Applied Impact Mechanics

This book is intended to help the reader understand impact phenomena as a focused application of diverse topics such as rigid body dynamics, structural dynamics, contact and continuum mechanics, shock and vibration, wave propagation and material modelling. It emphasizes the need for a proper assessment of sophisticated experimental/computational tools promoted widely in contemporary design. A unique feature of the book is its presentation of several examples and exercises to aid further understanding of the physics and mathematics of impact process from first principles, in a way that is simple to follow.

Table of Contents:
Preface v List of Figures xv List of Tables xix List of Symbols xxi Chapter 1: Introduction 1 18 1.1 General Introduction to Engineering Mechanics 2 1.2 General Introduction to Fracture Mechanics 3 1.3 Impact Mechanics Appreciating Impact Problems in Engineering 5 1.4 Historical Background 8 1.5 Percussion, Concussion, Collision and Explosion 10 1.6 Summary 11 Bibliography 12 Chapter 2: Rigid Body Impact Mechanics 19 34 2.1 Introduction 19 2.2 Impulse Momentum Equations 21 2.3 Coefficient of Restitution Classical Definitions 21 2.3.1 Kinematic Coefficient of Restitution 22 2.3.2 Measurement of Coefficient of Restitution 22 2.3.3 Relative Assessment of Various Impacts in Sports 23 2.4 Coefficient of Restitution Alternate Definition 24 2.4.1 Kinetic Coefficient of Restitution 24 2.4.1.1 Case Study: Rebound of Colliding Vehicles 25 2.4.2 Energy Coefficient of Restitution 27 2.4.2.1 Application in Vehicle Collisions 28 2.5 Oblique Impact Role of Friction 29 2.6 Limitations of Rigid Body Impact Mechanics 31 2.7 Summary 31 Exercise Problems 32 Bibliography 34 Chapter 3: One-Dimensional Impact Mechanics of Deformable Bodies 35 54 3.1 Introduction 35 3.2 Single Degree of Freedom Idealization of Impact Process 36 3.2.1 Governing Equations of Single Degree of Freedom (SDOF) System 37 3.2.2 Forced Vibrations due to Exponentially Decaying Loads 38 3.3 1-D Wave Propagation in Solids Induced by Impact 41 3.3.1 Longitudinal Waves in Thin Rods 42 3.3.1.1 The Governing Equation for Waves in Long Rods 42 3.3.1.2 Free Vibrations in a Finite Rod 46 3.3.2 Flexural Waves in Thin Rods 47 3.3.2.1 The Governing Equation for Flexural Waves in Rods 47 3.3.2.2 Free Vibrations of Finite Beams 48 3.3.3 The D Alembert s Solution for Wave Equation 50 3.4 Summary 51 Exercise Problems 52 Bibliography 54 Chapter 4: Multi-Dimensional Impact Mechanics of Deformable Bodies 55 78 4.1 Introduction 55 4.2 Analysis of Stress 56 4.2.1 Stress Components on an Arbitrary Plane 56 4.2.2 Principal Stresses and Stress Invariants 57 4.2.3 Mohr s Circles 58 4.2.4 Octahedral Stresses 58 4.2.5 Decomposition into Hydrostatic and Pure Shear States 59 4.2.6 Equations of Motion of a Body in Cartesian Coordinates 60 4.2.7 Equations of Motion of a Body in Cylindrical Coordinates 61 4.2.8 Equations of Motion of a Body in Spherical Coordinates 62 4.3 Analysis of Strain 63 4.3.1 Deformation in the Neighborhood of a Point 63 4.3.2 Compatibility Equations 64 4.3.3 Strain Deviator 65 