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Frost-resistant rubbers retain their highly elastic properties over a wide temperature range. They are used in various friction units (e.g., seals), but their high friction coefficient and low wear resistance lead to the need for frequent replacement. In this paper, we propose applying thin (several hundred microns) UHMWPE coatings to formed rubber rings. The application technology depends on the required coating thickness. Friction tests of the coatings and pure UHMWPE were performed using the ball-on-disk (unidirectional sliding) scheme for various loads and velocities. In the experiments, the friction coefficients and temperatures near the contact area were determined. Friction tracks were studied using microscopy methods. The sliding contact of the ball and the two-layer material was modeled to obtain the dependences of the deformation component of friction on the sliding velocity for coatings of different thicknesses. UHMWPE is sensitive to frictional heating, so the thermal problem of determining the temperature in the contact area was also solved. It is shown that the minimum friction coefficient occurs for coatings with a thickness of 600 μm. At the same time, in the case of the 300 μm coating, the surface of the friction track is practically no different from the initial one. Thus, the studied combination of polymers provides antifrictional properties and wear resistance to the surface layer while maintaining the damping properties of rubber. [ABSTRACT FROM AUTHOR]

The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed. [ABSTRACT FROM AUTHOR]

The problems of constructing an exact or approximate solution of system of singular integro‐differential equations related to the problems of adhesive interaction between elastic thin finite or infinite nonhomogeneous patch and elastic plate are investigated. For the patch loaded with horizontal and vertical forces, the usual model of beam bending in combination with the uniaxial stress state model is valid. Using the methods of theory of analytic functions, integral transformation, or orthogonal polynomials, the singular integro‐differential equations reduced to the different boundary value problems (Karleman type problem with displacements and Riemann problem) of the theory of analytic functions or to the infinite system of linear algebraic equations. The asymptotic analysis of problem is carried out. [ABSTRACT FROM AUTHOR]

Solution of sparse linear complementarity problem (LCP) has been widely discussed in many applications. In this paper, we consider the ℓ p regularization problem with nonnegative constraint for sparse LCP, and propose algorithms based on the iterative reweighted method to approach a sparse solution of the LCP, and then show the convergence to the stationary point of ℓ p regularization problem. Numerical results on simulated data exhibit an excellent performance of the proposed algorithms on approaching a sparse solution of the LCP. Finally, we apply this method to the frictional and frictionless contact problems. The numerical experiments demonstrate that the contact problems can be efficiently solved by the proposed algorithm. • The ℓ p regularization problem with nonnegative constraint is proposed for sparse LCP. • The iterative reweighted algorithms are proposed with convergence guarantees. • Different merit functions for LCP are discussed in numerical experiments. • Numerical simulations demonstrate the superiority of proposed algorithms. • Application to contact problem exhibits the effectiveness of proposed method. [ABSTRACT FROM AUTHOR]

We deal with the Signorini contact problem between two Timoshenko beams. In this work we use the theory of semigroups to show the existence of solutions that decay uniformly to zero. This method is new and more effective than the widely used energy method. This is because in particular we obtain uniform decay of the solutions to zero for any boundary condition. A second important point is that we can take advantage of stabilization results of others linear dynamic systems with different dissipative mechanisms and apply them through our method for Contact Problems (see Sect. 4). Finally, thanks to Lipschitzian perturbations we can generalize the Signorini problem to more general semi linear problems in a simple way (see Sect. 4.3). [ABSTRACT FROM AUTHOR]

A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

This study offers a comparative study of the analytical and numerical methods for investigating a contact problem. The contact problem comprises a functionally graded layer supported by a half‐plane and loaded with a distributed load from the top surface. First, the analytical and numerical solutions to the problem are acquired by utilizing a theory of elasticity and finite element method, respectively. The problem is transformed into a system of integral equations in which the contact stress is an unknown function. The solution of the integral equation was achieved with Gauss–Jacobi integration formulation. The finite element model of the problem is created using ANSYS software, and the two‐dimensional analysis of the problem is performed. Results were obtained from the samples for different material properties and loading conditions. The distributed load width and non‐homogeneity parameters significantly impact on contact mechanics. The results indicate that the contact area and the contact stress obtained from finite element method (FEM) are close to the analytical results. As a result, acceptable error rates were obtained. Finally, this study provides evidence of a good agreement between the two methods. [ABSTRACT FROM AUTHOR]

