Introduction to Impact Dynamics

Introduction to Impact Dynamics

T.X. YuThe Hong Kong University of Science and TechnologyKowloon, Hong Kong

XinMing QiuTsinghua UniversityBeijing, China

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Library of Congress Cataloging-in-Publication Data

Names: Yu, T. X. (Tongxi), 1941– author. | Qiu, XinMing, author.Title: Introduction to impact dynamics / by T.X. Yu, Prof. XinMing Qiu.Description: Hoboken, NJ ; Singapore : John Wiley & Sons, 2018. | Includesbibliographical references and index. |

Identifiers: LCCN 2017033421 (print) | LCCN 2017044755 (ebook) | ISBN9781118929858 (pdf) | ISBN 9781118929865 (epub) | ISBN 9781118929841(cloth)

Subjects: LCSH: Materials–Dynamic testing.Classification: LCC TA418.34 (ebook) | LCC TA418.34 .Y8 2018 (print) | DDC620.1/125–dc23

LC record available at https://lccn.loc.gov/2017033421

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Contents

Preface xiIntroduction xiii

Part 1 Stress Waves in Solids 1

1 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 11

Questions 1 17Problems 1 18

2 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 25

v

2.1.5 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 36

Questions 2 38Problems 2 38

Part 2 Dynamic Behavior of Materials under High Strain Rate 39

3 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 69

Problems 3 69

4 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 90

Problems 4 90

Contentsvi

Part 3 Dynamic Response of Structures to Impact and Pulse Loading 91

5 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 Examples of Inertia Force in Beams 975.3 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 112

Questions 5 112Problems 5 112

6 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 133

Problems 6 136

7 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 166

Contents vii

Questions 7 168Problems 7 168

8 Mode Technique, Bound Theorems, and Applicability of the Rigid-PerfectlyPlastic Model 169

8.1 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 183

Problems 8 186

9 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 209

Problems 9 210

10 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 228

Contentsviii

10.3 Dynamic Response of a Clamped Circular Sandwich Plate Subject to ShockLoading 233

10.3.1 An Analytical Model for the Shock Resistance of Clamped SandwichPlates 234

10.3.2 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 257

References 259Index 265

Contents ix

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 seriesof impact and explosion events. With the rapid development of land vehicles, ships, andaircraft, traffic accidents have become a serious concern of modern society. Landing ofspacecraft, safety in nuclear plants and offshore structures, as well as protection ofhuman bodies during accidents and sports, all require better knowledge regarding thedynamic 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 structuresunder 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). Theimpact velocity may vary from a few meters per second (as seen in ball games) to severalkilometers per second (as seen in military applications). Driven by the needs of scienceand engineering, the dynamic response, impact protection, crashworthiness, and energyabsorption capacity of materials and structures have attracted more and more attentionfrom researchers and engineers. Numerous research papers and monographs haveappeared in the literature, and it is not possible for anyone to condense the huge amountknowledge 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 toconsider 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. Theemphasis here is on theoretical models and analytical methods, which will help readers tounderstand the fundamental issues raised by various practical situations. Numericalmethods and software are not themain topic of this textbook. Readers who are interestedin numerical modeling related to impact dynamics will have to consult other sources forthe 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 anddesigning/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

xi

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 contentsto deliver a short course for graduate students in many universities, including TsinghuaUniversity, Zhejiang University, Wuhan University, Xi’an Jiaotong University, TaiyuanUniversity of Technology, Hunan University, and Dalian University of Technology.The second author (X.M. Qiu) has also taught Impact Dynamics as a credit graduatecourse 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 in2011. 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 beenrewritten, 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), includinga 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 inthe relevant places, and we would like to express our sincere gratitude to the originalauthors.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 QiuJuly 2017

