Variational and Quasi-Variational Inequalities in Mechanics

Abstract

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.