Wet assembly is the most common method of joining parts in the aircraft industry. In this case, a layer of liquid sealant is applied to the contact surface of one of the parts, after which the structure is bolted. Despite the practical importance of this issue, until now there has been no mathematical model or software tool that would allow simulating the wet assembly process. This study is devoted to modeling the wet assembly process of large-scale aircraft structures, taking into account the flow of sealant applied between the joined parts. The original mathematical model includes a full two-way interaction of deformable parts and a viscous liquid sealant. Such effects as contact interaction of parts, the presence of the free surface of sealant layer, the sealant curing, deformation and loosening of fasteners are taken into account. The developed mathematical model is applied to the simulation of the assembly process of wing to fuselage upper joint. A special full-scale model has been developed in order to reproduce the effects observed in practical assembly. The numerous numerical experiments demonstrate these effects and illustrate how different parameters of the process affect both intermediate and final results of the assembly. The performed investigation shows how mathematical modeling can contribute to the improvement of existing and development of new technologies on the assembly lines of aircraft manufacturing companies, where any experiments are extremely expensive or simply impossible. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory. [ABSTRACT FROM AUTHOR]

An effective numerical and analytical method is proposed for studying the force state inside the contact spots of elastic composite spherical shells resting on two flat surfaces inclined at an angle to each other. The method is based on constructing the influence function for the shell in an axisymmetric formulation, followed by its expansion into a Fourier series using the Legendre and Gegenbauer polynomials. A resolving system of equations for the corresponding coefficients is obtained. The developed methodology will allow for a more correct study of the dynamics of various robotic systems, the main controlled elements of which are elastic spherical shells that carry out complex kinematic movements on various surfaces. These systems, in particular, include a “butterfly robot”, the main controlled element of which is a thin spherical shell, rolling with simultaneous spinning and sliding along two parallel plates, as if on rails.

One of the current and widely used non-destructive testing methods for monitoring and determining the elastic properties of materials is nanoindentation. In this case, to interpret the test results, a non-trivial task arises of constructing an adequate mathematical model of the indentation process. As a rule, in many cases, analytical formulas are used that are obtained from an elastic linear formulation of problems about the introduction of a non-deformable stamp into a homogeneous elastic half-space. Currently, the numerical formulation of the problem makes it possible to obtain and use a numerical solution obtained taking into account the complete plastic nonlinear behavior of the material. In this work, a study of contact problems on the introduction of a spherical and conical indenter into an elastoplastic homogeneous half-space was carried out. To verify the numerical solution, the problem of introducing a spherical and conical indenter into an elastic homogeneous half-space was also solved and compared with known analytical solutions. Issues of convergence and tuning of numerical methods, the influence of plasticity and the applicability of analytical solutions are explored. Problems are solved numerically using the finite element method in the Ansys Mechanical software package.

The problem of the time-harmonic behavior of two deformable semi-infinite stamps lying on a deformable base is considered. It is assumed that the stamps converge with parallel ends in such a way that they form a crack, defect, or tectonic fault in the convergence zone. The deformable die material has a simple rheology described by the Helmholtz equation. To consider the cases of deformable stamps of complex rheologies, a new universal modeling method can be used. It allows solutions of vector boundary value problems for systems of partial differential equations describing materials of complex rheologies to be represented as decomposed by solutions of individual scalar boundary value problems. A high-precision solution to the boundary value problem is constructed, which makes it possible to obtain a dispersion equation describing resonant frequencies. The existence of resonant frequencies for deformable stamps was predicted in the works of I. I. Vorovich. The result remains valid for the case of absolutely solid semi-infinite stamps. Earlier, it was shown that resonances arise in the contact problem of the oscillation of two absolutely rigid stamps of finite dimensions on a deformable layer. However, the dynamic contact problem for the case of two semi-infinite stamps acting on a multilayer medium has not been studied before. The study is based on the block element method, which makes it possible to construct exact solutions to boundary value problems for partial differential equations. In addition, factorization methods are used and some subtle properties of the Wiener – Hopf equations are used, in particular, those belonging to the famous mathematician M. G. Crane. The proposed methods make it possible to conduct research for the entire frequency range and an arbitrary distance between the ends of semi-infinite plates. The results of the study can be used to evaluate the strength properties of structures with contact joints made of different types of materials in dynamic modes.