Prefacexii

Introduction

With the rapid development of all kinds of transport vehicles, the lives lost and high costof traffic accidents are of serious concern to modern society. The public is becomingincreasingly aware of the safe design of components and systems with the objective ofminimizing human suffering as well as the financial burdens on society. At the sametime, many other issues in modern engineering, e.g., nuclear plants, offshore structures,and safety gear for humans, also require us to understand the dynamic behavior of struc-tures and materials.Driven by the needs of engineering, the dynamic response, impact protection, crash-

worthiness, and energy absorption capacity of various materials and structures haveattracted more and more attention from researchers and engineers. As a branch ofapplied mechanics, impact dynamics aims to reveal the fundamental mechanisms oflarge dynamic deformation and failure of structures and materials under impact andexplosive loading, so as to establish analytical models and effective tools to deal withvarious complex issues raised from applications.In the classical theories of elasticity and plasticity, usually only static problems are of

concern, in which the external load is assumed to be applied to the material or structureslowly, and the corresponding deformation of the material or structure is also slow. Theacceleration of material is very small and thus the inertia force is negligible comparedwith the applied external load; hence the whole deformation process can be analyzedunder an equilibrium state.However, it is known that the material behavior and structural response under

dynamic loading are quite different from those under quasi-static loading. In engineeringapplications, the external load may be intensive and change rapidly with time, termedintense dynamic loading; consequently, the deformation of material or of a structurehas to be quick enough under intense dynamic loading. Some examples are given inthe following.The collision of vehicles. Cars, trains, ships, aircraft, and other vehicles may collide with

each other or with surrounding objects during accidents. These accidents will lead to thefailure or deformation of the structures as well as personnel casualties, resulting in seri-ous economic losses. As the number of cars has rapidly increased in many countries, caraccidents have become the number one cause of death in the world. Collisions betweenships and collisions between ships and rocks/bridges all cause huge economic loss aswell as environmental pollution. Along with the development of high-speed rail trans-portation, the safety of occupants is also of greater public concern. It is very dangerous

xiii

if a bird impinges on the cockpit or engine of an airplane, as the relative velocity betweenbird and airplane could be high even though the speed of the bird is not great. More andmore space debris has been produced as a result of human activities, and the relativevelocity of space debris to spacecraft can be as high as 10 km/s, so there would be greatdamage in the case of a collision.Damage effects of explosive. Buildings, bridges, pipelines, vehicles, ships, aircrafts, and

protective structures could be subjected to intensive explosion loading, due to industrialaccident, military action, or terrorist attack. Typically, these structures would be sud-denly loaded by a shock wave propagating in the air.The effects of natural disasters. Natural disasters, such as earthquakes, tsunamis,

typhoons, floods and so on will produce intensive dynamic loads to structures, e.g., dams,bridges, and high-rise buildings. These intense dynamic loads are likely to cause damageto the structures.The strong dynamic loads caused by the local rupture of the storage structures. In

nuclear power plants or chemical plants, if there is local damage to a pipeline, the jetof high-pressure liquid that would escape from the broken section exerts a lateral reac-tion force (the blowdown force) on the broken pipe, causing rapid acceleration and largedeformation, termed “pipe whip”. After local damage, the consequences from a pressurevessel or a dam could be disastrous.Load of high-speed forming.During a dynamicmetal forming process, such as explosive

forming and electromagnetic forming, the work-piece is subjected to intensive dynamicloading and deforms rapidly. Similar situations take place in the process of forging orhigh-speed stamping.Impact or collision in daily life and sports. For example, falling objects, falling on the

ice, collision between moving people, a football or golf ball hitting the head or body withhigh speed.

All kinds of the above-mentioned problems encountered in engineering or daily liferequire the understanding and study of the behavior of the solid materials and structuressubjected to intensive dynamic loads. First of all, why is the dynamic behavior of materi-als and structures usually different from the quasi-static behavior? This is the result ofthree major attributes in mechanics, as briefly illustrated here.Stress wave propagation in material and structure.When a dynamic load is applied to

the surface of a solid, the stress and generalized deformation will propagate in the form ofstress wave. If the disturbance is weak, it is an elastic wave; but if the stress level of wave ishigher than the yield strength of the material, it will be plastic wave.Suppose a solid medium has a characteristic scale of L, and the wave speed of its mate-

rial is c. It is subjected to an external dynamic loading that has a characteristic time tc, e.g.,the time period for the external load to reach its maximum value or time duration of theimpulse. If tc << L c, the stress and deformation distribution in this solid are not uniform;hence, the effect of stress wave propagation must be considered. For example, the char-acteristic scale of the crust is very large, so the effects of earthquake or undergroundexplosion are mainly presented in the form of stress waves.In a piling machine or a split Hopkinson pressure bar device (SHPB for short – an

important experimental technique in studying the dynamic properties of materials; referto Chapter 3 for more details), the perturbation is along the longitudinal (large scale)direction of a long bar rather than in the radial (small scale) direction. Hence wave

