Dynamic fracture

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Dynamic Fracture

Elsevier Internet Homepage-http://www.elsevier.com Consult the Elsevier homepage for full catalogue information on all books, journals and electronic products and services. Elsevier Titles of Related Interest ALLIX and HILD Continuum Damage Mechanics of Materials and Structures. ISBN: 008-043918-7 BLACKMAN ET AL. Fracture of Polymers, Composites and Adhesives II ISBN: 008-044195-9 CARPINTERI ET AL. Biaxial/Multiaxial Fatigue and Fracture. ISBN: 008-044129-7 ELICES and LLORCA Fiber Fracture. ISBN: 008-044104-1 FRANC¸OIS and PINEAU From Charpy to Present Impact Testing. ISBN: 008-043970-5

MOORE The Application of Fracture Mechanics to Polymers, Adhesives and Composites ISBN: 008-044205-6 MOORE ET AL. Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites. ISBN: 008-043689-7 MURAKAMI Metal Fatigue Effects of Small Defects and Nonmetallic Inclusions. ISBN: 008-044064-9 PAULINO Fracture of Functionally Graded Materials. ISBN: 008-044160-2

FUENTES ET AL. Fracture Mechanics: Applications and Challenges. ISBN: 008-043699-4

K.S. RAVICHANDRAN ET AL. Small Fatigue Cracks: Mechanics, Mechanisms & Applications. ISBN: 008-043011-2

JONES Failure Analysis Case Studies II. ISBN: 008-043959-4

RE´MY and PETIT Temperature-Fatigue Interaction. ISBN: 008-043982-9

JONES Failure Analysis Case Studies III. ISBN: 008-044447-4

TANAKA & DULIKRAVICH Inverse Problems in Engineering Mechanics IV. ISBN: 008-044268-4

MILNE ET AL. Comprehensive Structural Integrity. ISBN: 008-043749-4

WILLIAMS & PAVAN Fracture of Polymers, Composites and Adhesives. ISBN: 008-043710-9

Related Journals Free specimen copy gladly sent on request. Elsevier Ltd, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK Acta Metallurgica et Materialia Cement and Concrete Research Composite Structures Computers and Structures Corrosion Science Engineering Failure Analysis Engineering Fracture Mechanics European Journal of Mechanics A & B International Journal of Fatigue International Journal of Impact Engineering International Journal of Mechanical Sciences International Journal of Non-Linear Mechanics International Journal of Plasticity

International Journal of Pressure Vessels & Piping International Journal of Solids and Structures Journal of Applied Mathematics and Mechanics Journal of Construction Steel Research Journal of the Mechanics and Physics of Solids Materials Research Bulletin Mechanics of Materials Mechanics Research Communications NDT&E International Scripta Metallurgica et Materialia Theoretical and Applied Fracture Mechanics Tribology International Wear

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Dynamic Fracture

K. Ravi-Chandar Department of Aerospace Engineering and Engineering Mechanics The University of Texas, Austin, USA

2004

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v

Preface The aim of this book is to provide an overview of dynamic fracture in nominally brittle materials. Brittle fracture in solids has attracted much attention over the second half of the 20th century from engineers as well as physicists due both to its technological interest and inherent scientiﬁc curiosity. Early investigations into brittle fracture were performed under dynamic loads without the presence of a dominant crack in the body; this type of loading causes spalling of the material through the nucleation, growth and coalescence of multiple cracks in the spall plane. The advent of a fracture mechanical description of material failure that began with the pioneering effort of Grifﬁth has steered recent dynamic brittle fracture investigations away from the spall problem and towards a crack-dominated approach; this book focuses on the crack problem. The literature on the subject is vast; a quick search on databases with keywords “dynamic and fracture” yields results numbering in many thousands; and research in the ﬁeld continues at a signiﬁcant pace, especially in the area of numerical simulations. Coverage of the topic must therefore be selective. I have restricted attention for the most part to opening mode cracks in homogeneous nominally brittle materials; extensions to shearing mode cracks can be made easily by applying the criterion of local symmetry insisting that the crack choose a path so as to ensure a locally opening mode crack. Such a claim is not valid when considering interfaces or graded materials; there are many investigations of this problem, but these are not considered in this book. I have also chosen to describe the classical elastodynamic interpretation of the problem in detail and provide only glimpses of new approaches such as the discrete models and cohesive zone models; while such models are potentially powerful in providing detailed numerical simulations of dynamic fracture problems, fundamental considerations regarding the determination of appropriate material properties for inclusion in these models and the quantitative comparison of the results of the simulations to experimental observations remain open issues. Dynamic failure criteria and the limitations in their applicability in assessing structural integrity are the primary focus of this book. Analytical characterization of the crack tip state, methods of implementation of dynamic fracture experiments, diagnostic techniques and their limitations, and interpretation of dynamic failure criteria are discussed in great detail. The introductory chapter provides examples of problems where structural integrity assessment through the elastodynamic fracture theory is important; an outline of the topics covered is also presented there. This book should be accessible to graduate students with a background in solid mechanics. This book could be used as a resource in courses devoted to fracture mechanics or experimental mechanics. This book should also be useful to workers in the ﬁeld of structural integrity assessment. For example, implementation of test methodologies for determination of rate-dependent dynamic crack initiation toughness or crack growth criterion discussed in this book should be a simple task given modern test and measurement equipment and data acquisition systems.

vi

Preface

I am deeply indebted to the many contributors to this ﬁeld for numerous discussions that helped me in understanding the mechanics and mechanisms of dynamic fracture. It is indeed my pleasure to speciﬁcally acknowledge Wolfgang Knauss (California Institute of Technology) for support and guidance during my graduate studies in this ﬁeld and for his continued encouragement and friendship through all these years. I am also indebted to Wolfgang Knauss, Michael Marder (University of Texas at Austin), and Sridhar Krishnaswamy (Northwestern University) for reading drafts of all or portions of the manuscript. I thank Dean Eastbury, Senior Publisher, for encouraging me to pursue this book. I also thank Sharon Brown and Carol Cooper who coordinated the production of the book, for putting up with the many delays on my part. I thank my wife Hema and son Prakash, for their love, support and the many sacriﬁces they endured as I worked on this book. I dedicate this book to my parents, Padma and Sunder Krishnaswamy. K. Ravi-Chandar Austin, Texas 2004

vii

Contents Preface........................................................................................................................

v

1. Introduction........................................................................................................

1

1.1 1.2 1.3 1.4

Pressurized Thermal Shock in Nuclear Containment Vessels ................... Boiler and Pipeline Burst Problems ............................................................ Dynamic Fracture in Airplane Structures ................................................... Notched Bar Impact Testing of Metallic Materials....................................

2 3 4 6

2. Linear Elastodynamics......................................................................................

9

2.1 2.2 2.3 2.4 2.5 2.6

Fundamental Boundary-Initial Value Problems in Elastodynamics........... Bulk Waves................................................................................................ Lame´ Solution.............................................................................................. Plane Waves ................................................................................................ Propagation of Discontinuities: Wavefronts and Rays ............................... Two-Dimensional Problems in Elastodynamics ......................................... 2.6.1 Anti-Plane Shear.............................................................................. 2.6.2 Plane Strain...................................................................................... 2.6.3 Plane Stress...................................................................................... 2.7 Surface Waves ............................................................................................. 2.8 Half-Space Green’s Functions..................................................................... 2.9 Lamb’s Problem ..........................................................................................

9 11 12 13 14 15 15 16 17 18 19 23

3. Dynamic Crack Tip Fields................................................................................

27

3.1 3.2

Dynamically Loaded Cracks ....................................................................... Asymptotic Analysis of Crack Tip Fields................................................... 3.2.1 Anti-Plane Shear.............................................................................. 3.2.2 In-Plane Symmetric Deformation ................................................... 3.2.3 In-Plane Antisymmetric Deformation ............................................. 3.3 Asymptotic Analysis for Nonsteady Crack Growth ................................... 3.4 Intersonic Crack Growth .............................................................................

27 30 30 32 37 39 43

4. Determination of Dynamic Stress Intensity Factors......................................

49

4.1

Analysis of Stationary Cracks Under Dynamic Loading ........................... 4.1.1 Semi-Inﬁnite Crack Under Uniform Loading................................. 4.1.2 Semi-Inﬁnite Crack Under a Point Load ........................................

49 49 57

viii

Contents

4.2

Analysis of Moving Crack Problems .......................................................... 4.2.1 The Yoffe Problem .......................................................................... 4.2.2 Dynamic Stress Intensity Factors for Moving Cracks....................

60 60 63

5. Energy Balance and Fracture Criteria ...........................................................

71

5.1 5.2 5.3 5.4 5.5 5.6

Energy Balance Equation ............................................................................ Dynamic Failure Criterion........................................................................... Dynamic Crack Initiation Toughness.......................................................... Dynamic Crack Growth Toughness ............................................................ Dynamic Crack Arrest Toughness .............................................................. Application of Dynamic Failure Criteria ....................................................

71 74 75 75 77 78

6. Methods of Generating Dynamic Loading......................................................

81

6.1 6.2 6.3 6.4 6.5 6.6

Static Loading of Cracks ............................................................................. Drop-Weight Tower and Instrumented Impact Testing.............................. Projectile Impact.......................................................................................... Hopkinson Bar Impact Test ........................................................................ Explosives .................................................................................................... Electromagnetic Loading.............................................................................

81 82 87 91 93 93

7. Measurement of Crack Speed ..........................................................................

97

7.1 7.2 7.3 7.4

Wallner Lines .............................................................................................. Stress Wave Fractography......................................................................... Electrical Resistance Methods .................................................................... High-Speed Photography.............................................................................

8. Crack Tip Stress and Deformation Field Measurement ............................. 8.1 8.2

8.3

8.4

8.5 8.6

Jones Calculus ............................................................................................. Photoelasticity.............................................................................................. 8.2.1 Evaluation of the Dynamic Stress Intensity Factor using Photoelasticity.................................................................................. Method of Caustics...................................................................................... 8.3.1 Physical Principle of Formation of Caustics .................................. 8.3.2 Caustic in Reﬂection ....................................................................... 8.3.3 Mixed-mode Caustics ...................................................................... 8.3.4 Limitations on the Applicability of the Method of Caustics.......... Lateral Shearing Interferometry .................................................................. 8.4.1 Evaluation of the Dynamic Stress Intensity Factor using Shearing Interferometry................................................................... Strain Gages................................................................................................. Interferometry ..............................................................................................

98 100 102 104 107 107 109 112 116 116 121 123 124 128 132 135 138

ix

Contents

9. Dominance of the Asymptotic Field .............................................................. 9.1 9.2 9.3

Stationary Cracks......................................................................................... Propagating Cracks...................................................................................... Dominance of the Asymptotic Field for Propagating Cracks ....................

10. Dynamic Fracture Criteria............................................................................. 10.1

Criteria for Crack Initiation....................................................................... 10.1.1 Initiation of Cracks Under Short Duration Stress Pulses................................................................................. 10.1.2 Loading Rate and Temperature Dependence of Crack Initiation Toughness .................................................................... 10.2 Dynamic Crack Arrest Criterion ............................................................... 10.2.1 Development of the Crack Arrest Criterion................................ 10.2.2 ASTM Standard Method for Crack Arrest ................................. 10.2.3 Application of the Crack Arrest Criterion .................................. 10.3 Dynamic Crack Growth Criterion ............................................................. 10.3.1 Crack Growth Toughness in Nominally Brittle Materials........................................................................... 10.3.2 Crack Growth Toughness in Ductile Materials .......................... 11. Physical Aspects of Dynamic Fracture.......................................................... 11.1 11.2

Limiting Crack Speed................................................................................ Fracture Surface Roughness ...................................................................... 11.2.1 Real-Time Observations of Multiple Crack Fronts .................... 11.2.2 Fast Fracture Surfaces in Polymethylmethacrylate .................... 11.2.3 Origin of the Microcracks ........................................................... 11.2.4 Geometry of the Conic Markings................................................ 11.2.5 Statistics of Microcracks in PMMA............................................ 11.2.6 Growth of Microcracks................................................................ 11.2.7 Solithane 113 ............................................................................... 11.2.8 Polycarbonate............................................................................... 11.2.9 Homalite-100 ............................................................................... 11.2.10 Other Brittle Materials............................................................... 11.3 Crack Branching ........................................................................................ 12. Phenomenological Models of Dynamic Fracture ......................................... 12.1 12.2 12.3

Discrete Models—Molecular Dynamics and Lattice Models .................. Cohesive Zone Models .............................................................................. Continuum Damage Models......................................................................

141 141 144 146 155 155 157 159 168 168 173 175 177 178 184 189 190 193 195 196 200 201 203 204 204 204 207 207 208 217 217 220 224

x

Contents

References................................................................................................................

229

Further Reading ..................................................................................................

237

Appendix A. Dynamic Crack Tip Asymptotic Fields ........................................

239

A1 A2 A3 A4

Dynamic Crack Tip Stress Field for a Stationary Crack.......................... Steady-State Dynamic Crack Tip Stress Field: Singular Term ................ Steady-State Crack Tip Displacement and Stress Field: N Terms ........... Transient Crack Tip Displacement and Stress Field: Six Terms .............

239 239 241 243

Appendix B. Mechanical and Optical Properties of Selected Materials .........

245

Index.........................................................................................................................

247

1

Chapter 1 Introduction

Rapidly applied loads are encountered in a number of applications. In some cases such loads might be applied deliberately, as for example in problems of blasting, mining, and comminution or fragmentation; in other cases, such dynamic loads might arise from accidental conditions. Regardless of the origin of the rapid loading, it is necessary to understand the mechanisms and mechanics of fracture under dynamic loading conditions in order to design suitable procedures for assessing the susceptibility to fracture. Quite apart from its repercussions in the area of structural integrity, fundamental scientiﬁc curiosity has continued to play a large role in engendering interest in dynamic fracture problems. At the outset, it is essential to identify the range of loading rates in which the dynamic analysis based on wave propagation is important. We illustrate this with a simple example: consider a single-edge notched specimen illustrated in Fig. 1.1. Let the load be applied by a tup, falling at a speed v and impacting on the specimen. The impact generates a stress wave that travels into the specimen, interacts with the crack and reﬂects from the far end. The time scale of the ﬁrst interaction is w=Cd ; where w is the depth of the specimen and Cd is the longitudinal wave speed of the material. If the impact speed v is sufﬁciently high, it is possible to initiate the fracture event before the arrival of the stress waves at the supporting posts—i.e. at times t , l=Cd and before the wave reﬂection from the bottom travels back up to the top t , 2w=Cd : This is a truly transient dynamic condition and requires a full elastodynamic analysis of the problem. If the specimen does not break within this time scale, then the stress waves subsequently reﬂect between the top and bottom surfaces of the specimen and eventually put the beam into a vibratory motion at a frequency that corresponds to the natural frequency of the beam on its supports; note that this may occur only after several wave passes. In many impact tests, the measurements clearly indicate the arrival of different reﬂections. In cases where a steady vibratory state has been reached, it might be sufﬁcient to perform quasi-static analysis, where the time-dependent load at the tup is measured and used in the static calculation of the stress state at the crack tip. At long times, these oscillations decay by transmission of the waves into the supports at the bottom end of the specimen and the full weight of the tup is held by the equilibrium bending stresses in the beam. The last stage corresponds to the quasi-static condition and is the realm of elastostatics. Of course, in practice, whether these regimes are exhibited or not

2

Chapter 1

Figure 1.1 Geometry of an impact test conﬁguration.

depends on the nature of the specimen and the material properties. There has been a lively debate in the literature, especially with respect to the pressurized thermal shock problem discussed in Section 1.1 about the suitability of a static analysis or the necessity for using a fully dynamic analysis. The interest in this book is on problems that fall primarily into the ﬁrst group—fully elastodynamic problems. There are a number of examples of engineering structures that are subjected to dynamic loads and require a fully dynamic analysis. Here we describe a few examples to understand the topics considered in this book.

1.1 Pressurized Thermal Shock in Nuclear Containment Vessels One of the major problems motivating investigations of dynamic fracture is in pressurized water reactors (PWR), where the containment vessels may be subjected to signiﬁcant thermal shock loading during a loss of coolant accident (LOCA). These pressure vessels are assumed to contain crack-like ﬂaws that result from various processing, handling and use conditions such as welding, load cycling, and stresscorrosion. The initial design of the pressure vessel according to the prevailing standards and codes usually accounts for the presence of such ﬂaws. In case of a LOCA, a failure in the primary cooling system is compensated by emergency core coolant introduced into the inner wall of the vessel which drops the temperature of the hot vessel wall by more than 2508C; the vessel must be capable of sustaining this thermal shock (Cheverton et al., 1981). However, during operation of a PWR, additional concerns arise: ﬁrst, the toughness of the material reduces with time as a result of radiation-induced damage, particularly along the inner wall. Second, in a LOCA, the steep thermal gradient generates large stresses, sufﬁcient to cause growth of ﬂaws present along the inner wall. Third, it is also possible that the thermal transient occurs while the pressure loading has not yet been released. Safe design of the vessel requires that this LOCA is a survivable incident, without propagation of a crack through the wall thickness. While initial designs were

Introduction

3

based on quasi-static linear elastic fracture mechanics analysis and material characterization, experiments have indicated that the event is truly dynamic and that multiple crack initiation, rapid crack propagation and crack arrest events can occur during the course of a single thermal shock event. Experiments have been conducted on thick-walled cylinders under thermal shock conditions (Cheverton et al., 1981) as well as in wide plates simulating the thermal shock in the pressure vessels (Pugh et al., 1988). Analyses of the wide-plate test results using quasi-static and dynamic fracture methodology were provided by Jung and Kanninen (1983), Bass et al. (1985), Brickstad and Nilsson (1986a,b), and Pugh et al. (1988). While there has been much debate over whether quasi-static analyses are conservative or not, consensus was reached on the need for the determination of the rate and temperature-dependent crack initiation and crack arrest properties.

1.2 Boiler and Pipeline Burst Problems Pipelines transport oil and gas under high-pressure conditions over long distances. Millions of miles of such transmission and distribution pipelines exist around the world. Loss of revenue due to accidental rupture of these pipelines is in the order of tens of millions of dollars per year; in some instances, these accidents have led to loss of life as well. In the United States of America, according to the data collected by the Department of Transportation, in the last 25 years of the 20th century, such losses total about a billion dollars in revenue; in addition there have been too many fatalities. Pressurized boilers and pipelines are designed according to applicable codes of the American Society of Mechanical Engineers (ASME), American Petroleum Institute (API), American Gas Association (AGA), etc.; the failures observed in service are generally not due to poor or improper design, but mainly due to other causes: ﬁrst, buried pipelines are ruptured due to incidental damage induced by digging or other operations in the vicinity of the pipeline; reports compiled by the US National Transportation Safety Board indicate that work crew installing utility lines for electric, cable and/or water supply near a gas line sometimes inadvertently damage gas distribution pipelines. Second, corrosion damage may have accumulated over time to the extent that the residual strength of the pipe falls below the operating stress levels; this is exacerbated by the fact that much of the infrastructure is aging and hence quite susceptible to failure. Repeated pressurization cycles experienced by the pipelines as they are pumped with different ﬂuids for transmission along their length also induces fatigue cracks to grow to critical dimensions in some pipelines. Fractures or ruptures that initiate under such conditions may propagate at speeds that are a substantial fraction of the wave speed in the solid; cracks growing at speeds order of 500 m/s have been observed in steel pipes (Ives et al., 1974). Depressurization of the pipe as a result of escaping gases travels along the length of the pipe at the speed of sound in the medium, typically on the order of 300 m/s. Hence, the dynamically growing crack outruns the depressurization and the driving force for the crack is maintained over long distances. The loading conditions on these pipelines are quite complex; in addition to the internal pressure, the backﬁll provides an external loading. Furthermore, the leakage of the contents through the opening produced by the propagating fracture results in a decrease of loading that must be estimated through a coupled ﬂuid –structure interaction problem.

4

Chapter 1

A complete analysis of this problem requires consideration of the dynamics of all these interactions and has been attempted only under special conditions. The two main approaches to the problem have been empirical in nature; both rely upon burst tests and correlations between the burst energy and the Charpy impact energy.1 The ASME Boiler and Pressure Vessel Code imposes a requirement of a minimum impact energy for the material and circumvents analysis of dynamic fracture. On the other hand, for gas pipeline applications, for example, the following empirical expression has been used to determine susceptibility to failure, once again correlated to experimental database (O’Donoghue et al., 1997): 2 Rhsh2 ðCv Þmin ¼ 2:52 £ 1024 Rsh þ 1:245 £ 1025 3 d 2 28 R d 2 0:627h 2 6:8 £ 10 h

ð1:1Þ

where ðCv Þmin is the Charpy energy in joules, h the wall thickness, R the pipe radius, d the depth of the backﬁll, all in millimeters, and sh is the hoop stress in MPa. O’Donoghue et al. (1997) also show that expressions such as Eq. 1.1 are not useful outside the range of data used to obtain such empirical relationships and could be signiﬁcantly nonconservative, pointing to the need for a fracture mechanics-based analysis even though this analysis is quite expensive. In spite of some lingering uncertainties in the dynamic fracture theory described in this book, it is capable of providing a good engineering estimate of rapid crack growth and arrest in such pressure vessel and pipeline applications.

1.3 Dynamic Fracture in Airplane Structures Another class of problems that provide motivation for studies of dynamic loading arises in the evaluation of the integrity of airplane structures. Fuselages of airplanes are pressure vessels, designed to maintain a cabin pressure at a higher level than the ambient pressure, at the cruising altitude. Fatigue cracks emanate from regions of stress concentration and grow with repeated pressurization cycles associated with take-off and landing cycles. While the early examples of such fatigue cracks, such as the de Havilland Comet aircraft at the dawn of commercial jet aviation, resulted in catastrophic disintegration of the aircraft structure, structural design concepts have evolved to limit fatigue crack extension during the design lifetime of the structure. Recent accidents, such as the Aloha Airlines Boeing 737 aircraft incident in 1988, demonstrated that small cracks emanating from neighboring rivet holes can interact with each other and critical lengths sufﬁcient to trigger dynamic crack growth can be reached. This phenomenon is referred to as multi-site damage. Growth of such cracks is similar to the problem of dynamic crack growth in the pipeline described above; at high crack speeds, the decompression may not appear rapidly enough to arrest the crack. Crack arrest or deﬂection methodologies based on the concept of tear straps are used here to limit the extent of crack growth; some of these concepts are 1

The Charpy test and the impact energy are described further in Section 1.4.

Introduction

5

discussed in Chapter 10. Once again, as in pipeline applications, such tear straps are currently designed based on empiricism relying on scale model and full-scale tests on actual structures rather than predictive models grounded in fracture mechanics. There has been some recent progress in evaluating the tear straps using principles of dynamic fracture mechanics (see Kosai and Kobayashi, 1991). Additional motivation for analysis based on dynamic fracture mechanics is provided by examining the response of aircraft to blast. The terrorist attack on Pan Am Boeing 747 over Lockerbie, Scotland in 1988 resulted in signiﬁcant loss of life; reconstruction of the damage to the aircraft indicated that the initial blast from the explosive charge resulted in small structural damage, but a large build-up of pressure in the fuselage, and further that the disintegration of the airplane resulted not from the initial blast, but due to dynamic growth of cracks triggered from the blast site by the high pressure still contained within in the fuselage (UK Air Accidents Investigation Branch, 1990). Fig. 1.2 shows the pattern of damage sustained by the aircraft; the initial blast resulted in the star-burst and petaling pattern marked as Region A. Major cracks, labeled Fractures 1– 3 in the ﬁgure emanated from the star-burst and propagated along the length of the fuselage, not only from the blast over-pressure, but also likely due to cabin pressure. The turning of these cracks at the tearstraps is seen in Regions B, C, D and E. The investigators of the disintegration of PanAm 103 recommended that the survivability of aircraft to such small scale blasts may be increased by suitable design of the fuselage panels to inhibit rapid crack propagation over long lengths and by provision of panels speciﬁcally designed to break-off and vent the blast products and decrease the pressure quickly. Dynamic analysis of the fracture phenomena and appropriate characterization of material properties are essential for

Figure 1.2 Schematic diagram of the fracture patterns on the PanAm 103 747 airplane that was destroyed over Lockerbie, Scotland by a terrorist attack. Region A identiﬁes the initial blast damage; Fractures 1– 3 are thought to have propagated as a result of the service pressure inside the cabin. Regions B– E illustrate the role of tear-straps in deﬂecting the cracks. (Reproduced from Aircraft Accident Report No 2/90 (EW/C 1094), UK Air Accidents Investigation Branch.)

6

Chapter 1

successful incorporation of these ideas in design practice. Therefore, recent concerns of aircraft structural integrity motivated by the aging of the commercial and military aircraft ﬂeet and by terrorist threats have focused some attention on the analysis of dynamic crack growth and arrest in airplanes.

1.4 Notched Bar Impact Testing of Metallic Materials Impact resistance of materials has long been evaluated beginning with the Charpy test (Charpy, 1901). The data obtained from this and other similar tests are qualitative at best and not suitable for predictive analysis based on fracture mechanics. On the other hand, the test is quite easily performed and therefore quite useful for comparative ranking of different materials and in selection of materials for speciﬁc use. The ASTM E-23 standard presents a method for the determination of the energy absorbed during impact fracture in metallic materials. The basic idea behind the test is the following: a V-notched beam specimen is supported on an anvil and impacted by a mass moving with a sufﬁcient energy; for a test to be valid under the standard, the impact speed must be in the range of 3 –6 m/s. From a measure of the initial and ﬁnal value of the potential energy of the mass, the energy absorbed in the fracture event, called the Charpy V-notched energy, Cv ; is determined. Fig. 1.3 shows the temperature dependence of Cv typical of low-strength ferrous alloys. In these alloys Cv exhibits a remarkable transition from a brittle behavior at low temperatures (called the lower shelf) to a ductile behavior at high temperatures (called the upper shelf). The temperature at which the failure mechanism changes from brittle to ductile is called the nil-ductility temperature (NDT). Many variants of the Charpy test have been proposed; within the ASTM E-23 standard, there are other possible conﬁgurations for the impact test. The drop weight tear test (ASTM E-466), and the dynamic tear test (ASTM E-604, ASTM E-208) are other tests aimed at providing qualitative information regarding the nature of the fracture (brittle or ductile) and an estimate of the transition temperature. Since the Charpy energy Cv cannot be used in a predictive mode, design codes use the energy and the transition temperature in setting materials’ speciﬁcations for applications. For example, the ASME Boiler and Pressure Vessel Code Section VIII, Division 3

Figure 1.3 Variation of the Charpy V-notch energy with temperature for typical low-strength ferrous alloys. An abrupt transition to a high-energy ductile fracture mode appears at temperatures above the brittle – ductile transition temperature.

7

Introduction

speciﬁes the minimum required Charpy energy Cv at the minimum design metal temperature (MDMT). Empirical correlations of the Charpy V-notch energy to plane strain fracture toughness have been proposed in the brittle fracture region of the Charpy test. For example, the ASME Boiler and Pressure Vessel Code provides the following correlation when the upper shelf regime is above MDMT

KIC sY

2

C ¼ 5 v 2 0:05 sY

ð1:2Þ

where Cv is in ft lbf (1 ft lbf ¼ 1.35582 J), the yield p stress, sY ; ispgiven in ksi and the plane p strain fracture toughness KIC is given in ksi in (1 ksi in ¼ 6.89475 MPa m). According to the code, the value of the fracture toughness estimated using Eq. 1.2 can be used in the evaluation of fracture criticality under slow-loading KIC conditions. It should be noted that this approach does not take into account the possible rate dependence of crack initiation, growth and possible arrest of the cracks. Regardless of the attempts to tie the Charpy test results to quantitative fracture mechanics parameters, this class of fracture parameters remains empirical; the utility of these methods and results lies in correlating with past experience, and in qualitative ranking of different materials, and not in providing a predictive methodology for the evaluation of the integrity of structures containing cracks and loaded dynamically. The latter is clearly in the realm of dynamic fracture mechanics. The above discussion presents a sampling of the problems in which dynamic fracture plays a key role. These and other practical applications have driven the research into dynamic fracture problems. In this book, fundamental mechanics aspects of dynamic fracture are presented. In the ﬁrst segment, the basic concepts from the continuum theory of dynamic fracture are presented; in Chapter 2, a review of linear elastodynamics is presented. This is followed in Chapter 3 by a description of the stress ﬁeld in the vicinity of the crack tip and the idea of the dynamic stress intensity factor is discussed. A concise description of the analytical determination of the dynamic stress intensity factor is provided in Chapter 4. This is followed in Chapter 5 by a consideration of the dynamic energy rate balance criterion for the formulation of dynamic fracture criteria. In the second segment, experimental and practical aspects of dynamic fracture are addressed. Methods of generating wellcharacterized dynamic loads for investigations of dynamic fracture phenomena are described in Chapter 6. This is followed by a detailed exposition of the methods of measuring crack speeds in Chapters 7 and of the diagnostic tools for characterization of the crack tip stress and or deformation ﬁelds in Chapter 8. The applicability of the concept of the dynamic stress intensity factor in characterizing dynamic fracture is discussed in Chapter 9. Experimental investigations aimed at formulating dynamic fracture criteria—separated into criteria for crack initiation, growth and arrest—are discussed in Chapter 10; practical applications of these ideas in assessment of structural integrity are developed as well. In the last segment, the physical aspects and models of dynamic fracture are discussed. Chapter 11 is devoted to a discussion of the mechanisms of fracture in different materials; discrepancies between the continuum theory and its mechanistic resolution are described. Various models developed for incorporation of the effects of the fracture process zone in the simulation of dynamic fracture are discussed in Chapter 12.

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9

Chapter 2 Linear Elastodynamics

2.1 Fundamental Boundary-Initial Value Problems in Elastodynamics We begin with a brief description of the linear elastodynamic theory. Complete treatments of the topic including solution techniques and details of the classical solutions can be found in the monographs by Graff (1975), Achenbach (1973) and Miklowitz (1978). Consider a body occupying the region R with boundaries ›R: Let the displacement vector u depend on the position vector x and time t and be denoted by uðx; tÞ. The strain tensor 1(x,t) is the symmetric gradient of u 1ðx; tÞ ¼

1 ½7u þ ð7uÞT 2

ð2:1Þ

Here we consider the inﬁnitesimal strain tensor and therefore neglect higher order terms involving higher powers of the gradient of u. It is also assumed that u is a continuous function of x, but it is not required that its derivatives with respect to x and t be continuous as we shall see later. The material of the body is assumed to be homogeneous, isotropic and linearly elastic. Therefore, the stress tensor s(x,t) is given by1 sðx; tÞ ¼ l1kk 1 þ 2m1

ð2:2Þ

where 1 is the identity tensor and l and m are the Lame´ constants. The balance of linear momentum results in the following equation of motion 7·s þ f ¼ ru€

ð2:3Þ

where r is the mass density, f the body force per unit volume and the superdot indicates time derivatives. Symmetry of the stress tensor ensures the balance of angular momentum. Substituting Eq. 2.1 in Eq. 2.2 and the result in Eq. 2.3, the equations of motion can be obtained in terms of the displacements alone; these are the Navier’s equations of motion ðl þ mÞ7ð7·uÞ þ m72 u þ f ¼ ru€ 1

ð2:4Þ

Standard index notation will be used throughout this book. Latin subscripts take the range 1,2,3 while Greek subscripts take the range 1,2. Repeated index implies summation over the range of the index and an index following a comma indicates partial differentiation with respect to the coordinate identiﬁed by that index.

10

Chapter 2

This is a system of three partial differential equations governing the motion of points in the body. In this book we shall assume that the body forces vanish and remove them from consideration in subsequent equations. To the set of equations 2.4, we must add initial conditions as well as boundary conditions. As in the quasi-static problem, there are three fundamental problems that can be posed, depending on whether the displacements, tractions or some combination are prescribed on the boundaries. For the displacement boundary value problem uðx; tÞ ¼ up ðx; tÞ

ð2:5Þ p

on ›R for t . 0; where u ðx; tÞ is a prescribed function. For the traction boundary value problem, the traction vector, sðx; tÞ; is prescribed sðx; tÞ ¼ sðx; tÞn ¼ sp ðx; tÞ

ð2:6Þ p

on ›R for t . 0; where n is the unit outward normal and s ðx; tÞ a prescribed function. The third problem is the mixed-boundary value problem for which the displacements are prescribed in a part of the boundary, and tractions are prescribed over the remainder uðx; tÞ ¼ up ðx; tÞ on ›1 R sðx; tÞ ¼ sp ðx; tÞ on ›2 R

ð2:7Þ

For all three problems, initial conditions must be added to complete the formulation of the problems uðx; 0Þ ¼ u0 ðxÞ _ 0Þ ¼ u_ 0 ðxÞ uðx;

ð2:8Þ

on R, where u0 ðxÞ and u_ 0 ðxÞ are prescribed functions. It should be evident that even though we have written the governing equations in terms of displacements, the boundary conditions may be in terms of tractions, i.e. in terms of linear combinations of the derivatives of the displacement components and therefore complicating the solution of the problem. Finally, the principle of conservation of energy (or the theorem of power expended) may be written as ð ð ›u ›u d ½UðtÞ þ TðtÞ ð2:9Þ s· dR þ rf· dV ¼ dt › t › t ›R R where UðtÞ is the strain energy and TðtÞ the kinetic energy given by ð 1 UðtÞ ¼ s·1dV R 2 ð 1 _ udV _ TðtÞ ¼ ru· R 2

ð2:10Þ ð2:11Þ

For problems in classical elastodynamics the conservation of energy provides a convenient way of approaching solutions; however, since there are no dissipative processes, it is seldom necessary to introduce the energy conservation equations explicitly into the problem formulation. On the other hand, in fracture problems that are the focus of this book, dissipation is inherent in the problem. The fracture processes that occur in

Linear Elastodynamics

11

the crack tip region remove energy from the system and hence, we have to augment Eq. 2.9 to account for the dissipation that occurs in the fracture process regions as we shall discuss later. Boundary-initial value problems posed within the context of linear elastodynamics above possess unique solutions (see Wheeler and Sternberg, 1968).

2.2 Bulk Waves Eq. 2.4 represents a hyperbolic system of partial differential equations and hence admit propagating wave solutions. The character of these waves can be obtained by considering special deformations. The Laplacian of u in Eq. 2.4 can be replaced using the following vector identity 72 u ¼ 7ð7·uÞ 2 7 £ 7 £ u to yield ðl þ 2mÞ7ð7·uÞ 2 m7 £ 7 £ u ¼ ru€

ð2:12Þ

First, if we consider u to be an irrotational deformation, 7 £ u ¼ 0; Eq. 2.4 reduces to sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 l þ 2m 2 ð2:13Þ 7 u ¼ 2 u€ with Cd ¼ r Cd This deformation is seen to obey the standard wave equation with a characteristic speed Cd : Such waves are called irrotational or dilatational waves. Next, if we consider the dilatation to be zero, 7·u ¼ 0; we obtain the case of an equivoluminal deformation. In this case, Eq. 2.4 becomes rﬃﬃﬃﬃﬃﬃ 1 m 72 u ¼ 2 u€ with Cs ¼ ð2:14Þ r Cs Therefore, equivoluminal deformations also obey the standard wave equation but with a characteristic speed Cs : Such waves are called equivoluminal or shear waves. Clearly Cd . Cs ; an observer at some distance from a source (such as an earthquake) of these waves will ﬁrst receive the dilatational wave and then the equivoluminal wave; hence in the seismology literature these waves are called primary (P) waves and secondary (S) waves. While the wave speeds are expressed here in terms of the Lame´ constants, materials are usually characterized in terms of the engineering constants E and n, the modulus of elasticity and Poisson’s ratio, respectively. Conversion between these constants can be effected using the following relationships E¼

mð3l þ 2mÞ l ;n¼ lþm 2ðl þ mÞ

ð2:15Þ

Now, the wave speeds may be expressed in terms of E and n, but more importantly, the ratio of wave speeds is seen to depend only on the Poisson’s ratio Cd 2 2 2n 1=2 ¼ ;k ð2:16Þ Cs 1 2 2n Representative values of material properties are given in Table 2.1; these values are based on nominal values of modulus of elasticity, Poisson’s ratio and density in order to provide

12

Chapter 2 Table 2.1 Wave speeds in solids Modulus Poisson’s Density, Dilatational Distortional Plane stress Rayleigh of elasticity, ratio, n r (Mg/m3) wave speed, wave speed, dilatational wave speed, E (GPa) Cs (m/s) wave speed, CR (m/s) Cd (m/s) Cdp (m/s)

Material

High strength steel Tungsten and alloys Aluminum alloys Alumina Silicon nitride Silica glass Homalite-100 Plexiglas Polycarbonate Rubber

200 406 70 390 350 70 4.5 3.4 2.6 0.1 0.01

0.3 0.3 0.3 0.22 0.22 0.22 0.34 0.34 0.40 0.499 0.499

7.8 13.4 2.7 3.9 3.2 2.6 1.2 1.2 1.2 0.85 0.85

5875 6386 5908 10,685 11,175 5544 2402 2088 1826 4434 1402

3140 3414 3158 6402 6695 3322 1183 1028 899 198 63

5308 5770 5338 10,251 10,721 5319 2059 1790 1565 396 125

2913 3167 2929 5858 6127 3040 1104 960 839 189 60

an idea about the order of magnitude of the wave speeds. As can be seen from the values in the table, the shear wave speeds are typically about one-half of the dilatational wave speeds.

2.3 Lame´ Solution Since the problem considered here is linear, Eqs. 2.13 and 2.14 suggest that propagation of an arbitrary deformation that is a combination of dilatation and shear will be governed by both types of waves. This can be shown directly using the Lame´ solution of the displacement equations of motion (Eq. 2.4). Consider the following representation of the displacement vector u u ¼ 7w þ 7 £ c

ð2:17Þ

where wðx; tÞ is a scalar function and cðx; tÞ a vector-valued function with 7·c ¼ 0: The displacement components obtained from Eq. 2.17 will satisfy the differential equations (Eq. 2.4) if wðx; tÞ and cðx; tÞ are obtained as solutions of the following wave equations 72 w ¼

1 w€ Cd2

ð2:18Þ

72 c ¼

1 € c Cs2

ð2:19Þ

The scalar potential wðx; tÞ corresponds to the dilatational wave and the vector potential cðx; tÞ corresponds to shear waves. The completeness of the Lame´ decomposition of the displacement vector has been demonstrated by Clebsch, Somigliana and others. Sternberg (1960) provides a discussion of this decomposition.

Linear Elastodynamics

13

It should be noted that the Lame´ solution is really a reduction of the complicated hyperbolic system of equations for the displacement vector into two standard wave equations for the potentials wðx; tÞ and cðx; tÞ coupled through the boundary conditions. As Miklowitz points out, the advantage of reformulating Navier’s equations in terms of the Lame´ potentials is that solutions and solution procedures developed for the standard wave equation can now be used to address problems associated with elastic wave propagation in solids.

2.4 Plane Waves In order to gain insight into the wave character of the dynamic problem, consider the propagation of plane waves in a three-dimensional solid medium. A plane is deﬁned by x·n ¼ d where x represents the position vector of any point in the plane, n the normal to the plane and d the distance from the origin to the plane along the normal. If the plane is assumed to move in the direction of the normal at a wave speed c, then d ¼ d0 þ ct; where d0 is the location of the plane at time t ¼ 0; describes the propagation of the plane. Clearly, as the wave propagates, d0 remains constant and is called the phase; the surface with constant phase (in this case the plane) is the wavefront. Now, applying this idea to elastodynamics, plane waves corresponding to dilatational and shear deformations can be represented as

w ¼ wðx·n 2 Cd tÞ

ð2:20Þ

c ¼ cðx·n 2 Cs tÞ

ð2:21Þ

It is easily demonstrated by substitution that if wðx; tÞ and cðx; tÞ are represented as above, they automatically satisfy the wave equations 2.18 and 2.19, respectively. To examine the plane waves further, without loss of generality, the direction of propagation can be taken to be the x1 axis; then using Eqs. 2.20 and 2.21 in Eq. 2.17, we obtain the displacement components u1 ¼ w0 ðx1 2 Cd tÞ u2 ¼ 2c03 ðx1 2 Cs tÞ u3 ¼

c02 ðx1

ð2:22Þ

2 Cs tÞ

where the prime denotes differentiation with respect to the argument. The dilatational wave travels at the speed Cd and can only sustain a displacement u1 in the direction of wave propagation, x1; hence this is a longitudinal wave with the particle motion in the direction of the wave propagation. This is the P wave or dilatational; note that it could be a compressive or tensile wave depending on whether the particle motion is in the direction of motion or opposed to it. The shear wave travels at the speed Cs and can sustain displacement components u2 and u3, i.e. in the directions perpendicular to the wave propagation; hence these are transverse waves. As a consequence of our resolution of the vector along the Cartesian coordinates, the shear wave has been decomposed into two components, with particle motions in x2 and x3. These are commonly called the SH arid SV waves (for the horizontally polarized and vertically polarized shear waves where the x3 axis points in the vertical direction).

14

Chapter 2

2.5 Propagation of Discontinuities: Wavefronts and Rays The displacement vector uðx; tÞ need not possess continuous derivatives; the governing equations 2.4 allow discontinuities in the derivatives of uðx; tÞ to exist along certain planes (called wavefronts) and propagate along certain directions (called rays). Discontinuities in the spatial gradients of uðx; tÞ imply a discontinuity in the strains and stresses, and discontinuities in the temporal gradients of uðx; tÞ indicate jumps in the particle velocity and/or acceleration. While such discontinuities cannot be sustained physically, rapid changes in uðx; tÞ that occur over very short distances or time intervals are approximated as discontinuous jumps in the gradients. This representation is useful in characterizing the variations in the strains, stresses and velocities generated by suddenly applied loads. Love (1927) described the kinematic and dynamic conditions that must hold on a surface of discontinuity. With the normal to the discontinuity denoted by n, the kinematic and dynamic jump conditions are ½_ui ¼ 2c½ui;j nj

ð2:23Þ

2rc½_ui ¼ ½sij nj

ð2:24Þ

where r is the density, c the appropriate wave speed and the square bracket around a quantity indicates the jump in that quantity across the discontinuity. Eqs. 2.23 and 2.24 can be interpreted using the equations of motion 2.4. Introducing Eq. 2.4 in Eq. 2.24 yields

rc½_ui ¼ 2ldij ½uk;k nj 2 m½ui;j nj 2 m½uj;i nj

ð2:25Þ

which may be rearranged as follows: ðrc2 2 mÞ½_ui ¼ 2cðl þ mÞ½uk;k nj

ð2:26Þ

If we impose a velocity jump ½_ui with zero dilatation, ½uk;k ¼ 0; the jump propagates at a speed c ¼ Cs : In a similar manner, if we consider a velocity jump ½_ui with 7 £ u ¼ 0; Eq. 2.25 reduces to ðrc2 2 ðl þ 2mÞÞ½_ui ni ¼ 0

ð2:27Þ

which indicates that jumps in dilatation travel with the speed c ¼ Cd : Therefore, we might expect that if an arbitrary velocity jump is provided (through an external loading agent or from an internal source), both dilatational and shear waves propagate in the body carrying the appropriate jump discontinuities along both wavefronts. The construction of wavefronts and rays is useful in understanding and interpreting the development of stress ﬁelds in elastodynamic problems. So, we shall brieﬂy outline the construction of the equations for the wavefronts and rays. The surface of discontinuity may be written as Sðx; tÞ ¼ tðxÞ 2 t ¼ 0 or equivalently by t ¼ tðxÞ: At any point on the wave front, the ray is normal to the wavefront; thus, the governing equation for the rays is obtained dx 7tðxÞ ¼ cn ¼ c dt l7tðxÞl

ð2:28Þ

Linear Elastodynamics

15

But, dS ›tðxÞ ›xi ¼ 2 1 ¼ 0 or c7tðxÞ·n ¼ 1 dt ›x i ›t Using the second expression in Eq. 2.28 and the above, we get the equation for the ray as cl7tðxÞl ¼ 1

ð2:29Þ

This is called the eikonal equation, the terminology arising from geometrical optics. Using Eq. 2.29 in Eq. 2.28 results in the following equation for the rays dx ¼ c2 7tðxÞ dt

ð2:30Þ

In the linearly elastic solid, the wave speeds are constant and therefore the rays are straight lines and the corresponding wavefronts are parallel surfaces. Therefore, the knowledge of the wavefront at some time t can be used to construct the wavefront at a later time t0 simply by extending the rays along the normal to the wavefront by an amount cðt0 2 tÞ: In the optics literature, this is called Huygens’ principle and Huygens’ construction of wavefronts. This construction is quite useful in obtaining a quick, qualitative picture of the wave propagation event as we shall see in later examples.

2.6 Two-Dimensional Problems in Elastodynamics So far, the three-dimensional elastodynamic problem has been discussed. However, only two-dimensional dynamic fracture problems are considered in this book. Therefore, the governing equations will now be reduced to the case of two dimensions for three special cases: anti-plane shear, plane strain and plane stress. The ﬁrst two reductions are based on restrictions on the deformation while the third is based on an assumption regarding the stress tensor. 2.6.1 Anti-Plane Shear It is assumed that the only nonzero displacement component is in the x3 direction and further that it is a function only of x1 and x2 ua ¼ 0; u3 ¼ u3 ðx1 ; x2 ; tÞ

ð2:31Þ

Substituting in the governing equations 2.4 and setting the body forces to zero results in the following wave equation for u3 72 u3 ¼

1 u€ 3 Cs2

ð2:32Þ

The corresponding nonzero components of the stress tensor are given by

s3a ¼ mu3;a

ð2:33Þ

16

Chapter 2

Clearly, only shear waves arise in this problem; these are the horizontally polarized shear waves. The wave motion is in the x1 – x2 plane with the transverse particle motion directed towards x3. Note that the reduction in the equations of motion can also be effected at the level of the Lame´ potentials, but this is not really necessary since the equation in terms of the nonzero displacement component is already quite simple. 2.6.2 Plane Strain It is assumed that u3 is constant or linear in x3 and further that the remaining components are independent of x3 ua ¼ ua ðx1 ; x2 ; tÞ; u3 / x3

ð2:34Þ

As a consequence, the strain displacement relations in Eq. 2.1 yield 1ab ðx1 ; x2 ; tÞ ¼

1 ðu þ ub;a Þ; 133 ¼ const; 13a ¼ 0 2 a;b

ð2:35Þ

The stress – strain relations in Eq. 2.2 reduce to

sab ðx1 ; x2 ; tÞ ¼ l1gg dab þ 2m1ab s33 ðx1 ; x2 ; tÞ ¼ nsgg s3a ðx1 ; x2 ; tÞ ¼ 0

ð2:36Þ

It is simpler in this case to use the Lame´ solution; corresponding to the above assumption, we have

w ¼ wðx1 ; x2 ; tÞ and c ¼ cðx1 ; x2 ; tÞe3

ð2:37Þ

where e3 is the unit vector in the x3 direction. Therefore we have two scalar potentials in the plane strain problem that satisfy 72 w ¼

1 w€ Cd2

ð2:38Þ

72 c ¼

1 € c Cs2

ð2:39Þ

where 72 is the two-dimensional Laplacian operator. From these equations, it is clear that under the conditions of plane strain, there are still dilatational and shear waves that propagate with speeds Cd and Cs, respectively, as in the full three-dimensional problem. At this point, we record the relationship between the potential and the nonzero displacement and stress components

›w ›c þ ›x 1 ›x 2 ›w ›c u2 ¼ 2 ›x 2 ›x 1 u1 ¼

ð2:40Þ

Linear Elastodynamics

17

and

s11 s22 s12

2 ›w ›2 c ¼ l7 w þ 2m þ ›x1 ›x2 ›x21 2 ›w ›2 c 2 ¼ l7 w þ 2m 2 ›x1 ›x2 ›x22 2 2 ›w › c ›2 c ¼m 2 þ 2 þ 2 ›x 1 ›x 2 ›x 2 ›x 1 2

ð2:41Þ

2.6.3 Plane Stress In the last of the plane problems, the following assumption is imposed on the stress components

s3i ¼ 0 sab ¼ sab ðx1 ; x2 ; tÞ

ð2:42Þ

Introducing the above assumptions into Eq. 2.4 results in the following displacement equations for plane stress 4mðl þ mÞ 7ð7·uÞ 2 m7 £ 7 £ u ¼ ru€ ðl þ 2mÞ

ð2:43Þ

Comparing with Eq. 2.12, it can be shown that there are once again two waves as in the plane strain and three-dimensional cases: a dilatational wave, with a speed Cdp (the superscript ‘p’ is for the plane stress dilatational wave speed) and a shear wave with a speed Cs. The plane stress dilatational wave speed is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4mðl þ mÞ E p ¼ ð2:44Þ Cd ¼ rðl þ 2mÞ rð1 2 n2 Þ For problems involving thin plates, Cdp is the dilatational wave speed appropriate for plane stress conditions; Table 2.1 shows that this speed is slightly smaller than the bulk dilatational wave speed. Apart from this, the plane stress problem with the assumptions imposed in Eqs. 2.42 is indistinguishable from the plane strain problem. Of course, the strain component 133 is given in terms of the in-plane components of stress 13 a ¼ 0 133 ¼ 2

nsaa E

ð2:45Þ

Lateral inertia in the x3 direction has been ignored; this analysis is appropriate when the wavelengths of the disturbances are long in comparison to the plate thickness.

18

Chapter 2

2.7 Surface Waves So far we have considered bulk waves in the medium. Rayleigh discovered that when free surfaces are present a wave that travels along the surface and decays into the medium is generated. Consider the medium occupying the region x2 . 0; with a traction free boundary at x2 ¼ 0: A surface wave propagating along the x1 direction with a speed c and decaying in the x2 direction can be represented as

w ¼ f ðx2 Þexp{ikðx1 2 ctÞ} ð2:46Þ c ¼ gðx2 Þexp{ikðx1 2 ctÞ} pﬃﬃﬃﬃﬃﬃﬃ where i ¼ 21; k ¼ v=c is the wave number and v is the frequency of a time harmonic disturbance. Substituting this in the wave equations (Eqs. 2.38 and 2.39) results in two ordinary differential equations for f and g d2 f 2 k2 a2d f ¼ 0 dx22 d2 g 2 k2 a2s g ¼ 0 dx22

ð2:47Þ

where sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c2 c2 ad ¼ 1 2 2 and as ¼ 1 2 2 Cd Cs

ð2:48Þ

Solving these equations and retaining only the solution that decays as x2 ! 1; the solution becomes

w ¼ A expð2ad kx2 Þexp{ikðx1 2 ctÞ} c ¼ B expð2as kx2 Þexp{ikðx1 2 ctÞ}

ð2:49Þ

The stress components corresponding to this can be determined from Eqs. 2.41. Applying the traction free boundary conditions at x2 ¼ 0 ð1 þ a2s ÞA þ i2as B ¼ 0 2i2ad A þ ð1 þ a2s ÞB ¼ 0

ð2:50Þ

For a nontrivial solution the determinant of coefﬁcients must be zero RðcÞ ¼ 4ad as 2 ð1 þ a2s Þ2 ¼ 0

ð2:51Þ

The function RðcÞ is called the Rayleigh function and its variation with c is shown in Fig. 2.1. Letting CR =Cs ¼ kR ; where CR is the root of Eq. 2.51 and recalling from Eq. 2.16 that the ratio of the bulk wave speeds, k, depends only on the Poisson’s ratio, the above can be rewritten as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2:52Þ 4 1 2 kR2 =k2 1 2 kR2 2 ð2 2 kR2 Þ2 ¼ 0

Linear Elastodynamics

19

Figure 2.1 Variation of the Rayleigh function RðcÞ with speed.

This is a cubic equation for kR2 ; and depends only on the Poisson’s ratio. The roots of this equation indicate the propagation speed of the surface wave assumed in Eq. 2.46. Physically meaningful solutions to Eq. 2.51 must be in the range 0 , kR , 1: For Poisson’s ratio in the range of 0 , n , 0:5; at least one real solution exists. This solution corresponds to the Rayleigh surface wave. Viktorov (1967) developed an approximate representation for the Rayleigh wave speed, CR kR ¼

CR 0:862 þ 1:14n ¼ Cs 1þn

ð2:53Þ

Clearly, CR , Cs : Table 2.1 lists the values of Rayleigh wave speed for selected materials. Eqs. 2.48 and 2.49 indicate that the Rayleigh wave travels along the surface in the x1 direction, and experiences an exponential decay along the x2 direction. In this chapter, we have presented a formal statement for the fundamental problems and described the nature of propagating waves within this theory; solving boundary value problems under dynamical loading still requires enormous effort. Typically integral transform methods and Green’s function methods are used in obtaining solutions to speciﬁc boundary value problems. For propagating cracks, further complication of moving boundary conditions must be addressed. In the following sections, we shall describe some of the methods used and solutions obtained for a few crack problems.

2.8 Half-Space Green’s Functions We begin by deriving the half-space Green’s function for elastodynamics and then discuss the approaches for solving dynamic crack growth problems. The region of interest is x2 $ 0: The governing differential equations for the potentials w ¼ wðx1 ; x2 ; tÞ and

20

Chapter 2

cðx1 ; x2 ; tÞ are the wave equations in Eqs. 2.38 and 2.39. Transform techniques are employed for obtaining solutions to these equations. The Laplace transform of a function f ðx1 ; x2 ; tÞ is deﬁned as ð1 ^ 1 ; x2 ; sÞ ¼ fðx f ðx1 ; x2 ; tÞe2st dt ð2:54Þ 0

where s is the transform parameter, considered to be real and positive. The bilateral Laplace transform is deﬁned as ð1 ^ 1 ; x2 ; sÞe2szx1 dx1 Fðz; x2 ; sÞ ¼ ð2:55Þ fðx 21

where sz is the transform parameter and z is complex. There are variations in the deﬁnition of the bilateral Laplace transform; Kostrov, e.g. uses a parameter q instead of sz; the use of sz is a matter of convenience since by proper choice of notation s can be eliminated from the equations in some problems as can be observed in the following development. Applying the Laplace transforms indicated in Eq. 2.54 the wave equations are transformed into the following 72 w^ 2 a2 s2 w^ ¼ 0; 72 c^ 2 b2 s2 c^ ¼ 0

ð2:56Þ

where a ¼ 1=Cd and b ¼ 1=Cs are the longitudinal and shear wave slownesses, respectively. Next, the bilateral Laplace transform deﬁned in Eq. 2.55 is applied to yield two ordinary differential equations d2 F d2 C 2 2 2 a ð z Þs F ¼ 0; 2 b2 ðzÞs2 C ¼ 0 dx22 dx22

ð2:57Þ

where Fðz; x2 ; sÞ and Cðz; x2 ; sÞ are the transformed potentials and

að z Þ ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a 2 2 z 2 ; bð z Þ ¼ b 2 2 z 2

ð2:58Þ

In the following, we suppress the arguments of aðzÞ and bðzÞ and simply write a and b. Taking the Laplace transform in time and the bilateral Laplace transform in space, the stress-potential and displacement-potential relations in Eqs. 2.40 and 2.41 yield dC U1 ðz; x2 ; sÞ ¼ szF þ dx2 dF U2 ðz; x2 ; sÞ ¼ 2 szC dx2 2 b b2 d2 F dC 2 2 S22 ðz; x2 ; sÞ ¼ m 2 2 s z F þ 2 2s z dx2 a2 a2 dx22 2 dF dC S12 ðz; x2 ; sÞ ¼ m 2sz þ 2 s2 z2 C dx2 dx22

ð2:59Þ

21

Linear Elastodynamics

The solutions to the ordinary differential equations in Eq. 2.57 are

Fðz; x2 ; sÞ ¼ Aðz; sÞe2sax2 þ A1 ðz; sÞesax2 Cðz; x2 ; sÞ ¼ Bðz; sÞe2sbx2 þ B1 ðz; sÞesbx2

ð2:60Þ

The exponentially growing terms will be unbounded as x2 ! 1 and must therefore be rejected. The next task is to determine Aðz; sÞ and Bðz; sÞ by imposing the transformed boundary conditions. Let us consider that the tractions on the surface of the half-space are given by

s12 ðx1 ; 0þ ; tÞ ¼ s1 ðx1 ; tÞ; s22 ðx1 ; 0þ ; tÞ ¼ s2 ðx1 ; tÞ

for 2 1 , x1 , 1

ð2:61Þ

where s1 ðx1 ; tÞ and s2 ðx1 ; tÞ are prescribed functions. Note that we are considering a halfspace problem and not crack problem; thus the traction components are speciﬁed all along the half-space boundary. Taking the Laplace transform in time and the bilateral Laplace transform in space of the speciﬁed boundary conditions results in

S12 ðz; 0þ ; sÞ ¼ S1 ðz; sÞ S22 ðz; 0þ ; sÞ ¼ S2 ðz; sÞ

ð2:62Þ

Substituting from Eqs. 2.62 and 2.60 in Eq. 2.59 results in the following two equations for the unknown functions Aðz; sÞ and Bðz; sÞ s2 m½ðb2 2 2z2 ÞAðz; sÞ þ 2zbBðz; sÞ ¼ S2 ðz; sÞ s2 m½22zaAðz; sÞ þ ðb2 2 2z2 ÞBðz; sÞ ¼ S1 ðz; sÞ

ð2:63Þ

Solving for the unknowns yields 1 ½ðb2 2 2z2 ÞS2 ðz; sÞ 2 2zbS1 ðz; sÞ s2 mRðzÞ 1 Bðz; sÞ ¼ 2 ½ðb2 2 2z2 ÞS1 ðz; sÞ þ 2zaS2 ðz; sÞ s mRðzÞ Aðz; sÞ ¼

ð2:64Þ

where RðzÞ ¼ ðb2 2 2z2 Þ2 þ 4z2 ab is the Rayleigh function already encountered in Eq. 2.51. With Aðz; sÞ and Bðz; sÞ given above, it is now possible to write formally the general expressions for the stress and displacement ﬁeld components in x2 . 0 in terms of the applied tractions on x2 ¼ 0: Substituting Eq. 2.64 in Eq. 2.60 and then into Eq. 2.59 results in the following expressions for the displacement components 1 ½ze2sax2 ðb2 2 2z2 ÞS2 ðz; sÞsgnðx2 Þ 2 2zbS1 ðz; sÞ msRðzÞ 2 be2sbx2 ðb2 2 2z2 ÞS1 ðz; sÞ þ 2zaS2 ðz; sÞsgnðx2 Þ 1 U2 ðz; x2 ; sÞ ¼ 2 ½ae2sax2 ðb2 2 2z2 ÞS2 ðz; sÞ 2 2z2 bS1 ðz; sÞsgnðx2 Þ msRðzÞ þ ze2sbx2 ðb2 2 2z2 ÞS1 ðz; sÞsgnðx2 Þ þ 2zaS2 ðz; sÞ U1 ðz; x2 ; sÞ ¼

ð2:65Þ

22

Chapter 2

Eq. 2.65 provides the transform of the displacement ﬁelds in terms of the transforms of the applied tractions on x2 ¼ 0: Introduction of the sign function sgnðx2 Þ in Eqs. 2.65 allows the above formulas to be used for both x2 . 0 and x2 , 0: Similar expressions may be written for the stress components as well. Inversion of these transforms poses signiﬁcant challenges and can be accomplished only in some special cases. We specialize the above equations for x2 ¼ 0; with the anticipation that these are needed for solving crack problems; the displacement on the surface x2 ¼ 0 may then be written as U1 ðz; 0; sÞ ¼ G11 ðz; sÞS1 ðz; sÞ þ G12 ðz; sÞS2 ðz; sÞ

ð2:66Þ

U2 ðz; 0; sÞ ¼ G21 ðz; sÞS1 ðz; sÞ þ G22 ðz; sÞS2 ðz; sÞ

ð2:67Þ

where: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b2 2 z 2 G11 ðz; sÞ ¼ 2 msRðzÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b 2 a2 2 z 2 G22 ðz; sÞ ¼ 2 msRðzÞ b2

G12 ðz; sÞ ¼ 2G21 ðz; sÞ ¼

zðb2 2 2z2 2 2abÞ msRðzÞ

ð2:68Þ ð2:69Þ ð2:70Þ

Gab ðz; sÞ are the transforms of the half-space Green’s functions Gab ðx1 ; tÞ: In Eqs. 2.66 and 2.67, the transforms of the displacements are given as the product of two transforms; therefore the displacements must be the double convolution of the original functions ð1 ð1 ½G11 ðj 2 x1 ; t 2 tÞs1 ðj; tÞ þ G12 ðj 2 x1 ; t 2 tÞs2 ðj; tÞdtdj u1 ðx1 ; tÞ ¼ 21

0

; G11 p s1 þ G12 p s2 u2 ðx1 ; tÞ ¼

ð1 ð1 21

0

ð2:71Þ

½G21 ðj 2 x1 ; t 2 tÞs1 ðj; tÞ þ G22 ðj 2 x1 ; t 2 tÞs2 ðj; tÞdtdj

; G21 p s1 þ G22 p s2

ð2:72Þ

where the p stands for the double convolution integral. Thus, for any traction boundary value problem, since s1 ðx1 ; tÞ and s2 ðx1 ; tÞ are prescribed on x2 ¼ 0; 21 , x1 , 1; t . 0; the displacements on x2 ¼ 0 can be found from the convolution integrals in Eqs. 2.71 and 2.72; in fact, similar expressions can be obtained for the displacement components in the interior of the body. Thus, the task is to ﬁnd the inverse transforms of Gab ðz; sÞ corresponding to a unit impulse applied at the origin. This is one of the problems considered by Lamb and the solution is well known. The complete solution of the problem obtained using integral transform methods can be found in the books by Achenbach (1973) and Miklowitz (1978). Slepyan (2002) describes an alternate method for inversion of the Green’s functions Gab ðz; sÞ: Here we look only at the displacement on the free surface for the Lamb problem.

