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Contact mechanics is a complex and advanced field of engineering, with continuous research dedicated to modeling and examining contact problems in multilayered and multibody systems. Given that fatigue and fracture failures often result from contact loads, precise determination of subsurface stresses is crucial for the design of mechanical components. This study provides an exact analytical solution for the frictionless indentation of a functionally graded (FG) orthotropic layer by a rigid cylindrical punch. The FG orthotropic layer, which rests on a rigid, non-adhesive substrate, is treated as a nonhomogeneous medium with an orthotropic stress–strain relationship. Part of its top surface experiences normal tractions, while the rest remains traction-free. The analysis also incorporates the gravitational force affecting the FG orthotropic layer. Five distinct real orthotropic materials are used, each with individually graded stiffness constants in their principal directions, to define the FG orthotropic behavior of the material. The governing equations are formulated using the singular integral equation method and transformed into algebraic systems via the Gauss–Chebyshev integration technique. A comprehensive parametric study examines how variations in dimensionless punch radius, compressive force, inhomogeneity parameters, and body force influence contact widths, contact pressures, normal and shear stresses, critical load factor, and the initial point of interface separation. [ABSTRACT FROM AUTHOR]

Axisymmetric contact problems on the interaction of a rigid stamp and a poroelastic layer, the bottom boundary of which is located on a Winkler base, were considered based on the equations of the Cowin–Nunziato theory of poroelastic bodies. It was assumed that the base of the cylindrical stamp has the flat or parabolic shape and there is no friction in the contact area. With the help of the Hankel integral transformation, the problems posed were reduced to the integral equations for an unknown contact stresses, which were solved with the use of the collocation methods, while the transformants of the kernels of the integral equations were obtained in explicit form. The values of contact stresses and size of the contact area in the case of the parabolic stamp were determined. The relationship, which is one of the main characteristics, when determining the mechanical parameters of a material by the indentation method, between the force acting on the stamp and its displacement, was studied. A comparative analysis of the studied quantities for various values of the poroelastic parameters and the coefficient of subgrade resistance of the Winkler base was carried out. The model of contact interaction considered and the solution obtained can be used in calculation of the structures foundations located on ground bases. [ABSTRACT FROM AUTHOR]

In this article we present the existence, asymptotic behavior and numerical analysis for the one-dimensional Signorini’s problem. We prove the existence of at least one global solution of the semi linear model, using the penalized hybrid method introduced in Muñoz Rivera and da Costa Baldez (J Math Anal Appl 458:1274–1291, 2018. ). This method is based on approximate the problem through a dynamic boundary condition depending of a parameter ϵ (hybrid model). Using the Lipschitzian perturbation theory, we show the global existence for the semi linear problem. Additionally, we show the existence of solutions to the Signorini’s problem by taking limit ϵ → 0 . In this way we prove that the energy associated with the model decays exponentially to zero, as time approaches infinity. Furthermore, we show the exponential decay for the semi-discrete model and that the decay rate does not depend on the penalty parameter ϵ > 0 . Finally, we show the convergence of the approximate model. [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 study is carried out in general for the theory of large initial (finite) deformations and two variants of the theory of small initial deformations within the framework of the linearized elasticity theory with elastic potential of arbitrary structure. In the case of equal roots of the characteristic equation, the mixed contact problem is solved for prestressed two elastic coaxed finite cylinders and layer without taking into account frictional forces. To solve the problem, the Hankel transform, pair integral equations, and other methods of the theory of contact problems of the linearized elasticity theory are used. [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]

The problem of numerical simulation of the contact interaction of a metal tape moving at a speed of approximately 0.5 km/s with a fixed inclined support for a time of approximately 100 s is considered. A two-dimensional approximation is used; an elastoplastic body model for large deformations is used to describe the tape and support. To take into account the boundary conditions on contacting surfaces, an iterative algorithm related to Neumann–Dirichlet type methods is used in the calculations. The finite element method is used for spatial discretization. The calculation results are presented. A number of model one-dimensional problems are considered that make it possible to qualitatively evaluate the results obtained in the two-dimensional case. [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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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]

We consider the Timoshenko beam model which can be in contact to two rigid obstacles and subject to a pointwise damping. We analyze the existence in time and the asymptotic behavior of solutions by using the hybrid method. [ABSTRACT FROM AUTHOR]