This work deals with a saddle point formulation of parameter identification in linear elastic contact problems with friction. Using the primal–dual formulation of the constrained minimization problem and given observations, we estimate the Lamé coefficients through the penalization and dualization of the considered inverse problem. By Fenchel duality, we provide the dual energy function associated with the constraint. We prove the existence of a solution to the regularized parameter identification problem as well as the convergence of the penalized problem to the original one. An augmented Lagrangian formulation of the inverse problem and the existence of its saddle point are provided. By means of the alternating direction method of multipliers (ADMM) and a primal–dual active set strategy (PDAS), we solve the problem numerically and illustrate our approach.

A coupled contact problem involving the interaction of a punch with a half-space under a follower load has been considered. The punch is pressed against the elastic half-space by a force that acts parallel to the punch axis and is offset from the center of the punch base. The direction of the force remains unchanged relative to the punch as the force increases, thus defining it as a follower force.Amathematical model has been formulated, and an analytical solution has been obtained. The specific motion behavior of the punch under increasing force has been analyzed, including the determination of the maximum angle of rotation. The variation in the horizontal motion of the punch depending on the applied force has been studied. Additionally, having considered a modification of the problem where the force direction is reversed, significant differences in the punch displacement have been found. [ABSTRACT FROM AUTHOR]

This work focuses on the numerical performance of HHT- α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche's method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α ~ for the HHT- α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of a class of evolutionary variational–hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modeling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb's friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient μ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable α and the slip rate | u · τ |. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a coupled system which consists of a fractional differential hemivariational inequality (FDHVI, for short) combined with a fractional hemivariational inequality (FHVI, for short). By utilizing Rothe method and surjectivity result, the existence of coupled system is established. In addition, we apply the results to a new quasistatic contact problem, in which the constitutive equations are described by the fractional Kelvin-Voigt laws. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a new class of coupled variational–hemivariational inequalities with unilateral constraints and history‐dependent operators. First, by employing a fixed‐point theorem for history‐dependent operators combined with surjectivity theorem, we establish the existence and uniqueness of solution for coupled system. Second, we obtain a convergence result, namely, related penalized problems of solutions converge to the solution to the original system. Finally, we conclude with a new contact problem of the body pressure of two elastic materials. [ABSTRACT FROM AUTHOR]

Under antiplane deformation, the problem of the contact interaction of a thin infinite stringer of finite width with an elastic 3D wedge-shaped body of an arbitrary opening angle is considered. It is assumed that one face of the wedge is rigidly clamped while a stringer is attached to the other face, which does not extend to the top of the wedge. The deformation of a stringer is described by the well-known Melan model for antiplane deformation, and the stress–strain state of an elastic wedge is described by the equations of elasticity theory. Two formulations of the contact problem are discussed according to the two models of stringer deformation. In the first formulation, as usual, external tangential loads and concentrated forces acting on the stringer are specified, and it is required to determine the tangential contact stresses arising under the stringer and axial stresses in it. In the second formulation of the problem, the elastic displacements of the stringer points are predetermined; it is required to determine the force factors acting on the stringer that provide the specified mode of elastic displacements. Using the Mellin integral transform, solving the contact problem under consideration is reduced to solving a singular integral equation (SIE) with a kernel in the form of a sum of the Cauchy kernel and a regular kernel in the first formulation and to solving a Fredholm integral equation (IE) of the first kind with a symmetric logarithmic kernel in the second formulation. The solution of both integral equations is constructed by the method of mechanical quadratures, based on the use of well-known Gauss-type quadrature formulas for calculating singular integrals at Chebyshev nodes, as well as Gauss-type quadrature formulas for calculating integrals containing logarithmic and signum regular kernels. Ultimately, solving the equation is reduced to solving a finite system of linear algebraic equations (SLAE). A numerical analysis of the problem was carried out, and regularities in the change in the main mechanical quantities of the problem were revealed in a quite wide range of changes in the characteristic physical and geometric parameters. [ABSTRACT FROM AUTHOR]

