growth of squeezing forces, growth occurs, both the maximum radius of the contact area and the maximum contact pressure of Fig. 2.6. At the same time, analogously previously obtained result in pure fluoroplast with a smaller contact pressure of a larger radius.
Consider the distribution of contact pressure for three contact pairs of materials while increasing. The distribution of contact pressure is shown along the radius of the contact site (Fig. 2.7.).
Similarly, the previously obtained results with an increase in the rapprixing forces occur, both the radius of the contact site and the contact pressure, while the character of the distribution of contact pressure is the same in all options for calculations.
Perform the implementation of the task in the ANSYS software package. When creating a finite element grid, the type of elements of Plane182 was used. This type is the four nodal element and has the second procedure for approximation. The element is used for two-dimensional modeling of tel. Each element node has two degrees of freedom UX and UY. This element is also used to calculate the tasks: axisymmetric, with a flat deformed state and with a flat intense state.
In the studied classic tasks, the type of contact pair was used: "Surface - surface". One of the surfaces are assigned target ( Target.), and another contact ( Conta.). Since a two-dimensional task is considered, TARGET169 and CONTA171 end elements are used.
The task is implemented in an axisimaric formulation using contact elements without taking into account friction on mating surfaces. The design scheme of the problem is shown in Fig. 2.8.
Fig. 2.8. Estimated contact scheme of spheres
The mathematical formulation of tasks about the compression of two contacting spheres (Fig.2.8) is implemented within the framework of the theory of elasticity and includes:
equations equations
geometrical relations
, (2.7)
physical relations
, (2.8)
where and - the parameters of Lama, the stress tensor, the strain tensor, is the vector of movements, the radius-vector of an arbitrary point, the first invariant of the deformation tensor, is a single tensor, an area occupied by the sphere area 1, the area occupied by the sphere 2 area.
Mathematical formulation (2.6) - (2.8) is complemented by boundary conditions and conditions of symmetry on surfaces and. In the sphere 1 is the power
for the sphere 2
. (2.10)
The system of equations (2.6) - (2.10) is also complemented by the conditions of interaction on the contact surface, while the two bodies of the conditional numbers of which are 1 and 2. The following types of contact interaction are considered to contact.
- slipping with friction: for friction
, , , , (2.8)
wherein , ,
- for friction slip
, , , , , , (2.9)
wherein , ,
- Flowing
, 
, (2.10)
- Full clutch
, , , , (2.11)
where - the friction coefficient - the conventions of the coordinate axes lying in the plane tangent to the surface of the contact - move the normal to the corresponding contact boundary - movement in the tangent plane, is the voltage according to the normal boundary, - tangential voltages on the contact border, - The magnitude of the vector of tangent contact stresses.
The numerical implementation of the solution to the contacting task will be implemented on the example of a contact pair of materials steel-fluoroplastic, while the compressive forces N. Such a load selection is due to the fact that for a smaller load, a smaller breakdown of the gene's end elements is needed, which is problematic due to limited resource computing technology.
In the numerical implementation of the contact task, one of the paramount tasks is to assess the convergence of the finite-elemental solution to the contact parameters of the contact parameters. Below is a table 2.6. In which the characteristics of the finite-element models involved in estimating the convergence of the numerical solution of the breaking option.
Table 2.6.
The number of nodal unknowns with various sizes of elements in the task of contacting spheres
In fig. 2.9. Presented the convergence of the numerical solution of the contacting problem.
Fig. 2.9. The convergence of a numerical solution
You can see the convergence of a numerical solution, while the distribution of the contact pressure of the model from 144 thousand nodal unknown has not significant quantitative and qualitative differences from the model from 540 thousand nodal unknown. At the same time, the program's account time is different several times, which is a significant factor in numerical study.
In fig. 2.10. The comparison of the numerical and analytical solution of the problem of controversial areas is shown. The analytical solution of the problem is compared with a numerical solution of the model from 540 thousand nodal unknown.
Fig. 2.10. Comparison of analytical and numerical solutions
It can be noted that the numerical solution of the problem has small quantitative and qualitative differences from the analytical solution.
Similar results on the convergence of a numerical solution were obtained for two remaining contact pairs of materials.
At the same time, at the Institute of Mechanics of Solid Wednesdays Uro RAS D.F.-M.N. A.A.Adamov was performed by the cycle of experimental studies of the deformation characteristics of antifriction polymer materials of contact pairs with complex multi-stage stories of deformation with unloading. The cycle of experimental studies included (Fig. 2.11.): Tests to determine the hardness of the materials on the brinel; Studies under free compression conditions, as well as constrained compression by pressing in a special adaptation with a rigid steel cylindrical samples with a diameter and a long 20 mm. All tests were carried out on the ZWICK Z100SN5A testing machine at strain levels not exceeding 10%.
Tests for determining the hardness of the bringel materials occurred by pressing the ball with a diameter of 5 mm (Fig. 2.11., A). In the experiment, after installing the sample on the substrate to the ball, a preload is applied 9.8 H, which is maintained for 30 seconds. Next, at the speed of movement, the traverse of the machine 5 mm / min The ball is embedded in the sample until the load is 132 H, which is supported by constant for 30 seconds. Then unloading to 9.8 N. The results of the experiment to determine the hardness of the previously mentioned materials are presented in Table 2.7.
Table 2.7.
Material hardness
Cylindrical samples with a diameter and a height of 20 mm were studied under free compression conditions. To implement a homogeneous stress state in a short cylindrical sample on each end of the sample, three-layer gaskets made of fluoroplastic film with a thickness of 0.05 mm, lubricated with low viscosity grease. Under these conditions, the sample compression occurs without noticeable "barcode" during deformations up to 10%. The results of experiments on free compression are shown in Table 2.8.
Results of free compression experiments
Studies in conditions of cramped compression (Fig. 2.11., B) were carried out by pressing cylindrical samples with a diameter of 20 mm, a height of about 20 mm in a special device with a rigid steel rope with permissible limit pressures of 100-160 MPa. In manual mode, the machine control is loaded with a pre-low load (~ 300 H, the axial compression voltage of ~ 1 MPa) to select all the gaps and extrusion of excess lubricant. After that, the sample is maintained for 5 minutes to attense relaxation processes, then the development of a given sample loading program begins.
The experimental data obtained on nonlinear behavior of composite polymer materials is difficult to compare quantitatively. Table 2.9. The values \u200b\u200bof the tangent module M \u003d σ / ε reflecting the stiffness of the sample under conditions of a uniaxial deformed state are given.
Stiffness of the samples in the conditions of a uniaxial deformed state
From the test results, the mechanical characteristics of the materials are also obtained: the modulus of elasticity, the Poisson coefficient, deformation chart
Table 2.11
Deformation and stress in samples from antifriction composite material based on fluoroplastic with spherical bronze inclusions and molybdenum disulfide
| room | Time, sec. | Extension,% | SLV, MPa |
| 0,00000 | -0,00000 | ||
| 1635,11 | -0,31227 | -2,16253 | |
| 1827,48 | -0,38662 | -2,58184 | |
| 2196,16 | -0,52085 | -3,36773 | |
| 2933,53 | -0,82795 | -4,76765 | |
| 3302,22 | -0,99382 | -5,33360 | |
| 3670,9 | -1,15454 | -5,81052 | |
| 5145,64 | -1,81404 | -7,30133 | |
| 6251,69 | -2,34198 | -8,14546 | |
| 7357,74 | -2,85602 | -8,83885 | |
| 8463,8 | -3,40079 | -9,48010 | |
| 9534,46 | -3,90639 | -9,97794 | |
| 10236,4 | -4,24407 | -10,30620 | |
| 11640,4 | -4,92714 | -10,90800 | |
| 12342,4 | -5,25837 | -11,18910 | |
| 13746,3 | -5,93792 | -11,72070 | |
| 14448,3 | -6,27978 | -11,98170 | |
| 15852,2 | -6,95428 | -12,48420 | |
| 16554,2 | -7,29775 | -12,71790 | |
| 17958,2 | -7,98342 | -13,21760 | |
| 18660,1 | -8,32579 | -13,45170 | |
| 20064,1 | -9,01111 | -13,90540 | |
| 20766,1 | -9,35328 | -14,15230 | |
| -9,69558 | -14,39620 | ||
| -10,03990 | -14,57500 |
Deformation and voltage in modified fluoroplastic samples
| room | Time, sec. | Deformation axial,% | Conditional voltage, MPa |
| 0,0 | 0,000 | -0,000 | |
| 1093,58 | -0,32197 | -2,78125 | |
| 1157,91 | -0,34521 | -2,97914 | |
| 1222,24 | -0,36933 | -3,17885 | |
| 2306,41 | -0,77311 | -6,54110 | |
| 3390,58 | -1,20638 | -9,49141 | |
| 4474,75 | -1,68384 | -11,76510 | |
| 5558,93 | -2,17636 | -13,53510 | |
| 6643,10 | -2,66344 | -14,99470 | |
| 7727,27 | -3,16181 | -16,20210 | |
| 8811,44 | -3,67859 | -17,20450 | |
| 9895,61 | -4,19627 | -18,06060 | |
| 10979,80 | -4,70854 | -18,81330 | |
| 12064,00 | -5,22640 | -19,48280 | |
| 13148,10 | -5,75156 | -20,08840 | |
| 14232,30 | -6,27556 | -20,64990 | |
| 15316,50 | -6,79834 | -21,18110 | |
| 16400,60 | -7,32620 | -21,69070 | |
| 17484,80 | -7,85857 | -22,18240 | |
| 18569,00 | -8,39097 | -22,65720 | |
| 19653,20 | -8,92244 | -23,12190 | |
| 20737,30 | -9,45557 | -23,58330 | |
| 21821,50 | -10,00390 | -24,03330 |
According to the data presented in Tables 2.10.-2.12. Deformation diagrams (Fig. 2.2) are constructed.