23

Linear Elastodynamics

2.9 Lamb’s Problem Consider a half-plane x2 $ 0 with a point force applied at the origin as follows

s22 ðx1 ; 0; tÞ ¼ 2dðx1 ÞdðtÞ; s12 ðx1 ; 0; tÞ ¼ 0

for 2 1 , x1 , 1

ð2:73Þ

Applying the Laplace transforms to the above boundary conditions and substituting in Eqs. 2.66 and 2.67 results in the following equations for the transforms of the displacement components:

z½b2 2 2z2 2 2zab msRðzÞ 2 b að z Þ U2 ðz; sÞ ¼ msRðzÞ

U1 ðz; sÞ ¼

ð2:74Þ

Here we consider only the u2 component of displacement. Formally, the inverse of the bilateral Laplace transform is written as ð 1 jþi1 b2 aðzÞ szx1 e dz ð2:75Þ u^ 2 ðx1 ; sÞ ¼ 2pi j2i1 mRðzÞ where 2a , j , 0: Evaluation of this integral is accomplished by invoking the Cagniardde Hoop technique. The path of the integration is shown in Fig. 2.2. By completing the contour as indicated in the ﬁgure, the integral in Eq. 2.75 can be converted to an integral along the real axis. The integrand is analytic inside the closed contour and therefore by Cauchy’s theorem, the integral is zero. However, by the decay of the integrand as z ! 1; it is evident that the integral along the circular arcs is zero. Therefore, Eq. 2.75 can be

Figure 2.2 Complex z plane indicating the branch cuts and the contour of integration.

24

Chapter 2

rewritten as an integral along the real axis ð b2 21 a ð z Þ s zx1 Im e dz u^ 2 ðx1 ; sÞ ¼ 2 RðzÞ pm 2a

ð2:76Þ

The Laplace transform in time is inverted by rearranging the integral in such a manner that it represents the product of two Laplace transforms; the inversion is then immediate by the convolution theorem. This is accomplished by noting that Eq. 2.76 can be recast in the form of a Laplace transform if zx1 is redeﬁned as 2 h ðt b2 að2h=x1 Þ 2sh Im ð2:77Þ u^ 2 ðx1 ; sÞ ¼ 2 e dh Rð2h=x1 Þ pmx1 ax1 Evaluation of the above integral is straightforward, but attention should be paid to the time of arrival of the dilatational and distortional waves b2 að2t=x1 Þ Im ð2:78Þ u2 ðx1 ; tÞ ¼ 2 ; G22 ðx1 ; tÞ Rð2t=x1 Þ pmx1 From this equation, we obtain pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 > b2 ðb2 2 2j2 Þ2 j2 2 a2 > > > < pmx ½ðb2 2 2j2 Þ4 2 16j 4 a2 b2 1 G22 ðx1 ; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 2 > b j2 2 a 2 > > : pmx1 RðjÞ

for lax1 l , t , lbx1 l ð2:79Þ t . lbx1 l

where j ¼ t=x1 : G22 ðpmx1 =Cs Þ is plotted in Fig. 2.3 as a function of the normalized time t ¼ Cd t=x1 ; this is the normal displacement felt by an observer located at x1. For t , 1; the dilatational wave has not reached the observer and hence the displacement is zero.

Figure 2.3 Variation of G22 ðpm x1 =Cs Þ with normalized time t 5 Cd t=x1 :

Linear Elastodynamics

25

The dilatational wave brings a very small upward displacement of the surface; this is followed later by the distortional wave that arrives at t ¼ k and closely behind by the Rayleigh wave at t ¼ k=kR ; the singularity of RðCR Þ at this time is seen in Fig. 2.3. Other Green’s functions may be constructed in a similar manner; Slepyan (2002) has presented the complete set of expressions for the Green’s functions appropriate for the half-space problem considered here. With the Green’s function known, any traction distribution on the half-space can be imposed and introduced into Eqs. 2.71 and 2.72 to obtain the surface displacements.

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27

Chapter 3 Dynamic Crack Tip Fields

3.1 Dynamically Loaded Cracks Within the framework of the two-dimensional linear elastodynamic theory described in Chapter 2, we ﬁrst formulate general crack problems. Consider an unbounded linearly elastic medium containing a crack. Without loss of generality, we may consider a state of plane strain1 and assume that the crack lies initially along x1 , 0; x2 ¼ 0: As a consequence of the applied loading the crack may extend, but at ﬁrst we suppress the crack extension. Furthermore, we may assume that the far boundaries of the specimen are traction free and that tractions are applied only on the crack surfaces. Let the crack face boundary conditions be given in terms of the stress components as

s12 ðx1 ; 0^ ; tÞ ¼ 2sHðtÞ s22 ðx1 ; 0^ ; tÞ ¼ 2pHðtÞ

for x1 # 0

ð3:1Þ

^

s32 ðx1 ; 0 ; tÞ ¼ qHðtÞ for x1 # 0; where p; s; and q are prescribed and HðtÞ is the unit step function. These three loads correspond to conventional modes I, II and III of linear elastic fracture mechanics and are illustrated in Fig. 3.1. Determination of the stress and deformation ﬁelds near the crack tip requires the solution of the governing equations for the potentials w ¼ wðx1 ; x2 ; tÞ and cðx1 ; x2 ; tÞ given by the wave equations in Eqs. 2.38 and 2.39. By the application of Huygens’ principle the wavefronts emanating from the crack under this loading can be drawn and are shown in Fig. 3.2. Bulk dilatational and shear waves travel into the body from the crack surface; these wavefronts are parallel to the crack line at distances far from the crack tip. We note that these wavefronts simply carry a jump in the appropriate stress components. On the other hand, the crack tip appears more like a point source and radiates cylindrical dilatational and shear wavefronts in this two-dimensional problem. The headwave or von Schmidt wave generated from the compression wave interacting with the free surface is shown as the angled lines. 1

From the discussion in Chapter 2, it should be clear that the solution corresponding to the plane stress problem can be obtained by replacing the bulk wave speed with the plate wave speed.

28

Chapter 3

Figure 3.1 Semi-inﬁnite crack under uniform loading representing opening, in-plane shearing and anti-plane shearing loading conditions.

The Rayleigh surface wave also travels along the negative x1 direction just behind the cylindrical shear wavefront; this wave is indicated in the ﬁgure by the small dot. In order to determine the crack tip stresses the elastodynamic ﬁeld behind the cylindrical wavefronts must be determined. An alternate way of thinking about the wave loading is to consider the loads along the crack line as a distribution of point sources; a point source at x1 ¼ 2j radiates a cylindrical wavefront that reaches the crack tip at t ¼ j=Cd : Since the load is distributed uniformly over the entire crack line, sources at farther distances from the crack tip will inﬂuence the crack tip stress ﬁeld at later times continuously increasing the stress at the crack tip. For a stationary crack, the elastodynamic stress ﬁeld near a crack tip has the same structure as the quasi-static crack tip with the only difference being the time dependence of the stress intensity factor. Thus, KI ðtÞ Is K ðtÞ IIs fab ðuÞ þ pIIﬃﬃﬃﬃﬃﬃﬃﬃ fab sab ðr; uÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ ð uÞ þ · · · 2pr 2pr K ðtÞ s3a ðr; uÞ ¼ pIIIﬃﬃﬃﬃﬃﬃﬃﬃ f3IIIs a ð uÞ 2pr

as r ! 0 ð3:2Þ

where KI ðtÞ; KII ðtÞ and KIII ðtÞ are the dynamic stress intensity factors under modes I, II and III, respectively; they depend on time as well as the applied loading and geometry and must be determined through a solution to the appropriate initial-boundary value problem.

Figure 3.2 Waves emanating from a crack tip. The solid lines indicate the dilatational waves. The dotted lines indicate shear waves.

29

Dynamic Crack Tip Fields

Is IIs IIIs fab ðuÞ; fab ðuÞ and fab ðuÞ are the angular variation of the crack tip stress ﬁeld and are given by Is ðuÞ ¼ cos 12 u½1 2 sin 12 u sin 32 u f11 Is ðuÞ ¼ cos 12 u½1 þ sin 12 u sin 32 u f22 Is f12 ð uÞ

ð3:3Þ

¼ cos u sin u cos u 1 2

1 2

3 2

IIs f11 ðuÞ ¼ 2sin 12 u½2 þ cos 12 u cos 32 u IIs ðuÞ ¼ cos 12 u sin 12 u cos 32 u f22 IIs f12 ð uÞ

ð3:4Þ

¼ cos 12 u½1 2 sin 12 u sin 32 u

IIIs ðuÞ ¼ 2sin 12 u f31

ð3:5Þ

IIIs f32 ðuÞ ¼ cos 12 u

Since we have suppressed the possibility that the crack may extend, the problem can be solved by standard methods for mixed boundary value problems. As noted earlier, since no dissipative processes are involved, there is no need to invoke the energy balance equation. However, as we have seen above, the crack tip stresses increase continuously with time; these stresses must reach a critical state and the crack must begin to grow dynamically. Once a crack begins to propagate, the direction and speed with which it moves must be determined. Therefore, an equation for the motion of the crack tip must be obtained. Mott (1948) proposed a simple extension of the Grifﬁth criterion: the speed may be determined by including the kinetic energy in the energy balance equation. Writing this equation formally, the crack must extend along a suitable path at a suitable speed in order to obey the energy rate balance equation 2.9 rewritten here as ð ›u dD ð3:6Þ s· dR ¼ ½UðtÞ þ TðtÞ þ dt ›t ›R to include the dissipation at the crack tip. UðtÞ is the strain energy and TðtÞ is the kinetic energy deﬁned in Eqs. 2.10 and 2.11. D is the dissipation in the fracture process zone. Eq. 3.6 must provide the criterion for path selection as well as for the speed of the crack along this path. In addition, it must be ensured that the boundary condition in Eq. 3.1 is augmented with traction free conditions on the newly created crack surfaces

s22 ðx1 ; x2 ; tÞ ¼ 0 s12 ðx1 ; x2 ; tÞ ¼ 0 s23 ðx1 ; x2 ; tÞ ¼ 0

for ðx1 ; x2 Þ [ S

ð3:7Þ

where S is the unknown extension of the crack. While this is now a physically complete formulation of the dynamic crack problem, it is not quite practical. If a representation for dissipation as described in Eq. 3.6 is readily available or easily developed, numerical solution of the system of equations 2.38 and 2.39 with boundary conditions 3.1 and 3.7 can be considered. Recent developments in the formulation of cohesive zone models for the crack tip dissipative processes have enabled large-scale computational simulations of dynamic fracture problems, but due to the inherent limitations of the cohesive zone models, such

30

Chapter 3

simulations have not been able to capture all the observed dynamic fracture phenomena. We shall describe these efforts later. In this section, we focus our attention on what has been the most successful strategy in dynamic fracture analysis. The fracture criterion (or equivalently the energy rate balance condition in Eq. 3.6) is decoupled from the governing ﬁeld equations by prescribing the path and speed of the crack tip; typically the crack is assumed to grow along a straight line at a constant speed. While this decoupling makes the problem amenable to analysis, the restriction of rectilinear crack extension is so severe that it is generally appropriate only in situations where the applied load has a mode I symmetry or the crack is trapped by a weak plane in layered media. In this manner, a set of dynamically admissible solutions is obtained; from this set, the correct solution must be selected by imposing the energy balance equation, with suitable assumptions regarding the dissipation to make the problem manageable. We describe the analytical solutions to dynamic problems in two parts—the ﬁrst part dealing with stationary cracks under dynamic loads and the second part dealing with dynamically growing cracks. Failure criteria will be examined after a description of these analyses.

3.2 Asymptotic Analysis of Crack Tip Fields The dynamic crack tip ﬁeld exhibits a square-root singularity just as in the case of the quasi-static problem. In this section, we determine the structure of this singular ﬁeld for anti-plane shear, opening mode and in-plane shear loading. 3.2.1 Anti-Plane Shear Consider a traction free crack that is assumed to lie initially along x1 , 0; x2 ¼ 0 and to move along x2 ¼ 0 at a constant speed v , CR : The governing differential equation for the nonzero displacement component u3 is 72 u3 ¼

1 u€ 3 Cs2

ð3:8Þ

The traction-free boundary condition can be written as

s32 ðx1 ; 0^ Þ ¼ mu3;2 ðx1 ; 0^ Þ ¼ 0

ð3:9Þ

where x2 ¼ 0^ indicates approach to the crack surface from the positive or negative x2 direction. If we use a Galilean transformation to a coordinate system moving with the crack tip with j1 ¼ x1 2 vt; j2 ¼ x2 ; Eq. 3.8 becomes v 2 ›2 u3 ›2 u3 þ ¼0 ð3:10Þ 12 2 Cs ›j21 ›j22 where u3 ¼ u3 ðj1 ; j2 Þ: Introducing a coordinate scaling zs ¼ j1 þ ias j2 ; the governing equation reduces to 72 u3 ðrs ; us Þ ¼ 0

ð3:11Þ

31

Dynamic Crack Tip Fields

where

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ as j2 2 2 2 rs ¼ j1 þ as j2 ; us ¼ arctan ; as ¼ j1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v2 12 2 Cs

ð3:12Þ

Let us seek a separable form of the solution for u3 ; similar to the Williams (1957) expansion for the quasi-static crack problem u3 ðrs ; us Þ ¼ rsl f ðus ; lÞ ð3:13Þ Eq. 3.11 reduces to an ordinary differential equation for the unknown function f f 00 þ l2 f ¼ 0

ð3:14Þ

The general solution to Eq. 3.14 that obeys the anti-plane symmetry of the problem is f ðus ; lÞ ¼ A sin lus

ð3:15Þ

where A is a constant. Introducing this solution into the boundary condition in Eq. 3.9 results in the characteristic equation for the determination of l: As in the quasi-static case, after rejecting the trivial solution, the stress components are singular and the displacements are bounded only when l ¼ 1=2: Of course, for l . 1=2 both stresses and displacements go to zero as the crack tip is approached; these terms are not considered here, but we shall examine them for the plane strain problem. Thus, 1 ð3:16Þ u3 ðrs ; us Þ ¼ 2Ars1=2 sin us þ · · · 2 The amplitude parameter A is left undetermined in this local analysis and must be obtained from a complete solution of the problem; it can be re-deﬁned in terms of the mode III dynamic stress intensity factor, KIII pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ KIII ¼ lim 2pj1 s32 ðrs ; 0^ Þ ð3:17Þ j1 !0

The stress components are then written as KIII 1 1 s32 ðrs ; us Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ cos us 2 2pr gs KIII 1 1 s31 ðrs ; us Þ ¼ 2 pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ sin us 2 2pr as gs where

gs ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 ðv sin u=Cs Þ2 and tan us ¼ as tan u

ð3:18Þ

ð3:19Þ

We note that this is entirely analogous to the quasi-static problem: the stress components exhibit the inverse square root singularity, but the angular distribution is distorted by the speed of the moving crack tip. The stress and displacement ﬁelds reduce to the corresponding quasi-static ﬁelds in the limit of v ! 0: In the above analysis, it has been assumed that the crack was in steady-state motion; under this condition the dynamic stress intensity factor must be constant. Freund (1990) has shown that if the crack moves with a nonuniform speed, the result described above carries over completely, with the only change that the stress intensity factor can now be

32

Chapter 3

considered to be a function of time and crack speed, KIII ðt; vÞ: The effects of the nonuniform motion of the crack appear only in terms of higher order than the singular ﬁeld considered here. 3.2.2 In-Plane Symmetric Deformation We now turn to the in-plane problem. As discussed before, plane strain and plane stress problems are indistinguishable except for the difference in the P-wave speed; the asymptotic ﬁeld is developed for the case of plane strain. The procedure for obtaining the asymptotic stress and displacement ﬁelds is identical to the mode III problem described in Section 3.2.1. Again, a traction-free crack is assumed to lie initially along x1 , 0; x2 ¼ 0 and to move along x2 ¼ 0 at a constant speed v , CR : The governing differential equations for the potentials w ¼ wðx1 ; x2 ; tÞ and cðx1 ; x2 ; tÞ are the wave equations in Eqs. 2.38 and 2.39. Introducing the Galilean transformation j1 ¼ x1 2 vt; j2 ¼ x2 ; and rescaling the coordinates through zd ¼ rd eiud ¼ j1 þ iad j2 ; zs ¼ rs eius ¼ j1 þ ias j2 ; with sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a j v2 d 2 ud ¼ arctan ad ¼ 1 2 2 rd ¼ j21 þ a2d j22 ; ð3:20Þ ; j1 Cd and as ; rs and us as deﬁned in Eq. 3.12, the governing equations become 72 wðrd ; ud Þ ¼ 0;

72 cðrs ; us Þ ¼ 0

ð3:21Þ

Once again we seek a separable form of the solution for w ¼ wðrd ; ud Þ and c ¼ cðrs ; us Þ; similar to the Williams expansion for the quasi-static crack problem

wðrd ; ud Þ ¼ rdl f ðud ; lÞ;

cðrs ; us Þ ¼ rsl gðus ; lÞ

ð3:22Þ

Following the same procedure as in the mode III problem, the solution can be written as

wðrd ; ud Þ ¼ Ardl cos lud ;

cðrs ; us Þ ¼ Brsl sin lus

ð3:23Þ

Note that we have only considered solutions that are symmetric with respect to the crack—i.e. a mode I problem. The antisymmetric solution would lead to the crack tip ﬁeld for a mode II or in-plane shear mode crack; this is described in the next section. Imposing the traction-free boundary condition results in the following equations for the constants A and B ð1 þ a2s ÞA cosðl 2 2Þp þ 2as B cosðl 2 2Þp ¼ 0 2ad A sinðl 2 2Þp þ ð1 þ a2s ÞB sinðl 2 2Þp ¼ 0

ð3:24Þ

For nontrivial solutions, the determinant of the above system of equations must be zero; this results in the characteristic equation. The general solution for the characteristic equation is

l¼

1 2

n þ 1;

for n ¼ 1; 2; 3; …

ð3:25Þ

33

Dynamic Crack Tip Fields

Nonpositive values of n result in displacement singularities at the crack tip and are therefore rejected. In the case of the mode III, we examined only the term n ¼ 1: Here, we shall examine all positive values of n in order to develop the higher order terms in the crack tip stress ﬁeld, since such higher order terms inﬂuence experimental schemes commonly used in dynamic fracture investigations. From Eq. 3.24 it can be shown that the constants A and B are related by

B¼

8 2a d > > > < 2 1 þ a2s A

for n odd

> 1 þ a2s > > :2 A 2a s

for n even

ð3:26Þ

Using Eqs. 3.23, 3.25, and 3.26 in Eqs. 2.40 and 2.41, the displacement components can be determined to be n n=2 nud nus n=2 rd cos ¼ An 1 þ 2 x1 ðnÞrs cos 2 2 2

n n u nus n=2 d n n=2 u2 ðr; uÞ ¼ An ad 1 þ 2rd sin þ x2 ðnÞrs sin 2 2 2

ð3:27Þ

8 2 a d as > > > < 1 þ a2 s x1 ðnÞ ¼ > > 1 þ a2s > : 2

ð3:28Þ

un1 ðr; uÞ

where for n odd and for n even

x2 ðnÞ ¼

8 > > > <

2 1 þ a2s

for n odd

> 1 þ a2s > > : 2ad as

for n even

The corresponding stress components are mAn n

n n22 2 2 2 ðn=2Þ21 1þ ð1 þ as Þð1 þ 2ad 2 as Þrd ¼ cos ud 2 2 ð1 þ a2s Þ 2 n22 ðn=2Þ21 2kðnÞrs cos us 2 mAn n

n n22 2 2 ðn=2Þ21 1 þ sn22 ðr; uÞ ¼ a Þ r cos u 2ð1 þ d s d 2 2 ð1 þ a2s Þ 2 (3.29) n22 ðn=2Þ21 þkðnÞrs cos us 2

mAn n n n22 ðn=2Þ21 1þ 24ad as rd sn12 ðr; uÞ ¼ sin ud 2 2 2a s 2 n22 ðn=2Þ21 þkðnÞrs sin us 2

sn11 ðr; uÞ

34

Chapter 3

where

(

kðnÞ ¼

4ad as

for n odd

ð3:30Þ

ð1 þ a2s Þ2 for n even

Clearly the term corresponding to n ¼ 1 results in bounded displacements and inverse square-root singular stress components as the crack tip is approached. The amplitude parameters An are left undetermined in this local analysis and must be obtained from a complete solution of the problem; as in the case of the mode III problem, introducing the deﬁnition of the mode I dynamic stress intensity factor, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð3:31Þ KI ¼ lim 2pj1 s22 ðr; 0^ Þ j1 !0

the crack tip stress and displacement ﬁelds may be written as KI I sab ðr; uÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ fab ðu; vÞ þ sox ða2d 2 a2s Þda1 db1 þ · · · 2pr pﬃﬃ KI r I ua ðr; uÞ ¼ pﬃﬃﬃﬃﬃﬃ ga ðu; vÞ þ · · · 2p

ð3:32Þ

In the above, we have included the terms corresponding to n ¼ 1 and 2 for stresses and n ¼ 1 for displacements; the term corresponding to n ¼ 2 implies a stress component parallel to the crack and is typically denoted by sox in the literature on experimental investigations (Kobayashi and Mall, 1978) and is called the T-stress in the literature I on quasi-static fracture (Cotterell and Rice, 1980). The functions fab ðu; vÞ are given below ( ) 1 1 1 I 2 2 2 cos 2 ud 1 ð1 þ as Þð1 þ 2ad 2 as Þ 1=2 2 4ad as 1=2 cos 2 us f11 ðu; vÞ ¼ RðvÞ g gs d

I f22 ðu; vÞ

( ) 1 1 1 2 2 cos 2 ud 1 2ð1 þ as Þ ¼ þ 4ad as 1=2 cos 2 us 1=2 RðvÞ g gs

ð3:33Þ

d

I f12 ðu; vÞ

2ad ð1 þ a2s Þ ¼ RðvÞ

(

sin 12 ud 1=2

gd

2

sin 12 us

)

1=2

gs

where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ gd ¼ 1 2 ðv sin u=Cd Þ2

tan ud ¼ ad tan u

and

RðvÞ ¼ 4ad as 2 ð1 þ a2s Þ2

and

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ gs ¼ 1 2 ðv sin u=Cs Þ2

tanus ¼ as tan u

ð3:34Þ

ð3:35Þ ð3:36Þ

Dynamic Crack Tip Fields

35

Figure 3.3 Angular variation of f11 ðuÞ for a mode I crack. I I I The angular variation of the functions f11 ðu; vÞ; f22 ðu; vÞ; and f12 ðu; vÞ; are shown in Figs. 3.3 –3.5. The angular variation of the hoop component of the stress ﬁeld is given in Fig. 3.6; this component was examined by Yoffe (1951). The shift in the peak from u ¼ 08 to 608 as the crack speed increased to about v , 0:6CR was suggested as the cause of crack branching. The angular variation of the principal stress is shown in Fig. 3.7. It should be noted that the maximum principal stress component does not act

Figure 3.4 Angular variation of f22 ðuÞ for a mode I crack.

36

Chapter 3

Figure 3.5 Angular variation of f12 ðuÞ for a mode I crack.

I I normal to the prospective crack line; Rice (1968) observed that the f11 ðu; vÞ . f22 ðu; vÞ and hence paradoxical that the crack continues to grow along the x1 direction. It can be seen from Eq. 3.29 that for n ¼ 2 the only nonzero component is s11 and is given by 3mA2 ða2d 2 a2s Þ; where A2 is another constant to be determined from a complete analysis of the problem. However, if the crack surface is loaded by normal forces, the s22 component corresponding to n ¼ 2 will be equal to the crack face pressure.

Figure 3.6 Angular variation of fuu ðuÞ for a mode I crack.

Dynamic Crack Tip Fields

37

Figure 3.7 Angular variation of f1 ðuÞ for a mode I crack.

The displacement components corresponding to n ¼ 1 are given by 2KI u u pﬃﬃﬃﬃﬃﬃ ð1 þ a2s Þrd1=2 cos d 2 2ad as rs1=2 cos s u1 ðr; uÞ ¼ 2 2 mRðvÞ 2p 2ad KI u u pﬃﬃﬃﬃﬃﬃ ð1 þ a2s Þrd1=2 sin d 2 2rs1=2 sin s u2 ðr; uÞ ¼ 2 2 mRðvÞ 2p

ð3:37Þ

It should be noted that the stress and displacement ﬁelds reduce to the appropriate quasistatic ﬁelds when v ! 0; however, the limit must be taken appropriately since RðvÞ also tends to zero as v ! 0: Freund (1990) has shown that if the crack moves with a nonuniform speed, the result described above carries over completely, with the only change that the stress intensity factor can now be considered to be a function of time and the instantaneous crack speed, KI ðt; vÞ: The effects of the nonuniform motion of the crack do not become apparent in the singular term or the constant term, but only in terms of higher order. In some experimental methods crack tip ﬁeld information is extracted from distances that are far from the crack tip; in these applications, the higher order transient expansion is required to obtain an estimate of the stress intensity factor. We shall describe the transient ﬁeld in Section 3.3. 3.2.3 In-Plane Antisymmetric Deformation The stress and displacement ﬁeld corresponding to a mode II crack growing at a speed v can be determined in a similar manner. If the antisymmetric solutions to Eq. 3.21 are used

wðrd ; ud Þ ¼ Ardl sin lud cðrs ; us Þ ¼ Brsl cos lus

ð3:38Þ

38

Chapter 3

Then following the same procedure as described above, the mode II stress and displacement ﬁelds can be determined. They are given below

mAn n

n n ðn=2Þ21 1 þ ð1 þ a2s Þð1 þ 2a2d 2 a2s Þrd 2 ð1 þ a2s Þ 2 n22 n22 ðn=2Þ21 sin u 2 kðnÞrs sin u 2 d 2 s m An n

n n22 2 2 ðn=2Þ21 1 þ sn22 ðr; uÞ ¼ a Þ r sin u 2ð1 þ s d 2 2 d ð1 þ a2s Þ 2 n22 ðn=2Þ21 þkðnÞrs sin us 2 mAn n

n n22 ðn=2Þ21 n 1þ 4 ad as r d s12 ðr; uÞ ¼ cos u 2 2 d 2as 2 n22 ðn=2Þ21 2kðnÞrs cos us 2 sn11 ðr; uÞ ¼

ð3:39Þ

where (

kðnÞ ¼

4ad as

for n even

ð1 þ a2s Þ2

for n odd

ð3:40Þ

The corresponding displacement components are

n n=2 nud nus n=2 rd sin 1þ 2 x1 ðnÞrs sin 2 2 2

n nud nus n=2 n n=2 u2 ðr; uÞ ¼ ad An 1 þ rd sin 2 x2 ðnÞrs sin 2 2 2

un1 ðr; uÞ ¼ An

ð3:41Þ

The mode II dynamic stress intensity factor may be deﬁned in analogy with the corresponding deﬁnition for the quasi-static problem; thus KII ¼ lim

j1 !0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pj1 s12 ðr;0^ Þ

ð3:42Þ

For a general plane crack propagation problem, the crack tip stress ﬁeld may then be represented as a superposition of the opening mode or mode I and the in-plane shearing mode or mode II dynamic stress ﬁeld and characterized by the stress intensity factors KI and KII : The extent of the K-dominant ﬁeld near the crack tip depends on the transient nature of the crack tip history; we shall explore this in Chapter 9.

39

Dynamic Crack Tip Fields

3.3 Asymptotic Analysis for Nonsteady Crack Growth In the analysis of the crack tip stress ﬁeld described Section 3.2, it was assumed that the crack was moving in a steady ﬁeld. In a number of experimental situations, it appeared that due to the transients resulting from arbitrary crack motion histories the steady-state K-ﬁeld could not be established in the neighborhood of the crack tip for times that are quite large compared to the timescale of interest in the fracture problem. Freund (1990) considered the inﬂuence of transient crack motion on the asymptotic ﬁeld and presented an approach to the development of the transient ﬁeld. This ﬁeld was further analyzed by Rosakis et al. (1991) and Freund and Rosakis (1992). Liu and Rosakis (1994) derived the general expressions of the transient ﬁeld for a crack moving at a nonuniform speed along a generally curved path. In this section, we provide a brief sketch of the basis of the transient analysis and the resulting asymptotic ﬁeld. Again, a traction-free crack is assumed to lie initially along x1 , 0; x2 ¼ 0 and to move along x2 ¼ 0 at a speed vðtÞ , CR : The governing differential equations for the potentials w ¼ wðx1 ; x2 ; tÞ and cðx1 ; x2 ; tÞ are the wave equations in Eqs. 2.38 and 2.39. Introducing the Galilean transformation j1 ¼ x1 2 vt; j2 ¼ x2 ; and noting that the resulting ﬁeld is not a steady-state ﬁeld, the governing equations reduce to

v2 12 2 Cd v2 12 2 Cs

›2 w ›2 w v_ ›2 w v ›2 w 1 ›2 w þ þ þ 2 2 ¼0 Cd2 ›j1 Cd2 ›t ›j1 Cd2 ›t2 ›j21 ›j22 ›2 c ›2 c v_ ›2 c v ›2 c 1 ›2 c þ þ þ 2 2 ¼0 Cs2 ›j1 Cs2 ›t ›j1 Cs2 ›t2 ›j21 ›j22

ð3:43Þ

Clearly, the last three terms in each equation vanish in the case of a steady-state problem. Freund (1990) suggested that the potential might be expressed as

wðj1 ; j2 ; tÞ ¼ wð1h1 ; 1h2 ; tÞ ¼

1 X

1ðmþ3Þ=2 w^m ðh1 ; h2 ; tÞ

m¼0

cðj1 ; j2 ; tÞ ¼ cð1h1 ; 1h2 ; tÞ ¼

1 X

1

ðmþ3Þ=2

c^m ðh1 ; h2 ; tÞ

ð3:44Þ

m¼0

where 1 is a small parameter introduced to rescale the spatial coordinates; as 1 ! 0; the asymptotic ﬁeld near the crack tip is revealed. The exponents on 1 are written in anticipation of the nature of the crack tip singularity already known from the steady-state analysis of Section 3.2. Substituting Eq. 3.44 into Eq. 3.43 and collecting terms with the same powers of 1 results in the governing equations for w^m ðh1 ; h2 ; tÞ and c^m ðh1 ; h2 ; tÞ: The ﬁrst three equations for the potential

40

Chapter 3

w^m ðh1 ; h2 ; tÞ are given below a2d

›2 w^0 ›2 w^0 þ ¼0 ›h21 ›h22

a2d

›2 w^1 ›2 w^1 þ ¼0 ›h21 ›h22

a2d

ð3:45Þ

pﬃﬃﬃ ›2 w^2 ›2 w^2 2 v › pﬃﬃﬃ ›w^0 þ ¼2 2 v ›h1 ›h21 ›h22 C d ›t

.. . A similar set of equations are obtained for the potential c^m ðh1 ; h2 ; tÞ: Three main features of the above equations are worth noting. First, the homogeneous solutions to these equations correspond to the steady-state asymptotic expansion already discussed in Section 3.2; thus the transient analysis will simply provide additional terms that add on to the steady-state crack tip ﬁeld in Eqs. 3.29 for mode I. Second, the equations for m ¼ 0 and 1 do not depend on time explicitly and furthermore depend only on the instantaneous crack speed. Indeed, the governing equations for w^0 ðh1 ; h2 ; tÞ; w^1 ðh1 ; h2 ; tÞ; c^0 ðh1 ; h2 ; tÞ and c^1 ðh1 ; h2 ; tÞ are identical to the equations considered for the steady-state crack growth problem discussed in Section 3.2 (see Eq. 3.21). Thus, the ﬁrst two terms of the transient stress ﬁeld near the crack tip must be identical to the steady-state asymptotic ﬁeld; however, we have allowed for explicit time dependence of the potentials. Therefore, the amplitude of the zeroth order term when identiﬁed as the dynamic stress intensity factor through Eq. 3.31 may depend on time as well as the instantaneous crack speed, KI ¼ KI ðt; vÞ: Finally, the governing equations for all m . 1 involve terms on the right hand side that are dependent on time; but the time dependence is generated only by the crack speed variations and by the explicit time dependence of the amplitudes of the potentials for m # 1: Therefore, the transient correction to the higher order terms of the asymptotic ﬁeld is fully determined by the time variation of the lower order terms. In many experiments, it has been observed that the crack moves at a constant speed even though the crack tip motion may not be considered as a steady-state motion (see for example, Ravi-Chandar and Knauss, 1984c). Also, in the analysis of crack growth problems it is often assumed that the crack speed is constant as in the works of Broberg (1960), Baker (1962), Freund (1972b), and others. We restrict attention here to this case. The homogeneous solution to the governing equations may be written as 3=2

wHm ¼ Am ðt; vÞrd cos cHm

¼

Bm ðt; vÞrs3=2

3ud 2

3u sin s 2

ð3:46Þ

Dynamic Crack Tip Fields

41

where we have transformed back to the physical plane through Eq. 3.44 and then to the distorted polar coordinates introduced in Eqs. 3.12 and 3.20. The dependence of the coefﬁcients on time and crack speed is shown explicitly; these coefﬁcients must be determined from a complete solution of speciﬁc boundary-initial value problems. The dynamic stress intensity factor for the transient ﬁeld is deﬁned exactly as in the steady-state ﬁeld, so that pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2pmRðvÞ ^ KI ðt; vÞ ¼ lim 2pj1 s22 ðr; 0 Þ ¼ A0 ðt; vÞ j1 !0 4ð1 þ a2s Þ

ð3:47Þ

Note that the coefﬁcients A0 and B0 are related in exactly the same way as in the steady-state problem as indicated in Eq. 3.26 (note that m ¼ n 2 1). The ﬁrst nonzero term from the transient equation arises for the case m ¼ 2: Substituting the zeroth order solution into Eq. 3.45 and the corresponding ones for c^m ðh1 ; h2 ; tÞ; and transforming back to the physical plane, the differential equations for the second-order term in distorted polar coordinates is obtained as 3v _ ud 1=2 ðt; vÞr cos A d 2 a2d Cd2 0 3v u 1=2 72 c2 ðrs ; us Þ ¼ 2 2 2 B_ 0 ðt; vÞrd sin s 2 as Cs

72 w2 ðrd ; ud Þ ¼ 2

ð3:48Þ

The particular solution to these equations is easily shown to be 1 p u 5=2 A ðt; vÞrd cos d 6 2 1 p us P 5=2 c2 ¼ B ðt; vÞrs sin 6 2

wP2 ¼

ð3:49Þ

where 4vð1 þ a2 Þ _ Ap ¼ 2 pﬃﬃﬃﬃﬃﬃ 2 2s KI ðt; vÞ 2pmad Cd RðvÞ

ð3:50Þ

8va Bp ¼ 2 pﬃﬃﬃﬃﬃﬃ 2 d 2 K_ I ðt; vÞ 2pmas Cs RðvÞ K_ I ðt; vÞ is the rate of change of the dynamic stress intensity factor; thus, the ﬁrst nonzero transient term depends only on the rate of change of the dynamic stress intensity factor. In addition, because the crack surfaces must remain traction free, the constants A2 and B2 are related. Similar solutions to the higher order equations may be developed. Freund and Rosakis (1992) examined the ﬁrst six terms in the solution for the potential w^m ðh1 ; h2 ; tÞ: The complete stress and displacement ﬁeld (up to the second-order term) may be evaluated

42

Chapter 3

by substituting into Eqs. 2.40 and 2.41 (see Rosakis et al., 1991) pﬃﬃﬃﬃﬃﬃ s11 ðr; uÞ 2pRðvÞ u u 21=2 ¼ ð1 þ 2a2d 2 a2s Þrd cos d 2 4ad as rs21=2 cos s KI ðt; vÞ 2 2 þ 2ð1 þ 2a2d 2 a2s Þ

A1 ðtÞ B ðtÞ þ 4 as 1 KI ðt; vÞ KI ðt; vÞ

15 A ðtÞ 21=2 u ð1 þ 2a2d 2 a2s Þ 2 rd cos d 4 KI ðt; vÞ 2 B ðtÞ 1=2 u þ 2 as 2 r cos s KI ðt; vÞ s 2 Ap ð1 þ a2d Þða2d 2 a2s Þ þ ð1 þ a2s Þ2 u þ cos d KI ðt; vÞ 2 2ð1 2 a2d Þ 1 3u 1=2 þ ð1 þ 2a2d 2 a2s Þcos d rd 8 2 þ

þ

as Bp 3u r 1=2 cos s 4 KI ðt; vÞ s 2

ð3:51Þ

pﬃﬃﬃﬃﬃﬃ s22 ðr; uÞ 2pRðvÞ u u 21=2 ¼ 2ð1 þ a2s Þrd cos d þ 4ad as rs1=2 cos s KI ðt; vÞ 2 2 2 2ð1 þ a2s Þ

mA1 ðtÞ mB1 ðtÞ 2 4a s KI ðt; vÞ KI ðt; vÞ

15 mA2 ðtÞ 1=2 u 2ð1 þ a2s Þ r cos d þ 4 KI ðt; vÞ d 2 mB2 ðtÞ 1=2 u rs cos s 2 2 as KI ðt; vÞ 2

þ

m Ap KI ðt; vÞ

2

as mBp 1=2 3u rs cos s 4 KI ðt; vÞ 2

ð1 þ a2d Þða2d 2 a2s Þ 2 ð1 2 a2s Þ2 u cos d 2 2ð1 2 a2d Þ 1 3u 1=2 2 ð1 þ a2s Þcos d rd 8 2 ð3:52Þ

43

Dynamic Crack Tip Fields

pﬃﬃﬃﬃﬃﬃ s12 ðr;uÞ 2pRðvÞ u u 21=2 ¼ 2ad ð1þa2s Þ rd sin d 2rs21=2 sin s 2 2 KI ðt;vÞ 15 mA2 ðtÞ 1=2 ud 3us 2 mB2 ðtÞ 1=2 2 ad r sin þð1þas Þ r sin 2 4 KI ðt;vÞ d KI ðt;vÞ s 2 2 ad mAp 1=2 3ud 1 mBp u rd sin 2 ð12a2s Þsin s þ 4 KI ðt;vÞ 2 2 KI ðt;vÞ 2 2

ð1þa2s Þ 3us 1=2 3us sin rs sin 4 2 2

ð3:53Þ

Since the transient terms are fully determined in terms of the singular term, it is possible to estimate the importance of the transient terms. The range of variation of K_ I ðt;vÞ in experiments is typically of the order of ,1026 MPa m1/2; hence the importance of this term can be established a priori for each material and loading conﬁguration.

3.4 Intersonic Crack Growth The possibility that the crack speed might exceed the shear wave speed, Cs has been considered by Burridge et al. (1979), Freund (1979), Slepyan and Fishkov (1981), Broberg (1989), and others. The development considered for subsonic crack growth is followed closely in the following with the condition that Cs , v , Cd : Introducing the Galilean transformation, j1 ¼ x1 2 vt; j2 ¼ x2 ; and rescaling the coordinates through zd ¼ rd eiud ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j1 þ iad j2 ; zs ¼ rs eius ¼ j1 þ ia^s j2 ; with ad ¼ 1 2 ðv2 =Cd2 Þ; a^s ¼ ðv2 =Cs2 Þ 2 1; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rd ¼ j21 þ a2d j22 ; rs ¼ j21 þ a^2s j22 ; ud ¼ arctanðad j2 =j1 Þ and us ¼ arctanða^s j2 =j1 Þ; the equations of motion (Eqs. 2.18 and 2.19) simplify to the following

›2 f 1 ›2 f þ ¼0 a2d ›j22 ›j21

ð3:54Þ

›2 c 1 ›2 c 2 ¼0 a^2s ›j22 ›j21

Unlike the subsonic case where both potentials were governed by elliptic equations, for the intersonic crack, the equation governing the potential wðx; tÞ is elliptic while the equation governing cðx; tÞ is hyperbolic. The relationship between the stress components and

44

Chapter 3

the potentials given in Eq. 2.41 can now be written in the transformed coordinates as ›2 w ›2 c s11 ¼ m ð1 þ 2a2d þ a^2s Þ 2 þ 2a^s ›j1 ›j2 ›j 1 2 2 › c 2 › w ð3:55Þ s22 ¼ 2m ð1 2 a^s Þ 2 þ 2a^s ›j 1 ›j 2 ›j 1 ›2 w ›2 c s12 ¼ m 2ad 2 ð1 2 a^2s Þ 2 ›j 1 ›j 2 ›j 1 As in the case of subsonic crack growth, we seek a general solution to these potentials that describe the characteristic crack tip ﬁelds. The general solution for the potential wðx; rÞ can be taken to be

wðrd ; ud Þ ¼ rdl ðA cos ud þ B sin ud Þ

ð3:56Þ

where A and B are undetermined constants describing the symmetric (mode I) and antisymmetric (mode II) crack tip conditions. The general solution for the potential cðx; tÞ governed by the hyperbolic equation can be taken to be

cðrs ; us Þ ¼ CfL ðj1 þ a^s j2 Þ þ DfR ðj1 2 a^s j2 Þ

ð3:57Þ

where C and D are undetermined constants, and the arguments of the unknown functions fL and fR represent left and right characteristics of the wave equation. These characteristics are shown in Fig. 3.8. Consider the upper half-plane, x2 . 0; the right going characteristics cannot be used since they represent waves going towards the crack; so we take the constant D to be zero. In addition, fL ðj1 þ a^s j2 Þ ¼ 0 for j1 . 2a^s j2 : Similar comments apply to the lower half-plane. The structure of the crack tip solution can be extracted by imposing the boundary conditions

s22 ðj1 , 0; j2 ¼ 0Þ ¼ 0 s12 ðj1 , 0; j2 ¼ 0Þ ¼ 0

ð3:58Þ

Figure 3.8 Crack tip coordinate systems ðx1 ; x2 Þ and ðj1 ; j2 Þ are shown. The crack is moving in the positive x1 direction with a speed v > Cs : The Mach waves corresponding to the shear wave speed are also shown.

Dynamic Crack Tip Fields

45

Mode I. For mode I, substituting the general solution in Eqs. 3.56 and 3.57 with B ¼ 0 and D ¼ 0 into Eq. 3.55 and evaluating the boundary conditions results in ð1 2 a^2s Þcosððl 2 2ÞpÞA 2 2a^s Cf 00L ¼ 0 2 2ad sinððl 2 2ÞpÞA þ ð1 2 a^2s ÞCf 00L ¼ 0

ð3:59Þ

These are the intersonic counterpart to Eq. 3.24 for subsonic crack growth for mode I. For nontrivial solutions to exist, the determinant of the above system of equations must be zero; this results in the characteristic equation 1 ð1 2 a^2s Þ2 arctan þ n ; 1ðvÞ þ n p 4ad a^s

l22¼

ð3:60Þ

where n is an integer. From Eq. 3.60, 0 , 1ðvÞ , 1=2: The corresponding displacement and stress components can be calculated from Eq. 3.55. The components u2 and s22 to leading order are given below 1ðvÞþnþ1

u2 , Ard

1ðvÞþn

s22 , Ard

ð3:61Þ

For n # 22; the displacements are singular, for n . 0; the stress components go to zero as the crack tip is approached, and for n ¼ 21; the energy released (estimated2 roughly as s22 u2;j2 Þ becomes unbounded. This leaves the only possibility that n ¼ 0; however, while the displacement and stress components remain bounded, the energy released vanishes. Thus, intersonic crack growth is inadmissible under mode I loading conditions. Mode II. Substituting the general solution corresponding to the mode II antisymmetry ðA ¼ 0Þ in Eqs. 3.56 and 3.57 into Eq. 3.55 and evaluating the boundary conditions results in the following equations ð1 2 a^2s Þsinððl 2 2ÞpÞA 2 2a^s Cf 00L ¼ 0 2ad cosððl 2 2ÞpÞA þ ð1 2 a^2s ÞCf 00L ¼ 0

ð3:62Þ

These are the intersonic counterpart to Eq. 3.24 for subsonic crack growth for mode II. For nontrivial solutions, the determinant of the above system of equations must be zero; this results in the characteristic equation

l22¼2

1 4ad a^s 1 arctan ; dðvÞ þ n ¼ 1ðvÞ þ n 2 2 2 p 2 ð1 2 a^s Þ

ð3:63Þ

From Eq. 3.63, it follows that 21=2 # dðvÞ # 0; the variation of dðvÞ with the crack speed is shown in Fig. 3.9. The corresponding displacement and stress components can be calculated from Eq. 3.55. The components u1 and s12 to leading order are given by dðvÞþnþ1

u1 , Ard

dðvÞþn

s12 , Ard 2

The energy release rate is discussed in more detail in Chapter 5.

ð3:64Þ

46

Chapter 3

Figure 3.9 Variation of dðvÞ with crack speed in the intersonic range Cs < v < Cd :

Once again, only for the case n ¼ 0 do the solutions indicate a bounded displacement and a singular stress pﬃﬃﬃ ﬁeld. However, even in this case, the energy release rate goes to zero except when v ¼ 2Cs ; at this speed, the stress ﬁeld exhibits a square-root singularity similar to the subsonic case. The asymptotic ﬁeld corresponding to the intersonic shear crack is given by Broberg (1999). Experiments on weak interfaces under shear loading exhibit such intersonic crack growth (Rosakis et al., 1999); the test conﬁguration is shown in Fig. 3.10. Two plates of Homalite-100 were bonded together with a weak epoxy and a crack was introduced in the weak plane. Dynamic shear loading was applied by an asymmetric impact just below the crack line on the weak plane. The resulting crack growth generated an intersonic mode II crack. Isochromatic fringe patterns corresponding to such a crack are shown in Fig. 3.10. The Mach waves corresponding to j1 . ^as j2 are clearly visible in this ﬁgure. For a thorough review of intersonic crack growth and its applications to earthquake problems,

Figure 3.10 Isochromatic fringes at the tip of an intersonic crack ðCs < v < Cd Þ: The shear Mach waves are clearly visible; additional structure due to a ﬁnite contact zone leads to a double Mach wave. See Rosakis et al. for details of the experiment. (Reproduced from Rosakis et al., 1999.)

Dynamic Crack Tip Fields

47

Figure 3.11 Isochromatic fringes at the tip of an intersonic crack ðCs < v < Cd Þ in a homogeneous material. The crack was forced to grow as a shear crack by conﬁning its path with a side-groove in the specimen. The shear Mach waves are clearly visible; curved shock indicates that the crack slowed down from a much greater speed. (Reproduced from Ravi-Chandar, 2001.)

see the recent article by Rosakis (2002). Kavaturu et al. (1998) have also examined intersonic crack growth. Ravi-Chandar (2001) demonstrated that intersonic crack growth is also possible in homogenous materials, without weak interfaces, if cracks are trapped along the maximum shear plane through other constraints such as a groove. Isochromatic fringe patterns corresponding to an intersonic crack within a groove in a Homalite-100 plate is shown in Fig. 3.11.

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49

Chapter 4 Determination of Dynamic Stress Intensity Factors

4.1 Analysis of Stationary Cracks Under Dynamic Loading A limited number of solutions are available for the dynamic stress ﬁeld near stationary and moving crack tips under transient conditions. Freund (1990), Broberg (1999) and Slepyan (2002) present a complete discussion of most of the available solutions. Since our objective here is to present an introduction to the topic, we restrict our attention to one problem and extract as much as possible from it. The problem we consider corresponds to a semi-inﬁnite pressure-loaded crack in an unbounded medium described above; we will ﬁrst examine the case of a stationary crack and then discuss the moving crack problem. Some generalizations will be presented to extend the scope of the presentation. A brief discussion of the methods used in obtaining the solutions of dynamic fracture problems is also included. The development presented in this section follows the work of Freund (1990).

4.1.1 Semi-Inﬁnite Crack Under Uniform Loading Consider an unbounded linearly elastic body containing a semi-inﬁnite crack lying along the negative x1 axis with its tip at x1 ¼ 0: A uniform normal load of intensity sp is applied on the semi-inﬁnite crack surfaces at t ¼ 0 and persists all the time; the shear tractions on the crack surface are taken to be zero. The crack is assumed to remain stationary and therefore no failure criterion needs to be imposed. The governing equations are simply the equations of elastodynamics that are given in terms of the Lame´ potentials in Chapter 2; boundary and initial conditions must of course be added to these equations. Due to the symmetry of the problem, it is sufﬁcient to consider the region x2 . 0: The boundary and symmetry conditions along x2 ¼ 0 can be written as

s22 ðx1 ; 0; tÞ ¼ 2sp HðtÞ 2 1 , x1 , 0 s12 ðx1 ; 0; tÞ ¼ 0 2 1 , x1 , 1 u2 ðx1 ; 0; tÞ ¼ 0 0 , x1 , 1

ð4:1Þ

50

Chapter 4

Furthermore, we impose the condition that the solutions to the problem remain bounded as x2 ! 1: In addition, the initial conditions are taken to be quiescent conditions

›w ðx ; x ; 0Þ ¼ 0 ›t 1 2 ›c ðx ; x ; 0Þ ¼ 0 cðx1 ; x2 ; 0Þ ¼ ›t 1 2

wðx1 ; x2 ; 0Þ ¼

ð4:2Þ

While the potentials wðx1 ; x2 ; tÞ and cðx1 ; x2 ; tÞ are governed by the standard wave equations, the boundary conditions are quite complex, involving linear combinations of second derivatives of the potentials. The standard method of solving these equations is through the use of transform techniques. In order to accomplish this, the boundary conditions must be deﬁned over the entire x1 axis. This is typically accomplished by extending the boundary conditions through the introduction of unknown functions for that part of the boundary where certain quantities are not prescribed

s22 ðx1 ; 0þ ; tÞ ¼ sþ ðx1 ; tÞ 2 sp HðtÞHð2x1 Þ s12 ðx1 ; 0þ ; tÞ ¼ 0

21 , x , 1

ð4:3Þ

þ

u2 ðx1 ; 0 ; tÞ ¼ u2 ðx1 ; tÞ where the domain for sþ ðx1 ; tÞ is x1 . 0 and the domain for u2 ðx1 ; tÞ is x1 , 0; but both are unknown functions to be determined. The procedure for solving the system of equations was described in Chapter 2 in connection with the half-space Green’s functions. With the reformulation of the boundary conditions for the crack problem in Eqs. 4.3, the transformed solution can be written as

Fðz; x2 ; sÞ ¼ A1 ðz; sÞe2sax2 þ A2 ðz; sÞesax2 Cðz; x2 ; sÞ ¼ B1 ðz; sÞe2sbx2 þ B2 ðz; sÞesbx2

ð4:4Þ

where Fðz; x2 ; sÞ and Cðz; x2 ; sÞ are the transformed potentials and aðzÞ and bðzÞ are given in Eq. 2.58. The exponentially growing terms will be unbounded as x2 ! 1 and must therefore be rejected. A1 ðz; sÞ and B1 ðz; sÞ must be determined from the transformed boundary conditions; taking the Laplace transform in time and the bilateral Laplace transform in space, the boundary conditions in Eqs. 4.3 get transformed into the following 2 a 2 sp 2 2 2 2 2d F 2 dC 2 m ðb 2 2a Þs z F þ b 2 2s z a ¼ a S ð z ; sÞ þ þ dx2 x2 ¼0þ dx22 s2 z 2 dF d C m 2sz þ 2 s2 z2 C ¼0 dx2 dx22 x2 ¼0þ dF 2 szC ¼ U2 ðz; sÞ dx2 x2 ¼0þ

ð4:5Þ

Determination of Dynamic Stress Intensity Factors

51

Substituting for Fðz; x2 ; sÞ and Cðz; x2 ; sÞ from Eqs. 4.4 results in three algebraic equations for the four unknown functions: A1 ðz; sÞ; B1 ðz; sÞ; Sþ ðz; sÞ and U2 ðz; sÞ; note that the latter two functions are deﬁned only in speciﬁc domains in the z plane. The second and third of Eqs. 4.5 can be used to solve for A1 ðz; sÞ and B1 ðz; sÞ in terms of U2 ðz; sÞ b2 2 2z 2 U2 ðz; sÞ b2 aðzÞs 2z B1 ðz; sÞ ¼ 2 2 U2 ðz; sÞ bs A1 ðz; sÞ ¼ 2

ð4:6Þ

Then, from the ﬁrst of Eqs. 4.5, we obtain one algebraic equation relating the two remaining unknown functions Sþ ðz; sÞ and U2 ðz; sÞ 2s3

mRðzÞ sp U2 ðz; sÞ ¼ s2 Sþ ðz; sÞ þ 2 z b aðzÞ

ð4:7Þ

where RðzÞ ¼ ðb2 2 2z2 Þ þ 4zab

ð4:8Þ

is the same Rayleigh function encountered in Eq. 2.51, but rewritten in terms of the wave slownesses. As indicated in that section, the Rayleigh function has real roots at z ¼ ^1=CR ¼ ^c; with c indicating the slowness of the Rayleigh wave. For Eq. 4.7 to be homogeneous in s, the functions Sþ ðz; sÞ and U2 ðz; sÞ must be of a separable form as indicated below U2 ðz; sÞ ¼

1 1 U ðzÞ; Sþ ðz; sÞ ¼ 2 Sþ ðzÞ 3 2 s s

ð4:9Þ

This separability is a feature of the problems that lack characteristic length scales; we shall see an example in Section 4.1.2 where this separability is lost due to the presence of a speciﬁc length scale in the problem. It is important to evaluate the terms in Eq. 4.7 to determine the domain of analyticity. Sþ ðzÞ is analytic in ReðzÞ . 2a and U2 ðzÞ is analytic in ReðzÞ , 0; it is implicitly understood that Eq. 4.7 can only hold in the common strip of analyticity of these functions. The function aðzÞ has branch points at z ¼ ^a; a single valued function is deﬁned by introducing branch cuts along a # lReðzÞl # 1; ImðzÞ ¼ 0: Similarly, the function bðzÞ has branch points at z ¼ ^b; a single valued function is deﬁned by introducing branch cuts along b # lReðzÞl # 1; ImðzÞ ¼ 0: RðzÞ is analytic in the z plane, with the exception of the branch points at z ¼ ^a; z ¼ ^b and the poles at z ¼ ^c: The last term on the right hand side of Eq. 4.7 corresponding to the applied load contains a pole at z ¼ 0: The Wiener-Hopf factorization technique is used to determine the functions Sþ ðzÞ and U2 ðzÞ from Eq. 4.7. Discussion of the method and its applications may be found in the book by Noble (1958). The basis of the method is to factor the terms in such a manner that the region of analyticity of the factored parts overlap; then using analytic continuation, each part is obtained independently. Therefore,

52

Chapter 4

the functions aðzÞ in Eq. 2.58 and RðzÞ in Eq. 4.8 must be factored. By observation, we can write pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ aðzÞ ¼ aþ ðzÞa2 ðzÞ ¼ a þ z a 2 z ð4:10Þ The well-known factorization of the Rayleigh function (ﬁrst derived by Maue, 1953) is accomplished by deﬁning a function SðzÞ ¼

2ðb2

RðzÞ ¼ Sþ ðzÞS2 ðzÞ 2 a2 Þðc2 2 z2 Þ

ð4:11Þ

where the zeros of the Rayleigh function have been removed and SðzÞ ! 1 as z ! 1: The factors of SðzÞ are ( " ) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # ð 1 b 21 4j2 ðj2 2 a2 Þðb2 2 j2 Þ dj S^ ðzÞ ¼ exp 2 ð4:12Þ tan p a j^z ðb2 2 2j2 Þ2 With the above factorization, Eq. 4.7 can be rewritten as 2

2mðb2 2 a2 Þ U2 ðzÞ sp ¼ S ð z ÞF ð z Þ þ F ðzÞ þ þ F2 ð z Þ z þ b2

ð4:13Þ

where F^ ðzÞ ¼ a^ ðzÞ=ððc ^ zÞS^ ðzÞÞ: The term on the left hand side is analytic in ReðzÞ , 0 and the ﬁrst term on the right hand side is analytic in ReðzÞ . 2a: The second term on the right hand side must now be decomposed into two parts to complete the Wiener-Hopf factorization; the only singularity in this term is the pole at z ¼ 0: The required additive factorization can be obtained by adding and subtracting the residue Fþ ð0Þ at the pole pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cd ð1 2 2nÞ=2 Fþ ð0Þ ¼ ð1 2 nÞ

ð4:14Þ

Then, Eq. 4.7 is rearranged as follows 2

2mðb2 2 a2 Þ U2 ðzÞ sp sp 2 Fþ ð0Þ ¼ Sþ ðzÞFþ ðzÞ þ ðF ðzÞ 2 Fþ ð0ÞÞ ð4:15Þ 2 F2 ð z Þ z z þ b

In Eq. 4.15, the left hand side is analytic in ReðzÞ , 0 and the right hand side is analytic in ReðzÞ . 2a: From the analytic function theory, each side of Eq. 4.15 represents the analytic continuation of the other side; furthermore, the two sides deﬁne an entire function. Next, examine the behavior of the right hand side as z ! 1; clearly, the second term goes to zero. The approach of the ﬁrst term can be determined since the stress is expected to be square root pﬃﬃﬃﬃﬃ singular as s^þ ðx1 ; sÞ , 1= x1 as x1 ! 0þ : The Abel p theorem provides the connection to the ﬃﬃﬃ asymptotic behavior of the transform: Sþ ðz; sÞ , 1= z as z ! 1: Therefore the right hand pﬃﬃﬃﬃﬃ side goes to zero as z ! 1: Similarly, u^ 2 ðx1 ; tÞ , x1 as x1 ! 02 and therefore,

Determination of Dynamic Stress Intensity Factors

53

U2 ðz; sÞ , 1=z3=2 as z ! 21: By Liouville’s theorem, the bounded entire function must be a constant, which in this case is zero. Thus " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # Cd ð1 2 2nÞ=2 Sþ ðzÞðc þ zÞ sp Fþ ð0Þ sp pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 21 ¼ Sþ ðzÞ ¼ 21 z Fþ ðzÞ z ð1 2 nÞ aþz ð4:16Þ b2 sp Fþ ð0ÞF2 ðzÞ U2 ðzÞ ¼ 2 z 2mðb2 2 a2 Þ The nontrivial task of inverting the transforms to determine the stress and displacements remains. In dynamic fracture problems of the kind considered here, the interest is primarily in the time dependence of the dynamic stress intensity factor; this can be extracted from the transformed solution in Eq. 4.16, without inverting the transform. From the Abel theorem, the dynamic stress intensity factor can be extracted from the asymptotic behavior of the Laplace transform s^þ ðx1 ; sÞ as x1 ! 0 pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ ð4:17Þ lim px1 s^þ ðx1 ; sÞ ¼ lim szSþ ðs; zÞ x1 !0

z!1

Therefore, the Laplace-transformed dynamic stress intensity factor is given by pﬃﬃﬃ p 2s Fþ ð0Þ KI ðsÞ ¼ s3=2

ð4:18Þ

Now, the Laplace transform is easily inverted to yield the time-dependent dynamic stress intensity factor pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2sp Cd tð1 2 2nÞ=p ð4:19Þ KI ¼ ð1 2 nÞ Eq. 4.19 provides the time variation of the dynamic stress intensity factor for the problem of stationary semi-inﬁnite crack with uniform pressure loading on the crack surfaces, and the crack tip asymptotic ﬁeld in Eq. 3.2 completes the characterization of the crack tip state. The dynamic stress intensity factor result in Eq. 4.19 is applicable to the case when the crack surface pressure is imposed as a step function. In practical implementations of this kind of loading, the load is usually generated by a ramp-type loading with ﬁnite rise time. For this and other general time variations of the crack surface loading, the stress intensity factor may be determined using a superposition integral ðt _ tÞdt KI ¼ KIstep ðt 2 tÞfð ð4:20Þ 0

_ is the time derivative of the time variation of the applied load and K step ðtÞ the where fðtÞ I stress intensity factor variation for the step loading. For example, let the applied crack surface pressure be a terminated ramp 8 < t 0#t#T T ð4:21Þ f ðtÞ ¼ : 1 t$T

54

Chapter 4

where T is the rise time of the loading function. Then, from Eqs. 4.19 and 4.20, the dynamic stress intensity factor for the ramp loading problem is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 1 2 2n p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t 3=2 4 > > > s pCd T 0#t#T < 3p 1 2 n T KI ðtÞ ¼ ð4:22Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3=2 > 1 2 2n p pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t 3=2 t 4 > > : s pCd T 21 2 t$T 3p 1 2 n T T The dynamic stress intensity factor time histories for a step and terminated ramp loading are shown in Fig. 4.1. In some problems, it has been shown to be useful to obtain more information about the stress ﬁeld than just the stress intensity factor. For this, the Laplace transforms must then be inverted. Inversion of the bilateral Laplace transform is accomplished by invoking the Cagniard-de Hoop technique. Formally, the inverse of the bilateral Laplace transform is written as ð 1 jþi1 1 s^þ ðx1 ; sÞ ¼ S ðzÞeszx1 sdz ð4:23Þ 2pi j2i1 s2 þ where 2a , j , 0: The path of integration is shown in Fig. 4.2. By completing the contour as indicated in the ﬁgure, the integral in Eq. 4.23 can be converted to an integral along the real axis. The integrand is analytic inside the closed contour and therefore by Cauchy’s theorem, the integral is zero. However, by the decay of Sþ ðzÞ as z ! 1; it is evident that the integral along the circular arcs is zero. Therefore, Eq. 4.23 can be rewritten as an integral along the real axis ð 21 1 Im{Sþ ðzÞ}eszx1 dz s^þ ðx1 ; sÞ ¼ ð4:24Þ 2a ps

Figure 4.1 Variation of the dynamic stress intensity factor with time for a semi-inﬁnite crack with pressure loading. Step and ramp loadings are shown.

