INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS

Transcription

1 INTRODUCTION TO THE EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS SHEN R. WU and LEI GU WILEY A JOHN WILEY & SONS, INC., PUBLICATION

2 ! PREFACE xv PARTI FUNDAMENTALS 1 1 INTRODUCTION Era of Simulation and Computer Aided Engineering A World of Simulation Evolution of Explicit Finite Element Method., Computer Aided Engineering (CAE) :Opportunities and Challenges Preliminaries 6. ; 1, Notations. ' Constitutive Relations of Elasticity 8 2 FRAMEWORK OF EXPLICIT FINITE ELEMENT METHOD FOR NONLINEAR TRANSIENT DYNAMICS Transient Structural Dynamics Variational Principles for Transient Dynamics Hamilton's Principle Galerkin Method 15 vii

3 Vili 2.3 Finite Element Equations and the Explicit Procedures Discretization in Space by Finite Element System of Semidiscretization Discretization in Time by Finite Difference Procedure of the Explicit Finite Element Method " Main Features of the Explicit Finite Element Method Stability Condition and Time Step Size Diagonal Mass Matrix Corotational Stress Assessment of Explicit Finite Element Method About the Solution of the Elastodynamics A Priori Error Estimate of Explicit Finite Element Method for Elastodynamics About the Diagonal Mass Matrix 30 PART II ELEMENT TECHNOLOGY 37 3 FOUR-NODE SHELL ELEMENT (REISSNER-MINDLIN PLATE THEORY) Fundamentals of Plates and Shells Characteristics of Thin-walled Structures Resultant Equations Applications to Linear Elasticity Kirchhoff-Love Theory Reissner-Mindlin Plate Theory Linear Theory of R-M Plate * Helmholtz Decomposition for R-M Plate Load Scaling for Static Problem of R-M Plate Load Scaling and Mass Scaling for Dynamic Problem of R-M Plate Relation between R-M Theory and K-L Theory Interpolation for Four-node R-M Plate Element Variational Equations for R-M Plate Bilinear Interpolations Shear Locking Issues of R-M Plate Element Reduced Integration and Selective Reduced Integration Reduced Integration 56

4 ix Selective Reduced Integration Nonlinear Application of Selective Reduced Integration Hughes-Liu Element Perturbation Hourglass Control Belytschko-Tsay Element Concept of Hourglass Control 61, Four-node Belytschko-Tsay Shell Element Perturbation Hourglass Control Improvement of Belytschko-Tsay Shell Element About Convergence of Element using Reduced Integration Physical Hourglass Control Belytschko-Leviathan (QPH) Element Constant and Nonconstant Contributions Projection of Shear Strain Physical Hourglass Control by One-point Integration Drill Projection Improvement of B-L (QPH) Element 76 ' 3.7 Shear Projection Method Bathe-Dvorkin Element Projection of Transverse Shear Strain Convergence of B-D Element 78! 3.8 Assessment of Four-node R-M Plate Element Evaluations with Warped Mesh and Reduced Thickness About the Locking-free Low Order Four-node R-M Plate Element 85 4 THREE-NODE SHELL ELEMENT (REISSNER-MINDLIN PLATE THEORY) Fundamentals of a Three-node C Element * Transformation and Jacobian Numerical Quadrature for In-plane Integration Shear Locking with C Triangular Element Decomposition Method for C Triangular Element with One-point Integration A C Element with Decomposition of Deflection 92 ' A C 0 Element with Decomposition of Rotations Discrete Kirchhoff Triangular Element Assessment of Three-node R-M Plate Element Evaluations with Warped Mesh and Reduced Thickness About the Locking-free Low Order Three-node R-M Plate Element 105

5 X 5 EIGHT-NODE SOLID ELEMENT Trilinear Interpolation for the-eight-node Hexahedron Element Locking Issues of the Eight-node Solid Element One-point Reduced Integration and the Perturbed Hourglass Control Assumed Strain'Method and Selective/Reduced Integration Assumed Deviatoric Strain An Enhanced Assumed Strain Method Taylor Expansion of Assumed Strain about the Element Center Evaluation of Eight-node Solid Element TWO-NODE ELEMENT Truss and Rod Element 128 I 6.2 Timoshenko Beam Element 129,, 6.3 Spring Element One Degree of Freedom Spring Element Six Degrees of Freedom Spring Element 132 ; r Three-node Spring Element Spot Weld Element ; Description of Spot Weld Separation Failure Criterion Finite Element Representation of Spot Weld 137 PART III MATERIAL MODELS MATERIAL MODEL OF PLASTICITY Fundamentals of Plasticity Tensile Test Hardening Yield Surface Normality Condition Strain Rate Effect/Viscoplasticity Constitutive Equations Relations between Stress Increments and Strain Increments Constitutive Equations for Mises Criterion Application to Kinematic Hardening 158