According to the results of the experiment, it can be assumed that the description of the behavior of materials is possible within the framework of the deformation theory of plasticity. On test problems, the influence of the elastoplastic properties of the materials was not checked in view of the absence of an analytical solution.
The study of the impact of the physicomechanical properties of materials when working as a contact pair material is considered in Chapter 3 on the real design of the spherical reference part.
480 rub. | 150 UAH. | $ 7.5 ", Mouseoff, Fgcolor," #FFFFCC ", BGColor," # 393939 ");" Onmouseout \u003d "Return nd ();"\u003e Dissertation period - 480 rub., Delivery 10 minutes , around the clock, seven days a week and holidays
Kravchuk Alexander Stepanovich. The theory of contact interaction deformable solid tel With circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces: dis. ... Dr. Fiz.-Mat. Sciences: 01.02.04: Cheboksary, 2004 275 c. RGB OD, 71: 05-1 / 66
Introduction
1. Modern problems of contact interaction mechanics 17
1.1. Classic hypotheses applied when solving contacts for smooth tel 17
1.2. Effect of solid creep on their formation in contact area 18
1.3. Evaluation of the convergence of rough surfaces 20
1.4. Analysis of contact interaction of multilayer structures 27
1.5. Relationship of mechanics and friction and wear problems 30
1.6. Features of the application of modeling in tribology 31
Conclusions on the first chapter 35
2. Contact interaction of smooth cylindrical bodies 37
2.1. Solution of the contact problem for smooth isotropic disks and plates with cylindrical cavity 37
2.1.1. General formulas 38.
2.1.2. Conclusion of the regional condition for movements in the area of \u200b\u200bcontact 39
2.1.3. Integral equation and its decision 42
2.1.3.1. Study of the obtained equation 4 5
2.1.3.1.1. Bringing a singular integro differential equation to an integral equation with a core having a logarithmic feature 46
2.1.3.1.2. Rating of the norm of the linear operator 49
2.1.3.2. Approximate solution of equation 51
2.2. Calculation of a fixed connection of smooth cylindrical tel 58
2.3. Determination of movement in the moving connection of cylindrical tel 59
2.3.1. Solution of auxiliary problem for an elastic plane 62
2.3.2. Solving auxiliary task for elastic disk 63
2.3.3. Determination of the maximum normal radial movement 64
2.4. Comparison of theoretical and experimental data research of contact stresses with internal touch of cylinders of close radii 68
2.5. Modeling the spatial contact interaction of the system of coaxial cylinders of the final dimensions 72
2.5.1. Setting the problem 73.
2.5.2. Solution of auxiliary two-dimensional tasks 74
2.5.3. Solution of the original task 75
Conclusions and main results of the second chapter 7 8
3. Contact tasks for rough bodies and their solution by adjusting the curvature of the deformed surface 80
3.1. Spatial nonlocal theory. Geometrical assumptions 83.
3.2. Relative rapprochement of two parallel circles, determined by the deformation of roughness 86
3.3. Method of analytical assessment of the effect of roughness deformation 88
3.4. Definition of movements in the contact area 89
3.5. Determination of auxiliary coefficients 91
3.6. Determination of the sizes of the elliptic area of \u200b\u200bcontact 96
3.7. Equations for determining the contact area close to circular 100
3.8. Equations for determining the contact area close to line 102
3.9. Approximate determination of the coefficient A in the case of a contact area in the form of a circle or strip
3.10. Features of the averaging of pressures and deformations when solving a two-dimensional problem of internal contact of rough cylinders of close radii 1I5
3.10.1. The output of the integro-differential equation and its solution in the case of internal contact of the rough cylinders 10 "
3.10.2. Determination of ripping coefficients
Conclusions and main results of the third chapter
4. Solution of contact tasks of viscoelasticity for smooth bodies
4.1. Basic provisions
4.2. Analysis of the principles of conformity
4.2.1. Principle of Volterra
4.2.2. Permanent transverse expansion coefficient during creep deformation 123
4.3. Approximate solution of a two-dimensional contact problem of linear creep for smooth cylindrical bodies
4.3.1. Common case of viscoelastic operators
4.3.2. Solution for the monotonous increasing area of \u200b\u200bcontact 128
4.3.3. Solution for fixed connection 129
4.3.4. Simulation of contact interaction in case
uniformly aging isotropic plate 130
Conclusions and main results of the fourth chapter 135
5. Creeping surface 136
5.1. Features of contact interaction bodies with low yield strength 137
5.2. Construction of a model deformation model taking into account creep in the case of an elliptic area of \u200b\u200bcontact 139
5.2.1. Geometrical assumptions 140.
5.2.2. Model Creeping Surface 141
5.2.3. Determination of medium deformations of a rough layer and medium pressures 144
5.2.4. Determination of auxiliary coefficients 146
5.2.5. Determination of the size of the elliptic area of \u200b\u200bcontact 149
5.2.6. Determination of the size of the circular area of \u200b\u200bcontact 152
5.2.7. Determination of the width of the contact area in the form of a strip 154
5.3. Decision of a two-dimensional contact task for internal touch
rough cylinders, taking into account the creep of the surface 154
5.3.1. Setting the problem for cylindrical bodies. Integro-
differential equation 156.
5.3.2. Definition of reprimising coefficients 160
Conclusions and main results of the fifth chapter 167
6. Mechanics interaction of cylindrical bodies, taking into account the presence of coatings 168
6.1. Calculation of efficient modules in the theory of composites 169
6.2. Construction of a self-consistent method for calculating the effective coefficients of inhomogeneous media, taking into account the scatter of physicomechanical properties of 173
6.3. Solution of the contact task for disk and plane with elastic composite coating on the opening circuit 178
6.3. 1 Statement of the problem and the basic formulas 179
6.3.2. Conclusion of the regional condition for movements in the area of \u200b\u200bcontact 183
6.3.3. Integral equation and its decision 184
6.4. Solution of the problem in the case of orthotropic elastic coating with cylindrical anisotropy 190
6.5. Determination of the influence of viscoelastic aging coating on the change in contact parameters 191
6.6. Analysis of the features of contact interaction of multicomponent coating and disk roughness 194
6.7. Simulation of contact interaction taking into account thin metal coatings 196
6.7.1. Contact ball with plastic coating and rough semisimpace 197
6.7.1.1. The main hypotheses and model of interaction of solid bodies 197
6.7.1.2. Approximate solution of problem 200
6.7.1.3. Determination of the maximum contact convergence 204
6.7.2. Solution of the contact problem for a rough cylinder and a thin metal coating on the loop circuit 206
6.7.3. Determination of contact stiffness with internal contact of cylinders 214
Conclusions and main results of the sixth chapter 217
7. Solution of mixed boundary value problems, taking into account the wear of the surfaces of the dressed bodies 218
7.1. Features of the solution of the contact task, taking into account the wear of surfaces 219
7.2. Setting and solving the problem in the case of elastic deformation of roughness 223
7.3. Method of theoretical assessment of wear, taking into account the creep of the surface 229
7.4. Wear appraisal method based on the effect of coverage 233
7.5. Final comments on the formulation of flat tasks taking into account the wear 237
Conclusions and main results of the seventh chapter 241
Conclusion 242.
List of sources used
Introduction to work
The relevance of the topic of the thesis. Currently, the considerable efforts of engineers in our country and abroad are aimed at finding ways to determine the contact stresses of interacting bodies, since to transition from the calculation of the wear of materials to the problems of structural wear resistance, the contact tasks of the mechanics of the deformable solid body have a crucial role.
It should be noted that the most wide studies of contact interaction are made using analytical methods. In this case, the use of numerical methods is significantly expanding the possibilities of analyzing the stress state in the area of \u200b\u200bcontact, taking into account the properties of surfaces of rough bodies.