Determination of Dynamic Stress Intensity Factors

55

Figure 4.2 Complex z plane indicating the branch cuts and the contour of integration.

The Laplace transform with respect to time is inverted by rearranging the integral in such a manner that it represents the product of two Laplace transforms; the inversion is then immediate by the convolution theorem. This is accomplished by noting that Eq. 4.23 is a product of 1=s and an integral that can be recast in the form of a Laplace transform if zx1 is redeﬁned as 2h ð1 1 h s^þ ðx1 ; sÞ ¼ 2 Im Sþ 2 ð4:25Þ e2sh dh psx1 ax1 x1 The inversion of the Laplace transform is then the following convolution integral ð 1 t h sþ ðx1 ; tÞ ¼ 2 Im Sþ 2 ð4:26Þ Hðt 2 ax1 Þdh px1 ax1 x1 Substituting for Sþ ðzÞ from Eq. 4.16 results in the following expression for the normal stress ahead of the crack line pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð sp Cd ð1 2 2nÞ=2 t Sþ ð2h=x1 Þðc 2 h=x1 Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Hðt 2 ax1 Þdh ð4:27Þ sþ ðx1 ; tÞ ¼ 2 pð1 2 nÞ ax1 h a 2 h=x1 Rescaling the variable of integration h=ax1 ¼ j; Eq. 4.27 can be rewritten in the following form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð t=ax1 sp Cd ð1 2 2nÞ=2 Sþ ð2jaÞðc=a 2 jÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Hðt=ax1 2 1Þ sþ ðx1 ; tÞ ¼ 2 dj ð4:28Þ pð1 2 nÞ j j21 1 Eq. 4.28 expresses the fact that there is neither a characteristic length nor a characteristic time in the problem and hence the ﬁelds must depend only on the ratio t=x1 : Eq. 4.28 was evaluated by Liu et al. (1998) in an effort to determine the build-up of the normal stress ahead of the crack tip for a stationary crack tip. Their calculation is displayed

56

Chapter 4

in Fig. 4.3; in this ﬁgure sþ ðx1 ; tÞ=sp is plotted as a function of Cd t=x1 : Consider a ﬁxed position x1 ahead of the crack tip; Fig. 4.3 shows the time evolution of the stress component s22 at this point. Clearly, upon arrival of the dilatational wave at time t ¼ x1 =Cd (identiﬁed by the point A on the ﬁgure), a negative normal stress of increasing magnitude develops indicating that the initial wave to reach the point x1 is a compressive wave. However, at t ¼ kx1 =Cd (identiﬁed by the point B on the ﬁgure) the shear wave from the crack tip arrives at the point x1 and with it brings an increase in the normal stress; the normal stress does not become tensile until t , 4x1 =Cd : A ﬁnite rise time on the loading pulse as in Eq. 4.21 will increase this delay even more as discussed by Liu et al. (1998). Considering the terminated ramp loading as a sequence of steps, the normal stress ahead of the crack tip can be written as 8 ðt 1 t2t > > s > < T 0 þ ax1 dt 0 , t , T ð4:29Þ s22 ðx1 ; tÞ ¼ ð > 1 T t2t > > : s dt t . T T 0 þ ax1 where sþ ðjÞ is the stress ﬁeld corresponding to the step loading determined in Eq. 4.28. This behavior has important consequences for crack initiation; if a stress-based fracture criterion is used—crack growth will begin when the normal stress s22 at a certain distance, rc, from the crack tip reaches a critical fracture stress, sf—then the crack cannot initiate at least until after t $ 4rc =Cd : It has been suggested by Liu et al. (1998) that this leads to a loading rate dependence on crack initiation even in linearly elastic, brittle materials as described in Chapter 10.

Figure 4.3 Variation of s1(x1, t) ahead of the crack tip for a semi-inﬁnite crack under uniform pressure loading. (Reproduced from Liu et al., 1998.)

57

Determination of Dynamic Stress Intensity Factors

4.1.2 Semi-Inﬁnite Crack Under a Point Load We now consider a problem involving a characteristic length; consider again an unbounded linearly elastic body containing a semi-inﬁnite crack lying along the negative x1 axis with its tip at x1 ¼ 0: The loading is prescribed as a point force of unit magnitude applied at some distance l behind the crack tip as illustrated in Fig. 4.4; the corresponding boundary conditions are as follows

s22 ðx1 ; 0; tÞ ¼ sþ ðx1 ; tÞ 2 dðx1 þ lÞHðtÞ s12 ðx1 ; 0; tÞ ¼ 0 2 1 , x1 , 1 u2 ðx1 ; 0; tÞ ¼ u2 ðx1 ; tÞ 0 , x1 , 1

2 1 , x1 , 1 ð4:30Þ

where the domain for sþ ðx1 ; tÞ is x1 . 0 and the domain for u2 ðx1 ; tÞ is x1 , 0; but both are unknown functions to be determined. If the procedure outlined in the previous section is followed, an equation analogous to the Wiener-Hopf equation in Eq. 4.7 is obtained 2ms

mRðzÞ 1 U2 ðz; sÞ ¼ Sþ ðz; sÞ 2 elsz s b2 aðzÞ

ð4:31Þ

The Rayleigh function is factorized just as indicated in Eq. 4.11; then, the above equation can be rewritten as 2

2msðb2 2 a2 Þ U2 ðz; sÞ 1 ¼ Sþ ðz; sÞFþ ðzÞ 2 elsz Fþ ðzÞ F2 ðzÞ s b2

ð4:32Þ

Unlike the problem in the previous section, the transform of the loading on the right hand side is not of a form where the dependence on s and z may be separated. To circumvent this difﬁculty, Freund (1974b) used a superposition scheme. First, he determined the stress intensity factor corresponding to a moving dislocation corresponding to the following boundary conditions

s22 ðx1 ; 0; tÞ ¼ sþ ðx1 ; tÞ 2 1 , x1 , 1 s12 ðx1 ; 0; tÞ ¼ 0 2 1 , x1 , 1 u2 ðx1 ; 0; tÞ ¼ u2 ðx1 ; tÞ þ Hðvt 2 x1 Þ 0 , x1 , 1

ð4:33Þ

Then, superposing solutions to this problem, the displacements generated in the point load problem (the Lamb problem discussed in Chapter 2) on x1 . 0 were negated. The dynamic

Figure 4.4 Semi-inﬁnite crack with a point force of magnitude P applied at a distance l behind the crack tip.

58

Chapter 4

stress intensity factor was then extracted. Kuo and Chen (1992) re-examined this problem recently and provided a solution by factoring the last term on the right hand side directly. After deﬁning the branches of the function F^ ðzÞ exactly as in Section 4.1.1, it is apparent that the left hand side of Eq. 4.32 is analytic in ReðzÞ , 0 and the ﬁrst term on the right hand side of Eq. 4.32 is analytic in ReðzÞ . 2a: However, the exponential term makes the right hand side become unbounded as z ! 1 and hence this term must be split into two bounded analytical functions; this was accomplished by Kuo (1993). From this point onwards, the procedure for extraction of the stress intensity factor is identical to that described in the previous section. The analytic continuation arguments described in connection with the previous problem can again be applied to show that

Sþ ðz; sÞ ¼

1 Spþ ðz; sÞ s Fþ ð z Þ

ð4:34Þ

where Spþ ðz; sÞ

¼ Fþ ðzÞe

lsz

2

Sp2 ðz; sÞ

1 ¼2 2pi

ð Br

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ elsj a þ j dj ðj þ cÞSþ ðjÞ j 2 z

ð4:35Þ

and Br stands for the Bromwich contour. The transform of the dynamic stress intensity factor is obtained by applying the Abel theorem, as in Eq. 4.17; inverting this, the dynamic stress intensity factor for the point load problem posed in Eq. 4.29 is then found to be rﬃﬃﬃﬃﬃﬃﬃ 8 ð t=l 2 2 b2 dj t > > 2 a, ,b 21 fðjÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2 > p pl l a 1 t=l 2 j > > > > < rﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c2a 2 t ð4:36Þ KIF ðtÞ ¼ 1 2 b, ,c > > ðc 2 t=lÞS pl l ðcÞ 2 > > > rﬃﬃﬃﬃﬃﬃﬃ > > > 2 t > : .c pl l where pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j 2 1ðc=a 2 jÞð2j2 2 b2 =a2 ÞSþ ðajÞ fðjÞ ¼ ½16j2 ðj2 2 1Þðb2 =a2 2 j2 Þ þ ð2j2 2 b2 =a2 Þ4 KIF ðtÞ is the dynamic stress intensity factor per unit load; the superscript F is used to indicate that this solution may be used as a fundamental solution for application in superposition schemes. The stress intensity factor variation with time is shown in Fig. 4.5. Upon arrival of the dilatational wave, a negative stress intensity of small magnitude develops indicating the compressive stress from this wave. Upon arrival of the shear wave, this compression increases without bound, and after passage of the Rayleigh wave, the stress intensity factor settles down at the value equal to the static value. Clearly, the negative stress intensities are not physical and should not appear in sharp cracks. However, in machine-notched cracks, this would imply a closing of the gap.

Determination of Dynamic Stress Intensity Factors

59

Figure 4.5 Time variation of the dynamic stress intensity factor for a point load on a half space at x1 5 2l. (Reproduced from Freund, 1990.)

With the above solution, it is now possible to generate the dynamic stress intensity factors for semi-inﬁnite cracks with arbitrary load variations along the crack surface through superposition. Consider a semi-inﬁnite crack with a non-uniform load distribution behind the crack tip as shown in Fig. 4.6. The dynamic stress intensity factor may then be calculated as ð KI ðtÞ ¼ pðjÞKIF ðt; jÞdj ð4:37Þ where the range of integration is over the load distribution. Consider as an example, the loading condition where a uniform pressure loading is applied over a length L behind the crack tip pðx1 ; tÞ ¼ 2p0 Hðx1 þ LÞHðtÞ

2 1 , x1 , 0

Introducing this loading in Eq. 4.37, the stress intensity factor is obtained as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ KI ðtÞ ¼ 2p0 2L=p for t . cL

ð4:38Þ

ð4:39Þ

Note that while the dynamic stress intensity factor for aL , t , cL may be obtained, only the range given in Eq. 4.39 is of interest in most problems. Solutions to other problems may be constructed by superposition; e.g., consider a semi-inﬁnite crack that is unloaded

Figure 4.6 Semi-inﬁnite crack with a distributed force behind the crack tip.

60

Chapter 4

over a length L near the crack tip; the stress intensity factor is obtained by the superposition of Eqs. 4.19 and 4.39 " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃ# Cd tð1 2 2nÞ=p 2L 2 KI ðtÞ ¼ 2s ð1 2 nÞ p p

for t . cL

ð4:40Þ

Comparison of these theoretical estimates with experimental measurements is discussed in Chapter 9.

4.2 Analysis of Moving Crack Problems The asymptotic ﬁeld near moving cracks discussed in Section 3.2 provides the basis for characterizing stress ﬁeld in terms of a single parameter—the dynamic stress intensity factor. We now turn to an evaluation of the stress intensity factor for moving crack problems. We shall ﬁrst discuss an idealized problem considered by Yoffe, and then discuss a general method for the determination of the stress intensity factor for arbitrary loading. 4.2.1 The Yoffe Problem The ﬁrst analytical solution to the problem of a moving crack was provided by Yoffe (1951), who considered the problem of a crack of constant length moving along a straight line in an inﬁnite two-dimensional medium under remote tractions, symmetric with respect to the crack. This is not a physically realistic situation since it requires the crack to open and grow at one end, but close or heal at the other. However, since the elastodynamic equations are usually solved after decoupling them from the failure criterion, the structure of the crack tip stress ﬁeld is still appropriate as we shall observe later. Yoffe examined the asymptotic ﬁeld to identify possible criteria for crack branching; we shall discuss this aspect in Chapter 11. The solution to the Yoffe problem is presented below, but assuming a mode II loading instead of Yoffe’s original mode I problem. The solution procedure follows the development of Freund (1990) for the mode I problem. Broberg (1999) has presented a solution of the mode II Yoffe problem by requiring the stress intensity factor at the trailing or healing crack tip to go to zero; this model is believed to be a physically realistic picture of slip events in sliding interfaces such as in earthquakes. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Consider an inﬁnite body with stresses decaying to zero as r ¼ x21 þ x22 ! 1: A crack of length 2a is assumed to lie initially along x2 ¼ 0; and to move at a constant speed v , CR : It is under a uniform stress t 1 along the crack faces as indicated in Fig. 4.7; it is possible to convert this problem to one where the crack surfaces are traction-free and the remote loading is a uniform shear, simply by adding a uniform shear t 1 over the entire body. Introducing the Galilean transformation j1 ¼ x1 2 vt; j2 ¼ x2 ; the governing differential equations for the potentials wðx1 ; x2 ; tÞ and cðx1 ; x2 ; tÞ can be rewritten in terms of the distorted polar coordinates as 72 wðrd ; ud Þ ¼ 0;

72 cðrs ; us Þ ¼ 0

ð4:41Þ

Determination of Dynamic Stress Intensity Factors

61

Figure 4.7 The moving Grifﬁth crack considered by Yoffe. A crack of length 2a moves along the x1 axis at a constant speed n. The applied shear stress t 1 follows the crack.

where the coordinates have been rescaled through

zd ¼ rd eiud ¼ j1 þ iad j2 ; zs ¼ rs eius ¼ j1 þ ias j2 ; ð4:42Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ aj aj ð4:43Þ rd ¼ j21 þ a2d j22 ; ud ¼ arctan d 2 ; rs ¼ j21 þ a2s j22 ; us ¼ arctan s 2 j1 j1 The solution that provides the correct mode II symmetry with respect to the crack line can be obtained if we set

wðzd Þ ¼ Re{Fðzd Þ}; cðzs Þ ¼ Im{Gðzs Þ}

ð4:44Þ

where Fðzd Þ and Gðzs Þ are analytic functions in the interior of the body. Symmetry of the potentials require that Fðzd Þ ¼ 2Fðzd Þ and Gðzs Þ ¼ 2Gðzs Þ: The boundary conditions on the crack faces 2a , j1 , a can be written as follows

s22 ðj1 ;0^ Þ ¼ 0; s12 ðj1 ;0^ Þ ¼ 2t1

ð4:45Þ

Note that the stress components are deﬁned in terms of the potentials in Eq. 2.41. Introducing the general solution 4.44 into the boundary conditions 4.45 results in the following equations ð1þ a2s Þ½F 00þ ðj1 Þ2F 002 ðj1 Þþ2as ½G00þ ðj1 Þ2G002 ðj1 Þ ¼ 0 i2t1 2ad ½F 00þ ðj1 ÞþF 002 ðj1 Þþð1þ a2s Þ½G00þ ðj1 ÞþG002 ðj1 Þ ¼ m

ð4:46Þ

F 00^ ðj1 Þ are the limits of F 00 ðzd Þ as j2 !0^ and G00^ ðj1 Þ are the limits of G00 ðzs Þ as j2 !0^ : The ﬁrst equation in 4.46 is satisﬁed by setting G00 ðzÞ ¼ 2

ð1þ a2s Þ 00 F ðz Þ 2 as

ð4:47Þ

Using this in the second equation in 4.46 results in F 00þ ðj1 ÞþF 002 ðj1 Þ ¼ 2

i4as 1 t mRðnÞ

ð4:48Þ

62

Chapter 4

where RðnÞ is the Rayleigh function deﬁned in Eq. 2.51. The particular solution to Eq. 4.48 is F 00 ðzÞ ¼ 2

i2as 1 t mRðnÞ

ð4:49Þ

It is necessary to include the homogeneous solution to Eq. 4.48; note that this corresponds to a traction-free crack. Therefore the homogeneous solution should provide the characteristic stress ﬁeld near moving traction-free crack tips; in fact, crack tip asymptotic solution extracted by Yoffe was later shown to be universally true for growing cracks. Here, we assume that the stresses have a square root singularity at j1 ¼ ^a; and therefore, the homogeneous solution for F 00 ðzÞ in Eq. 4.48 is Az F 00 ðzÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðz þaÞðz 2aÞ

ð4:50Þ

where the form of the numerator is dictated by the fact that s12 !0 as z !1 and the stress ﬁeld must exhibit antisymmetry; A is determined from the condition F 00 ðzÞ!0 at inﬁnity. Using the particular solution in Eq. 4.49 and the homogeneous solution in Eq. 4.50, the function F 00 ðzÞ is found to be " # i2as t1 z 00 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 21 F ðzÞ ¼ ð4:51Þ mRðnÞ ðz þaÞðz 2aÞ The function GðzÞ can be obtained by substituting the above in Eq. 4.47. To obtain the displacement components, it is necessary to ﬁnd F 0 ðzÞ; this is accomplished easily by integrating Eq. 4.51, so that i2as t1 h pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃi z 2 ðz þaÞðz 2aÞ ð4:52Þ F 0 ðz Þ ¼ mRðnÞ The shear stress component ahead of the crack along the line j2 ¼ 0 can then be obtained by using Eqs. 4.51, 4.49 and 4.44 in Eq. 2.41 " # j1 1 ð4:53Þ s12 ðj1 ;0Þ ¼ t pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 21 ðj1 þaÞðj1 2aÞ The crack sliding displacement can then be obtained by using Eqs. 4.52, 4.47 and 4.44 in Eq. 2.40 u1 ðj1 ;0þ Þ ¼

n 2 as t1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ða2 j1 Þðaþ j1 Þ 2a , j1 , a mCs2 RðnÞ

ð4:54Þ

The stress intensity factor at the right crack tip (at j1 ¼ 0) can be determined from the deﬁnition pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ KII ¼ lim 2pðj1 2aÞs12 ðj1 ;0^ Þ ð4:55Þ j1 !a

Determination of Dynamic Stress Intensity Factors

63

pﬃﬃﬃﬃﬃﬃ to be KII ¼ t1 pa; which is the equivalent quasi-static value. From Eq. 4.54, it is clear that the crack sliding displacement is not the same as in the quasi-static problem, but is distorted by a velocity-dependent term. Other elements of the stress and deformation ﬁeld can be extracted from the solution developed above, but are usually not of much interest. Other investigators have examined steady-state crack growth problems. Craggs (1960) provided the solution to the steady-state problem of a semi-inﬁnite crack in an inﬁnite solid under crack face loading that moves along with the crack tip. Freund (1990) has presented a discussion of the Yoffe problem for mode I, the problem of a concentrated shear load at a point behind the crack and a cohesive zone model. A common feature of all these solutions is that the stress intensity factor is constant and independent of the crack speed n.

4.2.2 Dynamic Stress Intensity Factors for Moving Cracks Broberg (1960) obtained the solution to a self-similar crack extension problem; he considered the problem of a center crack extending symmetrically at constant velocity from zero length under remotely applied loads. Baker (1962) evaluated the dynamic stresses caused by a semi-inﬁnite crack that suddenly appeared and grew at a constant speed under a uniform pressure loading on the semi-inﬁnite crack faces. A bibliography at the end provides a long list of articles that deal with the evaluation of the dynamic stress intensity factor for propagating cracks. In this section, we conﬁne attention to the problem of a semi-inﬁnite pressurized crack and determine the stress intensity factor variation with time for arbitrary crack motion. Freund (1972a,b, 1973, 1974a) addressed this problem in a series of four papers. Kostrov (1975a), Burridge (1976), Slepyan (2002), Willis (1989) and others have also contributed to this problem. One of the general methods of solving the equations of elastodynamics for crack problems (mixed-boundary value problems) is the Wiener-Hopf method. The method was described earlier in connection with the problem of a stationary semi-inﬁnite crack with uniform pressure loading over the crack tip. In an effort to circumvent the difﬁculties associated with the application of the transform methods and the Wiener-Hopf method for these mixed-boundary value problems with moving boundaries, Freund (1972a,b, 1973, 1974a,b) constructed a sequence of solutions ﬁrst for a constant velocity crack under timeindependent loading, then for an arresting crack, and ﬁnally for a crack under timedependent loading. Then by considering a crack growing at a nonuniform speed to be the limit of a sequence of piecewise constant velocity segments, Freund showed the following remarkable result: The stress intensity factor for mode I extension of a half-plane crack is given by the universal function k(n) times the stress intensity factor appropriate for a crack of ﬁxed length, equal to the instantaneous length, subjected to the given applied loading, whether this loading is time-independent or time-dependent (Freund, 1973); this is expressed as ð1 2 n=CR Þ ﬃ KI ðt; l; _lÞ ¼ kðnÞKI0 ðt; l; 0Þ; kðnÞ < pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 n=Cd

ð4:56Þ

64

Chapter 4

Figure 4.8 Variation of the universal function kðnÞ:

where KI0 ðt; l; 0Þ is the stress intensity factor corresponding to a stationary crack of length l. The function kðnÞ is plotted in Fig. 4.8. Kostrov addressed the issue of a crack growing at a variable speed ﬁrst under mode III loading (Kostrov, 1966) and then for the modes I and II cases (Kostrov, 1975) using a Green’s function approach. Burridge (1976), Slepyan (2002) and Willis (1992) have also contributed to the development of the solution to this problem. We describe the Green’s function approach here. For the crack growth problems of interest here, the method described in Section 4.1 breaks down because tractions are prescribed only on the crack surfaces and in general we are faced with a mixed-boundary value problem. The general solution in the transform space given in Eqs. 2.65 are still appropriate, but the boundary conditions become complicated. For a mode I problem, the boundary conditions are

s22 ðx1 ; 0þ ; tÞ ¼ s2 ðx1 ; tÞ

for 2 1 , x1 , lðtÞ

þ

for x1 . lðtÞ

þ

for 2 1 , x1 , 1

u2 ðx1 ; 0 ; tÞ ¼ 0

s12 ðx1 ; 0 ; tÞ ¼ 0

ð4:57Þ

where lðtÞ is the position of the crack tip. Attention will be restricted to cases when _lðtÞ , CR : Substituting these boundary conditions into Eqs. 2.71 and 2.72 yields u1 ðx1 ; tÞ ¼ G12 p s2 u2 ðx1 ; tÞ ¼ G22 p s2

ð4:58Þ

where u2 is unknown for x1 , lðtÞ and s2 is unknown for x1 . lðtÞ: Similarly, for the mode II problem, the boundary conditions are

s22 ðx1 ; 0þ ; tÞ ¼ 0 þ

u1 ðx1 ; 0 ; tÞ ¼ 0 þ

for 2 1 , x1 , lðtÞ for x1 . lðtÞ

s12 ðx1 ; 0 ; tÞ ¼ s1 ðx1 ; tÞ for 2 1 , x1 , 1

ð4:59Þ

Determination of Dynamic Stress Intensity Factors

65

where lðtÞ is the position of the crack tip. Attention will again be restricted again to cases when _lðtÞ , CR : Substituting these boundary conditions into Eqs. 2.71 and 2.72 yields u1 ðx1 ; tÞ ¼ G11 p s1 u2 ðx1 ; tÞ ¼ G21 p s1

ð4:60Þ

where u1 is unknown for x1 , lðtÞ and s1 is unknown for x1 . lðtÞ: From Eqs. 4.58 and 4.60, it is clear that only the second of Eq. 4.58 and the ﬁrst of Eq. 4.60 need to be considered. We examine these together by simply considering the convolution integral of the form u¼Gps

ð4:61Þ

where the u2 and s2 components are considered for the mode I problem and the u1 and s1 components are considered for the mode II problem. Reiterating, in Eq. 4.61 u is unknown for x1 , lðtÞ; and s is unknown for x1 . lðtÞ: First, the stress component is factored into two parts

s ¼ s þ þ s2

ð4:62Þ

where sþ ¼ 0; and s2 is prescribed for x1 , lðtÞ and s2 ¼ 0; sþ is to be determined for x1 . lðtÞ: Next, the Green’s function is factored as G ¼ G2 p Gþ

ð4:63Þ

The details of this factorization may be found in the book by Slepyan (2002). Substituting Eqs. 4.62 and 4.63 into Eq. 4.61 and using the fact that u ¼ 0 for x1 . lðtÞ results in the following G2 p ½Gþ p ðsþ þ s2 Þ ¼ 0

for x1 . lðtÞ

ð4:64Þ

The supports for Gþ and G2 are in the range CR t , x1 , Cd t and 2CR t , x1 , 2Cd t; respectively; therefore, the convolution integrals have to be evaluated only in these ranges. Next, Eq. 4.64 yields Gþ p sþ ¼ 2Gþ p s2

for x1 . lðtÞ

ð4:65Þ

The solution for sþ is then given by

sþ ¼ 2G21 þ p ðGþ p s2 Þ

for x1 . lðtÞ

ð4:66Þ

Eq. 4.66 represents the solution to the unknown function sþ for x1 . lðtÞ: In evaluating the convolution integrals, it is important to consider the domain of interest for each of the constituents. First, the support for the Gþ factor is contained in CR t , j , Cd t (Willis, 1989; Slepyan, 2002); second, since s2 is zero for x1 . lðtÞ; only the region x 2 j . lðt 2 tÞ contributes to the integral. Introducing these supports and writing out the convolution integrals in Eq. 4.66 ð1 ðt sþ ¼ 2 G21 þ ðj; tÞQðx1 2 j; t 2 tÞHðj 2 CR tÞ 21

0

£ HðCd t 2 jÞHðx1 2 j 2 lðt 2 tÞÞdj dt

ð4:67Þ

66

Chapter 4

where Q ¼ Gþ p s2 represents the convolution of the applied loading with the Green’s function. Substituting the solution in Eq. 4.66 into Eq. 4.60 results in the following solution for the unknown displacement u2 ¼ 2G2 p ðGþ p s2 Þ

for

x1 , lðtÞ

ð4:68Þ

Introducing the supports for the functions in the region x1 , lðtÞ yields u2 ¼ 2

ð1 ðt 21

0

G2 ðj; tÞQðx1 2 j; t 2 tÞHð2j 2 CR tÞHðCd t þ jÞ

£ Hð2x1 þ j þ lðt 2 tÞÞdjdt

ð4:69Þ

The quantity of interest in these crack problems is typically the stress intensity factor. With this in mind, Eq. 4.66 is evaluated as x1 2 lðtÞ ! 0þ ; and hence as j ! 0þ and t ! 02 ðð

sþ , 2QðlðtÞ; tÞ

dþ

G21 þ ðj; tÞdj dt

ð4:70Þ

where d þ stands for the domain: j , Cd t; j . CR t; and j , x1 2 lðtÞ þ _lðtÞt; note that _lðt 2 tÞ has been set equal to _lðtÞ 2 vðtÞt as t ! 02 : Similarly, the solution for the displacement behind the crack tip can be written as ðð u2 , QðlðtÞ; tÞ G2 ðj; tÞdjdt ð4:71Þ d2

where d 2 stands for the domain bounded by the lines: j . 2Cd t; j , 2CR t; and j . x1 2 lðtÞ þ _lðtÞt; The ﬁnal evaluation of the solution requires that the Green’s function be factored as in Eq. 4.63; these are described in detail by Willis (1989) and Slepyan (2002). However, the crux of the main result is apparent in Eqs. 4.70 and 4.71; the dependence of the stress amplitude on the applied loading arises only through the instantaneous value of QðlðtÞ; tÞ; corresponding to the current crack length and current time. The integrals over the domains d þ and d 2 depend only on the distance from the crack tip ðx1 2 lðtÞÞ and the crack speed _lðtÞ; also note that the structure of the crack tip singularity is contained in this integral. The ﬁnal result of the evaluation is QðlðtÞ; tÞ sþ ðx1 ! lðtÞ; tÞ , 2kðnÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x1 2 lðtÞ

ð4:72Þ

where ð1 2 n=CR Þ ﬃ kðnÞ ¼ S2 ð1=nÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 n=Cd

ð4:73Þ

where S2 ð1=nÞ is deﬁned in Eq. 3.12. As Freund (1990) notes, S2 ð1=nÞ is close to 1 over most of range and may be replaced by unity in Eq. 4.73. It is worth reiterating the main

Determination of Dynamic Stress Intensity Factors

67

result already displayed in Eq. 4.56. The dynamic stress intensity factor for a crack propagating at an arbitrary speed n can be written as the product of a universal function of the crack speed and the instantaneous stress intensity factor of a stationary crack of equivalent crack length KI ðt; lðtÞ; nÞ ¼ kðnÞKI0 ðt; lðtÞ; 0Þ

ð4:74Þ

where KI0 ðt; l; 0Þ is the stress intensity factor corresponding to a stationary crack of length l at time t. The result above was obtained for a semi-inﬁnite crack in an unbounded medium; however, as Freund (1990) suggests this result can be generalized to any complex geometrical and boundary conditions through a superposition of solutions to problems involving semi-inﬁnite crack problems. For ﬁnite crack problems, the result remains valid until the waves from one crack tip arrives at the other crack tip as demonstrated by Kostrov (1975). A similar result may be derived for the problem of mode II crack growth; the expression corresponding to Eq. 4.74 is obtained by replacing the dilatational wave speed by the distortional wave speed. As an example of the application of this method of determining the dynamic stress intensity factor, consider a semi-inﬁnite crack, loaded by a step tensile stress wave of amplitude s p as illustrated in Fig. 4.9. When the stress wave reaches the crack plane at time t ¼ 0; the traction-free boundary condition on this plane would require that a uniform compression be generated on the crack plane, regardless of whether the crack is stationary or moving. For a stationary crack, this corresponds to the uniformly pressure-loaded semiinﬁnite crack considered in Section 4.1 and the stress intensity factor is therefore given in Eq. 4.19. If the crack begins to grow at a speed n at some time t ¼ t; the stress intensity factor is still given by Eq. 4.19, but with the multiplying factor kðnÞ; the complete stress

Figure 4.9 (a) Stress pulse incident on a semi-inﬁnite crack. (b) Equivalent semi-inﬁnite crack with uniform pressure loading on the crack surfaces.

68

Chapter 4

intensity factor history for a growing crack can be written as 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 2sp Cd tð1 2 2vÞ=p > > > < ð1 2 vÞ KI ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > > 2sp Cd tð1 2 2vÞ=p > > kð n Þ : ð1 2 vÞ

for 0 , t , t ð4:75Þ for t . t

Note that there is an abrupt drop in the stress intensity factor at crack initiation associated with the sudden transition from a stationary crack to a crack moving at a speed n. Next, consider that the loading is actually applied on the crack plane rather than by an impinging stress wave; then, as the crack extends, the newly created crack must be traction free; thus

s22 ðx1 ; 0þ ; tÞ ¼ 2sp

for 2 1 , x1 , 0

þ

for 0 , x1 , lðtÞ

þ

for x1 . lðtÞ

s22 ðx1 ; 0 ; tÞ ¼ 0 u2 ðx1 ; 0 ; tÞ ¼ 0

s12 ðx1 ; 0þ ; tÞ ¼ 0

ð4:76Þ

for 2 1 , x1 , 1

Let the crack be stationary for time t , t and further, let the crack extend at a constant speed, n along the x1 axis. This problem is analogous to that considered at the end of Section 4.1.2. We now have to superpose the solution to a semi-inﬁnite pressure-loaded crack to that where the pressure loading is applied only over the newly created crack surface, thus creating a traction free crack plane. Thus, we superpose the solution in

Figure 4.10 Variation of the dynamic stress intensity factor (normalized by its value at t 5 t) with time (normalized by the time at initiation, t) for a crack growing at a constant speed (v/Cd 5 0.1, v 5 0.3).

69

Determination of Dynamic Stress Intensity Factors

Eq. 4.19 to the solution in Eq. 4.40, letting L ¼ nðt 2 tÞ: In addition, the multiplicative factor kðnÞ must be introduced for the growing crack 8 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > Cd tð1 2 2vÞ=p p > > 2s > > ð1 2 vÞ < KI ¼ " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ# > > Cd tð1 2 2vÞ=p 2nðt 2 tÞ > p > 2 > : 2s kðnÞ ð1 2 vÞ p

t,t ð4:77Þ t.t

Note again that there is an abrupt drop in the stress intensity factor at crack initiation. A sketch of the time variation of the dynamic stress intensity factor for stress pulse loading and for the case of crack face loading over the original crack surfaces is shown in Fig. 4.10. Thus, the main advantage of Eq. 4.74 is obvious; we can take any known solution for a stationary crack and then determine the stress intensity factor variation for a dynamically growing crack. In the next chapter we examine the results of an experimental implementation of the problem. It is important to recognize that what we have at this stage are dynamically admissible solutions; given a uniform pressure loading on the original semi-inﬁnite crack surface and given that crack initiation occurs at t ¼ t; and furthermore, that the crack grows at a constant speed n, Eq. 4.74 yields the time variation of the dynamic stress intensity factor. A criterion for the dissipation must be imposed to establish the conditions at crack initiation and for establishing the crack tip equation of motion. However, from the nature of the function kðnÞ; it is clear that the dynamic stress intensity factor for a growing opening mode crack will vanish as the crack speed approaches the Rayleigh wave speed and thus from the elastodynamic point of view, such cracks cannot propagate at a speed larger than the Rayleigh wave speed.

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71

Chapter 5 Energy Balance and Fracture Criteria

The discussion so far has focused on stress analysis. Given a cracked body subjected to prescribed loading and boundary conditions (including moving boundary conditions for growing cracks), the nature of the crack tip stress ﬁeld, and the methods used for the characterization of these ﬁelds have been discussed. No discriminating criteria were posed to determine when a stationary crack under dynamic loading would begin to grow, a growing crack would arrest or to determine the crack growth speed, other than to describe that the crack growth must satisfy the energy balance equation posed in Eq. 3.6. In this chapter, the energy balance equation is examined and the crack tip equation of motion is extracted, thus obtaining the energetic basis of the dynamic fracture criterion.

5.1 Energy Balance Equation Failure criteria for dynamic fracture may be motivated by an extension of Grifﬁth’s ideas postulated for equilibrium cracks. Mott (1948) suggested that for rapidly growing cracks, kinetic energy must be incorporated in writing down the energy balance. However, the correct and complete formulation of the local energy rate balance equation was not provided until after the nature of the dynamic stress ﬁeld had been determined completely. The works of Atkinson and Eshelby (1968), Kostrov and Nikitin (1970), Freund (1972a) and Willis (1975) are especially important in this development. The energy rate balance may be obtained by adding the rate of dissipation in Eq. 3.6. With this in mind, we begin by considering the rate of change of the strain and kinetic energy. ˙ be the rate of change of stored elastic energy and T˙ the rate of change in the kinetic Let U energy, then ð

d 1 ˙ ˙ ð5:1Þ sab 1ab þ ru˙ a u˙ a dA UþT¼ dt R 2 The integral Eq. 5.1 could be applied on any closed contour within the body; in the application to the crack problem, R is any region near the crack tip bounded by the curve ›R: Let us assume crack propagation along the x1 direction at a constant speed v:

72

Chapter 5

We apply the energy rate balance equation to the contour around the crack tip ›R illustrated in Fig. 5.1; however, due to the singularity of the stress and strain ﬁelds, we must exclude a small region near the crack tip. This region is enclosed by the contour labeled G and is indicated by RG : Thus, the rate of change of the strain energy and kinetic energy can be written as ð d 1 ˙ ˙ ½s 1 þ ru˙ a u˙ a dA UþT¼ ð5:2Þ dt R2RG 2 ab ab It should be noted that in evaluating the integrals above, the limits of integration vary with time as well; hence Reynolds transport theorem should be invoked in taking the time derivatives indicated in the terms on the right-hand side of Eq. 5.2. Applying this to Eq. 5.2, ð ð 1 ˙ þ T˙ ¼ lim ½sab u˙ a;b þ ru˙ a u€ a dA þ lim ½ru˙ a u˙ a þ sab ua;b vn1 ds ð5:3Þ U G!0 R2RG G!0 2 G where v is the speed of translation of the contour G and n1 is the component of the outward normal to the contour in the direction of translation of the contour. The ﬁrst integral in Eq. 5.3 can be simpliﬁed by replacing ru€ a ¼ sab;b and then applying the divergence theorem to yield ð G!0

ð

ð

lim

R2RG

½sab u˙ a;b þ ru˙ a u€ a dA ¼

›R

sab u˙ a na ds þ lim

G!0

G

sab u˙ a na ds

ð5:4Þ

The fact that the newly created crack surfaces are traction free has been used in writing the above equation. The ﬁrst term on the right-hand side of Eq. 5.4 is the power of the external forces, P: Eq. 5.3 can now be rewritten as ð ˙ þ TÞ ˙ ¼ 2 lim P 2 ðU

G!0

G

sab u˙ a nb þ

1 ðru˙ a u˙ a þ sab ua;b Þvn1 ds 2

Figure 5.1 Crack tip contour for evaluation of the energy ﬂux integral.

ð5:5Þ

Energy Balance and Fracture Criteria

73

The integral in Eq. 5.5 represents the amount of energy ﬂowing out of the region R and into the crack tip region through the contour G: The crack tip energy ﬂux integral is then deﬁned as follows: ð 1 F¼ sab nb u˙ a þ ðrua ua þ sab ua;b Þvn1 ds ð5:6Þ 2 G This energy ﬂux is dissipated in the crack tip process zone as the crack propagates along the x1 direction at a constant speed v: While the energy ﬂux integral in Eq. 5.6 is in general dependent on the path of integration, for the case of steady-state crack growth assumed in the present analysis, it turns out to be independent of the path G: If D is the total dissipation in the fracture process, dD=dt ¼ vdD=da ¼ vg is the rate of energy dissipation at the moving crack tip, where g is the fracture energy per unit extension of the crack and a is the crack length. Note that the fracture energy g can incorporate the surface energy as in the Grifﬁth model as well as other dissipative processes as long as the idea of small scale yielding is appropriate under the dynamic conditions. Thus, F ¼ vg: In analogy with the quasi-static crack problems, it is possible to deﬁne a dynamic energy release, G by G ¼ v21 F ¼ g

ð5:7Þ

The dynamic energy release rate is the energy released into the crack tip process zone per unit crack extension and must be equal to the dissipation per unit extension. With this formal equivalence, application of the fracture criterion to dynamic problems can be contemplated in much the same way as in quasi-static fracture, but with the additional expectation that Eq. 5.7 will provide the crack growth law as opposed to just the condition for crack initiation in the quasi-static problem. Introducing the elastodynamic singular stress ﬁeld from Eq. 3.32 in Eq. 5.7, G can be related to the dynamic stress intensity factor: G¼

1 2 n2 1 2 ½AI ðvÞK12 þ AII ðvÞKII2 þ A ðvÞKIII 2m III E

v2 a d v 2 as 1 AI ðvÞ ¼ ðvÞ ¼ ; A ; and AIII ðvÞ ¼ II as ð1 2 nÞCs2 RðvÞ ð1 2 nÞCs2 RðvÞ

ð5:8Þ

ad and as are deﬁned in Eqs. 3.12 and 3.20 and RðvÞ is deﬁned in Eq. 2.51. E is the modulus of elasticity and n is the Poisson’s ratio. A number of important features should be borne in mind in using the energy balance equation. First, it has been assumed that the fracture energy is a material constant; we shall discuss its dependence on the temperature and rate of loading as well as problems involved in evaluating it from experiments in later chapters. Second, the functions AI ðvÞ and AII ðvÞ in Eq. 5.8 are singular as v ! CR and the function AIII ðvÞ is singular as v ! Cs : To satisfy the energy balance equation, the dynamic stress intensity factors KI ðt; vÞ and KII ðt; vÞ must tend to zero as v ! CR and KIII ðt; vÞ must tend to zero as v ! Cs This implies that the limiting crack speed in modes I and II is the Rayleigh wave speed and in mode III, the shear wave speed. Observed limiting speeds are signiﬁcantly smaller than this even in nominally brittle materials; the reasons for this discrepancy will be examined

74

Chapter 5

later. Under shear dominant loading (mode II), it has been found that cracks may grow faster than the shear wave speed, and particularly at square root of two times the shear wave speed. Such intersonic cracks have been observed in weak planes, and are discussed brieﬂy in Section 3.4. Third, while the energy release rate is written for an arbitrary mixed-mode loading condition, the crack does not grow along a straight path in the case of mixed-mode loading unless forced by the presence of weak planes, interfaces or other constraints; the above energy-based formulation does not provide the means for determination of the crack growth direction. The criterion of local symmetry—i.e. that the crack will choose to grow in that direction along which the crack experiences a locally symmetric deformation—is commonly invoked in fracture mechanics and hence it is sufﬁcient to consider the fracture criterion for mode I loading. Finally, in dynamic formulations, due to practical complications in evaluating Eq. 5.8, the fracture criterion is simply evaluated in terms of the stress intensity factors rather than in terms of energy as described in Section 5.2.

5.2 Dynamic Failure Criterion Restricting our attention to mode I loading, and substituting Eq. 5.8 into Eq. 5.7, and the general result in Eq. 4.56 the crack tip equation of motion can be written in terms of the dynamic stress intensity factor: 1 þ n v 2 ad ½kðvÞKI0 ðt; lðtÞ; 0Þ2 ¼ g E Cs2 RðvÞ

ð5:9Þ

This is an implicit equation for the crack speed; if the dynamic stress intensity factor variation is determined, and if the total energy dissipation per unit extension of the crack is known, Eq. 5.9 may be used to determine the crack speed. In an experiment aimed at the determination of the dissipation, independent and simultaneous measurement of the dynamic stress intensity factor and the crack speed may be used to determine the dissipation; this is the energetic basis of the dynamic crack growth criterion. There has been a long debate on the suitability and applicability of the above equation of motion in dynamic problems, motivated by experimental results that indicated signiﬁcant departures from the criterion embodied in Eq. 5.9. These departures are attributed to specimen geometry dependence, strain rate dependence, acceleration dependence, and in some cases to the lack of dominance of the singular term in the fracture process and are discussed further in Chapters 9 to 11. For practical use, the dynamic fracture criterion is traditionally separated into three parts—a dynamic crack initiation criterion, a dynamic crack growth criterion, and a dynamic crack arrest criterion—and imposed independently on the growing crack. Corresponding to each criterion, dynamic initiation toughness, dynamic propagation toughness and dynamic crack arrest toughness are then deﬁned as independent material properties. These criteria and practical application are discussed here; their experimental determination is discussed in Chapter 10.

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75

5.3 Dynamic Crack Initiation Toughness Since the state of stress near the crack tip is described in terms of the dynamic stress intensity factor, KIdyn ; crack initiation can be identiﬁed with the stress intensity factor reaching a critical value, just as in the case of quasi-static fracture. Therefore, the crack initiation criterion can be postulated as follows: KIdyn ðtf Þ ¼ KId ðT; K˙ Idyn Þ

ð5:10Þ

The right-hand side represents the dynamic initiation toughness, with the subscript d replacing the subscript C used for the critical stress intensity factor for the quasi-static plane strain fracture toughness. The dependence of the dynamic crack initiation toughness on the temperature and rate of loading is indicated through the arguments; this dependence must be determined through experiments covering the range of temperatures and rates of loading of interest. The temperature dependence of the dynamic crack initiation toughness arises from the increase in ductility with increasing temperature or from heating associated with inelastic deformation in the near tip zone or a combination of both. The rate dependence could have two possible sources: one arising from the rate dependence of the inelastic material response within the process zone and the other arising form the inertial nature of the development of the stress ﬁeld near the crack tip and in the fracture process zone. An important consequence of the inertial contribution is that even for nominally brittle materials (with constant fracture energy), the crack growth criterion expressed in the form of Eq. 5.10 must indicate rate dependence. The left-hand side of Eq. 5.10 represents the applied stress intensity factor at time tf when crack propagation commences; calculation of the applied stress intensity factor for dynamic problems is signiﬁcantly more difﬁcult than in the quasi-static problems as we have seen in Chapter 4; analytical methods of determining the time variation of the dynamic stress intensity factor are discussed by Freund (1990), Broberg (1999), Slepyan (2002) and others. However, only a few analytical solutions are available and the use of numerical simulations or the use of local measurements suitably interpreted in terms of the dynamic stress intensity factor is required to evaluate KIdyn for complicated geometric and loading conditions. Key factors inﬂuencing the proper characterization of the dynamic initiation toughness as a material property have not been fully explored; therefore, restrictions necessary for standardizing this as a fracture criterion have not yet been established. The technical committees of standards organizations such as the European Society for Structural Integrity are currently working on drafting a standard for the initiation toughness. Experimental investigations into the determination of dynamic crack initiation toughness are described in detail in Chapter 10.

5.4 Dynamic Crack Growth Toughness Once dynamic crack growth has been initiated as per the conditions of the dynamic initiation toughness, subsequent growth must be determined through a separate criterion

76

Chapter 5

that characterizes the energy rate balance during growth. The dynamic stress ﬁeld near a growing crack is still characterized by the dynamic stress intensity factor, but now this is a function of loading, time, crack position, and speed and is represented as KIdyn ðt; vÞ: The energy rate balance in Eq. 5.8 can be expressed as a relation between the instantaneous dynamic stress intensity factor and the toughness equivalent of g; called the dynamic crack growth toughness. Note that this toughness must be a function of the crack speed and possibly the rate of loading and temperature. Thus, the dynamic crack growth criterion is written as KIdyn ðt; vÞ ¼ KID ðv; K˙ Idyn ; TÞ

ð5:11Þ

The upper case subscript D is used to indicate the dynamic crack growth toughness instead of the lower case d used to indicate the dynamic initiation toughness. Once again, the right-hand side represents the material property to be characterized through experiments and the left-hand side represents the dynamic stress intensity factor calculated form the solution of the boundary initial value problem in elastodynamics. A schematic diagram of the dynamic crack growth toughness is shown in Fig. 5.2. In numerical simulations, the crack extension must be imposed in such a manner that Eq. 5.11 is satisﬁed at each increment in time. Some important caveats must be recognized in using Eq. 5.11 for dynamic crack growth problems. First, the crack initiation point is not on the curve characterizing the dynamic crack growth criterion. Thus, KID ðv ! 0; K˙ Idyn ; TÞ – KId ðK˙ Idyn ; TÞ

ð5:12Þ

This could be possibly due to bluntness of the initial crack, due to intrinsic rate dependence of the material, or due to inertial effects. An interesting consequence of this difference between crack initiation and growth toughness is that the crack will jump to a large ﬁnite speed immediately upon initiation as indicated by the arrow in Fig. 5.2. Second, experimental efforts to determine KID ðv; K˙ Idyn ; TÞ have had mixed success; variations in the measurements with different specimen geometries and loading schemes have not been resolved completely. Also, indications that the measurements

Figure 5.2 Dynamic crack growth criterion.

Energy Balance and Fracture Criteria

77

are hysteretic—meaning that accelerating and decelerating cracks exhibit different behavior—have been reported. These discrepancies have been more prevalent in nominally brittle materials than in ductile materials (still under small scale yielding). Third, crack arrest does not appear naturally in the limit KID ðv ! 0; K˙ Idyn ; TÞ; necessitating the postulate of a separate crack arrest criterion. This is attributed to the fact that as v ! 0 the crack speed is very sensitive to small changes in the stress intensity factor, rendering the dynamic crack growth criterion unable to treat crack arrest in a reliable manner. Experimental investigations into the determination of dynamic crack growth toughness are described in detail in Chapter 10.

5.5 Dynamic Crack Arrest Toughness The differences between the crack initiation toughness and the crack growth toughness also introduce complications in the characterization of crack arrest. Crack arrest is not the reversal of initiation and hence the initiation toughness is not relevant for crack arrest; also, the characterization of the crack growth criterion at very slow speeds is quite difﬁcult. This can be appreciated by examining the steep increase in the crack speed with a small change in the dynamic stress intensity factor, particularly in brittle materials. The large scatter in data obtained in this regime has led to the postulation of a separate criterion for crack arrest: the dynamic crack arrest toughness is deﬁned as the smallest value of the dynamic stress intensity factor for which a growing crack cannot be maintained; thus the crack arrests when KIdyn ðtÞ , KIa ðTÞ

ð5:13Þ

In applications, the most conservative approach to design would utilize Eq. 5.13, thus assuring that the dynamic stress intensity factor for all possible loading conditions never exceeds the crack arrest toughness. Thus a dynamically growing crack is never encountered in the lifetime of the structure and one avoids the complications in computing the dynamic stress intensity factors for growing cracks and in determining the dynamic crack growth criterion. The ASTM has developed a standard procedure E-1221-96 for the experimental determination of crack arrest toughness, KIa ; for ferritic steels. Interpretation of experimental data in E-1221 is through a static analysis although the phenomenon of crack growth and arrest are clearly dynamic; the static analysis is justiﬁed on the basis that it provides a conservative estimate of the crack arrest toughness. In a sense, this harkens back to the procedures based on empirical correlations based on measurements of ‘impact fracture energy’ and transition temperature. Codes such as the ASME Boiler and Pressure Vessel Code have also incorporated requirements on the characterization and use of the crack arrest toughness in the design and evaluation of pressure vessels and pipelines. Experimental investigations into the determination of dynamic crack arrest toughness are described in detail in Chapter 10.

78

Chapter 5

5.6 Application of Dynamic Failure Criteria Thus, with the formulation of the three separate criteria for initiation, growth and arrest, dynamic analyses may be performed to evaluate structural integrity allowing for rapid crack propagation. The process for imposition of the fracture criteria is then shown schematically in Fig. 5.2. Crack initiation occurs at a value of KId ðT; K˙ Idyn Þ; the crack then jumps to a point ðv=CR ; KID kðvÞ ¼ KId ðvÞÞ on the dynamic crack growth toughness curve and arrests if the stress intensity factor reaches the value KIa ðTÞ: In general, the variation of the dynamic stress intensity factor KIdyn ðt; vÞ under general loading for arbitrarily varying velocity of the crack is not known; this makes application of the criterion described above difﬁcult. We illustrate the application of the dynamic failure criteria described above through a speciﬁc example. Consider a semi-inﬁnite crack lying along x1 , 0; x2 ¼ 0: The load on the crack surfaces is given by two opposing point forces, P at a Point L behind the crack tip, as illustrated in Fig. 5.3; therefore, the boundary conditions are

s22 ðx1 ; 0; tÞ ¼ 2Pdðx1 þ LÞ; s12 ðx1 ; 0; tÞ ¼ 0 for x1 , 0

ð5:14Þ

dð·Þ is the Dirac delta function. Furthermore, let the crack grow along the line x2 ¼ 0 with an arbitrary speed given by ˙lðtÞ; where the superscript dot indicates differentiation with respect to time. The situation depicted here is quite similar to the test geometry of the ASTM E-1221 procedure for determination of crack arrest toughness for very short times when the effects of the ﬁnite geometry of the specimen are not felt at the crack tip. The stress intensity factor for a stationary crack is given by (see Eq. 4.36) sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 dyn ð5:15Þ KI ðt; 0Þ ¼ P pðL þ lÞ Therefore, for the case of the propagating crack, using Eq. 4.56, we get the following expressions for the time variation of the dynamic stress intensity factor: rﬃﬃﬃﬃﬃﬃﬃﬃ 2 dyn for t # 0 KI ðt; 0Þ ¼ P pL ð5:16Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 KIdyn ðt; ˙lÞ ¼ kð˙lÞP for t $ 0 pðL þ lðtÞÞ

Figure 5.3 Concentrated load applied at a distance L behind the initial crack.

Energy Balance and Fracture Criteria

79

Now, imposing the dynamic crack initiation criterion at t ¼ 0; crack growth begins when rﬃﬃﬃﬃﬃﬃﬃﬃ 2 ð5:17Þ ¼ KId ðT; K˙ Idyn Þ P pL If the load at crack initiation is measured, Eq. 5.17 provides a convenient way of determining the dynamic initiation toughness and its dependence on the loading rate and temperature. Since the load is applied quasi-statically, this experiment cannot be used to determine the loading rate dependence of the initiation toughness. On the other hand, if the dynamic initiation toughness is known, the load at failure can be estimated. Once crack growth has initiated, the crack speed must be obtained by equating the dynamic stress intensity factor calculated for the growing crack in Eq. 5.16 to the dynamic crack growth toughness. Thus, sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ˙ kðlÞP ¼ KID ði; K˙ Idyn ; TÞ ð5:18Þ pðL þ lðtÞÞ Eq. 5.18 is a nonlinear ordinary differential equation for the crack growth speed at initiation. If the dynamic crack growth toughness is known, this equation must be solved numerically in order to determine the variation of lðtÞ: On the other hand, if the load and the crack speed are measured independently, Eq. 5.18 can be used in the characterization of the dynamic crack growth toughness. It should be noted that there exists an abrupt jump in the crack speed at initiation since the stress intensity factor drop by a factor kðvÞ: Finally, in this loading arrangement, it is clear that if the load is constant, the stress intensity factor drops as the crack extends; thus crack arrest must appear at some time. Using the dynamic crack arrest criterion, the crack will come to a stop when sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ˙ ¼ KIa ðTÞ ð5:19Þ kðlÞP pðL þ lðtÞÞ Once again, if the crack arrest toughness is known, Eq. 5.19 provides the crack length at crack arrest. On the other hand, if the load and crack arrest length are measured, the arrest toughness can be determined.

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81

Chapter 6 Methods of Generating Dynamic Loading

From Eq. 3.32, it is clear that the stress state in the crack tip region that dominates the fracture process zone is proportional to the dynamic stress intensity factor; hence, the rate of loading or the strain rate near the crack tip is proportional to the rate of increase of the dynamic stress intensity factor K_ Idyn ðt; vÞ: In experiments, the initial rate of loading on the stationary crack has been varied in the range from essentially zero in quasi-static loading p situations to about 108 MPa m/s in projectile impact experiments. Such high strain rates are expected in many applications associated with impact, explosive loading or catastrophic failure in pressure vessels and pipelines. However, the rate of change of the dynamic stress intensity factor after the crack begins to grow is determined primarily by the crack growth rate itself and therefore by the material, and is typically of the order of p 105 –106 MPa m/s in most cases. Therefore, experiments aimed at characterizing crack initiation have to cover a larger range of loading rates than those aimed at evaluating crack growth or arrest criteria. Numerous methods have been used to generate reproducible loading for the purpose of examining dynamic crack growth, ranging from static or quasistatic loading of blunt cracks to impact with projectiles at speeds ranging from 5 to 100 m/s. Furthermore, novel methods such as electromagnetic loading, pulse loading derived from a Hopkinson bar, and explosive loading have been developed to generate high-intensity, short-duration pulse loading on cracks. An indication of the rate of loading achieved in each of the loading schemes as well as an estimate of the typical time taken to initiate dynamic crack growth is given in Table 6.1. A short description of each loading method and appropriate references to the literature are provided below.

6.1 Static Loading of Cracks This method of loading has by far been the most popular of all loading methods due to its simplicity. Typically, a pre-cracked specimen is loaded in a standard testing machine to initiate crack growth from blunt crack tips; specialized equipment is not necessary to generate dynamic crack growth. The load is applied slowly over a time ranging from a few seconds to a few minutes; therefore, elastodynamics analysis of the problem is not necessary until the onset of crack initiation. The stress intensity factor can be calculated

82

Chapter 6 Table 6.1 Range of loading rates and crack initiation times Quasi-static loading machines

Loading rate p K_ dyn I ðt; vÞ MPa m/s Time to fracture (ms)

1 .106

Drop-weight towers 104 ,100

Projectile impact

Explosives

Electromagnetic loading scheme

104 – 108

105

105

1–20

10–100

1– 100

through the static formulas found in handbooks. The crack tip must be introduced carefully so that bluntness of the crack does not affect the values of crack initiation toughness p obtained. However, the loading rate K_ Idyn ð0; vÞ in these tests is about 1 MPa m/s about four orders of magnitude less than in specimens that are loaded by other techniques described below. Furthermore, the initial loading cannot be varied over a meaningful range and thus, rate dependence of crack initiation cannot be examined using this loading scheme. The method is more suitable when the interest is on determining crack propagation or crack arrest toughness. In these cases, by controlling the bluntness of the initial crack, the stress intensity factor at initiation of a dynamic crack can be made much larger than the initiation toughness and hence crack growth can be initiated at high speeds. If such loads are applied in a displacement-controlled machine, it is possible to generate conditions for arrest of the dynamically propagating crack as described in Chapter 10. According the ASTM Standard E-1221 procedure for the determination of crack arrest toughness, the load is applied by forcing a wedge into a split pin that wedges open a crack in a modiﬁed compact tension specimen. While the ASTM Standard suggests the use of a static analysis for evaluating the crack arrest toughness, both for the propagating crack and the arresting crack, a dynamic analysis of the problem is clearly necessary in order to determine the variation of the stress intensity factor K_ Idyn ðt; vÞ with time and crack speed. Further discussion of crack arrest is deferred until Chapter 10. Methods of evaluating crack speed are discussed in Chapter 7 and methods of determining K_ Idyn ðt; vÞ are discussed in Chapters 8 and 9. Almost every imaginable specimen conﬁguration of quasi-static loading has been used by investigators—center-cracked panel (Beebe, 1966), compact tension specimen (Carlsson et al., 1973; Dally, 1979; Kobayashi et al., 1980), double cantilever beam (Kalthoff et al., 1980a,b; Dally, 1979), single-edge-notched plate (Bradley and Kobayashi 1970; Dally, 1979; Brickstad, 1983; Taudou et al., 1992), and inﬁnite strip specimens (Paxson and Lucas, 1973; Fineberg et al., 1991). These specimen geometries are illustrated in Fig. 5.1. The stress intensity factors for these specimens for quasi-static loading are tabulated in textbooks and handbooks.