4.4 Linearised Stress-Strain Relations 65 4.4.1 Stress-Strain Relations for Isotropic Materials 66 4.5 Waves in Infinite Medium 67 4.5.1 Longitudinal Waves (Primary/Dilatational/Irrotational Waves) 67 4.5.1.1 Longitudinal Waves 68 4.5.1.2 The Governing Equations for Longitudinal Waves 68 4.5.2 Transverse Waves (Secondary/Shear/Distortional/Rotational Wave) 69 4.5.2.1 Transverse Waves 69 4.5.2.2 The Governing Equations for Transverse Waves 70 4.6 Waves in Semi-Infinite Media 70 4.6.1 Surface Waves 71 4.6.2 Symmetric Rayleigh-Lamb Spectrum in Elastic Layer 74 4.7 Summary 76 Exercise Problems 76 Bibliography 78 Chapter 5: Experimental Impact Mechanics 79 131 5.1 Introduction 80 5.2 Quasi-Static Material Tests 81 5.3 Pendulum Impact Tests 87 5.4 About High Strain Rate Testing of Materials 90 5.5 Split Hopkinson s Pressure Bar Test 91 5.5.1 Historical Background and Significance 91 5.5.2 Improvements in SHPB Test Apparatus 92 5.5.3 Principle of SHPB Test 93 5.5.4 Theory Behind SHPB 95 5.5.5 Design of Pressure Bars for a SHPB Apparatus 97 5.5.6 Applications, Availability and Few Results 100 5.6 Taylor Cylinder Impact Test 103 5.6.1 Methodology 104 5.6.2 Strain Rates 107 5.6.3 Limitations and Improvements 107 5.6.4 Case Study-1: Experiments with a Paraffin Wax 109 5.6.5 Case Study-2: Experiments with Steel Cylinders 109 5.7 Drop Impact Test 110 5.7.1 Drop Specimen Test (DST) 111 5.7.1.1 Few Standards for DST by Free Fall 113 5.7.1.2 Experimental Setup for DST 113 5.7.1.3 DST Procedure 115 5.7.1.4 A Case Study: DST of a helicopter in NASA 116 5.7.2 Drop Weight Test (DWT) 118 5.7.2.1 Experimental Setup for DWT 119 5.7.2.2 Case Study-1: DWT to study fracture process in structural concrete 121 5.7.2.3 Case Study-2: DWT tower for applying both compressive and 124 5.8 Summary 125 Exercise Problems 126 References 127 Chapter 6: Modeling Deformation and Failure Under Impact 133 169 6.1 Introduction 133 6.2 Equation of State 135 6.2.1 Gruneisen Parameter 135 6.2.2 Shock-Hugoniot Curve 136 6.2.3 Rankine-Hugoniot Conditions 137 6.2.4 Mie-Gruneisen (Shock) Equation of State 139 6.2.4.1 Implementation of Mie-Gruneisen Equation of State 141 6.2.5 Murnaghan Equation of State 142 6.2.6 Linear Equation of State 142 6.2.7 Polynomial Equation of State 143 6.2.8 High Explosive Equation of State 143 6.3 Constitutive Models for Material Deformation and Plasticity 144 6.3.1 Plasticity 145 6.3.2 Plastic Isotropic or Kinematic Hardening Material Model 147 6.3.3 Thermo-Elastic-Plastic Material Model 148 6.3.4 Power-Law Isotropic Plasticity Material Model 148 6.3.5 Johnson Cook Material Model 149 6.3.5.1 Determination of Parameters in Johnson Cook Model 150 6.3.6 Zerilli-Armstrong Material Model 151 6.3.6.1 Modified Zerilli-Armstrong Material Model 151 6.3.6.2 Determination of Parameters in Zerilli-Armstrong Model 152 6.3.7 Combined Johnson-Cook and Zerilli-Armstrong Material Model 152 6.3.8 Steinberg-Guinan Material Model 153 6.3.9 Barlat s 3 Parameter Plasticity Material Model 153 6.3.10 Orthotropic Material Model 154 6.3.11 Summary of Material Models 154 6.4 Failure/Damage Models 155 6.4.1 Void