The need to take into account the structure of the surface is explained by the fact that the protrusions formed during technological processing have different height distribution and touch of microneria occurs only on separate sites forming the actual contact area. Therefore, when simulating the convergence of surfaces, it is necessary to use the parameters characterizing the real surface.
The bulky of the mathematical apparatus used in solving contact problems for rough bodies, the need to use powerful computing means is significantly contained by the use of existing theoretical developments in solving applied tasks. And, despite the progress achieved, while it is difficult to obtain satisfactory results, taking into account the peculiarities of the macro and microgeometry of the surfaces of the interacting bodies, when the surface element on which the characteristics of the roughness of solid bodies are installed, commensurate with the contact area.
All this requires the development of a single approach to solving contact problems, which is most fully taking into account both the geometry of interacting bodies, microgeometric and rheological characteristics of surfaces, characteristics of their wear resistance, and the possibility of obtaining an approximate solution to the problem with the least amount of independent parameters.
Contact problems for bodies with circular boundaries constitute the theoretical basis for calculating such elements of machines as bearings, hinged connections, connections with tension. Therefore, these tasks are usually selected as a model when conducting such studies.
Intensive work carried out in last years in Belarusian National Technical UniveuiII SI E. Dwishk iIіkishenya
on the solution of this problem and constitute the floor at the Motddododododod ^ s.
Communication of work with brupter scientific programs, themes.
Studies were carried out in accordance with the following themes: "Develop a method for calculating contact stresses with an elastic contact interaction of cylindrical bodies, not described by the theory of Hertz" (Ministry of Education of the Republic of Belarus, 1997, No. 19981103); "The influence of micronics of contacting surfaces on the distribution of contact stresses in the interaction of cylindrical bodies having close in the magnitude of the radii" (Belarusian Republican Foundation Fund, 1996, No. GR 19981496); "Develop a method for predicting the wear of the suspension supports, taking into account the topographic and rheological characteristics of surfaces of interacting parts, as well as the presence of antifriction coatings" (Ministry of Education of the Republic of Belarus, 1998, No. 2009929); "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 № GR2000G251)
The purpose and objectives of the study. Development of a single method of theoretical prediction of the effect of geometric, rheological characteristics of the roughness of the solid surfaces and the presence of coatings on the stress state in the contact area, as well as the establishment on this basis the patterns of changing the contact rigidity and wear resistance of conjugations on the example of the interaction of bodies with circular boundaries.
To achieve the goal required to solve the following problems:
Develop a method of approximate solution of problems of the theory of elasticity and viscoelasticity about Contact interaction of the cylinder and cylindrical cavity in the plate using a mupimal amount of independent parameters.
Develop a nonlocal model of contact interaction
Taking into account microgeometric, rheological characteristics
Surfaces, as well as the presence of plastic coatings.
Justify an approach that allows you to adjust the curvature
interacting surfaces due to the deformation of roughness.
Develop a method of approximate solution of contact problems for disk and isotropic, orthotropic from cylindrical anisotropy and viscoelastic aging coatings on the hole in the plate, taking into account their transverse deformability.
Build a model and determine the effect of microgeometric features of the surface of the solid body to contact interaction from Plastic coating on the counter.
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 antifriction coatings.
The object and subject of the study are non-classical mixed objectives of the theory of elasticity and viscoelasticity for bodies with circular boundaries, taking into account the nonlocality of the topographic and rheological characteristics of their surfaces and coatings, on the example of which in this paper developed a comprehensive method for analyzing the intense state in the area of \u200b\u200bcontact depending on the quality indicators their surfaces.
Hypothesis. When solving the boundary challenges, taking into account the quality of the surface of the bodies, a phenomenological approach is used, according to which the roughness deformation is considered as deformation of the intermediate layer.
Tasks with time-changing regional conditions are treated as quasistatic.
Methodology and methods of research conducted. When conducting research, the main equations of the mechanics of the deformable solid body, tribology, functional analysis were used. The method has been developed and substantiated to correct the curvature of loaded surfaces due to the deformations of micronovalities, which significantly simplifies the conducted analytical transformations and allows you to obtain analytical dependencies for the size of the contact area and contact voltages, taking into account the specified parameters without using the assumption about the smallness of the basic measurement of the roughness of the roughness of the roughness Contact area.
When developing a method of theoretical prediction of surface wear, observed macroscopic phenomena were considered as a result of the manifestation of statistically averaged relations.
The accuracy of the results obtained in the work is confirmed by comparisons of the resulting theoretical solutions and the results of experimental studies, as well as comparison with the results of some solutions found by other methods.
Scientific novelty and significance of the results obtained. For the first time, the example of the contact interaction of bodies with circular boundaries was summarizing, and a single method of comprehensive theoretical prediction of the influence of nonlocal geometric, rheological characteristics of rough surfaces of interacting bodies and the presence of coatings on the stress state, contact rigidity and wear resistance of conjugations were developed.
A comprehensive research complex allowed the theoretically substantiated method of solving the problems of solid mechanics, based on a consistent consideration of macroscopically observed phenomena, as the result of the manifestation of microscopic links statistically averaged at a significant portion of the contact surface.
As part of solving the problem:
A spatial nonlocal contact model has been proposed.
The interactions of solid bodies with isotropic surface roughness.
A method has been developed for determining the effect of the characteristics of the surface of solid bodies on the distribution of stresses.
An integro-differential equation obtained in contact problems for cylindrical bodies was investigated, which made it possible to determine the conditions for the existence and uniqueness of its solution, as well as the accuracy of the constructed approximations.
Practical (economic, social) significance of the results obtained. The results of theoretical studies are brought to acceptable techniques for practical use and can be directly applied during engineering calculations of bearings, sliding supports, gears. The use of the proposed solutions will reduce the time of creating new machine-building structures, as well as with great accuracy to predict their official characteristics.
Some results of the research performed were introduced on N P P P "Cycloprod", NGO "Altech".
The main provisions of the dissertation endowed with the defense:
Approximate decide the task of the deformed mechanics
solid body about the contact interaction of the smooth cylinder and
Cylindrical cavity in the plate, with sufficient accuracy
describing the studied phenomenon when using the minimum
The number of independent parameters.
The solution of nonlocal boundary value problems of the mechanics of the deformable solid, taking into account the geometric and rheological characteristics of their surfaces based on the method, allowing you to correct the curvature of interacting surfaces by deformation of roughness. The absence of assumptions about the smallness of the geometric sizes of the basic lengths of roughness measurement compared with the size of the contact area makes it possible to move to the development of multi-level models for deformation of the surface of solids.
Construction and substantiation of the method of calculating the movements of the boundaries of cylindrical bodies caused by deformation of the surface layers. The results obtained allow to develop a theoretical approach,
contact stiffener from By consideration of the joint influence of all features of the state of the surfaces of real bodies.
Modeling of viscoelastic disk interaction and cavity in
Plate of aging material, ease of implementation of results
which allows you to use them for a wide circle of applied
Tasks.
Approximate solution of contact problems for disk and isotropic, orthotropic from Cylindrical anisotropy, as well as viscoelastic aging coatings on the hole in the plate from By consideration of their transverse deformability. This makes it possible to estimate the effect of composite coatings. from Low modulus of elasticity on the loading of pairs.
Construction of a nonlocal model and determination of the effect of the characteristics of the roughness of the solid surface on the contact interaction with plastic coating on the counter.
Development of the method of solving boundary value problems from By consideration of the wear of cylindrical bodies, the quality of their surfaces, as well as the presence of antifriction coatings. On this basis, a methodology that focuses mathematical and physical methods has been proposed in the study of wear resistance, which makes it possible to make the main emphasis on the study of phenomena taking place instead of research in Contact area.
Personal contribution of the applicant. All results endowed with protection are obtained by the author personally.
Approbation of the dissertation results. The results of the research given in the thesis were presented at 22 international conferences and congresses, as well as conferences of the CIS countries and the Republican, among them: "Pontryaginian 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, Perspectives" (Gomel, 1998, Belarus), "Polymer Composites 98" (Gomel, 1998, Belarus), " Mechanika "99" (Kaunas, 1999, Lithuania), P of the Belarusian Congress on theoretical and applied mechanics (Minsk, 1999, Belarus), Internat. Conf. On Engineering Rheology, Icer "99 (Zielona Gora, 1999, Poland)," TRANSPORT STRENGROUT TRANSPORT TRANSFERS "(St. Petersburg, 1999, Russia), International Conference on MultiField Problems (Stuttgart, 1999, Germany).
Structure and scope of the dissertation. The thesis consists of introduction, seven chapters, conclusions, the list of used sources and applications. The full amount of the thesis is 2 minutes, including the volume occupied by illustrations - 14 pages, tables - 1 page. The number of sources used includes 310 items.
