Dynamic Contact Problem Research Articles

Boundary optimal control of a dynamic frictional contact problem

AbstractIn this paper we study boundary optimal control of an evolutionary system governed by a history‐dependent variational‐hemivariational inequality. The inequality is a weak formulation of a dynamic frictional contact problem for a viscoelastic body with a multivalued normal damped response condition and a simplified version of the Coulomb law of dry friction. A continuous dependence result for the solution map is proved and the existence of optimal solutions to the control problem is established.

A Mathematical Model for Prediction of the Tribological Properties of Oil-Filled Composites under Vibration

The tribological properties of an oil-filled composite material are studied. The dynamic contact problem of the vibration of a rigid punch on a heterogeneous half-space is solved taking into account the friction process in the contact area. The oscillation of the friction force during the period of vibration, taking into account tangential displacements in the contact area depending on the compound of the composite is established.

A Dynamic Contact Problem Between Elasto-Viscoplastic Piezoelectric Bodies with Adhesion

We consider a dynamic frictionless contact problem between two elasto-viscoplastic piezoelectric bodies with damage. The evolution of the damage is described by an inclusion of parabolic type. The contact is modelled with normal compliance condition. The adhesion of the contact surfaces is considered and is modelled with a surface variable, the bonding field whose evolution is described by a first order differential equation. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed-point arguments.

Optimal Control of History-Dependent Evolution Inclusions with Applications to Frictional Contact

In this paper, we study a class of subdifferential evolution inclusions involving history-dependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example of an optimal control of a dynamic frictional contact problem in mechanics, whose weak formulation is the evolution variational inequality.

On the transient planar contact problem in the presence of dry friction and slip

This article models a plane strain dynamic contact problem for an infinite elastic body. Contact is established for x > 0 under the action of a time-dependent remote compressive load, and the system is subjected to a time-dependent remote tangential load. The two faces of the contact interface can slide relative to one other to accommodate mismatches between the shear stresses and the existing Coulomb friction. This is shown to be a cause of interfacial waves of slip, which this article models by deriving the general expression for the corrective traction due to an arbitrary slip distribution. This is achieved using a variant of the Wiener-Hopf technique. Combined with existing closed-form expressions for the interfacial tractions due to compressive and shear loads, this enables the formulation of the elastodynamic extension to the stick-and-slip problem of the Cattaneo-Mindlin type. Exploiting self-similarity a simple numerical algorithm is detailed for solving the resulting Volterra integral equations of the first kind. The solution is shown to display a number of features entirely missed in static problems: slip waves are shown to exist irrespective of the magnitude of the applied loads and the friction coefficient; a regime of reverse and forward slip is also shown to exist for low friction coefficients, brought about by the interfacial Rayleigh waves. The practical implications of these solutions are discussed.

Numerical Solution of Contact Problems with Friction under Dynamic Loads

The article deals with the dynamic problem of deformable solid mechanics with unilateral constraints and Coulomb friction. For the numerical solution of the problem, the finite element method is used, the contact interaction is modeled by means of the contact finite elements (CFE) of the frame-rod type. The statement of dynamic contact problem for the interacting elastic bodies is given. On the basis of the proposed discrete contact model and the method of step-by-step analysis a numerical algorithm has been developed, which allows in one step-by-step (on time) process to perform the integration of the equations of motion and the implementation of the conditions for unilateral constraints with Coulomb friction on the contact simultaneously. Under the conditions of ultimate friction the method of compensating loads has been applied. With the help of the proposed approach, numerical solutions of the problem of contact of the structure with the base at different parameters of the dynamic load have been obtained and analyzed.

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Variational and numerical analysis of a dynamic viscoelastic contact problem with friction and wear

ABSTRACT In this paper, we consider a dynamic viscoelastic contact problem with friction and wear, and describe it as a system of nonlinear partial differential equations. We formulate the previous problem as a hyperbolic quasi-variational inequality by employing the variational method. We adopt the Rothe method to show the existence and uniqueness of solution for the hyperbolic quasi-variational inequality under mild conditions. We also give a fully discrete scheme for solving the hyperbolic quasi-variational inequality and obtain error estimates for the fully discrete scheme.

On dynamic contact problem with generalized Coulomb friction, normal compliance and damage

We formulate a dynamic problem which governs the displacement of a viscoelastic body which, on one hand, can come into frictional contact with a penetrable foundation, and, on the other hand, may undergo material damage. We formulate and prove the theorem on the existence and uniqueness of the weak solution to the formulated problem.

