Introduction to Impact Dynamics

T.X. Yu, PhD, is Professor Emeritus at the Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Hong Kong. XinMing Qiu, PhD, is Professor at the School of Aerospace Engineering, Tsinghua University, Beijing, China.

• Preface xiIntroduction to Impact Dynamics xiiiPart 1 Stress Waves in Solids 11 Elastic Waves 31.1 Elastic Wave in a Uniform Circular Bar 31.1.1 The Propagation of a Compressive Elastic Wave 31.2 Types of Elastic Wave 61.2.1 Longitudinal Waves 61.2.2 Transverse Waves 71.2.3 Surface Wave (Rayleigh Wave) 71.2.4 Interfacial Waves 81.2.5 Waves in Layered Media (Love Waves) 81.2.6 Bending (Flexural) Waves 81.3 Reflection and Interaction of Waves 91.3.1 Mechanical Impedance 91.3.2 Waves When they Encounter a Boundary 101.3.3 Reflection and Transmission of 1D Longitudinal Waves 11Questions 1 17Problems 1 182 Elastic-Plastic Waves 192.1 One-Dimensional Elastic-Plastic Stress Wave in Bars 192.1.1 A Semi-Infinite Bar Made of Linear Strain-Hardening Material Subjected to a Step Load at its Free End 212.1.2 A Semi-Infinite Bar Made of Decreasingly Strain-Hardening Material Subjected to a Monotonically Increasing Load at its Free End 222.1.3 A Semi-Infinite Bar Made of Increasingly Strain-Hardening Material Subjected to a Monotonically Increasing Load at its Free End 232.1.4 Unloading Waves 252.1.5 Relationship Between Stress and Particle Velocity 262.1.6 Impact of a Finite-Length Uniform Bar Made of Elastic-Linear Strain-Hardening Material on a Rigid Flat Anvil 282.2 High-Speed Impact of a Bar of Finite Length on a Rigid Anvil (Mushrooming) 312.2.1 Taylor’s Approach 312.2.2 Hawkyard’s Energy Approach 36Questions 2 38Problems 2 38Part 2 Dynamic Behavior of Materials under High Strain Rate 393 Rate-Dependent Behavior of Materials 413.1 Materials’ Behavior under High Strain Rates 413.2 High-Strain-Rate Mechanical Properties of Materials 443.2.1 Strain Rate Effect of Materials under Compression 443.2.2 Strain Rate Effect of Materials under Tension 443.2.3 Strain Rate Effect of Materials under Shear 473.3 High-Strain-Rate Mechanical Testing 483.3.1 Intermediate-Strain-Rate Machines 483.3.2 Split Hopkinson Pressure Bar (SHPB) 533.3.3 Expanding-Ring Technique 613.4 Explosively Driven Devices 623.4.1 Line-Wave and Plane-Wave Generators 633.4.2 Flyer Plate Accelerating 653.4.3 Pressure-Shear Impact Configuration 663.5 Gun Systems 673.5.1 One-Stage Gas Gun 673.5.2 Two-Stage Gas Gun 683.5.3 Electric Rail Gun 69Problems 3 694 Constitutive Equations at High Strain Rates 714.1 Introduction to Constitutive Relations 714.2 Empirical Constitutive Equations 724.3 Relationship between Dislocation Velocity and Applied Stress 764.3.1 Dislocation Dynamics 764.3.2 Thermally Activated Dislocation Motion 814.3.3 Dislocation Drag Mechanisms 854.3.4 Relativistic Effects on Dislocation Motion 854.3.5 Synopsis 864.4 Physically Based Constitutive Relations 874.5 Experimental Validation of Constitutive Equations 90Problems 4 90Part 3 Dynamic Response of Structures to Impact and Pulse Loading 915 Inertia Effects and Plastic Hinges 935.1 Relationship between Wave Propagation and Global Structural Response 935.2 Inertia Forces in Slender Bars 945.2.1 Notations and Sign Conventions for Slender Links and Beams 955.2.2 Slender Link in General Motion 965.2.3 A Summary of the Methodology 1025.3 Plastic Hinges in a Rigid-Plastic Free–Free Beam under Pulse Loading 1025.3.1 Dynamic Response of Rigid-Plastic Beams 1025.3.2 A Free–Free Beam Subjected to a Concentrated Step Force 1045.3.3 Remarks on a Free–Free Beam Subjected To A Step Force At Its Midpoint 1085.4 A