Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.     

Duality applied to contact problems with friction

The duality theory of Mosco, Capuzzo-Dolcetta, and Matzeu for variational and quasi-variational inequalities is extended. Then it is applied to two problems of contact with friction of an elastic body with a rigid foundation. The more realistic normal compliance condition is used in place of Signorini's condition on the contact surface.We (A.M. and M.S.) are grateful for the financial support and hospitality of the Department of Mechanical Engineering in the Linköping Institute for Technology during our two weeks stay in August 1988. In addition M.S. was partially supported by the Oakland University Faculty Research Award.     

Sensitivity analysis of contact problems with prescribed friction

This paper is concerned with the sensitivity analysis of a class of variational inequalities arising in the mathematical modeling of the contact of an elastic body with a rigid foundation. The sensitivity coefficient of the response of the elastic body with respect to the perturbations of the load is derived in the form of a unique solution of an auxiliary contact problem. The result obtained can be used in numerical methods of optimal design of elastic distributed parameter structures.The writing of this paper was completed while the author was visiting the Department of Mathematics, University of Florida, Gainesville, FL 32611, USA.     

Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

Abstract

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.     

Thermoelastic rolling contact problem with temperature dependent friction

The paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémond?s theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.     

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.     

Theoretical analysis of discrete contact problems with Coulomb friction

A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction F depending on the spatial variable is analysed. It is shown that a solution exists for any F and is globally unique if F is sufficiently small. The Lipschitz continuity of this unique solution as a function of F as well as a function of the load vector f is obtained. Furthermore, local uniqueness of solutions for arbitrary F > 0 is studied. The question of existence of locally Lipschitz-continuous branches of solutions with respect to the coefficient F is converted to the question of existence of locally Lipschitz-continuous branches of solutions with respect to the load vector f. A condition guaranteeing the existence of locally Lipschitz-continuous branches of solutions in the latter case and results for determining their directional derivatives are given. Finally, the general approach is illustrated on an elementary example, whose solutions are calculated exactly.     

A quasistatic viscoplastic contact problem with normal compliance and friction

We consider a quasistatic contact problem between a viscoplasticbody and an obstacle, the so-called foundation. The contactis modelled with normal compliance and the associated versionof Coulomb's law of dry friction. We derive a variational formulationof the problem and, under a smallness assumption on the normalcompliance functions, we establish the existence of a weak solutionto the model. The proof is carried out in several steps. Itis based on a time-discretization method, arguments of monotonicityand compactness, Banach fixed point theorem and Schauder fixedpoint theorem.     

Evolutional contact with Coulomb's friction on a periodic microstructure

We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε ≪ 1, including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile-like material. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in [2]. A time discretization scheme from [2] reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time-increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in [4] for closed cracks and can be extended to textile-like materials. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from [2], for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro-level lead to a kind of an evolutional plastic behavior of the homogenized medium. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization for contact problems with friction on rough interface

We consider a contact problem between a macroscopic solid with a smooth boundary and a technical textile, while the textile has a periodic microscopic structure and microscopically rough surface. Two–scale homogenization approach is applied to the problem. The microscopic solution is approximated in terms of macroscopic solution and some concentration factor, given as a solution of auxiliary boundary value or contact problems of elasticity on the periodicity cell. Local friction condition is represented as a continuous non–linear functional over the stress field. Two–scale convergence is used to prove the convergence of friction functional. The macroscopic initial frictional limit is found. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Qualitative analysis of 3D elastostatic contact problems with orthotropic Coulomb friction and solution-dependent coefficients of friction

This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented.     

One-sided contact problems with friction arising along the normal

We study a boundary contact problem for a micropolar homogeneous elastic hemitropic medium with regard of friction; in the considered case, friction forces do not arise in the tangential displacement but correspond to a normal displacement of the medium. We consider two cases: the coercive case (in which the elastic body has a fixed part of the boundary) and the noncoercive case (without fixed parts). By using the Steklov–Poincaré operator, we reduce this problem to an equivalent boundary variational inequality. Existence and uniqueness theorems are proved for the weak solution on the basis of properties of general variational inequalities. In the coercive case, the problem is unconditionally solvable, and the solution depends continuously on the data of the original problem. In the noncoercive case, we present closed-form necessary conditions for the existence of a solution of the contact problem. Under additional assumptions, these conditions are also sufficient for the existence of a solution.     

Homogenization of contact problem with Coulomb's friction on periodic cracks

We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H^1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth‐order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro‐polar elasticity.     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

A quasistatic contact problem for viscoelastic materials with friction and damage

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the frictional contact is modelled with subdifferential boundary conditions. The mechanical damage of the material is described by the damage function, which is modelled by a nonlinear partial differential equation. We derive the variational formulation of the problem, which is a coupled system of a hemivariational inequality and a parabolic equation. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract stationary inclusion and a fixed point theorem.