Contents
Preface vii
Introduction ix
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
Contents
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 Examples of Inertia Force in Beams 97
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
Contents
Questions 7 168
Problems 7 168
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
8.5.1 Upper Bound of Final Displacement
8.5.2 Lower Bound of Final Displacement
8.6 Applicability of an RPP Model 183 Problems 8 186
180
180
181
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
v
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 265
內容試閱：
Preface
Various impact events occur every day and everywhere in the physical world， in engineer-ing and in people’s daily lives. Our universe and planet were formed as a result of a series of impact and explosion events. With the rapid development of land vehicles， ships， and aircraft， traffic accidents have become a serious concern of modern society. Landing of spacecraft， safety in nuclear plants and offshore structures， as well as protection of human bodies during accidents and sports， all require better knowledge regarding the dynamic behavior of structures and materials.
Obviously， impact dynamics is a big subject， which looks at the dynamic behavior of all kinds of materials (e.g. metals， concrete， polymers， and composites)， and the structures under study range from small objects (e.g.， a mobile phone being dropped on the ground) to complex systems (e.g.， a jumbo jet or the World Trade Center before 9/11). The impact velocity may vary from a few meters per second (as seen in ball games) to several kilometers per second (as seen in military applications). Driven by the needs of science and engineering， the dynamic response， impact protection， crashworthiness， and energy absorption capacity of materials and structures have attracted more and more attention from researchers and engineers. Numerous research papers and monographs have appeared in the literature， and it is not possible for anyone to condense the huge amount knowledge out there into a single book.
This book is mainly meant as a textbook for graduate students (and probably also for senior undergraduates)， aiming to provide fundamental knowledge of impact dynamics. Instead of covering all aspects of impact dynamics， the contents are organized so as to consider only its three major aspects: (i) wave propagation in solids; (ii) materials’ behav-ior under high-speed loading; and (iii) the dynamic response of structures to impact. The emphasis here is on theoretical models and analytical methods， which will help readers to understand the fundamental issues raised by various practical situations. Numerical methods and software are not the main topic of this textbook. Readers who are interested in numerical modeling related to impact dynamics will have to consult other sources for the relevant knowledge.
The audience for this textbook may also include those engineers working in the auto-motive， aerospace， mechanical， nuclear， marine， offshore， and defense sectors. This text-book will provide them with fundamental guidance on the relevant concepts， models， and methodology， so as to help them face the challenges of selecting materials and designing/analyzing structures under intensive dynamic loading.
The contents of this textbook have been used in graduate courses at a number of uni-versities. The first author (T.X. Yu) taught Impact Dynamics as a credit graduate course
Preface
at Peking University， UMIST (now University of Manchester)， and the Hong Kong Uni-versity of Science and Technology. In recent years， he has also used part of the contents to deliver a short course for graduate students in many universities， including Tsinghua University， Zhejiang University， Wuhan University， Xi’an Jiaotong University， Taiyuan University of Technology， Hunan University， and Dalian University of Technology. The second author (X.M. Qiu) has also taught Impact Dynamics as a credit graduate course at Tsinghua University over recent years.
Using the content developed for these graduate courses， we authored a textbook in Chinese， entitled Impact Dynamics and published by Tsinghua University Press in 2011. Although the current English version is mainly based on this Chinese version， we have made many changes. For instance， some contents in Chapters 3 and 4 have been rewritten， and Chapter 10 containing case studies is entirely new for this English version.
As a textbook， we have adopted much content from relevant monographs， such as Meyer (1994) (for Chapters 3 and 4) and Stronge and Yu (1993) (for Chapter 6)， including a number of figures. This is because that content clearly elaborated the respective con-cepts and methods with carefully selected examples and illustrations， which are partic-ularly suitable for a graduate course. Those monographs have been cited accordingly in the relevant places， and we would like to express our sincere gratitude to the original authors.
We would also like to thank Ms. Lixia Tong of Tsinghua University Press， who gave us a great deal of help in preparing this book.
T.X. Yu and XinMing Qiu July 2017
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