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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 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]

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]

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]

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]

• 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 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]

A method for measuring the optical functions of a flat sample made of homogeneous and isotropic material, using attenuated total reflection spectroscopy when there is poor contact between the sample and the internal reflection element is presented. The approach consists in treating the spacing between the internal reflection element and the sample as an adjustable parameter, along with the dispersion model parameters, in the simultaneous fitting of s - and p -polarized spectra obtained when the gap distance is unknown. The method is tested with both synthetic and experimental (polystyrene) spectra. The results demonstrate the method's ability to accurately determine the optical functions even in the presence of a contact problem. Graphical Abstract This is a visual representation of the abstract. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

• For existence of weak solution in Signorini's contact problem in the previous works was used the Div-Curl lemma. • We follow a different and new approach applying the semigroup theory. • We prove the existence of at least one solution to the Signorini problem by using the hybrid-penalized method. • We prove uniform stabilization assuming equal wave speeds in the first and second equations of the system. • The uniqueness of the solution to Signorini's problem remains an open question. In this paper, we study Signorini's problem for a laminated Timoshenko beam with interfacial slip. We assume that the transverse displacements at the end of the beam are in the presence of two rigid obstacles (Signorini conditions). We prove the existence of solution and analyze the asymptotic behavior of the system. We use the hybrid-penalized method to show the global existence of at least one solution to Signorini's problem and finally, numerical experiments verify the theoretical results. [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]

The goal of this paper is to study a mathematical model which describes the quasistatic frictional contact between aviscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint, friction, damage andmemory term, which also includes foundation memory property. We present the classical formulation of the problem and derivevariational-hemivariational formulation. Finally we provide a unique weak solution of the problem. [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]

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established. [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]

Torsional deformation of an elastic half-space containing a vertical cylindrical cavity caused by a rotation of a bonded rigid bolt with a long shank is studied in this article by employing Weber integral transforms and integral equations. In the limiting case of the infinite shank, we suggest two different techniques reducing the problem to the regular integral equations of the second kind, which permit efficient solutions in distinct ranges of geometric parameters and allow us to derive a highly accurate approximate analytic solution of the problem for any values of the parameters. Almost exact approximations are suggested for the stress concentration at the edge of the bolt head. In the case of a bolt with a long finite shank, relationships between the torque and the angle of twist of the bolt are derived by utilizing solutions obtained for the bolt with the infinite shank as asymptotic approximations. The effect of the geometric parameters is investigated providing insight into the behavior of the bolt and contact stress in dependence on the bolt head radius and the length of the shank. [ABSTRACT FROM AUTHOR]

An exact solution of the problem of contact interaction of an absolutely rigid stamp with a flat base with an elastic half-plane is constructed, when the shear stresses acting in the contact zone are related to the normal pressure by dry friction law. It is assumed that the coefficient of friction depends on the coordinates of the contacting points of the contacting bodies and is directly proportional to them. A numerical calculation was carried out, the results of which were compared with the results obtained for a similar contact problem solved using Galin’s contact model. [ABSTRACT FROM AUTHOR]

One of the most common problems in engineering is contact problems. In recent years, researchers have turned to alternative methods that can offer effective solutions in a shorter time, instead of solutions containing complex and long mathematical expressions. This study focuses on the estimation of the contact lengths in a homogeneous elastic layer suppressed by two elastic punches with two solution methods. Firstly, a new model was designed for estimation using Deep Learning Neural Network (DNN), one of the deep learning structures. Estimation of contact lengths was provided with the output of the DNN model, which was fed with the homogeneous elastic layer, the ratio of shear modules of the punches and the input parameters of punch radii. The finite element method was used as the second solution method. The problem was modelled in the ANSYS programme, and the solution was made with the same parameters used in DNN modelled. The results obtained from both solutions were compared with the solutions obtained by the theory of elasticity and classical NN in the literature. It had been seen that the results obtained with DNN and ANSYS were compatible with the results obtained with analytical and classical NN and the margin of error was smaller. [ABSTRACT FROM AUTHOR]