Free Ring Subjected to a Radial Load 1095.4.1 Comparison between a Supported Ring and a Free Ring 112Questions 5 112Problems 5 1126 Dynamic Response of Cantilevers 1156.1 Response to Step Loading 1156.2 Response to Pulse Loading 1206.2.1 Rectangular Pulse 1206.2.2 General Pulse 1256.3 Impact on a Cantilever 1266.4 General Features of Traveling Hinges 133Problems 6 1367 Effects of Tensile and Shear Forces 1397.1 Simply Supported Beams with no Axial Constraint at Supports 1397.1.1 Phase I 1397.1.2 Phase II 1427.2 Simply Supported Beams with Axial Constraint at Supports 1447.2.1 Bending Moment and Tensile Force in a Rigid-Plastic Beam 1447.2.2 Beam with Axial Constraint at Support 1467.2.3 Remarks 1517.3 Membrane Factor Method in Analyzing the Axial Force Effect 1517.3.1 Plastic Energy Dissipation and the Membrane Factor 1517.3.2 Solution using the Membrane Factor Method 1537.4 Effect of Shear Deformation 1557.4.1 Bending-Only Theory 1567.4.2 Bending-Shear Theory 1587.5 Failure Modes and Criteria of Beams under Intense Dynamic Loadings 1617.5.1 Three Basic Failure Modes Observed in Experiments 1617.5.2 The Elementary Failure Criteria 1637.5.3 Energy Density Criterion 1657.5.4 A Further Study of Plastic Shear Failures 166Questions 7 168Problems 7 1688 Mode Technique, Bound Theorems, and Applicability of the Rigid-Perfectly Plastic Model 1698.1 Dynamic Modes of Deformation 1698.2 Properties of Modal Solutions 1708.3 Initial Velocity of the Modal Solutions 1728.4 Mode Technique Applications 1748.4.1 Modal Solution of the Parkes Problem 1748.4.2 Modal Solution for a Partially Loaded Clamped Beam 1768.4.3 Remarks on the Modal Technique 1798.5 Bound Theorems for RPP Structures 1808.5.1 Upper Bound of Final Displacement 1808.5.2 Lower Bound of Final Displacement 1818.6 Applicability of an RPP Model 183Problems 8 1869 Response of Rigid-Plastic Plates 1879.1 Static Load-Carrying Capacity of Rigid-Plastic Plates 1879.1.1 Load Capacity of Square Plates 1889.1.2 Load Capacity of Rectangular Plates 1909.1.3 Load-Carrying Capacity of Regular Polygonal Plates 1929.1.4 Load-Carrying Capacity of Annular Plate Clamped at its Outer Boundary 1949.1.5 Summary 1969.2 Dynamic Deformation of Pulse-Loaded Plates 1969.2.1 The Pulse Approximation Method 1969.2.2 Square Plate Loaded by Rectangular Pulse 1979.2.3 Annular Circular Plate Loaded by Rectangular Pulse Applied on its Inner Boundary 2019.2.4 Summary 2049.3 Effect of Large Deflection 2049.3.1 Static Load-Carrying Capacity of Circular Plates In Large Deflection 2059.3.2 Dynamic Response of Circular Plates with Large Deflection 209Problems 9 21010 Case Studies 21310.1 Theoretical Analysis of Tensor Skin 21310.1.1 Introduction to Tensor Skin 21310.1.2 Static Response to Uniform Pressure Loading 21310.1.3 Dynamic Response of Tensor Skin 21710.1.4 Pulse Shape 21810.2 Static and Dynamic Behavior of Cellular Structures 21910.2.1 Static Response of Hexagonal Honeycomb 22110.2.2 Static Response of Generalized Honeycombs 22310.2.3 Dynamic Response of Honeycomb Structures 22810.3 Dynamic Response of a Clamped Circular Sandwich Plate Subject to Shock Loading 23310.3.1 An Analytical Model for the Shock Resistance of Clamped Sandwich Plates 23410.3.2 Comparison of Finite Element and Analytical Predictions 23810.3.3 Optimal Design of Sandwich Plates 23910.4 Collision and Rebound of Circular Rings and Thin-Walled Spheres on Rigid Target 24110.4.1 Collision and Rebound of Circular Rings 24110.4.2 Collision and Rebound of Thin-Walled Spheres 24910.4.3 Concluding Remarks 257References 259Index 265

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