# Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics

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A systematic introduction to the theories and formulations of the explicit finite element method

As numerical technology continues to grow and evolve with industrial applications, understanding the explicit finite element method has become increasingly important, particularly in the areas of crashworthiness, metal forming, and impact engineering. Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is the first book to address specifically what is now accepted as the most successful numerical tool for nonlinear transient dynamics. The book aids readers in mastering the explicit finite element method and programming code without requiring extensive background knowledge of the general finite element.

The authors present topics relating to the variational principle, numerical procedure, mechanical formulation, and fundamental achievements of the convergence theory. In addition, key topics and techniques are provided in four clearly organized sections:

Fundamentals explores a framework of the explicit finite element method for nonlinear transient dynamics and highlights achievements related to the convergence theory

Element Technology discusses four-node, three-node, eight-node, and two-node element theories

Material Models outlines models of plasticity and other nonlinear materials as well as the mechanics model of ductile damage

Contact and Constraint Conditions covers subjects related to three-dimensional surface contact, with examples solved analytically, as well as discussions on kinematic constraint conditions

Throughout the book, vivid figures illustrate the ideas and key features of the explicit finite element method. Examples clearly present results, featuring both theoretical assessments and industrial applications.

Introduction to the Explicit Finite Element Method for Nonlinear Transient Dynamics is an ideal book for both engineers who require more theoretical discussions and for theoreticians searching for interesting and challenging research topics. The book also serves as an excellent resource for courses on applied mathematics, applied mechanics, and numerical methods at the graduate level.