6.2 Drop-Weight Tower and Instrumented Impact Testing The drop-weight tower has become standard laboratory equipment as a result of its use in characterizing the nil-ductility-temperature, TNDT ; through the ASTM Standard E-208. Commercial systems are readily available to implement this loading scheme. However, in dynamic fracture investigations, the interest is not in determining TNDT ; but in determining

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83

the initiation, growth and arrest criteria and hence additional instrumentation is required in order to obtain quantitative measurements during the test. The drop-weight tower is a simple apparatus in which a mass in the range of a few tenths of a kilogram to a few tens of kilograms is raised to a height less than 10 m, but typically in the range of 2– 3 m and dropped on the test specimen; the weight is typically provided with a rounded indenting tip or a sharp tip (dart) to inﬂuence the nature of contact generated. The impact speeds achieved are on the order of 5– 10 m/s. As a result of the impact, the crack tip experiences a reasonably modest rate of loading for a long duration (about 10 ms) that can be controlled by choosing the specimen dimensions. Typical loading rates achieved in dropp weight towers are on the order of K_ Idyn ðt; vÞ ¼ 104 MPa m/s. Modiﬁcations of the standard Charpy test apparatus have also been used to obtain loading rates in this range. Since the loads are applied rapidly, a naturally cracked specimen can be used; therefore the use of drop-weight towers for dynamic fracture toughness testing has now become standard practice. It is possible to design many different specimen geometries to be tested in the dropweight tower, but the most common specimen used in drop-weight and other instrumented impact tests is the edge-notched specimen in a three-point bend conﬁguration as shown in Fig. 1.1; the terminology of ‘bending’ is inappropriate because a typical bending stress ﬁeld may not be developed in the specimen under the dynamic loading conditions during the time of interest and one may not be able to use the quasi-static analysis of the same three-point bending conﬁguration. In this test, the impacting tup is instrumented in order to measure the time variation of the load, PðtÞ; and the displacement of the tup sðtÞ: By applying a strain gage to the tup and calibrating its response in terms of the load on the tup, the time variation of the impact load can be measured. The tup displacement may be determined through an independent measurement or estimated from the deceleration of

Figure 6.1 Loading conﬁgurations for quasi-static loading experiments.

84

Chapter 6

the tup under the measured load. Let us denote the mass of the tup by m and the speed of the tup at the onset of impact by v0 : Then the position at anytime t is given by ð t0 ðt PðjÞ dj dt0 v0 2 ð6:1Þ sðtÞ ¼ 0 0 m Typical variation of the measured impact load P with the calculated tup displacement s is shown in Fig. 6.2. Several key features are to be noted in this ﬁgure. First the load variation exhibits oscillations at the characteristic frequency of bending vibrations of the beam. Second, the load reaches a maximum at some value Pmax : It is generally assumed that crack initiation occurs at this value although the load maximum could simply be the plastic limit load for the geometry. Third, there is a rapid drop in the load at a critical value of the tup displacement; this is assumed to correspond to rapid crack growth and crack arrest. Finally, the remaining uncracked ligament deforms plastically and dissipates additional energy. Thus the instrumented impact test provides a breakdown of the events that typically occur in a Charpy type of impact test. If the objective of the test is simply to measure the energy absorbed in the impact test, then the oscillations pose no problem. The energy absorbed in the specimen, EðtÞ; is the area under the load – displacement curve in Fig. 6.2 and is given by ð sðtÞ EðtÞ ¼ Pðs0 Þds0 ð6:2Þ 0

The effect of the oscillations is integrated out in the calculation of the displacement and the energy. Note that the energy absorbed may be calculated at the peak load, at the crack

Figure 6.2 Variation of the tup load with tup displacement in an instrumented impact test. (Reproduced from Kalthoff, 1990a; axes labels have been redrawn to improve readability.)

Methods of Generating Dynamic Loading

85

propagation load or at the ﬁnal displacement. None of these estimates is equal to the Charpy V-notch energy, Cv ; which measures the difference between the potential energy difference in the striker before and after impact. It is not clear how the energies calculated above may be used; it appears that the instrumented drop-weight simply provides a different measure of impact toughness that the Charpy tests, but not readily interpretable in terms of the Charpy V-notch energy. On the other hand, the measurements from the drop-weight or instrumented impact test may be used to determine the dynamic crack initiation toughness. However, in order to accomplish this, one must determine the dynamic stress intensity factor at initiation of crack growth. Different methods have been used for the calculation of the dynamic stress intensity factor. Could the dynamic stress intensity factor be determined from the quasistatic analysis of the three-point bend conﬁguration with the dynamically measured load? Ireland (1974) indicated that this procedure could be used when the time to fracture tf was longer than about three times the time of the characteristic oscillation, t: The rationale is that after about the third oscillation, the amplitude of the load oscillations becomes small enough to be neglected. Thus, the dynamic stress intensity factor is estimated as KIstat ðtÞ ¼

PðtÞS a f ; W BW 3=2

t . 3t

ð6:3Þ

where a is the crack length, S the span of the beam, W the specimen width, and f ð aÞ ¼

pﬃﬃﬃ 3 a½1:99 2 að1 2 aÞð2:15 2 3:93a þ 2:7a2 Þ 2ð1 þ 2aÞð1 2 aÞ3=2

is the geometric factor for the three-point bend specimen given by Srawley (1976). Therefore, in the experiment, if the time for crack initiation and the tup load are measured independently, the dynamic crack initiation toughness can be calculated through Eq. 6.3. However, it is possible to induce crack initiation for times tf , 3t simply by altering the specimen dimensions or the impact speed. Under this condition, one must resort to a dynamic analysis. Two approaches have been used to determine the dynamic stress intensity factor: impact response curves (Kalthoff, 1990a) and key curves (Bo¨hme, 1990). Impact response curves are in fact calibration curves; the basic idea is that the true dynamic time variation of the stress intensity factor for a stationary crack is determined through analysis, numerical simulation or experimental measurements (using one of the methods described in Chapter 8)—this is the impact response curve. The crack is forced to remain stationary in this test by making a blunt initial crack. Then, to measure the dynamic crack initiation toughness, a fatigue-precracked specimen is then impacted at the same speed in the same geometry. Measuring only the time to initiate the crack, the dynamic stress intensity can be read-off from the impact response curve. The concept of a key curve was introduced by Bo¨hme (1990); in this approach, ﬁrst, the impact event is modeled with a spring-mass model—the tup of mass m and the specimen of compliance Csp (the support compliance Csup may also be included) constitute the spring mass system. An estimate of the stress intensity factor, KIqs ðtÞ; is obtained from this

86

Chapter 6

model. The dynamic stress intensity factor is then written in terms of this estimate KIdyn ðtÞ ¼ kkey ðtÞKIqs ðtÞ

ð6:4Þ

where kkey ðtÞ is called the dynamic key curve; it must be determined by a comparison of KIqs ðtÞ to KIdyn ðtÞ obtained from analysis, numerical simulation or experimental measurements. Bo¨hme (1990) made a comparison of the three methods described above—static analysis, key curves and direct measurements of the dynamic stress intensity factor; his result is reproduced in Fig. 6.3. The optical method of caustics (described in Section 8.3) was used to determine the dynamic stress intensity factor. KIdyn ðtÞ exhibits a small amplitude oscillation about a mean curve; the mean curve is well estimated by the quasi-static analysis suggesting that the spring-mass model is a reasonably good approximation to the actual dynamic variation. However, if one is interested in characterizing critical material properties, the discrepancy between KIqs ðtÞ and KIdyn ðtÞ is not acceptable. Also, the static estimate, KIstat ðtÞ; based on Eq. 6.3 is quite different from the actual variation whenever the time to failure is tf , 3t: Kalthoff (1990a) shows examples of impact tests where the static analysis is inadequate even for tf . 3t: From the wealth of test results that have been obtained on drop-weight or instrumented impact tests, it is clear that a dynamic interpretation—analytical, numerical or experimental—is necessary to evaluate stress intensity factor from impact tests. Thus, both the key curve and impact response curve concepts are useful in evaluating data from

Figure 6.3 Comparison of stress intensity factors for an impact test estimated using static, quasi-static and dynamic analysis. (Reproduced from Bo¨hme, 1990.)

Methods of Generating Dynamic Loading

87

impact tests. Two additional concerns arise in using these curves: ﬁrst, the nature of dynamic contact between the specimen and the tup is difﬁcult to control between different experiments. Second, only the load at the tup is measured, with the implicit assumption that the contact between the specimen and anvil is reproducible. In fact, there is loss of contact between the specimen and anvil (Kalthoff, 1990a) which makes the analysis less reliable. Hence, the procedures for the interpretation of drop-weight and instrumented impact tests are far from satisfactorily established. Direct measurement of the dynamic stress intensity factor in each test through one of the techniques discussed in Chapter 8 is the best strategy for the unambiguous evaluation of the dynamic stress intensity factor from the impact tests.

6.3 Projectile Impact Projectile impact experiments are just variations of the drop-weight or other instrumented impact tests; once again, specimens of different geometrical conﬁgurations are impacted by a projectile of mass m moving at a speed v: The only difference in these impact experiments is that the projectile (weighing anywhere from a few grams to a few kilograms) is launched from the barrel of a gun at speeds in the range of a 10 m/s to 1 km/s; the higher speeds lead to penetration, a much more complex problem of dynamic failure. For characterization of dynamic fracture, usually projectile speeds in the range of 10 –200 m/s is used (Kalthoff, 1990a; Ravichandran and Clifton, 1989; Taudou et al., 1992; Mason et al., 1992 and many others). The loading duration is governed by the length of the projectile and is typically much shorter than in the drop-weight test—about 10 –200 ms. The time to fracture is typically in the range of 1– 100 ms. With such short duration tests, dynamic analysis of the response of the specimen is clearly necessary for the interpretation of the test results; in many cases, crack initiation begins even before stress waves from the impact reach all the far boundaries of the specimen. Loading generated by such projectile impact onto a specimen can yield loading rates in the range of p p 5 8 _ dyn K_ dyn I ðt; vÞ ¼ 10 MPa m/s at the low impact speeds to about KI ðt; vÞ ¼ 10 MPa m/s at the high projectile speeds for a very short duration (less than 1 ms). Apparatus for generating such impact loads are not commercially available and must be designed and fabricated especially for each implementation. But with this wide range of loading rates, projectile impact tests are suitable for characterization of the loading rate dependence of the dynamic crack initiation toughness. Because the projectile velocity can be controlled to within close tolerances, this method of loading is quite repeatable. The projectiles used are typically circular cylinders with a ﬂat nose; impact of the ﬂat nose on the edge of the specimen generates a compression wave in the specimen that travels through the specimen at the speed Cd : The magnitude of the stress wave depends on the impedance mismatch between the projectile and the specimen. The duration of the loading pulse depends on the length of the projectile. If the impedance of the projectile and the specimen are matched, upon impact a compression wave of equal magnitude travels both into the specimen and into the projectile. The magnitude of the stress pﬃﬃﬃﬃﬃcompressive ﬃ may be estimated from one-dimensional wave theory to be sp ¼ v rE=2 where r is the density, and E the modulus of elasticity, but since the specimen is usually larger than

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the diameter of the projectile, this estimate of magnitude is unlikely to be accurate far from the impact point. The tensile pulse reﬂected from the far end of the projectile cannot pass through the impact surface and the loading of the specimen ends. Thus, the duration of the loading pulse is t ¼ 2Lp =C0proj ; where Lp is the length of the projectile and C0proj is the bar wave speed in the projectile. The compressive pulse traveling in the specimen can be used in different ways to generate a transient load history on the cracked specimen. Three different implementations are shown in the sketches in Fig. 6.4. In the ﬁrst implementation shown in Fig. 6.4a, the projectile is made to impact opposite to the side containing the edge crack. While this might appear to be similar to the ‘three-point bend’ specimen we refrain from using this terminology; the specimen is usually not supported by rigid supports or anvils, but is simply suspended by strings. The compressive stress wave travels to the free boundary and reﬂects as a tensile wave. When this wave reaches the crack tip, the mode I stress intensity factor grows until crack initiation and growth are generated. The second conﬁguration is shown in Fig. 6.4b where the projectile travels in the direction normal to the crack line; once again, impact at the top end of the specimen generates a compression wave that travels to the stress free boundary at the bottom, reﬂects as a tensile pulse and loads the crack in the middle of the specimen with a mode I loading symmetry. Closed form solutions of the elastodynamic problem for calculation of the dynamic stress intensity factor are not available for these conﬁgurations even for the case of stationary cracks except for very short times. Consider a tensile stress pulse of magnitude sp and duration t propagating with a speed Cd as indicated schematically in Fig. 6.5. Let the stress wave reach the crack tip at time t ¼ 0: Before waves from the corner A in Fig. 6.5 arrive at the crack tip—i.e. for t , a=Cdp the mode I stress intensity factor is given by (Freund, 1990): pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cd ð1 2 2nÞ=p pﬃ ¼ 2s t for t , t 12n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cd ð1 2 2nÞ=p pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ KIdyn ðtÞ ¼ 2sp ð t 2 t 2 tÞ 12n

KIdyn ðtÞ

p

ð6:5Þ for t , t , a=Cd

Figure 6.4 Conﬁgurations for projectile impact. (a) Impact parallel to the crack line. (b) Impact normal to the crack line. (c) Impact parallel to the crack line, but just below the crack line. The specimens are usually not supported rigidly, but simply suspended with strings. The dimensions of the specimen can be varied over quite a large range in experimental implementations.

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Figure 6.5 Interaction of a tensile stress pulse magnitude sp and duration t with a crack. The stress pulse travels with a speed Cd :

It is assumed that the duration of the loading pulse is shorter than the time for arrival of the wave from the corner: t , a=Cd : If this is not the case, the stress intensity factor for times 0 , t , a=Cd is given by the ﬁrst expression in Eq. 6.5. For t . a=Cd ; the waves from the corner A must be taken into account and this makes analysis very difﬁcult; then, one must resort to numerical simulations using ﬁnite element analysis or to direct experimental evaluation through the methods discussed in Chapter 8. In practice, departures from the ideal impact conditions described here such as ﬁnite rise times of the loading pulse, variations in the impedance between the projectile and the specimen, and departure from the one-dimensional theory considered here will force the need for numerical simulations or direct experimental measurements even for short times. While both the conﬁgurations in Fig. 6.4a and b generate a mode I loading on the crack, the main difference between the two is in how the nonsingular stress ﬁeld in the neighborhood of the crack develops. This is known to inﬂuence the crack growth direction (Cotterell and Rice, 1980), but its inﬂuence on crack initiation is not well established. Finally, in the conﬁguration shown in Fig. 6.4c, the projectile is made to impact the specimen just below the line of the crack. The compression wave generated from this impact travels along the bottom half of the specimen and diffracts around the crack; points below the crack line are made to displace to the right while points above are at rest suggesting that the loading is anti-symmetric. This loading method generates a mode II dynamic loading at the crack tip and has been used for evaluating the dynamic response of cracks under anti-symmetric loading (Kalthoff, 1990b; Mason et al., 1992; Ravi-Chandar, 1995). Lee and Freund (1990) determined the stress intensity factor variation with time for this loading conﬁguration KIdyn ðtÞ ¼

Ev 2Cd

rﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃ a Ev a gðtÞ; KIIdyn ðtÞ ¼ hðtÞ 2Cd p p

for t ,

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a2 þ d2 =Cd

ð6:6Þ

where d is the diameter of the projectile, E the modulus of elasticity of the specimen, and gðtÞ and hðtÞ are functions deﬁned by Lee and Freund (1990). For this loading, the mode I

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stress intensity factor turns out to be negative, which implies contact between the top and bottom crack surfaces; Eq. 6.6 may be used only when p the initial crack is a blunt crack ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ with a large opening that prevents such contact. For t . a2 þ d2 =Cd ; the ﬁnite diameter of the projectile brings about differences in the loading from that considered in the analysis and the analytical results are no longer applicable. Once again recourse must be made to numerical simulation or experimental measurements as in the work of Mason et al. (1992). In mode II experiments such as this, it is possible to determine the critical condition for crack initiation; in some cases, an opening mode crack is initiated at an angle to the initial crack while in the other a shear crack or a shear band extends along the extension of the original crack line; these problems are not addressed in this book. One ﬁnal variation of the projectile impact loading scheme merits additional description—the plate impact test. In this test, both the projectile and the specimen are extremely thin disks as illustrated in Fig. 6.6. In this ﬁgure, the projectile, called the ‘ﬂyer plate’ is propelled from a gas gun, usually mounted on a hollow ﬁber glass carrying tube, at speeds of up to about 1 km/s. This ﬂyer plate impacts the thin-disk specimen that contains a crack across half of its cross-section; the arrangement is very similar to that illustrated in Fig. 6.4b, except that the cross-section of the projectile and specimen are identical here. The pulse duration t ¼ 2Lp =C0proj is extremely short because the projectile length is small—for example, Lp ¼ 5 mm, C0proj ¼ 5000 m/s results in a pulse of duration t ¼ 1 ms. Details of how these are implemented for the mode I loading are given in Ravichandran and Clifton (1989). The loading rate achieved in these experiments is about p 8 K_ dyn I ðt; vÞ ¼ 10 MPa m/s. While the loading scheme generates a well-characterized onedimensional pulse loading, the diagnostics of the specimen response is quite difﬁcult; the normal, and tangential displacements of the specimen backplane are monitored with

Figure 6.6 Plate impact test conﬁguration. (From Ravichandran and Clifton, 1989.)

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the aid of interferometric velocimeters and interpreted in terms of the dynamic stress intensity factor. The alignment and set up of this apparatus are quite challenging tasks.

6.4 Hopkinson Bar Impact Test The Hopkinson pressure bar is a one-dimensional wave guide, usually of circularpcrossﬃﬃﬃﬃﬃﬃ section. One-dimensional compression stress wave of magnitude sp ¼ v rE=2 and duration t ¼ 2Lp =C0proj can be set up in this bar by projectile impact as discussed in Section 6.3. It is also possible to generate tensile waves by suitable arrangement of impact or through the use of explosives. Many variations of the Hopkinson bar loading scheme appear in the literature; for a detailed description of Hopkinson bar procedures, see the ASM Metals Handbook, Volume 8. Three versions of this method are illustrated in Fig. 6.7. The simplest arrangement is that used by Costin et al. (1977); as illustrated in Fig. 6.7a, the specimen is a long round bar, B; with a circumferential precrack introduced by fatigue cycling. An explosive charge is detonated at one end of the specimen resulting in a tensile pulse traveling down the length of the specimen. Strain gages are placed on either side of the crack to monitor the stress waves on either side of the crack; in addition, Costin et al. (1977) used a moire´ technique to determine the crack opening displacement. The rise time of the loading pulse was in the range of 35 –40 ms, and fracture occurred within 20 – 25 ms. The type of experimental measurement obtained in this apparatus is shown in Fig. 6.8, where the time variation of the measured load and crack opening displacement are shown. It should be noted that the quality of signals is extremely good in comparison to the noisy signals obtained in an instrumented impact test. From such measurements, the dynamic stress intensity factor may be estimated. Costin et al. (1977) used this method to determine

Figure 6.7 Hopkinson pressure bar arrangements for fracture tests. (a) Tension pulse generated by an explosive charge (Costin et al., 1977). (b) Impact generated on single edge notched specimen (Nicholas, 1975; Ruiz and Mines, 1985). (c) Compact compression specimen (Maigre and Rittel, 1995).

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Figure 6.8 Time variation of the load measured with the strain gage and the displacement measured with the moire´ technique. (Reproduced from Costin et al., 1977.)

the rate dependence of the dynamic crack initiation toughness at loading rates of the order p 6 of K_ dyn I ðt; vÞ ¼ 10 MPa m/s. The second implementation of the Hopkinson bar shown in Fig. 6.7b is a variation of the instrumented impact test; instead of relying on a falling weight or a projectile, a welldeﬁned one-dimensional compression stress wave is generated by the projectile A impacting the bar B; this wave travels down the length of the bar and impacts on a singleedge-notched specimen as in the typical instrumented impact test (Nicholas, 1975). The strain gage signal can be interpreted directly in terms of the load and displacement at the point of contact between the bar and the specimen. Dynamic analysis of the data is required as in the case of the instrumented impact test. Ruiz and Mines (1985) showed that in contrast to the noisy signal characteristic of the instrumented impact test, the strain signals observed in the Hopkinson bar are of much better quality. The last variation on this type of loading that we consider is due to Maigre and Rittel (1995). They considered a special type of specimen called the compact compression specimen; by sandwiching this specimen between two bars B and C as shown in Fig. 6.7c, and impacting bar B with a projectile A; they were able to generate high rate loading on the crack. Strain gages mounted on bars B and C were used to monitor the force and displacement at the points of contact between the specimen and the bars. These measurements were then used in a dynamic analysis to evaluate the dynamic stress intensity factor. They applied the method to the determination of the rate dependence of dynamic initiation toughness at rates in the p 5 range of K_ dyn I ðt; vÞ ¼ 10 MPa m/s (Rittel and Maigre, 1995). There are many other variations on this scheme of high strain rate testing; Klepaczko (1990) has provided a lengthy review of the use of Hopkinson pressure bars to dynamic fracture testing.

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6.5 Explosives Lead azide and pentaerythritol tetranitate (PETN) explosives have been used to apply high strain rate loading in the examination of dynamic fracture. However, issues related to the safe handling and operation of these explosives make it so difﬁcult to use that this method of loading is not commonly employed. Furthermore, since the loading rates obtained are comparable to that obtained with moderate speed impact of projectiles, p , K_ Idyn ðt; vÞ ¼ 105 MPa m/s, there are no signiﬁcant advantages in using explosives. Some examples of dynamic fracture under explosive loading may be found in the work of Dally and Barker (1988) and Shukla and Rossmanith (1995); here we describe the experiment of Dally and Barker. In order to determine the dynamic crack initiation toughness of A533-B reactor grade steel, Dally and Barker developed an explosive loading scheme shown in Fig. 6.9. A long bar, 25 – 50 mm wide and 400– 500 mm long, was cut into the dog-bone shape shown in the ﬁgure. A fatigue precrack was introduced at one edge in the middle of the bar. By detonating four explosive charges at the ends of the dog bone, a tensile stress pulse was sent towards the crack from both ends of the specimen. The resulting interaction at the crack tip generated a high rate loading at the crack tip; crack initiation was triggered within 10 ms; strain gages were used to determine the strain evolution near the crack tip; the time variation of the strain is shown in Fig. 8.16 and the results are discussed further in Chapter 8. The strain rate obtained in this experiment is p K_ Idyn ðt; vÞ ¼ 8 £ 106 MPa m/s.

6.6 Electromagnetic Loading The electromagnetic loading scheme is based on electromagnetic interaction between two current carrying conductors (Ravi-Chandar and Knauss, 1982). The specimen is made

Figure 6.9 Four explosive charges are detonated at the four tabs on the dog-bone specimen to generate two tensile pulses that travel towards the crack from both sides of the specimen. (Reproduced from Dally and Barker, 1988.)

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of a large plate (500 mm £ 300 mm) of thickness 4.76 mm; a crack is introduced parallel to the long side by machining a 3-mm-wide slit down the middle. A natural crack tip is introduced by wedging a razor blade inside the slit and applying a small impact force. A ﬂat copper strip, 4.76 mm £ 1.2 mm thick is folded back on itself and the space between the two layers is ﬁlled with a mylar insulating strip. This assembly is then introduced into the machined slit as indicated in the schematic diagram in Fig. 5.9a. When a current ﬂows through the copper loop, each leg generates a magnetic ﬁeld surrounding it, with the magnetic ﬁeld oriented normal to the current vector. The current vector in each leg interacts with the magnetic ﬁeld of the other leg to produce an electromagnetic repulsion that forces the conductors apart. Since the two legs of the copper strip are conﬁned in the slot of the machined crack, they do not move apart, but simply press upon the top and bottom surfaces of the crack with a uniform pressure. The magnitude of the pressure loading may be estimated easily from electromagnetic theory; for two idealized conductors of width b; carrying a current iðtÞ the pressure of repulsion, in the limit where the separation between the two conductors is negligibly small in comparison to the conductor width b; is given by 1 iðtÞ 2 pðtÞ ¼ m0 ð6:7Þ 2 b where m0 ¼ 4p £ 1027 Wb/A m is the permeability constant. The current in the copper strip is generated by a discharge from a capacitor bank. The time history of the current which dictates the magnitude and duration of the pressure applied on the crack surface may be controlled by suitable choice of capacitors and inductors that form the pulse shaping circuit; Ravi-Chandar and Knauss (1982) generated a nearly trapezoidal pulse, with a rise to the peak amplitude in about 25 ms, and a total duration of about 150 ms. For typical values of current used in the experiments, the crack surface pressures were in the range of 1– 20 MPa. For the large specimen, this loading conﬁguration is equivalent to

Figure 6.10 Principle of the electromagnetic loading scheme. (Reproduced from RaviChandar and Knauss, 1982.)

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95

an inﬁnite plate, with a pressurized semi-inﬁnite crack for the duration of the current pulse. Of course, waves from the edges of the plate arrive at the crack tip, but this may be delayed for as long as the duration of the loading pulse.pThe loading rate achieved in this loading device is on the order of K_ Idyn ðt; vÞ ¼ 105 MPa m/s. Of the many different loading schemes described here, the electromagnetic loading provides two major advantages. First, it provides a very repeatable, electrically synchronized loading that makes experiments easy. Therefore, this loading method is well suited for studies of crack initiation as well as continued growth. Results on experiments aimed at evaluating rate dependence of the dynamic initiation toughness are presented in Chapter 10. Second, and perhaps more important, this scheme provides crack surface loads in a conﬁguration that can be modeled as a pressurized semi-inﬁnite crack in an unbounded medium, a conﬁguration that is easily analyzed; Fig. 6.10b indicates the loading conﬁguration. The closed-form expression for the dynamic stress intensity factor given in Eq. 6.5 is appropriate to this loading conﬁguration. However, specialized equipment is necessary for implementation of this loading and is not readily available commercially.

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97

Chapter 7 Measurement of Crack Speed

In addition to the ability to apply a well-characterized, reproducible loading on the cracks, it is necessary to use diagnostic methods for observing the response of the cracks to the applied loading. In the test methods described above, the focus of the experimental measurement was on global measurements at the boundaries of the specimen or test apparatus; for instance, in the instrumented impact test, typically the tup load and position are measured. However, the response of the specimen must be characterized eventually in terms of the crack position, speed, and the dynamic stress intensity factor, KIdyn ðt; vÞ: Problems associated with the use of static, quasi-static and dynamic analyses of the measured data have already been described. Therefore, experimental investigations into dynamic fracture that are aimed at revealing fundamental characteristics must incorporate diagnostic methods to determine not only the global quantities at the specimen boundaries, but also the crack position, speed, and a measure of the crack tip stress or deformation ﬁeld. Several different methods of performing measurements of the stress and displacement ﬁeld near a dynamically loaded and dynamically growing crack tip have been developed and used. In this chapter we describe the methods commonly used to determine the time variation of the crack position and/or speed of crack growth. This is followed in Chapter 8 by a discussion of methods used for the evaluation of the stress and deformation ﬁelds— photoelasticity, the optical method of caustics, the method of coherent gradient sensing, strain gages and optical interferometry. One distinguishing feature of dynamic problems in fracture is the necessity for time-resolved observations and measurements of the fracture event. Cracks may grow in solids at speeds comparable to the characteristic wave speeds, typically in the range of about 500– 1500 m/s for most common materials. Therefore, a host of methods— some real time and others based on post-mortem observations—have been developed in order to extract the time variation of the crack position. While most of the experimental investigations have relied on the use of a high-speed camera to determine the crack speed, a few other techniques have also been used. The major source of concern in all of the methods of speed determination is in the spatial and temporal averaging that arises in the measurement scheme that inﬂuences the accuracy of the measurements.

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7.1 Wallner Lines Wallner (1939) showed that the interaction of the propagating crack with shear waves emanated from the fracture can lead to characteristic undulations on the fracture surface; he suggested that these lines—now called Wallner lines—can be used to determine the crack speed quite accurately. Such lines are easily recognized during post-mortem examination of the fracture surface and are illustrated in Fig. 7.1. The basic principle is as follows: consider a curved crack front propagating at a constant speed, v: In Fig. 7.2a the crack front position at four equal time intervals is shown by the thin, solid lines. Let a disturbance be introduced in the crack path at the location indicated by the asterisk. The shear waves generated by the disturbance radiates out at a speed Cs : The locus of intersection of the shear wave with the propagating crack front manifests itself on the fracture surface as an alteration in the local topography and hence visible in reﬂected light illumination. From the geometry of the Wallner line, it is possible to determine the crack speed; in order to accomplish this, the direction of crack growth as well as the location of the disturbance generating the Wallner line must be known. Consider a point C on a Wallner line as indicated in Fig. 7.2b; let O represent the source of the disturbance. The directions of crack growth and the shear disturbance are indicated by arrows. In a time increment Dt; the crack extends a distance CC0 ¼ vDt and the shear wave moves through a distance OQ 2 OC ¼ Cs Dt; hence, the crack speed can be obtained as: v ¼ Cs

CC0 OQ 2 OC

ð7:1Þ

An alternative way of considering the geometry of Wallner lines is shown in Fig. 7.2c. Let the instantaneous angle between the tangent to the Wallner line and the directions of the crack growth and shear wave direction be denoted by a and b; respectively; then

Figure 7.1 Wallner lines on a fracture surface in an inorganic glass. Crack initiation occurred at N and crack growth direction was in the direction indicated by the arrow. The disturbances causing the Wallner lines are taken to be from the sides of the specimen. Estimated crack speed at the location P is 0.26 Cs. (Reproduced from Field, 1971.)

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Figure 7.2 (a,b) Formation of Wallner lines through the interaction of the propagating crack front with a shear wave created from a perturbation along the crack. (c) Geometry indicating the relationship between the crack direction, the shear wave direction and the Wallner line. (d) Formation of two Wallner lines.

v ¼ Cs cosa=cosb: In some cases, perturbations from two different sources generate a pair of Wallner lines in the same spatial region; in this case, let us denote the angle between the tangent to the Wallner line and the direction of the shear waves by b1 and b2 ; respectively, and let the angle between the Wallner lines be denoted by w: Then the crack speed can be determined in terms of these angles: sinw v ¼ Cs pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 cos b1 þ cos b2 þ 2cosb1 cos b2 cosw

ð7:2Þ

A number of investigators have used this idea to determine crack speeds (see Congleton and Petch 1967; Anthony et al., 1970; Field, 1971; Hull, 1997a,b). Wallner lines have been observed in several different materials such as glass, epoxy resins, tungsten, and carbonﬁlled rubber fractured in a glassy state. It is worth noting that the correlation of the Wallner lines has been to the shear wave and not the dilatational wave. More recently, these surface features have been reinterpreted as evidence of crack front waves by Sharon et al. (2001). The crack front waves have recently been discovered theoretically by Ramanathan and Fisher (1997) and Morrissey and Rice (1998, 2000). However, the speeds of the crack front waves are so close to the shear wave speed that discrimination between the two has not yet been possible. Bonamy and Ravi-Chandar (2003) introduced ultrasonic waves

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(see Section 7.2 for a discussion of the effect of ultrasonic waves on propagating cracks) to perturb propagating surfaces with a mode III disturbance and demonstrated that indeed Wallner lines are the result of interaction between propagating cracks and the shear waves.

7.2 Stress Wave Fractography While the crack speed may be determined accurately from Wallner lines, the method relies on the availability of random disturbances on the crack surface to generate the patterns. Kerhkof (1973) demonstrated that by imposing a small-amplitude highfrequency stress wave, a predictable crack surface undulation may be produced. The scheme is illustrated in Fig. 7.3; typically stress waves generated by an ultrasonic transducer are made to interact with the growing crack. As a result of this interaction the crack tip experiences a mixed-mode loading, although the magnitude of mode II is quite small. Nevertheless, the crack tip attempts to follow the direction of local symmetry (Cotterell and Rice, 1980) and deﬂects by an angle g ¼ 22KII =KI : The periodic variation of incoming stress wave causes KII to vary periodically. Let the stress amplitude of the incident wave vary as

sw ¼ s0 cosðkxcos a þ kysina 2 vtÞ

ð7:3Þ

where k ¼ v=C is the wave number, v the frequency of the incident wave, C the wave speed of the incoming stress pulse (equal to Cd for a longitudinal wave and Cs for a shear wave), and a is the orientation of the propagation direction of the stress wave with respect to the crack normal as illustrated in Fig. 7.3. This wave creates a perturbation in the local loading; it can be assumed that the perturbations in the mode I stress intensity factor are negligible and hence that the crack speed will remain constant. Therefore, x ¼ vt: The perturbation in the mode II stress intensity factor can be estimated to be KII ¼ KI0 CII ða; vÞcosðkxcosa þ kysina 2 vx=vÞ KI0

ð7:4Þ

where is the mode I stress intensity factor of the crack without the effect of the stress wave. The function CII ða; vÞ must be determined from a dynamic analysis of the crackwave interaction problem; however, since our interest here is in the variation of the crack path, we do not have to estimate this factor. Imposing the criterion of local symmetry, and

Figure 7.3 Stress wave fractography.

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the crack deﬂection angle is given by

g ¼ 22

KII ¼ 22CII ða; vÞcosðkxcosa þ kysina 2 vx=vÞ KI0

ð7:5Þ

Therefore, the crack path oscillation will mirror this variation of the perturbation of the mode II stress intensity factor—a periodic modulation of the crack path is created as indicated schematically in Fig. 7.3. Noting that the amplitudes of the crack oscillations are likely to be very small, the second term in the argument of the sine function can be neglected. Then, the wave length of the ripple pattern observed on the crack surface is 2p lr ¼ C kcosa 2 v

ð7:6Þ

The ripple marks can be analyzed post-mortem to determine lr and used to determine the crack speed. Note that a longitudinal or a shear wave can be used in modulating the crack path; clearly, better spatial resolution is obtained with a shear wave. For a stress wave impinging at an angle a ¼ p=2; the crack speed is given by v ¼ lr f

ð7:7Þ

where f is the frequency in Hz of the incident wave. A micrograph of the crack surface modulations observed in reﬂected light microscopy is shown in Fig. 7.4. The accuracy of the spatial measurement can be very high since one relies on a post-mortem examination

Figure 7.4 Optical micrograph of an ultrasonically modulated fracture surface; the dark region is a capillary tube of diameter 0.06 mm in the direction perpendicular to the crack surface. Crack growth is from bottom to top at a speed of about 150 m/s below the tube and , 20 m/s above the tube. Fringes are produced by the undulations in the fracture surface, when illuminated obliquely. (Reproduced from Kerkhof, 1973.)

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Chapter 7

of the fracture surface modulation at high magniﬁcations; micron resolution in crack position identiﬁcation is possible. The temporal resolution is dictated by the ultrasonic modulation frequency—1 MHz in Kerhkof’s experiments. With typical crack speeds on the order of 103 m/s and frequency in the 1 MHz range, the wave length of the ripple pattern can be estimated to be 1 mm. At higher crack speeds the spacing increases and reduces the spatial resolution that may be obtained. At lower speeds, the ripple spacing decreases quickly and at a speed of about 0.02 m/s, one runs into limits of transmitting the stress wave into the material and observing the ripple spacing due to optical diffraction limit. One major limitation is the large attenuation of high-frequency waves as they propagate through short distances. Through suitable choice of the ultrasonic transducer, crack speeds in the range of 0.02 –2000 m/s could be measured by this technique called stress wave fractography; a recent review of this technique can be found in Richter and Kerkhof (1994). Field (1971) and his colleagues have demonstrated the application of the ultrasonic modulation to a number of amorphous and crystalline materials. Kerkhof (1973) made careful measurements of the limiting speed in inorganic glasses by systematically varying the composition and showed that the composition had a signiﬁcant inﬂuence on the terminal crack speed.

7.3 Electrical Resistance Methods The electrical resistance methods in different manifestations have been used to determine the crack speed: resistive grid methods and potential drop methods. In the grid method a number of electrical wires are laid across the path of the crack. As the crack propagates, it severs the wires sequentially and provides an electrical signal which can then be used to determine the crack position and speed with time (Dulaney and Brace, 1960; Cotterell, 1965, 1968; Anthony et al., 1970; Paxson and Lucas, 1973). Commercial suppliers of strain gages now provide such grids for crack speed measurements; these grids can be incorporated into standard strain gage bridge circuits to provide the history of wire breakage and hence the crack position as a function of time. While very good estimates of the crack speed can be obtained from such grid techniques, the discrete nature of the grids dictate that the sampling rate of the crack speed will typically be much lower than that obtained using other methods such as ultrasonic modulation. However, if a thin conducting ﬁlm is used instead of a grid, higher temporal resolution can be obtained; this is the basis of the potential drop technique. The resistance of the conducting layer changes as the crack length increases; it is also inﬂuenced by the ﬁlm thickness and the crack opening. If this resistance change is measured, it can be related to the crack length and speed, while the other inﬂuences are negligible or minimized. Carlsson et al., 1973, demonstrated this application to the measurement of crack speed in PMMA; they used a voltage divider circuit to measure the resistance change. Many other investigators have used this method to determine the speed of running cracks (Stalder et al., 1983; Fineberg et al., 1991, 1992). Commercial versions of this technique—such as the KrakGagee—are now available; it is also quite easily accomplished in the laboratory with thin ﬁlm coating methods. The discussion below follows the work of Hauch and Marder (1998) who used a Wheatstone bridge circuit for the measurement of resistance changes. The samples of PMMA,

Measurement of Crack Speed

103

Homalite, and glass were coated with a 20—50 nm thick coating of aluminum by evaporation in a vacuum of 1026 Torr. The coating process naturally results in a spatial variation of the thickness, with a greater thickness at the center of the plate. Hausch and Marder describe how one can eliminate the thickness dependence of the resistance by calibration of each coating. Since there is no frequency dependence of the resistance, a calibration of the variation of resistance with the crack length may be performed with stationary cracks and then used to interpret the dynamic data. Crack opening displacement inﬂuences the measurements signiﬁcantly; Carlsson et al. (1973) indicate that by choosing a proper frequency of the alternating current, the inﬂuence of crack opening may be decreased; they used a frequency of 3 MHz. On the other hand, Hauch and Marder used a direct current for measuring the resistance and considered the possibility of electrical discharge across an open crack; since the discharge results in clearly identiﬁable signals, they can be accounted for in the data analysis. See Hauch and Marder (1998) for a complete discussion of the factors that inﬂuence the potential drop technique. The variation of resistance with crack length is nearly linear and hence can be expressed as Rsp ðaÞ ¼ R0 þ

dRsp da da

ð7:8Þ

R0 and dRsp =da can be determined from a calibration experiment or from a solution of Laplace’s equation for the potential. Note that R0 depends on the contact resistance and is not measured with great accuracy. However, this does not pose a problem since the initial resistance can be measured and subtracted. The instrumentation required for the measurement of this resistance is quite simple. The leads from the conducting surfaces across the crack line are connected to a Wheatstone bridge circuit. The resistance can be expressed in the following form Vbat R1 þ Vout ðaÞðR1 þ Ra Þ Rsp ðaÞ ¼ Rb ð7:9Þ Vbat Ra 2 Vout ðaÞðR1 þ Ra Þ where Vbat is the input voltage derived from the battery, Vout is the measured output voltage, R1 ; Ra ; and Rb are the resistors used in the Wheatstone bridge circuit. From a measurement of Vout the crack length can be found through the calibration in Eq. 7.9. The change in output voltage with time can also be expressed in terms of the crack speed: v¼

ðRb þ Rsp Þ2 Vbat dVout da ¼ dRsp dt dt Rb da

ð7:10Þ

In order to increase the sensitivity of the method, the differentiation of the output voltage is accomplished through an analog circuit as indicated in Fig. 7.5. Both Vout and the analog differentiated dVout =dt should be recorded with high accuracy and sampling rate to obtain the crack position and speed. According to Fineberg et al. (1992), this method provides for an evaluation of the crack position to within 200 mm and the crack speed to within 10 m/s. A sampling rate 10 M samples per second can be achieved—thus resulting in a very high temporal resolution of the measurements.

104

Chapter 7

Figure 7.5 Potential drop technique for crack position and speed measurements. (Reproduced from Hauch and Marder, 1998.)

However, there remain some concerns still unaddressed. Crack tip process zone size is typically on the order of about 100 mm particularly at high load levels that tend to generate fast cracks. When the resistance is sampled at a rate of 10 M samples per second, a crack moving at 500 m/s moves about 50 mm between sampling intervals which is only about half of the process zone size; if the crack position is sampled at small time intervals such that the overall crack extension is less than the size of the fracture process zone, sampling introduces a random error—some spatio-temporal averaging may be desirable.

7.4 High-Speed Photography Perhaps the most important technique for crack speed measurement is the high-speed camera; this has been a very popular although expensive method for dynamic fracture investigations. One major advantage of high-speed photography is that the event is observed without imposing any preconceived models for the interpretation of the observations. Another advantage is that full-ﬁeld optical methods of stress and deformation analysis such as photoelasticity, method of caustics, shearing interferometry, moire´ interferometry, and other techniques can be used to augment the crack position data with additional information on the crack tip deformation and stress ﬁelds. While the earliest attempts produced a single image by using a single spark light source, advances in the technology of high-speed photography have enabled camera designs for obtaining multiple images with a high spatial and temporal resolution. Modern high-speed cameras are capable of obtaining high-resolution images at time intervals on the order of 10 ns,

Measurement of Crack Speed

105

with exposure times on the order of 2 ns. Field (1983) provides a good review of the techniques used in high-speed photography; modern high-speed cameras are available from many commercial vendors. The ﬁrst high-speed camera to obtain multiple images of crack growth was the CranzSchardin multiple-spark camera; in this camera system, multiple spark light sources are arranged in a square or rectangular array. The object or region of interest is imaged on a camera with multiple lenses, also arranged in the same square or rectangular array as the light source. Each spark then forms a discrete image through its matching lens. Riley and Dally (1969) provide a detailed description of the design, construction and operation of a Cranz-Schardin camera. Two major limitations exist in the Cranz-Schardin camera system: ﬁrst, the multiple light sources cannot be aligned with the optical axis and hence a parallax error is introduced in the observations. Second, the duration of the spark sources is about 0.3 ms; this long exposure time results in an image smear, but for typical crack speeds of about 103 m/s, the image quality is quite acceptable since the smear is only about 300 mm. Large ﬁelds of view are possible in this arrangement; the system described by Riley and Dally has an 45.7 cm ﬁeld of view. The timing between the sparks is controlled independently and hence the images can be obtained at different time intervals, focusing on times where higher data rates may be needed. Rotating mirror cameras and rotating drum cameras provide an improvement over the Cranz-Schardin cameras in some respects, but not others. The operating principle in these cameras is to transport the image to a different location on the ﬁlm either by moving the image with a rotating mirror or by moving the ﬁlm by rotating the drum that holds the ﬁlm and in some cases through a combination of the two mechanisms. Illumination from a spark light source and more recently from a pulsed laser is used to form discrete images. Parallax errors are eliminated in this arrangement since the light source and the imaging system are aligned along the optical axis. The number of images that can be obtained is increased signiﬁcantly to more than a 100 frames. On the other hand, the size of the frames is usually quite small; this is dictated by the fact that during the inter-frame time interval the image has to be translated by a distance equal to its size. The speed with which this can be accomplished is limited by resonance phenomena in the turbines used to rotate the mirror or the drum. Typical framing rates that are possible with the rotating mirror or drum cameras is in the range of 103 – 106 frames per second. More recently, two types of electronic high-speed imaging cameras have been developed—the image converter cameras and CCD cameras. In the image converter cameras, the image falls on a photo-cathode which converts the image to a stream of electrons. These streams are steered to different parts of a phosphorescent screen by deﬂector plates, forming discrete images; the phosphorescent images are retained for many seconds and are then photographed on standard photographic ﬁlm. Exposure time is controlled by the ﬂash that illuminates the photo-cathode, but the framing rate is controlled by the speed of switching of the deﬂector plates. Therefore, very high framing rates may be obtained with these cameras. The main drawback is that the number of frames that can be obtained is usually quite small—about 4 – 12. The spatial resolution is dictated by the phosphorescent screen and is inferior to that achieved with rotating mirror cameras. Multiple CCD cameras have also been introduced recently; these are not too different from the Cranz-Schardin cameras in that arrays of CCDs are used to obtain multiple images.

106

Chapter 7

However, the image is made to fall on the different sensors by using beam splitters thereby eliminating parallax errors. The CCDs are exposed at high speeds but are then read off-line at slower speeds. As in the image converters, while high speeds—on the order of 108 frames per second—are possible, the number of frames and the image resolution are limited. Typically about 8– 16 frames are available and the image resolution is about 1300 £ 1000 pixels over the image. New hybrid rotating mirror cameras with CCDs for image capture have also been developed with the capability to obtain large number of frames at extremely high framing rates. Crack speed measurements provide two important conclusions that are quite independent of the details related to the resolution of the different crack speed measuring techniques. First, cracks attain a limiting speed of propagation that is signiﬁcantly lower than the limit of the Rayleigh wave speed set by the continuum energy balance argument (see Section 5.1). Second, the limiting wave speed is not a ﬁxed fraction of the characteristic wave speeds in the material, ranging anywhere from 0.33 CR to 0.66 CR (see Section 11.1 for further details). These conclusions point to the fact that while a limiting speed is set by the continuum wave propagation theory, inherent material processes that govern fracture dictate a signiﬁcantly lower limit. In fact, Schardin (1959) suggested that the limiting crack speed be considered a new physical constant, perhaps related to other physical parameters that govern the fracture process. Along these lines, Kerkhof (1973) made an interesting observation: if the material ahead of the crack tip is breaking apart rapidly, then the relevant physical quantity is not the modulus, but the surface tension. In other words, if elastic energy propagation is characterized by the elastic modulus, then the surface energy must play an equivalent role in crack propagation (surface energy propagation?). Therefore, he suggested that the limiting speed must be proportional to ﬃﬃﬃﬃﬃﬃﬃﬃ p T=r where T is the surface tension and r the density. However, since it is difﬁcult to estimate the surface tension in solids, Kerkhof used the hardness sH in place of T and showed thatpthe limiting speed in many inorganic glasses of different compositions can be ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ related to sH =r: It must be noted that the motivation for this correlation is purely phenomenological, but the correlation does exist. More recently, Gao (1996) has described a nonlinear continuum model that is essentially similar to the Kerkhof idea; sH is equivalent to the maximum cohesive stress smax in the Gao model. We will return to this issue in Chapters 11 and 12.

107

Chapter 8 Crack Tip Stress and Deformation Field Measurement

We now provide a discussion of various methods that have been used for the evaluation of the crack tip stress, strain or displacement ﬁelds. These include optical methods such as photoelasticity, moire´ interferometry, the method of caustics and coherent gradient sensing developed specially for applications in fracture mechanics and strain gage methods.

8.1 Jones Calculus Propagation of a polarized light beam through optical components that exhibit refractive index changes can be analyzed with great ease through the matrix approach introduced by Jones (1941); Jones calculus is summarized here in order to facilitate discussions of photoelasticity and lateral shearing interferometry. Detailed account of the method can be found in the monograph by Srinath (1983). A polarized light beam can be represented by the amplitude of its electric ﬁeld, decomposed along two directions orthogonal to the propagation direction; let us take the propagation direction to be x3 and let ðx1 ; x2 ; x3 Þ form a right-handed coordinate frame. The electric vector components are then written in a vector form, called the Jones vector " # E1 eiðvt2w1 Þ ð8:1Þ E¼ E2 eiðvt2w2 Þ where E1 and E2 are the amplitudes resolved along the x1 and x2 directions, v ¼ 2pc=l; c the speed of light, l the wave length of light, and w1 and w2 are the absolute phase angles of the two components; in general, this Jones vector represents an elliptically polarized light beam. In evaluating the inﬂuence of any optical element on light propagation, we will be concerned with the relative phase difference introduced between these two components of the light vector and hence for convenience, we can set one of the phase changes to unity and carry the relative phase difference in the other component. Let the relative phase angle Ds ; w1 2 w2 : Each optical element in the path of the light beam is characterized by two parameters—the orientation, a; of its fast principal optical axis relative to the global x1 direction and the amount of relative phase change Ds that the optical element introduces;

108

Chapter 8

with these two parameters, the Jones vector of the light incident on the optical element can be transformed to determine the Jones vector of the light leaving the optical element. The transformation matrix is called the Jones matrix and is easily calculated. We will illustrate this for a general optical element and then apply it to particular cases. The transformation is performed in three steps: ﬁrst, the incoming wave Ein is resolved along the principal directions of the optical element: " # cos a sina 0 ð8:2Þ E ¼ Ein 2sina cosa Second, the relative phase change (a retardation along the slow axis relative to the fast axis) is introduced along the slow principal direction " # 1 0 00 E ¼ E0 ð8:3Þ 0 e2iDs pﬃﬃﬃﬃﬃﬃﬃ where i ¼ 21 and Ds the relative phase difference between the principal (the fast and slow) components of the light vector. Finally, the principal components are recombined along the global x1 ; and x2 directions. " # cosa 2sina 00 ð8:4Þ Eout ¼ E sina cosa Combining these operations, the relationship between the input Jones vector Ein and the output Jones vector Eout for this optical element is given by Eout ¼ JðDs; aÞEin

ð8:5Þ

where " JðDs; aÞ ¼

cos2 a þ e2iDs sin2 a

sina cosað1 2 e2iDs Þ

sina cosað1 2 e2iDs Þ

sin2 a þ e2iDs cos2 a

# ð8:6Þ

is the Jones matrix for the optical element. If the light beam passes through a collection of n optical elements each with Jones matrix Jk ; then the output of each element is the input for the succeeding element; hence, the output electric vector is given by Eout ¼ Jn ðDsn ; an ÞJn21 ðDsn21 ; an21 Þ· · ·J2 ðDs2 ; a2 ÞJ1 ðDs1 ; a1 ÞEin

ð8:7Þ

For future reference, we list below the Jones matrices of common optical elements: a linear polarizer, a quarter wave plate, and a half-wave plate. A linear polarizer allows only one plane of polarization; therefore, the relative phase change can be taken to be inﬁnite in the other component; let the orientation be at an angle a with respect to the global direction. Then the Jones matrix is given by " # sina cosa cos2 a Jlinear ð1; aÞ ¼ ð8:8Þ sina cosa sin2 a

Crack Tip Stress and Deformation Field Measurement

109

For a quarter wave plate, Ds ¼ p=2; it is usually placed at an angle of a ¼ ^p=4 with respect to the global directions; therefore the Jones matrix is given by 3 2 12i 1þi ^

p p 6 2 2 7 7 ð8:9Þ ;^ ¼6 Jl=4 4 1þi 2 4 12i 5 ^ 2 2 An isotropic phase retarder changes the phase in each component by the same amount; the corresponding Jones matrix is " # expð2iwÞ 0 Jiso ðw; aÞ ¼ ð8:10Þ 0 expð2iwÞ With the Jones calculus of polarized light, we can examine light propagation in a stressed specimen in speciﬁc optical conﬁgurations.

8.2 Photoelasticity Brewster (1814, 1815) found that in some materials the index of refraction was affected by the application of pressure; upon removal of the external load, the index of refraction returned to its original value. The basis of the method of photoelasticity is this temporary, stress-induced birefringence exhibited by many polymers; for example polymethylmethacrylate, Homalite-100 and polycarbonate are commonly used in dynamic fracture investigations because of their birefringent properties. Thus, a stressed specimen becomes a phase retarder (with spatially varying retardation induced by the spatial variation of the stress ﬁeld); the fast and slow principal optical directions coincide with the principal stress directions. The index of refraction at a point on a stressed specimen depends on the stress state at that point; hence a polarized light beam traveling through a stressed specimen will experience a relative phase retardation in the components of the light resolved along the local principal stress directions. This stress-induced birefringence can be exploited to reveal the stress ﬁeld in the specimen. In this section, we ﬁrst describe the phenomenon of temporary birefringence, then evaluate the Jones matrix for a stressed specimen and ﬁnally describe the circular polariscope that is commonly used for revealing the shear stress distribution in the specimen. Maxwell expressed the dependence of the index of refraction on the principal stresses in the following form n1 2 n 0 ¼ c 1 s1 þ c 2 ð s2 þ s3 Þ n2 2 n 0 ¼ c 1 s2 þ c 2 ð s3 þ s1 Þ

ð8:11Þ

n3 2 n 0 ¼ c 1 s3 þ c 2 ð s1 þ s2 Þ where n0 is the isotropic, unstressed index of refraction, c1 and c2 are the direct and transverse stress optic coefﬁcients (in units of m2/N, sometimes labeled brewsters), si the principal stress components at any point and ni the refractive indices in the principal stress directions at that point. For the plane stress conditions that are typical of the thin plates

110

Chapter 8 Table 8.1 Material stress fringe value fs for selected polymers (l 5 514 nm)

Material Homalite-100 Polycarbonate Polymethylmethacrylate

fs (kN/m)

Source

22.2 6.6 129

Dally and Riley (1978) Dally and Riley (1978) Kalthoff (1987)

used in most fracture experiments, s3 can be set to zero; furthermore, for light rays propagating in the x3 direction, only the refractive indices n1 and n2 are of interest. Hence, a polarized light beam that travels through a point on a stressed specimen will be decomposed into two components along the principal directions and these two components travel with different speeds; thus the light components that emerge from the specimen will have a relative angular phase difference given by Dsðx1 ; x2 Þ ¼

2ph 2phC 2ph ðn1 2 n2 Þ ¼ ð s1 2 s 2 Þ ¼ ð s1 2 s2 Þ fs l l

ð8:12Þ

where C ¼ ðc1 2 c2 Þ is the relative stress optic coefﬁcient, h the thickness of the specimen through which the light travels, and fs ¼ l=C is called the material stress fringe value. Methods for the determination of fs are discussed in books dealing with photoelasticity (see for example, Dally and Riley, 1978). Typical values of fs for different materials are listed in Table 8.1. The spatial dependence of the phase difference arises from the spatial dependence of the stress ﬁeld. Thus, the effect of a stressed birefringent specimen on light propagation is simply that a phase retardation is introduced between components resolved along the local principal directions. Let bðx1 ; x2 Þ denote the orientation of the local maximum principal direction with respect to the global x1 direction at any point in the specimen. The effect of the stressed specimen on the propagation of a polarized light beam at the point ðx1 ; x2 Þ may be represented using a Jones matrix: " # cos2 b þ e2iDs sin2 b sinb cosbð1 2 e2iDs Þ ð8:13Þ Jb ðDs; bÞ ¼ sinb cosbð1 2 e2iDs Þ sin2 b þ e2iDs cos2 b In order to exhibit the phase difference in a visual form, the stressed specimen is introduced into a circular polarizer; the arrangement of a circular polarizer is shown schematically in Fig. 8.1. Consider a light beam polarized along the global x2 direction. The incoming light is described by the Jones vector: " # 0 Ein ¼ ð8:14Þ keivt It is then made to pass through a quarter wave plate that introduces a relative retardation of one quarter wavelength between the two components of the incident light vector. The fast axis of the quarter wave plate is oriented at an angle p=4 with respect to the global x1 direction; the Jones matrix of the quarter wave plate is then evaluated from Eq. 8.6 with Ds ¼ p=2; a ¼ p=4 and is given in Eq. 8.9. The light vector exiting the quarter wave plate

111

Crack Tip Stress and Deformation Field Measurement

Figure 8.1 Optical arrangement of a circular polariscope in a dark ﬁeld conﬁguration.

is easily determined through Jones calculus: 2 3 1þi

p p 6 2 7 7 E0 ¼ Jl=4 ; Ein ¼ keivt 6 4 12i 5 2 4

ð8:15Þ

2 The light beam exiting the quarter wave plate is a circularly polarized light; this beam is then transmitted through the stressed specimen where it suffers a phase retardation in one component relative to the other through the stress-induced birefringence discussed above. The light vector exiting the specimen accumulates a phase difference given in Eq. 8.12 and is represented by

p p E00 ¼ Jb ðDs; bÞJl=4 ; ð8:16Þ Ein 2 4 The light beam exiting the specimen is then made to pass through a second quarter wave plate oriented at 2p=4 with respect to the global x1 axis and then through a second polarizer. The second polarizer may be oriented either along the x1 direction (called the dark ﬁeld arrangement) or along the x2 direction (called the bright ﬁeld arrangement). The light beam exiting the second polarizer in the dark ﬁeld arrangement can then be written as

p

p p p ;2 ; ð8:17Þ Eout ¼ Jlinear ð1; 0ÞJl=4 Jb ðDs; bÞJl=4 Ein 2 4 2 4

112

Chapter 8

Substituting for the input from Eq. 8.14 and for the appropriate Jones vectors from Eqs. 8.6, 8.8 and 8.9, the output can be shown to be " 2i2b # ð1 2 e2iDs Þ e keivt J Eout ¼ ð1; 0Þ ð8:18Þ 2 linear 2ið1 þ e2iDs Þ The linear polarizer placed along x1 direction allows only the x1 component of the light to pass through and therefore the output electric vector is given by " 2i2b # ð1 2 e2iDs Þ keivt e ð8:19Þ Eout ¼ 2 0 The intensity of this light beam is the time average of E2out ; averaged over a time signiﬁcantly longer than the period: k2 2 Dsðx1 ; x2 Þ 2 sin ð8:20Þ Iðx1 ; x2 Þ ¼ kEout l ¼ 2 2 Thus, the spatial variation of the light intensity is governed only by the phase retardation introduced by the stressed specimen; in this optical arrangement, the orientation of the principal directions bðx1 ; x2 Þ does not inﬂuence the intensity. Introducing the phase difference from Eq. 8.12, bright fringes corresponding to maximum light intensity are lines in the x1 2 x2 plane along which ð s1 2 s2 Þ ¼

Nfs ; with N ¼ 0; ^1; ^2; … h

ð8:21Þ

where N is called the fringe order. Therefore, placing the specimen between crossed circular polarizers reveals lines of constant intensity that are contours of constant in-plane shear stress; these lines are called isochromatic fringes. This is the basis of all photoelasticity; in applications to dynamic problems, the isochromatic fringe patterns are captured with a high-speed camera at short time intervals to provide a time history of the evolution of the shear stresses in the specimen. In applications to fracture mechanics, it is assumed that the crack tip asymptotic ﬁeld is applicable in the vicinity of the crack tip and the parameters of the asymptotic ﬁeld are extracted by ﬁtting the observed isochromatic fringes to theoretically predicted patterns in a least-squared error process. This procedure is discussed in Section 8.2.1. 8.2.1 Evaluation of the Dynamic Stress Intensity Factor using Photoelasticity Assuming that the fringe pattern formation is governed by the asymptotic stress ﬁeld near the crack tip, the geometry of the fringe pattern can be expressed as follows Nfs ¼ ðs1 2 s2 Þ ¼ gðr; u; KIdyn ; KIIdyn ; sox Þ h

ð8:22Þ

where N is the fringe order, fs the fringe sensitivity, h the specimen thickness, and s1 ; s2 are the principal stress components. gðr; u; KIdyn ; KIIdyn ; sox Þ is determined from the linear

Crack Tip Stress and Deformation Field Measurement

113

elastic crack tip stress ﬁeld (see Eq. 3.32); in this representation, only the singular term and the ﬁrst higher order term are indicated whereas in actual applications higher order terms in the crack tip stress ﬁeld are also taken into account in interpreting the experimental fringe pattern. Simulated isochromatic fringe patterns corresponding to assumed values of KIdyn and sox are shown in Fig. 8.2. An example of the time evolution of the fringe patterns obtained with a high-speed camera is shown in Fig. 8.3. The dynamic stress intensity factor KIdyn ; can be obtained at each instant in time by using a least-squares matching of the experimentally measured fringe pattern with simulations based on Eq. 8.22. Many investigators have contributed to the development of this method of determining the stress intensity factor. Details may be found in the Handbook of Experimental Mechanics. Here a brief summary is provided. First, the experimental isochromatic fringe pattern is

Figure 8.2 Simulated isochromatic fringe patterns corresponding to a dark ﬁeld circular polariscope arrangement. (a) KI 5 1 MPa m1/2, KII 5 0 and sox 5 0; (b) KI 5 1 MPa m1/2, KII 5 0 and sox 5 50 MPa; (c) KI 5 0 MPa m1/2, KII 5 1 MPa m1/2 and sox 5 0; (d) KI 5 1 MPa m1/2, KII 5 1 MPa m1/2 and sox 5 0: fs 5 7 kN/m corresponding to polycarbonate has been assumed. The ﬁeld of view shown in these ﬁgures 40 mm to a side and v=Cd 5 0:1:

114 Figure 8.3 Selected frames from a high-speed sequence showing isochromatic fringe patterns obtained in an experiment with a quasi-statically loaded single-edge notched specimen. Polycarbonate specimen observed in a dark ﬁeld circular polariscope arrangement. A conducting paint line was used to trigger the high-speed camera. Frames are 10 ms apart. The height of the paint line crossing the crack path is 25 mm. (Reproduced from Taudou et al., 1992.)