Fully history-dependent evolution hemivariational inequalities with constraints

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

Numerical analysis of a nonmonotone dynamic contact problem of a non-clamped piezoelectric viscoelastic body

<p style='text-indent:20px;'>We consider a contact process between a body and a foundation. The body is assumed to be viscoelastic and piezoelectric and the contact is dynamic. Unlike many related papers, the body is assumed to be non-clamped. The contact conditions has a form of inclusions involving the Clarke subdifferential of locally Lipschitz functionals and they have nonmonotone character. The problem in its weak formulation has a form of two coupled Clarke subdifferential inclusions, from which the first one is dynamic and the second one is stationary. The main goal of the paper is numerical analysis of the studied problem. The corresponding numerical scheme is based on the spatial and temporal discretization. Furthermore, the spatial discretization is based on the first order finite element method, while the temporal discretization is based on the backward Euler scheme. We show that under suitable regularity conditions the error between the exact solution and the approximate one is estimated in an optimal way, namely it depends linearly upon the parameters of discretization.

Modelling and numerical solution of problems of structural mechanics with unilateral constraints and friction

The article deals with the problems of calculation of elastically deformable systems with unilateral constraints and Coulomb friction. For the numerical solution of the problem, the finite element method is used, the contact interaction is modeled by means of the contact finite elements (CFE) of the frame-rod type. The basic relations and statements of static and dynamic contact problems for the interacting elastic bodies are given. To comply with the limitations under the conditions of ultimate friction-sliding, the method of compensating loads is applied. On the basis of the proposed discrete contact model and step-by-step loading schemes, the numerical algorithms are developed, which allows calculating structures with unilateral constraints and Coulomb friction, both under static and dynamic loads, as well as taking into account the different contact conditions that approximate the calculation scheme to the real operating conditions of a structure or construction. With the help of the proposed approach, numerical solutions of the problem of contact of the structure with the base under dynamic load are obtained and analyzed.

Учет трения в области контакта при колебаниях жесткого штампа на поверхности полуограниченной среды

The dynamic contact problem of oscillation of a rigid punch on a semi-infinite viscoelastic base is considered, moreover, friction in the contact area is taken into account. The base is provided with a microstructure wich is determinated in the framework of the micromechanics model. The problem is considered in a flat formulation for a steady-state oscillation regime. The base is modeled by a viscoelastic half-space. A rigid punch oscillates on the day surface of the elastic half-space. Normal and tangential stresses in the contact area are related by the Amonton-Coulomb law. Displacements satisfy the Lamaet equations. The connection of displacements and stresses is given by the generalized Hooke's law. The solution of this boundary value problem is constructed using the Fourier transform, which is applied to the Lamaet equations and boundary conditions. The base microstructure was taken into account in the framework of the micromechanics model. Mechanical characteristics corresponding to an equivalent elastic medium have been determined. The boundary-value problem is reduced to an integral equation of the first kind with a difference kernel. The numerical discretization of the integral equation is based on the collocation method. As a regularizer of the main part of the kernel, a function is used to isolate the logarithmic singularity, which coincides with it at infinity and has no singularities in the complex plane. As a result of discretization, the solution reduces to a finite system of equations with a quasi-diagonal matrix. The numerical analysis of the solution of the dynamic contact problem allowed us to draw the following conclusions. The change in contact stresses depending on the coefficient of friction in the contact region substantially depends on the oscillation frequency of the stamp. A significant effect on contact stresses is exerted by the coefficient of friction and the mechanical characteristics of the base material.

A dynamic Tresca's frictional contact problem with damage for thermo elastic-viscoplastic bodies

We consider a dynamic contact problem between an elastic-viscoplastic body and a rigid obstacle. The contact is frictional and bilateral, the friction is modeled with Tresca's law with heat exchange. We employ the elastic-viscoplastic with damage constitutive law for the material . The evolution of the damage is described b y an inclusion of parabolic type.We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniquness result on parabolic inequalities, differentiel equations and fixed point argument.

Analysis of a contact problem involving thermoelastic mixtures

In this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle. The classical normal compliance condition is used for modeling the contact. The variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations. An existence and uniqueness result is proved using the Faedo-Galerkin method. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the decay of the discrete energy and the linear convergence of the algorithm are deduced. Finally, some numerical simulations are presented to show the convergence and the behavior of the solution.

