Introduction to Impact Dynamics

Contents Preface xi Introduction to Impact Dynamics xiii Part 1 Stress Waves in Solids 1 1 Elastic Waves 3 1.1 Elastic Wave in a Uniform Circular Bar 3 1.1.1 The Propagation of a Compressive Elastic Wave 3 1.2 Types of Elastic Wave 6 1.2.1 Longitudinal Waves 6 1.2.2 Transverse Waves 7 1.2.3 Surface Wave (Rayleigh Wave) 7 1.2.4 Interfacial Waves 8 1.2.5 Waves in Layered Media (Love Waves) 8 1.2.6 Bending (Flexural) Waves 8 1.3 Reflection and Interaction of Waves 9 1.3.1 Mechanical Impedance 9 1.3.2 Waves When they Encounter a Boundary 10 1.3.3 Reflection and Transmission of 1D Longitudinal Waves 11 Questions 1 17 Problems 1 18 2 Elastic-Plastic Waves 19 2.1 One-Dimensional Elastic-Plastic Stress Wave in Bars 19 2.1.1 A Semi-Infinite Bar Made of Linear Strain-Hardening Material Subjected to a Step Load at its Free End 21 2.1.2 A Semi-Infinite Bar Made of Decreasingly Strain-Hardening Material Subjected to a Monotonically Increasing Load at its Free End 22 2.1.3 A Semi-Infinite Bar Made of Increasingly Strain-Hardening Material Subjected to a Monotonically Increasing Load at its Free End 23 2.1.4 Unloading Waves 25 2.1.5 Relationship Between Stress and Particle Velocity 26 2.1.6 Impact of a Finite-Length Uniform Bar Made of Elastic-Linear Strain-Hardening Material on a Rigid Flat Anvil 28 2.2 High-Speed Impact of a Bar of Finite Length on a Rigid Anvil (Mushrooming) 31 2.2.1 Taylor's Approach 31 2.2.2 Hawkyard's Energy Approach 36 Questions 2 38 Problems 2 38 Part 2 Dynamic Behavior of Materials under High Strain Rate 39 3 Rate-Dependent Behavior of Materials 41 3.1 Materials' Behavior under High Strain Rates 41 3.2 High-Strain-Rate Mechanical Properties of Materials 44 3.2.1 Strain Rate Effect of Materials under Compression 44 3.2.2 Strain Rate Effect of Materials under Tension 44 3.2.3 Strain Rate Effect of Materials under Shear 47 3.3 High-Strain-Rate Mechanical Testing 48 3.3.1 Intermediate-Strain-Rate Machines 48 3.3.2 Split Hopkinson Pressure Bar (SHPB) 53 3.3.3 Expanding-Ring Technique 61 3.4 Explosively Driven Devices 62 3.4.1 Line-Wave and Plane-Wave Generators 63 3.4.2 Flyer Plate Accelerating 65 3.4.3 Pressure-Shear Impact Configuration 66 3.5 Gun Systems 67 3.5.1 One-Stage Gas Gun 67 3.5.2 Two-Stage Gas Gun 68 3.5.3 Electric Rail Gun 69 Problems 3 69 4 Constitutive Equations at High Strain Rates 71 4.1 Introduction to Constitutive Relations 71 4.2 Empirical Constitutive Equations 72 4.3 Relationship between Dislocation Velocity and Applied Stress 76 4.3.1 Dislocation Dynamics 76 4.3.2 Thermally Activated Dislocation Motion 81 4.3.3 Dislocation Drag Mechanisms 85 4.3.4 Relativistic Effects on Dislocation Motion 85 4.3.5 Synopsis 86 4.4 Physically Based Constitutive Relations 87 4.5 Experimental Validation of Constitutive Equations 90 Problems 4 90 Part 3 Dynamic Response of Structures to Impact and Pulse Loading 91 5 Inertia Effects and Plastic Hinges 93 5.1 Relationship between Wave Propagation and Global Structural Response 93 5.2 Inertia Forces in Slender Bars 94 5.2.1 Notations and Sign Conventions for Slender Links and Beams 95 5.2.2 Slender Link in General Motion 96 5.2.3 A Summary of the Methodology 102 5.3 Plastic Hinges in a Rigid-Plastic Free-Free Beam under Pulse Loading 102 5.3.1 Dynamic Response of Rigid-Plastic Beams 102 5.3.2 A Free-Free Beam Subjected to a Concentrated Step Force 104 5.3.3 Remarks on a Free-Free Beam Subjected To A Step Force At Its Midpoint 108 5.4 A Free Ring Subjected to a Radial Load 109 5.4.1 Comparison between a Supported