| This book deals with approximation and numerical realization of variational inequalities of elliptic type, having applications in mechanics of solids. Emphasis is devoted to the study of contact problems of elastic bodies and problems of plasticity. The main feature of the book is that problems are treated in all their complexity - from the analysis of the continuous models, existence and uniqueness results, to finite element models and the study of their mutual relation, error estimates, convergence results. Special attention is given to contact problems with friction, where some new results are presented, concerning Coulomb`s model of friction. Contents
Preface
1 Unilateral Problems for Scalar Functions 1
1.1 Unilateral Boundary Value Problems for Second Order Equations 2
1.1.1 Primal and Dual Variational Problems 4
1.1.1.1 Dual Variational Formulation 7
1.1.1.2 Relation Between the Primal and Dual Variational Formulations 10
1.1.2 Mixed Variational Formulations 13
1.1.3 Solution of Primal Problems by the Finite Element Method and Error Bounds 18
1.1.3.1 Approximation of Problem P1 by the Finite Element Method 18
1.1.3.2 The General Theory of Approximations for Elliptic Inequalities 21
1.1.3.3 A Priori Bound for Problem P1 24
1.1.4 Solution of Dual Problems by the Finite Element Method and Error Bounds 27
1.1.4.1 Problems with Absolute Terms 27
1.1.4.1.1 A Priori Error Bounds 29
1.1.4.1.2 A Posteriori Error Bounds and the Two-Sided Energy Bound 34
1.1.4.2 Problems Without Absolute Terms 36
1.1.4.2.1 A Priori Error Bound 41
1.1.4.2.2 A Posteriori Error Bounds and the Two-Sided Energy Estimate 48
1.1.5 Solution of Mixed Problems by the Finite Element Method and Error Bounds 49
1.1.5.1 Mixed Variational Formulations of Elliptic Inequalities 51
1.1.5.2 Approximation of the Mixed Variational Formulation and Error Bounds 54
1.1.5.3 Numerical Realization of Mixed Variational Formulations 59
1.1.6 Semicoercive Problems 62
1.1.6.1 Solution of the Primal Problem by the Finite Element Method and Error Bounds 66
1.1.6.2 Solution of the Dual Problem by the Finite Element Method and Error Bounds 69
1.1.6.3 Convergence of the Dual Finite Element Method 75
1.1.7 Problems with Nonhomogeneous Boundary Obstacle 82
1.1.7.1 Approximation of the Primal Problem 85
1.1.7.2 Solution of the Dual Problem by the Finite Element Method 87
1.1.7.3 A Posteriori Error Bounds and Two-Sided Energy Estimate 88
1.2 Problems with Inner Obstacles for Second-Order Operators 89
1.2.1 Primal and Dual Variational Formulations 89
1.2.2 Mixed Variational Formulation 93
1.2.3 Solution of the Primal Problem by the Finite Element Method 94
1.2.4 Solution of the Dual Problem by the Finite Element Method 98
1.2.4.1 Approximation of the Dual Formulation of the Problem with an Inner Obstacle 98
1.2.4.2 Construction of the Sets Qrh and Their Approximate Properties 99
1.2.4.3 A Priori Error Bound of the Approximation of the Dual Formulation 99
1.2.5 Solution of the Mixed Formulation by the Finite Element Method 104
2 One-Sided Contact of Elastic Bodies 109
2.1 Formulations of Contact Problems 110
2.1.1 Problems with Bounded Zone of Contact 112
2.1.2 Problems with Increasing Zone of Contact 114
2.1.3 Variational Formulations 116
2.2 Existence and Uniqueness of Solution 121
2.2.1 Problem with Bounded Zone of Contact 121
2.2.2 Problem with Increasing Zone of Contact 130
2.3 Solution of Primal Problems by the Finite Element Method 134
2.3.1 Approximation of the Problem with a Bounded Zone of Contact 134
2.3.2 Approximation Problems with Increasing Zone of Contact 136
2.3.3 A Priori Error Estimates and the Convergence 138
2.3.3.1 Bounded Zone of Contact 138
2.3.3.1.1 Polygonal Domains 139
2.3.3.1.2 Curved Contact Zone 148
2.3.3.2 Increasing Zone of Contact 154
2.4 Dual Variational Formulation of the Problem with Bounded Zone of Contact 164
2.4.1 Approximation of the Dual Problem 170
2.4.1.1 Equilibrium Model of Finite Elements 173
2.4.1.2 Applications of the Equilibrium Model 175
2.4.1.3 Algorithm for Approximations of the Dual Problem 176
2.5 Contact Problems with Friction 182
2.5.1 The Problem with Prescribed Normal Force 187
2.5.2 Some Auxiliary Spaces 192
2.5.3 Existence of Solution of the Problem with Friction 194
2.5.4 Algorithms for the Contact Problem with Friction for Elastic Bodies 196
2.5.4.1 Direct Iterations 196
2.5.4.2 Alternating Iterations 207
2.5.4.2.1 Unilateral Contact with a Given Shear Force 208
2.5.4.2.2 Realizability of the Algorithm of Alternating Iterations 212
3 Problems of the Theory of Plasticity 221
3.1 Prandtl-Reuss Model of Plastic Flow 226
3.1.1 Existence and Uniqueness of Solution 229
3.1.2 Solution by Finite Elements 233
3.1.2.1 A Priori Error Estimates 235
3.2 Plastic Flow with Isotropic or Kinematic Hardening 238
3.2.1 Existence and Uniqueness of Solution of the Plastic Flow Problem with Hardening 241
3.2.2 Solution of Isotropic Hardening by Finite Elements 247
3.2.2.1 A Priori Error Estimates 250
3.2.2.2 A Priori Error Estimates for the Plane Problem 258
3.2.2.3 Convergence in the Case of Nonregular Solution 262
References 267
Index 273 |
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