Plane Contact Problem Research Articles

Calculation of the decomposition coefficients for plane contact problem kernel in the orthonormal basis

Analytical solutions of some contact problems are infinite functional series according to the system of basic functions. When constructing such solutions, it becomes necessary to represent the kernels of integral equations describing the process of interaction in the form of two-dimensional series on a given basis. Often the kernels have a rather complex appearance, therefore, the process of finding the decomposition coefficients is a rather complex and labor-intensive process, on which the accuracy and speed of obtaining final results depend. The paper proposes a calculation method that allows calculating the coefficients of decomposition of the kernels of plane contact problems according to a special orthonormal basis that takes into account the features of contacting bodies. Other approximate formulas are also derived for the special case when coating characteristics are constant. Based on the received presentation, conclusions and recommendations are formulated.

О резонансах академика И.И. Воровича в контактных задачах с деформируемым штампом в сейсмологии

In this paper, for the first time, methods for obtaining resonant frequencies in plane and spatial contact prob-lems for deformable stamps are presented in a complex. Their existence was predicted by academician I.I. Vorovich. He constructed one-dimensional models of de-formable stamps consisting of a rigid stamp and a spring, illustrating the occurrence of resonances. Currently, thanks to the use of block element methods, it has become possible to investigate real contact problems with a deformable stamp. Earlier in the works of the authors, the existence of a new type of earthquakes, called starting, was established. In the course of the study, some functionals remained uncomputed, the role of which was not clear. It is shown that they serve to construct dispersion curves giving resonant frequencies predicted by academician I.I. Vorovich. A study is carried out for different types of contact problems and shows how the method of constructing dispersion equations becomes more complicated, both due to the complication of the shape of stamps and the complication of the rheologies of deformable stamps. Semi-infinite and deformable stamps of simple rheology in the form of a quarter plane are considered. In addition, a semi-infinite stamp with the rheology of Kirchhoff plates is analyzed. The obtained results provide an answer to one of the methods of constructing dispersion equations for I.I.Vorovich resonances in contact problems for different deformable stamps. The study is closely related to the identification of new precursors of seismicity, as well as to solve some problems in the theory of strength.

Some remarks on the two-dimensional contact of dissimilar elastic solids under non-slipping boundary conditions

Spence's self-similarity approach is often preferred to an incremental method for solving plane, non-slipping normal contacts of dissimilar elastic solids under progressive loading. The number of plane contact problems solved in this way so far is nevertheless very limited, since explicit expressions especially for the pressure distribution and the tangential tractions within the contact area require the solution of complicated singular integral equations. In contrast, the application of the incremental method leads to much simpler integral equations, which are derived here and applied to solve some new non-slipping contact problems including the one characterized by an initial power-law gap profile function. The incremental method is also capable to determine the so-called interface mismatch eigenstrain. In Spence's method, however, the determination of its magnitude represents a mandatory step within the solution procedure. By means of a comprehensive comparison of both methods this work provides new insights into the solution of non-slipping plane normal contact problems including explicit analytical solutions for the interface mismatch eigenstrain.

On the Mechanical Concept of Self-Assembly of Nanomaterials

The article presents the mechanical concept of self-assembly of nanoparticles. It is assumed that nanoparticles are deformable stamps in a plane dynamic contact problem, lying on the boundary of a multilayer deformable medium. The constant vibration in the microcosm is caused by the oscillatory mode by the energy of phonons and magnons. Earlier, in the works of the authors, the mechanical concept of self-organization of nanoparticles was presented. It is based on high-frequency resonance, which causes the formation of standing waves. They localize the available aggregates of nanoparticles on the crest of standing waves. The self-assembly of nanoparticles is based on resonance, previously predicted by Academician I. I. Vorovich and inherent only in deformable dies in contact problems on a multilayer medium. Deformable nanoparticles are modeled by fractals representing packed block elements described by the Helmholtz equation. The resonance of the deformable dies allows the capture of nanoparticles, dictated by the Coulomb forces of attraction. It is shown that the combination of two fractals generates a new fractal with a combined carrier, and in the case of multiple association, a fragment of a nanomaterial is obtained. To implement the study, for the first time it was possible to construct a high-precision approximate solution of a plane contact problem on the action of a stamp of any finite size on a multilayer base. This result is dictated by the need for an analytical construction of the theory of self-assembly of nanomaterials.

Plane Contact Problems for an Elastic Foundation with Two Bedding Coefficients

We consider the problem of contact interaction of the punches of various shapes with an elastic foundation modeled by an elastic layer with two bedding coefficients. The application of this model makes it possible to determine not only the stress states of the punch and elastic foundation but also their displacements. We perform the numerical analyses and compare the results obtained for different models of elastic foundations and different shapes of the punches.

