Computational Continuum Mechanics

COVER......Page 1
HALF-TITLE......Page 3
TITLE......Page 5
COPYRIGHT......Page 6
CONTENTS......Page 7
PREFACE......Page 11
1 INTRODUCTION......Page 15
1.1 MATRICES......Page 16
1.2 VECTORS......Page 20
1.3 SUMMATION CONVENTION......Page 26
1.4 CARTESIAN TENSORS......Page 27
1.5 POLAR DECOMPOSITION THEOREM......Page 39
1.6 D’ALEMBERT’S PRINCIPLE......Page 41
1.7 VIRTUAL WORK PRINCIPLE......Page 48
1.8 APPROXIMATION METHODS......Page 51
1.9 DISCRETE EQUATIONS......Page 54
1.10 MOMENTUM, WORK, AND ENERGY......Page 57
1.11 PARAMETER CHANGE AND COORDINATE TRANSFORMATION......Page 59
PROBLEMS......Page 62
2 KINEMATICS......Page 65
2.1 MOTION DESCRIPTION......Page 66
2.2 STRAIN COMPONENTS......Page 74
2.3 OTHER DEFORMATION MEASURES......Page 81
2.4 DECOMPOSITION OF DISPLACEMENT......Page 83
2.5 VELOCITY AND ACCELERATION......Page 85
2.6 COORDINATE TRANSFORMATION......Page 89
2.7 OBJECTIVITY......Page 96
2.8 CHANGE OF VOLUME AND AREA......Page 99
2.9 CONTINUITY EQUATION......Page 103
2.10 REYNOLDS’ TRANSPORT THEOREM......Page 104
2.11 EXAMPLES OF DEFORMATION......Page 106
PROBLEMS......Page 114
3.1 EQUILIBRIUM OF FORCES......Page 117
3.2 TRANSFORMATION OF STRESSES......Page 120
3.3 EQUATIONS OF EQUILIBRIUM......Page 121
3.4 SYMMETRY OF THE CAUCHY STRESS TENSOR......Page 123
3.5 VIRTUAL WORK OF THE FORCES......Page 125
3.6 DEVIATORIC STRESSES......Page 134
3.7 STRESS OBJECTIVITY......Page 137
3.8 ENERGY BALANCE......Page 141
PROBLEMS......Page 143
4 CONSTITUTIVE EQUATIONS......Page 145
4.1 GENERALIZED HOOKE’S LAW......Page 146
4.2 ANISOTROPIC LINEARLY ELASTIC MATERIALS......Page 148
4.3 MATERIAL SYMMETRY......Page 149
4.4 HOMOGENEOUS ISOTROPIC MATERIAL......Page 151
4.5 PRINCIPAL STRAIN INVARIANTS......Page 158
4.6 SPECIAL MATERIAL MODELS FOR LARGE DEFORMATIONS......Page 160
4.7 LINEAR VISCOELASTICITY......Page 164
4.8 NONLINEAR VISCOELASTICITY......Page 178
4.9 A SIMPLE VISCOELASTIC MODEL FOR ISOTROPIC MATERIALS......Page 185
4.10 FLUID CONSTITUTIVE EQUATIONS......Page 187
4.11 NAVIER–STOKES EQUATIONS......Page 188
PROBLEMS......Page 189
5 PLASTICITY FORMULATIONS......Page 191
5.1 ONE-DIMENSIONAL PROBLEM......Page 193
5.2 LOADING AND UNLOADING CONDITIONS......Page 194
5.3 SOLUTION OF THE PLASTICITY EQUATIONS......Page 195
5.4 GENERALIZATION OF THE PLASTICITY THEORY: SMALL STRAINS......Page 204
5.5 J2 FLOW THEORY WITH ISOTROPIC/KINEMATIC HARDENING......Page 211
5.6 NONLINEAR FORMULATION FOR HYPERELASTIC–PLASTIC MATERIALS......Page 228
5.7 HYPERELASTIC–PLASTIC J2 FLOW THEORY......Page 239
PROBLEMS......Page 244
6 FINITE ELEMENT FORMULATION: LARGE-DEFORMATION, LARGE-ROTATION PROBLEM......Page 245
6.1 DISPLACEMENT FIELD......Page 247
6.2 ELEMENT CONNECTIVITY......Page 254
6.3 INERTIA AND ELASTIC FORCES......Page 257
6.4 EQUATIONS OF MOTION......Page 260
6.5 NUMERICAL EVALUATION OF THE ELASTIC FORCES......Page 264
6.6 FINITE ELEMENTS AND GEOMETRY......Page 270
6.7 TWO-DIMENSIONAL EULER–BERNOULLI BEAM ELEMENT......Page 277
6.8 TWO-DIMENSIONAL SHEAR DEFORMABLE BEAM ELEMENT......Page 281
6.9 THREE-DIMENSIONAL CABLE ELEMENT......Page 283
6.10 THREE-DIMENSIONAL BEAM ELEMENT......Page 284
6.11 THIN-PLATE ELEMENT......Page 286
6.12 HIGHER-ORDER PLATE ELEMENT......Page 288
6.13 ELEMENT PERFORMANCE......Page 289
6.14 OTHER FINITE ELEMENT FORMULATIONS......Page 294
6.15 UPDATED LAGRANGIAN AND EULERIAN FORMULATIONS......Page 296
PROBLEMS......Page 298
7 FINITE ELEMENT FORMULATION: SMALL-DEFORMATION, LARGE-ROTATION PROBLEM......Page 300
7.1 BACKGROUND......Page 301
7.2 ROTATION AND ANGULAR VELOCITY......Page 305
7.3 FLOATING FRAME OF REFERENCE......Page 310
7.4 INTERMEDIATE ELEMENT COORDINATE SYSTEM......Page 311
7.5 CONNECTIVITY AND REFERENCE CONDITIONS......Page 314
7.6 KINEMATIC EQUATIONS......Page 320
7.7 FORMULATION OF THE INERTIA FORCES......Page 321
7.8 ELASTIC FORCES......Page 325
7.9 EQUATIONS OF MOTION......Page 327
7.10 COORDINATE REDUCTION......Page 328
7.11 INTEGRATION OF FINITE ELEMENT AND MULTIBODY SYSTEM ALGORITHMS......Page 331
PROBLEMS......Page 333
References......Page 335
Index......Page 341