INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS......Page 2
CONTENTS......Page 8
PREFACE......Page 16
PART I FUNDAMENTALS......Page 18
1.1.1 A World of Simulation......Page 20
1.1.2 Evolution of Explicit Finite Element Method......Page 21
1.1.3 Computer Aided Engineering (CAE)—Opportunities and Challenges......Page 22
1.2.1 Notations......Page 23
1.2.2 Constitutive Relations of Elasticity......Page 25
2.1 Transient Structural Dynamics......Page 28
2.2.1 Hamilton’s Principle......Page 30
2.3 Finite Element Equations and the Explicit Procedures......Page 32
2.3.1 Discretization in Space by Finite Element......Page 33
2.3.3 Discretization in Time by Finite Difference......Page 36
2.3.4 Procedure of the Explicit Finite Element Method......Page 37
2.4 Main Features of the Explicit Finite Element Method......Page 38
2.4.1 Stability Condition and Time Step Size......Page 39
2.4.2 Diagonal Mass Matrix......Page 40
2.5.1 About the Solution of the Elastodynamics......Page 41
2.5.2 A Priori Error Estimate of Explicit Finite Element Method for Elastodynamics......Page 42
2.5.3 About the Diagonal Mass Matrix......Page 47
PART II ELEMENT TECHNOLOGY......Page 54
3 FOUR-NODE SHELL ELEMENT (REISSNER–MINDLIN PLATE THEORY)......Page 56
3.1.1 Characteristics of Thin-walled Structures......Page 57
3.1.2 Resultant Equations......Page 59
3.1.3 Applications to Linear Elasticity......Page 61
3.1.4 Kirchhoff–Love Theory......Page 63
3.2 Linear Theory of R-M Plate......Page 64
3.2.2 Load Scaling for Static Problem of R-M Plate......Page 65
3.2.3 Load Scaling and Mass Scaling for Dynamic Problem of R-M Plate......Page 66
3.2.4 Relation between R-M Theory and K-L Theory......Page 67
3.3.2 Bilinear Interpolations......Page 69
3.3.3 Shear Locking Issues of R-M Plate Element......Page 72
3.4.1 Reduced Integration......Page 73
3.4.2 Selective Reduced Integration......Page 74
3.4.3 Nonlinear Application of Selective Reduced Integration—Hughes–Liu Element......Page 75
3.5 Perturbation Hourglass Control—Belytschko–Tsay Element......Page 77
3.5.1 Concept of Hourglass Control......Page 78
3.5.2 Four-node Belytschko–Tsay Shell Element—Perturbation Hourglass Control......Page 80
3.5.3 Improvement of Belytschko–Tsay Shell Element......Page 85
3.5.4 About Convergence of Element using Reduced Integration......Page 87
3.6.1 Constant and Nonconstant Contributions......Page 88
3.6.2 Projection of Shear Strain......Page 89
3.6.3 Physical Hourglass Control by One-point Integration......Page 90
3.6.4 Drill Projection......Page 91
3.7.1 Projection of Transverse Shear Strain......Page 93
3.7.2 Convergence of B-D Element......Page 95
3.8.1 Evaluations with Warped Mesh and Reduced Thickness......Page 97
3.8.2 About the Locking-free Low Order Four-node R-M Plate Element......Page 102
4 THREE-NODE SHELL ELEMENT (REISSNER–MINDLIN PLATE THEORY)......Page 105
4.1.1 Transformation and Jacobian......Page 106
4.1.3 Shear Locking with C0 Triangular Element......Page 108
4.2.1 A C0 Element with Decomposition of Deflection......Page 109
4.2.2 A C0 Element with Decomposition of Rotations......Page 113
4.3 Discrete Kirchhoff Triangular Element......Page 114
4.4.1 Evaluations with Warped Mesh and Reduced Thickness......Page 119
4.4.2 About the Locking-free Low Order Three-node R-M Plate Element......Page 122
5.1 Trilinear Interpolation for the Eight-node Hexahedron Element......Page 124
5.2 Locking Issues of the Eight-node Solid Element......Page 128
5.3 One-point Reduced Integration and the Perturbed Hourglass Control......Page 130
5.4 Assumed Strain Method and Selective/Reduced Integration......Page 132
5.6 An Enhanced Assumed Strain Method......Page 135
5.7 Taylor Expansion of Assumed Strain about the Element Center......Page 137
5.8 Evaluation of Eight-node Solid Element......Page 140
6.1 Truss and Rod Element......Page 145
6.2 Timoshenko Beam Element......Page 146
6.3.1 One Degree of Freedom Spring Element......Page 148
6.3.2 Six Degrees of Freedom Spring Element......Page 149
6.3.3 Three-node Spring Element......Page 150
6.4.1 Description of Spot Weld Separation......Page 151
6.4.2 Failure Criterion......Page 152
6.4.3 Finite Element Representation of Spot Weld......Page 154
PART III MATERIAL MODELS......Page 156
7 MATERIAL MODEL OF PLASTICITY......Page 158
7.1.1 Tensile Test......Page 159
7.1.2 Hardening......Page 161
7.1.3 Yield Surface......Page 162
7.1.4 Normality Condition......Page 167