Introductionxiv

reflection, transmission, and dispersion are important factors that need to be analyzedcarefully.By contrast, some other structural components that are widely used in engineering,

such as beams, plates, and shells, are usually subjected to lateral loads along their thick-ness direction, which is the smallest scale direction of the structure. The elastic wavespeed in metals is usually in the order of several kilometers per second (e.g., 5.1 km/sfor steel). Therefore, in several micro-seconds all the particles in the thickness directionof the structure will be affected by the external disturbance, and then the entiresection of structure will be accelerated and then move together. This global motionof the whole cross-sections of the structure is classified as the elastic-plastic dynamicresponse of the structure; this is discussed in detail in Part 3 of this book. This subse-quent global structural response may last several milliseconds or even several seconds,depending on the type of structure and loading, before the structure reaches itsmaximum deformation.Because the effective time of stress wave propagation is usually several orders of mag-

nitude smaller than that of the long-term structural response, the total response of thestructure can be divided into two decoupled separate stages. That is, in the analysis ofwave propagation, the structure is assumed to remain in its original configuration, whichis regarded as the reference frame for geometric relations and equations of motion, whilein the analysis of structural response, the early time wave propagation is disregarded andonly its global deformation is considered.

Rate-dependency of a material’s properties. The material in a solid or structure willdeform rapidly under intensive dynamic loading. Depending on the microscopic defor-mation mechanism, the resistance of material to rapid deformation is generally higherthan that to slow deformation, as revealed by numerous experiments on materials.For example, the mechanism of plastic deformation of metals is mainly attributed tothe movement of dislocations. The resistance to the dislocation motion will be muchhigher when the dislocation passes through the metal lattice at a high speed than at alow speed, and this will lead to the higher yield stress and the high flow stress of metalsduring high-speed deformation.An important task in the study of dynamic properties of materials is to summarize the

effect of strain rate on the stress–strain relationship, based on the experimental data, soas to establish the strain rate-dependent constitutive relation of materials. As the strainhistory and instantaneous strain rate of the material elements inside a structure vary withposition and time, the dynamic constitutive relation has to be simplified to a large extentwhen it is applied to dynamics analysis of structures.

Inertia effect in structural response. In the analysis of dynamic response of a structure,usually both elastic deformation and plastic deformation exist, and the boundarybetween elastic-plastic regions changes with time. Therefore, different constitutive rela-tions should be employed in different regions. Further, the complicated moving bound-ary has to be dealt with. In order to reduce the complexity, the constitutive relation ofmaterial also needs to be simplified in the theoretical modeling of structural dynamics.Themost successful idealization commonly adopted in theoretical modeling is to assumethat the structure is made of a rigid-perfectly plastic material, which neglects all theeffects of elasticity, strain hardening and strain rate.The basis of the hypothesis is that the structure usually experiences considerable large

plastic deformation under intensive dynamic loading, and thus most of the work done by

Introduction xv

the external load will be dissipated by plastic deformation, while only a small amount ofexternal work will be stored in the form of elastic deformation energy. Therefore, toignore the elastic deformation and the corresponding energy will only result in minorinfluence over the final deformation and failure mode of the structure. As will be seenin the book, this idealization largely simplifies the analysis.Due to the similarity of material idealization, the dynamic analysis of rigid-perfectly

plastic structures is closely related to the limit analysis of structures, especially inits concept and methodology. For example, the widely accepted concepts in limitanalysis in the kinematically admissible velocity field can be successfully extendedto construct dynamic deformation mechanisms containing stationary or travelingplastic hinges.At the same time, it should be noted that the main difference between dynamic analysis

and limit analysis lies with the intervention of the inertia effect in dynamic analysis. Thelimit analysis based on the theory of plasticity reveals that a structure made of rigid-perfectly plastic material under external load must have a limit state, i.e., if the externalload reaches a certain limit value, the structure will become a mechanism and lose theload-carrying capacity. On the other hand, from the dynamic analysis of the same struc-ture, it is found that if the dynamic load exceeds the static limit load, i.e., the collapseload, the structure will be accelerated. According to the D’Alembert principle, it isthe inertia force of the structure that is in equilibrium with the external load and thatresists deformation. The greater the external load, the greater the acceleration, so thegreater the inertia force. Thus, the structure can bear a much higher external load thanthe static limit load in a short time, which is a notable feature of the structural dynamicresponse and is different from the static limit analysis.Generally speaking, dynamic loading and dynamic response always become significant