Chapter 8

115

Crack Tip Stress and Deformation Field Measurement

quantiﬁed by a collection of ðNi ; ri ; ui Þ; measured at M points. The distance r at which these measurements are taken should be appropriate for the application of the twodimensional asymptotic crack tip stress ﬁeld; this is typically interpreted to hold for r=h . 0:5 based on the experimental results of Rosakis and Ravi-Chandar (1986). The sum of the squared error in Eq. 8.22 at all measured points is then given by 2 M X Ni fs 2 gðri ; ui ; a1 ; a2 ; …; ak Þ ð8:23Þ eða1 ; a2 ; …; ak Þ ¼ h i¼1 where the stress ﬁeld parameters ðKIdyn ; KIIdyn ; sox ; …Þ are represented by the vector a ¼ ða1 ; a2 ; …; ak Þ: The stress ﬁeld parameters must be obtained by minimizing e with respect to the parameters. Near the minimum, the function e can be expanded as a quadratic form eða1 ; a2 ; a3 ; …Þ , g 2 2

M X k¼1

bk ak þ

M X M X

akl ak al

ð8:24Þ

k¼1 l¼1

where M X 1 ›e N i fs ›gi 2 gi bk ¼ 2 ¼ 2 ›ak h › ak i¼1 and

akl ¼

2 M X 1 ›2 e ›gi ›gi Ni fs › gi 2 gi ¼ 2 2 ›ak ›al ›ak ›al h ›ak ›al i¼1

and gi ¼ gðri ; ui ; a1 ; a2 ; a2 ; …Þ: Since ½ðNi fs =hÞ 2 gi is expected to be small, the second term in akl is usually neglected and only the ﬁrst derivative of g needs to be evaluated. The estimate for the increment in the parameters is obtained by setting 7e ¼ 0: This results in the following estimate for the increments in the parameters dak :

bl ¼

M X

akl dak

ð8:25Þ

k¼1

With the update of the parameters a ¼ ða1 ; a2 ; …; ak Þ obtained from the above, the procedure is repeated until the parameters converge. For a thorough discussion of the procedures for such least squared error ﬁtting, see Press et al. (1992). Curve ﬁtting routines based on the above are also implemented in Mathematica, Matlab and other commercial software. There appears to be multiple minima for the function in Eq. 8.23 and hence it is usually a good practice to simulate the isochromatic fringe patterns with the converged parameters and perform a visual comparison of the recreated fringes to the experimentally observed pattern. It is important to note that in following the procedure described above, it has been implicitly assumed that the stress ﬁeld is completely determined by the two-dimensional crack tip ﬁeld. However, there are signiﬁcant limitations on the application of the crack tip asymptotic ﬁeld. Clearly, this ﬁeld cannot hold very close to the crack tip; in some regions near the crack tip, the ﬁeld must exhibit a three-dimensional variation. Although most

116

Chapter 8

investigators have assumed that three-dimensional effects are important at distances less than about one half of the plate thickness (using observations of Rosakis and RaviChandar, 1986), Mahajan and Ravi-Chandar (1989) demonstrated that quasi-static K-dominance can be observed in photoelastic experiments at much smaller distances from the crack tip. In addition, the asymptotic ﬁeld cannot hold very far from the crack tip due to the dynamic nature of the stress ﬁeld. See Chapter 9 for a full discussion of the dominance of the asymptotic stress ﬁeld.

8.3 Method of Caustics The method of caustics was discovered accidentally by Schardin (1959), when the highspeed camera he used to photograph running cracks was slightly out of focus! It has been developed into a quantitative tool for the determination of static and dynamic stress intensity factors by Mannogg (1964), Theocaris (1970), Kalthoff (1987), Rosakis (1980) and others. A special volume of the International Journal of Lasers and Optics in Engineering in 1991 contains a summary of many of the advances in the method. Kalthoff (1987) and Rosakis (1993) have also provided an elaborate discussion of the method. The method of caustics has been particularly popular in dynamic fracture investigations due to its simplicity—sophisticated optical conﬁgurations and elaborate analyses of data are not required. However, the method also has signiﬁcant inherent limitations. Here we describe the essential ingredients of the method for application to dynamic crack problems and discuss the limitations of the method. 8.3.1 Physical Principle of Formation of Caustics The principle of formation is very simple as illustrated in Fig. 8.4: consider a parallel beam of light incident along the x3 direction, normal to the specimen free surface. Polymers such as Araldite, Homalite-100 and polymethylmethacrylate that have been popular as materials for dynamic fracture investigations are transparent. On the other hand, structural materials are opaque, but the method applies equally well to these opaque materials when light rays are reﬂected from the surface of the specimen; the surface must be polished to a mirror ﬁnish to ensure specular reﬂection of light. In a transparent material a light ray passing through a stressed plate is deviated from its path partly due to thickness variation generated by the deformation and partly due to the change in refractive index caused by stress-induced birefringence. In an opaque material, the light ray reﬂects with the deviation from parallelism dictated by the local slope or equivalently the thickness change. These deviations in the light path occur in any specimen that is under a nonuniform stress ﬁeld; if, however, the plate contains a crack, the rays are deviated from the region around the crack tip and these form a singular curve called ‘stress corona’, ‘shadow spot’ or ‘caustic’ on a reference plane at some distance away from the specimen. The size of the caustic curve can be related to the stress intensity factor by introducing an analysis based on geometrical optics and fracture mechanics. This analysis is described below.

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Figure 8.4 A schematic illustration of the principle of formation of the caustic curve; (a) transmission arrangement for transparent specimens; (b) reﬂection arrangement for opaque specimens.

First, we examine the path of the light rays traversing a transparent cracked plate. Consider a light ray traversing in the x3 direction as illustrated in Fig. 8.5. The Maxwell – Neumann stress-optic law describes variation of the refractive index with stress inside the specimen. For the case of an optically isotropic material, the stress optic law relating the refractive index nðxÞ at a point x in the material on the stress ﬁeld can be written as nðxÞ ¼ n0 þ cðs1 ðxÞ þ s2 ðxÞ þ s3 ðxÞÞ

ð8:26Þ

where n0 is the unstrained refractive index of the specimen material, c the stress-optic coefﬁcient and si ðxÞ the principal stresses at the point x. The path of the light ray in the specimen varies continuously, but can be determined through geometrical optics.

Figure 8.5 Path of a ray from the specimen plane to the screen plane.

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Chapter 8

The differential equation governing the ray is given by dR 7nðxÞ ¼ dl n

ð8:27Þ

where R is the ray vector and l the scalar length along the ray path. At the entry and exit interface with the deformed surface of the specimen, we have a discrete change in the refractive index and the ray paths at these points are determined by the discrete application of Snell’s law for refraction. To be able to apply this law, we must determine the equations of surfaces S1 and S2 (see Fig. 8.4). This is done using the two-dimensional asymptotic elastic crack tip stress ﬁeld. We note that this represents an approximation imposed on the mapping since the actual stress ﬁeld near the crack tip must have a three-dimensional variation; we shall return to this point later. The Cartesian components of the stress distribution at the tip of a dynamically growing crack under steady state speed v are expressed in Eq. 8.28 KIdyn I KIIdyn II fab ðu; vÞ þ pﬃﬃﬃﬃﬃﬃﬃﬃ fab ðu; vÞ þ Oð1Þ as r ! 0; a; b ¼ 1; 2 sab ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ 2pr 2pr

ð8:28Þ

KI and KII are the mode I and mode II stress-intensity factors, respectively. ðr; uÞ are the polar coordinates centered at the tip of the crack moving with a speed v. The angular I II ðu; vÞ and fab ðu; vÞ are given in Appendix A. variations fab Assuming that a condition of plane stress exists, the out-of-plane stresses s3a and s33 are zero and the out-of-plane displacement ﬁeld u3 is given by the expression u3 ¼

2nx3 ðs11 þ s22 Þ þ Oð1Þ E

as r ! 0

ð8:29Þ

where n is the Poisson’s ratio and E the modulus of elasticity. Eqs. 8.28 and 8.29 can be used to obtain the equations of the deformed surfaces, S1 and S2 h h S1;2 : ^ x3 ^ u3 r; u; ^ ¼0 ð8:30Þ 2 2 where h is the plate thickness. Once the deformed surfaces are determined, the mapping analysis is complete and one can determine the ray path using Eqs. 8.26 –8.30. Simulation of caustics through this technique was demonstrated by Mahajan and Ravi-Chandar (1989). An approximate evaluation of these equations can be obtained by assuming that an equivalent change in the optical path or in the wavefront is introduced at the specimen midplane; the surface of constant phase is then given by ð h=2 ð h=2 Ds ¼ dsðx1 ; x2 Þ ¼ h Dndx3 þ ðn 2 1Þ 133 dx3 2p=l 2h=2 2h=2 ðn 2 1Þn ¼h c2 0 ð8:31Þ ðs11 þ s22 Þ ; hct ðs11 þ s22 Þ E Note that plane stress conditions have been assumed in determining the 133 component of strain. The ﬁrst term in Eq. 8.31 represents the stress-optic effect and the second term

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the change in thickness due to Poisson contraction of the specimen. This approximation can then be used to determine the ray paths. The position vector of the rays in the screen plane can be expressed as R ¼ r 2 w ¼ r 2 z0 grad dsðr; uÞ

ð8:32Þ

where z0 is the distance from the specimen midplane to the screen plane. Eq. 8.32 represents the transformation relation for mapping points r ¼ x1 e1 þ x2 e2 in the specimen plane to points R ¼ X1 e1 þ X2 e2 in the screen plane. ðe1 ; e2 Þ are the unit vectors in the ðx1 ; x2 Þ; directions. If the Jacobian determinant of this transformation is zero, then the mapping is singular; this is the condition for the formation of a caustic curve on the screen. Before examining the details of the mapping, it is useful to examine the mapping qualitatively. As can be seen from the illustration in Fig. 8.4, far away from the crack tip, the light rays pass through the transparent specimen and maintain their parallel propagation; the inﬂuence of the stress ﬁeld on the wavefront is small enough to be neglected. On the other hand, in the region near the crack tip, where the specimen exhibits a concave surface due to the Poisson contraction, the light rays deviate signiﬁcantly from parallelism. As a result, a dark region called the shadow spot forms on the screen at z0 where there are no light rays at all. This shadow region is surrounded by a bright curve, called the caustic curve. The line on the specimen plane whose image is the caustic curve on the specimen is called the initial curve. Light rays from outside the initial curve fall outside the caustic, rays from inside the initial curve fall on or outside the caustic curve and rays from the initial curve fall on the caustic curve. Hence the caustic curve is a bright curve that surrounds the dark region. Therefore, the equation for the initial curve is obtained by setting the Jacobian determinant of Eq. 8.32 to zero and the caustic curve is obtained by evaluating the mapping in Eq. 8.32 on the initial curve. The details of these calculations are shown below. Using Eqs. 8.28 and 8.31 in Eq. 8.32 and restricting attention to mode I loading condition results in the following mapping equation (

) 5=2 2 r0 3ud cos X1 ¼ rd cosu d þ 3 ad r d 2 ( ) rd 2ad r0 5=2 3ud X2 ¼ sinu d þ sin ad 3 rd 2

ð8:33Þ

where 3hct z0 KI 2ð1 þ a2s Þða2d 2 a2s Þ and FðvÞ ¼ r0 ¼ pﬃﬃﬃﬃﬃﬃ 4ad as 2 ð1 þ a2s Þ2 2 2pFðvÞ

ð8:34Þ

ad and as are deﬁned in Appendix A. A simulation based on the mapping equations in Eq. 8.33 is shown in Fig. 8.6a. The ‘shadow spot’ and the ‘caustic curve’ are clearly seen in this simulation. We can now evaluate the Jacobian determinant of the optical mapping

120

Chapter 8

Figure 8.6 Simulated bitmap image of the transformation in Eq. 8.32. The dark region is the ‘shadow spot’ and is surrounded by the bright ‘caustic’ curve. The ﬁeld of view represents a square, 20 mm to a side. (a) KI 5 1 MPa m1/2, KII 5 0; (b) KI 5 0; KII 5 1 MPa m1/2, (c) KI 5 1 MPa m1/2, KII 5 1 MPa m1/2.

in Eq. 8.33; this yields the equation for the initial curve: 5 5=2 rd rd 5u 21 þ ða2d 2 1Þcos d ¼ 0 r0 r0 2

ð8:35Þ

The Jacobian determinant is zero at points ðx1 ; x2 Þ or ðrd ; ud Þ that satisfy Eq. 8.35. Kalthoff (1987) and Rosakis (1980) showed that the last term in Eq. 8.35 is small when crack velocities are less than about 40% of the shear wave speed; since mode I cracks seldom move at speeds greater than this value, they suggested that this term could be neglected. Ignoring the last term, the Jacobian determinant goes to zero when rd ¼ r0 ; therefore, the initial curve is given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð8:36Þ rd ¼ r 1 2 ðvsinu =Cd Þ2 ¼ r0 This represents a circle in the distorted coordinates ðx1 ; ad x2 Þ: Substituting this result into Eq. 8.34 yields the equation of the caustic curve 2 3u cos d X1c ¼ r0 cosu d þ 3 ad 2 ð8:37Þ r 2 a 3 u d sin d X2c ¼ 0 sinu d þ ad 3 2

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where r0 is given in Eq. 8.35. Eqs. 8.36 indicate that the size of the caustic depends only on r0 ; which in turn depends directly on the plate thickness h, the optical coefﬁcient ct , the distance between the specimen midplane and the screen z0 , the dynamic stress intensity factor KI , and only weakly on the crack velocity. Eqs. 8.37 are the equations of an epicycloid, with r0 playing the role of a scale parameter. The maximum extent of the caustic curve in the x2 direction, D (marked on Fig. 8.6 and usually referred to as the transverse diameter of the caustic curve) is taken to be the a measure of the size of the caustic curve. Then, from Eqs. 8.37, the following relationship can be obtained: D ¼ ½X2 ðu ¼ 2p=2Þ 2 X2 ðu ¼ p=2Þ ¼ 3:17r0 Inserting this in Eq. 8.35 and rearranging establishes the following equation for the determination of the dynamic stress intensity factor: pﬃﬃﬃﬃﬃﬃ 2 2p D 3=2 KI ¼ 2ct hz0 FðvÞ 3:17

ð8:38Þ

Thus, from measurements of the transverse diameter of the caustic curve, the dynamic stress intensity factor can be determined. Application of the method to quasi-static and dynamic problems has been demonstrated by a number of investigators. If these measurements are performed by obtaining high-speed photographs of the images in the screen plane, the variation of the dynamic stress intensity factor with time may be obtained. An example of the caustics observed in a dynamically propagating crack is shown in Fig. 8.7. 8.3.2 Caustic in Reﬂection The analysis described above is for transmission of light through a transparent specimen; the analysis can be repeated for the formation of caustics in an opaque specimen by reﬂection. The following changes need to be taken into account. First, the screen plane is behind the specimen and therefore the screen plane is a virtual plane obtained by focusing the camera to this plane; the optical arrangement is shown in Fig. 8.4. Second, the stress optic effect is not relevant and only the thickness change contributes to the caustic formation. Therefore, in Eq. 8.31, the surface of constant phase is replaced by ð h=2 Ds h ¼ dsðx1 ; x2 Þ ¼ 2u3 x1 ; x2 ; 2 133 dx3 ¼2 2p=l 2 2h=2 ¼2

hn ðs þ s22 Þ ; hcr ðs11 þ s22 Þ E 11

ð8:39Þ

where cr ¼ 2n=E: Replacing ds in Eq. 8.32 with the above expression, all equations corresponding to caustics by reﬂection may be obtained. In particular, the relationship between the transverse diameter and the dynamic stress intensity factor is still given by Eq. 8.38, by simply replacing ct with cr :

122

Chapter 8

Figure 8.7 High-speed photographs of caustics at the tip of a dynamically propagating crack. (Reproduced from Ravi-Chandar and Knauss, 1984c.)

The analysis presented above corresponds to a crack growing at a constant speed v. Taking a limit as v ! 0 in Eq. 8.28, the derivation of the caustic curve appropriate for a stationary crack may be obtained. Note that the limit must be taken carefully because the I ðu; vÞ becomes indeterminate in the limit as v ! 0: In this case, the relationship function fab between the transverse diameter and the dynamic stress intensity factor is still given by Eq. 8.38, with Fðv ! 0Þ equal to 1.

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8.3.3 Mixed-mode Caustics The analysis presented here has considered only the problem of mode I; an entirely analogous procedure, but with the mode II asymptotic ﬁeld added to the mapping equations will result in the following equation for the caustic curve: 2 3ud 3ud c 2 msin X1 ¼ r0 cosu d þ cos 3 ad 2 2 ð8:40Þ r 2 a 3 u 3 ud 0 d d c sinu d þ X2 ¼ sin þ mcos ad 3 2 2 where m is the ratio of the stress intensity factors KII =KI : Fig. 8.6 illustrates the shape of the caustic curve for the case of pure mode II and an equal mix of modes I and II. The caustic curve is still an epicycloid—with r0 playing the role of a scale parameter—but now the factor m determines the orientation of the epicycloid. The maximum and minimum extents of the caustic curve in the x1 direction, Dmax and Dmin (marked in Fig. 8.6) are taken to be the measure of the size of the caustic curve; these two parameters are used to determine relationship for the determination of the mode I and mode II dynamic stress intensity factors. From Eq. 8.40, Dmax and Dmin ; can be related to the radius of the initial curve as follows: Dmax ¼ ½X1 ðu ¼ 2pÞ 2 X1 ðu ¼ 0Þ ¼ gðmÞr0 Dmin ¼ ½X1 ðu ¼ pÞ 2 X1 ðu ¼ 0Þ ¼ g1 ðmÞr0 It is clear that the ratio ðDmax 2 Dmin Þ=Dmax depends only on m; this dependence is shown in Fig. 8.8. The function gðmÞ is shown in Fig. 8.9. Therefore, m can be determined ﬁrst

Figure 8.8 Relationship between the caustic dimensions and the stress intensity factor ratio m. (Reproduced from Kalthoff, 1987.)

124

Chapter 8

Figure 8.9 Relationship between the numerical factor g and the stress intensity factor ratio m. (Reproduced from Kalthoff, 1987.)

from the measurements of Dmax and Dmin ; and then, KI and KII are determined from the following: pﬃﬃﬃﬃﬃﬃ 2 2p Dmax 3=2 KI ¼ ð8:41Þ 2ct hz0 FðvÞ gðmÞ KII ¼ mKI

ð8:42Þ

While the method has been used by a number of investigators in evaluating mixed-mode stress intensity factors in quasi-static problems, dynamically growing cracks typically follow a locally mode I path and hence there are very few examples of the evaluation of the mixed mode stress intensity factors. 8.3.4 Limitations on the Applicability of the Method of Caustics There are limitations on the applicability of the method of caustics that need attention in any application. Before examining these limitations, it is useful to recall that the caustic curve on the screen plane results from the light rays that go through the initial curve in the specimen plane. Therefore, the position at which the stress ﬁeld, and therefore the dynamic stress intensity factor, is measured is located at a distance r0 from the crack tip. In interpreting the mapping through Eq. 8.38 it is implicitly assumed that the stress ﬁeld at a distance r0 from the crack tip is governed by the square root singular K-ﬁeld corresponding to plane stress. The limitations in the interpretation of the caustic curve in terms of the dynamic stress intensity factor arise from two key effects—threedimensional and transient; we will discuss these in the following.

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Three-dimensional effects. Rosakis and Ravi-Chandar (1986) examined the inﬂuence of the three-dimensionality of the stress ﬁeld on the applicability of the method of caustics under quasi-static loading of a stationary crack. They tested a number of single-edge notched specimens made of PMMA (in transmission) and steel (in reﬂection). Specimens with the same in-plane dimensions but thickness in the range 1.59 – 25.4 mm were used. Furthermore, caustic diameters were measured by systematically varying the distance z0 between the specimen midplane and the screen plane while maintaining all other parameters constant. From Eq. 8.34, it can be seen that varying z0 is equivalent to varying the initial curve radius, r0 : Through this combination of specimens with different thicknesses and variation of z0 for each specimen, the range of 0 , r=h , 2 was covered in the experiments. The resulting caustics measurements were interpreted in terms of the stress intensity factor through Eq. 8.38; the measured values of the stress intensity factor were denoted K EXP : Simultaneously, the load applied on the specimen was also monitored, therefore, the stress intensity factor was calculated from the two-dimensional theory, and denoted by K2D : If the plane stress singular ﬁeld is appropriate, then, interpreting these caustic dimensions through Eq. 8.38 should provide the quasi-static stress intensity factor, K2D : The ratio of K EXP =K2D is plotted as a function of r=h in Fig. 8.10 for the PMMA specimen. The most striking feature of the experimental results displayed in Fig. 8.10 is the r=h similarity. Clearly, for r=h . 0:5; K EXP approaches K2D ; suggesting that the planestress conditions prevail at these distances and furthermore that the interpretation of the caustics in terms of Eq. 8.38 is appropriate. For r=h , 0:5; three-dimensional effects seem to take over completely and the validity of the two-dimensional analysis questionable. However, this should not just be interpreted as an invalidation of the singular ﬁeld itself because the method of caustics depends on the gradients of the ﬁeld and hence may be susceptible to large errors even if the deviation in the ﬁeld itself is quite small.

Figure 8.10 Ratio of stress intensity factor inferred from local shadow spot measurements (transmitted caustics) to analytical two-dimensional value vs the distance from the crack tip normalized by the plate thickness. (Reproduced from Rosakis and Ravi-Chandar, 1986.)

126

Chapter 8

Mahajan and Ravi-Chandar (1989) explored dominance of the singular ﬁeld in two different ways: First, recognizing that the method of caustics is sensitive to both the stressoptic effect and the Poisson effect (as seen in Eq. 8.31) and that the gradients of the surface displacement u3 might depart more signiﬁcantly from the plane stress prediction, they performed an experiment where the two effects were separated. By immersing the specimen in a liquid medium with a refractive index identical to that of the unstressed specimen, they eliminated the second term in Eq. 8.31; thus the caustics observed were sensitive only to the stress optic effect. In this arrangement, they repeated the measurements of Rosakis and Ravi-Chandar (1986) and concluded that the singular ﬁeld may dominate at distances as small as r=h , 0:25: Second, they analyzed photoelastic fringe patterns ðNi ; ri ; ui Þ to extract the stress intensity factor (see Section 8.2.1 for the procedure) in the same specimens; evaluating the stress intensity factor at different distances r, they extracted the r=h dependence; the variation of K EXP =K2D with r=h is shown in Fig. 8.11. Clearly, the plane stress singular ﬁeld is appropriate at distances as small as r=h , 0:1: It can be concluded that in interpreting caustics in terms of stress intensity factors, one must ensure that the initial curve is larger than r=h . 0:5: On the other hand, in interpreting photoelastic isochromatic fringe patterns in terms of the stress intensity factor, data can be gathered at distances as small as

Figure 8.11 Ratio of stress intensity factor inferred from analysis of isochromatic fringes vs the distance from the crack tip normalized by the plate thickness. (Reproduced from Mahajan and Ravi-Chandar, 1989.)

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127

r=h , 0:1: These limitations are easily overcome in quasi-static problems, but in dynamic problems, transient effects play a signiﬁcant role as discussed next. Dynamic effects. Application of caustics to dynamic problems was pioneered by Kalthoff et al. (1980a,b). However, the ﬁrst and still only comparison of theoretical predictions with experimental results was provided by Ravi-Chandar and Knauss (1982). Interpreting the caustics obtained from a high-speed photographic sequence through Eq. 8.38, the time variation of the dynamic stress intensity factor can be obtained. In using Eq. 8.38, it is important to recognize that there are two opposing requirements in the experiments and analyses: on one hand, due to the three-dimensional effects discussed above, the measurements must be made at least at a distance of one half of the plate thickness away from the crack tip. On the other hand, the transient nature of the stress ﬁeld requires that the measurements be made as close to the crack tip as possible to account for wave propagation effects; the dynamic instantaneous K-dominant ﬁeld (the region where the dynamic stress intensity factor is adequate to describe the total stress ﬁeld) is established only in a small region near the crack tip. These two opposing requirements render measurement of the dynamic stress intensity factor difﬁcult and may indeed bring into question the usefulness of the dynamic stress intensity factor in situations where the transient effects are large. The issue of dominance of the dynamic asymptotic ﬁeld for growing cracks is very important and is discussed in detail in Chapter 9. Ravi-Chandar and Knauss (1987) systematically varied the specimen to screen distance, z0 ; in repeat experiments performed with the electromagnetic loading method; these are the dynamic analogs of the quasi-static experiments of Rosakis and RaviChandar (1986). Varying the ratio of r=h in the range 0.1– 0.76 and Ravi-Chandar and Knauss (1987) found that the dynamic K-ﬁeld was not dominant even at very small distances. Hence, in transient problems, an additional limitation arises due to the wave character of the problem. While three-dimensionality requires that r=h . 0:5; the establishment of a dynamic K-ﬁeld requires that r p ðCs 2 vÞt: However, at the lower end, r=h q rp =h; where rp is the process zone size in order to establish the crack tip singular ﬁeld. Thus, the radius of the initial curve must satisfy the following inequality, bounded from below by the three-dimensional effects and from above by the dynamic and process zone effects: rp p ðCs 2 vÞt . r . 0:5h In order to get around the limitations imposed by the imposition of the K-ﬁeld, Liu et al. (1993) suggested using a transient asymptotic ﬁeld near the crack tip, and including higher order terms in the expansion to represent the stress ﬁeld, to obtain a more accurate measure of the dynamic stress intensity factor. While they demonstrate this through a numerical simulation, this approach has not been used in experimental procedures for stress intensity factor determination. We emphasize that for dynamically growing cracks, Eq. 8.38 presents a valid method of determining the dynamic stress intensity factor as long as the transient effects are small; this could be interpreted as a crack that grows at a speed of about 20% of the shear wave speed.

128

Chapter 8

A variant of the method of caustics was developed by Kim (1985a,b). This method— called the stress intensity factor tracer—can be used even in the absence of a high-speed camera; all one needs is a photodetector and an oscilloscope. The main principle of the method is the same as in the method of caustics; however, instead of capturing the caustic in a screen plane, the light rays are brought to a focus through a lens. The parallel rays come to a focus, but the deviated rays do not go through the focal point. If a focal plane mask is used to block out all the parallel rays, then only the deviated rays will pass through the mask; the intensity of the deviated light rays can be measured with a photodetector and related to the stress intensity factor. The location of the crack tip is irrelevant in this technique and hence another method must be used to determine the crack position and or speed.

8.4 Lateral Shearing Interferometry Shearing interferometers have been in use in a number of applications over the last century; the earliest application was apparently demonstrated by Waetzmann (1912). The principle of the method is very similar to other two-beam interferometric techniques (see Born and Wolf (1999) for a discussion of interference); the main idea in shearing interferometry is that rather than using a reference wavefront to interfere with the wavefront of interest, the wavefront is made to interfere with a copy of itself after introducing a predetermined (selectable) shear between the two components. One dominant use of the method continues to be in identifying and measuring aberrations in optical components. In microscopy, lateral shearing interferometry has been very useful in examining specimens by introduction of phase changes (Nomarski, 1955). In mechanics, the method was used to characterize the slope and curvatures of deﬂected plates by a number of investigators (see, for example, Assa et al., 1979; Chao et al., 1982; Subramaninan and Nair, 1985). The method of introducing the lateral shear varied in these different investigations, but the basic physical principle described above was common to all. Tippur et al. (1990) developed a variant based on a diffraction grating that enabled them to examine the surface deformations near a dynamically propagating crack; they termed their implementation the coherent gradient sensor or CGS. Lee and Krishnaswamy (1996) later introduced a simpler implementation of the experiment based on a calcite crystal. In this section, we discuss an even simpler experimental arrangement and then discuss its application to dynamic fracture. The analysis of the light propagation through Jones calculus makes the manipulations simple. The optical arrangement is shown in Fig. 8.12. Consider a light beam polarized at an angle of p=4 with respect to the global x1 axis. The electric vector components are " # 1 ð8:43Þ E ¼ keivt 1 If this light beam passes through a stressed specimen, it will accumulate phase retardation with respect to free-space propagation in both of its components. Thus, the stressed specimen is analogous to an isotropic phase retarder. The light emerging from the

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129

Figure 8.12 Optical arrangement for the lateral shearing interferometry or the coherent gradient sensor.

specimen is given by " 2iDsðx ;x Þ # 1 2 i vt e E ¼ ke e2iDsðx1 ;x2 Þ

ð8:44Þ

where Ds is the absolute phase angle of the components that depends on the state of stress of the specimen at the point ðx1 ; x2 Þ: If the specimen is birefringent, it behaves like a wave plate and the light beam will suffer different phase retardations along the two principal directions; this is not considered here. On the other hand, if the specimen is specularly reﬂecting, the distortions of an incident plane wavefront due to surface curvature may be expressed in terms of the phase retardation as described in Section 8.3. The main idea in shearing interferometry is to introduce a shear (or displacement) in the phase of one of the components relative to the other in Eq. 8.44; this is accomplished by introducing a uniaxial optical crystal (for example, calcite) into the path of the light beam. Uniaxial crystals produce two refracted beams—the ordinary ray with a refractive index n0 and the extraordinary ray with a refractive index n00 —for each incident beam, each polarized in mutually orthogonal planes. For a light beam incident at an arbitrary angle u1 with respect to the crystal axis, the angles of the refracted beams are u02 and u002 ; respectively, obtained from Snell’s law; after passing through the crystal of thickness h, the two beams are displaced by Dx1 with respect to each other along a direction speciﬁed by the orientation of the crystal. The beam displacement depends

130

Chapter 8

on the anisotropy of the crystal, ðn0 2 n00 Þ; the orientation of the crystal with respect to the incident light and the thickness of the crystal. Through careful choice of these parameters, a beam displacement on the order of a few millimeters may be obtained (See Born and Wolf (1999) for a discussion of uniaxial crystals). Let us consider the uniaxial crystal to be aligned such that the ordinary ray is polarized along x1 ; the extraordinary ray is polarized along the x2 axis and that these rays are sheared by an amount Dx1 along the x1 axis. At every point in the ﬁeld, there are now two beams: the ordinary ray that entered the specimen at the point ðx1 ; x2 Þ and the extraordinary ray that entered the specimen at the point ðx1 þ Dx1 ; x2 Þ: Thus the light beam leaving the crystal may be written as " 2iDsðx ;x Þ # 1 2 e i vt ð8:45Þ E ¼ ke e2iDsðx1 þDx1 ;x2 Þ Next, these two components are brought together by a polarizer oriented at an angle of p=4 with respect to the global x1 axis. The output electric vector is given by 2 3 1 2iDsðx1 ;x2 Þ 2iDsðx1 þDx1 ;x2 Þ þe Þ7 6 2 ðe 7 ð8:46Þ Eout ¼ keivt 6 41 5 2iDsðx1 ;x2 Þ 2iDsðx1 þDx1 ;x2 Þ ðe þe Þ 2 The intensity of this light beam is the time average of the electric vector E2out ; averaged over a time signiﬁcantly longer than the period: k2 Dsðx1 þ Dx1 ; x2 Þ 2 Dsðx1 ; x2 Þ cos2 Iðx1 ; x2 Þ ¼ kE2out l ¼ ð8:47Þ 2 2 If the beam displacement Dx1 ; is small, Eq. 8.47 may be rewritten as k2 2 Dx1 ›Ds cos Iðx1 ; x2 Þ ¼ 2 2 ›x1

ð8:48Þ

Thus light intensity variation observed in this optical arrangement depends on the gradients of the phase difference in the x1 direction. In applying this method to problems in mechanics, it remains to evaluate the angular phase difference Ds; this was already evaluated in our discussion of the method of caustics (see Eq. 8.31); for a transparent optically isotropic specimen, this is given by Ds ðn0 2 1Þn ¼h c2 ð8:49Þ ðs11 þ s22 Þ ; hct ðs11 þ s22 Þ 2p=l E For an optically opaque specimen, if the surfaces are ﬁnished to be specularly reﬂecting, the phase difference is given by (see Eq. 8.39) Ds hn ¼ 2 ðs11 þ s22 Þ ; hcr ðs11 þ s22 Þ 2p=l E

ð8:50Þ

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Therefore, bright fringes corresponding to maximum light intensity are lines in the x1 2 x2 plane along which hce

›ðs11 þ s22 Þ ml ¼ ; with m ¼ 0; ^1; ^2; … ›x 1 Dx1

ð8:51Þ

where m is the fringe order, ce ¼ ct in the transmission mode and ce ¼ cr in the reﬂection arrangement. Therefore, fringes observed in the shearing interferometer or the coherent-gradient-sensor are lines of constant gradient ›ðs11 þ s12 Þ=›x1 : Obviously, by reorienting the ordinary axis of the calcite crystal with the x2 direction, fringes representing lines of constant ›ðs11 þ s12 Þ=›x2 can be obtained. The above description of the shearing interferometer is quite general and can be applied to many problems in mechanics. Also, from Eq. 8.50, it is clear that the gradient of u3 ðx1 ; x2 ; 2h=2Þ may be determined directly, without relating it to the plane stress calculation of the stress component; therefore, the method can also be used to determine surface proﬁles of objects; examples of this application can be found in the work of Assa et al. (1979) and more recently, Rosakis et al. (1998). Our interest is in the application of the method to dynamic crack problems and this is discussed further below. The explicit equations for the light intensity can be obtained through the introduction of the asymptotic crack tip stress ﬁeld in Eq. 8.48. The singular term, the higher order terms in the steady state asymptotic expansion, and the transient asymptotic ﬁeld, have all been used in interpreting the fringe pattern observed near dynamically growing cracks. We will describe these in turn. Introducing the singular term from Eq. 3.32 into Eqs. 8.51 and 8.48 results in the equation for bright fringes: hce FðvÞ 3ud 3ud ml þ KII sin KI cos ¼ pﬃﬃﬃﬃﬃﬃ 3=2 2 2 Dx1 2pðrd Þ

ð8:52Þ

where rd and ud are deﬁned in Eq. 8.28 and FðvÞ is deﬁned in Eq. 8.34. The entire analysis presented above carries over if the image shearing is introduced in the x2 direction; in this case, the x2 gradient of the ﬁeld is obtained. The bright fringes are then given by hc FðvÞ 3ud 3ud ml þ K ¼ K sin cos pﬃﬃﬃﬃﬃﬃe I II 3=2 2 2 Dx2 2pðrd Þ

ð8:53Þ

Clearly, the symmetry between the mode I and mode II patterns with respect to the x1 and x2 gradients is obvious in Eqs. 8.51 and 8.53. Simulated interference fringe patterns corresponding to the x1 gradient for assumed values of the stress intensity factors are shown in Fig. 8.13 for pure mode I, pure mode II and mixed mode loading. An example of the time evolution of the fringe patterns obtained with a high-speed camera is shown in Fig. 8.14. In comparing the images in these ﬁgures, it should be noted that only the singular term in the crack tip asymptotic ﬁeld has been used in the simulations, whereas the complete ﬁeld will inﬂuence the patterns observed in the experiments.

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Figure 8.13 Simulated shearing interferometric fringe patterns corresponding to the optical arrangement in Fig. 8.12. (a) KI 5 1 MPa m1/2, KII 5 0; (b) KI 5 0 MPa m1/2, KII 5 1 MPa m1/2; (c) KI 5 1 MPa m1/2, KII 5 0:5 MPa m1/2. The ﬁeld of view shown in these ﬁgures is 40 mm to a side and v=Cd 5 0:1: The lack of clarity in the region near the crack tip is a numerical artifact.

8.4.1 Evaluation of the Dynamic Stress Intensity Factor using Shearing Interferometry In the discussion above, only the singular term was introduced in the evaluation of the fringe patterns. However, since the asymptotic ﬁeld is not expected to establish dominance at large distances from the crack tip where the fringe patterns are typically analyzed, higher order nonsingular terms in the expansion in Eq. 3.32 must be introduced as we discussed in the case of photoelasticity. The dynamic stress intensity factor KI can be obtained at each instant in time by using a least-squares matching of the experimentally measured fringe pattern with simulations based on Eqs. 8.48 and 8.49 for transparent specimens in the transmitted mode and Eqs. 8.48 and 8.50 in the reﬂected mode. First,

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133

Figure 8.14 Selected sequence of interference fringe patterns corresponding to the coherent gradient sensing scheme. (Reproduced from Krishnaswamy et al., 1992.)

the experimental fringe pattern is quantiﬁed by a collection of ðmi ; ri ; ui Þ; measured at M points. The distance r at which these measurements are taken should be appropriate for the application of the two-dimensional asymptotic crack tip stress ﬁeld; based on caustics the experiments of Rosakis and Ravi-Chandar (1986), it should be recognized that the distance r must be larger than 0:5h to be away from the zone of three-dimensional deformations. Then, the sum of the squared error in Eq. 8.51 at all measured points is given by 2 M X mi pl eðA0 ; A1 ; …; Ak21 Þ ¼ 2 hce gðr1 ; ui ; A0 ; A1 ; …; Ak21 Þ ð8:54Þ Dx1 i¼1 In the above equation, the stress ﬁeld parameters ðKI ; KII ; …Þ are represented by the vector A ¼ ðA0 ; A1 ; …; Ak21 Þ and gðr; u; A0 ; A1 ; …; Ak21 Þ is used to represent ›ðs11 þ s22 Þ=›x1 determined from the k-term description of the steady state or transient asymptotic crack tip stress ﬁeld given in Appendix A. Also, since shearing interferometry or coherent gradient sensing method depends on the gradient of the stress ﬁeld, the constant nonsingular term sox in the asymptotic expansion does not contribute to fringe formation and cannot be determined in the analysis. The remaining stress ﬁeld parameters must be obtained by

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minimizing e with respect to the parameters. The least-squared error method described in connection with photoelastic data analysis in Section 8.2.1 can be used here as well to evaluate the best ﬁt coefﬁcients A ¼ ðA0 ; A1 ; …; Ak21 Þ: Rosakis and co-workers have examined the suitability of using the method of coherent gradient sensing to dynamic problems and outlined its potential and limitations (Rosakis, 1993). Since this method is based on the gradients of the crack tip ﬁeld, one would automatically infer that limitations that apply to the method of caustics would appear here as well. We summarize their observations in the following. First, under quasi-static loading, Tippur et al. (1991) attempted to determine the stress intensity factor using the method of CGS. Three point bend specimens of PMMA in the transmission mode and AISI 4340 steel in reﬂection mode were used in the experiments. Interpreting the observed fringe patterns in terms of the singular term in the crack tip stress ﬁeld, they obtained an estimate of the stress intensity factor; this estimate was obtained by evaluating the fringe patterns at different distances from the crack tip. Simultaneously, the load applied on the specimen was also monitored; therefore, the stress intensity factor was calculated from the two-dimensional theory, and denoted by K2D : Their results are shown in Fig. 8.15; in this ﬁgure, the stress intensity factor estimated from CGS fringes is denoted by Y1static ðr; uÞ since the estimates correspond to fringe data from speciﬁc points ðr; uÞ: pﬃﬃﬃﬃﬃﬃ ml 2p r 3=2 static ð8:55Þ Y1 ðr; uÞ ¼ KI ¼ Dx1 hce cosð3u=2Þ

Figure 8.15 Y1static ðr; uÞ vs r=h for a PMMA specimen, indicating that under quasi-static conditions, CGS fringes can be interpreted in terms of the singular stress ﬁeld for r=h > 0:5: (Reproduced from Tippur et al., 1991.)

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135

The dotted line in Fig. 8.15 indicates K2D : Two observations can be made from this result: the r=h similarity observed in caustics measurement is seen here as well and for r=h . 0:5; K EXP approaches K2D ; suggesting that three-dimensional effects are important only when r=h , 0:5: Furthermore, dynamic limitations discussed in Section 8.3, regarding the use of the method of caustics should also apply to this technique. However, because the method of lateral shearing interferometry provides a “full-ﬁeld” optical picture, the fringe patterns may be analyzed with higher order terms in the steady-state or transient stress ﬁeld (see Eqs. 8.28 and 3.51 –3.53) in order to determine the dynamic stress intensity factor from experiments. This evaluation provides important insight into the dominance of the dynamic stress ﬁeld and is discussed further in Chapter 9.

8.5 Strain Gages It should be clear from the discussions in the previous sections that the dynamic stress intensity factor can be determined through the measurement and interpretation of any ﬁeld quantity. Much of the early work on dynamic fracture relied on optical methods; while optical methods yield measurements of the stress ﬁeld components over the complete ﬁeld of observation, they also require elaborate instrumentation, in particular, a high-speed camera for recording the patterns to be interpreted. On the other hand, multiple strain gages can be used more readily with simpler instrumentation requirements; this method was used very successfully by Kinra and Bowers (1981), Shukla et al. (1989), and many other investigators. Recent progress on this area was summarized by Dally and Berger (1993) who describe the application of strain gages to quasi-static as well as dynamic fracture problems. Here we provide a brief description as applied to dynamic problems. The crack tip strain ﬁeld corresponding to the asymptotic stress ﬁeld in Eq. A1 is given by 1ab ðr; uÞ ¼

KIdyn ðt; vÞ I K dyn ðt; vÞ pﬃﬃﬃﬃﬃﬃﬃﬃ Fab ðu; v; nÞ þ IIpﬃﬃﬃﬃﬃﬃﬃﬃ FIIab ðu; v; nÞ þ · · · E 2pr E 2pr

ð8:56Þ

where FIab ðu; v; nÞ and FIIab ðu; v; nÞ are known functions of u; v and n; given in Appendix A. In general, strain gages may be placed at different positions ðr; uÞ to measure one or more components of the strain tensor. These measurements can be used to determine the dynamic stress intensity factors, KIdyn ðt; vÞ and KIIdyn ðt; vÞ; as well as the higher order terms by ﬁtting in a least-square sense to Eq. 8.56. It should be noted that the crack velocity is not known and must either be obtained as part of the ﬁtting procedure or through an independent measuring scheme. The strain gage-based method can provide an estimate of KIdyn ðt; vÞ and KIIdyn ðt; vÞ only over a small time interval during which the gage is positioned within the enhanced strain ﬁeld near the crack tip ﬁeld. Practical considerations on the type, dimensions, and sensitivity of the strain gage are extremely important and are addressed in many standard textbooks (see, for example, Dally and Riley, 1978). Dally and Berger (1993) introduced a very simple idea for the use of strain gages in the evaluation of dynamic stress intensity factors. Consider a strain gage mounted at a point

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Figure 8.16 Location and orientation of strain gage relative to the crack tip.

oriented at an angle u with respect to the crack tip as illustrated in Fig. 8.16. Furthermore, let the strain gage itself be oriented to measure the strain at an angle a with respect to the global x1 axis; the strain gage will measure the extensional strain 1011 in the direction x01 indicated in Fig. 8.16. For thepcase ﬃﬃ of a stationary crack, v ¼ 0; under mode I loading, retaining up to terms of order r in Eq. 8.56 we get E K dyn ðtÞ 1011 ðr; uÞ ¼ pI ﬃﬃﬃﬃﬃﬃﬃﬃ 2ð1 þ nÞ 2pr 1 1 3 1 3 £ k cos u 2 sinu sin u cos2a þ sin u cos u sin 2a 2 2 2 2 2 pﬃﬃ 1 þ A1 ðk þ cos 2aÞ þ A2 r cos u 2 1 2 £ k þ sin u cos 2a 2 sinu sin 2a þ · · · ð8:57Þ 2 where k ¼ ð1 2 nÞ=ð1 þ nÞ: Through a proper choice of u and a the second and third terms in Eq. 8.57 can be made to vanish; this is assured by the following conditions: 1 cos2a ¼ 2k and tan u ¼ 2cot 2a 2

ð8:58Þ

For a material with n ¼ 1=3; we get k ¼ 1=3 and a ¼ u ¼ p=3: Introducing these values in Eq. 8.57, we get E1011

p r; u ¼ ¼ 3

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 K dyn ðtÞ 8pr I

ð8:59Þ

Thus, by placing a strain gage aligned along a line oriented at an angle u ¼ p=3; the measured strain can be related to the dynamic stress intensity factor directly. It should be noted that the distance r at which the strain gage is placed is still open, but this can be farther from the crack tip than the K-dominant region, since Eq. 8.57 is based on a threeterm representation of the strain ﬁeld. This arrangement can be used in the evaluation of dynamic initiation toughness, KID ðT; K_ dyn I Þ: Examples of application of this method were demonstrated by Dally and Barker (1988) who evaluated the dynamic initiation toughness

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137

Figure 8.17 Strain measured by a strain gage located at an initial distance r 5 6 mm and an angle u 5 608 with respect to the crack line; orientation of the strain gage a 5 608: (Reproduced from Dally and Berger, 1993.)

of a Homalite 100 specimen at high loading rates by imposing an explosively driven stress wave on a crack. The test specimen was the dog-bone specimen loaded by explosives as shown in Fig. 6.9. The strain history measured in this experiment is shown in Fig. 8.17. From this strain measurement, they evaluated the time variation of the dynamic stress p intensity factor through Eq. 8.57. They then determined KID ðT; K_ dyn m I Þ to be 0.598 MPa p at a strain rate of 0.076 MPa m=s; the plane-strain fracture toughness for this material, p obtained from quasi-static tests is 0.445 MPa m. This method was also used by Owen et al. (1998), who determined both the dynamic initiation toughness for an aluminum 2024-T3 alloy; they found the initiation toughness to be independent of the rate of loading p up to K_ dyn ¼ 105 MPa m=s, but then to increase three-fold as the rate of loading increased I p ¼ 106 MPa m=s. Owen et al. (1998) also determined the dynamic propagation to K_ dyn I toughness from the strain gage measurements, but using Eq. 8.57, justifying its use by the fact that the crack speeds were quite low, about 4% of the shear wave speed. We will discuss these and other results on crack initiation toughness in Chapter 10. If the crack moves at a signiﬁcant fraction of the wave speed, two major errors are encountered in using Eq. 8.57; the ﬁrst is due to the inertial distortion of the crack tip strain ﬁeld which is ignored in developing Eq. 8.57. While this error is likely to be small at low crack speeds, it cannot be ignored when the crack speed is high; this error can be removed completely by using the appropriate asymptotic strain ﬁeld. The second and perhaps a more important contribution to the error arises from the fact that the distance r and the orientation u of the position of the strain gage relative to the moving crack tip is continuously changing

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as a result of crack growth. This must be taken into account in the analysis. Therefore, the simpliﬁcation introduced in Eq. 8.57 is not appropriate and one must use Eq. 8.56, with the additional feature that both r and u are now functions of time, given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rðtÞ ¼ r02 þ v2 t2 2 2r0 vt cos u 0 ð8:60Þ r sinu 0 uðtÞ ¼ arcsin 0 ð8:61Þ rðtÞ where r0 and u0 are the distance and orientation at time t ¼ 0: Of course, it has been assumed that the crack extension is straight along the x1 direction. Ravi-Chandar (1983) used the above analysis to evaluate the measurements of Kinra and Bowers (1981). More recently, Berger et al. (1990) have used measurements from multiple strain gages to set up an overdetermined system of equations and obtained improved estimates of the dynamic propagating toughness. In summary, strain gage-based methods are just as powerful as full-ﬁeld optical techniques for crack tip ﬁeld characterization while at the same time requiring only minimal investment in measuring equipment. This is particularly useful in the development of standard methods of measuring the initiation, propagation and arrest toughness.

8.6 Interferometry Optical interferometry (Pfaff et al., 1995), and high-resolution moire´ interferometry (Epstein and Dadkhah, 1993) have also been applied to dynamic fracture problems. In classical interferometry, coherent light reﬂected from the deformed specimen surface is superposed on light reﬂected from a reference ﬂat surface resulting in optical interference fringes that are interpreted in terms of the out-of-plane displacement of the specimen. The arrangement of a Twyman –Green interferometer to accomplish this is shown in Fig. 8.18. In high-resolution moire´ interferometry, a ﬁne grating is coated on to the specimen and its deformations are then compared with a reference grid to extract the components of the in-plane or out-of-plane displacement components depending on the implementation of the experimental method. The crack tip displacement components corresponding to

Figure 8.18 Schematic illustration of the Twyman – Green interferometer used for determination of out-of-plane deformation near a crack.

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139

Figure 8.19 Fringes obtained in a Twyman – Green interferometer displaying the out-of-plane displacement ﬁeld near a rapidly growing crack in a PMMA specimen. Post-processing was used to subtract out the initial fringe pattern. Note that a region near the crack tip is obscured due to aperture limitations and this region has been removed during image processing. Stress waves radiating from the crack tip damage zone can be seen as perturbations in the fringe pattern, especially behind the crack tip. The crack speed was 0.52 mm/ms, (about 0:522Cs ). (Reproduced from Pfaff et al., 1995.)

the singular term of the asymptotic stress ﬁeld is given by:

nKIdyn ðt; vÞ pﬃﬃ I pﬃﬃﬃﬃﬃﬃ r ga ðu; vÞ E 2p nKIdyn ðt; vÞ I pﬃﬃﬃﬃﬃﬃﬃﬃ faa ðu; vÞ u3 ðr; uÞ ¼ E 2pr

ua ðr; uÞ ¼

ð8:62Þ

I ðu; vÞ are known functions of u and v given in Appendix A. The fringe where gIa ðu; vÞ and fab patterns observed in the interferometry experiments are contours of constant displacement component: u3 in the case of classical interferometry, and u1 or u2 in the case of in-plane moire´ interferometry. An example of the fringes obtained in a Twyman–Green interferometer indicating the variation of the out-of-plane displacement ﬁeld is shown in Fig. 8.19. The fringe patterns in Fig. 8.19 indicate that stress waves emanate from the crack tip at discrete times; these are identiﬁed by the perturbations in the fringe contours and are caused by the intermittency in the crack propagation process. Fitting the observed fringe patterns ahead of the crack to a prediction based on Eq. 8.62 using a least-square error method, the dynamic stress intensity factor can be determined, in much the same way as for the methods of photoelasticity and coherent gradient sensing. A major difﬁculty in using these methods arises from the fact that while the measurements of the displacement components are themselves very accurate, the application of the plane elastodynamic ﬁelds in the interpretation of the crack tip ﬁeld parameters introduces signiﬁcant errors, particularly when one approaches close to the crack tip. Furthermore the interferometric methods indeed pose a challenging task in dynamic situations and very few successful attempts have been reported in the literature.

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141

Chapter 9 Dominance of the Asymptotic Field

The basic theory of elastodynamic fracture, the experimental methods used in generating dynamically growing cracks, the diagnostic tools for use in the evaluation of the crack position and crack tip stress ﬁeld have been discussed in the preceding chapters. We now turn to a critical analysis of the theory through a comparison of the theory with experimental measurements. The limitations of the experimental methods in the determination of the dynamic stress intensity factor as well as the validity of characterizing the dynamic crack tip stress ﬁeld through this parameter are discussed in this chapter. This is then followed by a discussion of the experimental determination of dynamic failure criteria in Chapter 10. The mechanisms that govern dynamic fracture and phenomenological models that have been developed in order to capture these mechanisms are presented in Chapters 11 and 12.

9.1 Stationary Cracks The semi-inﬁnite crack geometry with uniform pressure loading is amenable to experimental implementation with the use of an electromagnetic loading scheme (Ravi-Chandar and Knauss, 1982) described in Chapter 6. The specimen is made of a large Homalite-100 plate (500 mm £ 300 mm) of thickness 4.76 mm; selected mechanical and optical properties of this material are given in Table B.1 in Appendix B. A crack is introduced parallel to the long side by machining a 3 mm wide slit down the middle. A natural crack tip is introduced by wedging a razor blade inside the slit and applying a small impact force. A ﬂat copper strip, 4.76 mm £ 1.2 mm thick is folded back on itself and the space between the two layers is ﬁlled with a mylar insulating strip. This assembly is then introduced into the machined slit as indicated in the schematic diagram in Fig. 6.9. Insertion of this assembly introduced a small static loading on the crack tip, which was evident in the experimental measurements as we shall discuss later. When a current ﬂows through the copper loop, each leg generates a magnetic ﬁeld surrounding it, with the magnetic ﬁeld oriented normal to the current vector. The current vector in each leg interacts with the magnetic ﬁeld of the other leg to produce an electromagnetic repulsion that forces the conductors apart. Since the two legs of the copper strip are conﬁned in the slot of the machined crack, they press upon the top and

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bottom surfaces of the crack with a uniform pressure. The current in the copper strip is generated by a discharge from a capacitor bank. The magnitude of the pressure loading may be estimated easily from electromagnetic theory; the time history of the current, which dictates the magnitude and duration of the pressure applied on the crack surface, may be controlled by suitable choice of capacitors and inductors that form the pulseshaping circuit. Ravi-Chandar and Knauss (1982) generated a nearly trapezoidal pulse, with a rise to the peak amplitude in about 25 ms and a total duration of about 150 ms. For typical values of current used in the experiments, the crack surface pressures were in the range of 1– 20 MPa. For the large specimen, this loading conﬁguration is equivalent to an inﬁnite plate, with a pressurized semi-inﬁnite crack for the duration of the current pulse, conforming to the boundary-initial value problems discussed in Chapter 4. The crack tip response to the loading was monitored to determine the time variation of the dynamic stress intensity factor. A high-speed camera capable of capturing images with a 15 ns exposure time and 5 ms time interval between frames was used; the optical method of caustics (described in Chapter 8) was used to determine the dynamic stress intensity factor. A typical sequence of high-speed photographs obtained in this experimental arrangement is shown in Fig. 8.7. From a series of these experiments, Ravi-Chandar and Knauss (1982, 1984c) evaluated the dynamic stress intensity factor as a function of time and compared it with the theoretical estimates in Eq. 4.19. Two sets of results from their experiments are reproduced here. First, Ravi-Chandar and Knauss (1982) considered a partially loaded semi-inﬁnite crack. The time history of the dynamic stress intensity factor shown in Fig. 9.1 corresponds to a pressure load distributed over a length l at a distance L behind the crack tip. The theoretical variation of the dynamic stress intensity factor was obtained using the superposition integral in Eq. 4.37. Clearly, the experimental measurements and theoretical predictions agree well within the experimental accuracy. The slight drop in the stress intensity factor at the arrival of the dilatational wave that is indicated in the theoretical prediction is not easily observed in the experiment due to limitation in the resolution. Kim (1985a,b) used the same loading apparatus, but a different optical diagnostic method for the evaluation of the dynamic stress intensity factor. Instead of photographing the caustics, he brought the light rays to a focus and ﬁltered the focal volume; therefore, the only rays that pass through are the rays that are deviated by the crack tip deformation. Kim was then able to analyze the intensity of these rays collected at a photodetector in terms of the dynamic stress intensity factor. Due to the higher time resolution of the photodetector compared with the high-speed camera, he was able to obtain stress intensity factor measurements with a better time resolution; in his experiments, a length L near the crack tip was left without the pressure loading. His results also indicated a good agreement between the measured dynamic stress intensity factor and that calculated from Eq. 4.40, including the initial reduction in the stress intensity factor from the compressive dilatational wave. In the second series of tests, a uniform pressure of magnitude in the range of 0.63 –15.4 MPa was applied over the entire crack corresponding to a pressurized semiinﬁnite crack. Experimental measurements of the dynamic stress intensity factor corresponding to a stationary crack for four different pressure levels are shown in Fig. 9.2.

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Figure 9.1 Comparison of the measured time history of the dynamic stress intensity factor for a semi-inﬁnite crack in an unbounded medium with the theoretical prediction. (Reproduced from Ravi-Chandar and Knauss, 1982.)

Figure 9.2 Comparison of the measured time history of the dynamic stress intensity factor for a semi-inﬁnite crack in an unbounded medium with the theoretical prediction. (Reproduced from C.C.Ma, 1991.)

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These measurements are compared to the theoretical prediction from Eq. 4.19. Clearly, very good agreement is demonstrated; note that in evaluating the theoretical predictions, a trapezoidal ﬁt to the actual time history of the pressure loading was used. Ma (1991) made a further improvement in this by considering the exact stress ﬁeld ahead of the crack indicated in Eq. 4.28 rather than by assuming that the singular term alone was sufﬁcient to represent the stress ﬁeld. In all these experiments, the dynamic stress intensity factor reached a sufﬁciently high value to cause crack growth to occur. The results corresponding to crack initiation and growth will be examined later. The experiments described here are the only ones where a comparison of theoretical estimates of the dynamic stress intensity factor could be obtained directly; however, there are many other experimental investigations where the dynamic stress intensity factors have been evaluated through different diagnostic methods—such as photoelasticity, shearing interferometry or CGS, and strain gauges—and compared to numerical simulations (we will not delve into these here, but see for example the works of Kobayashi, Dally, Kalthoff, Shukla, Rosakis, Takahashi and others listed in the bibliography).

9.2 Propagating Cracks The experiments described above also provided the means of evaluating the dynamic stress intensity factor history for dynamically propagating cracks. We examine the case of propagating cracks in this section. In this series of experiments, crack initiation occurred in the time interval: 15ms , t , 110ms: Continued growth of the crack was observed to occur at a constant speed with acceleration to the ﬁnal speed occurring within the time resolution of the measurement (less than 5 ms). Fig. 9.3 shows a comparison of the experimentally measured time variation of the dynamic stress intensity factor with the dynamic stress intensity factor estimated from the analysis described in Section 4.2. The experimentally observed variation of the crack position with time is also shown in Fig. 9.3. The crack surface pressure in this experiment was 1.10 MPa. Upon loading, the dynamic stress intensity factor increased gradually until 56 ms when the crack began to propagate. Crack extension was observed to occur at a constant speed of 240 m/s ( ¼ 0.22CR), with a corresponding drop in the stress intensity factor. Beyond 150 ms, waves from the ﬁnite boundaries of the specimen arrived at the crack tip to load it further with a stress pulse; it is possible to analyze the results in this time range, but this is not of immediate interest. As pointed out earlier, the analysis provides dynamically admissible solutions; in order to pick the correct solution, the experimentally measured time of crack initiation and crack speed were used. Thus, imposing the experimental observations that the crack initiated at time t and propagated with a speed v, the dynamic stress intensity factor was calculated; this expression is given in Eq. 4.77. The comparison between the theoretical estimate and the experimental measurement shown in Fig. 9.3 is remarkably good, conﬁrming the validity of the elastodynamic stress analysis of cracks under the conditions presented here.

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145

Figure 9.3 Time history of the dynamic stress intensity factor for a crack growing at a constant speed v 5 240 m/s. sp 5 1:1 MPa. (Reproduced from Ravi-Chandar and Knauss, 1982.)

When the crack surface pressure exceeded 5 MPa, crack initiation occurred at shorter times, with a larger stress intensity factor at initiation and a larger speed of the resulting crack. Experimental measurements from a test with a crack surface pressure of 5.55 MPa is shown in Fig. 9.4. Crack initiation occurred at 18 ms and the resulting crack speed was about 410 m/s ( ¼ 0.38CR). In this case, the calculated dynamic stress intensity factor for the growing crack did not agree well with the experimental measurements as can be observed in Fig. 9.4. In interpreting this discrepancy, it is important to recognize that there are two opposing requirements in the experiments and analyses: on the one hand, the experimental measurements were obtained with the method of caustics; inherent limitations in the method due to the three-dimensional nature of the stress ﬁeld require that measurements be made at a distance of one-half of the plate thickness away from the crack tip (see discussion of the method of caustics in Section 8.3). On the other hand, the transient nature of the stress ﬁeld requires that the measurements be made as close to the crack tip as possible to account for wave propagation effects; the dynamic instantaneous K-dominant ﬁeld (the region where the dynamic stress intensity factor is adequate to describe the total stress ﬁeld) is established only in a small region near the crack tip. These two opposing requirements render measurement of the dynamic stress intensity factor difﬁcult and may indeed bring into question the usefulness of the dynamic stress intensity factor in situations where the transient effects are large. The issue of dominance of the dynamic asymptotic ﬁeld for growing cracks is considered in the next section.