Carbon nanoadditives are widely used as modifiers for various materials. An important issue is the effect of modification on the material properties, including the sensitivity to temperature changes. In this study, we performed fixed-temperature indentation of polyurethane samples produced by mortar technology with fullerenes and carbon nanotubes. It was found that the addition of modifiers not only changes the mechanical and rheological properties of the material, but also makes these properties more temperature dependent. Based on solving an axisymmetric contact problem of constant loading rate indentation of a viscoelastic half-space, a method was developed for determining material properties from experimental indentation curves obtained at different rates. The properties of the original and modified polyurethanes were determined at three fixed temperatures. The modifiers produced different effects: nanotubes increased stiffness, while fullerenes reduced it. The effect of ion plasma surface treatment, leading to the formation of a hard carbonized nanolayer, on the indentation results at different temperatures was also investigated. [ABSTRACT FROM AUTHOR]

Nanoindentation at temperatures of 23 and 150°C is used to study the coating of molybdenum disulfide doped with silver and calcium fluoride. The Nanoscan-4D scanning nanohardness tester was used for the experiments. A method is presented for determining the elastic properties of a coating from elastic indentation curves, taking into account the real shape of the indenter head, which is determined by optical profilometry. The elasticity modulus of the coating is determined based on the exact solution of the contact problem for a two-layered elastic foundation, taking into account the calculated compliance of the measurement system. Newton's method is used for the inverse problem solution. The input parameters of the problem, in addition to the geometry of the indenter head and the load, are the elastic properties of the head and substrate materials. The elastic type of indentation was provided at maximal load of 10 mN for both temperatures. The loading–unloading curves at room temperature and at 150°C turned out to be close (within the experimental error), which proves the stability of the elastic properties in the considered temperature range. The calculated elastic modulus of the coating was 326 GPa. Using the same device, equipped with a lateral force sensor, the sliding friction coefficient of the coating was determined under different loads (5, 10, and 20 mN). Such a study can be considered as a physical model of the contact of the coating with a single asperity. The experiments were carried out on straight tracks 1 mm long at a speed of 11 µm/s. It is shown that the coatings are antifrictional (with friction coefficients in the range 0.033—0.078). The coefficient of friction increases with increasing load, which may be due to the dissipation of energy for plastic deformation of the coating material at relatively high loads. The conclusion about the presence of plastic deformation is based on the results of optical profilometry, which showed plastically deformed and pushed material along the edges of the friction track under relatively high loads. At low loads, this phenomenon is not observed. This coating can be used in sliding friction units that require one or two applications with a low friction coefficient. [ABSTRACT FROM AUTHOR]

A new mathematical model describing an equilibrium of an elastic Kirchhoff-Love plate is proposed. We suppose that the plate may come into contact on the side edge with a rigid obstacle. On a given part of the closed curve corresponding to the plate side edge we impose clamping conditions and on the rest part of the curve we consider a nonpenetration condition of the Signorini type. The rigid obstacle is defined by cylindrical surface with generators normal to the plate midplane. In addition, the plate has a flat rigid inclusion (stiffener) on the side edge. The inclusion also defined by a cylindrical surface. The peculiarity of the problem is related to the partial delamination of the inclusion on the contact surface. Boundary conditions of an inequality type ensure nonpenetration of the plate points and the obstacle as well as for points of an elastic matrix and inclusion points. Solvability of the problem is proved. Under the assumption that the solution of the variational problem is smooth enough, optimality conditions on the contact boundary are found. [ABSTRACT FROM AUTHOR]

This paper considers the size-dependent plane frictional contact problem of a homogeneous coated half-plane indented by a rigid punch based on the couple stress elasticity. Using the Fourier integral transform technique in addition to the boundary and compatibility conditions, the mixed-boundary value problem is converted into a singular integral equation of the second kind. The integral equation is further derived and normalized for the cylindrical and flat punch case profiles. Applying the Gauss–Jacobi integration formula, the resulting singular integral equation is reduced to a system of algebraic equations. The obtained results are first validated based on those published for the case of a frictionless contact problem of a half-plane indented by rigid cylindrical and flat punches and solved based on the couple stress theory. A detailed parametric study is then performed to investigate the effect of the characteristic material length, the friction coefficient, the layer height, the shear modulus, the indentation load and Poisson's ratio on the contact and in-plane stresses. [ABSTRACT FROM AUTHOR]

The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich's differential calculus to compute the needed subgradient information. The state problem is solved using successive approximations combined with the Total FETI (TFETI) method. The latter is based on tearing the bodies into "floating" subdomains, discretization by finite elements, and solving the resulting quadratic programming problem by augmented Lagrangians. The presented numerical experiments demonstrate our method's power and the importance of the proper modelling of 3D frictional contact problems. The state problem solution and the sensitivity analysis process were implemented in parallel. [ABSTRACT FROM AUTHOR]