when accidents occur. Nowadays, alongside the development of computing capabilityand software, different kinds of numerical tools and methods have been developed veryrapidly, so they have wider and wider applications. Therefore, some researchers thinkthat it is appropriate to employ numerical simulations in handling problems of impactdynamics. However, even for numerical simulations, a proper understanding of the basicprinciples, concepts, and theoretical models used in dynamic analyses is crucial if sim-ulation methods and models are to give reliable results. Furthermore, the numericalsimulations will result in a huge amount data, so our understanding of impact dynamicswill greatly help us to digest the data and discover the underlying physical significanceand engineering implications.This textbook aims to demonstrate the fundamental features of the dynamic behavior

of materials and structures, to clearly illustrate the widely applicable theoretical modelsand analytical methods, and to highlight the most important factors that affect dynamicbehavior. The textbook is highly relevant to education programs at both graduate andsenior undergraduate levels. For those engineers who are working in the automotive, aer-ospace, mechanical, nuclear, marine, offshore, and defense sectors, the book will alsoprovide fundamental guidance on relevant concepts, models, and methodology, to helpthem face the challenges of understanding the dynamic behavior of materials and of ana-lyzing and designing structures under various types of intensive dynamic loading.

Introductionxvi

Part 1

Stress Waves in Solids

1

1

Elastic Waves

In a deformable solid medium, the disturbance to mechanical equilibrium is representedby the change in particle velocity and the corresponding changes in stress and strainstates. When some parts of a solid are first disturbed, finite time durations are requiredfor this disequilibrium to be felt by other parts of the body, due to the deformable proper-ties of the body. This kind of propagation as a result of the disturbance in stress andstrain through a solid body is termed a stress wave.

1.1 Elastic Wave in a Uniform Circular Bar

1.1.1 The Propagation of a Compressive Elastic Wave

Consider a uniform circular bar made of isotropic material, as shown in Figure 1.1. Let xdenote the longitudinal coordinate measured from an origin O, which is fixed in thespace; and let u(x) denote the displacement undergone by a plane AB in the bar, which

is initially at a distance x from O. Then u+∂u∂x

δx is the displacement of plane A B which

is parallel to AB but is initially at a distance x+ δx from O.A force applied rapidly at time t = 0, over the end plane at x = 0, will cause a disturbance

to propagate elastically along the bar, so that a compressive normal stress, −σ0, will passthrough plane AB at time t.It should be noted that the slender bar assumption is adopted here, i.e. the pulse

length is at least six times the typical cross-sectional dimension of the bar. In this case,the strain and inertia in the transverse direction can be neglected. The gravitational forceand damping of the material are also ignored in the following analysis.The equilibrium of a representative element of the bar is illustrated in Figure 1.2. Here

A0 is the initial cross-section area of the bar, ρ0 is the initial density of the material, and−σ0 is the stress transmitted, with the negative sign reflecting the fact that the stress iscompressive, as shown in the figure.From Newton’s second law, the equation of motion of the representative element is

−∂σ0∂x

δx A0 = ρ0A0δx∂2u∂t2

, which could be simplified as

∂σ0∂x

= −ρ0∂2u∂t2

1 1

3

Introduction to Impact Dynamics, First Edition. T.X. Yu and XinMing Qiu.© 2018 Tsinghua University Press. All rights reserved.Published 2018 by John Wiley & Sons Singapore Pte. Ltd.

To analyze the deformation of the representative element of length δx, it is clear thatthe strain of the element is

ε=∂u∂x

1 2

Assuming the solid material has Young’s modulus E, then, according to Hooke’s law, inthe linear elastic stage we have

−σ0 =E∂u∂x

1 3

The stress variation over the element is obtained from the partial differential ofEq. (1.3) with respect to x:

∂σ0∂x

= −E∂2u∂x2

1 4

Substituting Eq. (1.1) into Eq. (1.4) leads to

ρ0∂2u∂t2

=E∂2u∂x2

1 5

With the notation of cL = E ρ0, Eq. (1.5) is rewritten as

∂2u∂t2

= c2L∂2u∂x2

1 6

Wave

u

x

B

A

δx

B′

A′

O x

Unstrained configuration

u +∂u∂x

−σ0

δx

Figure 1.1 The propagation of a compressive elastic wave in a uniform circular bar.