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Figure 9.4 Time history of the dynamic stress intensity factor for a crack growing at a constant speed v 5 410 m/s. sp 5 5:55 MPa. (Reproduced from Ravi-Chandar and Knauss, 1987.)

9.3 Dominance of the Asymptotic Field for Propagating Cracks The lack of agreement between the experimentally measured time history of the dynamic stress intensity factor and the theoretically estimated one displayed in Fig. 9.4 has signiﬁcant implications on dynamic fracture mechanics. This disagreement may be due to (i) intrinsic errors in the measurement technique used to determine the dynamic stress intensity factor and therefore concerns the adequacy of the experimental measurement technique or (ii) the lack of a well-established stress ﬁeld characterized by the asymptotic ﬁeld assumed or (iii) a combination of both. The dominance of the dynamic singular ﬁeld was examined theoretically and experimentally by a number of investigators (Ma and Freund, 1986; Ravi-Chandar and Knauss, 1987; Krishnaswamy and Rosakis, 1990; Rosakis, 1993; and others). We review this work here since it provides important clues to the use of the dynamic stress intensity factor as a fracture-characterizing parameter. Ma and Freund (1986) examined the problem from the perspective of the establishment of a K-dominant stress and deformation ﬁeld near the crack tip. They considered the pressure-loaded semi-inﬁnite crack problem. In order to evaluate the development of a K-dominant ﬁeld, they extracted two pieces of information from the analysis. First, the exact variation of the normal stress s22 ðj1 ; 0; tÞ along the crack line was obtained in the form of an integral; this is the expression analogous to the one shown in Eq. 4.28 for the case of a stationary crack. Second, the asymptotic behavior of this integral was used to evaluate the stress intensity factor KI ðt; vÞ (Eq. 4.76) and the elastodynamic stress ﬁeld was

Dominance of the Asymptotic Field

147

obtained from the singular ﬁeld alone. If the singular ﬁeld is dominant at a ﬁxed distance j1 from the moving crack tip, then the stress at that distance calculated from the exact stress variation should be equal to that obtained from the singular term alone. Such a comparison pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ is shown Fig. 9.5, where the ratio s22 ðj1 ; 0; tÞ 2pj1 =KI ðt; vÞ is plotted as a function of normalized time, Cd t=j1 : In Fig. 9.5a, results are shown corresponding to crack growth beginning at a time t ¼ 50j=Cd and for different crack speeds; in Fig. 9.5b, results corresponding to a ﬁxed crack speed of 0.2Cd and varying initiation times are shown. From these ﬁgures it is seen that the stress estimated from the singular term alone is larger than the actual stress, and that as time increases, the singular ﬁeld approaches the exact ﬁeld. Consider a speciﬁc example: if Cd ¼ 2000 m/s, and j ¼ 2 mm, then the normalized time can be interpreted directly in microsecond. From these ﬁgures, it is seen that for a delay time t ¼ 20 ms and a crack speed of 0.2Cd (approximately equal to the results displayed in Fig. 9.4) the time taken for the K-ﬁeld to approach the exact ﬁeld is quite large—about 100 ms; most of the fracture events of interest in this experiment occur in this time scale. Clearly, the parameter of interest in these comparisons is the nondimensional delay time, h ¼ Cd t=j: From Fig. 9.5, it is obvious that a K-ﬁeld dominates the crack tip when h becomes large, say on the order of 100. For a given material, this condition may be achieved either by letting the time for crack initiation t to be large or by permitting the distance from the crack tip j to be small. The results in Figs. 9.3 and 9.4 may be interpreted in light of the results in Fig. 9.5. We note that the data in Fig. 9.3, which corresponds well with the analytical calculation of the stress intensity factor, involves t ¼ 56 ms, crack speed v ¼ 0:1Cd and 2:1mm ,pjﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ , 2:2 mm: Thus 50 , h , 51 and one sees from Fig. 9.5 that the ratio of s22 ðj; 0; tÞ 2pj1 =KI ðt; vÞ is approximately 0.8 at initiation and reaches 0.9 within the following 30 ms. Considering the ^ 10% accuracy inherent in the experimental measurements, it is clear that within this time frame, the analytical computations should agree with the experimental measurements of the stress intensity factor. Next, considering the data in Fig. 9.4, the following conditions apply speed v ¼ 0:2Cd and j , 2:2 mm. Thus, just after crack initiation: t ¼ 18 ms, crack pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h , 20 so that the stress ratio s22 ðj; 0; tÞ 2pj1 =KI ðt; vÞ is about 0.7; one could expect the experimentally derived stress intensity factor to be about 40% larger than the true value. Moreover, one observes that because the value of j increases during the subsequent rise of the stress intensity factor, h decreases further and thus accentuates the discrepancy. One may also arrive at the above conclusion through a simple argument based on the time taken for stress waves to propagate. Let the crack start to grow at a time t ¼ t with a constant velocity v. In the experiments, measurements are made at a distance j from the moving crack tip. At any time t p, at this point, the stress ﬁeld information corresponding to a tip location at t ¼ tp 2 j=ðCs 2 vÞ is obtained. If this information is to reﬂect the true stress ﬁeld at t p, then j=ðCs 2 vÞ must be small in comparison to other signiﬁcant time scales, in particular, the time elapsed from the beginning of the transient event tp 2 t: This results in the following inequality: ðtp 2 tÞðCs 2 vÞ p1 j

ð9:1Þ

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Chapter 9

Figure 9.5 Comparison of the exact stress ﬁeld and the singular ﬁeld. (Reproduced from Ma and Freund, 1986.)

Dominance of the Asymptotic Field

149

This inequality may be satisﬁed at very small distances from the crack tip, for very small crack speeds and at long times after crack initiation. Ravi-Chandar and Knauss (1987) evaluated the dominance of the dynamic K-ﬁeld by performing the experiment reported in Fig. 9.4 repeatedly. In each implementation, the size of the initial curve that formed the caustic was varied by varying the distance between the specimen midplane and the screen plane. The results from two of these experiments are shown in Fig. 9.6; the position of measurement normalized by the plate thickness ðj=hÞ and the nondimensional delay time h are indicated in the ﬁgure. For j=h , 0:4 there is a tendency for the experimentally measured stress intensity factors to approach the analytically predicted values. However, distortions of the crack tip ﬁeld due to process zone effects were found to be signiﬁcant as j=h ! 0:1: Thus the conditions for K-dominance may be posed as follows: to be away from three-dimensional effects, r=h . 0:5: In order to establish a dynamic K-ﬁeld, r=h ! 0; and ﬁnally in order to eliminate the effects of the crack tip process zone, r=h q rp =h; where rp is the process zone size. It is thus possible that a K-dominant zone may not exist in the vicinity of the crack tip. Krishnaswamy and Rosakis (1990) examined this problem further by simultaneously observing caustics corresponding to two different initial curves using a bifocal apparatus. Splitting of the light beam along two different paths enabled them to photograph two caustics corresponding to different initial curves simultaneously. Dynamic crack growth in

Figure 9.6 Stress intensity factor measurement at different distances from the crack tip. (Reproduced from Ravi-Chandar and Knauss, 1987.) [Note: Plate thickness is denoted by d .]

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Figure 9.7 Stress intensity factor measurement at different distances from the crack tip. (Reproduced from Krishnaswamy and Rosakis, 1990.)

an AISI 4340 steel specimen was examined through this bifocal arrangement. The two caustics were both evaluated to yield the instantaneous stress intensity factor; one result is reproduced in Fig. 9.7. In this experiment, crack growth was initiated at 620 ms after impact from a drop weight tower, and the crack grew at speeds in the range of 1000 –2000 m/s. The initial curve radii were kept larger than about 0.5 of the plate thickness in order to reduce three-dimensional effects on the measurements. The divergence between the estimates of the dynamic stress intensity factor from the two simultaneous measurements of caustics further reinforces the lack of dominance of the dynamic K-ﬁeld observed in the experiments of Ravi-Chandar and Knauss (1987). From these experiments, the conclusion at hand is that the dynamic stress intensity factor alone does not govern the stress ﬁeld in the region where experimental measurements are typically obtained in most of the techniques described in Chapter 8. A more important conclusion is regarding the validity of the stress intensity factor as a fracture-characterizing parameter: does the dominance of K apply at least over the fracture process zone in order to use this as the fracture parameter? Freund and Rosakis (1992) and Krishnaswamy et al. (1992) examined this issue of K-dominance further by considering the inﬂuence of the transient terms in the asymptotic ﬁeld. Freund and Rosakis (1992) developed an analysis of the crack tip stress ﬁeld that included the transient effects of nonsteady crack motion. This ﬁeld is described in Section 3.3. Krishnaswamy et al. (1992) applied this ﬁeld in an attempt to determine

Dominance of the Asymptotic Field

151

the dynamic stress intensity factor from experiments. Three-point-bend specimens of PMMA and AISI 4340 steel were loaded in a Dynatup 8100A drop-weight tower. The CGS fringe patterns resulting from the growing crack was captured using a high-speed camera (see Section 8.4 for details regarding the method). The crack speed was measured to be 0.25Cs. Fringe patterns corresponding to a typical test are shown in the selected sequence of high-speed photographs in Fig. 8.14. Fringes observed in this technique are lines of constant gradient ›ðs11 þ s22 Þ=›x1 : The main advantage of the method over caustics is that it is a full-ﬁeld technique; pointwise measurements over the complete ﬁeld of view can be obtained; however, it suffers from the same drawback that due to the three dimensionality of the stress ﬁeld, measurements must be made at r=h . 0:5: Introducing the dynamic stress ﬁeld into Eq. 8.51 and rearranging the terms, it can be shown that if a K-dominant ﬁeld is generated, then the ratio Y1d ðr; uÞ

pﬃﬃﬃﬃﬃﬃ 2p ðrd Þ3=2 ml ¼ 3u Dx1 hcFðvÞ cos d 2

ð9:2Þ

must be independent of r and be equal to the dynamic stress intensity factor. Krishnaswamy et al. (1992) evaluated this ratio along different radial lines from the crack tip and their results are shown in Fig. 9.8. The discrete symbols indicate Y1d calculated at a fringe order m and location ðr; uÞ: Clearly, there is no region where the dynamic stress intensity factor dominates. The conclusion that the dynamic singular ﬁeld does not dominate at r=h . 0:5 is not surprising in the wake of the analytical results

Figure 9.8 Y1d ðr; uÞ vs r=h for a PMMA specimen, indicating that under dynamic loading conditions, CGS fringes can not be interpreted in terms of the singular stress ﬁeld for r=h > 0:5: (Reproduced from Krishnaswamy et al. 1992.)

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Figure 9.9 Y1d ðr; uÞ vs r=h for a PMMA specimen, indicating that under dynamic loading conditions, CGS fringes can be interpreted in terms of the higher order transient stress ﬁeld for r=h > 0:5: (Reproduced from Krishnaswamy et al., 1992.)

of Ma and Freund (1986) and the experimental results of Ravi-Chandar and Knauss (1987) and Krishnaswamy and Rosakis (1990) obtained with the method of caustics. However, the CGS fringes contain much more information than do the corresponding caustics. If the fringes are interpreted using the transient asymptotic ﬁeld, a different result emerges. Interpreting the CGS fringes shown in Fig. 8.14 again, but this time with the transient crack tip stress ﬁeld given in Appendix A, the equation for the bright fringes can be written as:

Y1d ¼

pﬃﬃﬃﬃﬃﬃ 2p ðrd Þ3=2 ml ¼ Gd1 ðr; u; KId ; A1 ; A2 …A5 Þ 3ud Dx1 hce FðvÞ cos 2

ð9:3Þ

Krishnaswamy et al. extracted the parameters of the ﬁrst six orders of the asymptotic expansion, ðKId ; A1 ; …A5 Þ; by ﬁtting the experimentally measured fringe data (left hand side of Eq. 9.3). The experimental results are shown in Fig. 9.9. Comparison of the radial variation of Y1d to the best theoretical estimate Gd1 is shown in this ﬁgure for various radial lines. The agreement is quite good, suggesting that interpretation of the CGS fringe patterns through a transient asymptotic expansion is appropriate. Since the analytical solution for the stress intensity factor is not known, comparison to theory is not possible. In an attempt to verify the efﬁcacy of the data extraction procedure, Krishnaswamy et al. regenerated the fringe pattern from the estimated parameters. The comparison of the reconstructed fringes to the experimental fringes is shown in Fig. 9.10. Clearly, while the K-dominant analysis does not ﬁt the experimental observation well, the higher order transient analysis appears to have captured the experimental data quite well, at least ahead

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153

Figure 9.10 Comparison of CGS fringes observed in the experiment with fringe patterns reconstructed using the best estimates for the ﬁeld parameters. (Reproduced from Krishnaswamy et al., 1992.)

of the crack. The conclusion from these experiments is that even though the dominance of the singular ﬁeld is not established, by taking into account the development of the transient stress ﬁeld over a large region near the crack tip, a consistent estimate of the ﬁeld parameters can be obtained, and hence the dynamic stress intensity factor may be used as a fracture characterizing parameter.

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155

Chapter 10 Dynamic Fracture Criteria

The energetic basis of fracture criterion to be used in dynamic problems was discussed in Chapter 5. It was also indicated that for practical applications, this fracture criterion is typically formulated in terms of separate criteria for initiation, growth and arrest of cracks. In this chapter, we describe experimental determination of these fracture criteria in different materials.

10.1 Criteria for Crack Initiation Here attention is focused on experimental measurements of the dynamic crack initiation toughness in various materials. Recall that for a stationary crack under timedependent loading, for small-scale yielding conditions, the crack tip stress and deformation ﬁeld are determined by the dynamic stress intensity factor and that the initiation toughness is the critical value at initiation of crack growth, denoted by KId. The dynamic initiation toughness may depend on the loading rate and temperature. The since all ﬁeld loading rate appropriate to the crack tip region is characterized by K_ dyn I parameters near the crack tip are proportional to KIdyn ðtÞ: The main experimental task is then to characterize this material property, KId ðK_ dyn I ; TÞ; the dynamic crack initiation toughness, under controlled loading and environmental conditions. We provide a survey of the many attempts made in characterizing dynamic crack initiation toughness. Experiments aimed at characterizing KId ðK_ dyn I ; TÞ require a sharp initial crack generated by fatigue precracking or by arrest of a dynamically growing crack, a repeatable stress wave loading scheme with a well-characterized load history, and the ability to vary the rate of loading over a wide range. In addition, diagnostic schemes for monitoring the time variation of crack position and load are required, in order to determine the instant of crack initiation and the calculation of the dynamic stress intensity factor. While it was recognized quite early in the development of fracture mechanics that the onset of crack initiation would depend on the rate of loading, quantitative measurements had to wait for the development of experimental methods meeting the requirements listed above and analytical methods to identify the appropriate crack tip asymptotics for stationary and growing cracks under dynamic loads. The earliest attempt to determine

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the loading rate dependence of the crack initiation toughness was reported by Eftis and Kraft (1965); their results are reproduced in Fig. 10.1. They used single-edgenotched specimens of a carbon steel alloy, and determined the crack initiation toughness over a temperature range of rate in the range p 2195 to 808C and a loading dyn 6 _ 1 £ 10 # K_ dyn # 1 £ 10 MPa m/s. They found that K ð K ; TÞ was a decreasing Id I I over this range. In order to extend the range of the data, they reinterpreted function of K_ dyn I the experimental results on wide-plate tests performed by Videon et al. (1963) and obtained estimates of the possible rate dependence of initiation toughness over the higher rates of loading; the experimental measurements of Videon et al. were obtained for rapidly growing crack. Therefore, Eftis and Kraft (1965) displayed the estimated stress intensity factors as a function of the crack speed and not the loading rate. While the analysis based on elastostatics predated a complete understanding of the dynamic crack problem, and therefore, was not quantitatively correct, the trends described in their results—that the dynamic crack initiation toughness would ﬁrst decrease slightly as the loading rate increased and then subsequently increase dramatically—are indeed intriguing and important. An accurate evaluation of the loading rate dependence of the dynamic crack initiation toughness and an understanding of the microstructural mechanisms responsible for such rate dependence are essential for the assessment of the safety of structures subjected to dynamic loading. In this section, we present a survey of investigations in dynamic crack initiation, its rate dependence and the fracture mechanisms responsible for rate dependence.

Figure 10.1 Effect of loading rate on the yield and fracture behavior of a high-strength steel. (Reproduced from Eftis and Kraft, 1965.)

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Dynamic Fracture Criteria

10.1.1 Initiation of Cracks Under Short Duration Stress Pulses Shockey and Curran (1973) evaluated crack initiation toughness at high strain rates in an ingenious experiment that utilized pulse loading. Thin disks of polycarbonate of 38 mm diameter and 3 mm thickness were impacted by a ﬂyer plate arrangement (see Chapter 6 for a description of the experimental apparatus) at a speed of about 140 m/s; the shortduration compression loading pulse reﬂected as a tensile pulse from the specimen free surface and created numerous internal microcracks. The specimen was recovered and the dimensions of the microcracks were measured—this impact resulted in a number of penny-shaped interior cracks; they were able to measure 48 microcracks ranging in diameter from 14 microns to 5.5 mm. The same specimen was impacted once again at different speeds (14.3, 32.6 and 57 m/s) with a stress pulse of duration t ¼ 2:8 ms and stress amplitude (sp ¼ 27:8, 60.3, 94.5 MPa) in order that only some of these microcracks may grow under the second loading. They found that none of the cracks grew at the lower impact speeds; on the other hand, at the highest speed, all microcracks with an initial diameter greater than about 1 mm grew, but those with a diameter smaller than 0.71 mm did not grow. From a measurement of the tensile stress generated during the loading and the dimensions of the penny-shaped cracks that were initiated into growth, they estimated the crack initiation toughness, using a quasi-static analysis. Kalthoff and Shockey (1977) and Shockey et al. (1983a,b) re-examined this initiation problem by subjecting the results to a dynamic analysis. The analyzed conﬁguration is represented in a meridional section in Fig. 10.2; the stress pulse is of magnitude sp and duration t propagating with a speed Cd and interacting with a penny-shaped crack of radius a: The stress intensity factor for the axisymmetric problem of a penny-shaped crack under a ﬃquasi-static stress of pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ magnitude sp was derived by Sneddon (1946): KIstat ¼ sp 2a=p: The stress intensity factor for a penny-shaped crack under pulse loading has been evaluated by Chen and Sih (1977). Two different situations must be considered in the dynamic problem: ‘short’ cracks and ‘long’ cracks. The distance traveled by the dilatational wave in the duration t of the loading pulse deﬁnes a characteristic length j ¼ Cd t: If the characteristic length is very large, i.e. 2a p j; we have a short crack or a long-pulse duration. In this case, the ﬁnite diameter of the penny-shaped crack inﬂuences deformation; the dynamic stress intensity factor quickly rises and overshoots the quasi-static value by about 25%, oscillates about and settles down at the quasi-static value as a result of repeated wave interactions with the ﬁnite crack boundaries. Thus, for the short cracks or long-pulse loading, the maximum stress intensity factor attained during the loading history is: KIdyn lmax

¼

1:25KIstat

rﬃﬃﬃﬃﬃﬃﬃ 2a ¼ 1:25s p p

ð10:1Þ

On the other hand, if the characteristic length is very small, i.e. 2a q j; we have a long crack or a short-pulse loading. During the time of loading pulse, the stress waves from any point on the crack do not reach the diametrically opposite points before termination of the loading pulse; the ﬁnite diameter of the penny-shaped crack is never felt completely by the crack and the situation corresponds more closely to an unbounded medium. The stress intensity factor increases with time as t1=2 (see Eq. 4.19 for the plane-strain

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Figure 10.2 Interaction of a tensile stress pulse of magnitude sp and duration t with a crack. The stress pulse travels with a speed Cd.

equivalent problem) and the peak stress intensity factor occurs not because of the interactions between the ﬁnite boundaries of the crack but because the loading pulse is of ﬁnite duration; Chen and Sih (1977) found the maximum stress intensity factor to be: pﬃﬃﬃﬃﬃﬃ KIdyn lmax ¼ 0:59sp pj ð10:2Þ Note that the maximum stress intensity factor is independent of the crack length and depends only on the duration of the loading pulse. It is now possible to determine the critical stress amplitude sc required for crack initiation by applying the crack initiation criterion: KIdyn lmax ¼ KId : Then, from Eqs. 10.1 and 10.2, we get: KId pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ for long cracks or short pulses 0:59pﬃﬃﬃ pC ﬃ dt KId p pﬃﬃﬃﬃﬃ for short cracks or long pulses sc ¼ 1:25 2a

sc ¼

ð10:3Þ

This analysis provided a rational interpretation of the experimental observations of Shockey and Curran (1973) and Kalthoff and Shockey (1977). Their results are shown in Fig. 10.3; in this ﬁgure, the stress amplitude, sc, is plotted on the ordinate and the crack length is plotted on the abscissa. Open symbols represent cracks that were not initiated and ﬁlled symbols indicate cracks that grow during the pulse loading. The lines drawn at the boundary between no growth and growth of cracks are based on the maximum stress intensity factor reaching the initiation toughness; the results also provide a good estimate of the dynamic initiation toughness of the polycarbonate material: KId ¼ 2:2 ^ 0:2 MPa m1/2. This value is about 40% lower than the plane-strain fracture toughness determined from quasi-static experiments. Shockey et al. (1983a,b) estimated the rate of loading in this short-pulse projectile impact experiment to be about ¼ 107 MPa m1/2/s. The short-pulse loading experiment, while difﬁcult to implement K_ dyn I because of the equipment requirements, provides an effective method for the determination of the dynamic crack initiation toughness. However, it is difﬁcult to

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159

Figure 10.3 Data from Shockey et al. (1983a,b) indicating the variation of the critical stress amplitude required for crack initiation as a function of the crack radius. (Reproduced from Shockey et al., 1983a,b.)

vary the rate of loading over a large range and so its utility is primarily in the very high rate of loading conditions. Shockey et al. (1983a,b) interpreted the loading rate dependence of the dynamic initiation toughness by arguing that while it was necessary for the stress intensity factor to reach a critical value, it was not sufﬁcient; they postulated that the critical value of the stress intensity factor must be maintained for a minimum time for the fracture processes to develop completely. This idea can be justiﬁed only if speciﬁc kinetic processes that occur within the process zone on the same time scale as the applied loading are postulated. Such processes must depend on the fracture mechanisms that dictate crack growth and hence on the material, and even for the same material depend on whether a brittle or ductile fracture mechanism is triggered. 10.1.2 Loading Rate and Temperature Dependence of Crack Initiation Toughness Costin et al. (1977) developed another experimental scheme to examine the loading rate dependence of dynamic crack initiation toughness. The experimental arrangement— an implementation of the Hopkinson bar apparatus—is shown in Fig. 6.7a. A tensile wave generated by detonation of an explosive charge traveled down the length of the specimen to reach the fatigue precrack and load it to failure within 25 ms. The load, PðtÞ; across the cracked specimen was measured with strain gages mounted on the specimen and the load

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point displacement, dðtÞ; was monitored with a moire´ grid technique; the time variations of the load and crack opening displacement are shown in Fig. 6.8. Costin et al. (1977) then used the quasi-static analysis to evaluate the fracture toughness. Two different conditions were identiﬁed: ﬁrst, when the condition of small-scale yielding was fulﬁlled, i.e. R $ 2:5ðKI =sY Þ2 ; the stress intensity factor is given by KIstat ¼

P pﬃﬃﬃﬃﬃﬃﬃ pRf ðR=DÞ pR2

ð10:4Þ

where R is the remaining uncracked ligament, D the diameter of the bar and f ðR=DÞ the geometric factor for the round-notched bar (Tada, 1973). Second, if the specimen size requirements for the plane-strain fracture toughness test were not met, an estimate of the J-integral for cracked round bars by Rice et al. (1973) was used to evaluate the stress intensity factor at crack initiation: ðd c 1 2 n2 2 1 J¼ KI ¼ 3 Pð d Þd d 2 P d ð10:5Þ c E 2pR2 0 This quasi-static analysis was justiﬁed because the ligament dimensions in the cracked specimen were small; this claim was later shown to be acceptable by Nakamura et al. (1986) through a full-scale numerical simulation of the experiment. Costin et al. (1977) examined the initiation toughness and its dependence on temperature in a 1018 cold-rolled steel. Wilson et al. (1980) determined the temperature dependence of a 1020 hot-rolled steel. The loading rate in these experiments was ﬁxed at a value of K_ dyn . 2 £ 106 MPa m1/2/s. Their I measurements of the temperature dependence of the dynamic crack initiation toughness are shown in Fig. 10.4. The fracture toughness obtained from quasi-static tests are also shown in this ﬁgure. There are a number of trends in these data that needs to be discussed further. First, the quasi-static experiments indicate p brittle fracture by cleavage at temperatures below about 2 1508C with KId ¼ 40 MPa m for both steels. Ductile fracture by void nucleation and growth was p observed at temperatures above the transition temperature, with p KIC ¼ 80 MPa m and TNDT ¼ 2608C for the hot-rolled steel and KIC ¼ 110 MPa m and TNDT ¼ 21108C for the cold-rolled steel. Second, the temperature dependence of p the 6 . 2 £ 10 MPa m/s dynamic initiation toughness evaluated at a strain rate of about K_ dyn I was similar to the quasi-static toughness, but with the transition temperature for both alloys increasing to TNDT ¼ 308C: Third, the dynamic initiation toughness was dramatically lower than the quasi-static toughness below the transition temperature, but exceeded the low strain rate toughness at temperatures above 208C. Finally, dynamic initiation toughness of both alloys was nearly the same over the entire temperature range tested. From an examination of the fracture surfaces, Costin et al. and Wilson et al. identiﬁed that, associated with the transition temperature was a switch in the fracture mechanism from brittle cleavage to ductile ﬁbrous fracture. Brittle fracture dictated by cleavage was induced at the carbide particles. Wilson et al. (1980) determined that the mean thickness of the grain boundary carbide plates was the same in both alloys and the mean spacing between carbide plates, favorably oriented to act as triggers for cleavage fracture, was also the same in both alloys. These microstructural features were used to justify the observed similarity in the dynamic initiation toughness between the two alloys in the brittle fracture

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161

Figure 10.4 Dependence of the dynamic crack initiation toughness on loading rate and temperature. (Reproduced from Wilson et al., 1980.)

regime. On the other hand, ductile fracture is governed by void nucleation that occurs at inclusions and grain boundaries. Cracks in pearlite colonies were also thought to nucleate voids in the surrounding ductile ferrite phases. The hot-rolled steel was shown to contain larger pearlite colonies at a higher volume fraction. Furthermore, the two alloys were also shown to have the same ﬂow stress at high strain rates in spite of differences in the volume fraction of pearlite in the two alloys. Thus, the strain ﬁeld near the crack tip at crack initiation would have been similar in both alloys and hence the two alloys are expected to indicate similar trends in the fracture initiation toughness with temperature. The higher dynamic initiation toughness observed above 208C was attributed to viscoplastic effects on void nucleation, growth and coalescence. One major limitation of the loading method used in this study is that the rate of loading could not be varied signiﬁcantly and therefore the detailed variation of the dynamic initiation toughness with loading rate could not be obtained. The ﬁrst direct experimental measurement of the loading rate dependence of dynamic crack initiation toughness was provided by Ravi-Chandar and Knauss (1984a). The electromagnetic loading scheme described in Chapter 6 was used. The specimen conﬁguration was a semi-inﬁnite crack in an unbounded medium, with uniform pressure over the crack surfaces. Since the time history of the loading is controlled by the choice of capacitors and inductors used in the current generation circuit and the magnitude of

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the crack surface pressure is controlled independently by the charge stored in the capacitors, time rate of increase of pressure could be varied simply by charging the capacitors to different voltage levels. Furthermore, since the load application is triggered electrically, synchronization of the loading with a high-speed camera with the application of load was achieved easily enabling the tracking of crack initiation. Using this scheme, Ravi-Chandar and Knauss (1984a) explored the loading rate dependence of the initiation fracture toughness in Homalite-100, a brittle polyester. Selected mechanical and optical properties of this material are given in Appendix B. A series of experiments were 1/2 4 5 performed at different loading rates in the range K_ dyn I ðt; vÞ ¼ 10 – 10 MPa m /s. The optical method of caustics described in Chapter 8 was used to determine the dynamic stress intensity factor. A rotating mirror high-speed camera capable of 200,000 frames per second was used to obtain the time variation of the caustic diameter and crack position. A selected sequence of high-speed photographs from such experiments is shown in Fig. 8.7. An example of the type of experimental data obtained in such pictures is shown in Figs. 9.2 – 9.4. The dynamic stress intensity factor at the onset of crack growth—the dynamic crack initiation toughness, KId ðK_ dyn I ; TÞ—was determined by interpolating the measured stress intensity factor time history at the time of onset indicated by the crack position history. The results are shown in Fig. 10.5; in this ﬁgure, the measured value of the dynamic stress intensity factor at crack initiation is plotted against the time to fracture, tf. The rate of loading, which is inversely related to the time to fracture, is also indicated in the ﬁgure at the two extremes. Furthermore, the plane-strain fracture toughness, KIC, obtained under quasi-

Figure 10.5 Dynamic crack initiation toughness for Homalite-100. (Data from Ravi-Chandar and Knauss, 1984a– d.)

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static loading conditions p at 208C is also indicated in the ﬁgure. Clearly, at low loading rates, 4 ðt; vÞ # 10 MPa m/s, the initiation toughness is equal to the plane-strain fracture K_ dyn I p 4 toughness of the material. At higher loading rates, K_ dyn m/s, I ðt; vÞ . 10 MPa dyn KId ðK_ I ; TÞ increases rapidly. In contrast to the conjecture of Eftis and Kraft (1965) that there might be a minimum in the crack initiation toughness at some loading rate, these results displayed a monotonic increase with increasing loading rate. Obviously, this is a material-dependent property as the results of Costin et al. (1977) and Wilson et al. (1980) on steels indicate that at the loading rate used in their work KId ðK_ dyn I ; TÞ decreased with an increase in loading rate. Ravi-Chandar and Knauss (1984a) interpreted the increase in the dynamic crack initiation toughness with increasing loading rate to crack tip separation processes such as nucleation, growth and coalescence of microcracks that might be governed by the nonlinear viscoelastic behavior of the polymer material. Liu et al. (1998) provided an alternate explanation derived simply from inertial constraints on the development of the stress ﬁeld. They considered the development of the stress ahead of the crack tip for the problem of a pressure-loaded semi-inﬁnite crack located in an unbounded plane; this is the geometry corresponding to the experiment of RaviChandar and Knauss (1984a) shown in Fig. 6.9. Rather than imposing a fracture criterion based on the stress intensity factor reaching a critical value, Liu et al. postulated that the crack will grow when the normal stress component s22, evaluated at a distance d from the crack tip, reached a critical value, sc. This criterion may be motivated for a brittle material by considering that nucleation, growth and coalescence of microcracks within the crack tip process zone required a certain critical stress level to be reached. Then, in this model, d is the distance from the crack tip to the critical ﬂaw and sc the stress required to grow this microcrack. From the solution of Freund (1990) for the problem of a semi-inﬁnite pressurized crack, Liu et al. evaluated the magnitude of the normal stress component s22 ðx1 ¼ d; x2 ¼ 0; tÞ; the formal expression is given in Eq. 4.29. They determined that corresponding to the transient loading, the stress component s22 ðx1 ¼ d; x2 ¼ 0; t # d=Cs Þ was initially compressive and developed large positive values only after t q d=Cs (see Fig. 4.3). The magnitude of the compression and the time delay, before s22 ðx1 ¼ d; x2 ¼ 0; tÞ attained the level of sc, increased with increasing loading rate; this time delay was proposed as the reason for the higher dynamic crack initiation stress intensity factor that is observed at higher loading rates. Liu et al. reinterpreted the experimental measurements of RaviChandar and Knauss (1984a) through the following argument: ﬁrst, the time to attain the failure stress was determined from the equation s22 ðx1 ¼ d; x2 ¼ 0; tf Þ ¼ sc ; subsequently, the stress intensity factor at this time, KI ðtf Þ; was calculated from Eq. 4.20 and taken to be the dynamic initiation toughness. Their estimates of the critical stress intensity factor based on different critical distances d are shown in Fig. 10.6 for comparison with the experimental measurements. Thus, for a simple stress based criterion for crack initiation, the results of Liu et al. demonstrate that nominally brittle materials may exhibit an apparent rate dependence, driven only by the inertial effect. A similar investigation on the loading rate dependence of crack initiation toughness was conducted by Kalthoff (1986), who evaluated the dynamic crack initiation toughness behavior of Araldite B and high-strength steel. Kalthoff used a cylindrical projectile ﬁred from an air-gun to impact a specimen as indicated in Fig. 6.4b. The evolution of the dynamic stress intensity factor was monitored through high-speed photography of caustic

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Figure 10.6 Dynamic crack initiation toughness for Homalite-100. (Data from Ravi-Chandar and Knauss, 1984a– d, model from Liu et al., 1998.)

patterns. Kalthoff’s results are reproduced in Figs. 10.7 and 10.8. These results reinforce the idea of material dependence of the rate effects on the crack initiation toughness. The rate of 3 – 1 £ 105 MPa loading used in these experiments was in the range K_ dyn I ðt; vÞ ¼ 3 £ 10 p p dyn 6 7 m/s for the Araldite specimens to K_ I ðt; vÞ ¼ 1 £ 10 – 1 £ 10 MPa m/s for the steel specimens. In this range, the Araldite specimens did not exhibit any rate dependence, while the steel specimens exhibited an initial drop in the dynamic initiation toughness followed by a sharp increase at even higher loading rates. Kalthoff (1986) interpreted the increase once again in terms of the minimum time criterion described above in connection with the stress amplitude necessary for crack initiation under extremely short-duration pulse loading. The Hopkinson bar test has been used by a number of investigators in the evaluation of the dynamic crack initiation toughness. Klepaczko (1990) has performed a large number of tests to determine the loading rate dependence of the dynamic crack initiation toughness in metallic materials (Klepaczko, 1982, 1985). He used the Hopkinson bar apparatus to generate impact loads on cracked specimens (see Chapter 6) at loading rates varying in the 1/2 0 5 range of K_ dyn I ðt; vÞ ¼ 3 £ 10 – 1 £ 10 MPa m /s. Two graphs illustrating the rate and temperature dependence of the initiation toughness in A533 reactor grade steel and two different aluminum alloys are shown in Figs. 10.9 and 10.10, respectively. In all these experiments, the strain gage signals captured in the incident and transmitter bars of the split-Hopkinson bar were interpreted in terms of the loading on the specimen. Time variation of the dynamic stress intensity factor was determined through a static analysis of

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Figure 10.7 Dynamic crack initiation toughness of Araldite. (Reproduced from Kalthoff, 1986.)

Figure 10.8 Dynamic crack initiation toughness of high-strength steel. (Reproduced from Kalthoff, 1986.)

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Figure 10.9 Dependence of the dynamic initiation toughness on the loading rate and temperature for A533B reactor grade steel. (Reproduced from Klepaczko, 1990.)

Figure 10.10 Dependence of the dynamic initiation toughness on the loading rate and temperature for two different aluminum alloys. (Reproduced from Klepaczko, 1990.)

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the stress intensity factor, although the dynamic load measured from the Hopkinson bar apparatus was used. As seen in the ﬁgures, in all these tests, the initiation toughness was found to decrease with increasing loading rate and increase with increasing temperature, for the steel as well as the aluminum alloys. In contrast, in another investigation, Rittel and Maigre (1995) used the compact compression specimen described in Chapter 6 (see Fig. 6.7c) to examine the strain rate dependence of the initiation toughness. The applied load and boundary displacements were measured with strain gages mounted on the incident and transmitter bars of the Hopkinson pressure bar apparatus. The strain gage signals were used to determine the time variation of the dynamic stress intensity factor through a path-independent integral described by Bui et al. (1992). The initiation and growth of the crack was monitored with two single wire fracture gages glued across the crack line, one on each side of the specimen. Rittel and Maigre (1995) evaluated the rate dependence of polymethylmethacrylate (PMMA); the rate of loading in these p experiments was determined to be in 4 5 the range: K_ dyn ðt; vÞ ¼ 1 £ 10 – 2 £ 10 MPa m/s, typical of the range used in the I determination of initiation toughness in polymers. Their results p indicated that the dynamic initiation toughness in PMMA increased from about 2 MPa m for quasi-static loading to p 4 ðt; vÞ , 1 £ 10 MPa m/s), to about 13.5 MPa m1/2 under low rates of loading ðK_ dyn I p dyn 5 dynamic loading at rates of about K_ I ðt; vÞ , 2 £ 10 MPa m/s; while these values are twice what is normally reported for this material p (typically the quasi-static fracture toughness of PMMA is quoted to be around 1 MPa m although this depends signiﬁcantly on the molecular weight), the trend of increasing toughness with loading rate corresponds well with observations in this and other polymers. Owen et al. (1998) examined the rate dependence of crack initiation toughness in thin sheets (1.63, 2.03 and 2.54 mm) of 2024-T3 aluminum alloy. Single-edge-notched specimens (see Fig. 6.1) were loaded dynamically by a tension pulse in a split-Hopkinson tension apparatus. The strain gages in the incident and transmitted bars were used to evaluate the stress state in the specimen; assuming that the characteristic length j was very small, Owen et al. suggested that a quasi-static analysis was adequate for the determination of the dynamic stress intensity factor. They veriﬁed that this assumption was appropriate for small specimens by measuring the crack opening displacement directly and comparing it to the calculations based on the estimate of the dynamic stress intensity factor obtained from the strain gage measurements. The loading rates p imposed in this experiment were in 4 6 ðt; vÞ ¼ 1 £ 10 – 2 £ 10 MPa m/s. The results of Owen et al. are the range of K_ dyn I shown inpFig. 10.11. The plane-strain fracture toughness of this aluminum alloy was about 30 MPa m. The aluminum alloy exhibited a monotonic p increase from this quasi-static 4 value atp low loading rates, K_ dyn ðt; vÞ , 1 £ 10 MPa m/s, to ap value of about I ðt; vÞ ¼ 2 £ 106 MPa m/s. This is in 77 MPa m at the highest loading rate K_ dyn I contrast to the data shown in Fig. 10.10 from Klepaczko (1990); presumably both alloys are nearly the same and therefore exhibit a similar fracture response. Also shown in Fig. 10.11 is the rate of loading dependence of the crack initiation toughness in brittle polyester Homalite-100 (Ravi-Chandar and Knauss, 1984a); the plot is in normalized form of the data shown in Fig. 10.5 for comparison. The similarity between the rate dependence of these two vastly different materials is indeed striking. However, the underlying fracture mechanisms are signiﬁcantly different in the metallic and polymeric materials;

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Figure 10.11 Variation of the crack initiation toughness with time-to-fracture for 2024-T3 aluminum alloy; comparison to the behavior of Homalite-100 is also shown. The time to fracture is inversely related to the loading rate. (Reproduced from Owen et al., 1998.)

hence the similarity must arise from some other common attribute. It appears that the inertial effect in the build-up of stress ﬁelds near the crack could be the main contributor to the loading rate dependence of the initiation toughness as discussed in the model by Liu et al. (1998).

10.2 Dynamic Crack Arrest Criterion The general formulation of the dynamic crack arrest criterion has been described in Chapter 5. In this section, a discussion of the experimental investigations aimed at characterizing the crack arrest behavior is presented. Recall that dynamic crack arrest toughness, KIa ðTÞ; was deﬁned as the smallest value of the dynamic stress intensity factor for which a growing crack cannot be maintained. As described earlier, a conservative design can be implemented in practice based on crack arrest criteria alone. If the dynamic stress intensity factor of a crack in any structure never exceeds KIa ðTÞ; a growing crack can never be sustained and hence the design is not susceptible to dynamic crack growth! So, the main focus of experiments has been the determination of the crack arrest toughness. In this section, we describe ﬁrst the studies aimed at identifying the crack arrest criterion, and then follow this with a description of the standard test method for the determination of the arrest toughness. Finally, we describe strategies adopted for the design of fracture critical structures. 10.2.1 Development of the Crack Arrest Criterion Early attempts at characterizing crack arrest toughness as a material property were actually motivated by difﬁculties in determining the dynamic initiation toughness

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(Crosley and Ripling, 1969). In their experiments aimed at determining the loading rate dependence of the dynamic crack initiation toughness of A533B steel, Crosley and Ripling found that the state of crack arrest was more easily reproduced than the state of crack initiation. This is readily understood by considering that while crack initiation toughness is inﬂuenced signiﬁcantly by the bluntness of the initial crack, crack arrest always proceeds by the deceleration of a natural crack and is independent of how the crack was produced. Hence from an experimental point of view, it is easy to create appropriate conditions for crack arrest than for initiation. However, the crack arrest experiment is inherently a dynamic experiment and time-resolved measurements of the crack position and load are generally required in order to determine the crack arrest toughness. As was the case of the crack initiation problem, early investigations on crack arrest relied upon a static analysis for the interpretation of experimental measurements in terms of the stress intensity factor at arrest. However, with the development of the various diagnostic techniques described in Chapters 7 and 8, many careful studies have examined the determination of the crack arrest criterion. Many different types of specimen and loading have been used to generate arrest of a rapidly growing crack; these have been interpreted based on static or dynamic analysis. ASTM Publications STP 627 (1977) and 711 (1980) contain detailed discussion of the development of the many techniques. Here we provide a discussion of two experiments that provide an overview of the determination of crack arrest toughness. The key ingredient necessary in crack arrest experiments is that at some time during the test history the crack arrest condition is satisﬁed KIdyn ðtÞ # KIa ðTÞ for t . ta

ð10:6Þ

where ta is the time of crack arrest. This can be accomplished in one of the three ways: ﬁrst, by performing the test in a conﬁguration where KIdyn ðtÞ is a decreasing function of time; a number of different conﬁgurations have been used to accomplish this. From Eq. 5.15, it is seen that if a wedge-loaded specimen is used, as the crack extends, the stress intensity factor will decrease and eventually satisfy Eq. 10.6; this is the basis of the ASTM Standard test. Crosley and Ripling (1980), Hoagland et al. (1977), Kalthoff et al. (1977) and others also used this approach. Kobayashi et al. (1977) used a single-edge-notched specimen but with loading that decreased linearly away from the crack tip resulting in a decreasing KIdyn ðtÞ with crack extension. Ravi-Chandar and Knauss (1984b) studied crack arrest in the electromagnetic loading scheme by applying a short-duration pulse loading. The second method of generating crack arrest is by increasing the crack arrest toughness along the crack path; this is the basis of the Robertson test (1953) where the cracked end of the specimen is held at a very low temperature while the other end is kept at a signiﬁcantly higher temperature. This idea has also been used in the wide-plate tests at the National Institute of Standards and Technology (Pugh et al., 1988). As the crack extends, it encounters a material with a higher crack arrest toughness and once again inequality in Eq. 10.6 is satisﬁed at some point along the crack path. Lastly, duplex crack arrest specimens have also been used: here a fast crack is started in a material with a low toughness that then grows into the second material of higher crack arrest toughness and hence arrests. Hoagland et al. (1977), Dally and Kobayashi (1978) and others have used this approach to

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crack arrest studies. All these investigations have been instrumental in identifying that the dynamic crack arrest toughness can be established as a material property, dependent on temperature, and in establishing that a dynamic analysis is necessary to interpret the experimental measurements reliably in terms of the arrest toughness. Kalthoff et al. (1977) performed an illuminating series of experiments to determine the crack arrest toughness as well as the necessity of performing a dynamic evaluation. Specimens of Araldite B, a brittle epoxy, were used in a rectangular double-cantilever beam conﬁguration. The loading was generated by wedging the crack as indicated in Fig. 10.12. The initial crack was made blunt, with varying radius. The wedge opening displacement at the instant of crack initiation was determined through a clip gage; since dynamic effects are not involved prior to crack initiation the stress intensity factor at the onset of crack growth, labeled KIq to distinguish it from the initiation toughness, can be determined by static analysis. While the dynamic crack initiation toughness for Araldite B p is about 0.79 MPa m (see Fig. 10.7), by using blunt notches of different notch radii, p Kalthoff et al. (1977) were able to vary KIq in the range 0.7– 2.5 MPa m. For the propagating crack, Kalthoff et al. recorded caustic patterns from the crack tip on a CranzSchardin type camera. The high-speed camera was triggered by interruption of a laser beam by the crack as it grew from the initial blunt notch; thus the initial phase of crack growth was not captured and the early time history of the dynamic stress intensity factor was not determined. The variations of the dynamic stress intensity factor, KIdyn ; and the crack speed, v, obtained from the caustic measurements with crack position are shown in Fig. 10.13. Also indicated in this ﬁgure is the estimate of the stress intensity factor, KIstat ; calculated from the measured crack position and crack opening displacement. In particular, the static calculation of the stress intensity factor with arrested crack length and measured crack opening displacement yields the static estimate for the crack arrest toughness: KIastat : A number of important features in the experimental results must be noted. First, KIdyn exhibits a rapid drop not captured in the data recorded, but settles down at a constant value for much of the crack growth period and then gradually drops down to the arrest value. Second, the crack speed is nearly constant during the period that KIdyn is constant; deceleration of the crack appears as KIdyn decreases, but always lags the latter. The reasons for this are not completely understood. Third, for all the experiments shown, the crack comes to a stop when KIdyn reaches a constant value of about 0.7 MPa m1/2; thus, this value must be considered to be the crack arrest toughness, KIa. Beyond crack arrest, KIdyn continues to drop; the time variation of KIdyn for the arrested crack is shown in

Figure 10.12 Wedge-loaded rectangular double-cantilever beam conﬁguration.

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Figure 10.13 Variation of the dynamic stress intensity factor and crack speed with crack position in a wedge-loaded double-cantilever beam specimen. (Reproduced from Kalthoff et al., 1977.)

Fig. 10.14. In this ﬁgure, KIdyn normalized KIastat is plotted as a function of time. Clearly, KIdyn for the arrested crack oscillated about KIastat and eventually settled down at this value; however, crack arrest did not occur at KIastat ; but at KIa as indicated in Fig. 10.13. KIastat is always lower than KIa in all the experiments and furthermore, KIastat depends on the initial notch bluntness! Finally, from the comparison shown in Fig. 10.13 between KIdyn and KIstat ; the static analysis is never appropriate in this test, except at very long times when the crack has been arrested and all the oscillations have died out. In another series of experiments, Ravi-Chandar and Knauss (1984a) used the electromagnetic loading method to examine crack initiation and arrest in the same test. In this conﬁguration of the pressurized semi-inﬁnite crack geometry the crack was initiated, arrested and reinitiated, all before the arrival of waves reﬂected from the far boundaries of the specimen; thus, in this experiment there are no geometrical inﬂuences on crack arrest. Crack arrest was achieved by loading the crack faces dynamically with a load just sufﬁcient to start crack growth. The shape of the loading pulse was designed to be long enough to generate crack growth, yet so short that after about a few millimeters of crack growth, the crack tip was unloaded. A double-trapezoidal pulse with each pulse lasting

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Figure 10.14 Variation of the dynamic stress intensity factor and crack speed with time after crack arrest. (Reproduced from Kalthoff et al., 1977.)

about 70 ms was used in the loading. The results from this series of tests on Homalite-100 are shown in Fig. 10.15; the variation of KIdyn and the crack position with time are shown in the ﬁgure. The starter crack was made using a razor blade as a wedge, which led to variations in the initiation stress intensity factor. From Fig. 10.15 it is evident that when KIdyn reached the dynamic initiation toughness KId, the crack began to grow at a constant speed; associated with the crack extension is a drop in the stress intensity factor (as dictated by Eq. 4.56). As soon as the arrest condition in Eq. 10.6 is satisﬁed, the crack is arrested. Note that, in all the experimental results shown in Fig. 10.15, at around 70 ms the stress intensity factor is indeed always the same within the accuracy of the measurement; the crack arrest appears quite consistently at KIdyn ¼ 0:4 MPa m1/2. This is about 11% lower than KIC for this material (see Fig. 10.5). Since these tests were conducted in an inﬁnite specimen geometry, there are no reﬂected stress waves and hence no oscillations in the stress intensity factor. Clearly, the crack arrest toughness can be taken to be KIa ¼ 0:4 MPa m1/2. Continued loading from the second pulse of the loading reinitiated the arrested crack at KId. From these careful sets of experiments described above, it is easy to conclude that KIa is an appropriate measure of the crack arrest toughness and that a dynamic analysis is essential in order to determine this parameter from experiments. However, for standard tests, it is essential to have a simple experimental procedure that utilizes conventional laboratory apparatus. Kalthoff et al. (1977) showed that in some specimen conﬁgurations such as the compact tension specimen, the dynamic effects may be small and a quasi-static analysis may be used conservatively; this geometrical conﬁguration has been implemented in the ASTM standard test procedure described in Section 10.2.2.

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Figure 10.15 Variation of the dynamic stress intensity factor and crack speed with time from crack initiation and arrest experiments. (Reproduced from Ravi-Chandar and Knauss, 1984a.)

10.2.2 ASTM Standard Method for Crack Arrest Based on round-robin tests and an accumulation of data, a standard test procedure, the ASTM E-1221 Standard, has been established that describes the determination of crack arrest toughness in ferritic steels. A brief summary of the test method is provided here. The schematic diagram of the test arrangement under this standard is shown in Fig. 10.16. In this arrangement, the geometry of a compact tension specimen is used; however, loading is generated by introducing a split pin into the hole in the specimen and forcing the pins apart

Figure 10.16 Specimen and loading conﬁguration for the determination of crack arrest toughness. (Reproduced from ASTM E-1221 Standard.)

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by a wedge with a force P; this conﬁguration is called the compact crack arrest (CCA) specimen. Side-grooves are introduced to guide the crack along the plane of symmetry. A rapid crack growth arrest sequence is generated by forcing a wedge; as the crack grows away from the hole, the stress intensity factor drops quickly (see the discussion in Section 5.6.) and hence results in arrest of the crack. The preparation of the starting notch is quite important to the successful performance of this standard test. In low to intermediate strength alloys, it is suggested that the crack line be embrittled by depositing a weld along the crack line; other forms of the starter notch include a quench-embrittled Chevron notch, and a fatigue-precracked and overloaded crack. The starter notch dictates the level of the stress intensity factor at which crack initiation occurs; the extent of subsequent crack growth segment is dictated by this value. If the stress intensity factor at initiation is low enough, arrest of the crack can occur at an appropriate length within the compact specimen. The dynamic run arrest sequence that this specimen experiences under the wedge load clearly indicates the need for a dynamic analysis of the problem. However, according to the ASTM standard, a static analysis is considered to be appropriate; thus from a recording of the maximum load, P, and the crack mouth opening displacement, d, the stress intensity factor can be determined. The load and crack mouth opening displacement at the onset of crack initiation are labeled P0 and d0, while the corresponding values soon after crack arrest are labeled Pa and da; typically the arrest values oscillate with time, but the ASTM standard assumes that the values measured at 2 ms after crack arrest do not differ signiﬁcantly from the values measured at 100 ms after crack arrest. The stress intensity factors corresponding to crack initiation K0 and arrest Ka are then calculated from a static analysis through the following equation sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

a B KI ¼ Edf W BN W

ð10:7Þ

where d is the appropriate crack mouth opening displacement and E the modulus of elasticity; other geometrical quantities are deﬁned in Fig. 10.16. The geometric correction factor f ðaÞ is given by: f ðaÞ ¼ ð1 2 aÞ0:5 ð0:748 2 2:179a þ 3:56a2 2 2:55a3 þ 0:62a4 Þ

ð10:8Þ

The stress intensity factor at arrest is taken to be the crack arrest toughness, KIa, if the following conditions are met: (i) the crack growth during the run arrest segment must be at least greater than the plane-stress plastic zone corresponding to K0 and greater than twice the side-groove slot width; (ii) the thickness B $ ðKa =syd Þ2 ; and (iii) the unbroken ligament must be larger than 0.15W and 1:25ðKa =syd Þ2 : The dynamic yield strength, syd, is taken to be 205 MPa (30 ksi) larger than the static yield strength of the material. The standard test procedure also restricts the maximum crack mouth opening that can be attained in one continuous loading; if this value is reached, the specimen must be unloaded and reloaded with the maximum crack opening displacement increased to a speciﬁed level at each reloading. Further details can be found in the ASTM E-1221 Standard.

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10.2.3 Application of the Crack Arrest Criterion Once the crack arrest criterion has been postulated as in Eq. 10.6, and the crack arrest toughness determined as an appropriate material property in the range of temperatures of interest, application of the criterion is quite straightforward. For example, according to Milne et al. (1988) the R6 procedures for failure assessment could be used to evaluate ﬂaw criticality under dynamic loading as long as the material properties are evaluated at the appropriate rates and quasi-static analysis of the stress intensity factors is appropriate; however, if inertial effects become important, such procedures become invalid. But, if the dynamic stress intensity factor variation is calculated analytically or through numerical simulations, evaluation of crack arrest is simply through the use of Eq. 10.6. Two different design approaches have been used in industrial practice. The ﬁrst one is based on the selection of a material with a large enough crack arrest toughness KIa that the structure should never experience a stress intensity factor (dynamically) that exceeds this value; hence crack initiation and growth do not occur in the structure. This is similar to the minimum speciﬁed fracture toughness in the ASME Boiler and Pressure Vessel Code, however, a requirement based on crack arrest toughness will be even more conservative since KIa , KId : While this stringent requirement is suitable for structures that must be designed to be damage-resistant, high performance structures and aging structures that have already accumulated ﬂaws in service must be evaluated with a damage-tolerant approach. Therefore, the second approach to the critical design problem has been to perform a dynamic analysis of any particular design; in this approach crack initiation and growth of pre-existing cracks are allowed, but the extent of their growth is controlled through judicious design of the structure and by the placement of crack arresters, deﬂectors or tear straps; this has been the practice in ship and airplane industries. The design of the stiffeners or arrestor plates, however, requires a detailed fracture mechanics analysis of the dynamic loading and the resulting crack growth. While dynamic analysis of the crack arrest strategies such as these have been shown for a long time (see for example, Wade and Kobayashi, 1970), in most industrial practice, this is currently accomplished through scale model and full-scale tests. Two examples are provided in the following discussion in Figs. 10.17 and 10.18. In Fig. 10.17, the use of an arrester plate is indicated, typical of applications in ship building and more recently in pipelines. For a hull made of material A with a crack arrest toughness KIaA ; a strip of material B with a crack arrest toughness KIaB q KIaA is inserted by welding in a patch as indicated in Fig. 10.17; material B may simply be an alloy of slightly different composition than material A. Alternatively, a patch of material B or even the same material may be superposed on material A, much as a stringer attached to an aircraft skin; the stiffener can be welded or fastened with rivets, but essentially plays the same role as the welded plate in Fig. 10.17. The concept behind this method of crack arrest is the following: as the crack extends, let the stress intensity factor increase monotonically as indicated in the inset graph in Fig. 10.17; note that it is the dynamic stress intensity factor that must be calculated, either in some closed form approximation or through a numerical simulation of the problem. As the crack extends into material B, it suddenly encounters a material with a much higher crack arrest toughness; since KIdyn , KIaB ; the condition in

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Figure 10.17 Crack arrest through an arrestor strip. A crack running from zone A into the arrestor zone encounters a material with higher arrest toughness in zone B and therefore arrests. In practice, the strap of material B is welded to material A.

Eq. 10.6 is satisﬁed and the crack arrests within material B. It must be noted that a dynamic analysis is necessary, for if a subsequent loading is applied on the same crack, perhaps from stress waves reﬂected from other parts of the body, it is possible to reinitiate the crack if these waves provide a stress intensity factor history indicated by the dotted line in Fig. 10.17. The second example we describe is from applications in aircraft design. A typical aircraft fuselage is made of a thin metallic skin and reinforced by riveted longitudinal stringers and circumferential frames. The skin itself contains a riveted longitudinal lap splice joint. Longitudinal cracks appear in such structures from fatigue loading; small cracks grow at rivet sites in close proximity to each other and as they approach each other this multi-site damage can result in triggering catastrophic axial crack propagation along the lap splice joint which can then grow in an uncontrolled manner; the circumferential stiffeners do carry some of the load as well as provide additional resistance to crack growth as discussed in the previous paragraph, but the stiffeners eventually break. An effective method of arresting these axial cracks is through the use of tear straps as shown schematically in Fig. 10.18. Tear straps are of the same thickness as the skin, but are narrow strips that are attached to the skin at every circumferential frame and midway between the frames. The action of the tear strap can be considered in the following manner: analysis of the problem of an axial crack under internal pressure indicates that as the crack extends, the stress intensity factor must increase; however, as the crack approaches the tear strap/frame, since these parts share some of the load, the crack tip gets unloaded as shown in the inset in Fig. 10.18. At the same time, due to the increased crosssection to be cracked at the tear strap, the crack arrest toughness increases sharply. Thus, the crack cannot penetrate through the tear strap; however, due to pressure loading, a bulge develops on the skin, which results in a mixed-mode loading on the arrested crack. Under suitable conditions, the crack turns to the circumferential direction. This leads to an opening of a ﬂap on the fuselage and hence depressurization of the cabin and a decrease in the driving force for crack extension. In order to implement this concept, the practice has

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Figure 10.18 Crack arrest at a stiffener. A crack running towards a stiffener experiences a drop in the driving force as a result of the load sharing in the stiffener. In some cases, the crack can be made to turn by 908 and grow parallel to the tear strap. This tearing can result in a depressurization of the container and a decrease in the driving force growing the crack. These so-called tear-straps have been used effectively in designing for damage tolerance.

been to perform scale model and full-scale tests; recently dynamic fracture mechanics has been applied to this problem in order to determine the effectiveness of tear strap design (Kosai et al., 1999).

10.3 Dynamic Crack Growth Criterion The general formulation of the crack growth criterion was discussed in Section 5.4. While the fundamental criterion for crack growth is provided by the energy rate balance equation given in Eq. 5.7, for convenience in applications, the criterion based on dynamic stress intensity factors is preferred; this criterion, given in Eq. 5.11, is the following: KIdyn ðt; vÞ ¼ KID ðv; K_ dyn I ; TÞ

ð10:9Þ

The crack must grow at a speed v such that the above equality is maintained. The dynamic crack growth toughness, KID ðv; K_ dyn I ; TÞ; is a function of the crack speed, the loading rate and the temperature and must be determined experimentally or calculated from models

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governing fracture processes. Over the last three decades, many investigators have examined this issue through experiments on different materials. However, many questions still remain unanswered although a working theory based on Eq. 10.9 has been developed. In recent years, models of the fracture process based on atomistic, molecular and lattice dynamics simulations, and cohesive zones have been used to examine the growth of cracks under dynamic loading. The theoretical models are discussed in Chapter 12. In this section, we present a survey of experimental investigations aimed at determining the crack growth criterion in brittle and ductile materials. 10.3.1 Crack Growth Toughness in Nominally Brittle Materials For a nominally brittle material, it is easy to motivate the assumption that the fracture energy must be independent of the rate of its creation; thus, if we take the fracture energy per unit extension, g; to be constant, then the dynamic crack growth criterion described in Eq. 5.8 provides a one-to-one relationship between the instantaneous stress intensity factor and the crack speed; this is commonly referred to as the K – v relationship and is merely an expression of the dynamic crack growth toughness. Numerous investigations were undertaken over the last three decades with the goal of determining the crack growth toughness experimentally. Every loading conﬁguration and diagnostic technique discussed in Chapter 6, and some others that have not been discussed here, have been used in these investigations. The early studies relied on measuring the crack speed and identifying the driving force through calculations. For example, Paxson and Lucas (1973) used an inﬁnite strip conﬁguration illustrated in Fig. 6.1 and assumed that the energy release rate was a constant in this conﬁguration. Using this Paxson and Lucas were able to determine the relationship between the dynamic energy release rate and the measured crack speed in PMMA; interestingly, Hauch and Marder (1998) repeated the same experiments many years later and found a similar variation of the dynamic energy release rate with crack speed, although Hauch and Marder observed intermittency in the crack growth and related this to a microbranching instability. Do¨ll (1976a,b) examined dynamic fracture in single-edge-notched PMMA specimens. The heating of the specimen was measured using 40 mm thick thermocouples close to the anticipated crack path and interpreted in terms of the energy dissipation at the crack tip. The crack speed was measured with an opto-electronic measurement. With this arrangement Do¨ll investigated the molecular weight dependence of dynamic crack growth. Other investigations into the dynamic crack growth criterion have used direct measurements of the crack tip stress intensity factor through optical methods described in Chapter 8, and the results of some of these investigations are described here. Beebe (1966), Bergkvist (1974), Mall and Kobayashi (1978), Dally (1979 and references contained therein), Kalthoff (1983), Ravi-Chandar and Knauss (1984c), Arakawa and Takahashi (1991), Hauch and Marder (1998), and many others have examined the dynamic crack growth criterion through direct experimental measurements and/or numerical simulations coupled with experiments. While Schardin’s early experiments were primarily on glass, due to the high crack speeds encountered in glass, in almost all the more subsequent experiments, polymeric materials have been used as examples of nominally brittle materials. Araldite B, Homalite-100, and PMMA have been

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the most popular materials. In Figs. 10.19 –10.24 we present data from these investigations of the dynamic crack growth criterion. Dally (1979) and co-workers performed a number of experiments on Homalite-100, and an epoxy resin. For the Homalite-100, single-edgenotched specimens were loaded in a manually operated hydraulic loading frame. The initial cracks were introduced by a saw cut followed by a scribing with a razor blade. The experiments were performed in a dynamic photoelastic apparatus coupled to a Cranz – Schardin multi-spark camera; triggering of the camera was achieved with a conducting wire placed across the prospective crack path. The isochromatic fringe patterns were analyzed through an overdetermined curve-ﬁtting algorithm (see Section 8.2.1 for a discussion of one such method) to extract the dynamic stress intensity factor. Their results are shown in Fig. 10.19. The crack driving force, represented by the instantaneous value of the dynamic stress intensity factor, KIdyn ; is plotted in the abscissa; the response of the crack, represented by the instantaneous crack speed v, is plotted in the ordinate. These experimental measurements are then the experimental characterization of the dynamic crack growth toughness: KID ðv; K_ dyn I ; TÞ: As is typical of such experiments, the initiation event is not captured in the high-speed photographs and hence the low crack

Figure 10.19 Crack growth toughness in Homalite-100. (Reproduced from Dally, 1979.)

