Nonsmooth dynamic frictional contact of a thermoviscoelastic body

Abstract

This paper studies a system of two hemivariational inequalities modeling a dynamic thermoviscoelastic contact problem with general nonmonotone and multivalued subdifferential boundary conditions. Thermal effects are included in the Kelvin–Voigt thermoviscoelastic constitutive law and in the boundary conditions, and so in frictional heat generation, which takes place on the boundary and enters the condition for the temperature. The existence of a weak solution to the problem is established using a recent surjectivity result for differential inclusions associated with pseudomonotone operators.     

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.     

Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction

In this article we examine an evolution problem, which describes the dynamic contact of a viscoelastic body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. First we derive a formulation of the model in the form of a multidimensional hemivariational inequality. Then we establish a priori estimates and we prove the existence of weak solutions by using a surjectivity result for pseudomonotone operators. Finally, we deliver conditions under which the solution of the hemivariational inequality is unique.     

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.     

A quasistatic viscoplastic contact problem with normal compliance,unilateral constraint,memory term and friction

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance, unilateral constraint and memory term. We present the classical formulation of the problem together with the list of assumptions on the data. Then we derive the variational–hemivariational formulation of the model and we prove its unique weak solvability. The proof is based on a recent abstract result of a class of history-dependent variational–hemivariational inequalities.     

One-dimensional dynamic thermoviscoelastic contact with damage

The problem of thermoviscoelastic dynamic contact between a rod and a rigid obstacle, when the material damage is taken into account, is modeled and analyzed. The contact is modeled by the normal compliance condition and the stress-strain constitutive equation is of Kelvin-Voigt type. The damage, which describes the reduction of the load carrying capacity of the rod, evolves because of the opening of microcracks as a result of tension or compression. When the damage reaches a critical value at a point on the rod the material cannot carry any load and the system breaks down. Mathematically, this is expressed by the quenching of the solution. The existence of a local weak solution is established using penalization and a priori estimates.     

Homogenization of boundary hemivariational inequalities in linear elasticity

In this paper we consider a mathematical model describing static elastic contact problems with the Hooke constitutive law and subdifferential boundary conditions. We treat boundary hemivariational inequalities which are weak formulations of contact problems. We establish existence and uniqueness of solutions to hemivariational inequalities. Using the notion of H-convergence of elasticity tensors we investigate the limit behavior of the sequence of solutions to hemivariational inequalities.     

Existence for a thermoviscoelastic beam model of brakes

The existence of a weak solution to a model for the dynamic thermomechanical behavior of a viscoelastic beam, which is in frictional contact with a rigid rotating wheel, is established. The model describes a simple braking system in which a rotating wheel comes to a stop as a result of the frictional traction generated by the beam. The classical model consists of a system of coupled equations for the beam temperature and displacement, the wear of the beam's contacting end, the wheel temperature and its angular velocity. The weak formulation is an abstract differential inclusion involving set-valued pseudomonotone operators, The existence is proved by using recent results for such operators. Uniqueness is shown to hold when the wheel's angular velocity and temperature are known.     

Existence results for dynamic adhesive contact of a rod

The existence and uniqueness of the weak solution to the model for the dynamics of a viscoelastic rod which is in adhesive contact with an obstacle is established. The model consists of a hyperbolic equation for the vibrations of the rod coupled with a nonlinear ordinary differential equation (ODE) for the evolution of the bonding function. The model allows for failure, i.e., complete debonding, in finite time. The existence of the weak solution is established by using an existence result for ODEs and the Schauder fixed-point theorem. The limit of an elastic rod when the viscosity vanishes is studied, too.     

A class of dynamic frictional contact problems governed by a system of hemivariational inequalities in thermoviscoelasticity

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the nonlinear constitutive viscoelastic law with a long-term memory, which includes the thermal effects and considers the general nonmonotone and multivalued subdifferential boundary conditions for the contact, friction and heat flux. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using recent results from the theory of hemivariational inequalities and a fixed point argument.     

A class of implicit variational inequalities and applications to frictional contact

This paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of an elasto-piezoelectric system of hemivariational inequalities with thermal effects

In this paper we prove the existence and uniqueness of a weak solution for a dynamic electo-viscoelastic problem that describes a contact between a body and a foundation. We assume the body is made from thermoviscoelastic material and consider nonmonotone boundary conditions for the contact. We use recent results from the theory of hemivariational inequalities and the fixed point theory.     

On a doubly nonlinear model for the evolution of damaging in viscoelastic materials

We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.     

A viscoelastic frictionless contact problem with adhesion

We consider a model for the quasistatic, bilateral, adhesive and frictionless contact between a viscoelastic body and a rigid foundation. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and an integro-differential equation for the bonding field. The existence of a unique weak solution for the problem is established by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space. We also consider the problem describing the bilateral contact between two viscoelastic bodies, and establish similar results.     

Existence and asymptotic stability of periodic solution for evolution equations with delays

In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H

     

A dynamic viscoelastic contact problem with normal compliance,finite penetration and nonmonotone slip rate dependent friction

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modelled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solution of the regularized problem is obtained. Next, we prove the convergence of the weak solution of the regularized problem to the weak solution of the initial nonregularized problem. Then, we introduce a fully discrete approximation of the variational problem based on a finite element method and on a second order time integration scheme. The solution of the resulting nonsmooth and nonconvex frictional contact problems is presented, based on approximation by a sequence of nonsmooth convex programming problems. Finally, some numerical simulations are provided in order to illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence result.     

A class of history-dependent systems of evolution inclusions with applications

This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, we prove that the system has a unique solution. We conclude with two examples of an evolutionary differential variational–hemivariational inequality and of a dynamic frictional contact problem in mechanics, which illustrate the abstract results.     

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born-Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.