A0σ0 A0(σ0+ ∂σ0 / ∂x · δx)

A A′

B B′

δx Figure 1.2 The equilibrium of arepresentative element of the bar.

Introduction to Impact Dynamics4

Obviously, Eq. (1.6) is a typical one-dimensional (1D) wave equation of thefollowing form:

∂2u∂t2

= c2∂2u∂x2

1 7

Considering the general solution of Eq. (1.7), u(x, t), in the following form:

u x, t = f1 x−ct + f2 x+ ct 1 8

and substituting Eq. (1.8) into the wave equation (Eq. 1.7), we find

∂u∂t

= −cf1 x−ct + cf2 x+ ct ,∂2u∂t2

= c2f1 x−ct + c2f2 x+ ct

∂u∂x

= f1 x−ct + f2 x+ ct ,∂2u∂x2

= f1 x−ct + f2 x+ ct

Thus it can be verified that Eq. (1.8) satisfies the wave equation, so it gives a generalsolution to Eq. (1.7). In order to understand the mechanical meaning of Eq. (1.8), onlyone term is studied here, u x, t = f1 x−ct , i.e., f2 = 0 (see Figure 1.3).At t = t1, the particle at position x = x1 has a displacement u = s, and at t = t2, the particle

at position x = x2 also has a displacement u = s. Thus from Eq. (1.8), the displacementshould satisfy s= f1 x1−ct1 = f1 x2−ct2 , which results in x1−ct1 = x2−ct2, leading to thefollowing speed of wave propagation:

c=x2−x1t2− t1

1 9

This confirms that for the wave propagation governed by Eq. (1.6), cL = E ρ0precisely represents the speed of the longitudinal waves (compressive or tensile).There are two terms on the right-hand side of the general solution of wave propaga-

tion, Eq. (1.8). The term f1(x − ct) denotes the wave traveling in the +x direction, i.e., aforward-traveling wave, and the term f2(x + ct) denotes the wave traveling in the −x direc-tion, i.e., a backward-traveling wave. Both the traveling waves in Eq. (1.8), f1(x − ct) andf2(x + ct), have the following characteristics: the waves are traveling at a constantspeed with no change in their shape or magnitude, i.e. the 1D longitudinal waves arenon-dispersive.

s

u t = t1 t = t2

O x1 x2 x

Figure 1.3 Particle displacements produced by a forward wave.

Elastic Waves 5

The speed of longitudinal (compressive or tensile) waves in four typical materialsare given in Table 1.1. It should be emphasized that the wave speed depends onboth Young’s modulus and density of the material. Therefore, although the density ofaluminum is only about one-third of that of steel, the wave speeds are similar.For a 1D longitudinal wave, there are two ways in which you can distinguish between a

compressive and a tensile wave:

• Look at the sign of the stress. A compressive wave produces a negative normal stress anda tensile wave produces a positive stress.

• Look at the directions of the particle velocity and the wave propagation. For a compres-sive wave, the particle velocity is in the same direction as the wave propagation,whereas for a tensile wave, the particle velocity is in the opposite direction to the wavepropagation.

1.2 Types of Elastic Wave

Different types of elastic wave can propagate in solids. These waves are classified accord-ing to the relationship between the motion of the particles and the direction of propa-gation of the waves and also according to the boundary conditions. The most commontypes of elastic wave in solids are:

• Longitudinal (irrotational) waves

• Transverse (shear) waves

• Surface (Rayleigh) waves

• Interfacial (Stoneley) waves

• Bending (flexural) waves (in beams and plates).

1.2.1 Longitudinal Waves

Longitudinal waves are those in which the particle velocity is parallel to the direction oftravel of the wave. In particular, longitudinal waves are called compressional wavesor compression waves, because they produce compression as the particle velocity andwave velocity are in the same direction. By contrast, a tensile wave produces tensionas the particles and waves travel in opposite directions.

Table 1.1 Typical longitudinal wave speed in solid materials

Steel Aluminum Glass Polystyrene

E (GPa) 205 75 95

ρ0 (g/cm3) 7.8 2.7 2.5

cL (m/s) 5100 5300 6200 2300

E, Young’s modulus; ρ0, density; cL, wave speed.