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speed, low-Kdyn I , regime can only be attained as the crack decelerates. A number of points are worth noting from this result. As KIdyn increases above the initiation toughness, there is a rather sharp increase in the crack speed; the crack speed changes from close to quasistatic to about 300 m/s (roughly 30% of the shear wave speed) for an increase in the stress intensity factor of about 20% Beyond this, the crack speed increases much more slowly with increase in the driving force: between 300 and 400 m/s, the stress intensity factor more than doubles. Even though the energy balance in Eq. 5.8 suggests that the Rayleigh wave speed is the upper limit of the crack speed, practically, the crack does not attain a speed much greater than about 45% of this speed; this fact has been observed in many earlier investigations, particularly those of Schardin (1959). As marked in Fig. 10.19, the crack branches into two or more branches when KIdyn becomes quite large. Similar experiments were performed on an epoxy resin (Dally, 1979). The K –v relationship corresponding to this material is shown in Fig. 10.20. This material exhibits a hysteretic response: an accelerating crack follows the right branch in Fig. 10.20, while a decelerating crack follows the left branch! Also, cross-overs from the right to left branches were observed with changes in crack speed; these features remain unexplained. Furthermore,

Figure 10.20 Crack growth toughness in KTE epoxy, exhibiting hysteresis. Accelerating cracks go along the right branch while decelerating cracks follow the left branch. CPL, EPL and CLL all refer to different loading schemes for a SEN specimen. CDCB is a contoured DCB specimen. (Reproduced from Dally, 1979.)

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Figure 10.21 Crack growth toughness in Homalite-100. The toughness was shown to depend on whether the cracks were accelerating or decelerating. Contrary to the observations of Dally (1979) decelerating cracks experience a higher stress intensity factor. (Reproduced from Arakawa and Takahashi, 1991.)

there is a limiting speed at 335 m/s; at this speed, even though the stress intensity factor increases by more than a factor of 2, the crack speed does not change within the precision of the measurement; this is contrary to the one-to-one relationship exhibited in Eq. 5.8. Arakawa and Takahashi (1991) also found that the K – v relationship depended on whether the crack was accelerating or decelerating, but their trend was exactly the opposite of that found by Dally. The results of Arakawa and Takahashi (1991) are shown in Fig. 10.21. Kobayashi and co-workers performed numerous crack growth experiments on specimens of Homalite-100; they used SEN, DCB, three-point bending and other conﬁgurations with both quasi-static and dynamic (drop-weight tower) loading conditions. They also used dynamic photoelasticity and a Cranz – Schardin multi-spark camera to determine the crack growth criterion. Their results for the K –v relationship are shown in Fig. 10.22. The scatter in the experimental data was extremely large and attributed to variations in the material and inﬂuence of specimen geometry and errors in crack speed measurement; it is also possible that a K-dominant ﬁeld was not established in the region where the isochromatic fringe data were analyzed. Mall and Kobayashi (1978) concluded that it was difﬁcult to establish a unique K– v relationship through interpretation of their results. Kalthoff (1983) used the method of caustics to determine the dynamic crack growth criterion in Araldite B. His earlier experiments indicated large scatter in the K –v relationship. In order to sort this out, he performed three sets of experiments on the same

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Figure 10.22 Crack growth toughness in Homalite-100. Large scatter in the data are observed in specimens of different geometry. (Reproduced from Mall and Kobayashi, 1978.)

batch of the specimen material in order to remove any concerns regarding material variability. In each set of experiments, different specimen geometry was used: one set of experiments was performed on a rectangular DCB specimen; the second set was on a single-edge-notched specimen. In the third set the specimen had two segments: a starting segment that had the same dimensions as the DCB and a second segment that had the dimensions of the SEN specimen. The K – v relationship corresponding to these experiments are shown in Fig. 10.23; the specimen geometry corresponding to all three kinds of specimens are also shown in the inset. The upper curve in Fig. 10.23 corresponds to the DCB specimen and the lower curve corresponds to the SEN specimen. Kalthoff found that for the DCB/SEN specimens of the third set, the data from the DCB segment fell on the upper curve and that from the SEN segment fell on the lower curve. Thus Kalthoff demonstrated that there was a clear specimen geometry dependence on the K –v relationship. Ravi-Chandar and Knauss (1984c) used the electromagnetic loading device to investigate the dynamic crack growth criterion in Homalite-100. Since the loading corresponds to a pressure-loaded semi-inﬁnite crack in an unbounded medium, this loading scheme should provide the dynamic crack growth criterion without the complications of specimen geometry dependence. They used a rotating mirror highspeed camera and captured the crack tip caustics right from crack initiation. In a startling departure from the previous measurements, they found that while the dynamic stress

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Figure 10.23 Crack growth toughness in Araldite B. The two sets of data were obtained on the same material, but in different loading geometries. Upper curve corresponds to the rectangular doublecantilever beam or the rectangular double-cantilever beam with an enlarged end and the lower curve corresponds to a single-edge-notched specimen. (Reproduced from Kalthoff, 1983.)

intensity factor continued to vary subsequent to crack initiation, the crack speed remained constant within experimental accuracy. Their results are reproduced in Fig. 10.24. The open symbols in the ﬁgure identify the stress intensity factor at crack initiation and the horizontal lines indicate the range of KIdyn attained by the crack running at the same speed. At the highest load levels, they observed crack branching as indicated in Fig. 8.7. Also shown in Fig. 10.24 is the trend data from Dally (1979) for Homalite-100. From these experimental observations on the dynamic crack growth criterion, a few things are apparent. First, the driving force required to propagate a dynamic crack rapidly increases as the crack speed increases; however, this relationship may not be unique. Dependence of the K – v relationship on specimen geometry, crack tip acceleration and loading as well as initial conditions have been shown through many experiments. Second, a limiting crack speed that is roughly about half the Rayleigh surface wave speed has been observed in almost all experimental investigations. Third, as the driving force is increased, cracks typically split into two or more branches, with each propagating at a high speed along its own path. Ravi-Chandar and Knauss (1984b,d) provided a mechanistic explanation for the nonuniqueness in the K– v relationship, the appearance of the limiting crack speed and the appearance of crack branching based on a microcracking model of the fracture process; physical aspects of fracture process are discussed in Chapter 11. Notwithstanding the nonuniqueness addressed above, for practical purposes, one can introduce an ‘averaged’ K –v relationship, such as the one suggested by the dotted line in

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Figure 10.24 Crack growth toughness in Homalite-100. Note the nonunique response observed: while the stress intensity factor changes, the crack speed remains unaltered. The solid line (identiﬁed as Reference 38) is from Dally (1979). (Reproduced from Ravi-Chandar and Knauss, 1984c.)

Fig. 10.24 in order to evaluate the extent of crack growth in any particular experiment. While such a representation is a gross approximation, it is useful in obtaining an estimate of the extent of crack growth that might occur for use in conservative design practice. However, further work is needed in this area to determine the appropriate dynamic crack growth criterion for brittle materials. Some recent work on generating atomistic and lattice models as well as cohesive zone models of the process zone effects are described in Chapter 12. 10.3.2 Crack Growth Toughness in Ductile Materials There have been fewer investigations of the dynamic crack growth toughness in ductile materials, possibly due to the difﬁculties associated with diagnostic methods that must be used on these materials. Kobayashi and Dally (1980) investigated the K – v relationship in 4340 steel specimens; the specimens were heat treated for 1 h at 9008C, followed by an oil quench and tempering at 3708C. Wedge-loaded compact tension specimens similar in design to the CCA specimen were used. The crack tip state was determined with a

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Figure 10.25 Crack growth toughness in a 4340 steel. (Reproduced from Kobayashi and Dally, 1980.)

photoelastic coating placed on the steel specimen. The variation of the dynamic crack growth toughness with crack speed determined from three experiments is shown in Fig. 10.25. The general trends in the K – v relationship are somewhat similar to that observed in the nominally brittle materials. In the early stages, a very small change in KIdyn results in a large change in the crack speed. Apparently, the specimen marked 375 in this ﬁgure was heat treated differently and hence exhibited a signiﬁcantly lower initiation threshold and a lower limiting crack speed. Dahlberg et al. (1980) and Brickstad (1983) measured crack position with a high-speed camera or potential drop method, but calculated KIdyn through ﬁnite element analysis incorporating boundary measurements of loads. Dahlberg et al. (1980) examined 0.5 mm thick sheets cold-rolled and hardened carbon steel and observed signiﬁcantly larger experimental scatter than Kobayashi and Dally (1980). They concluded that the large scatter is due to large errors in crack speed measurements and that if the measurements were performed in experiments where KIdyn did not vary much, the scatter would be reduced signiﬁcantly. Brickstad (1983) made

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Figure 10.26 Crack growth toughness in a high-strength carbon steel. No obvious dependence on acceleration was noted. (Reproduced from Brickstad, 1983.)

Figure 10.27 Crack growth toughness in 4340 steel. (Reproduced from Rosakis et al., 1984.)

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measurements on a high-strength carbon steel. Single-edge-notched specimens were used; the crack position was monitored with a potential drop measurement (see Section 6.3). The boundary forces and displacements were measured and used as input to a ﬁnite element simulation in order to calculate KIdyn : Brickstad’s estimate of the K – v relationship is shown in Fig. 10.26. The general trend is similar to that shown by Kobayashi and Dally (1980). A large scatter in the data is observed; however, unlike the case of nominally brittle materials, results for accelerating and decelerating cracks are indistinguishable. Rosakis et al. (1984) evaluated the dynamic crack growth toughness of 4340 steel, heat treated to 8438C, followed by an oil quench and tempering at 3168C for 1 h. They used a DCB specimen and the method of caustics to measure the instantaneous KIdyn and crack position. Their results are shown in Fig. 10.27. A monotonic increase in KIdyn with crack speed v was observed. Thus, unlike the case of brittle materials, experimental characterization of the dynamic crack growth toughness, through both direct measurements and combinations of measurements and analysis have resulted in suggesting that a unique K –v relationship is indeed possible in metallic materials. The contrast in response between nominally brittle and ductile materials in the K –v relationship must be attributed to the deformation and failure mechanisms that occur in the fracture process zone. In ductile materials, dissipation associated with plasticity is a signiﬁcant fraction of the overall energy expended by the propagating crack and must be taken into account in dynamic fracture modeling. Lam and Freund (1985) performed a numerical simulation of a dynamically growing crack in a non hardening material, imposing a critical crack tip opening angle as the criterion for crack growth. From this simulation, they obtained a K –v relationship that was quite similar to the results exhibited in Fig. 11.27; clearly, the rapid increase in dynamic fracture toughness with crack speed is attributed to the work of plastic deformation. For brittle materials, on the other hand, one must look into the particular details of the fracture process. These are addressed in the next chapter.

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Chapter 11 Physical Aspects of Dynamic Fracture

In the previous chapters, analysis and experiments of dynamically loaded and propagating cracks have been discussed. Formulations of uniﬁed as well as ad hoc failure criteria have also been described. While the theory has been able to provide the time variation of the dynamic stress intensity factor as well as other ﬁeld quantities, the ad hoc failure criteria discussed in Chapter 10 are not fully quite adequate in capturing the dynamic response of nominally brittle materials. The reason for this is quite simple: continuum fracture mechanics theory has focused on analyzing the ‘outer problem’ of determining the stress, deformation and energy ﬂow in a region near the crack tip; this has been a fruitful exercise in that engineering structures at the large scale can be designed to be fracture-critical. However, in order to determine the fundamental failure characteristics, one needs to examine the ‘inner problem’ of understanding, characterizing and modeling the failure processes that actually lead to energy dissipation. The fact that the fracture processes have dynamics of their own—as seems to be the case with nominally brittle materials—leads to an interesting array of response of these materials. Careful experimental observations form the basis for developing an understanding of the fundamental physics and mechanics of the inner problem in dynamic fracture. From the pioneering experiments of Hopkinson (1901) and Schardin and Struth (1938), to the spurt of dynamic fracture activities of the 1970s and 1980s, there is a wealth of experimental observations of dynamic fracture phenomena at various scales. There are a number of crucial experimental observations and theoretical models and numerical simulations of the fracture process have to be constructed and performed with due attention to these observations. In this chapter, we shall be concerned primarily with the materials that are characterized as nominally brittle—inorganic glasses, ceramics, and organic polymers in their glassy state would be included in this class of materials; the elastodynamic theory works well for ductile materials that possess a unique KI – v relation and do not exhibit crack branching. Early dynamic fracture experiments were performed mostly in inorganic glasses (most of this work by Schardin and co-workers is summarized by Schardin (1959) and Kerhkof (1973)), but much of the recent experiments exploring dynamic fracture have been conducted in organic polymers (Kobayashi and Mall, 1978; Dally, 1979; Kalthoff, 1983; Ravi-Chandar and Knauss, 1984a – d). There have been very few dynamic fracture

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investigations in brittle crystalline materials (Bowden et al., 1967; Field, 1971; Cramer et al., 1997), probably due to difﬁculties associated with conducting the experiments. The concentration of effort on brittle organic glasses has been driven mainly by the possibility which these materials offer to measure crack tip stress or deformation ﬁelds through the standard techniques of experimental mechanics. Results of these experiments pertaining to the physics of dynamic fracture are summarized here to provide a mechanistic basis for understanding the fracture process.

11.1 Limiting Crack Speed The energy balance described in Eq. 5.8 can be rewritten by combining all of the velocity dependence into one term so that: gðv=CR ÞKI0 ðt; lðtÞ; 0Þ

v < 12 KI0 ðt; lðtÞ; 0Þ ¼ g CR

ð11:1Þ

where KI0 ðt; lðtÞ; 0Þ is the dynamic stress intensity factor at time t for a stationary crack of length lðtÞ: Regardless of the nature of the fracture energy g the continuum limit for the crack speed is the Rayleigh wave speed since the left-hand side of the equation, representing the dynamic energy release rate, would approach zero as this speed is approached. Long before the establishment of the above equation, a remarkable experimental observation was made by Schardin and Struth (1938): cracks set in motion by rapid impact loading quickly attained a maximum speed that was characteristic of the material, but signiﬁcantly smaller than the Rayleigh wave speed; they showed this conclusively by taking time-resolved photographs of cracks propagating in inorganic glasses with a Cranz – Schardin multiple spark camera. This result has since been reinforced through hundreds of measurements by numerous research groups working on different materials. A survey of measured limiting crack speeds for nominally brittle materials is shown in Table 11.1 for crack growth in noncrystalline materials and in Table 11.2 for cracks trapped on cleavage planes in crystalline materials or along weakened planes in amorphous materials. From these results, it is clear that cracks growing on cleavage planes of crystals grow at speeds that are close to the Rayleigh wave speed. The fracture energy g is associated primarily with the surface energy and this is expected to be independent of the speed of generation of the surface. Thus g is constant and the crack grows at the limit set by the continuum energy release rate. In contrast, for the noncrystalline materials, the crack has to ‘ﬁnd’ its way through the disordered material; the experimental conclusion is that the limiting crack speed is signiﬁcantly lower than the Rayleigh wave speed, lying in the range of 0:4 – 0:7CR : In fact, Schardin (1959) performed an illuminating investigation: he measured the limiting crack speed in 29 inorganic glasses obtained by systematically varying the composition. The limiting crack speeds measured in these glasses are shown in Fig. 11.1. This investigation showed that the limiting crack speed, while a constant for each material, was not the same fraction of the characteristic wave speeds for all the materials examined; the liming speed varied in the range from 0.347 to 0:614Cs : This suggests that the continuum formulation (outer problem) provides

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Physical Aspects of Dynamic Fracture Table 11.1 Limiting crack speeds for noncrystalline materials Author

n

v=Cd

v=Cs

v=C0

v=CR

Bowden et al. (1967) Edgerton and Bartow (1941) Schardin and Struth (1938) Anthony et al. (1970) Cotterell (1965) Paxson and Lucas (1973) Dulaney and Brace (1960) Fineberg et al. (1992) Beebe (1966) Kobayashi and Mall (1978) Dally (1979) Ravi-Chandar and Knauss (1984a– d) Hauch and Marder (1998)

0.22 0.22 0.22 0.22 0.35 0.35 0.35

0.27 0.26 0.28 0.36 0.26 0.28 0.28

0.42 0.43 0.47 0.6 0.54 0.58 0.58

0.29 0.28 0.30 0.39 0.33 0.36 0.36

0.31 0.345 0.31 0.31

0.16 0.17 0.22 0.24

0.31 0.35 0.35 0.41

0.19 0.22 0.24

0.51 0.47 0.52 0.66 0.58 0.62 0.62 0.58–0.62 0.33 0.37 0.38 0.45 0.37

Material Glass

Plexiglas

Homalite-100

no clues to identiﬁcation of the limiting speed and that one must look towards models of the fracture process (inner problem) in order to determine the velocity dependence of the fracture energy. Schardin suggested that the limiting crack speed be considered a new physical constant, perhaps related to other physical parameters that govern the fracture process. It is important to state the dilemma that faces us: in nominally brittle amorphous solids, and in the absence of any rate-dependent dissipative processes, g should be independent of the crack speed, v; why then does the crack not accelerate to the Rayleigh wave speed? Early suggestions that the speed was limited due to the onset of crack branching (discussed in Section 11.3) beg the question because branching was itself not well understood; Cotterell (1965) suggested that the maximum speed must depend on the material properties even though continuum analysis sets an upper limit of the Rayleigh wave speed. Based on an extensive investigation of the microscale aspects of fracture, Ravi-Chandar and Knauss (1984c) suggested that even in nominally brittle solids, fracture proceeds with a signiﬁcant process zone in which nucleation, growth and coalescence of microcracks occur; they suggested that the dynamics of evolution of these processes and the microscopic path instabilities that this triggers provide a rate and state-dependent character to the fracture energy; thus, g ¼ gðv; KI ðtÞÞ: The microstructural details of the fracture process are discussed in Section 11.2. More recently, Fineberg et al. (1991, 1992) observed that cracks growing faster than about 0:36CR in polymethylmethacrylate Table 11.2 Limiting crack speeds for crystalline materials and trapped cracks Material LiF MgO Silicon Weak bond in PMMA

Author Gilman et al. (1958) Field (1971) Cramer et al. (1997) Washabaugh and Knauss (1993)

n

v=Cs

v=CR 0.63 0.88

0.75 0.90

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Figure 11.1 Limiting crack speeds in inorganic glasses. (Data from Schardin, 1959.)

(PMMA) exhibited rapid oscillations in the crack velocity, triggered primarily by small microbranches that are issued off the main crack; they suggested that this dynamic crack path instability was the reason for the observed limiting speed. Strong evidence that the process zone was responsible for determining the limiting crack speed was provided by Washabaugh and Knauss (1994); they prepared a specimen by diffusion bonding two plates of PMMA, but controlling the bonding process to yield a bonding plane with very low toughness. Essentially, the fracture process zone was conﬁned to the width of the diffusion bond layer; cracks in this specimen were found to travel at about 0:90CR : Contrasting this with a limiting speed of 0:60CR in a completely bonded specimen, it is apparent that the fracture process dictates the limiting speed. Hull (1997a,b) has expanded on these explanations by suggesting that twisting and tilting of the crack front due to microscale variations in the symmetries results in a structure to the process zone in thermoset plastics that may not be able to nucleate independent microcracks. All these models of the crack growth process are supported through different experimental measurements and observations and we shall review them in Section 11.2. The resolution of our dilemma is, in fact, physically complete: while the continuum analysis dictates that the only way a crack can respond to additional inﬂux of energy is by accelerating until it eventually outruns the energy at the Rayleigh wave speed, the fact that a fracture process zone has structure and dynamics associated with its evolution— regardless of the particular model—provides for rate and state dependence for the fracture energy and presents other possibilities for explaining the lower observed limiting speeds. Models of the fracture process range from idealized atomistic and lattice models to phenomenological models to mechanism-based nucleation and growth models; while none

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of these models have advanced to the stage of providing quantitative explanation of the limiting speed or for the dependence of the fracture energy on the crack speed, there are qualitative features of the models that appear promising. We shall ﬁrst examine the overall development of crack surface roughness and the mechanisms of fracture in this chapter and then examine speciﬁc models of dynamic fracture in Chapter 12.

11.2 Fracture Surface Roughness The fast fracture surfaces in nominally brittle materials clearly indicate a varying surface roughness—this is evident in naked-eye observations of the fracture surface. However, this is not always the case; gem cutters would be out of business were it not for the fact that in crystalline materials, fast fracture can run along cleavage planes leaving— if not atomically ﬂat—at least optically ﬂat surfaces. In addition, experimental measurements indicate that a crack speed approaching the Rayleigh surface wave speed is observed in crystalline materials (see Bowden et al., 1967; Field, 1971). In this case, dynamic fracture develops in such a manner that the continuum theory of dynamic fracture is appropriate. Clearly, continuum wave propagation sets the limiting speed and the crack tip fracture processes develop in a very small spatial domain at much faster rates and are not the rate limiting processes in governing the crack speed. On the other hand, in nominally brittle glasses that have been the focus of much of dynamic fracture investigations, the fracture processes that occur over a spatial domain comparable to the surface roughness dominate the dynamics of crack growth. It is the rate of development of these fracture processes that appears to set the limiting speed rather than the rate of delivery of elastic energy through stress waves. Typical characterization of the fracture surface roughness is through its appearance to the unaided eye under normal lighting conditions; the fracture surface of the specimen from the experiment described in Fig. 8.7 is shown in Fig. 11.2. Three distinct zones are usually identiﬁed on the fracture surface and labeled as ‘mirror’, ‘mist’, and ‘hackle’. pﬃ In fact, in the literature on fractography of glass, the so-called mirror constant M ¼ sf l is often deﬁned where sf is the macroscopic stress at fracture and l is the length of the mirror zone; clearly this can be translated in terms of a stress intensity factor criterion. However, these demarcations are really an artifact introduced by the observing tool; in the mirror region the scale of roughness is small compared to the wavelength of light and as a result the fracture surface reﬂects light specularly appearing mirror-like. RaviChandar and Knauss (1984b) measured the surface roughness in a Homalite-100 fracture surface (also shown in Fig. 11.2) and found that the roughness varied almost continuously along the crack path and that there was nothing special about the mirror– mist – hackle boundaries other than that these can be discriminated based on a visual observation. A similar variation in roughness was found by Hull (1997a,b) in Araldite, a thermosetting epoxy. Furthermore, the crack surface roughness was found to be independent of the crack speed—the speed was constant over the segment of the fracture surface shown—and depended on the stress intensity factor. Thus, the mirror-like surface appearance deﬁnitely does not indicate that the fracture surface in this region is smooth; only that the roughness has not been resolved in the observation. Higher resolution

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Figure 11.2 Fracture surface in Homalite-100. Crack speed was 0:38CR : Typical ‘mirror’, ‘mist’ and ‘hackle’ regions are identiﬁed. The graph indicates the maximum depth of the fracture surface as a function of crack position and also the stress intensity factor. (From Ravi-Chandar and Knauss, 1984b.)

measurements with an atomic force microscope yield morphological details of the surface roughness at this scale as well. The topological as well as morphological features on the fracture surface present evidence of the events that occurred during crack propagation. Many different interpretations of these features have been presented in the literature; Ravi-Chandar and Knauss (1984b) and Ravi-Chandar and Yang (1997) present a picture that is based on nucleation, growth and coalescence of microcracks in the fracture process zone. Fineberg and Marder (1999) consider a dynamical instability that is triggered at a critical crack speed to contribute to the development of surface roughness. Hull (1997a,b) also suggested that local microbranching and tilting of microbranches results in surface roughening. It should be noted that these interpretations are not mutually exclusive, but may in fact be different interpretations of the same or similar events. Mandelbrot et al. (1984) showed that fracture surfaces have a fractal character. Bouchaud (1997) has studied the fracture surfaces of different materials and suggested that the roughness scales in a self-afﬁne manner; thus the fracture surface structures are

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invariant under an afﬁne transformation {x1 ; x2 ; x3 } ! {bx1 ; bx2 ; bz x3 }

ð11:2Þ

where b is a constant and z is called the roughness index or Hurst exponent. This afﬁne scaling implies that the height at any point is given by the following: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃz hðx1 ; x2 Þ , x21 þ x22 ð11:3Þ Bouchaud showed that under both quasi-static and dynamic crack growth conditions the exponent z was around 0.8 whenever the scale of the measurement of roughness was greater than a material-dependent length jc : This exponent appears to be universal in the sense that it is independent of material, and the conditions that generated the fracture surface. For roughness at a scale below jc ; the roughness index was about 0.5. jc is identiﬁed as a cross-over length scale at which mechanisms of roughness generation changes; it has been suggested that this crossover length scale depends on the largest heterogeneity in the material. More work remains to be done in this area to relate the roughness index to fracture energy and processes as well as to evaluate the inﬂuence of crack speed. 11.2.1 Real-Time Observations of Multiple Crack Fronts In an attempt to obtain real-time observations of dynamic crack growth mechanisms, Ravi-Chandar and Knauss (1984b) used the repeatability of the electromagnetic loading scheme to generate high-speed photomicrographs of the crack front. These photographs were obtained in an experiment with Homalite-100 specimens repeated three times with the same loading, but with the center of the camera’s ﬁeld of view located in the mirror, mist and hackle regions, respectively, in each of the experiments. In the mirror zone, from Fig. 11.3a, one notes that the crack front exhibits a thumb-nail shape reminiscent of quasi-static crack propagation; this crack front curvature is due to a change from a plane strain to plane stress constraint. The crack front in the middle of the specimen leads the crack front at the plate ends by about 0.5 mm and there is a smooth variation in the crack front between the middle and face of the plate. At the faces of the plate the crack front forms a caustic and this leads to the crack front appearing to be wider than the plate width (see Section 8.3 for a discussion of caustics and Ravi-Chandar and Knauss (1984b) for a discussion of this particular experimental

Figure 11.3 High-speed photomicrographs of a moving crack front in Homalite-100. Crack front is located in the (a) ‘mirror’ zone; (b) ‘mist’ zone and (c) ‘hackle’ zone. Photographs were obtained with the crack front observed obliquely. (Reproduced from Ravi-Chandar and Knauss, 1984b.)

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arrangement). Fig. 11.3b shows the crack front in the mist zone and it appears distinctly different from that in the mirror zone. Again, caustics are formed at the faces of the plate and these are larger because of the higher stress intensity factors associated with the mist zone. The most striking difference is that the ‘crack front’ is nearly straight but exhibits a number of small caustics. This indicates that there is no longer a single crack front, but an ensemble of cracks propagating along the apparent crack front and generating multiple caustics. Note that this image may also be interpreted as a single crack front splitting into multiple fronts as suggested by Hull (1997a,b), and not necessarily into multiple microcracks. The image in Fig. 11.3c shows the appearance of the crack front in the hackle zone which seems to be similar to that in the mist zone but represents the coarser structure of the fracture surface in the process of forming. This might have been expected in view of the fact that post-mortem examinations of the mist and hackle surfaces exhibit similarities. However, the caustics appear larger indicating still higher loading; furthermore, along the crack front fewer but larger caustics are observed. Microbranches are also visible in these photographs; clearly these frustrated branches do not develop through the entire thickness of the specimen. From these photographs and from post-mortem examination of the corresponding fracture surfaces, Ravi-Chandar and Knauss (1984b) suggested the following picture of dynamic crack growth: initially, in the mirror zone a ‘single crack’ propagates with a curved crack front similar to that observed in quasi-static crack propagation. In the mist zone several small cracks propagate simultaneously and the ensemble crack front is nearly straight. Further crack propagation is really governed by the details of the interaction between these microcracks. In the hackle zone crack growth occurs by the same physical process, except that the size scale of the microfracturing increases. Real-time photomicrographs of a crack branching event were also captured; these are described in Section 11.3. 11.2.2 Fast Fracture Surfaces in Polymethylmethacrylate A number of investigators have reported observations of parabolic markings (Smekal, 1953; Irwin and Kies, 1952; Leeuwerik, 1962) and a periodic morphology in the dynamic fracture surface of PMMA (Green and Pratt, 1974; Do¨ll, 1976a,b; Fineberg et al., 1991; Washabaugh and Knauss, 1993). Fineberg et al. (1991) suggest that the periodicity appears at a critical velocity and that the surface roughness is well correlated with measured velocity ﬂuctuations, while Washabaugh and Knauss (1993) indicate that the periodicity correlates well with the stress intensity factor. Fineberg et al. (1991) also suggest that this periodic phenomenon is responsible for limiting crack speeds to about 50% of the theoretical limit of the Rayleigh surface wave speed. Here, we ﬁrst describe the experimental observations of Fineberg et al. and then follow this with fractographical examinations of Ravi-Chandar and Yang (1997). Fineberg et al. (1991) performed dynamic fracture experiments in a narrow strip conﬁguration shown in Fig. 11.4a. Typical specimen dimensions were 100– 200 mm wide and 200 –250 mm tall, with thickness of either 1.6 or 3.2 mm. The initial crack length was only about 3 –4 mm. Load was applied by moving the top grip in steps of 1024 of the height and holding for 10 –20 s in order to allow crack initiation with very

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Figure 11.4 (a) Strip specimen geometry; length ,250 mm, width ,200 mm, initial crack length , 3– 4 mm. (b) Crack speed measured with the potential drop technique. The lower curve shows the overall time history while the upper curves show details near the initiation time and the time at onset of dynamic instabilities. (c) Fracture surface of the PMMA specimen in the region of crack speed oscillations. x – z is the nominal plane of the fracture surface; signiﬁcant roughness and periodic bands can be seen. (Reproduced from Fineberg and Marder, 1999.)

little time dependence in the loading. In this conﬁguration, the energy release rate remains constant at a value that can be determined from the load measured at the onset of fracture propagation. Fineberg et al. used a potential drop method to determine the crack position to an accuracy of about 0.2 mm and crack speed to an accuracy of about ^ 5 m/s. A typical measurement of the crack speed as a function of time from one of their experiments is shown in Fig. 11.4b. In these experiments they found that the crack speed typically jumped to a value of about 200 m/s within 1 ms. Then, the crack accelerated smoothly until reaching a critical speed vc ; beyond vc large oscillations in the crack speed were observed. Through observations of the fracture surface features, these crack speed oscillations were associated with a microbranching instability; the structure of the fracture surface is shown in Fig. 11.4c. Fineberg et al. (1991) have suggested that this microbranching instability is responsible for large energy dissipation (through the creation of extra surface area in the microbranches). In fact, through careful measurements of the proﬁle and the density of the microbranches, Sharon and Fineberg (1996) have correlated the increase in area generated by the microbranches to the increase in fracture energy. Questions regarding the origin of the crack path instability (microbranches) and the periodic band formation remain to be answered: What are the micromechanisms governing the fast fracture of PMMA? What triggers the periodicity in the fracture surface morphology? We discuss the fracture mechanisms next.

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Figure 11.5 Fracture surface in a PMMA specimen. The density of conic markings increases from about 350 mm2 in (a) to about 2600 mm2 in (c) and (d). The periodic bands seen in Fig. 11.4c can be seen in (d) at a larger magniﬁcation. (Reproduced from Ravi-Chandar and Yang, 1997.)

An examination of the fracture surface of PMMA is shown in Fig. 11.5. The surface is tiled with a pattern of conic1 marks (Fig. 11.5a); these markings are well known in the fracture of amorphous materials. They are usually interpreted as being level differences resulting from an encounter between a microcrack and a main crack. Along the crack path, the number and density of the conic marks increase (see Fig. 11.5b – d). In addition, their eccentricity appears to change perhaps indicating a difference in the velocity of approach between the microcrack and the main crack(s) or equivalently the ratio of the nucleation distance and the nuclei spacing. At higher load levels, a version of the periodic morphology observed by Fineberg and Marder (1999) appears to emerge from this state of increasing density of conic marks; this is clearly observable in Fig. 11.5d. The spacing, size and depth of these periodic bands also evolve along the crack path—the period of these bands increases from about 300 mm to about 500 mm; in fact, the periodic bands begin independently at several different sites across the plate thickness as very small spots and then gradually grow in width as the crack propagates, indicating signiﬁcant threedimensionality in its evolution. Washabaugh and Knauss (1993) observed that coincident with the appearance of the periodic surface roughness, stress waves are visible in high-speed photographs; they associated these periodic bands with intermittent growth of the crack. 1

In the literature, these are referred to as parabolic markings. In general they are conic markings; some are parabolic while most are hyperbolic. Some elliptic marks have also been identiﬁed. We will refer to these collectively as conic markings.

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Figure 11.6 Fracture surface in a PMMA specimen. (a) The region just ahead of periodic bands seen in Fig. 11.5d; note that the conic marks are terminated abruptly. (b) A close-up view of the rough regions inside the periodic bands showing multiple microcracking. ‘A’ is the ﬁrst microcrack that was nucleated by the ‘main crack’; other microcracks were nucleated by this microcrack as it grew. This is evident from the fact that all the conic marks appear to focus at the point A. (Reproduced from Ravi-Chandar and Yang, 1997.)

The fracture surface shown in Fig. 11.5 indicates that the periodicity of the fracture surface is not really due to a change in the mechanism of crack growth: dynamic crack growth is always governed by microcrack growth and coalescence. The ﬂat surface tiled with conic marks of different sizes is generated by a continuous (perhaps) sequential nucleation, growth and coalescence of microcracks whereas in the periodic bands a rather rough surface is created because microcracks or microcrack clusters formed far ahead of the main ensemble crack coalesce with it, with surface asperities in the range 20– 30 mm. In fact, the conic marks in the ﬂat region before the band end abruptly and give way to the rough surface as can be seen from the high-magniﬁcation photograph shown in Fig. 11.6a. In Fig. 11.6b, the surface of one of the periodic bands is shown: that this is at a different plane from the ‘main’ crack plane is obvious from the defocusing of other areas in the optical micrograph. Clearly, the surface of these bands is also tiled with conic marks. Notice that the ﬂaw identiﬁed by A in this ﬁgure nucleated ﬁrst by the ‘main crack’ and its growth has led to the nucleation of the other microcracks in this surface. This cluster of microcracks develops in the enhanced stress ﬁeld of the main crack, but is quite independent of the main crack. The two cracks merge by breaking the ligament if the conditions are favorable. This process of forming off-axis microcrack clusters and coalescing them with the main crack could be repeated leading to the periodic banding; on the other hand, if the load levels are sufﬁciently high, the off-axis microcrack clusters could outrun the main crack and become fullﬂedged crack branches. While the above discussion gives compelling evidence of brittle microcrack-based fracture mechanism that results in the periodic morphology and crack branching, it is insufﬁcient to identify the period; this will have to wait until the appropriate crack growth equations for the microcrack model of crack propagation are identiﬁed.

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11.2.3 Origin of the Microcracks Smekal (1953) postulated a very simple mechanism for the formation of the conic markings on the fracture surface. His model is simply that in the enhanced stress ﬁeld of a primary crack, inhomogeneities triggered the initiation of a secondary crack ahead of the primary crack front; the secondary fracture may not be in the same plane as the primary front and when these two fronts intersect in space and time, the ligament separating the two cracks breaks up leaving a conic marking on the fracture surface. The conic marking thus indicates a level difference boundary, marking the common space –time interaction of the two fracture fronts with the focus of the conic identifying the origin of the secondary fracture front. Given that such conic markings are observed in a wide range of materials with very different microstructures, the microcrack interaction model appears to be the most appropriate characterization. There is, however, some disagreement over the origin of the secondary microcracks: what is the nucleus of the microcracks? Irwin and Kies (1952) pointed to small cavities or inhomogeneities at the focus of the conic markings in cellulose acetate and steels. Leeuwerik (1962) observed a bowl shaped cavity with a diameter of 0.3 mm at the foci of the conic markings. All these authors speculated that these are voids (ﬂaws) that are naturally randomly distributed throughout the material. Cotterell (1968) suggested that perhaps these voids are formed inside the crack tip craze region. Matsushinge et al. (1984) examined the formation of the secondary cracks using acoustic emission and post-mortem X-ray microanalysis. While contamination in the form of Si was found in a small percentage of the markings that were examined, they could not conclude that such foreign objects were the nuclei for all the secondary fractures. It is quite possible that randomly distributed ﬂaws acting as the origin for the microcracks is the appropriate model; it is also possible that voids are nucleated in the high-triaxial tensile ﬁeld near the crack tip. A simple estimate of the stress ﬁeld in which the microcracks nucleate may be obtained by considering the K-dominant ﬁeld near the crack tip. For the specimen shown in Fig. 11.4, the dynamic stress intensity factor at crack p p initiation was 1.03 MPa m and gradually increased to about 1.2 MPa m. The yield tress (determined from quasi-static experiments) for PMMA is approximately 100 MPa and is likely to be higher at the high strain rates encountered at the dynamically growing crack tip. In an effort to estimate the size of the plastic zone, let us assume that PMMA obeys a power law hardening behavior of the type 1 ¼ sn ; with n ¼ 3: The size of the plastic zone2 under plane strain conditions that are appropriate to the interior of the specimen is then estimated to vary from about 2.8 to 3.8 mm as the stress intensity factor increases from p 1.03 to 1.2 MPa m. Note that the plastic zone is very small, and in particular smaller than the average nucleation distance (see Section 11.2.5 for the deﬁnition of nucleation distance); thus conditions of small-scale yielding seem to be appropriate. Assuming then that the linear elastodynamic stress ﬁeld estimates are appropriate at these distances, the stress-state at the nucleation of the ﬂaws can be determined. The triaxial stress state at the p nucleation distance of 5.5 mm, corresponding to a stress intensity factor of 1.03 MPa m 2

This is a quasi-static estimate; under dynamic conditions, since the yield stress is likely to be higher, and also due to the shortening of the zone caused by inertia effects, the plastic zone will typically be smaller than this estimate. Hence we have an upper bound estimate of the plastic zone size.

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and a crack velocity of about 0:25CR can then be calculated as: s11 ¼ 189 MPa, s22 ¼ 175 MPa and s33 ¼ 125 MPa; similarly at p the nucleation distance of 8.5 mm, corresponding to a stress intensity factor of 1.2 MPa m, and a velocity of about 0:25CR ; the triaxial stress state is given by: s11 ¼ 172 MPa, s22 ¼ 164 MPa and s33 ¼ 118 MPa; note that these stresses are much higher than typical values quoted for nucleation of crazing in PMMA under uniaxial tension (Argon and Salama, 1977), which is in the range of 60 –100 MPa. It appears that at such high-triaxial tensile stress levels, extensive cavitation occurs and that these cavities are the nuclei for further development of the dynamic fracture process; in this sense of cavitation, these are stress-induced or stressactivated ﬂaws or nuclei and hence the increase in the number density of microcracks with the crack tip stress level. Of course, these cavities will develop preferentially at sites where inhomogeneities in the nanometer scale exist and facilitate the initiation of the cavities and hence the randomness in the spatial distribution of ﬂaws on the surface.

11.2.4 Geometry of the Conic Markings By applying Smekal’s model to the conic markings in the micrographs, we can obtain information concerning the size, shape (eccentricity), and critical distance of the secondary microcracks. The interaction of straight fronted main crack with a radially growing microcrack and the further interaction of this microcrack with another microcrack is depicted in Fig. 11.7a. Consider a planar crack front approaching a microcrack nucleus with a velocity vc ; when the distance between the crack front and the nucleus is dn ; the microcrack nucleates (i.e. it begins to grow) and grows radially symmetrically at a velocity vc1 ; dn is the critical nucleation distance and is presumably dictated by the inherent characteristics of the material and the stress ﬁeld. A second nucleus is at a spacing s from the ﬁrst nucleus; the spacing s will in general follow some statistical distribution, perhaps dictated by the local stress ﬁeld and the material microstructure; when the distance between the growing front of the ﬁrst microcrack and the second nucleus is equal to dn — the nucleation distance—the second nucleus begins to grow radially symmetrically with a velocity vc2 : The microcrack nuclei are usually not on the same plane as the main crack, but offset by about a few microns; thus the interaction between the main crack and the microcracks leaves a trail of their common space – time interaction. If the spacing, s; the nucleation distance, dn ; and the velocities, vc ; vc1 ; and vc2 ; are known, the resulting conic marking on the fracture surface may be calculated as indicated by the solid line in Fig. 11.7a. Such an interaction of a planar crack front with two microcrack nuclei is observed in experiments as shown in Fig. 11.7b. Alternatively, from the micrographs, one might extract details regarding s; dn ; vc ; vc1 ; and vc2 : For simplicity, assume that the velocities of the main crack and microcracks are identical and equal to v: Then the equation describing conic marking on the fracture surface is given by ð2x1 þ s 2 dn Þ2 4x22 2 ¼1 2 ð2s 2 dn Þdn ðs 2 dn Þ

ð11:4Þ

By comparing the measured conic with Eq. 11.4, one can either verify that the microcracks grow with the same speed or determine the appropriate speed ratio. Note that assuming

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Figure 11.7 (a) Geometry of formation of conic markings on the fracture surface due to nucleation and growth of microcracks. (b) Experimentally observed conic marks on the fracture surface of PMMA. (c) AFM image of the topography of a conic mark. (d) Topography of the conic mark along a radial line. ((a) and (b) reproduced from Ravi-Chandar and Yang, 1997.)

the velocities of the two microcracks to be equal leads to a shape of the conic that is independent of the velocity; Kies et al. (1950) demonstrated this very nicely by performing a displacement-controlled experiment where the crack propagated only a very short distance at each displacement increment. However, the growth was dynamic, and secondary cracks were nucleated ahead of the main crack; conic marks were observed on the fracture surface, decorated with the arrest lines corresponding to each displacement increment. Thus it appears reasonable to assume that the microcracks grow at the same speed; the value of this speed must still be controlled by the applied stress ﬁeld and the local stress state in the fracture process zone. The variation of height in a conic marking is shown in Fig. 11.7c; a trace along one radial line of the conic marking is shown in Fig. 11.7d. Clear level differences across the conic marking of the order of a micron can be observed. Note the contrast between this level difference and the difference between the different planes when the periodic features appear on the fracture surface. Also, the boundary between the two microcracks is seen to have been stretched during the cracking process.

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11.2.5 Statistics of Microcracks in PMMA The contours of a large number of conic markings were digitized and then ﬁt with a general second-order equation to determine both the eccentricity and the distance to the focus. The eccentricity values were close to unity in region corresponding to Fig. 11.5a but were slightly higher in subsequent regions (Fig. 11.5b –d); assuming that the velocities of the microcracks are all equal, the secondary microcracks were nucleated by nearly planar crack fronts in the early stages of crack growth, but in later stages, since the spacing between the microcrack nuclei is small, nucleation of microcracks is triggered by the curved crack fronts of other microcracks. This is supported by the fact that the average spacing between nuclei decreases from about 53 mm in Fig. 11.5a to 33 mm in Fig. 11.5b and 20 mm in Fig. 11.5c. The nucleation distance can be obtained from measurements of the distance to the foci; the average focal distance increases from about 2.75 mm in Fig. 11.5a to 3.75 mm in Fig. 11.5b to 4.25 mm in Fig. 11.5c; this indicates that the nucleation distance dn increases from about 5.5 mm in Fig. 11.5a to 7.5 mm in Fig. 11.5b to 8.5 mm in Fig. 11.5c. It is clear that the conic markings on the fracture surface indicate an increase in the number of nuclei activated into growing along the crack path and an increase in the nucleation distance at which the secondary microcracks begin to grow. Since the average crack speed is constant in these regions, the increasing stress intensity factor (actually the increasing near tip stress ﬁeld) is the driving force in generating this large density of microcracks. Thus, as the stress intensity factor increases, more ﬂaws are nucleated and they are nucleated farther ahead of the main crack; this appears to be the primary mechanism of crack growth in PMMA. Note that this is the case regardless of whether one observes steady-state crack growth, periodic banding and crack branching. A number of investigators have examined the population of the microcracks and attempted to correlate them to the velocity and/or the stress intensity factor or fracture toughness (Cotterell, 1968; Carlsson et al., 1973; Matsushinge et al., 1984). From Fig. 11.4, we can infer that while the average crack velocity is nearly constant, and the stress intensity factor increases slowly, the density of conic markings increase dramatically; thus, the density of conic marking should be correlated to the stress or the stress intensity factor rather than the velocity. We note that the density of conic markings observed by the investigators listed above is on the order of 1 –50 per mm2; in contrast, the density of conic markings in Fig. 11.5 is 1– 2 orders of magnitude higher! This could not only be due to differences in the molecular weight of the PMMA, but also perhaps due to differences in the stress level applied and the transient nature of the applied loading. To interpret the conic markings in Fig. 11.5a – c quantitatively, the number of conic markings was counted and the density was determined. This was achieved by breaking up each micrograph into six regions and counting the number density of conic markings in each region. The results can be summarized as follows: ﬁrst, the density of conic markings in Fig. 11.5a is about 350 per mm2, an order of magnitude larger than in the observations in the references cited above. Secondly, there is an increase in the density of conic markings from Fig. 11.5b (900 per mm2) to Fig. 11.5c (1500 – 2600 per mm2); along the crack path the stress intensity factor increases at nearly constant average crack speed, suggesting that the stress-induced nucleation of secondary microcracks is the appropriate mechanism. Finally, in Fig. 11.5c, a rapid increase in density of conic markings is experienced just

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prior to the onset of the periodic banding morphology, indicating that a large density of ﬂaws activated into nucleation might be the trigger for the periodic banding. 11.2.6 Growth of Microcracks The growth rate of the microcracks cannot be easily characterized from post-mortem examinations of the fracture surface. While a description of the geometry of the conic markings can be provided if the spacing between nuclei, the nucleation distance and the respective velocities of the microcracks are given, the inverse problem is not unique: given the spacing, nucleation distance and the form of the conic marking, it is not possible to determine the velocity uniquely, especially if there is signiﬁcant variation. The simplest approach is to assume that the microcracks grow with a constant velocity, but this may not be appropriate when the microcrack density becomes large. Also, we have interpreted fracture surface markings as being the result of a sequential nucleation and growth of ﬂaws; it is conceivable that many nuclei ahead of the crack may be activated simultaneously and this possibility needs to be investigated further. Indeed, as remarked earlier, the nucleation, growth and coalescence of microcrack clusters away from the main crack tip is the primary cause for the rough crack surface, periodic banding and crack branching. Given the criterion for nucleation and growth of microcracks, the microcrackdominated dynamic crack growth model that is described above can be simulated numerically to reveal the features observed in experiments. A simple implementation of this model was descried by Ravi-Chandar and Yang (1997). A damage model that mimics the loss of stiffness of microcracked materials was developed by Johnson (1992) and more recently by Klein and Gao (2001). These models, discussed in Chapter 12, appear to do well in predicting the experimental observations. 11.2.7 Solithane 113 Solithane 113 is a polyurethane elastomer manufactured by Thiokol Corporation. The glass transition temperature is around 2 208C and hence the tests on Solithane 113 were performed at 2 908C, well into the glassy regime; thus, this material also exhibits a brittle fracture response. The crack speed in these experiments was around 400 m/s. Precise measurements of the Rayleigh wave speed in this material at 2 908C are not available, but one might expect that the crack speed is roughly the same fraction of the Rayleigh wave speed as in other materials. Fig. 11.8 shows the fracture surface in Solithane. The similarity to PMMA, in spite of the microstructural differences, in all aspects of the fracture surface morphology, is complete and striking. This similarity suggests the possibility that a microcrack-based model might be capable of capturing the behavior of many different brittle materials. 11.2.8 Polycarbonate Polycarbonate (PC) is a noncrosslinked thermoplastic polymer; it is capable of signiﬁcant inelastic deformation due to the relatively high mobility of the carbonate segments of its structure. However, at high rates of loading, it does exhibit brittle dynamic

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Figure 11.8 Fracture surface in the elastomer Solithane 113 indicating tiling with conic marks. (Reproduced from Ravi-Chandar and Yang, 1997.)

fracture and this is the regime considered here. The fracture surface from a polycarbonate specimen, loaded by a projectile impact as in the experiments of Taudou et al. (1992) is shown in Fig. 11.9. The corresponding crack velocity and stress intensity factor were again monitored by high-speed photography; the crack propagated at about 480 m/s which is roughly 50% of the Rayleigh wave speed, similar to the observed limiting speed in other materials. The stress intensity factor, measured using the method of caustics, increased p only slightly from the value at initiation (KIC , 4:9 MPa m). The fracture surface exhibits changes in its morphology that are very different from those seen in PMMA. In Fig. 11.9a, during the early stages of crack growth, the surface is remarkably uniform; conic markings typical of PMMA and Solithane 113 are not seen at all. Further along the crack path, a banded morphology develops as shown in Fig. 11.9b, but the bands are very shallow, and their period is only about 60 mm; these appear to be at about the same scale as the conic marks in PMMA, but apparently are not generated by microcracking. Doyle (1983) observed a similar banded structure in polystyrene (PS) at 808C with roughly the same morphology and spacing; he interpreted these as being formed by a dominant craze at the crack tip. As the crack extends, the banded morphology is interspersed with another periodic structure with a period of roughly 300 mm and a larger surface roughness, as shown in Fig. 11.9c. The roughness of the fracture surface along the crack path and the size scale of the periodicity appears to increase. Although not clearly visible in Fig. 11.9c, contained within each 300 mm band are 60 mm bands similar to the ones observed in Fig. 11.9b. Remarkably, the period of this large periodic surface feature is about the same as the periodicity in PMMA and Solithane 113; Doyle (1983) also observed this larger periodic banding with a spacing again in the range of 300 mm in polystyrene at room temperature, and attributed it to multiple crazes being generated at the crack tip due to rapid stress build up under the dynamic loading. The larger periodic surface features in Fig. 11.9c are clearly a manifestation of level changes in the fracture surface; even though clear evidence of microcracking (through the conic markings) is not seen on the fracture

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Figure 11.9 Fracture surface in polycarbonate showing the absence of conic marks. (a) Highmagniﬁcation image in the low stress intensity region. (b) Periodic striations with a spacing of about 60 mm. (c) Periodic bands with a spacing of 300 mm. (d) Variation in the fracture surface across the specimen width. (Reproduced from Ravi Chandar and Yang, 1997.)

surface of PC, these level differences can be interpreted as being caused by the progression of fracture caused at different levels by microcracks. If a craze is initiated ﬁrst and then forms a microcrack as it breaks down, one might not observe the characteristic conic marks. Thus, one might consider that in the early stages of the crack growth process, a single craze is formed across the entire plate thickness and grows by advancing both the craze and crack tips; hence the absence of any remarkable features in Fig. 11.9a. As the stress level continues to rise, the crack tip catches up with the craze tip and a sequence of new craze development and breakdown follows leading to the formation of the 60 mm banded surface feature. As this process continues under increasing stress levels, the single craze spanning the entire width of the specimen is not stable and breaks up into many independent craze cracks along the width of the specimen. These are the level differences observed in the 300 mm bands. This model might provide a link between the fracture mechanisms in PMMA, PC and PS. We have observed under different loading conditions periodic bands similar in structure to those in Fig. 11.9b, but with periods ranging from 4 to 60 mm; these bands are still under investigation.

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There appears to be a signiﬁcant three-dimensional inﬂuence on the nucleation and growth of these features. Near the plate surfaces, the fracture surface always displays a boundary layer where a transition is observed from a smooth surface to a 60 mm periodic striation and to the 300 mm periodic feature as shown in Fig. 11.9d. This threedimensional nature of the surface variation is a clear demonstration that the crack speed is not the driving factor in determining the periodicity since the speed variations along the crack front across the width are not expected to be signiﬁcant; the real-time photographs of the crack fronts shown in Fig. 11.3 support this claim. However, stress variations are quite signiﬁcant across the thickness due to changes in the constraint in the thickness direction from a plane strain state in the interior to a plane stress state in the boundary layer near the surface of the plate. A similar boundary layer is also observed in PMMA, where the periodic banding caused by the microcrack clusters does not develop in a thin layer near the plate surfaces and in other materials. 11.2.9 Homalite-100 Homalite-100 is a thermoset polyester that has been used in dynamic fracture studies by many investigators because of its nominally brittle behavior. The evolution of the fracture surface in this material has been considered in detail by Ravi-Chandar and Knauss (1984b). The maximum observed crack speeds in this material are again around 50% of the Rayleigh wave speed. The fracture surface morphology appears to be quite different from that in PMMA, Solithane and polycarbonate; individual microcracks identiﬁable by their traces on the fracture surface in the form of conic marks are only rarely seen. Also, Homalite-100 does not exhibit tractable periodicity on the fracture surface in any velocity range up to the limiting speed. The surface roughness, however, appears to evolve rather continuously along the crack path; Ravi-Chandar and Knauss (1984b,d) proposed that this was due to the evolution of the microcrack-dominated process zone. In recent experiments in the strip conﬁguration, Hauch and Marder (1998) observed crack surface roughening similar to that shown in Fig. 11.2; in addition to the parabolic marks, attempted microbranches, they showed a periodic microbranching phenomenon similar to that shown in Fig. 11.4. 11.2.10 Other Brittle Materials The fast fracture surface morphology in many other brittle materials exhibit similarities to the features presented here. Among amorphous materials, conic markings were observed in silicate glasses by Smekal (1953), in polystyrene by Regel (1951), and in cellulose acetate by Kies et al. (1950). Leeuwerik (1962) observed conic markings in spherulitic nylon, a semicrystalline polymer. Irwin and Kies (1952) report conic markings in polycrystalline materials such as steel, copper molybdenum and lead – tin alloys and in a large (single?) crystal of potassium bromide. Thus, it appears that the microcrack-based crack growth mechanism might be appropriate for a whole class of brittle materials. The origin, nucleation and growth mechanisms and kinetics, in each material could be quite different, but one might expect a common framework in their modeling.

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Hull (1997a,b) provided a different interpretation of the evolution of the fracture surface roughness. He discounted the possibility that microcracking can occur in brittle thermosetting polymers suggesting that the ﬂaws required to nucleate such microcracks must be very small. Instead, Hull proposed that roughening is generated by tilting of the crack front out of the plane of the main crack. The tilting was associated with nucleation of steps on the fracture surface and the associated mixed-modes I and III generated at the crack tip. This model of crack tilting is used to generate multiple crack fronts, not multiple microcracks, with each crack front developing along different planes and resulting in roughness. The level differences between the different crack fronts was shown to increase with increasing stress level and not correlated to the crack speed. While there are speciﬁc differences between this model and the microcrack model discussed above, the similarity lies in the use of multiple crack fronts or cracks associated with the break-up of a single propagating crack as the source of roughening.

11.3 Crack Branching Branching of cracks in glass was recorded by Schardin (1959); other investigators have observed crack branching in crystalline as well as amorphous materials. Spectacular patterns are formed when dynamically growing cracks break up into multiple cracks; an example obtained in an experiment using the electromagnetic loading is shown in Fig. 11.10. Dally (1979) observed multiple branches emanating from a single crack in an explosively loaded crack in Homalite-100. Kobayashi et al. (1973) examined branching in quasi-statically loaded specimens; cracks were driven into an increasing stress ﬁeld and branching was promoted. From isochromatic fringe patterns observed near the propagating and branching cracks, these authors determined the stress intensity factor and crack speed histories. They concluded that the critical stress intensity factor at

Figure 11.10 Branching pattern in a Homalite-100 specimen.

