INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS -- CONTENTS -- PREFACE -- PART I FUNDAMENTALS -- 1 INTRODUCTION -- 1.1 Era of Simulation and Computer Aided Engineering -- 1.1.1 A World of Simulation -- 1.1.2 Evolution of Explicit Finite Element Method -- 1.1.3 Computer Aided Engineering (CAE)-Opportunities and Challenges -- 1.2 Preliminaries -- 1.2.1 Notations -- 1.2.2 Constitutive Relations of Elasticity -- 2 FRAMEWORK OF EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS -- 2.1 Transient Structural Dynamics -- 2.2 Variational Principles for Transient Dynamics -- 2.2.1 Hamilton's Principle -- 2.2.2 Galerkin Method -- 2.3 Finite Element Equations and the Explicit Procedures -- 2.3.1 Discretization in Space by Finite Element -- 2.3.2 System of Semidiscretization -- 2.3.3 Discretization in Time by Finite Difference -- 2.3.4 Procedure of the Explicit Finite Element Method -- 2.4 Main Features of the Explicit Finite Element Method -- 2.4.1 Stability Condition and Time Step Size -- 2.4.2 Diagonal Mass Matrix -- 2.4.3 Corotational Stress -- 2.5 Assessment of Explicit Finite Element Method -- 2.5.1 About the Solution of the Elastodynamics -- 2.5.2 A Priori Error Estimate of Explicit Finite Element Method for Elastodynamics -- 2.5.3 About the Diagonal Mass Matrix -- PART II ELEMENT TECHNOLOGY -- 3 FOUR-NODE SHELL ELEMENT (REISSNER-MINDLIN PLATE THEORY) -- 3.1 Fundamentals of Plates and Shells -- 3.1.1 Characteristics of Thin-walled Structures -- 3.1.2 Resultant Equations -- 3.1.3 Applications to Linear Elasticity -- 3.1.4 Kirchhoff-Love Theory -- 3.1.5 Reissner-Mindlin Plate Theory -- 3.2 Linear Theory of R-M Plate -- 3.2.1 Helmholtz Decomposition for R-M Plate -- 3.2.2 Load Scaling for Static Problem of R-M Plate -- 3.2.3 Load Scaling and Mass Scaling for Dynamic Problem of R-M Plate.

3.2.4 Relation between R-M Theory and K-L Theory -- 3.3 Interpolation for Four-node R-M Plate Element -- 3.3.1 Variational Equations for R-M Plate -- 3.3.2 Bilinear Interpolations -- 3.3.3 Shear Locking Issues of R-M Plate Element -- 3.4 Reduced Integration and Selective Reduced Integration -- 3.4.1 Reduced Integration -- 3.4.2 Selective Reduced Integration -- 3.4.3 Nonlinear Application of Selective Reduced Integration-Hughes-Liu Element -- 3.5 Perturbation Hourglass Control-Belytschko-Tsay Element -- 3.5.1 Concept of Hourglass Control -- 3.5.2 Four-node Belytschko-Tsay Shell Element-Perturbation Hourglass Control -- 3.5.3 Improvement of Belytschko-Tsay Shell Element -- 3.5.4 About Convergence of Element using Reduced Integration -- 3.6 Physical Hourglass Control-Belytschko-Leviathan (QPH) Element -- 3.6.1 Constant and Nonconstant Contributions -- 3.6.2 Projection of Shear Strain -- 3.6.3 Physical Hourglass Control by One-point Integration -- 3.6.4 Drill Projection -- 3.6.5 Improvement of B-L (QPH) Element -- 3.7 Shear Projection Method-Bathe-Dvorkin Element -- 3.7.1 Projection of Transverse Shear Strain -- 3.7.2 Convergence of B-D Element -- 3.8 Assessment of Four-node R-M Plate Element -- 3.8.1 Evaluations with Warped Mesh and Reduced Thickness -- 3.8.2 About the Locking-free Low Order Four-node R-M Plate Element -- 4 THREE-NODE SHELL ELEMENT (REISSNER-MINDLIN PLATE THEORY) -- 4.1 Fundamentals of a Three-node C0 Element -- 4.1.1 Transformation and Jacobian -- 4.1.2 Numerical Quadrature for In-plane Integration -- 4.1.3 Shear Locking with C0 Triangular Element -- 4.2 Decomposition Method for C0 Triangular Element with One-point Integration -- 4.2.1 A C0 Element with Decomposition of Deflection -- 4.2.2 A C0 Element with Decomposition of Rotations -- 4.3 Discrete Kirchhoff Triangular Element -- 4.4 Assessment of Three-node R-M Plate Element.