The influence of solid creep on their forming in the area of \u200b\u200bcontact
The practical preparation of analytical dependencies 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 tasks, it is customary to resort to idealization. So, it is believed that if the dimensions of the bodies themselves are sufficiently large compared to the size of the contact area, the voltages in this zone are weakly dependent on the configuration of the bodies away from the contact area, as well as the method of their consolidation. At the same time, the voltage 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 body is assumed to topographically smooth on micro and macro levels. At the micro-level, this means the absence or invoking the micronetics of contacting surfaces, which would determine the incomplete fit of the contact surfaces. Therefore, the real area of \u200b\u200bcontact that is formed on the tops of the protrusions is significantly less theoretical. On the macro level, surface profiles are considered continuous in the contact zone along with the second derivatives.
These assumptions were first used by Herz when solving the contact task. The results obtained on the basis of its theory satisfactorily describe the deformed state of ideally elastic bodies in the absence of friction on the contact surface, but not applicable, in particular, to low-modulus materials. In addition, the conditions in which the theory of hertz is used is violated when considering the contact of the agreed surfaces. This is due to the fact that due to the application of the load, the size of the contact area is rapidly growing and can reach the values \u200b\u200bcomparable to the characteristic sizes of contacting bodies, so that the bodies cannot be considered as elastic half-space.
Of particular interest in solving contact tasks causes taking into account the forces of friction. At the same time, the last on the surface of the section of the two bodies of the agreed form, which are in normal contact;, plays a role only at relatively high friction coefficient values.
The development of the theory of contact interaction of solid bodies is associated with the refusal to the hypotheses listed above. It was carried out in the following main areas: the complication of the physical model of deformation of solids and (or) the refusal to the hypotheses of smoothness and the homogeneity of their surfaces.
Interest in creep sharply increased due to the development of technology. Among the first researchers, found the phenomenon of deformation of materials in time at a constant load, were Vika, Weber, Kollarush. Maxwell first presented the law of deformation in time in the form of a differential equation. A somewhat later Bolygman created a common apparatus for describing linear creep phenomena. This device, significantly developed subsequently Volterra, is currently a classic section of the theory of integral equations.
Until the middle of the last century, elements of the theory of deformation of materials in time have found small use in the practice of calculating engineering structures. However, with the development of energy installations, chemical-technological devices operating at higher temperatures and pressures, the phenomenon of creep is needed. Requests of mechanical engineering led to a huge scope of experimental and theoretical studies in the field of creep. Due to the need for accurate calculations, the phenomenon of creep began to take into account even in such materials as wood and soils,
The study of creep in contact with the contact of solid bodies is important for a number of causes of applied and principled nature. So, even with constant loads, the form of interacting bodies and their intense state, as a rule, it changes, which must be considered when designing machines.
A qualitative explanation of the processes occurring during creep can be given, based on the main representations of the theory of dislocations. So, in the structure of the 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 the dislocation movement is shift to one interatomic distance. The intense state of the body facilitates the movement of dislocations, reducing potential barriers.
Temporary laws of creep depends on the structure of the material that changes over the creep. Experimentally obtained an exponential dependence of the speeds of the steady creep against voltages at relatively high voltages (-10 "and more from the modulus of elasticity). In a significant voltage interval, the experimental points on the logarithmic grid are usually grouped at some straight line. This means that in the considered voltage interval (- 10 "-10" from the modulus of elasticity) There is a power dependence of the velocities of strain deformations. It should be noted that at low stresses (10 "and less from the elastic module) this dependence is linear. A number of works are given various experimental data on the mechanical properties of various materials in a wide range of temperatures and deformation rates.
Integral equation and its decision
Note that if the elastic permanent disks and plates are equal, then Wow \u003d O and this equation becomes an integral equation of the first kind. Features of the theory of analytic functions allow in this case using additional conditions, to obtain a single solution. These are the so-called formulas for the treatment of singular integral equations, allowing to obtain a solution to the task explicitly. The peculiarity is that in the theory of boundary value problems, three cases are usually considered (when V is part of the border of the border): the solution has a feature at both ends of the integration region; The solution has a feature at one of the ends of the integration area, and on the second it turns into zero; The solution is drawn to zero at both ends. Depending on the choice of one or another embodiment, a general type of solution is built, which in the first case includes a general solution of a homogeneous equation. Setting the behavior of the solution on infinity and angular points of the contact area, based on physically reasonable assumptions, a single solution is built that satisfies the specified restrictions.
Thus, the uniqueness of the solution of this task is understood in the sense of the restrictions received. It should be noted that when solving contact problems of the theory of elasticity, the most common restrictions are the requirements of circulation to zero solutions at the ends of the contact area and the assumption of the disappearance of stresses and rotations on infinity. In the case when the area of \u200b\u200bintegration is the entire border of the region (body), the uniqueness of the solution is guaranteed by Cauchy formulas. In this case, the easiest and most common method of solving applied tasks in this case is the view 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 regions are not specified, because In this case, it is known that the arc of the circle (the curve along which integration is performed) satisfies the Lyapunov condition. Generalization of the theory of two-dimensional boundary value problems In case of more general assumptions on the smoothness of the boundaries of areas can be found in the monograph of the AI. Danilyuk.
The greatest interest is the total case of the equation, when 7i 0. The absence of methods for constructing an accurate solution in this case leads to the need to apply numerical analysis methods and approximation theory. In fact, as it has already been noted, the numerical methods for solving integral equations are usually based on approximation of solving the equation by the functionality of a certain species. The amount of accumulated results in this area allows you to highlight the main criteria for which these methods are usually compared when using them in applied tasks. First of all, the simplicity of the physical analogy of the proposed approach (usually it is in one form or another 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 that makes the feature of its structure and, accordingly, allowing the numerical result with the highest speed. It should be noted that the last criterion plays an essential role only in the case of large-order linear equations. All this determines the effectiveness of the approach used. At the same time, it should be stated that by now there are only separate studies on comparative analysis and possible simplifications in solving practical problems with the help of various approximations.
It should be noted that the integro-differential equation can be given to the form: V arc of the circumference of a single radius concluded between two points with the angular coordinates -ss0 and a0, a0 є (0, l / 2); U1 is a real coefficient determined by the elastic characteristics of interacting bodies (2.6); F (T) is a known function defined by the applied loads (2.6). In addition, we recall that the STG (T) appeals to zero at the ends of the integration segment.
Relative rapprochement of two parallel circles determined by the deformation of roughness
The task of the internal compression of circular cylinders of close radii was considered first considered by I.Y. Stanman. When solving the problem supplied by it, it is customary that the external load acting on the internal and external cylinders on their surfaces is carried out in the form of a normal pressure, diametrically opposite contact pressure. When the equation is derived, the solution is used to compress the cylinder with two opposite forces and the solution of a similar problem for the appearance of the circular opening in an elastic medium. It was obtained an explicit expression for moving the cylinder loop points and holes through the integral operator from the voltage function. This expression was used by a number of authors to assess the contact stiffness.
Using the heuristic approximation for the distribution of contact stresses for the scheme I.Ya. Stanman, A.B. Milov received simplified dependence for maximum contact movements. However, it was found that the resulting theoretical assessment differs significantly from experimental data. Thus, the movement determined from the experiment turned out to be less theoretical 3 times. This fact is explained by the author of the essential influence of the features of the spatial loading scheme and proposes the transition coefficient from a three-dimensional task to flat.
A similar approach used M.I. Warm, setting the approximate solution somewhat different. It should be noted that in this work, in addition, a linear second-order linear differential equation was obtained to determine the contact movements in the case of the circuit shown in Figure 2.1. The specified equation follows directly from the method of obtaining an integro differential equation to determine normal radial stresses. In this case, the complexity of the right part determines the bulky of the resulting expression for movements. In addition, in this case there are unknown values \u200b\u200bof the coefficients in solving the corresponding homogeneous equation. At the same time, it is noted that without establishing the values \u200b\u200bof permanent, one can determine the amount of radial movements of the diametrically opposite points of the holes and shaft.
Thus, despite the relevance of the task of determining the contact stiffness, the analysis of literary sources did not allow to identify the method of its solution, which makes it possible to reasonably establish the magnitudes of the largest normal contact movements due to the deformation of the surface layers without taking into account the deformations of the interacting bodies as a whole, which is due to the lack of formalized definition of the concept "Contact Stiffness ".
When solving the task, we will proceed from the following definitions: movements under the action of the main vector of forces (excluding the features of contact interaction) will be called closer (removal) of the center of the disk (holes) and its surface that does not lead to a change in the shape of its border. Those. This is the rigidity of the body as a whole. Then the contact stiffness is the maximum movement of the center of the disk (holes) without taking into account the movement of the elastic body under the action of the main vector of forces. This system of concepts allows you to divide the movement; obtained from the solution of the problem of the theory of elasticity, and shows that the assessment of the contact rigidity of cylindrical bodies obtained by A.B. Milovish from Solution Il. Stanman, Verne only for this loading scheme.