PREFACE xv

PART I FUNDAMENTALS 1

1 INTRODUCTION 3

1.1 Era of Simulation and Computer Aided Engineering 3

1.1.1 A World of Simulation 3

1.1.2 Evolution of Explicit Finite Element Method 4

1.1.3 Computer Aided Engineering (CAE)—Opportunities and Challenges 5

1.2 Preliminaries 6

1.2.1 Notations 6

1.2.2 Constitutive Relations of Elasticity 8

2 FRAMEWORK OF EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS 11

2.1 Transient Structural Dynamics 11

2.2 Variational Principles for Transient Dynamics 13

2.2.1 Hamilton’s Principle 13

2.2.2 Galerkin Method 15

2.3 Finite Element Equations and the Explicit Procedures 15

2.3.1 Discretization in Space by Finite Element 16

2.3.2 System of Semidiscretization 19

2.3.3 Discretization in Time by Finite Difference 19

2.3.4 Procedure of the Explicit Finite Element Method 20

2.4 Main Features of the Explicit Finite Element Method 21

2.4.1 Stability Condition and Time Step Size 22

2.4.2 Diagonal Mass Matrix 23

2.4.3 Corotational Stress 24

2.5 Assessment of Explicit Finite Element Method 24

2.5.1 About the Solution of the Elastodynamics 24

2.5.2 A Priori Error Estimate of Explicit Finite Element Method for Elastodynamics 25

2.5.3 About the Diagonal Mass Matrix 30

PART II ELEMENT TECHNOLOGY 37

3 FOUR-NODE SHELL ELEMENT (REISSNER–MINDLIN PLATE THEORY) 39

3.1 Fundamentals of Plates and Shells 40

3.1.1 Characteristics of Thin-walled Structures 40

3.1.2 Resultant Equations 42

3.1.3 Applications to Linear Elasticity 44

3.1.4 Kirchhoff–Love Theory 46

3.1.5 Reissner–Mindlin Plate Theory 47

3.2 Linear Theory of R-M Plate 47

3.2.1 Helmholtz Decomposition for R-M Plate 48

3.2.2 Load Scaling for Static Problem of R-M Plate 48

3.2.3 Load Scaling and Mass Scaling for Dynamic Problem of R-M Plate 49

3.2.4 Relation between R-M Theory and K-L Theory 50

3.3 Interpolation for Four-node R-M Plate Element 52

3.3.1 Variational Equations for R-M Plate 52

3.3.2 Bilinear Interpolations 52

3.3.3 Shear Locking Issues of R-M Plate Element 55

3.4 Reduced Integration and Selective Reduced Integration 56

3.4.1 Reduced Integration 56

3.4.2 Selective Reduced Integration 57

3.4.3 Nonlinear Application of Selective Reduced Integration—Hughes–Liu Element 58

3.5 Perturbation Hourglass Control—Belytschko–Tsay Element 60

3.5.1 Concept of Hourglass Control 61

3.5.2 Four-node Belytschko–Tsay Shell Element—Perturbation Hourglass Control 63

3.5.3 Improvement of Belytschko–Tsay Shell Element 68

3.5.4 About Convergence of Element using Reduced Integration 70

3.6 Physical Hourglass Control—Belytschko–Leviathan (QPH) Element 71

3.6.1 Constant and Nonconstant Contributions 71

3.6.2 Projection of Shear Strain 72

3.6.3 Physical Hourglass Control by One-point Integration 73

3.6.4 Drill Projection 74

3.6.5 Improvement of B-L (QPH) Element 76

3.7 Shear Projection Method—Bathe–Dvorkin Element 76

3.7.1 Projection of Transverse Shear Strain 76

3.7.2 Convergence of B-D Element 78

3.8 Assessment of Four-node R-M Plate Element 80

3.8.1 Evaluations with Warped Mesh and Reduced Thickness 80

3.8.2 About the Locking-free Low Order Four-node R-M Plate Element 85

4 THREE-NODE SHELL ELEMENT (REISSNER–MINDLIN PLATE THEORY) 88

4.1 Fundamentals of a Three-node C0 Element 89

4.1.1 Transformation and Jacobian 89

4.1.2 Numerical Quadrature for In-plane Integration 91

4.1.3 Shear Locking with C0 Triangular Element 91

4.2 Decomposition Method for C0 Triangular Element with One-point Integration 92

4.2.1 A C0 Element with Decomposition of Deflection 92

4.2.2 A C0 Element with Decomposition of Rotations 96

4.3 Discrete Kirchhoff Triangular Element 97

4.4 Assessment of Three-node R-M Plate Element 102

4.4.1 Evaluations with Warped Mesh and Reduced Thickness 102

4.4.2 About the Locking-free Low Order Three-node R-M Plate Element 105

5 EIGHT-NODE SOLID ELEMENT 107

5.1 Trilinear Interpolation for the Eight-node Hexahedron Element 107

5.2 Locking Issues of the Eight-node Solid Element 111

5.3 One-point Reduced Integration and the Perturbed Hourglass Control 113

5.4 Assumed Strain Method and Selective/Reduced Integration 115

5.5 Assumed Deviatoric Strain 118

5.6 An Enhanced Assumed Strain Method 118

5.7 Taylor Expansion of Assumed Strain about the Element Center 120

5.8 Evaluation of Eight-node Solid Element 123

6 TWO-NODE ELEMENT 128

6.1 Truss and Rod Element 128

6.2 Timoshenko Beam Element 129

6.3 Spring Element 131

6.3.1 One Degree of Freedom Spring Element 131

6.3.2 Six Degrees of Freedom Spring Element 132

6.3.3 Three-node Spring Element 133

6.4 Spot Weld Element 134

6.4.1 Description of Spot Weld Separation 134

6.4.2 Failure Criterion 135

6.4.3 Finite Element Representation of Spot Weld 137

PART III MATERIAL MODELS 139

7 MATERIAL MODEL OF PLASTICITY 141

7.1 Fundamentals of Plasticity 142

7.1.1 Tensile Test 142

7.1.2 Hardening 144

7.1.3 Yield Surface 145