A variational inequality describing an elastic body with a finite set of rigid inclusions is considered. The Signorini condition is imposed on a part of the boundary of the body. On the other part a homogeneous Dirichlet boundary condition is specified. The inclusions are arranged such that distance between any two inclusions is not less than a given positive number. All inclusions are located at a nonzero distance from the outer boundary. An inverse problem is investigated, which consists in identification of positions of the rigid inclusions from the measurement of displacements on an observation boundary. Continuous dependency of the solution of the forward problem on parameters of inclusions' location and rotation is established. This provides existence of a solution for the inverse identification problem. [ABSTRACT FROM AUTHOR]

An exact solution to the contact problem of indentation of an absolutely rigid stamp with a straight base, taking into account friction, into one of the edges of a finite crack located in a homogeneous elastic plane is constructed. It is assumed that tangential contact stresses are directly proportional to normal contact pressure. In this case, it is believed that the friction coefficient is directly proportional to the coordinates of the contacting points of the contacting surfaces. The defining system of equations for the problem is derived in the form of an inhomogeneous Riemann problem for two functions with variable coefficients and its closed solution in quadratures is constructed. Simple formulas for contact stresses and the normal dislocation component of displacements of crack edge points are obtained. The patterns of changes in contact stresses and crack opening depending on the maximum value of the friction coefficient have been studied. [ABSTRACT FROM AUTHOR]

A stochastic model of the wear process of a shaft lip seal that takes into account random changes in the temperature and external load is described. The results of a numerical analysis of the process of seal wear in relation to the conditions of operation in open space at near-Earth orbital stations are presented. The importance of taking into account random changes in temperature and external load for predicting the seal durability is evaluated. [ABSTRACT FROM AUTHOR]

In this study, moving contact problem for a rigid punch and a functionally graded orthotropic layer bonded to an isotropic homogeneous layer is solved, semi-analytically. The cylindrical punch moves steadily with a constant subsonic velocity over the upper layer and transmits concentrated normal and tangential forces. Using Fourier transform and Galilean transformation, the contact problem is converted to a Cauchy-type singular integral equation of the second kind, in which the contact stress and contact width are the unknowns. The numerical solution of the singular integral equation is obtained by using Gauss–Jacobi integration formulas. Numerical results for the contact stress, in-plane stress, and the contact width are given. [ABSTRACT FROM AUTHOR]

The novelty of the presented results is the deriving of exact solutions of axisymmetric poroelasticity problems for finite solid and hollow cylinders. The statement of these contact problems is given under the assumptions of Biot's model for poroelastic materials. A simulation of the poroelastic response of a cylinder is implemented using the apparatus of boundary problems. By the method of integral transforms, the original boundary problem is reduced to a one-dimensional boundary problem in the transform domain. Based on the exact solution of the latter one, it is manageable to derive the explicit formulas for displacements, stress, and pore pressure. The investigation of contact stress and pore pressure distribution inside the cylinder regarding its geometrical ratios, type of applied loading, and poroelastic material is carried out. [ABSTRACT FROM AUTHOR]

A two-dimensional problem of the theory of elasticity is solved for a circular disk with an inclined edge crack, on a finite section of the boundary of which uniformly distributed pressure and tangential forces (friction forces) are applied. The contact load is balanced by concentrated forces and moment applied in the center of the disk. Contact with the friction of the crack faces is allowed. The problem is reduced to solving a system of two complex singular integral equations of the first kind with the Cauchy kernel. Numerical values of the stress intensity factors are obtained, depending on the location of the contact load on the disk boundary, as well as on the crack length and friction coefficients at the contact load application section and between the crack faces. The solution is presented using a railroad wheel as an example. [ABSTRACT FROM AUTHOR]

Frost-resistant rubbers retain their highly elastic properties over a wide temperature range. They are used in various friction units (e.g., seals), but their high friction coefficient and low wear resistance lead to the need for frequent replacement. In this paper, we propose applying thin (several hundred microns) UHMWPE coatings to formed rubber rings. The application technology depends on the required coating thickness. Friction tests of the coatings and pure UHMWPE were performed using the ball-on-disk (unidirectional sliding) scheme for various loads and velocities. In the experiments, the friction coefficients and temperatures near the contact area were determined. Friction tracks were studied using microscopy methods. The sliding contact of the ball and the two-layer material was modeled to obtain the dependences of the deformation component of friction on the sliding velocity for coatings of different thicknesses. UHMWPE is sensitive to frictional heating, so the thermal problem of determining the temperature in the contact area was also solved. It is shown that the minimum friction coefficient occurs for coatings with a thickness of 600 μm. At the same time, in the case of the 300 μm coating, the surface of the friction track is practically no different from the initial one. Thus, the studied combination of polymers provides antifrictional properties and wear resistance to the surface layer while maintaining the damping properties of rubber.