Consider the task set in paragraph 2.1. (Figure 2.1) with boundary condition (2.3). Given the properties of analytical 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 concentrated strength in the infinite region. This explains the presence of a logarithmic feature. The second terms (2.30), (2.32) are determined by the lack of tangent stresses on the disk circuit and the opening;, as well as the condition for the analytical behavior of the corresponding components of the complex potential in zero and at infinity. On the other hand, superposition (2.26) and (2.29) ((2.27) and (2.31)) gives the zero main vector of forces acting on the contour of the hole (or disk). All this allows you to express through the third term of the radial movement in an arbitrary fixed direction C, in the plate and in the disk. To do this, we find the difference between the FPD (g), (z) and FP 2 (2), 4v2 (z):
Approximate solution of a two-dimensional contact problem of linear creep for smooth cylindrical bodies
The idea of \u200b\u200bthe need to take into account the microstructure of the surface of compressible bodies belongs to I.Ya. Stanman. It was introduced a 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 tasks of the theory of elasticity, additional normal movements arise, due to the purely local deformations, depending on the microstructure of contacting surfaces. I.Ya.Sterman suggested that additional movement is proportional to normal pressure, and the proportionality coefficient is for this material The magnitude of constant. As part of this approach, they were first obtained by an equation of a flat contact problem for an elastic rough body, i.e. Body having a layer of high adhesiveness.
In a number of work, it is assumed that additional normal movements due to the deformation of the microprifurizers of contacting bodies are proportional to the macro-strand into some extent. This is based on an equalization of the averaged values \u200b\u200bof movements and stresses within the base length of the surface roughness measurement. However, despite the fairly well developed apparatus of solving problems of a similar class, a number of methodical difficulties are not overcome. So, the hypothesis of the power supply of the stresses and movements of the surface layer, taking into account the real characteristics of microgeometry, is true at small baseline lengths, i.e. The high purity of the surface, and, therefore, with the justice of the hypothesis about the topographic smoothness on micro and macro levels. It should also be noted a significant complication of the equation when using a similar approach and the impossibility of describing with its help of the effect of waviness.
Despite the fairly well-developed device for solving contact problems, taking into account the layer of increased advantage, a number of issues of a methodical nature remained, impede its application in the engineering practice of calculations. As already noted, the surface roughness has a probabilistic height distribution. The commensity of the size of the surface element on which the characteristics of roughness are determined, with the size of the contact area, it is the main difficulty in solving the task and determines the incorrect of applying by some authors of the direct connection between the diameters and the strain of roughness in the form: where S is the surface point.
It should also be noted the solution of the task using the assumption about the transformation of the type of pressure distribution into parabolic, if the deformations of the elastic half-space in comparison with the deformations of the rough layer can be neglected. This approach leads to a significant complication of the integral equation and allows you to receive only numerical results. In addition, the authors have been used by the already mentioned hypothesis (3.1).
It is necessary to mention the attempt to develop an engineering method of accounting for the effect of roughness in the internal touch of cylindrical bodies based on the assumption that elastic radial movements in the contact area caused by the deformation of micro-irregularities are constant and proportional to the average contact voltage T to some extent to. However, Despite its obvious simplicity, the disadvantage of this approach is that with this method of accounting for roughness, its influence gradually increases with an increase in the load, which is not observed in practice (Figure 3 l,).
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Contact interaction mechanics
Introduction
mechanics Contact Roughness Elastic
Contact interaction mechanics is a fundamental engineering discipline, extremely useful in designing reliable and energy-saving equipment. It will be useful in solving many contact problems, such as wheel-rail, when calculating couplings, brakes, tires, sliding and rolling bearings, gear wheels, hinges, seals; Electrical contacts, etc. It covers a wide range of tasks, ranging from calculating the strength of the interface elements of the tribosystems, taking into account the lubricant and structure of the material and ending with the use of micro and nanosystems.
The classic mechanics of contact interactions are primarily associated with the name of Heinrich Hertz. In 1882, Hertz solved the problem of contacting two elastic bodies with spontaneous surfaces. This classic result and today underlies the mechanics of contact interaction.
1. Classic tasks of mechanics of contact interaction
1. Contact between a ball and elastic half-space
The solid ball of the radius R is pressed into the elastic half-space to the depth D (the depth of penetration), forming the contact area of \u200b\u200bthe radius
Necessary for this force is equal
Here, E1, E2 - elastic modules; H1, H2 - Poisson coefficients of both bodies.
2. Contact between two balls
When contacting two balls with radius R1 and R2, these equations are valid, respectively for R radius
Pressure distribution in contact area is determined by the formula
with maximum pressure in the center
The maximum tangent voltage is achieved below the surface for H \u003d 0.33 at.
3. Contact between two cross-cylinders with the same Radius R
The contact between two crossed cylinders with the same radii is equivalent to the contact between the ball R radius and the plane (see above).
4. Contact between a solid cylindrical indenter and elastic half-space
If the solid cylinder with radius A is pressed into the elastic half-space, the pressure is distributed as follows:
The connection between the penetration depth and normal force is determined
5. Contact between a solid conical indenter and elastic half-space
When the elastic half-span of the solid cone-shaped indenter, the penetration depth and the contact radius are determined by the following ratio:
Here and? The angle between the horizontal and the side plane of the cone.
Pressure distribution is determined by the formula
The voltage at the top of the cone (in the center of the contact area) varies according to the logarithmic law. The total force is calculated as
6. Contact between two cylinders with parallel axes
In case of contact between two elastic cylinders with parallel axes, the force is directly proportional to the depth of penetration
The radius of curvature in this ratio is not at all present. The semi-width of contact is determined by the following relationship.
as in the case of contact between two balls.
Maximum pressure equals
7. Contact between rough surfaces
When two bodies with rough surfaces interact with each other, then the real area of \u200b\u200bcontact A is much smaller than the geometric area A0. When contacting between the plane with randomly distributed roughness and elastic half-space, the actual contact area is proportional to the normal force F and is determined by the following approximate equation:
At the same time RQ? The rms value of the irregular surface of the rough surface and. Average pressure in real contact area
it is calculated in a good approximation as half of the elastic module E *, multiplied by the mean square value of the irregularity of the RQ surface profile. If this pressure is more HB material hardness and thus
the micronether is completely in plastic condition.
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 contacting the sphere with a half-space, taking into account the presence of a rough layer, it was concluded that the calculated parameters depend not so much from the deformation of the rough layer, as from the deformation of individual irregularities.
When developing a model of contacting a spherical body with a rough surface, previously obtained results were taken into account:
- at low loads, the pressure for the rough surface is less calculated on the theory of the city of Hertz and is distributed for the larger area (J. Greenwood, J. Williamson);
- the use of a widely used model of a rough surface in the form of an ensemble of the proper geometric shape, whose heights are subject to a specific distribution law leads to significant errors in assessing contacting parameters, especially at low loads (N.B. Demkin);
- There are no useful expressions suitable for calculating contacting parameters and the experimental base is not sufficiently developed.
This paper proposes an approach based on fractal ideas about the rough surface as a geometric object with fractional dimension.
We use the following ratios, reflecting the physical and geometric features of the rough layer.
The elastic modulus of the rough layer (and not the material from which the part is and, accordingly, the grungy layer) is EEFF, being a variable value, is determined by the dependence:
where E0 is the modulus of the elasticity of the material; e - relative deformation of the irregularities of the rough layer; f - constant (g \u003d 1); D - fractal dimension of a rough surface profile.
Indeed, the relative rapprochement characterizes in a certain sense, the distribution of material in the height of the rough layer and, thus, the effective module characterizes the features of the porous layer. At e \u003d 1, this porous layer is degenerated into a solid material with its own modulus of elasticity.
We believe that the number of touch spots is proportional to the size of the contour area having an AC radius:
Rewrite this expression in the form
Find the ratio of the proportionality of C. Let n \u003d 1, then the AC \u003d (SMAX / P) 1/2, where Smax is the area of \u200b\u200bone contact span. From
Substitting the obtained value with in equation (2), we get:
We believe that the cumulative distribution of contact spots with an area, more s, is subordinate to the following law
Differential (module) The distribution of the number of spots is determined by the expression
Expression (5) allows you to find the actual area of \u200b\u200bcontact
The resulting result shows that the actual area of \u200b\u200bcontact depends on the structure of the surface layer, determined by the fractal dimension and the maximum area of \u200b\u200ba separate tangency spot located in the center of the contour area. Thus, to assess the parameters of contacting, it is necessary to know the deformation of a separate irregularity, and not the whole rough layer. The cumulative distribution (4) does not depend on the state of contact spots. It is fair when the touch spots may be in elastic, elastoplastic and plastic states. The presence of plastic deformations determines the effect of adaptability of the rough layer to external influence. This effect is partially manifested in leveling pressure on the touch area and increasing the contour area. In addition, the plastic deformation of multiple protrusions leads to the elastic state of these protrusions with a small number of re-loading, if the load does not exceed the initial value.