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branching was between 2.4 and 3 times the initiation toughness. Congleton and Petch (1967) attempted to estimate the stress intensity factor at branching by considering a small Grifﬁth crack—a microcrack in the crack tip process zone—placed ahead of the main crack in order to explain crack branching; this appears to be the ﬁrst attempt to tie the fracture evolution in terms of microscale fracture processes. However, this phenomenon has eluded analytical description, even though the search for an explanation to branching triggered early interest in dynamic fracture analysis. Yoffe (1951) attempted to explain the branching of cracks from an analysis of the problem of a crack of constant length that translates with a constant velocity in an unbounded medium. From this solution she found that the maximum of the hoop stress acted normal to lines that make an angle of 608 with the direction of crack propagation when the crack speed exceeded 0:60Cs ; the angular variation of suu is shown in Fig. 3.6. Therefore, Yoffe suggested that this stress ﬁeld rearrangement might lead to crack branching. However, as was reviewed in Section 11.1, cracks seldom reach this speed, but branch nevertheless; so the macroscopic stress ﬁeld rearrangement, is attractive as it is because of its simplicity, is not likely to explain macroscopic crack branching. Eshelby (1970) argued that since at least twice as much area is to be created after branching, and since the energy available cannot change discontinuously, the crack would not branch unless the factor gðvÞ in Eq. 11.1 could be doubled with a corresponding reduction in the crack speed; while it is difﬁcult to determine the pre-branching crack speed that would result in this doubling, estimates based on zero branching angle indicate that this speed is around 0:50CR : While this argument is appealing from an energetic point of view, and must be correct in principle, it is deﬁcient in two aspects. First, there is an inherent assumption that the energy available during the pre-branching growth is sufﬁcient to propagate only one crack; second this argument would require the branched cracks propagate with a signiﬁcantly reduced speed at least in the initial stages of branching. Both are contrary to experimental observations. The process zone of the crack prior to branching is quite large, and as the crack grows at the limiting speed it dissipates many times the energy required for the growth of a crack with a mirror surface. Based on measurements, the dynamic stress intensity factor near crack tips growing at about 0:50CR could be as large as three times the dynamic initiation toughness; furthermore, the dynamic stress intensity factor changes signiﬁcantly for very small changes in the crack speed (see Figs. 10.19– 10.24). Thus, the crack does not have to change speed signiﬁcantly in order to make available additional energy for growing two or more cracks; it simply has to ﬁnd a mechanism to break up into two or more cracks and can continue to grow at the same speed, but perhaps with a slightly smaller process zone. In fact, experimental measurements bear this out; as we shall see later, the crack speed does not change upon branching. We describe here some experiments performed to examine the underlying macroscopic and microscopic aspects of crack branching. In the ﬁrst experiment, a narrow strip specimen, 500 mm long and 50 mm wide and 4.76 mm thick, with a notch machined parallel to the long side of the rectangle to simulate the crack was used. The electromagnetic loading scheme described in Section 6.6 was used to generate a uniform pressure loading over the crack surfaces. Selected sequence of high-speed photographs from this test are shown in Fig. 11.11. The time variation of the dynamic stress intensity

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Figure 11.11 Selected frames from a high-speed sequence showing caustics at the tip of a branching crack. The vertical dimension is 25 mm.

factor measured with caustics and crack position are shown in Fig. 11.12. Crack initiation p occurred at about 20 ms, as the stress intensity factor increased to about 0.45 MPa m. The crack began to grow at a constant speed of about 430 m/s. The stress intensity factor continued to increase (caustics interpreted with the assumption of a K-dominant ﬁeld) to p about 1.2 MPa m at about 90 ms. At this time, the crack branched into three distinct cracks, with one branch continuing along the original crack direction and the other two moving at an angle of 658 and 708 from the main crack for the top and bottom branches, respectively. Beyond branching, the three cracks continued to grow without any measurable change in the speed for the next 50 ms; however, the dynamic stress intensity factor at the continuation of the main crack dropped signiﬁcantly to a value that is closer to the initiation toughness. The newly developed branches are also at a stress intensity factor that is comparable to the crack initiation level. Continued loading from the pressure loading on the parent crack and the arrival of reﬂected stress waves in the narrow strip conﬁguration cause additional increase in the stress intensity factor. Examination of the fracture surface revealed that the pre-branching crack roughness was in the ‘hackle’ range while the fracture surfaces of the three branches were in the ‘mirror’ range. While in this particular example reﬂected waves arrived rather quickly, the only role of these waves was to generate a larger loading at the crack tip; crack branching was also seen in the experiment shown in Fig. 8.7 in the absence of reﬂected waves, with similar characteristics. Measurements from different investigators indicate that branching occurs when the stress intensity factor reaches a critical value that is between two and three times the quasi-static fracture toughness of the material, but the crack speed at branching varied in different experiments. Variations in the crack branching angle were also found; for the test in Fig. 8.7, the angle included between the two branches is 628 while each branch is inclined at 658 to 708

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Figure 11.12 Time history of the stress intensity factor (open symbols) and crack position (ﬁlled symbols) for the experiment shown in Fig. 11.11. The vertical arrow indicates the branching point obtained from crack position measurement.

in the test in Fig. 11.11. Clearly, the macroscopic stress ﬁeld also inﬂuences the angle and number of branches that appear. Ravi-Chandar and Knauss (1984d) explored this by altering the crack parallel compressive stress at the branch location; the following loading was generated. With the electromagnetic loading device, a pressure of magnitude P1 was applied on the crack surfaces. A second loading pulse with a pressure P2 was applied to the specimen from a second electromagnetic loading device, with the pressure parallel to the crack. This situation is illustrated in Fig. 11.13. The repeatability of the loading scheme and the electrical synchronization of the events made it possible to time the events such that the secondary stress pulse ðP2 Þ arrived at the point of anticipated branching just prior to the instant of crack tip arrival at that point. In this arrangement, the magnitude P2 of the secondary pulse was varied systematically to examine its effect on the macroscopic aspects branching. It was observed that depending on the magnitude of the dynamic crack parallel stress (compression), crack branching was suppressed for some time and the angle of subsequent branching was changed. Two images from such tests are shown in Fig. 11.13, showing clearly the delay in branching and the decrease in the branching angle. The marker in the photographs indicates the length at branching Lbr for the case P2 ¼ 0: The change in Lbr and the angle of branching ubr with P2 are shown in Fig. 11.14. In contrast, in the experiment shown in Fig. 11.11, many reﬂected waves bounce to the crack tip region and during the time of branching, it is expected that the crack parallel stress should be

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Figure 11.13 Assembled plates from tests where crack-parallel compressive stress wave was imposed to delay branching. The marker in the photographs indicates the length at branching in the absence of the compressive stress wave. (Reproduced from Ravi-Chandar and Knauss, 1984d.)

Figure 11.14 Variation of the branching angle with compressive crack parallel stress component. (Reproduced from Ravi-Chandar and Knauss, 1984d.)

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Figure 11.15 Variation of the branching length and branching angle with compressive crack parallel stress component. (Reproduced from Ravi-Chandar and Knauss, 1984d.)

tensile; the resulting branch angle ubr increases by a large amount, with an included angle of about 1358. The secondary stress pulse also causes a temporary decrease in the stress intensity factor, which is illustrated in Fig. 11.15 for the case when P1 ¼ 10:35 MPa and P2 ¼ 5:5 MPa. This delays the onset of branching. Furthermore, the compressive stress parallel to the crack axis would impede off-axis microcrack growth and decrease the size of the process zone. Indeed, examination of the fracture surface in Fig. 11.16 shows that

Figure 11.16 Inﬂuence of crack parallel compressive stress on fracture surface roughness evolution. (Reproduced from Ravi-Chandar and Knauss, 1984d.)

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the surface roughness decreases upon arrival of the secondary stress pulse. Under the primary loading, the mirror, mist, hackle type fracture surface develops, but associated with the small drop in the stress intensity factor with the arrival of the secondary stress wave, the surface roughness decreases into the mist zone. Eventually, the loading from the primary pulse increases the stress intensity factor and results in branching; associated with this, the fracture surface roughness increases just prior to branching. At this point, the phenomenology of branching seems quite clear: when a crack reaches a critical stage identiﬁed macroscopically by its stress intensity factor, it splits into two or more branches, each propagating with the same speed as the parent crack, but with a much reduced process zone. This is a clear indication that the process of branching is governed by the inner problem and not the outer problem that is treated by the continuum elastodynamics. In order to reveal the mechanisms underlying crack branching, RaviChandar and Knauss (1984b) performed another experiment in which a high-speed camera was trained on a 3 mm diameter ﬁeld of view located at the anticipated crack branching location; since the electromagnetic loading scheme was extremely repeatable, it was possible to estimate this location with reasonable accuracy. Fig. 11.17 shows a high-speed photomicrograph of the branching process—captured in the act of branching! Many

Figure 11.17 Real-time micrograph of crack branching in Homalite-100. The ﬁeld of view is 3 mm. Note the large number of microbranches that are generated; three successful macrobranches emanated from this region. (Reproduced from Ravi-Chandar and Knauss, 1984b.)

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Figure 11.18 Mechanism of crack branching. (Reproduced from Ravi-Chandar and Knauss, 1984c.)

attempted branches are observed in this micrograph; these emanate from the main crack plane and continue to turn more or less smoothly away from the main direction of the crack. The micrographs indicate that (i) the branching process starts from microcracks that are initially parallel to the main crack propagation direction; (ii) these microcracks are not located preferentially through the plate thickness, thus giving branching a threedimensional character; and (iii) the growth of successful microbranches is along a path smoothly turning away from the direction of the main crack. This micrograph together with the real-time micrographs of the crack fronts reveal that the mechanism of crack growth and branching is multiple-microcracking. Ravi-Chandar and Knauss (1984c) used these observations to propose a mechanism for crack branching illustrated in Fig. 11.18. Initially, a crack propagates at the level of the initiation stress intensity factor generating a mirror-like fracture surface. The crack cuts through voids that may be present or nucleated by the crack tip stress ﬁeld, with some of the voids diverting the crack to propagate along different planes; these are the origins of surface roughening. When the stress intensity level becomes sufﬁciently high, the voids grow into microcracks well ahead of the arrival of the main crack; this interaction leads to the well-known conic markings on the fracture surface. The idea of a single crack front is no longer applicable at the scale of the fracture process zone. The microcracks within the fracture process zone interact with each other and under suitable conditions repel each other; these deviated microcracks then appear as microbranches; these microbranches are observed in the realtime micrograph shown in Fig. 11.18 and have been studied in great detail recently by Fineberg and Marder (1999) and Hauch and Marder (1998).

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217

Chapter 12 Phenomenological Models of Dynamic Fracture

Physical aspects of the generation of crack surface roughening considered in the previous section required postulating different models or mechanisms of crack growth. These mechanistic models have been developed further in the literature through models of the fracture phenomena; while some of these are derived from the micromechanics, others are approximate representations of the fracture process zones that do not require any mechanistic motivation. Such models range from discrete models of molecular dynamics and lattices to cohesive zone models of the fracture process to continuum damage models, thus covering scales from the atomic to continuum. While these models have been shown to be quite powerful, none of these models has been developed to the extent of the continuum theory. Speciﬁcally, these models are not yet capable of fully capturing the essence of the physical aspects of fracture discusses in the previous chapters. In this section we describe brieﬂy some of the phenomenological models.

12.1 Discrete Models—Molecular Dynamics and Lattice Models Discrete models of fracture have been considered at many length scales. At the smallest length scale, fracture—dynamic or static—results from the breakage of bonds between atoms. Thus, the hope is that simulations of cracking in a regular lattice arrangement of atoms with a known interaction potential should provide an indication of how fast fracture evolves. Thus, in molecular dynamics (MD) simulations, typically, a large number of atoms (about a million or two) are arranged usually in a perfect two-dimensional crystalline arrangement, and the motion and interaction of the atoms are calculated numerically using the assumed interaction potential. Abraham et al. (1994, 1997, 2003) used an idealized Lennard-Jones solid; Nakano et al. (1995) considered a porous silica and a silicon nitride in their simulations and included more complex interactions between the atoms. Since the numerical computation is time-consuming and expensive, the simulations are usually performed only over the size scale of a few nanometers spatially and only over a few picoseconds temporally. These computations are not yet extendable to realistic length and time scales; proper scaling of the results of the simulations requires mesoscopic models of the fracture process zone that are not yet available. Linking of the MD

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simulations directly to macroscale models through ﬁnite element methods is not an appropriate way to scale the results of the MD simulations since mesoscale structure and its inﬂuence on the fracture process evolution are then ignored; while this may be acceptable in determining global structure-independent properties, it is not likely to work for structure and scale-dependent phenomenon, like fracture. The results of MD simulations exhibit many of the features observed in experiments: – the average crack tip speed reaches a limiting value of about 0.57CR, – the instantaneous crack tip exhibits erratic oscillations due to crack path deﬂections beyond a crack speed of about 0.32CR, and – the crack surface exhibits signiﬁcant roughness, caused initially by crack deﬂection from the tips and later by secondary cracks forming away from the main crack at an angle and linking with the main crack. While these models appear to exhibit some similarities to experimentally observed behavior, these results are, however, somewhat paradoxical. The simulation is based on a regular arrangement of the atoms and hence should be predictive of cleavage fracture along crystallographic planes; but experiments indicate that the crack surface in single crystals is smooth and the crack speed reaches a signiﬁcant fraction of the Rayleigh wave speed without exhibiting branching; of course, anisotropy of the fracture energy plays a crucial role in crystalline materials and could be a signiﬁcant factor. On the other hand, in noncrystalline materials fast fracture exhibits crack surface roughening, limiting crack speed of about 50% of the Rayleigh wave speed and crack branching, all of which are present in the atomistic simulation. Thus, the simulation presents features that are observed at a much larger scale in experiments with noncrystalline materials! An issue still to be resolved is the role of discretization on the results; the discretized equations have a dynamics of their own, and their relationship to fast fracture in amorphous materials needs to be examined carefully. For instance, is the crack path instability simply a manifestation of the Yoffe (1951) stress ﬁeld rearrangement in the discretized problem? Lattice dynamics models of fast crack propagation have been presented by Slepyan and Fishkov (1981) and Slepyan (2002); Marder and Gross (1995) examined this model further to evaluate steady-state solutions and their stability. In these models, growth of a crack in an idealized spring mass lattice is considered, with a prescribed model for the spring behavior. We illustrate lattice dynamics with a mode III problem in a square lattice shown in Fig. 12.1.

Figure 12.1 Square lattice with a crack. The lattice spacing is a.

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The governing equation for the lattice can be obtained as a ﬁnite difference approximation of the anti-plane shear problem discussed in Eq. 3.8. Thus, umþ1;n 2 2um;n þ um21;n um;nþ1 2 2um;n þ um;n21 r d2 um;n þ ¼ m dt2 a2 a2

ð12:1Þ

If the mass is considered to be concentrated at the nodes, the equivalent spring-mass model is obtained by setting ra2 ¼ M; thus the square lattice is governed the equation:

mðumþ1;n þ um21;n þ um;nþ1 þ um;n21 2 4um;n Þ ¼ M

d2 um;n dt2

ð12:2Þ

This equation is valid for all rows except for the two rows immediately on the crack. Marder and Gross (1995) added a damping term and provided a fracture criterion for the lattice—that the lattice will break when the displacement reached a failure value, 2uf : These equations are then transformed into Fourier space and solved through a WienerHopf procedure; see Slepyan (2002) and Marder and Gross (1995) for details. Marder and Gross used this model to examine the existence and stability of steady-state solutions. Their main result is shown in Fig. 12.2 where the crack speed normalized by the lattice speed is plotted as a function of the applied load, D. The main conclusions from the lattice solution are: – There exist linearly stable lattice-trapped states corresponding to no crack growth, even though the load has exceeded the strength of the lattice. Such states were ﬁrst discussed by Thompson (1971). – Steady-state crack growth was not possible in the lattice at slow speeds; Marder and Gross called this the ‘velocity gap’ or the range of ‘forbidden velocities’. – Steady-state crack growth was possible in a range of speeds from about 0.3 to about 0.7 of the lattice wave speed. – Steady-state crack growth was path unstable at very high speeds; Marder and Gross interpreted this as an indication of a limiting speed, since the path instabilities will lead eventually to crack branching.

Figure 12.2 Crack velocity vs loading parameter D in a lattice. The lines indicate possible steadystate solutions. (From Fineberg and Marder, 1999.)

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Lattice dynamics solutions present an interesting array of possibilities in modeling dynamic fracture; however, appropriate interpretation of the implications of these predictions to the nominally brittle amorphous materials that have been discussed here requires additional physical understanding and analytical modeling efforts. For example, how does the existence of a range of forbidden velocities in the lattice correlate with experimental observations? Dynamic fracture experiments indicate that in crack arrest experiments the crack speed can decrease smoothly from a signiﬁcant fraction of the Rayleigh wave speed to zero (Dally, 1979; Kalthoff et al., 1980a,b), apparently going through the range of unstable speeds. Lattice trapping has also not been observed or demonstrated. Also, the lattice dynamics model reproduces many of the features of the MD simulations, and the paradox discussed earlier remains.

12.2 Cohesive Zone Models Barenblatt (1959) posed the idea of a crack tip cohesive zone to account for the ‘inner’ problem of the fracture process; a similar model was suggested by Dugdale (1960) to model the line plastic zone at a crack tip. Since then generalizations of the cohesive zone ideas to craze failure in polymers (Knauss, 1974; Schapery, 1975) and general cohesive failure in weakening solids such as concrete by Hillerborg et al. (1976) have been suggested. While the physical motivations for postulating a cohesive model might be quite different in these different applications—ductile void growth and associated softening in metallic materials to microcracking in brittle materials such as concrete—the form of the cohesive model is similar in all cases: separation of the cohesive surfaces is described by imposing a constitutive relation between the traction vector connecting the cohesive surfaces and the displacement jump across the surfaces. Xu and Needleman (1994) adapted the cohesive zone model for simulations of dynamic crack growth problem in brittle solids and incorporated it into a ﬁnite element formulation. They considered the cohesive model to be given in terms of a potential fðDÞ; where D ¼ wn n þ wt t is the cohesive surface separation; n and t the unit vectors in the normal and tangential directions, respectively; and wn and wt the normal and tangential cohesive surface separations, respectively. The traction-separation relation is then expressed as T ¼ ›f=›D; where T ¼ Tn n þ Tt t and Tn, Tt are the normal and tangential components of the traction vector, respectively. The advantage of this formulation is that the energy balance in Eq. 3.6 may now be written incorporating the energy of the cohesive surfaces ð ð ð ›u › › s· dR ¼ ½UðtÞ þ TðtÞdV þ f dS ð12:3Þ ›t R ›t S ›t ›R where S is the area of the cohesive surfaces. The power supplied by the external tractions is then contained in the kinetic, potential and cohesive energies. Also, the cohesive surface model may be incorporated easily into the virtual work equation and used in formulating the discretized equations. In order to complete the model, the particular form of fðDÞ must be deﬁned; Xu and Needleman (1994) assumed a form that resulted in a tractionseparation relation that mimics interatomic separation processes. Their model is deﬁned by four constants: smax and tmax, the normal and tangential strengths of the cohesive

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Figure 12.3 Cohesive surface traction-separation relationship; smax 5 E=10 5 324 MPa, tmax 5 755:5 MPa, fn 5 ft 5 352:3 J/m2. (From Xu and Needleman, 1994.)

surface, respectively, and fn and ft, the work of separation under normal and shear tractions, respectively. The forms of the traction-separation law for crack opening and crack sliding are shown in Fig. 12.3. It should be noted that the form of the tractionseparation relation derived from a potential of the type indicated above is conservative and thus this cohesive model is reversible; however, when it is used in simulations of crack growth, unloading occurs as the crack grows and hence healing of cohesive surfaces occur in this model. Irreversible models of the cohesive zone have been proposed, all based on the maximum attained crack surface separation (Geubelle and Rice, 1995; Yang and RaviChandar, 1996; Ortiz and Pandolﬁ, 1999). For example, Yang and Ravi-Chandar used the following form T ¼ kðwd ÞD

ð12:4Þ

where wd is the maximum separation distance between two originally coincident points on the crack over the entire loading history, and is used as a damage parameter. The stiffness of the cohesive zone material kðwd Þ is assumed to depend on the current state of damage. In their numerical simulations, Xu and Needleman (1994) discretized the elastic body into triangular elements, arranged in square or quadrilateral patterns. In order to model the cracking process, all element boundaries were connected to each other by cohesive surface elements. As a result of prescribing the cohesive surfaces, crack separation appears naturally in this simulation, just as in the case of the atomic and lattice models. However, as a result of the assumed form of fðDÞ; no clear crack tip exits. The locations where a speciﬁc value of wn is attained are identiﬁed as the crack tips; the time variation of the crack tips are tracked to determine the crack speed. Crack speeds determined from one set of Xu and Needleman’s simulations are shown in Fig. 12.4. The cohesive zone parameters used in these simulations are the following: smax ¼ E=10 ¼ 324 MPa, tmax ¼ 755:5 MPa, fn ¼ ft ¼ 352:3 J/m2. The elastic properties of the material were taken to correspond to that of PMMA: E ¼ 3:24 GPa, n ¼ 0:35; r ¼ 1190 Mg/m3, Cd ¼ 2090 m/s, Cs ¼ 1004 m/s and CR ¼ 938 m/s. Since cohesive

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Figure 12.4 Crack speed variation with time obtained in different simulations. ‘No branching’ indicates simulation where cohesive surfaces were placed only along the prospective crack plane; the resulting crack accelerates to the Rayleigh wave speed. ‘Crack arrest’ identiﬁes simulation in which the cohesive strength was increased abruptly resulting in an abrupt arrest of the crack. The other two lines indicate simulations in which the cohesive surfaces were introduced at all element boundaries. (From Xu and Needleman, 1994.)

surfaces were provided in all element boundaries, arbitrary crack path evolution could be obtained; crack path prediction from one simulation that indicated crack branching is shown in Fig. 12.5. The main results from the numerical simulations are: – The crack accelerates quickly to the Rayleigh wave speed if the cohesive surfaces are restricted to the initial crack plane; this is shown by the line identiﬁed by the label ‘no branching’ in Fig. 12.4. – The instantaneous crack tip speed exhibits erratic oscillations due to crack path deﬂections beyond a crack speed of about 0:45CR ; cohesive surfaces off the main crack plane were observed to separate and close as the main crack propagated. The oscillations appear to be related to the off-axis cohesive surface development, similar to the microbranching-induced oscillations observed by Fineberg et al. (1992). – Very large stress levels are found to occur over some region near the crack; as a result, cohesive surfaces not connected with the main crack were found to exhibit opening displacement jumps reminiscent of microcrack nucleation and growth. – Successful crack branching appears in the simulations. The results of these simulations are also remarkably similar to the results from the MD simulations. However, this is not surprising if one examines the underlying similarities in the models. In both cases, one has a discrete set of (nodal or lattice) points, each with two displacement degrees of freedom, ua. The evolution equation for these displacement degrees of freedom is described by the simple equation: Mu¨a ¼ F(ua), where M is the mass associated with the nodal or lattice point and F the force of interaction that depends on the

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Figure 12.5 Crack path selected in a simulation with cohesive surfaces at all element boundaries and with symmetric loading; crack branching appears automatically. (From Xu and Needleman, 1994.)

displacement. The difference arises in the computation of the force: in the MD simulations, the force arises from the interaction with lattice points in a certain neighborhood and is calculated using the assumed interaction potential. In the ﬁniteelement simulations, the force arises from the elastic interaction with the surrounding nodes, as determined from the usual elastic analysis and from the cohesive law. From the similarity of the governing equation, one expects a similarity in the result from the two models. Xu and Needleman (1994) also examined the inﬂuence of the discretization on the development of fast fracture. The results of this examination are perhaps the most useful in interpreting the results of all the discrete models discussed in this section. Xu and Needleman performed a number of simulations for a center crack in a strip of width 10 mm and height 2 mm; a velocity boundary condition of ^ 10 m/s on the top and bottom boundaries was prescribed, with material properties appropriate for PMMA as described above. The resulting crack growth was examined in simulations with meshes where the cohesive boundaries off the initial crack plane were oriented at angles of ^ 158, ^ 308, ^ 458 and ^ 608. They made the following observations from the results of the simulations: – when the cohesive surfaces are at ^ 158 and ^ 308, the crack grows in a zigzag mode with crack speed oscillations appearing right from crack initiation; – crack speed oscillations appear to be most pronounced when the cohesive surfaces are at ^ 458;

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– crack speed oscillations are almost completely absent when the cohesive surfaces are at ^ 608; – onset of crack branching is affected signiﬁcantly by the angle of the cohesive surfaces; the crack speed at branching varies nonmonotonically with the angle of the cohesive surfaces; – in almost all the cases, the crack speed continues to increase and reaches values very close to the Rayleigh wave speed; for cohesive surfaces at ^ 608, at the onset of branching, the calculated crack speed is 0:91CR : These observations are consistent with the Yoffe type instability of the crack path; at high crack speeds, the maximum tensile stresses arise off the initial crack plane. If the cohesive surface is oriented at the appropriate angle, its development will be enhanced as the crack speed increases; the crack branches and velocity oscillations occur. If, on the other hand, the cohesive surface orientation is not at the appropriate angle, its development will be suppressed and the crack continues to grow straight ahead at high speeds. An important point to note is that crack extension occurs only along the element boundaries; the underlying assumption in performing such simulations is that by making the element size small, the simulation can model arbitrary evolution of the crack path. In a recent evaluation of this strategy Falk et al. (2001) have questioned this proposition by pointing out that, in simulations aimed at analyzing crack branching, the mesh size introduced a characteristic length scale into the problem that inﬂuenced the results.

12.3 Continuum Damage Models Macroscopic or continuum level modeling of the fracture process zone has also been attempted. Johnson (1992) considered a continuum damage mechanics approach following a cell model for fracture developed by Broberg (1982); in the region near the crack tip the material was considered to be of an elastic-softening type and the damage was assumed to be triggered by the dilatational strain. Thus, in the crack tip region, the elastic stiffness of each element was decreased by a factor that depended on the current level of dilatation, q 8 1 0 , q , q1 > > > < ðq =qÞn 2 1 2 ð12:5Þ vðnÞ ¼ q1 , q , q2 > ð q =q1 Þn 2 1 2 > > : 0 q . q2 where q1 is a threshold dilatation below which no damage occurs, q2 the dilatation at which the damage is complete and n a parameter. Irreversibility of damage was also incorporated into the model; if the volume dilatation decreased temporarily after exceeding the damage threshold, v was maintained at its previous maximum value. This model was implemented in a very ﬁne-scale ﬁnite-element simulation of the pressurized semi-inﬁnite crack problem using the same conditions that were attained in the experiments of Ravi-Chandar and Knauss described in Section 9.3. While the scale of the elements is large compared to the atomic scale fracture processes, the mesh size was

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large enough to capture the structure of the overall fracture process zone. Because of computational cost considerations, Johnson introduced the damage model only in the 12 rows immediately next to the crack surface; this restriction results in some artifacts in the numerical simulations, but these are discriminated easily. The main results from this numerical simulation of the continuum damage model are: – In each simulation, the crack grew at a constant speed that depended on the applied load; concomitant with this constant speed was an appropriate expansion of the process zone, identiﬁed as cells in which the dilatation has exceeded either q1 or q2 ; qualitative correspondence between the simulations and experiments described in connection with Figs. 9.2, 9.3 and 9.4 was demonstrated. In Fig. 12.6 the evolution of the damage in the cells as a result of crack growth is shown. At a crack surface

Figure 12.6 Evolution of the fracture process zone from three simulations of the continuum damage model. q1 5 0:00075; q2 5 0:018 and n 5 1:5; the crack surface pressure was 0:001E in (a), 0:002E in (b) and 0:003E in (c). The circles indicate elements that have begun to accumulate damage (max q > q1 ) and the crosses indicate elements in which damage is complete (max q > q2 ). (Reproduced from Johnson, 1992.)

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pressure of 0:001E; the process zone developed only along the ﬁrst three rows of elements. As the crack surface pressure was increased to 0:003E; the process zone began to develop faster in the direction normal to the crack line and left a trail of partially or fully damaged material. – With increased crack surface loading, the crack accelerated to a limiting speed that is signiﬁcantly smaller than the Rayleigh wave speed. – If the spread of the damaged elements near the crack tip is identiﬁed as crack surface roughening, increase in surface roughness was correlated with energy supply rather than velocity. – If the development of damage concentrated along speciﬁc directions other than the initial crack line is considered as indicative of branching, this simulation also showed the development of branching; however, whether this is driven by a Yoffetype instability or not is not clear. If should be recognized that the local properties of the damaged cells vary signiﬁcantly from the initial properties and hence the crack speed might be large enough to trigger the Yoffe-type instability. Recent work by Klein and Gao (2001) has suggested that this might be the case. The results exhibit many of the characteristics observed in experiments. This type of a model is quite attractive since it is not computationally as intensive as the discrete models such as MD simulations and furthermore, the simulation could be performed easily in standard ﬁnite element codes. However, additional work is necessary to determine the inﬂuence of discretization on the results and the appropriate form of the damage law to be used. For instance, the damage law could be derived from homogenized description of the microcracked crack tip material. Such damage models based on nucleation and growth models for crack growth were considered ﬁrst by Zhurkov (1965) for thermally activated crack processes such as creep. Curran et al. (1973) applied these ideas to the stress-induced nucleation and growth processes in the dynamic spalling problem. The description of dynamic crack growth provided by Ravi-Chandar and Knauss (1984a – d) and RaviChandar and Yang (1997) is also based on nucleation and growth of cavities and microcracks. In these models, the nucleation rate of microcracks is given by N_ ¼ N_ 0 exp½ðs 2 sn0 Þ=s1

ð12:6Þ

where sn0 is the nucleation threshold, and s1 and N_ 0 the constants to be determined through calibration experiments. Note that this form of the nucleation rate is typically applied in many models, not necessarily related to fracture. Zhurkov (1965) applied this to the rate of bond breakage and Curran et al. (1973) applied it to nucleation of microcracks. Based on experimental observations, Curran et al. also assumed an exponential distribution of initial radii of the nucleated microcracks: DNðRÞ ¼ DN0 exp½2R=R1 ; where DNðRÞ is the number of microcracks per unit volume with radius larger than R that are nucleated in a time interval Dt and R1 is another parameter in the model. Finally, a growth rate is imposed on the microcracks such that s 2 s0 ðRÞ R_ ¼ ð12:7Þ 4h R where s0 ðRÞ is the critical stress for a penny-shaped ﬂaw of radius R to be fracture critical and h the viscosity; this form was derived by Curran et al. from viscosity-limited growth in

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ductile materials, but was also applied successfully to brittle materials. With this nucleation and growth model, for any load history, the current state of damage can be determined and used in degrading the material properties for simulating the overall response of the material. Such models of the mechanical properties of the damaging material could then be incorporated into numerical simulations of the fracture problem in the sense of the continuum simulations of Johnson (1992) or the cohesive zone simulations of Xu and Needleman (1994) and others. In this chapter, we have reviewed some of the phenomenological models that are aimed at providing an analysis of the “inner problem” associated with the dynamics of the fracture process zone. While there have been signiﬁcant advances in the computational methods used, comparison to experimental results remain qualitative at best. Fundamental considerations regarding the determination of appropriate material properties for inclusion in these phenomenological models and the quantitative comparison of the results of the simulations to experimental observations remain open issues.

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Further Reading H. Coque, C.P. Leung and S.H. Hudak, Jr.: Effect of Planar Size and Dynamic Loading Rate on Initiation and Propagation Toughness of a Moderate-Roughness Steel, Engng Fract. Mech., 47 (1994), 249 –267. F. Costanzo and J.R. Walton: Numerical Simulations of a Dynamically Propagating Crack with a Nonlinear Cohesive Zone, Int. J. Fracture, 91 (1998), 373–389. L.S. Costin and J. Duffy: The Effect of Loading Rate and Temperature on the Initiation of Fracture in Mild, RateSensitive Steel, J. Engng Mater. Technol., 101 (1979), 258 –264. J.W. Dally and A. Shukla: Dynamic Crack Behavior at Initiation, Mech. Res. Commun., 6 (1979), 239– 244. J.W. Dally and A. Shukla: Inﬂuence of Late-Breaking Ligaments on Crack-Propagation in Compact Specimens – A Photo-Elastic Study, Exp. Mech., 23 (1983), 298–303. A.T. de Hoop: Representation Theorems for the Displacement in an Elastic Solid and Their Application to Elastodynamic Diffraction Theory, Ph.D. Thesis, Technical University, Delft, 1958, p. 72. L.B. Freund: Crack Propagation in an Elastic Solid Subjected to General Loading. II. Nonuniform Rate of Extension, J. Mech. Phys. Solids, 20 (1972c), 141–150. S.K. Khanna and A. Shukla: On the Use of Strain Gages in Dynamic Fracture Mechanics, Engng Fract. Mech., 51 (1995), 933–948. W.G. Knauss and K. Ravi-Chandar: Some Basic Problems in Stress Wave Dominated Fracture, Int. J. Fracture, 27 (1985), 127–143. A.S. Kobayashi: Handbook on Experimental Mechanics, Prentice Hall, New York, 1993. T. Kobayashi, I. Yamamoto and M. Niinomi: Evaluation of the Dynamic Toughness Parameters by Instrumented Charpy Impact Test, Engng Fract. Mech., 24 (1986), 773–782. T.L. Leise and J.R. Walton: A Method for Solving Dynamically Accelerating Crack Problems in Linear Viscoelasticity, SIAM J. Applied Math., 64 (2003), 94–107. J.J. Mason, A.J. Rosakis and G. Ravichandran: Full Field Measurements of the Dynamic Deformation Field Around a Growing Adiabatic Shear Band at the Tip of a Dynamically Loaded Crack or Notch, J. Mech. Phys. Solids, 42 (1994), 1679–1697. K. Ravi-Chandar: An Experimental Investigation into the Mechanics of Dynamic Fracture, Ph.D. Thesis, California Institute of Technology, Pasadena, 1982.

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239

Appendix A Dynamic Crack Tip Asymptotic Fields

A1 Dynamic Crack Tip Stress Field for a Stationary Crack For a stationary crack loaded with a time-dependent load, the stress ﬁeld in the vicinity of the crack tip is characterized through the following equation KI ðtÞ Is K ðtÞ IIs sab ðr; uÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ ðuÞ þ s0x da1 db1 þ · · · as r ! 0 fab ðuÞ þ pIIﬃﬃﬃﬃﬃﬃﬃﬃ fab 2pr 2pr

ðA1:1Þ

where KI ðtÞ and KII ðtÞ are the mode I and mode II dynamic stress intensity factors and s0x is the ﬁrst nonsingular term in the asymptotic expansion. The angular variation of the Is IIs ðuÞ and fab ðuÞ are given below: functions fab Is f11 ðuÞ ¼ cos 12 u½1 2 sin 12 u sin 32 u; Is f22 ðuÞ ¼ cos 12 u½1 þ sin 12 u sin 32 u; Is ð uÞ f12

ðA1:2Þ

¼ cos u sin u cos u 1 2

1 2

3 2

IIs f11 ðuÞ ¼ 2sin 12 u½2 þ cos 12 u cos 32 u;

ðA1:3Þ

IIs f22 ðuÞ ¼ cos 12 u sin 12 u cos 32 u; IIs ðuÞ f12

¼ cos u½1 2 sin u; sin u 1 2

1 2

3 2

A2 Steady-State Dynamic Crack Tip Stress Field: Singular Term For easy reference, the asymptotic dynamic crack tip stress ﬁeld is written down here for different cases. The general form of the steady-state stress ﬁeld is given by KIdyn I sab ðr; uÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ fab ðu; vÞ þ 2pr pﬃﬃ K dyn r ua ðr; uÞ ¼ Ipﬃﬃﬃﬃﬃﬃ gIa ðu; vÞ þ m 2p

KIIdyn II pﬃﬃﬃﬃﬃﬃﬃﬃ fab ðu; vÞ þ s0x da1 db1 þ · · · as r ! 0; 2pr KIIdyn II pﬃﬃﬃﬃﬃﬃﬃﬃ ga ðu; vÞ þ · · · as r ! 0 ðA2.1Þ m 2pr

where KIdyn and KIIdyn are the mode I and mode II dynamic stress intensity factors and s0x is

240

Appendix A

the ﬁrst nonsingular term in the asymptotic expansion. The angular variation of the I II ðu; vÞ; fab ðu; vÞ; gIab ðu; vÞ; and gIIab ðu; vÞ are given below functions fab ( ) 1 1 1 I 2 2 2 cos 2 ud ð1 þ as Þð1 þ 2ad 2 as Þ 1=2 2 4ad as 1=2 cos 12 us ; f11 ðu; vÞ ¼ RðvÞ g gs d

I f22 ðu; vÞ

1 ¼2 RðvÞ

( ð1 þ

a2s Þ2

cos 12 ud 1=2

gd

2 4 ad a s

1 1=2

gs

) cos us ; 1 2

ðA2.2Þ

( ) sin 12 us 2ad ð1 þ a2s Þ sin 12 ud ¼ 2 1=2 1=2 RðvÞ gd gs ( ) 1 sin u 2 a 1 d s II 2 ð1 þ 2a2d 2 a2s Þ 1=2 f11 ðu; vÞ ¼ 2 2 ð1 þ a2s Þ 1=2 sin 12 us ; RðvÞ g gs

I ðu; vÞ f12

d

II f22 ðu; vÞ

2as ð1 þ a2s Þ ¼ RðvÞ

II f12 ðu; vÞ

1 ¼ RðvÞ

(

( 4a d as

sin 12 ud 1=2

gd

cos 12 ud 1=2

gd

2

)

1 1=2

gs

2 ð1 þ

ðA2:3Þ

sin us ;

a2s Þ

1 2

cos 12 us

)

1=2

gs

2 u u 1=2 ð1 þ a2s Þrd cos d 2 2ad as rs1=2 cos s ; RðvÞ 2 2 2a u u 1=2 gI2 ðr; uÞ ¼ 2 d ð1 þ a2s Þrd sin d 2 2rs1=2 sin s RðvÞ 2 2 2as u u 1=2 2rd sin d 2 ð1 þ a2s Þrs1=2 sin s gII1 ðu; vÞ ¼ ; RðvÞ 2 2 2 u u 1=2 gII2 ðu; vÞ ¼ 2ad as rd sin d 2 ð1 þ a2s Þrs1=2 sin s RðvÞ 2 2 gI1 ðr; uÞ ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ad x2 2 2 2 rd ¼ x1 þ ad x2 ; tanud ¼ tan x1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ as x 2 2 2 2 rs ¼ x1 þ as x2 ; tanus ¼ tan x1 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v2 v2 ad ¼ 1 2 2 ; a s ¼ 1 2 2 Cd Cs RðvÞ ¼ 4ad as 2 ð1 þ a2s Þ2

ðA2:4Þ

ðA2:5Þ

ðA2:6Þ ðA2:7Þ

ðA2:8Þ ðA2:9Þ

Dynamic Crack Tip Asymptotic Fields

241

Cd is the dilatational wave speed and Cs the shear wave speed. For thin plates, the appropriate dilatational wave speed is the plate wave speed given by: Cdp

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E ¼ rð1 2 n2 Þ

ðA2:10Þ

A3 Steady-State Crack Tip Displacement and Stress Field: N Terms The steady-state crack tip displacement and stress components for symmetric loading (mode I) about the crack line are given below:

n n=2 nud nus n I n=2 rd cos u1 ðr; uÞ ¼ An 1 þ 2 x1 ðnÞrs cos ; 2 2 2 ðA3:1Þ

n nud nus n=2 2rd sin þ xI2 ðnÞrsn=2 sin un2 ðr; uÞ ¼ An ad 1 þ 2 2 2 n n=221 n22 n22 I n=221 cos ud 2 x1 ðnÞrs cos us ; rd 2 2 2 n n22 n=221 cos ud 2a2d rd 2 2 n22 I n=221 þ x1 ðnÞrs cos us ; ðA3:2Þ 2 A n

n n22 n=221 1þ 24ad as rd sin ud 1n12 ðr; uÞ ¼ n 2 2 4 as 2 n22 n=221 þ kI ðnÞrs sin us 2

n

1þ ¼ An 2 n

1þ 1n22 ðr; uÞ ¼ An 2

1n11 ðr; uÞ

mAn n

n n22 2 2 2 n=221 1þ ð1 þ as Þð1 þ 2ad 2 as Þrd ¼ cos ud 2 2 ð1 þ a2s Þ 2 n22 2kI ðnÞrsn=221 cos us ; 2 mAn n

n n22 2 2 n=221 1 þ sn22 ðr; uÞ ¼ a Þ r cos u 2ð1 þ d s d 2 2 ð1 þ a2s Þ 2 n22 þkI ðnÞrsn=221 cos us ; ðA3.3Þ 2 mAn n

n n22 n=221 1þ 24ad as rd sn12 ðr; uÞ ¼ sin ud 2 2 2a s 2 n22 þkI ðnÞrsn=221 sin us 2

sn11 ðr; uÞ

242

Appendix A

where 8 2a d as > > > < 1 þ a2 ; s xI1 ðnÞ ¼ > 1 þ a2 > s > : ; 2 ( 4a d as kI ðnÞ ¼ ð1 þ a2s Þ2 ;

for n odd and for n even for n odd for n even

xI2 ðnÞ

¼

8 > > > <

2 ; 1 þ a2s

> 1 þ a2s > > : ; 2 ad as

for n odd ðA3:4Þ for n even

ðA3:5Þ

For the case of antisymmetry (mode II), the displacement and stress components are

n n=2 nud nus un1 ðr; uÞ ¼ An 1 þ rd sin 2 xII1 ðnÞrsn=2 sin ; 2 2 2 ðA3:6Þ

n n=2 nud nus rd cos 2 xII2 ðnÞrsn=2 cos un2 ðr; uÞ ¼ ad An 1 þ 2 2 2 n n=221 n22 n22 sin ud 2 xII1 ðnÞrsn=221 sin us ; rd 2 2 2 n n22 n=221 2a2d rd sin ud 2 2 n22 þxII1 ðnÞrsn=221 sin us ; ðA3.7Þ 2 A n

n n22 n=221 1þ cos ud 1n12 ðr; uÞ ¼ n 4a d as r d 2 2 4a s 2 n22 us 2kII ðnÞrsn=221 cos 2

n

1þ 2 n

1þ 1n22 ðr; uÞ ¼ An 2 1n11 ðr; uÞ ¼ An

mAn n

n n22 2 2 2 n=221 1þ ð1 þ as Þð1 þ 2ad 2 as Þrd ¼ sin ud 2 2 ð1 þ a2s Þ 2 n22 2kII ðnÞrsn=221 sin us ; 2 mAn n

n n22 2 2 n=221 1 þ sn22 ðr; uÞ ¼ a Þ r sin u 2ð1 þ d s d 2 2 ð1 þ a2s Þ 2 n22 þkII ðnÞrsn=221 sin us ; ðA3:8Þ 2 mAn n

n n22 n=221 1þ 4ad as r d sn12 ðr; uÞ ¼ cos ud 2 2 2a s 2 n22 2kII ðnÞrsn=221 cos us 2

sn11 ðr; uÞ

243

Dynamic Crack Tip Asymptotic Fields

where

8 2ad as > > > < 1 þ a2 ; for n even s and xII1 ðnÞ ¼ > > 1 þ a2s > : ; for n odd 2 ( 4a d as ; for n even kII ðnÞ ¼ 2 2 ð1 þ as Þ ; for n odd

xII2 ðnÞ ¼

8 > > > <

2 ; for n even 1 þ a2s

> 1 þ a2s > > : ; for n odd 2ad a s

ðA3:9Þ

ðA3:10Þ

A4 Transient Crack Tip Displacement and Stress Field: Six Terms s11 þ s22 3v2 ud 21=2 2v2 A cos þ 2 A1 ¼ rd 0 2 2rðCd2 2 Cs2 Þ 4Cd2 cd 15v2 v2 ud 1 þ A þ 1 2 ðA Þ D 2 0 cos 2 4Cd2 2Cd2 v2 3u 1=2 þ 2 D1 ðA0 Þcos d rd 2 8Cd 2 6v v2 1 þ A3 þ 1 2 D ðA1 Þ cos ud rd Cd2 4Cd2 35v2 v2 1 v2 1 1 2 þ A þ 1 2 ðA Þ þ D D2 ðA0 Þ 4 2 9 4Cd2 2Cd2 4Cd2 2 v2 3ud 3v 1 € A0 cos þ 12 D ðA2 Þ þ 2 2Cd2 8Cd2 1 v2 3v2 € ud 2 þ A 12 ðA Þ þ D 0 0 cos 6 2 4Cd2 8Cd2 2 v 5ud 3=2 þ D2 ðA0 Þ cos rd 2 2 96Cd 12v2 v2 D2 ðA1 Þ þ A þ 1 2 D1 ðA3 Þ þ 5 2 2 16 Cd 2Cd 2 v2 € 1 cosð2ud Þ þ v D1 ðA3 Þ þ 12 A 2Cd2 2Cd2 1 v2 v2 € 2 þ 12 A D rd2 þ oðrd2 Þ; ðA Þ þ 1 1 8 4Cd2 2Cd2 ðA4:1Þ

244

Appendix A

where D1 ðAk Þ ¼ 2

ðk þ 3Þv d ðAk Þ; Cd2 a2d dt

D2 ðAk Þ ¼ D1 bD1 ðAk Þc

and

k ¼ 0; 1; 2; … A€ k ¼

1 Cd2 a2d

d2 Ak dt2

245

Appendix B Mechanical and Optical Properties of Selected Materials

Tables B.1 – B.3.

Table B.1 Homalite-100 (reproduced from Ravi-Chandar and Knauss, 1982) Modulus of elasticity (dynamic) Poisson’s ratio Density Plate wave speed Shear wave speed Rayleigh wave speed Index of refraction Direct stress-optic coefﬁcient Transverse stress-optic coefﬁcient Plane strain fracture toughness

E n r Cpd Cs CR n C1 C2 KIC

4550 MPa 0.31 1230 kg/m3 2057 m/s 1176 m/s 1081 m/s 1.5 20.906 £ 10210 m2/N 21.140 £ 10210 m2/N 0.44 MPa m1/2

Table B.2 Polymethylmethacrylate (reproduced from Kalthoff, 1987) Modulus of elasticity (dynamic) Poisson’s ratio Density Plate wave speed Shear wave speed Rayleigh wave speed Index of refraction Direct stress-optic coefﬁcient Transverse stress-optic coefﬁcient Plane strain fracture toughness

E n r Cpd Cs CR n C1 C2 KIC

3240 MPa 0.35 1230 kg/m3 1816 m/s 1035 m/s 967 m/s 1.491 20.530 £ 10210 m2/N 20.570 £ 10210 m2/N 1.05 MPa m1/2

246

Appendix B Table B.3 Araldite B (reproduced from Kalthoff, 1987)

Modulus of elasticity (dynamic) Poisson’s ratio Density Plate wave speed Shear wave speed Rayleigh wave speed Index of refraction Direct stress-optic coefﬁcient Transverse stress-optic coefﬁcient Plane strain fracture toughness

E n r Cpd Cs CR n C1 C2 KIC

3660 MPa 0.392 1230 kg/m3 1931 m/s 1065 m/s 1001 m/s 1.592 20.056 £ 10210 m2/N 20.620 £ 10210 m2/N 1.05 MPa m1/2

247

Index

A A533B reactor grade steel 164 –7 airplane structures 4 –6, 176 –7 AISI 4340 steel 151 – 2, 185 – 7 aluminum 137, 164 –8 amorphous materials 190 –1, 207 –8 angular variation crack branching 210 –13 crack tip stress ﬁelds 29, 34– 7, 239 –41 stress wave fractography 100– 1 Wallner lines 98 –100 anti-plane shear 15 –16, 30 –2, 219 antisymmetric deformation 37 –8 antisymmetric loadings 242 –3 Araldite B crack toughness 163 – 5, 170 – 1, 178, 182 –3 fracture surface roughness 193 mechanical properties 246 optical properties 246 arrester plates 175– 6 ASTM Standard method 173– 4 asymptotic ﬁelds crack tip stress 30– 43, 131– 2, 135 –8, 239 –44 dominance 141 –53 lateral shearing interferometry 131–2 method of caustics 123– 4 moving cracks 144 –53 propagating cracks 144– 53 stationary cracks 141 –4 strain gages 135 –8

B bifocal apparatus 149 –50 bilateral Laplace transforms 54– 6

19 –25,

birefringence 109– 16 blast response 5– 6 boiler bursting 3 –4 branch lengths 213 –4 bright ﬁeld fringes 111– 12, 131– 2, 152 –3 brittle materials 189– 215 cohesive zone models 220 crack growth 178 –84, 187 crack initiation 160 –1 fracture surface roughness 193 –208 limiting crack speed 190– 3 bulk waves 11– 13

C Cagniard-de Hoop technique 54 cameras see high-speed photography cantilever beams 170– 1, 180– 3 carbon steels 185– 7 caustics formation 112 –21 see also method of caustics cavities 226 –7 CCD cameras 105– 6 CGS see coherent gradient sensors Charpy energy 4, 6 –7 circular polariscopes 109 –16 cleavage 160 –1, 190 –1, 218 –20 coherent gradient sensors (CGS) 128 –35, 151 –3 cohesive zone models 220 –4 compact compression 91, 167, 172 –4 conic markings 198– 203 conservation of energy 10 continuum damage models 224 –7 crack arrest 74, 77 – 9, 168 – 77 crack branching 208 –15 crack front waves 99 –100

248 crack growth cohesive zone models 220 –1 continuum damage models 226 –7 criterion/toughness 74 – 9, 177– 87 energy balance equation 71– 4 intersonic 43– 7 nonsteady 39 –43 crack initiation asymptotic ﬁelds 145 criterion 74– 9, 155– 68 toughness 85, 136 –7, 155 –69 crack paths 222 – 3 crack propagation 60–9, 144–53, 218–20 crack speed 97– 106 cohesive zone models 221 –4 dynamic crack growth 177 – 87 electrical resistance 102– 4 high-speed photography 104 –6 limiting 73 –4, 190 –3 stress intensity factors 170 – 3 stress wave fractography 100– 2 ultrasonic transducers 101 –2 Wallner lines 98– 100 crack tip energy ﬂux integral 72 –3 crack tip equation of motion 71– 4 crack tip ﬁelds 27 –47, 241 – 4 asymptotic 30 – 43, 131 – 2, 135 – 8, 239– 44 dynamic loading 27– 30 intersonic crack growth 43 – 7 see also dynamic... crack tip stress ﬁelds 107– 39 angular variation 29, 34– 7 asymptotic 30 – 43, 131 – 2, 135 – 8, 239– 44 interferometry 128– 35, 138– 9 lateral shearing interferometry 128–35 method of caustics 116– 28 photoelasticity 109– 16 strain gages 135– 8 transient 243– 4 crack velocity 177 –87, 203 –4, 219 –20

Index

Cranz-Schardin multiple-spark cameras 105 creep models 226– 7

D DCB see double-cantilever beams deformation ﬁelds 146 –53 dilatational waves 11– 13 discontinuity propagation 14– 15 discrete models 217– 20 displacement ﬁelds 107– 39 dynamic crack tips 37– 8 interferometry 128– 35, 138– 9 Jones calculus 107 –9 lateral shearing interferometry 128– 35 method of caustics 116– 28 photoelasticity 109– 16 stationary cracks 53– 6 strain gages 135– 8 distortional wave speed 12 double-cantilever beams (DCB) 170 –1, 180– 3 drop-weight tower 82 – 91, 151 – 3, 181– 2 drum cameras 105 ductility 161, 184 – 7 dynamic crack arrest 74, 77 –9, 168 –77 dynamic crack growth criterion 74– 9, 177– 87 toughness 177 –87 dynamic crack initiation criterion 74– 9, 155– 68 toughness 85, 136 –7, 155 –69 dynamic crack tip ﬁelds 27– 47 asymptotic analysis 30 –43 intersonic crack growth 43 – 7 loaded cracks 27 –30 dynamic energy release 73 dynamic fracture criteria 71, 74– 9, 85, 136– 7, 155– 87 crack arrest 74, 77 –9, 168 – 77 crack growth 74– 9, 177– 87

249

Index

crack initiation 74 –5, 78 –9, 155 –68, 203 –4 energetic basis 71, 74 –9 dynamic loading 27 – 30, 49 –60, 81 –95 drop weight towers 82 – 91 electromagnetic loading 82, 93 –5 explosive charges 82, 91– 3 Hopkinson bar impact tests 91 –2 impact testing 82 – 93 pressure loading 94– 5 projectile impact 82, 87 – 91 dynamic propagation 137 dynamic singular ﬁeld dominance 146–53 dynamic stress ﬁelds 239 dynamic stress intensity factors asymptotic ﬁelds 142 –53 crack arrest 170– 4 crack speed 170– 3 crack tips 27 –31, 34, 38, 41– 3 determination 49 –69 energy balance equation 73 –4 impact tests 86– 90 lateral shearing interferometry 132 –5 moving cracks 60 –9 photoelasticity 112– 16 point loads 57 –60 projectile impact 88– 90 semi-inﬁnite cracks 49 –60 stationary cracks 49 –60 strain gages 135 –8

E elastodynamics 144 see also linear elastodynamics electric vectors 128 – 9 electrical resistance 102– 4 electromagnetic loading 82, 93 –5 electronic high-speed cameras 105 –6 energy absorption 84

balance 29, 71 –4, 190 conservation 10 ﬂux integral 72 –3 fracture criterion 71, 74– 9 release rates 45, 73, 178 equations of motion 9– 10, 71– 4 equivoluminal waves 11– 13, 43– 7, 98– 100 explosive charges 82, 91– 3

F failure criteria 71, 74 – 9 ﬂaw criticality 175– 7 formation of caustics 116– 21 fractography 100 – 2, 194– 5 fracture criteria see dynamic fracture criteria fracture surface roughness 193– 208 undulations 98– 100 full-scale models 176 –7

G Galilean transformations 39 geometry conic markings 201– 2 correction factors 174 glass crack branching 208 crack speed 103 inorganic 98– 100, 189– 92 Green’s function 19 –25, 64 –6 Grifﬁth cracks 60 –3

H hackle 193 half-space Green’s function

19 –25

250 high-resolution moire interferometry 138– 9 high-speed photography caustics 121 – 2 crack branching 209 –10, 214 crack speed measurements 104 –6 fracture surface roughness 195 –6 high-strength steels 156, 163– 5 Homalite crack branching 208 –9, 214 –15 crack growth 178– 84 crack initiation 162 –4, 167 –8 crack speed 103 fracture surface roughness 193 –6, 204, 208 limiting crack speed 191 mechanical properties 245 optical properties 245 strain gages 137 Hopkinson pressure bar tests 91 – 2, 159– 61, 164– 8

I image converter cameras 105 impact response curves 85 – 7 impact tests 6–7, 82–93, 159–61, 164–8 in-plane antisymmetric deformation 37–8 in-plane symmetric deformation 32 –7 initial curves 119 inorganic glasses 98 –100, 189 –92 instrumented impact tests 6 –7, 82 –93, 159– 61, 164– 8 interferometry 128– 35, 138– 9 intersonic crack growth 43 – 7 inverse bilateral Laplace transforms 23 inverse Laplace transforms 54 –6 irrotatational waves 11 –13 isochromatic fringes 46 – 7, 112 – 16, 126– 7

Index

J J-integral 160 Jacobian determinant 119 –21 Jones calculus 107 –9, 128 Jones matrix 108– 9 Jones vectors 107 –12 jump conditions 14 –15

K key curves 85 –7 KTE epoxy 180– 1

L Lamb’s problem 23– 5 Lame constants 11– 13 Lame potentials 13, 16, 49 Laplace transforms 19– 25, 50– 1, 53– 6 lateral shearing interferometry 128– 35 lattice models 217 –20 lead azide explosives 93 light intensity 130 –5 limiting crack speed 73– 4, 190– 3 linear elastodynamics 9– 25 boundary-initial value problems 9– 11 bulk waves 11 – 13 discontinuity propagation 14– 15 half-space Green’s function 19 –25 Lamb’s problem 23– 5 Lame constants 11– 13 plane waves 12 –13 rays 14 – 15 surface waves 18– 19 two-dimensional problems 15– 17 wave fronts 14– 15 loss of coolant accidents (LOCA) 2 –3

251

Index

M Mach waves 43– 7 macroscopic crack branching 209 macroscopic damage models 224– 7 material stress fringe values 110 Maxwell –Neumann stress optic law 117 –18 MD see molecular dynamics mechanical properties Araldite B 246 Homalite 245 Polymethylmethacrylate 245 method of caustics 116– 28 asymptotic ﬁelds 149 –50 crack branching 209 –11 crack growth toughness 181 –2 curve size 116 – 28 dynamic effects 127– 8 opaque specimens 117, 121– 3 reﬂection 121– 3 stress intensity factors 116– 28 three-dimensional effects 124– 7 microcracks 200 –4, 226 –7 microscopic crack branching 209 mirror 105, 193 mist 193 mixed-mode caustics 123 – 4 mode I cracks 35– 7, 116 –23 mode II cracks 123– 4 Modulus of elasticity 11– 12 moire technique 92, 138 –9 molecular dynamics (MD) 217 –18, 222 –3 moving cracks 60 –9, 144 –53, 218 – 20 multiple crack fronts 195 –6

N Navier’s equations of motion 9 – 10 nil-ductility temperatures (NDT) 6

nominally brittle materials 178 – 84, 187, 189– 215 non crystalline materials 190 –1 noncrosslinked thermoplastic polymers 113– 15, 157– 9, 204 –7 nonsteady crack growth 39 –43 notched bar impact testing 6– 7 nuclear containment vessels 2 –3 nucleation rates 226 –7

O optical interferometry 128– 35, 138– 9 optical mapping 119 –21 optical properties Araldite B 246 Homalite 245 Polymethylmethacrylate 245 organic polymers 189 – 90

P P-waves 11 –13, 43 –7, 98 –100 PanAm 103 747 airplane 5 –6 parabolic markings 196, 198 PC see polycarbonate pentaerythritol tetranitate (PETN) explosives 93 periodic morphology 196 – 7 PETN see pentaerythritol tetranitate phenomenological models 217– 27 photoelasticity 109– 16 physical aspects of dynamic fracture 189 –215 crack branching 208 –15 fracture surface roughness 193 –208 limiting crack speed 190– 3 pipelines 3 – 4, 175 – 6 plane strain 16– 17, 32– 7 plane stress 12, 17 plane waves 12, 13 plate impact tests 90 – 1

252

Index

Plexiglas see polymethylmethacrylate PMMA see polymethylmethacrylate point loads 57 –60 Poisson‘s ratio 11 – 12 polarization 110 – 16 polycarbonate (PC) 113 –15, 157 –9, 204– 7 polymers 113 – 15, 157 – 9, 189 – 90, 204– 8, 220 polymethylmethacrylate (PMMA) crack growth 178 crack speed 102 fracture surface roughness 196– 201, 203– 7 interferometry 134– 5, 138– 9 lateral shearing interferometry 134– 5 limiting crack speed 191– 2 mechanical properties 245 microcracks 203– 4 optical properties 245 stress intensity factors 151 – 2 polyurethane elastomers 204– 5 post-mortem X-ray analysis 200 potential drop methods 102 –4 pressure bar impact tests 91– 2, 159– 61, 164– 8 pressure loading 94– 5, 141– 4, 163– 4 pressurized thermal shock 2– 3 projectile impact 82, 87 –91 propagating cracks 60 – 9, 144– 53, 218– 20 pulse loading 89 –92, 157 –9, 213 –14

Q quasi-static loading

82– 7

R Rayleigh function 18– 19, 21– 2, 51– 2 Rayleigh wave speed 12 rays 14 – 15, 117 – 18 reactor grade steel 164 –7

rectangular double-cantilever beams 170– 1, 180– 3 refraction index 109 resistive grid methods 102 – 4 ripple pattern wave lengths 101 – 2 rotating cameras 105 roughness 193 – 208

S S-waves 11 –13, 43 –7, 98 –100 scale-models 175– 6 secondary stress pulses 213– 14 self afﬁne roughness scaling 194– 5 semi-inﬁnite cracks 49 –69, 141 –4 SEN see single edge notched SH and SV waves 13 shadow spots 119, 125 shape of conic markings 201 – 2 shear Mach waves 43– 7 shear waves 11 –13, 43 –7, 98 –100 shearing interferometry 128 –35 short duration stress pulses 157– 9 single edge notched (SEN) specimen 91– 2, 180– 82 singular ﬁeld dominance 146 – 53 singular terms, steady-state crack stress 239– 41 size of conic markings 201– 2 Snell‘s law 118 Solithane 113 polyurethane elastomer 204– 5 split-pin loading 173 – 4 static loading 81– 95 stationary cracks 49– 60, 141– 4 stationary ﬁelds 239 steady-state ﬁelds 239– 43 steels 151 – 2, 156, 163 – 7, 184 – 7 step tensile stress loads 67– 9 strain gages 91– 2, 135– 8, 167 plane 16 –17, 32 – 7 rate testing 91 –2 tensors 9

253

Index

strain ﬁelds 107– 39 interferometry 128– 35, 138– 9 Jones calculus 107 –9 lateral shearing interferometry 128 –35 method of caustics 116– 28 photoelasticity 109– 16 strain gages 135 –8 stress amplitude 100, 158– 9 crack branching 211 –14 dynamic crack tips 33– 4 fractography 100 – 2 induced birefringence 109 –16 intensity-crack velocity relationship 177 –87 nonsteady crack growth 41 –3 optic law 117– 18 plane 12, 17 tensors 9 waves 100 –2, 147 – 53 stress ﬁelds asymptotic 146 –53 dynamic crack tips 37– 8 inertial constraints 163 stationary cracks 53 –6 see also crack tip... stress intensity factors asymptotic ﬁelds 142 –53 caustic curve size 116– 28 fractography 100 microcracks 203 –4 see also dynamic... surface roughness 193 –208 surface traction-separation relationship 221 surface waves 18– 19 symmetric deformation 32 –7 symmetric loading 241 – 3

T tear straps

176– 7

temperature dependence 159– 68, 177 –8 tensile stress 67– 9, 89– 90, 157– 9 tension pulse impact tests 91 – 2 thermoplastic polymers 113 – 15, 157 –9, 204 –7 thermoset plastics 192 thermosetting epoxies see Araldite thermosetting polymers 208 see also Homalite three-dimensional effects 124– 7, 145 tilting 192 time-dependent stress intensity factors 53– 4, 59– 60 toughness crack arrest 170– 7, 203– 4 crack growth 74 –9, 177 –87, 203 –4 crack initiation 85, 136– 7, 155 –69, 203 –4 traction free cracks 30 –2, 39 –43 separation relationship 221 vectors 10 transient crack tip ﬁelds 243 –4 transient effects 127 –8, 145, 147– 9 transmitted caustics 119, 125 transparent specimens 116 –21 transverse diameter of the caustic curve 121 tup loads 84 twisting 192 two-dimensional linear elastodynamics 15 – 17 Tyman-Green interferometers 138 – 9

U ultrasonic transducers 101 –2 unbounded media 60– 6 undulations 98– 100 uniform loading 49 – 56, 141 – 4, 209 –10 universal function 63 –4, 66 –7

254

Index

W

Wiener-Hopf factorization

Wallner lines 98– 100 wave fronts 14– 15 wave speeds 11 – 13, 43– 7 Wheatstone bridge circuits 102 –4

Y

52, 63

Yoffe problem of moving cracks 60– 3