6 Xi 7.3 Software Implementation Explicit Finite Element Procedure with Plasticity Normal (Radial) Return Scheme A Generalized Plane Stress Model Stress Resultant Approach Evaluation of Shell Elements with Plastic Deformation CONTINUUM MECHANICS MODEL OF DUCTILE DAMAGE Concept of Damage Mechanics Gurson's Model Damage Variables and Yield Function Constitutive Equation and Damage Growth Chow's Isotropic Model of Continuum Damage Mechanics Damage Effect Tensor Yield Function and Constitutive Equation Damage Growth Application to Plates and Shells Determination of Parameters Chow's Anisotropic Model of Continuum Damage Mechanics MODELS OF NONLINEAR MATERIALS Viscoelasticity Spring-Damper Model A General Three-dimensional Viscoelasticity Model Polymer and Engineering Plastics Fundamental Mechanical Properties of Polymer Materials A Temperature, Strain Rate, and Pressure Dependent Constitutive Relation i', '' A Nonlinear Viscoelastic Model of Polymer Materials Rubber Mooney-Rivlin Model of Rubber Material Blatz-Ko Model Ogden Model Foam A Cap Model Combining Volumetric Plasticity and Pressure Dependent Deviatoric Plasticity 205, A Model Consisting of Polymer Skeleton and Air A Phenomenological Uniaxial Model 207

7 Xii Hysteresis Behavior Dynamic Behavior 209 \ 9.5 Honeycomb Structure of Hexagonal Honeycomb Critical Buckling Load A Phenomenological Material Model of Honeycomb Behavior of Honeycomb under Complex Loading Conditions Laminated Glazing Application of J-integral Application of Anisotropic Damage Model A Simplified Model with Shell Element for the Laminated Glass 216 PART IV CONTACT AND CONSTRAINT CONDITIONS THREE-DIMENSIONAL SURFACE CONTACT Examples of Contact Problems Uniformly Loaded String, with a Flat Rigid Obstacle Hertz Contact Problem Elastic Impact of Two Balls Impact of an Elastic Rod on the Flat Rigid Obstacle Impact of a Vibrating String to the Flat Rigid Obstacle Description of Contact Conditions Contact with a Smooth Rigid Obstacle Signorini's Problem Contact between Two Smooth Deformable Bodies Coulomb's Law of Friction Conditions for "In Contact" Domain Contact Variational Principle for the Dynamic Contact Problem Variational Formulation for Frictionless Dynamic Contact Problem Variational Formulation for Frictional Dynamic Contact Problem Variational Formulation for Domain Contact 250

8 xiii 10.4 Penalty Method and the Regularization of Variational Inequality Concept of Penalty Method Penalty Method for Nonlinear Dynamic Contact Problem Explicit Finite Element Procedure with Penalty Method for Dynamic Contact NUMERICAL PROCEDURES FOR THREE-DIMENSIONAL SURFACE CONTACT A Contact Algorithm with Slave Node Searching Master Segment Global Search Bucket Sorting Method Local Search Penalty Contact Force A Contact Algorithm with Master Segment Searching Slave Node Global Search with Bucket Sorting Based on Segment's Capture Box Local Search with the Projection of Slave Point Method of Contact Territory and Defense Node Global Search with Bucket Sorting Based on Segment's Territory Local Search in the Territory Defense Node and Contact Force Pinball Contact Algorithm The Pinball Hierarchy Penalty Contact Force Edge (Line Segment) Contact Search for Line Contact Penalty Contact Force of Edge-to-Edge Contact Evaluation of Contact Algorithm with Penalty Method KINEMATIC CONSTRAINT CONDITIONS Rigid Wall A Stationary Flat Rigid Wall A Moving Hat Rigid Wall Rigid Wall with a Curved Surface 293

9 xiv Rigid Body 296 V' 12.3 Explicit Finite Element Procedure with Constraint Conditions 298 ' r 12.4 Application Examples with Constraint Conditions 300 REFERENCES 305 INDEX 325