By analogy with the expression (4), we write the integral function of the distribution of the area of \u200b\u200bcontact spots in the form
The differential form of expression recording (7) seems to be the following expression:
Then the mathematical expectation of the contact area is determined by the following expression:
As the actual contact area is equal
and, given the expressions (3), (6), (9), we write:
Considering 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.
Determine Smax from the famous expression
where b is a coefficient equal to 1 for the plastic state of the contact of the spherical body with a smooth half-space, and b \u003d 0.5 - for elastic; R is the radius of the roundabout of the vertices of irregularities; Dmax - deformation of irregularities.
We assign that the radius of circular (contour) area of \u200b\u200bthe AC is determined by the modified formula of Hertz
Then, substituting the expression (1) in formula (11), we obtain:
Equating the right parts of expressions (10) and (12) and solving the equality obtained relative to the deformation of the maximum loaded irregularity, we write:
Here, R is the radius of the roundabout of the vertices of irregularity.
In the output of equation (13), it was taken into account that the relative deformation of the most loaded irregularity is equal
where DMAX is the greatest deformation of irregularities; RMAX is the highest profile height.
For a Gaussian surface, the fractal dimension of the profile d \u003d 1.5 and at T \u003d 1 expression (13) has the form:
Considering the deformation of the irregularities and the sediment of their foundation additive values, write:
Then the total convergence will be found from the following relationship:
Thus, the obtained expressions allow you to find the main parameters of contacting the spherical body with a half-space, taking into account the roughness: the radius of the contour area was determined by expressions (12) and (13), rapprochement? By formula (15).
3. Experiment
The tests were carried out on the installation for the study of the contact stiffness of fixed joints. The accuracy of measurement of contact deformations was 0.1-0.5 microns.
The test diagram is shown in Fig. 1. The technique of carrying out the experiment provided for smooth loading and unloading of samples that have a certain roughness. There were three balls with a diameter of 2r \u003d 2.3 mm between the samples.
Samples have been investigated, having the following roughness parameters (Table 1).
In this case, the upper and lower samples had the same roughness parameters. Sample material - steel 45, heat treatment - improvement (HB 240). Test results are shown in Table. 2.
Here is also a comparison of experimental data with the calculated values \u200b\u200bobtained on the basis of the proposed approach.
Table 1
Roughness parameters
|
Sample number |
Roughness parameters of steel samples |
||||||
|
Approximation parameters of the reference curve |
|||||||
table 2
Rapid spherical body with rough surface
|
Sample No. 1. |
Sample number 2. |
||||||||
|
dOSOS, MKM. |
Experiment |
dOSOS, MKM. |
Experiment |
||||||
A comparison of experimental and calculated data showed their satisfactory compliance, which indicates the applicability of the considered approach to assessing the parameters of contacting spherical bodies, taking into account the roughness.
In fig. 2 shows the dependence of the ratio of the AC / AU (H) of the contour area, taking into account the roughness to the area, calculated on the theory of the city of Hertz, from fractal dimension.
As can be seen in fig. 2, with an increase in fractal dimension reflecting the complexity of the profile structure of the rough surface, the ratio of the contour contact area to the area, calculated for smooth surfaces on the theory of Gersi, is growing.
Fig. 1. Test scheme: A - loading; B - the location of the balls between the test samples
The dependent dependence (Fig. 2) confirms the fact of increasing the area of \u200b\u200btouching a spherical body with a rough surface compared to the area calculated on the theory of Gersi.
When evaluating the actual touch area, it is necessary to take into account the upper limit equal to the ratio of the load to the sulk element brinell.
The area of \u200b\u200bcontour area is based on roughness using formula (10):
Fig. 2. The dependence of the radius ratio of the contour area, taking into account the roughness to the radius of the Geretse Square from the fractal dimension D
To estimate the relationship of the actual area of \u200b\u200bcontact to the contour we divide the expression (7.6) on the right side of equation (16)
In fig. 3 shows the dependence of the actual contact area of \u200b\u200bthe AR to the contour area of \u200b\u200bthe AC from the fractal dimension D. with an increase in fractal dimension (increasing roughness) The AR / AC ratio decreases.
Fig. 3. Dependence of the attitude of the actual area of \u200b\u200bcontact AR to the contour area of \u200b\u200bthe AC from fractal dimension
Thus, the plasticity of the material is considered not only as a property (physical and mechanical factor) of the material, but also as a carrier of the effect of adaptability of the discrete multiple contact to external influence. This effect is manifested in some leveling of pressures on the contour touch area.
Bibliography
1. Mandelbrot B. Fractal Geometry of Nature / B. Mandelbrot. - M.: Institute of Computer Research, 2002. - 656 p.
2. Voronin N.A. The patterns of contact interaction of solid topopositional materials with a rigid spherical stamp / N.A. Voronin // friction and lubrication in machines and mechanisms. - 2007. - №5. - P. 3-8.
3. Ivanov A.S. Normal, angular and tangent contact stiffness of a flat joint / A.S. Ivanov // Bulletin of Mechanical Engineering. - 2007. - №1. P. 34-37.
4. Tikhomirov V.P. Contact ball interaction with a rough surface / friction and lubrication in machines and mechanisms. - 2008. - №9. -FROM. 3-
5. Demkin N.B. Contact roughwater wavy surfaces, taking into account the mutual influence of irregularities / N.B. Demkin, S.V. Udalov, V.A. Alekseev [and others] // friction and wear. - 2008. - T.29. - Number 3. - P. 231-237.
6. Bulanov E.A. Contact problem for rough surfaces / E.A. Bulanov // Engineering Engineering. - 2009. - №1 (69). - P. 36-41.
7. Lannkov, A.A. The probability of elastic and plastic deformations in compressing metal rough surfaces / A.A. Lakkov // friction and lubrication in machines and mechanisms. - 2009. - №3. - P. 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. - №1422. - P. 300-319.
9. Majumdar M. Fractal model of elastic-plastic contact of rough surfaces / M. Majumdar, B. Bhushan // Modern engineering. ? 1991.? Number? P. 11-23.