In the paper we deal with a new class of systems of differential quasi-hemivariational inequalities. The system consists of a time-dependent constrained elliptic quasi-hemivariational inequality, an evolution quasi-hemivariational inequality and an ordinary differential equation in Banach spaces. The system is characterized by a strong coupling between the inequalities and the equation, presence of the unilateral constraint and history-dependent operators which leads to a complex time dependent dynamics. The result on the well posedness of the system is established via a surjectivity of multivalued pseudomonotone operators combined with a fixed point approach. The results find an application to a frictional quasistatic contact problem in mechanics with wear and thermal effects. [ABSTRACT FROM AUTHOR]

The contact problem of the interaction of two prestressed half-spaces with a prestressed elastic cylindrical punch without friction forces is solved in the case of unequal roots of the characteristic equation. The study is conducted in a common form for the theory of large initial (final) strains and two variants of the theory of small initial strains within the framework of the linearized theory of elasticity for an arbitrary elastic potential. The problem is solved using the Hankel transform, paired integral equations, orthogonal polynomials, and other methods of the theory of contact problems of linearized elasticity. The numerical analysis is conducted for the Treloar potential in the form of curves. [ABSTRACT FROM AUTHOR]

A stochastic model of the wear process of a thin coating in a radial plain bearing that takes into account random changes in the bearing temperature and external load is described. The results of a numerical analysis of the process of bearing 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 bearing wear and, in particular, its durability is evaluated. [ABSTRACT FROM AUTHOR]

A new model describing the equilibrium of a Kirchhoff–Love plate which may come into mechanical contact with a rigid non-deformable obstacle is proposed. Unlike previous works, we consider the case when a contact is possible on curves located at the intersection of the side face of the plate and two obstacle surfaces that bound the plate from above and below. A boundary condition of Signorini's type is imposed in the form of two inequalities restricting plate displacements. An equilibrium problem describing the contact of the plate with the rigid obstacle is formulated as a minimization problem for an energy functional over a suitable convex and closed set of admissible displacements. It is proved that the problem has a unique solution. [ABSTRACT FROM AUTHOR]

Dual analysis, requiring a pair of compatible and equilibrated solutions, has been well used for strict error estimation of many problems, including contact problems. Compatible solution is easily obtained by conventional displacement-based finite element method (FEM), while getting the equilibrated solution is not straightforward. To this end, an equilibrium finite element method (EFEM) is developed in this paper for equilibrated solution of contact problems. In doing so, there are two key steps. The first is the establishment of complementary energy principle for contact problems, along with inequality pressure constraint on contact force. In the second step, a traction-based equilibrium element is developed to construct the discrete equilibrated stress field, with which the complementary energy formulation turns into a quadratic programming problem. How to get strict bounds for the discretization error in compatible or equilibrated solution by dual analysis is also addressed. Numerical examples are finally conducted to see the performance and the availability to strict error estimation of the proposed EFEM. [ABSTRACT FROM AUTHOR]

The development trend of functional materials and devices from block to thin film is inevitable. The elastic field of coating structure loaded with concave indenter is studied in this paper. Based on the mirror method and potential theory method, the analytical solutions of the elastic field of coating structure loaded with parabolic concave indenter are obtained. The convergence problem caused by the mirror image method is analyzed. The displacements and stresses with explicit formulation in the coating/substrate are obtained. The stress distribution inside the coating and at the interface is explored. The influence of coating thickness on interfacial stresses and the problem of the interface failure are discussed. The solution is degraded to the existing theory by numerical method for verifying its accuracy, where let the contact radius approach zero for approximating to Green's function of coated structure. The analytical solution given in this paper is expressed as elemental functions, which is convenient for further application. It can used as a benchmark for other solution for indentation contact problem of coating structure. [ABSTRACT FROM AUTHOR]

Based on the solution to plane contact problems for one and two contact zones, the influence of the geometry of a wavy cylinder on the length of the actual contact region when it is introduced into an elastic half-plane is studied. Analytical expressions are obtained to determine the boundaries of two contact zones and the total load for an arbitrary gap function between the surfaces. The study establishes a relationship between the boundaries of the contact zone and the fictitious zone of negative (tensile) contact stresses that arise when the solution for a simply connected region is used. It is shown that the waviness of the cylinder surface has a significant effect on the load dependence of the length of the real contact area. A nonmonotonic behavior of this dependence is revealed in the transition from a doubly connected contact region to a simply connected one. [ABSTRACT FROM AUTHOR]

