Indentation Fracture E-Book

Robert F Cook

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 1076

Veröffentlichungsjahr: 2025

Beschreibung

A comprehensive reference with demonstrations of advanced indentation fracture techniques in practical applications to enable better design and more efficient manufacturing

Indentation Fracture describes and illustrates advanced applications of indentation fracture in evaluating strength, toughness, and related properties of brittle materials. The book enables better design, manufacture, performance, and reliability of brittle materials as elements in structural components. Emphasizing the practical applications of indentation fracture techniques, especially those of indentation-strength, the book builds on a development of indentation fracture mechanics to demonstrate clear quantitative comparisons and rankings of material fracture behavior, including the behavior of toughened materials.

The book includes several case studies linking indentation fracture analyses and measurements to other phenomena including fragmentation cracking, edge chipping and dicing damage, scratching and the Mohs scale, and semiconductor grinding damage. Each case study highlights a different aspect of controlled flaw fracture measurements.

The book enables readers to increase the utilization of glasses, ceramics, semiconductors, and similar brittle materials in load-bearing applications by demonstrating experimental indentation techniques to better characterize fracture. A primary focus of the book is demonstration of controlled flaw indentation-strength measurements.

Written by an independent scientist with significant research contributions to the field, Indentation Fracture explores a range of topics including:

• Brittle fracture fundamentals, discussing equilibrium fractures on uniform and localized loading leading to the Griffith and Roesler equations
• Indentation strength variation, covering the effects of contact angle, spherical and flat punch contacts, and contacts in stressed and layered components
• Toughened materials strength observations in alumina, silicon nitride, glass ceramics, ferroelastics, and zirconia
• Strengths of silicon devices determined by wafer backgrinding, covering linear flaws in silicon, including single and multiple scratches, as well as linear, general, controlled, and grinding flaws

Indentation Fracture is an essential reference on the subject for engineers across disciplines, nanotechnologists, geophysicists, and environmental scientists, along with graduate students in materials science and related programs of study.

Leseprobe

Indentation Fracture

Strength and Toughness for Brittle Materials Design

Robert F. Cook

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Cover Design: WileyCover Image: © Robert F. Cook

To Brian and Shelley,

who gave me every opportunity to learn about fracture

Preface

This is a book for experimentalists. In particular, it is for all those, including me, who obtain experimental results using the techniques of indentation fracture and ask themselves the same questions I have often asked myself: How do my results compare with those of other researchers? Are the values the same? Are the uncertainties the same? How can I use the results? How can I analyze the results? The goal of this book is to provide answers to these questions and to advance the field of fracture within the broader fields of mechanics and physics.

Indentation fracture techniques are one of the most effective means of determining the mechanical properties of brittle materials. Brittle materials—glasses, ceramics, semiconductors, rocks, minerals, bones, teeth, and seashells—typically provide many advantageous engineering properties: lightweight, resistant to elastic and plastic deformation, electrically and thermally insulating, often optically transparent, usually corrosion resistant, and formable from raw materials commonly found on earth. The uses of brittle materials thus range over domestic, architectural, industrial, cyber, and biological applications. The major limitation, however, is that they are brittle. On tensile loading, components of brittle materials fail by fracture after limited elastic deformation—they are usually not very tough. Consequently, the fracture strengths of such materials are typically small, often variable, and often time dependent, significantly limiting structural applications, despite the other, superior, properties. This book describes and illustrates advanced applications of indentation fracture in evaluating strength, toughness, and related properties of brittle materials.

In the 1921 fracture study of Griffith, the physics of brittle fracture was shown to be a manifestation of the first law of thermodynamics: Work performed on a component by applied loading is converted into internal elastic strain energy within the component that is then converted by fracture into surface energy on newly formed surfaces of the component. The mechanism by which the final conversion to surface energy takes place is crack propagation, the expansion of a plane of ruptured bonds in a component—a crack—leading to component failure. Griffith showed that the critical stress required to propagate a crack was an inverse function of crack size. Hence, variation of fracture strengths within a group of components is explained by variation in the size of the strength controlling cracks or flaws.