10. Varadi K. Evaluation of the Real Contact Areas, Pressure Distribution 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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1. Modern problems of contact mechanics
Interaction
1.1. Classic hypothesis used when solving contact problems for smooth bodies
1.2. The influence of solid creep on their forming in the area of \u200b\u200bcontact
1.3. Evaluation of ragged rough surfaces
1.4. Analysis of contact interaction of multilayer structures
1.5. Relationship of mechanics and friction and wear problems
1.6. Features of the application of modeling in tribology 31 conclusions on the first chapter
2. Contact interaction of smooth cylindrical bodies
2.1. Solution of the contact problem for smooth isotropic disks and plates with cylindrical cavity
2.1.1. General formulas
2.1.2. Conclusion of the regional condition for movements in the area of \u200b\u200bcontact
2.1.3. The integral equation and its decision 42 2.1.3.1. Study of the obtained equation
2.1.3.1.1. Bringing a singular integrational equation to an integral equation with a kernel with a logarithmic feature
2.1.3.1.2. Rate of the norm of the 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 movement in the movable connection of cylindrical bodies
2.3.1. Solving auxiliary problem for an elastic plane
2.3.2. Solving auxiliary task for an elastic disk
2.3.3. Determination of the maximum normal radial movement
2.4. Comparison of theoretical and experimental data of the study of contact stresses with internal tapping cylinders of close radii
2.5. Simulation of the spatial contact interaction of the system of coaxial cylinders of final sizes
2.5.1. Formulation of the problem
2.5.2. Solution of auxiliary two-dimensional tasks
2.5.3. Solution of the original problem 75 Conclusions and basic results of the second chapter
3. Contact tasks for rough bodies and their solution by adjusting the curvature of the deformed surface
3.1. Spatial nonlocal theory. Geometrical assumptions
3.2. Relative rapprochement of two parallel circles determined by the deformation of roughness
3.3. Method of analytical assessment of roughness deformation
3.4. Determination of movements in the contact area
3.5. Determination of auxiliary coefficients
3.6. Determination of the size of the elliptic area of \u200b\u200bcontact
3.7. Equations for determining the contact area close to circular
3.8. Equations for determining the contact area close to line
3.9. Approximate definition of coefficient A in the case of a contact area in the form of a circle or strip
3.10. Features of the averaging of pressures and deformations when solving a two-dimensional problem of internal contact of rough cylinders of close radii
3.10.1. The output of the integro-differential equation and its solution in the case of internal contact of the rough cylinders
3.10.2. Determination of reprimising coefficients ^ ^
3.10.3. Stressed landing of rough cylinders ^ ^ Conclusions and main results of the third chapter
4. Solution of contact tasks of viscoelasticity for smooth bodies
4.1. Basic provisions
4.2. Analysis of the principles of conformity
4.2.1. Principle of Volterra
4.2.2. Permanent transverse expansion coefficient during creep deformation
4.3. Approximate solution of a two-dimensional contact problem of linear creep for smooth cylindrical tel ^^
4.3.1. Common case of viscoelastic operators
4.3.2. Solution for the monotonous increasing area of \u200b\u200bcontact
4.3.3. Fixed connection solution
4.3.4. Simulation of contact interaction in the case of a uniformly aging isotropic plate
Conclusions and main results of the fourth chapter
5. Creeping surface
5.1. Features of contact interaction bodies with low yield strength
5.2. Construction of a model of deformation of the surface, taking into account the creep in the case of the elliptic area of \u200b\u200bthe contact
5.2.1. Geometrical assumptions
5.2.2. Surveys model model
5.2.3. Determination of medium deformations of the rough layer and medium pressures
5.2.4. Determination of auxiliary coefficients
5.2.5. Determination of the size of the elliptic area of \u200b\u200bcontact
5.2.6. Determination of the size of the circular area of \u200b\u200bcontact
5.2.7. Determination of the width of the contact area in the form of a strip
5.3. The solution of a two-dimensional contact task for the internal touch of rough cylinders, taking into account the creep surface
5.3.1. Setting the problem for cylindrical bodies. INTEGRO-DIFFERENTIAL EQUATION
5.3.2. Determination of reprimising coefficients 160 Conclusions and main results of the fifth chapter
6. Mechanics of the interaction of cylindrical bodies, taking into account the presence of coatings
6.1. Calculation of efficient 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 scatter of physicomechanical properties
6.3. Solution of the contact task for disk and plane with elastic composite coating on the hole circuit
6.3.1. Statement of the task and basic formulas
6.3.2. Conclusion of the regional condition for movements in the area of \u200b\u200bcontact
6.3.3. Integral equation and its decision
6.4. Solution of the problem in the case of orthotropic elastic coating with cylindrical anisotropy
6.5. Determination of the influence of viscoelastic aging coating on changing contact parameters
6.6. Analysis of the features of contact interaction of multicomponent coating and disk roughness
6.7. Modeling contact interaction with taking into account thin metal coatings
6.7.1. Contact ball with plastic coating and rough half-space
6.7.1.1. The main hypotheses and the model of interaction of solid bodies
6.7.1.2. Approximate solution of the task
6.7.1.3. Determination of the maximum contact convergence
6.7.2. Solution of the contact problem for a rough cylinder and a thin metal coating on the opening circuit
6.7.3. Determination of contact hardness with internal contact of cylinders
Conclusions and main results of the sixth chapter
7. Solution of mixed boundary value problems taking into account the wear of surfaces
Interacting tel
7.1. Features of the solution of the contact problem, taking into account the wear of the surfaces
7.2. Setting and solving the problem in the case of elastic deformation of roughness
7.3. Method of theoretical wear assessment, taking into account the creep surface
7.4. Method of wear evaluation, taking into account the effects of coverage
7.5. Final comments on the formulation of flat tasks taking into account the wear
Conclusions and main results of the seventh chapter
Recommended list of dissertations
On the contact interaction between thin-walled elements and viscoelastic bodies when cutting and axisymmetric deformation, taking into account the aging factor 1984, Candidate of Physical and Mathematical Sciences Davtyan, Zavena Azibekovich
Static and dynamic contact interaction of plates and cylindrical shells with rigid bodies 1983, Candidate of Physical and Mathematical Sciences Kuznetsov, Sergey Arkadyevich
Technological support for the durability of machines based on hardening processing with simultaneous application of antifriction coatings 2007, Doctor of Technical Sciences Bersudsky, Anatoly Leonidovich
Thermoplastic contact tasks for coatings 2007, Candidate of Physical and Mathematical Sciences of Gubareva, Elena Aleksandrovna
Methods of solving contact problems for arbitrary bodies, taking into account the surface roughness by the end element method 2003, Candidate of Technical Sciences Olshevsky, Alexander Alekseevich
The dissertation (part of the author's abstract) on the topic "The theory of contact interaction of deformable solids with circular boundaries, taking into account the mechanical and microgeometric characteristics of surfaces"
The development of technology puts new challenges in the field of the study of the operability of cars and their elements. The increase in their reliability and durability is the most important factor determining the increase in competitiveness. In addition, the elongation of the service life of machinery and equipment, even to a small extent with a large saturation of technology, is equivalent to entering significant new production facilities.
The current state of the work processes of machines in combination with extensive experimental technique to determine the workloads and a high level of development of the applied theory of elasticity, with the existing knowledge of the physicomechanical properties of materials, allow the overall strength of the parts of the machines and devices with a sufficiently large warranty from breakdowns in normal conditions Services. At the same time, the tendency to reduce the largerness of the latter simultaneously with the simultaneous increase in their energy saturation is forced to revise well-known approaches and assumptions when determining the intense state of details and require the development of new settlement models, as well as the improvement of experimental research methods. Analysis and classification of the failures of mechanical engineering products showed that the main reason for the failure under operating conditions is not a breakdown, but wear and damage to their working surfaces.
Increased wear of parts in articulations in some cases violates the tightness of the working space of the machine, in others - the normal lubrication mode, in the third, leads to the loss of kinematic accuracy of the mechanism. Wear and damage to the surfaces reduce the fatigue strength of the parts and can cause their destruction after a certain service life with minor structural and technological concentrators and low rated voltages. Thus, elevated wear violate the normal interaction of parts in nodes, can cause significant additional loads and cause accidental destruction.
All this attracted to the problem of increasing the durability and reliability of cars a wide range of scientists of various specialties, designers and technologists, which allowed not only to develop a number of activities to improve the service life of the machine and create rational methods of care for them, but also on the basis of physics, chemistry, and Metal science to lay the foundations of teaching on friction, wear and lubrication in conjugation.
Currently, the 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, because To move on the calculation of the wear of materials to the tasks of structural wear resistance, the contact tasks of the mechanics of the deformable solid body have a decisive role. Significant importance for engineering practice are the solutions of contact problems of the theory of elasticity for bodies with circular boundaries. They constitute the theoretical basis for calculating such elements of machines as bearings, hinged connections, some types of gears, connections with tension.
The widest studies are made using analytical methods. It is the presence of fundamental bonds of modern integrated analysis and the theory of potential with such a dynamic region, as a mechanic, determined their rapid development and use in applied studies. The use of numerical methods is significantly expanding the ability to analyze the stress state in the contact area. At the same time, the bulky of the mathematical apparatus, the need to use powerful computing means significantly restrains the use of existing theoretical developments in solving applied tasks. Thus, one of the actual directions of development of mechanics is to obtain explicit approximate solutions of the tasks of the tasks that provide simplicity of their numerical implementation and with sufficient to practice the accuracy of the described phenomenon. However, despite the progress achieved, while it is difficult to obtain satisfactory results, taking into account local design features and microgeometry of interacting bodies.
It should be noted that the properties of contact have a significant impact on the wear processes, since due to the discreteness of contact with the touch of micronether, only on the individual sites forming the actual area. In addition, the protrusions formed during technological processing are diverse in shape and have different allocation 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 single approach to solving contact problems, taking into account the wear, the most fully taking into account both the geometry of interacting parts, microgeometric and rheological characteristics of surfaces, characteristics of their wear resistance and the possibility of obtaining an approximate solution with the least amount of independent parameters.
Communication of work with major scientific programs, themes. Studies were carried out in accordance with the following themes: "Develop a method for calculating contact stresses with an elastic contact interaction of cylindrical bodies, not described by the theory of Hertz" (Ministry of Education of the Republic of Belarus, 1997, No. 19981103); "The influence of micronics of contacting surfaces on the distribution of contact stresses in the interaction of cylindrical bodies having close in the magnitude of the radii" (Belarusian Republican Foundation Fund, 1996, No. GR 19981496); "Develop a method for predicting the wear of the suspension supports, taking into account the topographic and rheological characteristics of surfaces of interacting parts, as well as the presence of antifriction coatings" (Ministry of Education of the Republic of Belarus, 1998, No. 2009929); "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. 20001251)
The purpose and objectives of the study. Development of a single method of theoretical prediction of the effect of geometric, rheological characteristics of the roughness of the solid surfaces and the presence of coatings on the stress state in the contact area, as well as the establishment on this basis the patterns of changing the contact rigidity and wear resistance of conjugations on the example of the interaction of bodies with circular boundaries.