Introduction to Impact Dynamics6

Longitudinal waves are also known as irrotational waves. In seismology, they areknown as P-waves. This is because a longitudinal wave travels at the fastest speedsand so arrives at seismic stations first (primary waves); the rock moves forward andbackward in the same direction as the wave is traveling (push–pull wave, parallel topropagation), and the wave is able to pass through solids and liquids. In infinite andsemi-infinite media, longitudinal waves are also known as dilatational waves, due tothe changes in the volume of the media.

1.2.2 Transverse Waves

Transverse waves are another common type of wave, in which the particle velocity isperpendicular to the wave’s propagation. An example of a transverse wave is shownin Figure 1.4. In this arrangement a circular bar is clamped at a given position and a tor-que is applied to the left end of the bar. Thus there is shear stress on the left side of theclamp and zero stress on the right-hand side. When the clamp is suddenly released, thestress disturbance will propagate, i.e., a wave will travel from the left side to the rightside of the bar. The particle velocity is within the cross-sectional plane of the bar, whilethe wave propagation direction is along the bar; hence, as the particle velocity is perpen-dicular to the wave velocity, this torsional wave is a transverse wave. The normal strainsare all zero, with no resulting change in density, while the shear strains are non-zero,producing a change in shape. Thus, transverse waves are called shear waves, and are alsoknown as distortional or equivolumal waves.For an elastic material with shear modulusG and density ρ0, the speed of the transverse

wave is derived as cS = G ρ0, which is slower than that of the longitudinal wave in the

same solid, ascScL

=GE=

1

2 1+ v< 1.

In seismology, transverse waves are known as S-waves, because they do not travel asquickly as P-waves (slow wave) and will arrive at a seismic station second (secondarywave); the rocks move from side to side (shear wave) and the waves only travel throughsolids.

1.2.3 Surface Wave (Rayleigh Wave)

Surface waves are analogous to gravitational waves on the surface of water. As shown inFigure 1.5, the material particles move up and down as well as back and forth, tracingelliptical paths. The surface wave is restricted to the region adjacent to the surface.The particle velocity decreases very rapidly (exponentially) as one moves away from

TorqueClamp release

Shear stress wave

Figure 1.4 The transverse wave in a circular bar produced by suddenly releasing the clamp.

Elastic Waves 7

the surface. Surface waves in solids (called Rayleigh waves) are a particular case of inter-facial waves where one of the materials has negligible density and elastic wave speed.If a hammer hits the surface of a semi-infinite solid, several waves are propagated after

the blow – among these the speed of the longitudinal wave (P-wave) is greater than thatof the transverse wave (S-wave), and the surface wave (Rayleigh wave) only affects thesolid for a finite distance under the surface.

1.2.4 Interfacial Waves

When two semi-infinite media with different material properties are in contact, specialwaves form at their interface in the case of a disturbance. The surface wave in a solid(Rayleigh wave) could be regarded as a special case of the interfacial wave – that is,the density and elastic wave in one of the contacting media, such as air, could be omitted.

1.2.5 Waves in Layered Media (Love Waves)

The earth is composed of layers with different properties, and so special wave patternsemerge. This was first studied by Love. As a result of Love waves, the horizontal com-ponent of displacement produced by earthquakes can be significantly larger than the ver-tical component, which is a behavior that is not consistent with Rayleigh waves.

1.2.6 Bending (Flexural) Waves

These waves involve propagation of flexure in 1D (beams and arches) or 2D (plates andshells) configurations. Made from a material of density ρ0 and elastic modulus E, astraight beam with cross-sectional area A0 and principal moment of inertia I is shownin Figure 1.6. A coordinate system with the x-axis along the beam length and the z-axisin the direction of deflection is adopted. When a bending moment M and shear force Qare applied, a transverse deflection w is produced in the beam.By considering the equilibrium of a small element of length δx as shown in Figure 1.6

(b), the equation of motion of this element gives

− ρ0A0δx∂2w∂t2

=∂Q∂x

δx 1 10

Reyleigh wave

Figure 1.5 Diagram of a Rayleigh wave.