Alternatively, deliberate introduction of controlled flaws into components can circumvent strength variability. This obvious method to enable assessment of material strengths was implemented from the earliest studies. Griffith used annealed scratches. In the 1970s, Marshall and Lawn developed experimental techniques and fracture mechanics analyses to describe the cracks formed by sharp diamond indentation contacts on brittle surfaces. Subsequently, thousands of studies have implemented indentation techniques to evaluate fracture properties of brittle materials. The vast majority of these studies make use of the simplicity and economy of indentations to generate measurable stable surface cracks from which material toughness is estimated. Very few make use of the controlled flaw properties to investigate material strengths and the underlying flaw and toughness variations or make use of the myriad additional applications. Design, manufacturing, and application of brittle materials is impeded by this limited use of indentation fracture.

This book demonstrates a diversity of indentation fracture techniques with an emphasis on indentation strength tests. In these tests indentations are the controlled flaws in strength measurements of components. The indenter used in the experiments described throughout the book is primarily the four‐sided Vickers diamond pyramid, although other indenters such as the cone and three‐sided Berkovich and cube corner are considered. The book begins with an overview of the fundamentals of brittle fracture, clearly defining strength and toughness and making connections with broader physical concepts. This background is followed by a study of the lengths of cracks generated by indentations prior to strength testing. This in turn is followed by detailed and extensive considerations of indentation strengths. The considerations are initially for materials that exhibit invariant toughness (primarily glass) and then for materials that exhibit increases in toughness with crack length (primarily alumina). The two chapters concerned with these topics are the longest in the book and are the foundation for much that follows. Measurement of toughness of brittle materials at strength relevant length scales is then described. The measurement is based on the simultaneous determination of crack length and failure stress at fracture instability in indentation strength tests. This is followed by consideration of static and dynamic fatigue—the time‐dependent failure of brittle materials in reactive environments. For many brittle materials this environment is simply water or moisture. Indentation techniques are demonstrated for determination of fatigue kinetics and the critical threshold stress for fatigue failure. The next three chapters demonstrate the power of the indentation technique to control the size, location, and geometry of flaws in studies of the common phenomena of fragmentation, chipping, and scratching, including shedding light on the well‐used mineralogical Mohs scale. The three following chapters have a materials focus, demonstrating application of indentation techniques to characterize the fracture properties of semiconductors, ceramic composites, and toughened materials such as the chemically treated glasses used in mobile phone screens. The final chapter brings all the indentation strength measurements together to illustrate the power of indentation fracture techniques in designing with brittle materials.

Some elements of the book may be surprising. Very little attention is given to measurements of indentation crack lengths. This oft‐debated subject is discussed in many publications. Much attention is given to the effects of lateral cracks on strength. Lateral cracks markedly influence the indentation strength behavior of nearly all brittle materials over the most commonly used indentation load domains. Failure to account for lateral crack‐related relaxation of indentations stress fields can lead to overestimation of toughness and misidentification of indentation fracture data as indicating microstructural effects, crack blunting, crack healing, phase transformations, densification, or surface stresses. As will be noted it is almost impossible to discuss the effects of microstructure on the fracture strength of polycrystals without considering the effects of lateral cracks.

Many of the ideas presented and developed here have formed over the years from discussions and interactions with colleagues, including S. J. Bennison, T. P. Dabbs, E. R. Fuller Jr, B. R. Lawn, D. B. Marshall, M. L. Oyen, G. M. Pharr IV, M. V. Swain, R. Tandon, and S. M. Wiederhorn.

Raw data for many of the results contained in Chapters 4, 6–8, and 10–12 were obtained with the extremely able experimental assistance of K. L. Donsbach, M. P. Hughey, E. G. Liniger, R. L. Mendelson, D. J. Morris, R. Tandon, and A. M. Vodnick. Dr M. V. Swain is thanked for provision of the raw data for Figure 9.10.

The thinking and writing for this book were conducted on the Atlantic Ocean coast and in the Appalachian mountains of North Carolina—in Duck, Manteo, Wilmington, and Asheville—with the patient and much appreciated encouragement of my partner Michelle.