To achieve the goal required to solve the following problems:
Develop a method of approximate solution of problems of the theory of elasticity and viscoelasticity on the contact interaction of the cylinder and the cylindrical cavity in the plate using the minimum number of independent parameters.
Develop a nonlocal model of contact interaction of bodies, taking into account the microgeometric, rheological characteristics of surfaces, as well as the presence of plastic coatings.
Enough an approach that allows you to adjust the curvature of interacting surfaces by deformation of roughness.
Develop a method of approximate solution of contact problems for a disk and isotropic, orthotropic with cylindrical anisotropy and viscoelastic aging coatings on the hole in the plate, taking into account their transverse deformability.
Build a model and determine the effect of microgeometric features of the solid surface on the contact interaction with plastic coating on the counter.
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 antifriction coatings.
The object and subject of the study are non-classical mixed objectives of the theory of elasticity and viscoelasticity for bodies with circular boundaries, taking into account the nonlocality of the topographic and rheological characteristics of their surfaces and coatings, on the example of which in this paper developed a comprehensive method for analyzing the intense state in the area of \u200b\u200bcontact depending on the quality indicators their surfaces.
Hypothesis. When solving the boundary challenges, taking into account the quality of the surface of the bodies, a phenomenological approach is used, according to which the roughness deformation is considered as deformation of the intermediate layer.
Tasks with time-changing regional conditions are treated as quasistatic.
Methodology and methods of research conducted. When conducting research, the main equations of the mechanics of the deformable solid body, tribology, functional analysis were used. The method has been developed and substantiated to correct the curvature of loaded surfaces due to the deformations of micronovalities, which significantly simplifies the conducted analytical transformations and allows you to obtain analytical dependencies for the size of the contact area and contact voltages, taking into account the specified parameters without using the assumption about the smallness of the basic measurement of the roughness of the roughness of the roughness Contact area.
When developing a method of theoretical prediction of surface wear, observed macroscopic phenomena were considered as a result of the manifestation of statistically averaged relations.
The accuracy of the results obtained in the work is confirmed by comparisons of the resulting theoretical solutions and the results of experimental studies, as well as comparison with the results of some solutions found by other methods.
Scientific novelty and significance of the results obtained. For the first time, the example of the contact interaction of bodies with circular boundaries was summarizing, and a single method of comprehensive theoretical prediction of the influence of nonlocal geometric, rheological characteristics of rough surfaces of interacting bodies and the presence of coatings on the stress state, contact rigidity and wear resistance of conjugations were developed.
A comprehensive research complex allowed the theoretically substantiated method of solving the problems of solid mechanics, based on a consistent consideration of macroscopically observed phenomena, as the result of the manifestation of microscopic links statistically averaged at a significant portion of the contact surface.
As part of solving the problem:
A spatial nonlocal model of contact interaction of solid bodies with an isotropic surface roughness is proposed.
A method has been developed for determining the effect of the characteristics of the surface of solid bodies on the distribution of stresses.
An integro-differential equation obtained in contact problems for cylindrical bodies was investigated, which made it possible to determine the conditions for the existence and uniqueness of its solution, as well as the accuracy of the constructed approximations.
Practical (economic, social) significance of the results obtained. The results of theoretical studies are brought to acceptable techniques for practical use and can be directly applied during engineering calculations of bearings, sliding supports, gears. The use of the proposed solutions will reduce the time of creating new machine-building structures, as well as with great accuracy to predict their official characteristics.
Some results of the research performed were introduced on the NLP "Cycloprod", Altech NGOs.
The main provisions of the dissertation endowed with the defense:
Approximate solution of the problem of the mechanics of the deformed solid body on the contact interaction of the smooth cylinder and the cylindrical cavity in the plate, with sufficient accuracy of the described phenomenon when using the minimum number of independent parameters.
The solution of nonlocal boundary value problems of the mechanics of the deformable solid, taking into account the geometric and rheological characteristics of their surfaces based on the method, allowing you to correct the curvature of interacting surfaces by deformation of roughness. The absence of assumptions about the smallness of the geometric sizes of the basic lengths of roughness measurement compared with the size of the contact area makes it possible to move to the development of multi-level models for deformation of the surface of solids.
The construction and substantiation of the method for calculating the movements of the boundaries of cylindrical bodies caused by the deformation of the surface layers. The results obtained allow you to develop a theoretical approach that determines the contact stiffness of the conjugation, taking into account the joint influence of all the features of the state of the surfaces of real tel.
Modeling the viscoelastic disk interaction and cavity in a plate of aging material, simplicity of the implementation of the results of which allows them to use them for a wide range of applied tasks.
Approximate solution of contact problems for disk and isotropic, orthotropic with cylindrical anisotropy, as well as viscoelastic aging coatings on the hole in the plate, taking into account their transverse deformability. This makes it possible to assess the effect of composite coatings with a low modulus of elasticity on the loading of conjugates.
Construction of a nonlocal model and determination of the effect of the characteristics of the roughness of the solid surface on the contact interaction with plastic coating on the counter.
Development of the method of solving boundary value problems, taking into account the wear of cylindrical bodies, the quality of their surfaces, as well as the presence of antifriction coatings. On this basis, a methodology that focuses mathematical and physical methods is proposed in the study of wear resistance, which makes it possible instead of research of real friction units to make the main emphasis on the study of the phenomena occurring in the area of \u200b\u200bcontact.
Personal contribution of the applicant. All results endowed with protection are obtained by the author personally.
Approbation of the dissertation results. The results of the research given in the thesis were presented at 22 international conferences and congresses, as well as conferences of the CIS countries and the Republican, among them: "Pontryaginian 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, Perspectives" (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)," TRANSPORT STRENGROUT TRANSPORT TRANSFERS "(St. Petersburg, 1999, Russia), International Conference on MultiField Problems (Stuttgart, 1999, Germany).
Published results. According to the dissertation materials published 40 printed works, among them: 1 monograph, 19 articles in magazines and collections, including 15 articles under personal authorship. The total number of pages of published materials is 370.
Structure and scope of the dissertation. The thesis consists of introduction, seven chapters, conclusions, the list of used sources and applications. A full amount of the thesis is 275 pages, including the volume occupied by illustrations - 14 pages, tables - 1 page. The number of sources used includes 310 items.
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Conclusion of dissertation on the topic "Mechanics of the deformable solid body", Kravchuk, Alexander Stepanovich
Conclusion
In the course of the studies, a number of static and quasistatic problems of the mechanics of the deformable solid were also resolved. This allows you to formulate the following conclusions and indicate the results:
1. Contact voltages and surface quality are one of the main factors determining the durability of machine-building structures, which, in combination with a tendency to reduce the mass-darkened machines, the use of new technological and structural solutions leads to the need to revise and clarify the approaches and assumptions used in determining the stress state. , movements and wear in pairing. On the other hand, the bulky of the mathematical apparatus, the need to use powerful computing means is significantly contained by the use of existing theoretical developments in solving applied tasks and determine as one of the main directions of development of mechanics to obtain explicit approximate solutions of the tasks delivered, providing simplicity of their numerical implementation.
2. The approximate solution of the problem of the mechanics of the deformable solid body on the contact interaction of the cylinder and the cylindrical cavity in the plate with a minimum number of independent parameters is constructed, with sufficient accuracy describing the studied phenomenon.
3. For the first time, nonlocal boundary value problems of the theory of elasticity were solved, taking into account the geometric and rheological characteristics of roughness based on a method, allowing to correct the curvature of interacting surfaces. The absence of assumptions about the smallness of the geometric sizes of the basic measurement lengths of roughness compared with the size of the contact area makes it possible to correctly supply and solve the problem of the interaction of solid tel, taking into account the microgeometry of their surfaces with relatively small contact sizes, as well as go to the creation of multi-level roughness deformation models.
4. A method for calculating the highest contact displacements in the interaction of cylindrical tel is proposed. The results obtained allowed us to construct a theoretical approach that determines the contact rigidity of the conjugation taking into account the microgeometric and mechanical features of the surfaces of real tel.
5. Modeling the viscoelastic disk interaction and cavity in the plate made of aging material, the simplicity of the implementation of the results of which allows them to use them for a wide range of applied tasks.
6. Contact problems for the disk and isotropic, orthotropic, with cylindrical anisotropy and viscoelastic aging coatings on the hole in the plate, taking into account their transverse deformability, are solved. This makes it possible to assess the effect of composite antifriction coatings with a low modulus of elasticity.
7. The model was constructed and the effect of the surface microgeometry of one of the interacting bodies and the presence of plastic coatings on the surface of the counter. This makes it possible to emphasize the leading effect of the characteristics of the surface of real composite bodies in the formation of the area of \u200b\u200bcontact and contact stresses.
8. A general method of solving cylindrical bodies, the quality of their antifriction coatings has been developed. boundary value problems with the wear of surfaces, as well as availability
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