A contact between an infinitely extended plane indenter and a viscoelastic layer in the scope of a self-consistent approach according to Derjaguin under the surface application (traditional formulation) and bulk application (refined formulation) of intermolecular interaction forces is considered. Using the first law of thermodynamics, the problem of determining the energy dissipation in a viscoelastic layer is solved for a preset indenter approach/retraction law. Based on this solution, friction force under sliding a rough counterbody over a viscoelastic layer has been calculated. The results of the calculations indicate a significant effect of abrupt changing in the contact gap over time exerted on the energy dissipation and the friction force. [ABSTRACT FROM AUTHOR]

The structure of foam materials is analyzed. It is noted that foam materials have different strain diagrams in uniaxial tension and compression. A simplified physical model in the form of a steel ring is proposed for multimodulus foam materials. The model was tested on a Walter + Bai AG LFM-L-1 testing machine. The deformation of the model was analyzed using the ANSYS Workbench software package taking into account the contact interaction between the elements of the model. The calculation results were compared with experimental data. [ABSTRACT FROM AUTHOR]

Computational modelling of contact problems is still one of the most difficult aspects of non-linear analysis in engineering mechanics. The article introduces an original efficient explicit algorithm for evaluation of impacts of bodies, satisfying the conservation of both momentum and energy exactly. The algorithm is described in its linearized 2-dimensional formulation in details, as open to numerous generalizations including 3-dimensional ones, and supplied by numerical examples obtained from its software implementation. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study a class of differential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We first construct a penalized problem for such a nonlinear system and then derive the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the obtained approximating sequence to the solution of the original nonlinear system when the penalty parameter converges to zero. Finally, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics. First part title: Introduction Second part title: Preliminaries Third part title: Convergence result for (1.1) Fourth part title: An application [ABSTRACT FROM AUTHOR]

A mathematical model that describes the bending of the cantilever and takes into account its imperfection factors (initial deflection, change in thickness), as well as the compliance of the substrate in relation to the adhered cantilever method is proposed. On the basis of the proposed model, a numerical analysis of the cantilever bending has been performed and estimates of the adhesion energy are given. The obtained results indicate the importance of taking into account the non-ideal factors of the cantilever, as well as the compliance of the substrate. [ABSTRACT FROM AUTHOR]

The paper considers the contact bending problem for a multilayer composite plate. Each layer of the composite is a material reinforced with thin parallel fibers. The mathematical model is constructed based on the assumptions of the existence of a neutral surface in the plate and the validity of Kirchhoff's hypotheses. Using the Lagrange variational principle, we obtain a bending equation generalizing the Sophie Germain equation. An elastic energy functional taking into account the different resistance of the material to tension and compression is obtained. The contact problem of plate bending with the aid of a rigid die is considered. To solve the contact problem of plate bending by a rigid die, a Lagrangian with an inequality constraint is constructed. The finite element method using a triangular Bell element is applied for the numerical solution of the problem. The results of calculations for the bending of laminated rectangular plates with various directions of fiber laying and various die shapes are presented. [ABSTRACT FROM AUTHOR]

Sticking of some elements of microelectromechanical systems is a serious problem, which may arise at the stages of producing or exploiting such devices. The equilibrium distances and adhesion energies between surfaces with a normal distribution of roughness have been studied with the purpose of controlling this process. Characteristics of the studied materials, such as dielectric functions, roughness, and plasticity, which are necessary for the calculations, have been discussed. The basic force balance equation has been briefly described. Detailed calculations have been performed for a Si–Au system in air. It has been shown that the plastic properties of gold very weakly affect the adhesion energy as a function of the interfacial distance; however, they markedly influence the establishment of the equilibrium distance. The long-range dispersion forces dominate in the adhesion energy at a roughness with a root-mean-square deviation of larger than 1 nm; therefore, the energy can be calculated within the framework of the Lifshits theory. [ABSTRACT FROM AUTHOR]

Based on the equations of the Cowin-Nunziato poroelastic bodies theory, the article considers an axisymmetric contact problem on the interaction of a rigid stamp with a poroelastic layer fixed on an elastic half-space. It is assumed that the base of the stamp has a flat shape, there is no friction in the contact zone. With the help of the Hankel integral transformation, the problem posed is reduced to an integral equation for an unknown contact stress, which is solved using the collocation method. The values of contact stresses are found and the deformation of the surface outside the stamp is investigated. The relationship between the force acting on the stamp and its displacement, which is one of the main characteristics in determining the mechanical parameters of the material by the indentation method, has been studied. A comparative analysis of the studied quantities for various values of the parameters of the poroelastic layer and the elastic foundation has been carried out. Numerical results are presented in the form of graphs. [ABSTRACT FROM AUTHOR]