June 2024

Robert F. CookWilmington, NC

Abbreviations and Symbols

ASG

aluminosilicate glass

BSG

borosilicate glass

CTE

coefficient of thermal expansion

FSG

fused silica glass

pdz

plastic deformation zone

SIF

stress‐intensity factor

SLG

soda lime silicate glass

TZP

tetragonal zirconia polycrystal

contact impression semi‐diagonal; radius of inclusion; radius of force application or contact

component dimension; characteristic pdz dimension

crack length

function; microstructural line force; frequency

component dimension; distance to edge; Planck's constant

function

indices

component stiffness

Boltzmann's constant

pressure; partial pressure

radial coordinate

scallop dimension

mass

time

displacement

crack affected volume; crack velocity; particle velocity

spatial coordinate

crack area

cracking susceptibility

strength degradation; scallop dimension

Young's modulus

force applied to component

mechanical energy release rate

hardness

stress‐intensity factor

component length

elastic modulus

Mohs number

indentation load

empirical distribution function

heat

component radius; fracture resistance

entropy; contact strength

toughness

energy

energy density

component volume; pdz volume

work

linear CTE; numerical coefficient

numerical coefficient

surface energy

load‐point displacement; indentation depth; microstructural traction zone length

strain

thermal energy/area

bulk elastic modulus

component compliance; material grain size; crack branching distance

chemical potential; microstructural geometry term

Poisson's ratio

residual stress geometry term

temperature; angular coordinate

indenter included semi‐angle

angular coordinate

material mass density

stress; strength

residual stress geometry term

applied stress geometry term

relative crack length

finite change in quantity

adsorption coverage

entropy production rate

thermodynamic state function

crack nucleus geometry term

“varies as”

“approximately equal to”

Acronyms include the plural. Subscripts, superscripts, indices, dummy variables, bold variables, annotated variables, and less frequent uses of these and other symbols are defined in context.

1Brittle Fracture Fundamentals

This chapter defines and describes in clear, simple mathematical terms the physical, thermodynamic, and mechanical basis and associated terminology of brittle fracture. The transformation on fracture of mechanical energy (including work and elastic energy) into surface energy is shown as the fundamental brittle fracture process. From an energy viewpoint, fracture equilibrium is defined by the balance between configurational forces associated with these energies: the mechanical energy release rate and the fracture resistance. From an equivalent stress viewpoint, fracture equilibrium is defined by the balance between the stress‐intensity factor and the toughness. The limiting cases of uniform and localized loading of cracks are shown to correspond to mechanical energy release rates that increase and decrease with crack area, respectively, leading to the well-established Griffith and Roesler equations of fracture. The limiting cases are similarly described by stress‐intensity factors that increase and decrease with crack length. Nonequilibrium fracture is described by the addition of a kinetic energy term into analysis of adiabatic fast fracture and the addition of an entropy term into analysis of isothermal slow fracture. Fracture in a spatially varying loading field is considered through the example of crack initiation at a misfitting inclusion. The example is based on stress‐intensity factor behavior and highlights considerations of fracture stability.

1.1 Brittle Components and Materials

Many of the objects encountered in everyday life are composed of brittle materials. On mechanical loading such materials are characterized by several distinctive responses—of particular distinction is the nature of failure on tensile loading. On exposure to excess tension brittle materials exhibit limited deformation followed by instability and failure in the form of abrupt fracture and separation into two or more fragments. After failure the fragments recover their initial unloaded and undeformed shapes. The fragments include newly formed surfaces on which the fracture and separation took place. Prior to failure, deformation is elastic and reversible. In a related phenomenon, on excess compressive loading at surface contacts brittle materials exhibit a mixture of localized reversible and irreversible deformation followed by adjacent cracking and chipping. After contact the surface recovers its unloaded shape, usually incompletely, and newly formed fracture surfaces remain in the form of stable cracks and chips. These two characteristic responses are often related: components formed of brittle materials often fail in applied tension from cracks generated by prior compressive contacts.

This book combines the two related responses just discussed in a deliberate way, showing how indentation fracture (intentionally introduced contact cracks) can be used to both evaluate parameters for brittle materials design (and avoid component failure) and understand everyday brittle fracture phenomena (such as fragmentation and chipping). This chapter develops and summarizes the fundamental mechanics and thermodynamics of brittle fracture as a basis for Chapters 2–12 to follow. The chapter begins with phenomenological descriptions of brittle fracture and failure and the transformations of energy that characterize such behavior. This is followed by development from an energy viewpoint of analyses describing equilibrium fracture in limiting stable and unstable configurations, and how such analyses are implemented in practice from a stress viewpoint. The limiting stability configurations are shown to correspond to the two characteristic brittle fracture responses discussed earlier. Experimental data from the last 100 years that exemplify stable and unstable equilibrium fracture are given. Nonequilibrium fracture is then considered, also in two limits, as adiabatic and isothermal behaviors that pertain to these responses. Finally, a practical example of fracture mechanics is given, that of cracking in a brittle material at a misfitting inclusion. The example highlights the stress‐based analysis method used throughout the book and demonstrates unstable, stable, adiabatic, and isothermal fracture behavior in a single system. The inclusion system exhibits the phenomenon of crack initiation and provides a basis for description of many other fracture systems. Although intended as a basis for the analysis and application of indentation fracture, the material in this chapter is general and applies to all fracture.