Growth and Fracture Strain Model 156 6.4.1.1 Void Growth Model 156 6.4.1.2 Fracture Strain Model 157 6.4.2 Johnson Cook Failure Model 158 6.4.3 Unified Model of Visco-plasticity and Ductile Damage 159 6.4.4 Johnson-Holmquist Concrete Damage Model 160 6.4.4.1 Determination of Parameters in Johnson-Holmquist Model 161 6.4.5 Chang-Chang Composite Damage Model 161 6.4.6 Orthotropic Damage Model 162 6.4.7 Plastic Strain Limit Damage Model 162 6.4.8 Material Stress/Strain Limit Damage Model 162 6.4.9 Implementation of Damage 163 6.4.9.1 Discrete Technique 163 6.4.9.2 Operator Split Technique 163 6.5 Temperature Rise During Impact 164 6.6 Summary 165 Exercise Problems 166 References 167 Chapter 7: Computational Impact Mechanics 171 219 7.1 Introduction 171 7.2 Principles of Numerical Formulations 174 7.2.1 Classical Continuum Methods: Lagrangean, Eulerian and 174 7.2.1.1 Lagrangean Formulation 174 7.2.1.2 Eulerian Formulation 176 7.2.1.3 Arbitrary Lagrangean- Eulerian Coupling (ALE-Formulation) 177 7.2.2 Particle Based Methods 179 7.2.2.1 Smooth Particle Hydrodynamics Method 180 7.2.2.2 Discrete Element Method 183 7.2.3 Meshless Methods 185 7.2.4 Hybrid Particle and Mesh based Methods 187 7.3 Numerical Simulation Using Finite Element Methods 189 7.4 Numerical Integration Methods 192 7.4.1 Implicit Integration 192 7.4.2 Explicit Integration 193 7.4.3 Application of Integration Schemes and Material Response 194 7.5 Computational Aspects in Numerical Simulation 196 7.5.1 Hour Glass Deformations and Control 196 7.5.1.1 Hour Glass Deformations 196 7.5.1.2 Hour Glass Control 197 7.5.2 Shockwaves, Numerical Shockwaves and Artificial Viscosity 198 7.5.2.1 Shockwaves 198 7.5.2.2 Numerical Shockwaves 198 7.5.2.3 Artificial Viscosity 199 7.5.3 Acoustic Impedance 200 7.5.4 Adaptive Meshing 200 7.5.5 Contact-Impact Considerations 201 7.5.5.1 Kinematic Constraint Method 201 7.5.5.2 Penalty Method 202 7.5.5.3 Distributed Parameter Method 202 7.5.5.4 Automatic Surface to Surface Contact 202 7.5.5.5 Initial Contact Interpenetrations 203 7.5.5.6 Friction in Sliding Interfaces 203 7.6 Case Studies in Numerical Simulation 203 7.6.1 Case-1: Simulation of Ballistic Impact on a Plate with 203 7.6.2 Case-2: Simulation of Plugging Failure with a Unified 206 7.6.3 Case-3: Simulation of Ballistic Impact of a Steel Bullet on a GFRP Plate 209 7.6.4 Case-4: Discrete Element Method for Simulation of Ballistic 212 7.7 Summary 214 Exercise Problems 216 References 216 Chapter 8: Vehicle Collision 221 267 8.1 Introduction 221 8.2 Mechanics of Vehicle Collision 223 8.3 Crash Impact Tests for Safety Regulations 225 8.3.1 Crash Impact Tests 227 8.3.1.1 Frontal Crash Impact Test 227 8.3.1.2 Side Crash Impact Test 229 8.3.1.3 Rear Crash Impact Test 230 8.3.1.4 Pedestrian Impact Test 231 8.3.1.5 Roll-over Crash Impact Test 231 8.3.2 Data Acquisition and Filtering in Crash Impact Tests 232 8.3.3 Vehicle Safety Regulations in India 233 8.4 Concepts in Analysis of Vehicle/Occupant Systems 234 8.4.1 Introduction 234 8.4.2 Analysis of Frontal Rigid Barrier Collision (Frontal Impact Crash) 236 8.4.3 Vehicle Response in