7.1.5 Strain Rate Effect/Viscoplasticity......Page 169
7.2.1 Relations between Stress Increments and Strain Increments......Page 170
7.2.2 Constitutive Equations for Mises Criterion......Page 174
7.2.3 Application to Kinematic Hardening......Page 175
7.3 Software Implementation......Page 176
7.3.2 Normal (Radial) Return Scheme......Page 177
7.3.3 A Generalized Plane Stress Model......Page 180
7.3.4 Stress Resultant Approach......Page 181
7.4 Evaluation of Shell Elements with Plastic Deformation......Page 186
8.1 Concept of Damage Mechanics......Page 192
8.2 Gurson’s Model......Page 194
8.2.1 Damage Variables and Yield Function......Page 195
8.2.2 Constitutive Equation and Damage Growth......Page 196
8.3 Chow’s Isotropic Model of Continuum Damage Mechanics......Page 197
8.3.1 Damage Effect Tensor......Page 198
8.3.2 Yield Function and Constitutive Equation......Page 200
8.3.3 Damage Growth......Page 202
8.3.4 Application to Plates and Shells......Page 204
8.3.5 Determination of Parameters......Page 205
8.4 Chow’s Anisotropic Model of Continuum Damage Mechanics......Page 206
9.1.1 Spring–Damper Model......Page 209
9.1.2 A General Three-dimensional Viscoelasticity Model......Page 213
9.2.1 Fundamental Mechanical Properties of Polymer Materials......Page 214
9.2.2 A Temperature, Strain Rate, and Pressure Dependent Constitutive Relation......Page 215
9.2.3 A Nonlinear Viscoelastic Model of Polymer Materials......Page 216
9.3.1 Mooney–Rivlin Model of Rubber Material......Page 217
9.3.2 Blatz–Ko Model......Page 219
9.4 Foam......Page 220
9.4.2 A Model Consisting of Polymer Skeleton and Air......Page 222
9.4.3 A Phenomenological Uniaxial Model......Page 224
9.4.4 Hysteresis Behavior......Page 225
9.5 Honeycomb......Page 226
9.5.2 Critical Buckling Load......Page 227
9.5.3 A Phenomenological Material Model of Honeycomb......Page 228
9.5.4 Behavior of Honeycomb under Complex Loading Conditions......Page 230
9.6.1 Application of J-integral......Page 231
9.6.2 Application of Anisotropic Damage Model......Page 232
9.6.3 A Simplified Model with Shell Element for the Laminated Glass......Page 233
PART IV CONTACT AND CONSTRAINT CONDITIONS......Page 236
10.1 Examples of Contact Problems......Page 238
10.1.1 Uniformly Loaded String with a Flat Rigid Obstacle......Page 239
10.1.2 Hertz Contact Problem......Page 242
10.1.3 Elastic Impact of Two Balls......Page 243
10.1.4 Impact of an Elastic Rod on the Flat Rigid Obstacle......Page 245
10.1.5 Impact of a Vibrating String to the Flat Rigid Obstacle......Page 248
10.2.1 Contact with a Smooth Rigid Obstacle—Signorini’s Problem......Page 250
10.2.2 Contact between Two Smooth Deformable Bodies......Page 254
10.2.3 Coulomb’s Law of Friction......Page 257
10.2.5 Domain Contact......Page 259
10.3.1 Variational Formulation for Frictionless Dynamic Contact Problem......Page 260
10.3.2 Variational Formulation for Frictional Dynamic Contact Problem......Page 264
10.3.3 Variational Formulation for Domain Contact......Page 267
10.4.1 Concept of Penalty Method......Page 269
10.4.2 Penalty Method for Nonlinear Dynamic Contact Problem......Page 273
10.4.3 Explicit Finite Element Procedure with Penalty Method for Dynamic Contact......Page 275
11 NUMERICAL PROCEDURES FOR THREE-DIMENSIONAL SURFACE CONTACT......Page 278
11.1 A Contact Algorithm with Slave Node Searching Master Segment......Page 279
11.1.1 Global Search......Page 280
11.1.2 Bucket Sorting Method......Page 281
11.1.3 Local Search......Page 283
11.1.4 Penalty Contact Force......Page 285
11.2.1 Global Search with Bucket Sorting Based on Segment’s Capture Box......Page 289
11.3 Method of Contact Territory and Defense Node......Page 290
11.3.2 Local Search in the Territory......Page 291
11.3.3 Defense Node and Contact Force......Page 292
11.4.1 The Pinball Hierarchy......Page 294
11.4.2 Penalty Contact Force......Page 295
11.5.1 Search for Line Contact......Page 296
11.5.2 Penalty Contact Force of Edge-to-Edge Contact......Page 298
11.6 Evaluation of Contact Algorithm with Penalty Method......Page 299
12.1 Rigid Wall......Page 306
12.1.1 A Stationary Flat Rigid Wall......Page 307
12.1.2 A Moving Flat Rigid Wall......Page 308
12.1.3 Rigid Wall with a Curved Surface......Page 310
12.2 Rigid Body......Page 313
12.3 Explicit Finite Element Procedure with Constraint Conditions......Page 315
12.4 Application Examples with Constraint Conditions......Page 317
REFERENCES......Page 322
INDEX......Page 342