ПЛОСКА ЗАДАЧА ПРО КОЛИВАННЯ БАЛКОВОЇ ПЛИТИ НА ПРЯМОКУТНІЙ ОСНОВІ

he plane contact problem of a beam’s oscillations on a rectangular elastic base is considered. The vertical edges of the rectangular base are in conditions of the nonfriction contact, the lower edge is fixed, and a beam (beam plate) with free ends is attached to the upper edge. The normal loading is applied to the beam and harmonically changes in time. To solve the boundary valued problem for the elastic base the integral transform method is applied. The apparatus of the Green’s function is used to construct the solution for the boundary valued problem for a beam. The displacements of the rectangular base and the deflection of the beam were found. The interface condition between the base and the plate is used to derive the integral singular equation relatively the dynamical contact stress. The orthogonal polynomial method was used to solve the integral equation. The investigation of the oscillations’ frequency influence on the deflection of beam and elastic rectangular base’s displacements and stress was conducted.

Contact problems for a transversely isotropic layer

Two spatial, one axisymmetric and two plane contact problems are considered for a transversely isotropic elastic layer with one face subjected to sliding support. In the spatial and plane contact problems, the planes of isotropy may be either parallel or perpendicular to the layer faces. In the case of axial symmetry, the planes of isotropy are parallel to the layer faces. By using Fourier integral transforms, the contact problems are reduced to integral equations with respect to the contact pressure, the limiting cases of which are the well-known equations of the corresponding problems for an isotropic layer. For solving the spatial problems with unknown contact domains, the nonlinear boundary integral equations method is used, which make it possible to determine the contact pressure and the contact domain simultaneously. To extract the kernel principal part of the spatial problem integral equation when the isotropy planes are perpendicular to the layer faces, it is used the kernel of the integral equation of the corresponding contact problem for a transversely isotropic half-space obtained earlier without quadratures. The integral equation of the axially symmetric problem is reduced to a Fredholm integral equation of the second kind with the help of the method of pair equations, and the method of mechanical quadratures is used for numerical solutions. Plane problems are solved in a closed form based on special approximations of the kernel symbols. The approximations accuracy grows as anisotropy increases. Here, the anisotropy level can be characterized by the difference between ratio of a characteristic equation roots and unit because the unit value corresponds to the isotropic case. Mechanical characteristics as well as errors of the approximations are calculated for well-known transversely isotropic materials.

Contact problems for an inhomogeneous elastic wedge with variable Poisson’s ratio

Plane contact problems of the elasticity theory are investigated for a wedge when Poisson’s ratio is an arbitrary smooth function with respect to the angular coordinate while shear modulus is constant. For this case Young’s modulus is also variable with respect to the angular coordinate. A finite contact domain is given on one wedge face, it does not include the wedge apex, while the other wedge face is rigidly fixed (problem A) or stress-free (problem B). To reduce the problems to integral equations with respect to the contact pressure, we use the general Freiberger type representation for the solution of elastic equilibrium equations written in polar coordinates with variable Poisson’s ratio. Exact solutions of auxiliary problems are constructed with the help of Mellin integral transforms. The regular asymptotic method employed is effective for contact domains relatively distant from the wedge apex. It is shown that logarithmic terms appear in the asymptotic solutions for the inhomogeneous material which are missing in the well-known asymptotics for the homogeneous one. In contact problem C which is corresponding to problem A, the friction and roughness are taken into account in the contact region. The roughness of the wedge surface is simulated by a Winkler type coating. The collocation method is used for solving integral equations of the second kind. Unlike problem A, in problem C the contact pressure does not have square root singularities at end-points where it takes finite values. Calculations are made for the cases when Poisson’s ratio and Young’s modulus increase or decrease from the surface of the wedge.

Contradictions in the Plane Contact Problem of the Theory of Elasticity on the Compression of Cylinders in Contact with Parallel Generators

The inapplicability of the reference formula for determining the convergence of two statically compressed parallel cylinders made of a homogeneous, isotropic and physically linear material has been proved due to a well-known logarithmic feature in the plane classical problem of mechanics of elastic solids. In the special case of the elastic interaction of a cylinder with a half-plane, when one of the radii has an infinite length, it has been found that the convergence also becomes equal to infinity. This paradoxical result contradicts not only the physical and mechanical meaning of the process under study, but also confirms the inadequacy of Flamant model of a simple radial stress state in determining displacements. The authors have proposed an algorithm for eliminating the contradictions based on the solution of Fredholm integral equation of the first kind. In the future, it can be considered as a new fundamental and applied problem of the theory of elasticity, which is of a great importance for a refined assessment of the contact strength and stiffness of the cylindrical parts of load-bearing structures taking into account the general and local deformations (cylindrical rollers, gears, road surfaces, when they are compacted with steel rollers, etc.) on the basis of a flat Flamant calculation scheme considering three stress components and the width of the cylinder contact area previously developed and mathematically approximated by the authors.