4.4.1 Evaluations with Warped Mesh and Reduced Thickness -- 4.4.2 About the Locking-free Low Order Three-node R-M Plate Element -- 5 EIGHT-NODE SOLID ELEMENT -- 5.1 Trilinear Interpolation for the Eight-node Hexahedron Element -- 5.2 Locking Issues of the Eight-node Solid Element -- 5.3 One-point Reduced Integration and the Perturbed Hourglass Control -- 5.4 Assumed Strain Method and Selective/Reduced Integration -- 5.5 Assumed Deviatoric Strain -- 5.6 An Enhanced Assumed Strain Method -- 5.7 Taylor Expansion of Assumed Strain about the Element Center -- 5.8 Evaluation of Eight-node Solid Element -- 6 TWO-NODE ELEMENT -- 6.1 Truss and Rod Element -- 6.2 Timoshenko Beam Element -- 6.3 Spring Element -- 6.3.1 One Degree of Freedom Spring Element -- 6.3.2 Six Degrees of Freedom Spring Element -- 6.3.3 Three-node Spring Element -- 6.4 Spot Weld Element -- 6.4.1 Description of Spot Weld Separation -- 6.4.2 Failure Criterion -- 6.4.3 Finite Element Representation of Spot Weld -- PART III MATERIAL MODELS -- 7 MATERIAL MODEL OF PLASTICITY -- 7.1 Fundamentals of Plasticity -- 7.1.1 Tensile Test -- 7.1.2 Hardening -- 7.1.3 Yield Surface -- 7.1.4 Normality Condition -- 7.1.5 Strain Rate Effect/Viscoplasticity -- 7.2 Constitutive Equations -- 7.2.1 Relations between Stress Increments and Strain Increments -- 7.2.2 Constitutive Equations for Mises Criterion -- 7.2.3 Application to Kinematic Hardening -- 7.3 Software Implementation -- 7.3.1 Explicit Finite Element Procedure with Plasticity -- 7.3.2 Normal (Radial) Return Scheme -- 7.3.3 A Generalized Plane Stress Model -- 7.3.4 Stress Resultant Approach -- 7.4 Evaluation of Shell Elements with Plastic Deformation -- 8 CONTINUUM MECHANICS MODEL OF DUCTILE DAMAGE -- 8.1 Concept of Damage Mechanics -- 8.2 Gurson's Model -- 8.2.1 Damage Variables and Yield Function -- 8.2.2 Constitutive Equation and Damage Growth.

