VARIATIONAL AND QUASI-VARIATIONAL INEQUALITIES IN MECHANICS

By PATRICIA GARCIA

CONTENTS NOTATIONS AND BASICS NOTATIONS AND CONVENTIONS SPACE OF INDEPENDENT VARIABLES TENSORS AND VECTORS SUMMATION NOTATION ALGEBRAIC OPERATION DI?ERENTIAL OPERATOR NOTATION FUNCTIONAL SPACES OPEN, CLOSED, AND COMPACT SETS IN RN METRIC SPACES SETS IN A METRIC SPACE NORMED LINEAR SPACES INNER PRODUCT SPACES, HILBERT SPACES, AND LEBESGUE SPACES GENERALIZED DERIVATIVES, SOBOLEV SPACES SOME EMBEDDING THEOREMS IN A SOBOLEV SPACE DUAL (CONJUGATE) SPACES AND WEAK CONVERGENCE QUOTIENT SPACE BASES AND COMPLETE SYSTEMS EXISTENCE THEOREM BASES AND COMPLETE SYSTEMS EXISTENCE OF A SOLUTION OF A SET OF LINEAR EQUATIONS TRACE THEOREM LAWS OF THERMODYNAMICS BASIC DE?NITIONS FIRST LAW OF THERMODYNAMICS SECOND LAW OF THERMODYNAMICS VARIATIONAL SETTING OF LINEAR STEADY-STATE PROBLEMS PROBLEM OF THE EQUILIBRIUM OF SYSTEMS WITH A ?NITE NUMBER OF DEGREES OF FREEDOM EQUILIBRIUM OF THE SIMPLEST CONTINUOUS SYSTEMS GOVERNED BY ORDINARY DI?ERENTIAL EQUATIONS SECOND-ORDER PROBLEMS PROBLEMS FOR FOURTH-ORDER EQUATIONS 3D AND 2D PROBLEMS ON THE EQUILIBRIUM OF LINEAR ELASTIC BODIES STRAIN STRESSES STRAIN–STRESS RELATION (THE GENERALIZED HOOKE LAW) FORMULATION OF BOUNDARY VALUE PROBLEMS 2D PROBLEMS OF THE LINEAR THEORY OF ELASTICITY TRANSITION TO THE VARIATIONAL FORMULATION POSITIVE DE?NITENESS OF THE POTENTIAL ENERGY OF LINEAR SYSTEMS UNIQUENESS OF THE MINIMUM POINT POSITIVE DE?NITENESS INEQUALITY FOR SCALAR FUNCTIONS APPLICATIONS TO LINEAR ELASTICITY PROBLEMS VARIATIONAL THEORY FOR NONLINEAR SMOOTH SYSTEMS EXAMPLES OF NONLINEAR SYSTEMS DI?ERENTIATION OF OPERATORS AND FUNCTIONALS EXISTENCE AND UNIQUENESS THEOREMS OF THE MINIMAL POINT OF A FUNCTIONAL CONDITION FOR THE POTENTIALITY OF AN OPERATOR BOUNDARY VALUE PROBLEMS IN THE HENCKY–ILYUSHIN THEORY OF PLASTICITY WITHOUT DISCHARGE PROBLEMS IN THE ELASTIC BODIES THEORY WITH ?NITE DISPLACEMENTS AND STRAIN VECTORS AND STRAINS IN SYSTEMS OF CURVILINEAR COORDINATES STRAINS AND STRESS EQUILIBRIUM (MOTION) EQUATIONS GOVERNING EQUATIONS PRINCIPLE OF VIRTUAL WORK UNILATERAL CONSTRAINTS AND NONDI?ERENTIABLE FUNCTIONALS INTRODUCTION: SYSTEMS WITH ?NITE DEGREES OF FREEDOM EXAMPLE COMMENT OF THE DEVELOPMENT OF THE VARIATIONAL INEQUALITIES METHOD VARIATIONAL METHODS IN CONTACT PROBLEMS FOR DEFORMED BODIES WITHOUT FRICTION CONTACT PROBLEMS FOR A BEAM AND MEMBRANE CONTACT BETWEEN AN ELASTIC BODY AND A RIGID STAMP CONTACT OF A SYSTEM OF DEFORMED BODIES GEOMETRICALLY NONLINEAR THEORY OF ELASTICITY COMMENTS VARIATIONAL METHOD IN CONTACT PROBLEM WITH FRICTION GENERALITIES QUASI-VARIATIONAL INEQUALITY FOR THE CONTACT PROBLEM WITH FRICTION INTERPRETATION