Introduction to Impact Dynamics8

From the relation given by elastic mechanics,

EI∂3w∂x3

=Q 1 11

The combination of Eqs (1.10) and (1.11) leads to the wave equation of a bending wave:

ρ0A0∂2w∂t2

= −EI∂4w∂x4

1 12

Or in an equivalent form:

∂2w∂t2

= −c2Lk2 ∂

4w∂x4

1 13

where cL = E ρ0 is the speed of the longitudinal wave, and k denotes the radius ofgyration of the cross-section about the neutral axis, i.e., I =A0k2 holds.Obviously, solutions in the form of w(x, t) = f1(x − ct) or w(x, t) = f1(x + ct), which are

the general solutions of the regular wave equation, do not satisfy the bending waveequation (Eq. 1.13). This implies that flexural disturbance of arbitrary form alwayspropagates with dispersion.

1.3 Reflection and Interaction of Waves

1.3.1 Mechanical Impedance

Let us focus on the longitudinal waves again. As shown in Sections 1.1 and 1.2, for a for-ward longitudinal wave, i.e., the wavemoving in the positive x-direction, from the generalsolution of the wave equation, Eq. (1.8), the displacement of particle is

u x, t = f x−ct 1 14

Differentiating Eq. (1.14) with respect to time t leads to the particle velocity:

v0 =∂u∂t

= −cf x, t 1 15

(a) (b)

δx

Ox A0

z, wElement

Element

δx Q + ∂Q∂x

Q δxM + ∂M∂x

M

δx

Figure 1.6 A straight beam under bending. (a) Beam under bending; (b) free body diagram of anelement.

Elastic Waves 9

A partial differentiation of Eq. (1.14) with respect to particle position x leads to thestrain of this element:

ε=∂u∂x

= f x, t 1 16

From the property of an elastic material, the stress on this element is given by

σ = −Eε= −Ef =Ev0c

1 17

Using the expression of wave speed, c= E ρ0, Eq.(1.17) can be rewritten as

σ =Ev0c

= ρ0cv0 = v0 Eρ0 1 18

where the quantity ρ0c is termed themechanical impedance, or sonic/sound impedance,of thematerial. This expression can also be applied to the propagation of the tensile wave:

v0 =σ

Eρ0=cEσ =

σ

ρ0c1 19

In Eq. (1.19), the particle velocity is related to the current stress. For example, for steel,if the stress is 100 MPa, then the particle velocity is

v0 =cEσ =

5100m s × 100MPa205GPa

≈2 49m s 1 20

The mechanical impedance of steel is

ρ0c= 7800kg m3 × 5100m s≈4 × 107Ns m3 1 21

Wave speed and mechanical impedance are two very important concepts for stresswaves. The wave speed indicates the velocity of the disturbance propagating in a deform-able solid, while the mechanical impedance represents the degree of resistance of thedeformable solid to the disturbance.

1.3.2 Waves When they Encounter a Boundary

We will briefly describe the interaction of waves when they encounter a boundary.Figure 1.7 shows the longitudinal waves that are reflected and refracted at the boundaryas well as the two transverse waves that are generated at the interface. These effects,reflection and refraction, occur when the wave encounters a medium with differentmechanical impedance, which is defined as the product of the medium density andits elastic wave speed. These refraction and reflection angles are given by a simplerelationship of the form:

sinθ1cL

=sinθ2cS

=sinθ3cL

=sinθ4cL

=sinθ5cS

1 22

The interactions of a wave with an interface are very simple when the incidence is nor-mal (θ1 = 0). In this case, a longitudinal wave refracts/transmits and reflects longitudinal

Introduction to Impact Dynamics10

waves, and a shear wave refracts/transmits shear waves. In this way, it becomes a 1Dwavereflection and transmission problem.

1.3.3 Reflection and Transmission of 1D Longitudinal Waves

Figure 1.8(a) shows the fronts of a wave propagating along a cylinder in a medium inwhich the wave speed is cA. The particle velocity is v and the stress is σ. Figure 1.8(b)illustrates the stresses at the interface related to incident, transmitted, and reflected

A BMaterial boundary

cA

vI

dx

(a)

cAcB

σT

σR

σI

(b)

(c)

cA cB

vI

vR

vT

Figure 1.8 The reflection and transmission of a longitudinal wave in a one-dimensional cylinder.

cL, cS c′L, c′S

A B

Incident longitudinal

Refracted longitudinal

Refracted transverse

Reflected longitudinal

Reflected transverse

θ5

θ4θ3

θ2

θ1

Figure 1.7 Reflection and refraction when a longitudinal wave encounters an interface.

Elastic Waves 11