Preface ........................................................ xi
I Introduction to variational inequalities ..................... 1
1 Preliminaries on functional analysis ......................... 3
1.1 Normed spaces ........................................... 3
1.1.1 Basic definitions ................................ 3
1.1.2 Linear continuous operators ...................... 6
1.1.3 Fixed point theorems ............................. 8
1.2 Hilbert spaces ......................................... 11
1.2.1 Projection operators ............................ 11
1.2.2 Orthogonality ................................... 15
1.2.3 Duality and weak convergence .................... 17
1.3 Elements of nonlinear analysis ......................... 19
1.3.1 Monotone operators .............................. 19
1.3.2 Convex lower semicontinuous functions ........... 24
1.3.3 Minimization problems ........................... 29
2 Elliptic variational inequalities ........................... 33
2.1 Variational inequalities of the first kind ............ 33
2.1.1 Existence and uniqueness ........................ 34
2.1.2 Penalization .................................... 35
2.2 Variational inequalities of the second kind ............ 40
2.2.1 Existence and uniqueness ........................ 40
2.2.2 A convergence result ............................ 42
2.2.3 Regularization .................................. 43
2.3 Quasivariational inequalities .......................... 49
2.3.1 The Banach fixed point argument ................. 49
2.3.2 The Schauder fixed point argument ............... 51
2.3.3 A convergence result ............................ 54
3 History-dependent variational inequalities .................. 57
3.1 Nonlinear equations with history-dependent operators ... 57
3.1.1 Spaces of vector-valued functions ............... 58
3.1.2 Two examples .................................... 61
3.1.3 The general case ................................ 65
3.2 History-dependent quasivariational inequalities ........ 67
3.2.1 A basic existence and uniqueness result ......... 67
3.2.2 A convergence result ............................ 73
3.3 Evolutionary variational inequalities .................. 75
3.3.1 Existence and uniqueness ........................ 75
3.3.2 Convergence results ............................. 78
II Modelling and analysis of contact problems .................. 81
4 Modelling of contact problems ............................... 83
4.1 Function spaces in contact mechanics ................... 84
4.1.1 Preliminaries ................................... 84
4.1.2 Spaces for the displacement field ............... 85
4.1.3 Spaces for the stress field ..................... 88
4.1.4 Spaces for piezoelectric contact problems ....... 89
4.2 Physical setting and constitutive laws ................. 91
4.2.1 Physical setting ................................ 91
4.2.2 Elastic constitutive laws ....................... 92
4.2.3 Viscoelastic constitutive laws .................. 95
4.2.4 Viscoplastic constitutive laws .................. 98
4.2.5 The von Mises convex ........................... 100
4.3 Modelling of elastic contact problems ................. 103
4.3.1 Preliminaries .................................. 104
4.3.2 Contact conditions ............................. 104
4.3.3 Friction laws .................................. 107
4.4 Modelling of elastic-viscoplastic contact problems .... 111
4.4.1 Preliminaries .................................. 111
4.4.2 Contact conditions and friction laws ........... 112
4.5 Modelling of piezoelectric contact problems .......... 114
4.5.1 Physical setting and preliminaries ............. 114
4.5.2 Constitutive laws .............................. 117
4.5.3 Contact conditions ............................. 119
5 Analysis of elastic contact problems ....................... 123
5.1 The Signorini contact problem ......................... 123
5.1.1 Problem statement .............................. 123
5.1.2 Existence and uniqueness ....................... 126
5.1.3 Penalization ................................... 128
5.1.4 Dual variational formulation ................... 131
5.1.5 Minimization ................................... 137
5.1.6 One-dimensional example ........................ 139
5.2 Frictional contact problems ........................... 143
5.2.1 Statement of the problems ...................... 144
5.2.2 Existence and uniqueness ....................... 147
5.2.3 A convergence result ........................... 148
5.2.4 Regularization ................................. 149
5.2.5 Dual variational formulation ................... 155
5.2.6 Minimization ................................... 160
5.3 A frictional contact problem with normal compliance ... 162
5.3.1 Problem statement .............................. 162
5.3.2 The Banach fixed point argument ................ 164
5.3.3 The Schauder fixed point argument .............. 166
5.3.4 Convergence results ............................ 167
6 Analysis of elastic-visco plastic contact problems ......... 173
6.1 Bilateral frictionless contact problems ............... 173
6.1.1 Contact of materials with short memory ......... 174
6.1.2 Contact of materials with long memory .......... 176
6.2 Viscoelastic contact problems with long memory ........ 178
6.2.1 Frictionless contact with unilateral
constraint ..................................... 178
6.2.2 Frictional contact with normal compliance ...... 181
6.2.3 A convergence result ........................... 183
6.3 Viscoelastic contact problems with short memory ....... 185
6.3.1 Contact with normal compliance ................. 186
6.3.2 Contact with normal damped response ............ 189
6.3.3 Other frictional contact problems .............. 192
6.3.4 Convergence results ............................ 196
6.4 Viscoplastic frictionless contact problems ............ 200
6.4.1 Contact with normal compliance ................. 200
6.4.2 Contact with unilateral constraint ............. 205
6.4.3 A convergence result ........................... 208
7 Analysis of piezoelectric contact problems ................. 217
7.1 An electro-elastic frictional contact problem ......... 217
7.1.1 Problem statement .............................. 218
7.1.2 Existence and uniqueness ....................... 220
7.1.3 Dual variational formulation ................... 223
7.2 An electro-viscoelastic frictional contact problem .... 227
7.2.1 Problem statement .............................. 227
7.2.2 Existence and uniqueness ....................... 231
7.3 An electro-viscoplastic frictionless contact problem .. 237
7.3.1 Problem statement .............................. 237
7.3.2 Existence and uniqueness ....................... 241
Bibliographical notes ...................................... 251
List of symbols ............................................ 257
References ................................................. 262
Index ......................................................... 275
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