8.3 Chow's Isotropic Model of Continuum Damage Mechanics -- 8.3.1 Damage Effect Tensor -- 8.3.2 Yield Function and Constitutive Equation -- 8.3.3 Damage Growth -- 8.3.4 Application to Plates and Shells -- 8.3.5 Determination of Parameters -- 8.4 Chow's Anisotropic Model of Continuum Damage Mechanics -- 9 MODELS OF NONLINEAR MATERIALS -- 9.1 Viscoelasticity -- 9.1.1 Spring-Damper Model -- 9.1.2 A General Three-dimensional Viscoelasticity Model -- 9.2 Polymer and Engineering Plastics -- 9.2.1 Fundamental Mechanical Properties of Polymer Materials -- 9.2.2 A Temperature, Strain Rate, and Pressure Dependent Constitutive Relation -- 9.2.3 A Nonlinear Viscoelastic Model of Polymer Materials -- 9.3 Rubber -- 9.3.1 Mooney-Rivlin Model of Rubber Material -- 9.3.2 Blatz-Ko Model -- 9.3.3 Ogden Model -- 9.4 Foam -- 9.4.1 A Cap Model Combining Volumetric Plasticity and Pressure Dependent Deviatoric Plasticity -- 9.4.2 A Model Consisting of Polymer Skeleton and Air -- 9.4.3 A Phenomenological Uniaxial Model -- 9.4.4 Hysteresis Behavior -- 9.4.5 Dynamic Behavior -- 9.5 Honeycomb -- 9.5.1 Structure of Hexagonal Honeycomb -- 9.5.2 Critical Buckling Load -- 9.5.3 A Phenomenological Material Model of Honeycomb -- 9.5.4 Behavior of Honeycomb under Complex Loading Conditions -- 9.6 Laminated Glazing -- 9.6.1 Application of J-integral -- 9.6.2 Application of Anisotropic Damage Model -- 9.6.3 A Simplified Model with Shell Element for the Laminated Glass -- PART IV CONTACT AND CONSTRAINT CONDITIONS -- 10 THREE-DIMENSIONAL SURFACE CONTACT -- 10.1 Examples of Contact Problems -- 10.1.1 Uniformly Loaded String with a Flat Rigid Obstacle -- 10.1.2 Hertz Contact Problem -- 10.1.3 Elastic Impact of Two Balls -- 10.1.4 Impact of an Elastic Rod on the Flat Rigid Obstacle -- 10.1.5 Impact of a Vibrating String to the Flat Rigid Obstacle -- 10.2 Description of Contact Conditions.

10.2.1 Contact with a Smooth Rigid Obstacle-Signorini's Problem -- 10.2.2 Contact between Two Smooth Deformable Bodies -- 10.2.3 Coulomb's Law of Friction -- 10.2.4 Conditions for "In Contact" -- 10.2.5 Domain Contact -- 10.3 Variational Principle for the Dynamic Contact Problem -- 10.3.1 Variational Formulation for Frictionless Dynamic Contact Problem -- 10.3.2 Variational Formulation for Frictional Dynamic Contact Problem -- 10.3.3 Variational Formulation for Domain Contact -- 10.4 Penalty Method and the Regularization of Variational Inequality -- 10.4.1 Concept of Penalty Method -- 10.4.2 Penalty Method for Nonlinear Dynamic Contact Problem -- 10.4.3 Explicit Finite Element Procedure with Penalty Method for Dynamic Contact -- 11 NUMERICAL PROCEDURES FOR THREE-DIMENSIONAL SURFACE CONTACT -- 11.1 A Contact Algorithm with Slave Node Searching Master Segment -- 11.1.1 Global Search -- 11.1.2 Bucket Sorting Method -- 11.1.3 Local Search -- 11.1.4 Penalty Contact Force -- 11.2 A Contact Algorithm with Master Segment Searching Slave Node -- 11.2.1 Global Search with Bucket Sorting Based on Segment's Capture Box -- 11.2.2 Local Search with the Projection of Slave Point -- 11.3 Method of Contact Territory and Defense Node -- 11.3.1 Global Search with Bucket Sorting Based on Segment's Territory -- 11.3.2 Local Search in the Territory -- 11.3.3 Defense Node and Contact Force -- 11.4 Pinball Contact Algorithm -- 11.4.1 The Pinball Hierarchy -- 11.4.2 Penalty Contact Force -- 11.5 Edge (Line Segment) Contact -- 11.5.1 Search for Line Contact -- 11.5.2 Penalty Contact Force of Edge-to-Edge Contact -- 11.6 Evaluation of Contact Algorithm with Penalty Method -- 12 KINEMATIC CONSTRAINT CONDITIONS -- 12.1 Rigid Wall -- 12.1.1 A Stationary Flat Rigid Wall -- 12.1.2 A Moving Flat Rigid Wall -- 12.1.3 Rigid Wall with a Curved Surface -- 12.2 Rigid Body.

12.3 Explicit Finite Element Procedure with Constraint Conditions.