ВЕРТИКАЛЬНИЙ ІМПЕДАНС ФУНДАМЕНТУ НА ШАРІ ВОДОНАСИЧЕНОГО ҐРУНТУ

Один з методів динамічного аналізу відповідальних споруд – застосування імпедансних чи передаточних функцій частоти, які можуть бути включені до динамічних розрахункових схем будівель, що проектуються. На основі аналізу традиційних та сучасних методів визначення характеристик динамічної взаємодії фундаментів споруд з ґрунтовою основою пропонується для оцінки залежності реакції по підошві фундаменту від частоти у випадках водонасичення пористого незв’язного ґрунту в основі та горизонтально-шаруватої його неоднорідності використовувати хвильові рівняння руху ґрунтової пористопружної насиченої стисливою і в’язкою рідиною основи (модель Біо двофазного середовища). Методом інтегральних перетворень визначаються символьні вирази точного розв’язку для переміщень фаз на границі основи (під підошвою фундаменту) від розподілених вертикальних гармонічних навантажень на фази. При вертикальних коливаннях малозаглибленого фундаменту (смуги) розглядаються складові реакції з боку твердої пористої та рідинної порової фаз. Функції імпедансу для жорсткої полоси з непроникною для порової рідини підошвою на шаруватій пористопружній насиченій рідиною (ППНР) основі знаходяться з розв’язку динамічної контактної задачі методомортогональних поліномів (при поліноміальних розкладаннях реакцій фаз з урахуванням особливостей на контакті) і оригінального програмного забезпечення по заданих геометричним і фізико-механічним параметрам фундаментів та моделі основи. На числових прикладах показано, як реакція (імпеданс) ППНР основи відрізняється від реакції пружного півпростору, а взаємодія між фундаментом з недренованою підошвою і водонасиченим ґрунтом неоднобічна внаслідок змінного (до знаку) тиску порової рідини у пружній пористій матриці під підошвою. Визначаютьсярезонансні частоти для моделі одношарової основи з затисненою тильною гранню в залежності від висоти шару, ширини фундаменту і властивостей матеріалу двофазної основи.

On the theory of collisions of elastic bodies

This paper investigates the dynamic contact problem. In particular, there are considered two methods for determining the reduction coefficient, which is an integral energy characteristic of the impact process.It is shown that the value of this coefficient, obtained in the presence of the influence of Rayleigh surface waves, is closer to its value which is obtained experimentally.

On the transient dynamic antiplane contact problem in the presence of dry friction and slip

This article models the elastodynamic transient contact between two elastically similar half planes under antiplane loading and in the presence of friction. Contact is maintained along the positive real line under the presence of a certain remote contact pressure. An antiplane shear load is applied, which entails interfacial shear traction that opposes the frictional force entailed by the contact pressure. In order to balance the surface tractions, the surface must be allowed to slip. We derive the closed form solution of the interfacial traction due to a general antiplanar displacement distribution using a variant of the Wiener-Hopf technique. We also find closed-form expressions for the interfacial shear traction due to this remote antiplane load. In combination with the frictional force, this leads to an integral equation the solution to which is the distribution of relative slip. We quantify both this and the magnitude of the interfacial shear tractions under diverse loading, showing that transient loading leads to partial reverse slip of the contact surfaces. We show that the reverse slip tends to vanish over time, and that it is ameliorated if the friction coefficient is reduced.

Numerical analysis of an evolutionary variational–hemivariational inequality with application to a dynamic contact problem

In this paper, we consider the numerical solution of an evolutionary variational–hemivariational inequality arising in a dynamic contact problem. The material is assumed to be viscoelastic with short memory. The contact is featured by a normal damped response in the normal direction and by the Tresca friction law in the tangential direction. The linear finite elements are used to discretize the spatial variable. Optimal order error estimates are derived for the discrete velocity and discrete displacement under suitable solution regularity assumptions.

Existence of a class of variational inequalities modelling quasi‐static viscoelastic contact problems

AbstractThis work concerns the existence for a class of variational inequalities arising in quasi‐static frictional contact problems for viscoelastic materials. We first consider dynamic contact problems described by evolutionary variational inequalities with a small parameter in the inertial term. Then we study the asymptotic behavior of their solutions when the parameter tends to zero. Based on a time‐discretization technique and monotone operators theory, the inequalities are solved in the form of evolutionary inclusions in the framework of evolution triples. We show that the limit functions of dynamic problems are solutions to the quasi‐static variational inequalities. Applications of the abstract result to frictional contact problems with multivalued constitutive law and normal damped response are given. Moreover, some comments on nonmonotone boundary problem are presented.