Predictive Estimation of Sliding Bearing Load-Carrying Capacity and Tribological Durability

A computational method is presented as a method for solving a plane contact problem of the theory of elasticity to determine the contact strength and tribological durability of sliding bearings. The effect of load and radial clearance on the initial contact pressures and their reduction due to wear is studied. The durability of the bearing is estimated. Qualitative and quantitative regularities of changes in contact parameters and durability from the factors under study are established. In particular, it has been shown that both contact angles and maximum contact pressures are approximately linearly dependent on the load, and the durability decreases nonlinearly with increasing load.

A squircle-shaped punch design for efficient contact performance of elastic materials

A new curvature, called the squircle, is proposed as a punch profile to provide superior contact performance for bulk elastic materials. Plane contact problems of a rigid squircle-shaped punch pressed onto an elastic half-plane are treated via an integral equation (IE) approach which is evidently substantiated by a finite element (FE) approach. Compared to the piecewise-defined curvatures, the continuous surface gradient of the squircle profile enables the IE formulations more tractable. In the presence of sliding friction, the consistency condition in the IE approach is implemented through a new iteration procedure. Subsurface stresses are demonstrated via the FE approach. The squircle punch profile is shown to induce a low-magnitude and smoother pressure distribution on an elastic surface, promising to cause less wear under the action of tangential oscillations.

RELATIVE PORE VOLUME UNDER INDENTATION OF POROUS MATERIALS

Investigation of the function of the relative volume of pores under the load action is carried out on the base of the solution of the static contact problem of the indentation of a layer made of a material with voids or unfilled pores. A rigid strip indenter with a flat base is pressed into a porous layer that is adhered to a non-deformable base along the lower boundary. The formulated 3D problem of the indentation of a porous layer is reduced to solving the plane contact problem of the indentation of a porous strip. The plane contact problem is reduced to solving an integral equation for unknown contact stresses, the solution of which is constructed by the method of successive approximations in the form of an asymptotic expansion in the dimensionless parameter of the problem. The obtained contact stresses and the force acting on the indenter made it possible to study the influence of the nonclassical moduli of the layer porous material (the connectivity modulus and pore rigidity modulus) on the main contact characteristics and on the distribution of the function of the relative pore volume. The connectivity modulus increase leads to an increase in the compliance of the layer porous material, the pore rigidity modulus increase leads to an increase in the rigidity of the layer porous material. The maximum value of the distribution function of the relative pore volume in the porous material of the layer is achieved under the indenter base centre, regardless of the change in the porous material non-classical moduli.

Predviđanje kontaktnog pritiska i resursa metalno-polimernih linearnih cilindričnih kliznih ležajeva

Different types of linear bearings (guides) are widely used in practice in various equipment, as in many other areas of human activity. In particular, this applies to cylindrical guides (linear plain bearings) of reciprocating motion. However, despite their considerable distribution, in fact, there are no reasonable methods for calculating the wear and service life of not only metal but also metal-polymer linear bearings. According to the author's method, the influence of load, base diameter and radial clearance on the maximum initial pressures in this bearing was investigated within the plane contact problem of the theory of elasticity. Further, using the computational method according to the author's tribokinetic model of wear during sliding friction, the effect of the composite bushing wear on the change of the initial contact characteristics (contact pressures and contact angle) was evaluated. The forecast calculation of the service life of the bearing depending on the above factors is also carried out. Quantitative and qualitative regularities of dependence of contact pressures and a resource on the accepted factors are established.

Load Transfer from an Infinite Inhomogeneous Stringer to a Prestressed Elastic Strip Clamped at One Edge*

The plane contact problem of the transfer of a horizontal load from a weakly inhomogeneous infinite stringer to a prestressed elastic strip clamped at one edge is solved using the linearized theory of elasticity. The solution has a general form for the theory of large initial deformations and different theories of small initial deformations for an arbitrary elastic potential. The problem for the normal and tangential contact stresses is reduced to a governing system of recurrent systems of integro-differential equations, which is solved in powers of a small parameter. The zero-order approximation of the inhomogeneous problem is derived using the Fourier transform. The contact stresses are represented by Fourier integrals.