Ring and a Free Ring 112 Questions 5 112 Problems 5 112 6 Dynamic Response of Cantilevers 115 6.1 Response to Step Loading 115 6.2 Response to Pulse Loading 120 6.2.1 Rectangular Pulse 120 6.2.2 General Pulse 125 6.3 Impact on a Cantilever 126 6.4 General Features of Traveling Hinges 133 Problems 6 136 7 Effects of Tensile and Shear Forces 139 7.1 Simply Supported Beams with no Axial Constraint at Supports 139 7.1.1 Phase I 139 7.1.2 Phase II 142 7.2 Simply Supported Beams with Axial Constraint at Supports 144 7.2.1 Bending Moment and Tensile Force in a Rigid-Plastic Beam 144 7.2.2 Beam with Axial Constraint at Support 146 7.2.3 Remarks 151 7.3 Membrane Factor Method in Analyzing the Axial Force Effect 151 7.3.1 Plastic Energy Dissipation and the Membrane Factor 151 7.3.2 Solution using the Membrane Factor Method 153 7.4 Effect of Shear Deformation 155 7.4.1 Bending-Only Theory 156 7.4.2 Bending-Shear Theory 158 7.5 Failure Modes and Criteria of Beams under Intense Dynamic Loadings 161 7.5.1 Three Basic Failure Modes Observed in Experiments 161 7.5.2 The Elementary Failure Criteria 163 7.5.3 Energy Density Criterion 165 7.5.4 A Further Study of Plastic Shear Failures 166 Questions 7 168 Problems 7 168 8 Mode Technique, Bound Theorems, and Applicability of the Rigid-Perfectly Plastic Model 169 8.1 Dynamic Modes of Deformation 169 8.2 Properties of Modal Solutions 170 8.3 Initial Velocity of the Modal Solutions 172 8.4 Mode Technique Applications 174 8.4.1 Modal Solution of the Parkes Problem 174 8.4.2 Modal Solution for a Partially Loaded Clamped Beam 176 8.4.3 Remarks on the Modal Technique 179 8.5 Bound Theorems for RPP Structures 180 8.5.1 Upper Bound of Final Displacement 180 8.5.2 Lower Bound of Final Displacement 181 8.6 Applicability of an RPP Model 183 Problems 8 186 9 Response of Rigid-Plastic Plates 187 9.1 Static Load-Carrying Capacity of Rigid-Plastic Plates 187 9.1.1 Load Capacity of Square Plates 188 9.1.2 Load Capacity of Rectangular Plates 190 9.1.3 Load-Carrying Capacity of Regular Polygonal Plates 192 9.1.4 Load-Carrying Capacity of Annular Plate Clamped at its Outer Boundary 194 9.1.5 Summary 196 9.2 Dynamic Deformation of Pulse-Loaded Plates 196 9.2.1 The Pulse Approximation Method 196 9.2.2 Square Plate Loaded by Rectangular Pulse 197 9.2.3 Annular Circular Plate Loaded by Rectangular Pulse Applied on its Inner Boundary 201 9.2.4 Summary 204 9.3 Effect of Large Deflection 204 9.3.1 Static Load-Carrying Capacity of Circular Plates In Large Deflection 205 9.3.2 Dynamic Response of Circular Plates with Large Deflection 209 Problems 9 210 10 Case Studies 213 10.1 Theoretical Analysis of Tensor Skin 213 10.1.1 Introduction to Tensor Skin 213 10.1.2 Static Response to Uniform Pressure Loading 213 10.1.3 Dynamic Response of Tensor Skin 217 10.1.4 Pulse Shape 218 10.2 Static and Dynamic Behavior of Cellular Structures 219 10.2.1 Static Response of Hexagonal Honeycomb 221 10.2.2 Static Response of Generalized Honeycombs 223 10.2.3 Dynamic Response of Honeycomb Structures 228 10.3 Dynamic Response of a Clamped Circular Sandwich Plate Subject to Shock Loading 233 10.3.1 An Analytical Model for the Shock Resistance of Clamped Sandwich Plates 234 10.3.2 Comparison of Finite Element and Analytical Predictions 238 10.3.3 Optimal Design of Sandwich Plates 239 10.4 Collision and Rebound of Circular Rings and Thin-Walled Spheres on Rigid Target 241 10.4.1 Collision and Rebound of Circular Rings 241 10.4.2 Collision and Rebound of Thin-Walled Spheres 249 10.4.3 Concluding Remarks 257 References 259 Index