# Variational and Quasi-Variational Inequalities in Mechanics

## Beschreibung

### Beschreibung

The essential aim of the present book is to consider a wide set of problems arising in the mathematical modelling of mechanical systems under unilateral constraints. In these investigations elastic and non-elastic deformations, friction and adhesion phenomena are taken into account. All the necessary mathematical tools are given: local boundary value problem formulations, construction of variational equations and inequalities, and the transition to minimization problems, existence and uniqueness theorems, and variational transformations (Friedrichs and Young-Fenchel-Moreau) to dual and saddle-point search problems.Important new results concern contact problems with friction. The Coulomb friction law and some others are considered, in which relative sliding velocities appear. The corresponding quasi-variational inequality is constructed, as well as the appropriate iterative method for its solution. Outlines of the variational approach to non-stationary and dissipative systems and to the construction of the governing equations are also given.Examples of analytical and numerical solutions are presented. Numerical solutions were obtained with the finite element and boundary element methods, including new 3D problems solutions. TOC:From the contents 1. Notation and Basics.- 2. Variational Setting of Linear Steady-state Problems.- 3.Variational Theory for Nonlinear Smooth Systems.- 4. Unilateral Constraints and Non-Differentiable Functionals.- 5. The Transformation of Variational Principles.- 6. Non-Stationary Problems and Thermodynamics.- 7. Solution Methods and Numerical implementation.- 8. Concluding Remarks.- References.- Index.### Inhaltsverzeichnis

1. Notation and Basics:1.1. Notations and Conventions; 1.2. Functional spaces; 1.3. Bases and complete systems. Existence theorem; 1.4. Trace Theorem; 1.5. The laws of thermodynamics;

2. Variational Setting of Linear Steady-state Problems:

2.1. Problem of the equilibrium of system with a finite number of degrees of Freedom; 2.2. Equilibrium of the simplest continuous systems governed by ordinary differential Equations; 2.3. 3D and 2D problems on the equilibrium of linear elastic bodies; 3.4. Positive definiteness of the potential energy of linear systems;

3.Variational Theory for Nonlinear Smooth Systems:

3.1. Examples of nonlinear systems; 3.2. Differentiation of operators and functionals; 3.3. Existence and uniqueness theorems of the minimal point of a functional; 3.4. Condition for the potentiality of an operator; 3.5. Boundary value problems in the Hencky-Ilyushin theory of plasticity without discharge; 3.6. Problems in the elastic bodies theory with finite displacements and strain;

4. Unilateral Constraints and Non-Differentiable Functionals:

4.1. Introduction: systems with finite degrees of freedom; 4.2. Variational methods in contact problems for deformed bodies without friction; 4.3. Variational method in contact problem with friction;

5. The Transformation of Variational Principles:

5.1. Friedrichs Transformation; 5.2. Equilibrium, mixed and hybrid variational principles in the theory of elasticity; 5.3. The Young-Fenchel-Moreau duality transformation; 5.4. Applications of duality transformations in contact problems;

6. Non-Stationary Problems and Thermodynamics:

6.1. Traditional principles and methods; 6.2. Gurtin's method; 6.3. Thermodynamics and mechanics of the deformed solids; 6.4. The variational theory of adhesion and crack initiation;

7. Solution Methods and Numerical implementation:

7.1. Frictionless contact problems: finite element method (FEM); 7.2. Friction contact problems: boundaryelement method (BEM);

8. Concluding Remarks:

8.1. Modelling. Identification problem. Optimization; 8.2. Development of the contact problems with friction, wear and adhesion; 8.3. Numerical implementation of the contact interaction phenomena;

References; Index.

### Pressestimmen

From the reviews:

"The main idea of the presented monograph is to deal with mathematical models connected with mechanical systems under unilateral constraints. ' Examples of analytical and numerical solutions are presented. Numerical solutions were obtained using the finite element and boundary element methods. ' The text contains a big amount of latest results achieved in mathematical modeling of contact problems in mechanics together with applications. It can be recommended both to graduate students and the researchers in applied mathematics and mechanics." (Igor Bock, Zentralblatt MATH, Vol. 1131 (9), 2008)

"The aim of this interesting book is the study of problems in the mathematical modelling of mechanical systems ' . The work is intended for a wide audience: this would include specialists in contact processes in structural and mechanical systems ' as well as those with a background in the mathematical sciences who seek a self-contained account of the mathematical theory of contact mechanics. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics." (J''n Lovišek, Mathematical Reviews, Issue 2009 e)