• The indentation behaviors of multiferroic composite materials under an axisymmetric power-law shaped punch is investigated. • The full analytical solutions for the coupled physical fields at the surface of multiferroic half-space are obtained. • The explicit analytical expressions of load-displacement relationship are derived for the first time. • The shape index has a significant effect on both the load-displacement relationship and distribution of generalized stress fields. Nanoindentation technique has been widely employed to characterize various material properties of multiferroic composite medium, where the indenter tip may have more general profile rather than just three common simple shapes (flat, cone and sphere). In this paper, the frictionless indentation problem of a transversely isotropic multiferroic half-space punched by an axisymmetric power-law shaped indenter is investigated. By virtue of the Hankel integral transform, the stated mixed boundary value problem is reduced to dual integral equations, which are then solved analytically. Closed-form solutions for the coupled physical fields at the surface of multiferroic half-space under the action of the power-law shaped indenters with four different electromagnetic properties are obtained, which can include the existing results in literature as special cases. The explicit analytical expressions of load-displacement relationship for the power-law shaped indenter are derived for the first time, which provide the theoretical foundation for characterizing material properties of multiferroic composite medium. The effects of the shape index and electromagnetic properties of the indenter on indentation behaviors are discussed in detail. It is found that the shape index has a significant effect on both the load-displacement relationship and the distribution of the generalized stress fields. The analytical solutions derived from this work can serve as benchmarks for experimental test and characterization of multiferroic composite materials. [ABSTRACT FROM AUTHOR]

A new model is proposed for contacting a stamp with a flat base with an elastic half-plane in the presence of friction between them. As in the model proposed by L.A. Galin for such a problem, it is assumed that in the central part of the contact zone there is adhesion of the points of the half-plane with the base of the stamp, which is caused by the presence of friction between the stamp and the half-plane, and sliding occurs at the edges of the contact zone. The proposed model differs from the model of L.A. Galin in that the distribution of friction forces in the sliding zone is considered uniform, and not according to the law of dry friction. A system of governing equations has been constructed, and the features of the behavior of the sought functions at the point of separation of the adhesion and sliding zones, as well as at the end point of the contact zone, have been found. The problem was solved by the method of mechanical quadratures. Numerical analysis was carried out. [ABSTRACT FROM AUTHOR]

The article considers the plane stress state of a homogeneous elastic space, weakened by a periodic system of finite tunnel cracks. It is assumed there are also resting shear stresses under the punches, apart from the normal stresses. The shear stresses are directly proportional to the normal contact pressure with a proportionality coefficient that depends on the coordinates of the points of the contacting surfaces. Based on discontinuous solutions for the equations of the theory of elasticity for a plane with a periodic system of cracks, a governing system of equations for the problem is obtained in the form of a system of singular integral equations. The solution of system is constructed by the numerical-analytical method of mechanical quadratures. Numerical calculations were carried out and patterns of changes in important physical and mechanical characteristics were studied depending on the physical, mechanical and geometric parameters of the problem. [ABSTRACT FROM AUTHOR]

The variation analysis is a key tool for ensuring the high quality assembly in the process of developing the technology for manufacturing of aircraft parts. One of the main factors in variations is the deviations in the positioning procedure. This paper is devoted to the development of an approach that allows taking into account the variations during positioning and merging it with the special algorithm of contact problem solving. The impact of varied boundary conditions is incorporated into an additional vector of forces that can be interpreted as reactions to the shift of supports. The obtained results are illustrated with a case of wing-to-fuselage assembly. [ABSTRACT FROM AUTHOR]