7.1.4 Normality Condition 150

7.1.5 Strain Rate Effect/Viscoplasticity 152

7.2 Constitutive Equations 153

7.2.1 Relations between Stress Increments and Strain Increments 153

7.2.2 Constitutive Equations for Mises Criterion 157

7.2.3 Application to Kinematic Hardening 158

7.3 Software Implementation 159

7.3.1 Explicit Finite Element Procedure with Plasticity 160

7.3.2 Normal (Radial) Return Scheme 160

7.3.3 A Generalized Plane Stress Model 163

7.3.4 Stress Resultant Approach 164

7.4 Evaluation of Shell Elements with Plastic Deformation 169

8 CONTINUUM MECHANICS MODEL OF DUCTILE DAMAGE 175

8.1 Concept of Damage Mechanics 175

8.2 Gurson’s Model 177

8.2.1 Damage Variables and Yield Function 178

8.2.2 Constitutive Equation and Damage Growth 179

8.3 Chow’s Isotropic Model of Continuum Damage Mechanics 180

8.3.1 Damage Effect Tensor 181

8.3.2 Yield Function and Constitutive Equation 183

8.3.3 Damage Growth 185

8.3.4 Application to Plates and Shells 187

8.3.5 Determination of Parameters 188

8.4 Chow’s Anisotropic Model of Continuum Damage Mechanics 189

9 MODELS OF NONLINEAR MATERIALS 192

9.1 Viscoelasticity 192

9.1.1 Spring–Damper Model 192

9.1.2 A General Three-dimensional Viscoelasticity Model 196

9.2 Polymer and Engineering Plastics 197

9.2.1 Fundamental Mechanical Properties of Polymer Materials 197

9.2.2 A Temperature, Strain Rate, and Pressure Dependent Constitutive Relation 198

9.2.3 A Nonlinear Viscoelastic Model of Polymer Materials 199

9.3 Rubber 200

9.3.1 Mooney–Rivlin Model of Rubber Material 200

9.3.2 Blatz–Ko Model 202

9.3.3 Ogden Model 203

9.4 Foam 203

9.4.1 A Cap Model Combining Volumetric Plasticity and Pressure Dependent Deviatoric Plasticity 205

9.4.2 A Model Consisting of Polymer Skeleton and Air 205

9.4.3 A Phenomenological Uniaxial Model 207

9.4.4 Hysteresis Behavior 208

9.4.5 Dynamic Behavior 209

9.5 Honeycomb 209

9.5.1 Structure of Hexagonal Honeycomb 210

9.5.2 Critical Buckling Load 210

9.5.3 A Phenomenological Material Model of Honeycomb 211

9.5.4 Behavior of Honeycomb under Complex Loading Conditions 213

9.6 Laminated Glazing 214

9.6.1 Application of J-integral 214

9.6.2 Application of Anisotropic Damage Model 215

9.6.3 A Simplified Model with Shell Element for the Laminated Glass 216

PART IV CONTACT AND CONSTRAINT CONDITIONS 219

10 THREE-DIMENSIONAL SURFACE CONTACT 221

10.1 Examples of Contact Problems 221

10.1.1 Uniformly Loaded String with a Flat Rigid Obstacle 222

10.1.2 Hertz Contact Problem 225

10.1.3 Elastic Impact of Two Balls 226

10.1.4 Impact of an Elastic Rod on the Flat Rigid Obstacle 228

10.1.5 Impact of a Vibrating String to the Flat Rigid Obstacle 231

10.2 Description of Contact Conditions 233

10.2.1 Contact with a Smooth Rigid Obstacle—Signorini’s Problem 233

10.2.2 Contact between Two Smooth Deformable Bodies 237

10.2.3 Coulomb’s Law of Friction 240

10.2.4 Conditions for “In Contact” 242

10.2.5 Domain Contact 242

10.3 Variational Principle for the Dynamic Contact Problem 243

10.3.1 Variational Formulation for Frictionless Dynamic Contact Problem 243

10.3.2 Variational Formulation for Frictional Dynamic Contact Problem 247

10.3.3 Variational Formulation for Domain Contact 250

10.4 Penalty Method and the Regularization of Variational Inequality 252

10.4.1 Concept of Penalty Method 252

10.4.2 Penalty Method for Nonlinear Dynamic Contact Problem 256

10.4.3 Explicit Finite Element Procedure with Penalty Method for Dynamic Contact 258

11 NUMERICAL PROCEDURES FOR THREE-DIMENSIONAL SURFACE CONTACT 261

11.1 A Contact Algorithm with Slave Node Searching Master Segment 262

11.1.1 Global Search 263

11.1.2 Bucket Sorting Method 264

11.1.3 Local Search 266

11.1.4 Penalty Contact Force 268

11.2 A Contact Algorithm with Master Segment Searching Slave Node 272

11.2.1 Global Search with Bucket Sorting Based on Segment’s Capture Box 272

11.2.2 Local Search with the Projection of Slave Point 273

11.3 Method of Contact Territory and Defense Node 273

11.3.1 Global Search with Bucket Sorting Based on Segment’s Territory 274

11.3.2 Local Search in the Territory 274

11.3.3 Defense Node and Contact Force 275

11.4 Pinball Contact Algorithm 277

11.4.1 The Pinball Hierarchy 277

11.4.2 Penalty Contact Force 278

11.5 Edge (Line Segment) Contact 279

11.5.1 Search for Line Contact 279

11.5.2 Penalty Contact Force of Edge-to-Edge Contact 281

11.6 Evaluation of Contact Algorithm with Penalty Method 282

12 KINEMATIC CONSTRAINT CONDITIONS 289

12.1 Rigid Wall 289

12.1.1 A Stationary Flat Rigid Wall 290

12.1.2 A Moving Flat Rigid Wall 291

12.1.3 Rigid Wall with a Curved Surface 293

12.2 Rigid Body 296

12.3 Explicit Finite Element Procedure with Constraint Conditions 298

12.4 Application Examples with Constraint Conditions 300

REFERENCES 305

INDEX 325

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