We consider the plane-strain state of a piecewise homogeneous half plane formed by dissimilar infinite strip and half plane with interface finite cracks in the case where the half plane is deformed by a finite deformable patch acting upon the free boundary of the half plane. A governing system of integral and integrodifferential equations for the dislocations of displacements of points on the crack edges and the tangential contact stresses acting under the patch is obtained. The solution of this system is constructed with the help of the numerical-analytic method of mechanical quadratures. The numerical analysis is performed, the regularities of the behavior of contact stresses are analyzed, and the specific features of variations of the Cherepanov–Rice J -integral are determined depending on the physicomechanical and geometric parameters of the problem. [ABSTRACT FROM AUTHOR]

We consider a fixed point problem S u = u where S : C (R + ; X) → C (R + ; X) is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator S . Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations. [ABSTRACT FROM AUTHOR]

Computational algorithms based on the two-level additive Schwarz method (a version of the domain decomposition method with overlapping) and the Neumann–Dirichlet method (a version of the domain decomposition method without overlapping) for solving the problem of contact interaction of a system of bodies are proposed. Results of applying these algorithms for the numerical simulation of a stress-strain state of a part of a fuel element that includes 2–100 fuel pellets are presented. An axially symmetric thermoelastic approximation is used. The dependence of the number of iterations needed to ensure the desired accuracy on the number of bodies in the system, on the grid step, and on the boundary conditions on contact surfaces is investigated. [ABSTRACT FROM AUTHOR]

The problem of loading a textured two-layer elastic foundation with a rigid smooth indenter is considered. At the interface between the textured layer and the half-space, the conditions for complete adhesion are specified. Relief elements are elastic cylinders, characterized by height and radius, located on the surface of the base with a given period. The one-dimensional Winkler model is used to describe the mechanical properties of relief elements. The contact problem is solved using the boundary element method. Pressure, displacements, and also the shape of the indenter are approximated by piecewise constant functions. The influence coefficients are constructed using a method based on double integral Fourier transforms. The analysis of the influence of the texture elements arrangement density and their compliance on the distribution of the contact pressure is carried out. It was found that in most cases contact is made only with the elements of the texture. For a relatively stiff textured layer, the combined effect of layer curvature and additional pliability imparted by texture elements is important. For relatively stiff and pliable textured layers, load-penetration curves were obtained depending on the texture period. Within the framework of the proposed setting, the limiting case of contact of an indenter with a Winkler layer completely covering the surface of a two-layer elastic half-space is also considered. Load-penetration curves are obtained and analyzed for an indenter in the form of a Berkovich pyramid. [ABSTRACT FROM AUTHOR]

In this paper, we consider the contact of an infinitely extended plane indenter and a viscoelastic layer within the framework of the Deryagin self-consistent approach with surface (traditional formulation) and volumetric (refined formulation) application of intermolecular interaction forces. Equations that describe the time change in the contact gap for such a contact are obtained. A condition, under which an abrupt time change in the contact gap is possible, is formulated. Calculations that indicate the nonmonotonic nature of the time dependences of the contact gap and pressure in the indenter approach and retract modes are performed. It is shown that the traditional and refined problem formulations can lead to significantly different results of calculating the characteristics of the contact between the indenter and the viscoelastic layer. [ABSTRACT FROM AUTHOR]

The physical and mechanical properties of antifriction oil-filled composites with a viscoelastic nanomodified matrix are investigated on the basis of mathematical models. The identification and prediction of their mechanical properties is performed using a micromechanics model with account of experimental data obtained in nanoindentation. Solutions of new contact problems in a quasi-static formulation of the motion of a punch with a flat base into a heterogeneous oil-saturated half-space with allowance for friction in the contact area are constructed. The multiphase heterogeneous medium is described, first, within the framework of the Biot–Frenkel model and second, using the concept of effective homogeneity. The contact problem for the Biot medium is reduced to an integral equation of the first kind with a differential kernel and a logarithmic singularity. After regularization, the numerical solution of the integral equation is constructed by the boundary element method. The solution to the boundary value problem was found by the finite-element method in the ANSYS software package for an equivalent homogeneous medium. A comparative analysis of two approaches to modeling the microstructure of a heterogeneous medium is presented. The influence of mechanical properties of the composite on its stress-strain state is investigated. The magnitude of the friction force arising in the contact area of the medium is studied. Such studies are of great practical importance in investigating new nanomodified antifriction composite materials. For this purpose, numerical calculations for an oil-filled composite with a phenylone matrix and nanosize additives are presented. The influence of porosity, fluid saturation, and friction coefficient on the tangential contact stresses is also examined. [ABSTRACT FROM AUTHOR]