In view of the contact problem of functionally graded piezoelectric material (FGPM) coating under different kinds of conducting indenters, effects of the gradient index on the contact mechanical behavior of the FGPM coating were investigated. A model for the multilayer FGPM coating was established. The contact problem of the FGPM coating was transformed into singular integral equations by means of the Fourier integral transform technology and the transfer matrix method. The Gauss-Chebyshev quadrature formula was used to obtain the surface stress distribution and the charge distribution in the FGPM coating-substrate system under a rigid conducting flat indenter and a conducting cylindrical indenter. According to the numerical results, the effects of variations of the FGPM coating parameters on the indentation and electrical potential were analyzed. The distributions of stress and electrical displacement in the FGPM coating were obtained. The results show that, the variations of the FGPM coating parameters have an important influence on the contact behavior of the system. [ABSTRACT FROM AUTHOR]

In mechanical engineering, damage or fatigue due to relative motion between contacting bodies has always been the focus of research. However, in relative motion, the zone of the contact area is constantly changing. To accurately simulate the stress in the contact area, the mesh of all potential contact areas needs to be refined. To avoid this problem, a mesh updating method was adopted. Employing this method, only the mesh of the potential contact area at the current moment needs to be refined, and the matching meshes can be ensured. Besides, the rotating coordinate system method was used. By this method, the equivalence of boundary integral equations before and after sliding or rolling can be derived, avoiding repeated integral calculations. Based on the above methods, the solution method for 2D sliding or rolling contact problems was proposed, and the effectiveness of the proposed method was verified by numerical examples. [ABSTRACT FROM AUTHOR]

The work considers the plane stress state of a piecewise-homogeneous plane, of two heterogeneous half-planes, which is reinforced by a deformable thin inclusion at the junction of the half-planes and is deformed under action of distributed loads applied to the inclusion. It is assumed the inclusion undergo both tension and bending. A governing system of integro-differential equations with respect to jumps of contact stresses acting on the long sides of the inclusion is derived. The solution of the governing system in the general case is constructed by numerical-analytical method of mechanical quadratures. In the case of an infinite inclusion, an exact solution to the problem is obtained. [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]

We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-disks with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an optimal control problem of a quasistatic frictional contact model with history-dependent operators. The contact boundary of the model is divided into two parts where different contact conditions are specified. For the contact problem, we first derive its weak formulation and prove the existence and uniqueness of the solution to the weak formulation. Then we give a priori estimate of the unique solution and prove a continuous dependence result for the solution map. Finally, an optimal control problem that contains boundary and initial condition controls is proposed, and the existence of optimal solutions to the control problem is established. [ABSTRACT FROM AUTHOR]

In this study, the plane thermoelastic frictional contact problem between a magneto-electro-elastic (MEE) layer and a rigid cylindrical or flat punch is discussed. Punches are assumed to be perfect electromagnetic and thermal insulators; they slide over the layer with a small constant speed, and due to the friction, heat flux occurs. The general stress, displacement, and electromagnetic expressions for the contact problem are derived using the theory of thermoelasticity. Applying the boundary conditions, the contact problem is reduced to a Cauchy-type singular integral equation, which is then solved numerically using the Gauss-Jacobi integration formula. [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]

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 paper, we are concerned with numerical analysis on optimal control of a dynamic contact model. Our work builds on some existing results of this model. We derive a weak formulation, and present an approximate problem for the weak formulation by discretization in space. We prove that the approximate problem has a unique solution, which converges to the unique solution of the original problem. The existence of optimal solutions to the control problem is established based on a known continuous dependence result. We consider the numerical solutions of the optimal control problem and prove a convergence result of numerical solutions. [ABSTRACT FROM AUTHOR]

The GSCP (Gauss–Seidel based projection‐contraction) algorithm has been well established in solving elastoplastic mixed complementarity problems (MiCPs) or other problems which have similar mathematical form. However, the stability problems in the GSPC algorithm may cause nonconvergence or program crashes. Thus, the method of constructing and analyzing counterexamples is adopted to study the reasons for the instability of the GSPC algorithm. The R‐GSPC algorithm was developed by modifying the part of the GSPC algorithm that caused the crash to make the algorithm stable. In addition, this paper analyzes the convergence, compatibility, and initial value impact of R‐GSPC, and finds that the R‐GSPC algorithm greatly improves the robustness and stability without losing accuracy. This paper allows the GSPC algorithm to be better extended to more fields and to be suitabled for being introduced into commercial programs. [ABSTRACT FROM AUTHOR]