Plane contact problem on the interaction of a pre-stressed strip with an infinite inhomogeneous stringer

The article is devoted to the study of problems of contact interaction of an infinite elastic inhomogeneous stringer with a prestressed strip clamped along one edge. As a result of the research, we have obtained a resolving system of recurrent systems of integro-differential equations. In general, the studies were carried out for the theory of large initial and various versions of the theory of small initial deformations within the framework of the linearized theory of elasticity with an elastic potential of an arbitrary structure. Integral integer differential equations are obtained using the integral Fourier transform. Their solution is presented in the form of quasiregular infinite systems of algebraic equations. The article also investigates the influence of the initial (residual) stresses in strips on the distribution law of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed form using the Fourier transform. The stress expressions are represented by Fourier integrals with a fairly simple structure. The influence of the initial stress on the distribution of contact stresses has been studied and mechanical effects have been found under the action of concentrated loads.

Plane contact problems for an elastic foundation with two bedding coefficients

Plane contact problems for an elastic foundation with two bedding coefficients

On the contact problem of a moving rigid cylindrical punch sliding over an orthotropic layer bonded to an isotropic half plane

In this study, the frictional moving contact problem for an orthotropic layer bonded to an isotropic half plane under the action of a sliding rigid cylindrical punch is considered. Boundary conditions of the problem include the normal and tangential forces applied to the layer with a cylindrical punch moving on the surface of the layer in the lateral direction at a constant velocity V. It is assumed that the contact area is subjected to the sliding condition where Coulomb’⣙s law is used to relate the tangential traction to the normal traction. Using the Fourier integral transform technique and Galilean transformation, the plane contact problem is reduced to a singular integral equation in which the unknowns are the contact stress and the contact width. The singular integral equation is solved numerically using Gauss–Jacobi integration formulae. Numerical results for the contact widths and the contact stresses are given as a solution.

Fretting fatigue under variable amplitude loading considering partial and gross slip regimes: Numerical analysis

The main aim of this work was to numerically investigate the effect of sequences of variable amplitude shear loading on contact tractions/stresses and on the development of life estimation methodologies for these more challenging loading histories. In this setting, a finite element (FE) model has been implemented in order to simulate a cylinder on plane contact problem for an aeronautical Al 7075-T651 alloy. High-Low (H-L) and Low-High (L-H) shear amplitude loading sequences under partial and gross slip conditions were considered in three different case studies. For the case studies where gross slip was involved, the proposed life estimation procedure was enhanced to include the effect of material loss due to wear. More explicitly, such procedure was based on the use of (i) the SWT (Smith-Watson-Topper) multiaxial fatigue model, (ii) the Theory of Critical Distances, (iii) Miner's cumulative damage rule and (iv) the numerical update of the contact surfaces profiles (for loading blocks under gross slip). The numerical analyses for the case studies under partial slip revealed that, when the presence of wear is neglected in the modelling, the Miner's rule provided a divergence between the expected life for the H-L and L-H loading blocks. This difference was considerably larger when gross slip took place. For the case study involving the presence of gross slip in one of the shear loading blocks, the calculated life tended to infinite when the damage generated by the high amplitude block is greater than a certain critical value. These studies are the basis for an experimental programme which will be carried out in the 4 actuators fretting fatigue rig of the University of Brasilia.

Plane contact problem between a rigid punch and a bidirectional functionally graded medium

Analytical and computational solutions are presented for the sliding frictional contact problem between a bidirectional functionally graded material (BFGM) half-plane and an arbitrarily shaped rigid punch. The plane-strain elasticity is regarded in the solutions. Independent shear modulus variations are imposed through the lateral and the thickness directions of the elastic medium. Fourier transformation techniques are utilized in the derivation of the field quantities. The quartic characteristic equation is solved using the Ferrari's method. A singular integral equation (SIE) of the second kind is then formulated considering a known displacement gradient on the contact surface. Solution of the SIE is performed for the flat and triangular punch profiles via an expansion-collocation method, quadrature integration techniques and a recursive integration method for the Cauchy integral. Finite element analyses (FEA) the same contact problems are also implemented choosing the augmented Lagrange method as the iterative algorithm. The parametric analyses indicate the reliability and validity of both procedures developed in this study. Extra numerical results are generated to show the influences of the problem parameters on the surface stresses and contact forces. It is observed that the contact responses can be influentially controlled upon imposing a bidirectional gradation in an elastic material.

The plane contact problem for an initially stressed elastic strip reinforced by an infinite inhomogeneous stringer

The plane contact problem for an initially stressed elastic strip reinforced by an infinite inhomogeneous stringer