A plane contact problem on the interaction of a rigid stamp with a poroelastic strip lying on a Winkler base was investigated. The deformation of the strip was modeled based on the equations of the Cowin–Nunziato poroelastic bodies theory. It was assumed that the base of the stamp has a flat or parabolic shape and there is no friction in the contact zone. With the help of the Fourier integral transformation, the problems stated were reduced to an integral equation for an unknown contact stress, which was solved with the use of the collocation and the asymptotic methods. The values of contact stresses, sizes of the contact area in the case of a parabolic stamp and the relative deformation of the surface outside the stamp were determined. A comparative analysis of the studied quantities for various values of the poroelastic strip parameters and the coefficient of subgrade resistance of the Winkler base was carried out. Numerical results are presented in the form of tables and graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study a contact problem between a viscoelastic body with long memory and an obstacle. The contact is modelled with a normal compliance condition with unilateral penetration. Also, we assume that the contact is frictional and we model the friction with a time-dependent Coulomb's friction law. We provide a variational formulation to the model and we prove the existence of a unique weak solution. The proof is based on results concerning quasivariational inequalities and fixed point. Finally, we consider a penalization of the variational problem in order to study the dependence of the solution with respect to the data and to prove a convergence result. [ABSTRACT FROM AUTHOR]

A numerical-analytical method is proposed for calculating contact pressures and internal stresses that arise in a two-layer elastic half-space when a smooth indenter of arbitrary shape is indented, taking into account adhesion forces. The Maugis-Dugdale adhesion interaction model is used. The influence of layer thickness and the presence of adhesive attraction on the penetration of the indenter, as well as on internal stresses, was analyzed. The simulation results can be used to solve the inverse problem – determining the elastic modulus of the coating based on the results of indentation, taking into account adhesion and deformation of the substrate. The sliding of a smooth indenter along the boundary of a two-layer half-space is considered, taking into account the rheological properties of materials. The case of a viscoelastic layer associated with a rigid half-space is analyzed, as well as the opposite case – a viscoelastic half-space covered with a rigid layer. The problem is formulated as quasi-static. The numerical-analytical solution method is based on the boundary element method and an iterative procedure. An analytical relationship for calculating influence coefficients is obtained. Contact pressure, energy dissipation and internal stresses are analyzed as a function of sliding speed, layer thickness and Poisson's ratio. [ABSTRACT FROM AUTHOR]

A periodic contact problem for a system of indenters and a two-layer elastic half-space is considered. An algorithm for solving the problem is developed, and a study is carried out of the influence of the geometric and mechanical parameters of the problem, especially the density of indenters, on the solution of the contact problem and the stress distribution in the layer and half-space. It is shown that the multiple nature of loading has a particularly significant effect on stresses in relatively rigid thin coatings. A formulation and method for solving a two-level contact problem that models the contact of a body with a coating and a rough counter-body is proposed. The method is based on the use of the additional displacement function obtained by solving the periodic contact problem when formulating the contact problem at the macro level. It is shown that microgeometry affects the distribution of nominal contact pressure at the macro level. For the contact of a rough coating and a smooth indenter, an approximate method for solving the contact problem is proposed, which is used to obtain macro-characteristics of contact interaction. [ABSTRACT FROM AUTHOR]

Numerical and analytical methods are proposed for solving the axisymmetric contact problem and the spatial problem of determining the stress state for bodies with elastic coatings. In both cases, complicated conditions are considered at the interfaces between layers, consisting of the introduction of a linear relationship connecting tangential stresses and displacements. As an example of axisymmetric contact problems, the contact of a sphere, as well as an indenter with a flat base and rounded edges, with a two-layer elastic half-space is studied. It is shown that in the second case there are both circular and annular contact areas. An analysis was carried out of the influence of the degree of adhesion of the coating to the substrate on the distribution of contact pressure and internal stresses. The results of calculating the stress state in a coating with a large number of layers are also presented. An analysis of the influence of friction on stresses in coatings and substrates was carried out, including with input parameters corresponding to the conditions of the experiments. [ABSTRACT FROM AUTHOR]

This paper is the first ever rigorous investigation of a two-dimensional dynamic contact problem of the action of a deformable die on a quadrant of a multilayer medium. Compared to an absolutely rigid die, its deformable counterpart introduces additional features. They consist in the potential occurrence of discrete resonances predicted by Academician I. I. Vorovich. The paper shows that a method based on using brick-level elements makes it possible to derive an equation describing resonant frequencies to study contact problems with a deformable die made of complex-rheology materials, including smart materials. First of all, the case of a deformable die made of a simple-rheology material described by the Helmholtz equations is considered. Solutions of the boundary problems for complex-rheology dies are represented as a combination of solutions of boundary problems for simple-rheology dies. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with optimal control of a frictionless contact problem with multiple unilateral constraints for a two-dimensional bar. The existence of an optimal trajectory-control pair is firstly proven under the framework of general cost functional. The Pontryagin maximum principle is then established for the investigational system equipped with many equality and inequality constraints in fixed final horizon case, owing to the Dubovitskii and Milyutin functional analytical approach. A remark concludes the article with the discussion, which address the utilization of obtained necessary optimality condition. [ABSTRACT FROM AUTHOR]

