Mircea Sofonea

Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity

Nonlinear variational problems and numerical approximation: Preliminaries of functional analysis Function spaces and their properties Introduction to finite difference and finite element approximations Variational inequalities Mathematical modelling in contact mechanics: Preliminaries of contact mechanics of continua Constitutive relations in solid mechanics Background on variational and numerical analysis in contact mechanics Contact problems in elasticity Contact problems in viscoelasticity: A frictionless contact problem Bilateral contact with slip dependent friction Frictional contact with normal compliance Frictional contact with normal damped response Other viscoelastic contact problems Contact problems in visocplasticity: A Signorini contact problem Frictionless contact with dissipative potential Frictionless contact between two viscoplastic bodies Bilateral contact with Trescas friction law Other viscoelastic contact problems Bibliography Index.

Analysis and approximation of contact problems with adhesion or damage

Preface List of Symbols Modeling and Mathematical Background Basic Equations and Boundary Conditions Physical Setting and Evolution Equations Boundary Conditions Contact Processes with Adhesion Constitutive Equations with Damage Preliminaries on Functional Analysis Function Spaces and Their Properties Elements of Nonlinear Analysis Standard Results on Variational Inequalities and Evolution Equations Elementary Inequalities Preliminaries on Numerical Analysis Finite Difference and Finite Element Discretizations Approximation of Displacements and Velocities Estimates on the Discretization of Adhesion Evolution Estimates on the Discretization of Damage Evolution Estimates on the Discretization of Viscoelastic Constitutive Law Estimates on the Discretization of Viscoplastic Constitutive Law Frictionless Contact Problems with Adhesion Quasistatic Viscoelastic Contact with Adhesion Problem Statement Existence and uniqueness Continuous Dependence on the Data Spatially Semidiscrete Numerical Approximation Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Adhesion Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Adhesion Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Contact Problems with Damage Quasistatic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Dynamic Viscoelastic Contact with Damage Problem Statement Existence and Uniqueness Fully Discrete Numerical Approximation Quasistatic Viscoplastic Contact with Damage Problem Statement Existence and Uniqueness for the Signorini Problem Numerical Approximation for the Signorini Problem Existence and Uniqueness for the Problem with Normal Compliance Numerical Approximation of the Problem with Normal Compliance Relation between the Signorini and Normal Compliance Problems Notes, Comments, and Conclusions Bibliographical Notes, Problems for Future Research, and Conclusions Bibliographical Notes Problems for Future Research Conclusions References Index

Quasistatic Viscoelastic Contact with Normal Compliance and Friction

We prove the existence of a unique weak solution to the quasistatic problem of frictional contact between a deformable body and a rigid foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated version of Coulombs law of dry friction. We establish the continuous dependence of the solution on the normal compliance function. Moreover, we prove the existence of a unique solution to the problem of sliding contact with wear.

Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage

We consider a model for quasistatic frictional contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. The contact is modeled with the normal compliance condition and the associated version of Coulombs law of dry friction. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions.

Evolutionary Variational Inequalities Arising in Viscoelastic Contact Problems

We consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials. Under the smallness assumption of a certain coefficient, we prove an existence and uniqueness result using Banachs fixed point theorem. We then study two numerical approximation schemes of the problem: a semidiscrete scheme and a fully discrete scheme. For both schemes, we show the existence of a unique solution and derive error estimates. Finally, all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem, with the finite element method used to discretize the spatial domain.

Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion

A model for the adhesive, quasistatic and frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modelled by a bonding field on the contact surface, and contact is described by a modified normal compliance condition. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. The existence of a unique weak solution for the problem and its continuous dependence on the adhesion parameters are established. Then, the numerical analysis of the problem is conducted for the fully discrete approximation. The convergence of the scheme is established and error estimates derived. Finally, representative numerical simulations are presented, depicting the evolution of the state of the system and, in particular, the evolution of the bonding field.

A quasistatic viscoelastic contact problem with friction

We consider a mathematical model which describes the bilateral frictional contact of a viscoelastic body with an obstacle. The viscoelastic constitutive law is assumed to be nonlinear and the friction is described by a nonlocal version of Coulombs law. A weak formulation of the model is presented and an existence and uniqueness result is established when the coefficient of friction is small. The proof is based on classical results for elliptic variational inequalities and fixed point arguments. We also consider a model for the wear of the contacting surface due to friction. In the case of sliding contact we obtain a new nonstandard contact boundary condition. We prove the existence of the unique weak solution to the problem with the same restriction on the friction coefficient.

A piezoelectric contact problem with slip dependent coefficient of friction

Abstract We consider a mathematical model which describes the static frictional contact between a piezoelectric body and an obstacle. The constitutive relation of the material is assumed to be electroelastic and involves a nonlinear elasticity operator. The contact is modelled with a version of Coulombs law of dry friction in which the coefficient of friction depends on the slip. We derive a variational formulation for the model which is in form of a coupled system involving as unknowns the displacement field and the electric potential. Then we provide the existence of a weak solution to the model and, under a smallness assumption, we provide its uniqueness. The proof is based on a result obtained in [14] in the study of elliptic quasi‐variational inequalities.

A frictionless contact problem for elastic-viscoplastic materials with normal compliance and damage

Abstract We study a quasistatic frictionless contact problem with normal compliance and damage for elastic–viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modelled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

INTEGRODIFFERENTIAL HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO VISCOELASTIC FRICTIONAL CONTACT

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.