Frontal Barrier Collision 237 8.4.4 Equivalent Square Wave and Pulse Waveform Efficiency 240 8.4.4.1 Equivalent Square Wave (ESW) 240 8.4.4.2 Pulse Waveform Efficiency () 241 8.4.5 Occupant Response in Frontal Barrier Collision 242 8.4.5.1 Occupant Response in a General Braking Vehicle 243 8.4.5.2 Unrestrained Occupant Response in a Braking Vehicle 244 8.4.5.3 Unrestrained Occupant Response in a Crashing Vehicle 245 8.4.5.4 Restrained Occupant Response in a Crashing Vehicle 246 8.4.5.5 Effect of Occupant Restraint in a Crashing Vehicle 246 8.4.6 Guidelines for Design and Evaluation of a Good Occupant 247 8.4.7 Side Impact Analysis 248 8.4.8 Compatibility between Restraint System and Vehicle Front Structure 250 8.5 Standard Restraint Systems 253 8.5.1 Airbag Restraint System (ARS) 253 8.5.2 Safety (Seat) Belts 255 8.5.2.1 Case-1: Occupant with a Non-Stretching Seat Belt 255 8.5.2.2 Case-2: Occupant with a Stretchable Seat Belt 255 8.5.2.3 Case-3: Occupant with No Seat Belt 256 8.5.2.4 Response in all Cases 257 8.5.3 Collapsible Steering Columns 257 8.6 Crashworthiness and Crash Energy Management 258 8.6.1 Crashworthiness 258 8.6.2 Crash Energy Management 259 8.6.2.1 Parameters Adopted in Quantifying Crash Energy 260 8.6.2.2 Typical Structural Members for Crash Energy Management 261 8.7 Summary 264 Exercise Problems 265 References 267 Chapter 9: Ballistic Impact 269 312 9.1 Introduction 269 9.1.1 Classification of Ballistic Impact, Projectile Shape and Target 272 9.1.1.1 Classification of Ballistic Impact 272 9.1.1.2 Classification of Projectile Shape 273 9.1.1.3 Classification of Targets 273 9.1.2 Impact Response of Materials to Ballistic Impact at 274 9.2 Mechanics of Penetration and Perforation 276 9.2.1 Physics of Impact Phenomena in Penetration and Perforation 276 9.2.2 Elastic, Plastic and Hydrodynamic Limit Velocities and 277 9.2.2.1 Elastic Limit Velocity (VEL) 278 9.2.2.2 Plastic Limit Velocity (VPL) 278 9.2.2.3 Hydrodynamic Limit Velocity (VHL) 279 9.2.3 Ballistic Limit Velocity, Impact Regime Phase Diagram and Aerial Density 279 9.2.3.1 Ballistic Limit 280 9.2.3.2 Impact Regime Phase Diagram for Ballistic Limit 281 9.2.3.3 Aerial Density 282 9.3 Failure Modes and Mechanisms in Impacted Targets 282 9.4 Ballistic Impact Models 286 9.4.1 Methods Adopted in Developing Ballistic Impact Models 287 9.4.1.1 Analytical Methods 287 9.4.1.2 Empirical or Quasi-Analytical Methods 287 9.4.1.3 Numerical Methods 288 9.4.2 Select Ballistic Impact Models 288 9.4.2.1 Penetration Models 288 9.4.2.2 Residual Velocity Models 297 9.4.2.3 Models for Fragmentation 300 9.5 Ballistic Testing 302 9.5.1 Different Stages in Ballistic Experiments 302 9.5.2 A Simple Test Setup for Ballistic Impact 302 9.5.3 An Actual Test Setup for Ballistic Impact 304 9.5.4 Developments in Imaging Systems 306 9.5.5 Open Range Test Setup for Ballistic Impact 306 9.6 Summary 307 Exercise Problems 308 References 310 Chapter 10: Concluding Remarks 313 323 10.1 Introduction 314 10.2 Summary 315 10.3 Future Research Directions for Applied Impact Mechanics 318 10.4 Epilogue 321 Index 325 334 Colour Plate 335 350