In this work, for the first time, an exact solution of the contact problem of interaction with a multilayer base of two semi-infinite stamps, the ends of which are parallel to each other, is constructed. The stamps are assumed to be absolutely rigid, and the distance between them can have any finite value. The task is an important stage in the algorithm for constructing models of a new type of crack in materials of different rheologies. The mechanism of destruction of the medium by cracks of a new type is radically different from the mechanism of destruction of the medium by Griffiths cracks, and has so far been little studied. Griffiths formed his cracks with a smooth border as a result of compression from the sides of an elliptical cavity in the plate. Cracks of a new type have a piecewise smooth border, resulting from the replacement of an ellipse with a rectangle compressed from the sides. The problem considered in this article can be considered as the result of the formation of a new type of crack with absolutely rigid banks and a deformable lower boundary. Thanks to it, after the solution, it becomes possible to switch to deformable stamps and a crack of a new type in the rheological medium. The solution of this problem turned out to be possible due to the construction of exact solutions of the Wiener–Hopf integral equations on a finite segment. This paper shows how the solution of one of the previously unsolved problems allows us to investigate and solve exactly other problems and to identify previously unknown properties and resonances. As a result of constructing an exact solution to the problem, the fact that the solution of dynamic contact problems for stamp systems is not unique was confirmed and a dispersion equation for finding resonant frequencies was constructed. [ABSTRACT FROM AUTHOR]

In this study, the frictionless contact and crack problem of an elastic homogeneous semi-infinite plane has been investigated according to the elasticity theory. The problem has been solved as a superposition of the separate solutions of the contact and crack problem. The aim of this study is to find sub-punch stress distributions and stress intensity factors due to opening mode and shear mode for different loading conditions and geometric sizes. There are two rigid punches on the semi-infinite plane and P and Q loads are transferred to the semi-infinite plane by these punches. Problem has been considered as plain strain because of the geometry of the problem. The effect of the mass forces has not been included, the stress and displacement expressions to be used for the contact problem have been obtained by using Navier equations and Fourier integral transformation technique, and the boundary conditions determined for the problem has been applied. The equations to be used for the crack problem have been specified and the boundary conditions for the crack problem have been applied to these equations. The problem has been reduced to an integral equation system consisting of four singular integral equations where contact stresses and crack displacements are unknown. Numerical solution of the integral equation system has been realized by using Jacobi polynomials. Numerical results on sub-punch stress distributions and stress intensity factors have been obtained for different loading conditions, geometric sizes and presented by graphics. [ABSTRACT FROM AUTHOR]

The dynamical contact problem for viscoelastic half-space which is reinforced by an elastic inclusion in the form of a strip, is considered. The solution of the problem is reduced to the integro-differential equation. Using the method of orthogonal polynomials, the integral equation is reduced to an infinite system of linear algebraic equations. The quasi-completely regularity of the obtained system is proved and the method of reduction for approximate solution is developed. [ABSTRACT FROM AUTHOR]

In this study, the realm of three-dimensional geometrical contact problems is delved into, with a specific focus on the scenario involving two C 1 gear surfaces undergoing rotational motion about two fixed axes in the spatial domain. The development of a novel mathematical model for this contact problem is encompassed by our endeavor. The foundation of our model is rooted in a new parameterization technique, which enables the reduction of the conventional system of five generally non-linear equations, each with five unknowns and one independent parameter, into a more manageable system. This streamlined approach results in a system consisting of just two equations, two unknowns, and one independent parameter. Subsequently, a back-substitution methodology is employed to explicitly derive the values for the remaining three unknowns. In a bid to ascertain the efficacy of the newly proposed model, a comparative analysis is conducted with the well-established model formulated by Litvin. Our findings demonstrate that the accuracy and stability of numerical solutions in the context of this intricate geometrical contact problem are not only enhanced by our model but also that it represents a significant advancement in the understanding and computational treatment of such scenarios. [ABSTRACT FROM AUTHOR]