Normal Damped Response Research Articles

Well-posedness of evolutionary differential variational–hemivariational inequalities and applications to frictional contact mechanics

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 | .

On a dynamic frictional contact problem with normal damped response and long-term memory

A dynamic frictional contact problem between a viscoelastic body and a foundation is studied. The contact is modeled with normal damped response and a friction law. The constitutive law with long memory is assumed to be nonlinear. The existence result is proved using nonlinear monotone operators, fixed point argument, and extension procedure. Moreover, the exponential stability of the energy solution is established using the multiplier method.

Exponential stabilization of a dynamic frictional contact problem for viscoelastic materials with normal damped response and infinite memory

Exponential stabilization of a dynamic frictional contact problem for viscoelastic materials with normal damped response and infinite memory

Variational and numerical analysis of a dynamic elasto-viscoplastic piezoelectric frictional contact problem

Abstract This work studies a mathematical model involving a dynamic contact between two elasto-viscoplastic piezoelectric bodies with damage. The contact is modeled on a combination of a normal compliance and a normal damped response law associated with friction. We derive a variational formulation of the problem and we prove the existence and uniqueness result of the weak solution. The proof is based on the classical existence and uniqueness result for parabolic inequalities, differential equations and fixed-point arguments. This paper examines a friction problem between two piezoelectric bodies that are elastic- viscoplastic. The study uses both numerical and analytical methods. The contact surface is assumed to have a thin layer of lubricant, and a damped response contact condition is considered. The problem also takes into account electrical and frictional effects. The paper proves the existence of a unique weak solution to the problem by using variational inequalities and a fixed point argument. Additionally, a fully discrete approximation is investigated using the Euler scheme and finite element method for the spatial variable and time derivatives. Error estimates are then calculated, leading to convergence results under certain regularity conditions.

A frictional contact problem with normal damped response conditions and thermal effects for a thermo-electro-viscoelastic material

PurposeThe paper presents a mathematical problem involving quasistatic contact between a thermo-electro-viscoelastic body and a lubricated foundation, where the contact is described using a version of Coulomb’s law of friction that includes normal damped response conditions and heat exchange with a conductive foundation. The constitutive law for the material is thermo-electro-viscoelastic. The problem is formulated as a system that includes a parabolic equation of the first kind for the temperature, an evolutionary elliptic quasivariational inequality for the displacement and a variational elliptic equality for the electric stress. The author establishes the existence of a unique weak solution to the problem by utilizing classical results for evolutionary quasivariational elliptic inequalities, parabolic differential equations and fixed point arguments.Design/methodology/approachThe author establishes a variational formulation for the model and proves the existence of a unique weak solution to the problem using classical results for evolutionary quasivariational elliptic inequalities, parabolic difierential equations and fixed point arguments.FindingsThe author proves the existence of a unique weak solution to the problem using classical results for evolutionary quasivariational elliptic inequalities, parabolic difierential equations and fixed point arguments.Originality/valueThe author studies a mathematical problem between a thermo-electro-viscoelastic body and a lubricated foundation using a version of Coulomb’s law of friction including the normal damped response conditions and the heat exchange with a conductive foundation, which is original and requires a good understanding of modeling and mathematical tools.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

VARIATIONAL ANALYSIS OF AN ELASTIC-THERMO-VISCOPLASTIC CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE

We consider a quasistatic frictional contact problem between an elastic-viscoplastic body and an obstacle. The contact is modelled with normal damped response and a local friction law. The material is elastic-viscoplastic with two internal variables which may describe a temperature parameter and the damage of the contacting surface. We provide a variational formulation of the problem and prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed point.

A QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE FOR THERMO-ELECTRO-ELASTIC-VISCOPLASTIC BODIES

We consider a mathematical problem for quasistatic contact between a thermo-electro--elastic-viscoplastic body and an obstacle. The contact is modeled by a general normal damped response condition with friction law and heat exchange. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution. The proof is based on the formulation of four intermediate problems for the displacement field, the electric potential field and the temperature field, respectively. We prove the unique solvability of the intermediate problems, then we construct a contraction mapping whose unique fixed point is the solution of the original problem.