SOLUTION OF THE QUASI-VARIATIONAL INEQUALITY A PRIORI ESTIMATE OF THE DYNAMIC PROBLEM SOLUTION DYNAMIC CONTACT OF AN ELASTIC SOLID AND A SYSTEM OF RIGID MOVING INDENTERS LOCAL POTENTIAL METHOD PROPORTIONAL PROCESSES FURTHER GENERALIZATION 4 3 10 COMMENTS TRANSFORMATION OF VARIATIONAL PRINCIPLES FRIEDRICHS TRANSFORMATION INTRODUCTION BOUNDARY VALUE PROBLEM FOR THE ORDINARY DI?ERENTIAL EQUATION DIRICHLET PROBLEM FOR THE POISSON EQUATION EQUILIBRIUM, MIXED AND HYBRID VARIATIONAL PRINCIPLES IN THE THEORY OF ELASTICITY CASTIGLIANO PRINCIPLE AND THE REISSNER PRINCIPLE HYBRID PRINCIPLES KINEMATIC CONSTRAINTS IN THE DOMAIN (HERRMANN PRINCIPLE) YOUNG–FENCHEL–MOREAU DUALITY TRANSFORMATION DE?NITION OF DUALITY TRANSFORMATION GENERAL DE?NITION OF SUBDI?ERENTIAL AND SUBGRADIENT METHOD OF SOLVING MINIMIZATION (MAXIMIZATION) PROBLEMS TRANSITION TO THE SADDLE-POINT PROBLEM SPECIAL CASES APPLICATIONS OF DUALITY TRANSFORMATIONS IN CONTACT PROBLEMS DISTURBANCE OF THE ARROW–HURWITZ FORM IN THE PROBLEM (P) DISTURBANCE OF THE CASTIGLIANO FORM COMBINED DISTURBANCE (LAGRANGIANS) GENERALIZATION FOR THE DEFORMATION THEORY OF PLASTICITY CONTACT PROBLEM FOR SEVERAL ELASTIC BODIES NONSTATIONARY PROBLEMS AND THERMODYNAMICS TRADITIONAL PRINCIPLES AND METHODS DI?ERENTIAL VARIATIONAL PRINCIPLES INTEGRAL PRINCIPLES STUDY OF DISSIPATIVE SYSTEMS WITH THE ?NITE NUMBER OF DEGREES OF FREEDOM CONTINUOUS CONSERVATIVE SYSTEMS EXAMPLE OF A DISSIPATIVE CONTINUOUS SYSTEM GURTIN METHOD WAVE EQUATION HEAT CONDUCTION EQUATION DYNAMIC PROBLEM OF THE LINEAR THEORY OF ELASTICITY THEORY OF LINEAR VISCOELASTICITY THERMODYNAMICS AND MECHANICS OF THE DEFORMED SOLIDS EXTREMUM PRINCIPLES FOR DISSIPATION AND ENTROPY EXTREMUM PRINCIPLES IN THE THEORY OF PLASTIC ?OW THEORY OF NORMAL DISSIPATIVE MECHANISMS VARIATIONAL THEORY OF ADHESION AND CRACK INITIATION THEORY OF ADHESIVE JOINTS FORMATION AND DESTRUCTION COROLLARIES I AND II OF THE LAWS OF THERMODYNAMICS STATE EQUATIONS JOHNSON–KENDALL–ROBERTS THEORY OF ADHESION (JKR THEORY) EXAMPLES IN THE JKR THEORY MODELS OF ACCUMULATION OF DAMAGES ON THE SURFACE MODEL OF THE VISCOUS CRACK MOTION SOLUTION METHODS AND NUMERICAL IMPLEMENTATION FRICTIONLESS CONTACT PROBLEMS: ?NITE ELEMENT METHOD GENERALITIES: CONTINUOUS PROBLEM FINITE ELEMENT METHOD: EXAMPLES FRICTION CONTACT PROBLEMS: BOUNDARY ELEMENT METHOD BOUNDARY ELEMENT METHOD NUMERICAL EXAMPLES TO 2D CONTACT PROBLEMS SOLUTION TO 3D CONTACT PROBLEMS WITH THE BOUSSINESQ AND CERRUTI FORMULAE CONCLUDING REMARKS MODELING, AND IDENTI?CATION PROBLEM, AND OPTIMIZATION DEVELOPMENT OF THE CONTACT PROBLEMS WITH FRICTION, WEAR, AND ADHESION NUMERICAL IMPLEMENTATION OF THE CONTACT INTERACTION PHENOMENA REFERENCES