Dynamic History-Dependent Hemivariational Inequalities Controlled by Evolution Equations with Application to Contact Mechanics

This paper is devoted the study of a generalized hybrid dynamical system, which consists of a history-dependent hemivariational inequality of parabolic type and a nonlinear evolution equation. The unique solvability for the system is established via applying surjectivity of multivalued pseudomonotone operators, fixed point theorem, and properties of the Clarke generalized gradient. As an illustrative application, a dynamic frictional contact problem for viscoelastic materials with history-dependent and adhesion is investigated, in which the friction condition is described by the Clarke generalized gradient of a nonconvex and nonsmooth function involving adhesion, and the normal damped response condition is expressed by a given nonnegative function depending on the normal velocity and adhesion.

Mathematical and asymptotic analysis of thermoelastic shells in normal damped response contact

The purpose of this paper is twofold. We first provide the mathematical analysis of a dynamic contact problem in thermoelasticity, when the contact is governed by a normal damped response function and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Under suitable hypotheses on data and using a Faedo-Galerkin strategy, we show the existence and uniqueness of solution for this problem. Then, we study the particular case when the deformable body is, in fact, a shell and use asymptotic analysis to study the convergence to a 2D limit problem when the thickness tends to zero.

Dynamic Contact Problem with Normal Damped Response, Friction and Adhesion

This work considers a mathematical model describing a dynamic frictional contact between a viscoelastic body and an adhesive foundation. The contact is modeled with normal damped response condition associated with a new version of Coulomb’s law of dry friction with adhesion introducing a new term which gives a better transition from adhesion to friction. We present a variational formulation of the problem which is given as a system coupling an evolution inequality of the second order for the displacement and a differential equation of the first order for the bonding field. We establish the existence and uniqueness of the weak solution. The proof is based on parabolic variational inequalities of the second kind, differential equations and fixed point theorem.

A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

A new class of quasistatic frictional contact problems governed by a variational–hemivariational inequality

A quasistatic nonsmooth frictional contact problem for a viscoelastic material is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality coupled with an operator equation for the stress field. A result on the unique weak solvability to this coupled system is proved through a recent contribution on history-dependent subdifferential inclusions and a fixed point approach.

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.

Variational analysis and optimal control of dynamic unilateral contact models with friction

The target of the present work is to consider the optimal control for a class of evolutionary variational–hemivariational inequalities, which modeling the dynamic viscoelastic unilateral contact problems with normal damped response and friction. After studying the weak solvability of the unilateral contact model, we consider its optimal control by three aspects, the optimal control via external forces and initial conditions, the time optimal control problem and the maximum stay control problem. Finally, the conditions which guarantee the existence of optimal solutions to the corresponding control problems are delivered.

Variational–hemivariational inequality for a class of dynamic nonsmooth frictional contact problems

In this paper, a dynamic frictional contact problem for viscoelastic materials with long memory is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality for the velocity. A result on the unique weak solvability to this inequality is proved through a recent contribution on evolutionary subdifferential inclusions and a fixed point approach.

Numerical nodelling of a dynamic contact problem with normal damped response and unilateral constraint

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 damped response and unilateral constraint for the velocity field, associated to a version of Coulomb’s law of dry friction. Our aim is to present a detailed description of the numerical modelling of the problem. To this end, we use a penalty method to approximate the constraint. Then, we provide numerical simulations in the study of a two-dimensional example and compare the penalty model with the original one.

Numerical treatment of contact problems with thermal effect

The paper deals with the formulation and the finite element approximation of a quasi-static thermoviscoelastic problem which describes frictional contact between a deformable body and a rigid foundation. The contact is modeled by normal damped response condition whereas the friction is described by the Coulomb law of dry friction. The weak formulation of the model consists of a coupled system of the variational inequality for the displacement and the parabolic equation for the temperature. The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.