1.1.1 Components

Figure 1.1 illustrates a brittle bar loaded in tension. The bar is gripped at each end and loading consists of forces applied at each grip. The directions of the forces are outward, tensile, and parallel to the bar axis. The bar is rectangular in cross section with dimensions and and has length between the grips. Under the actions of the forces, the bar extends by a load‐point displacement . and are taken as positive in tension and directed as drawn. The force–displacement – response of the bar is shown in Figure 1.2a. At a maximum force , the bar fractures and separates into fragments, and the force supported by the bar decreases to zero with negligible further increase in displacement. The fragments recover to their unloaded, undeformed configurations. Schematic diagrams of the unloaded bar (dashed outline), loaded bar (shaded), and unloaded fragments (solid outline) are also shown in Figure 1.2a. This process is brittle failure.

Figure 1.1 Schematic diagram of a bar under uniform tensile loading. On application of forces the bar ends between the grips undergo relative displacement .

The force–displacement behavior of most brittle components is linear elastic, most noticeably prior to component failure. The behavior is linear, as the – relationship is given by , where is a displacement‐independent stiffness. In Figure 1.2a, is the stiffness of the bar. The behavior is elastic, as is also time‐invariant, such that the relation represents reversible equilibrium bar and fragment configurations. Prior to failure these are the loaded configurations of the bar. After failure, these are the unloaded configurations of the fragments. Macroscopically, the prefailure equilibrium states are balances between the applied forces at the grips. Microscopically, these equilibrium states are balances between the applied forces and the restoring forces exerted by atomic interactions within the bar. The parameters , , , and are indicated in Figure 1.2a. Linear elastic behavior for most components usually requires to be very small. After failure, the bar ceases to function as a structural component capable of supporting a mechanical force. Equivalently, linear elastic behavior and failure may also be caused by an imposed displacement , in which the applied loading generates an associated force such that , where is the compliance of the bar. For linear elastic behavior . A structural component is thus also capable of supporting an imposed deformation. In Figure 1.1 the applied forces are shown as dark shaded arrows and loading fixtures indicated by hatching—unless otherwise stated, this scheme is used throughout the book. Terms in italics are mostly definitions to be used throughout—some are specific to this book.

Figure 1.2 Brittle failure behavior of the bar and beam shown in Figures 1.1 and 1.3: (a) as a component, shown in a force–displacement plot and (b) as the underlying material, shown in a stress–strain plot. On application of loading, the force or stress increases linearly with displacement or strain. At a maximum force or stress, the supported load abruptly decreases to zero. The resulting fragments are undeformed. Schematic diagrams of the bar responses are shown in (a).

To perform as a structural component the condition must pertain for all forces and displacements imposed in application. To include an engineering “safety factor,” this operational condition is usually extended to . However, it is often not obvious that a functional brittle component is also acting as a structural component and that comparisons of and are thus necessary to guarantee component performance. Atomic interactions within brittle materials often impart useful material properties that lead to more obvious material and component functions. For example, the thermal insulating nature and small manufacturing costs of ceramics lead to their common use as house bricks and tableware; the controllable electrical conductivity of silicon leads to its extensive use as a semiconductor in myriad microelectronic devices; the optical transparency of glasses leads to their wide use as building windows, camera lenses, and mobile device screens; the electrical insulating nature of glasses and ceramics is also critical in microelectronic devices; the chemical and wear resistance of ceramics and glasses lead to their use as coatings in corrosive and abrasive environments; and, finally, most brittle materials are also resistant to elastic deformation and have small densities, rendering them ideal for applications in which dimensional stability is required, such as telescope mirrors.

The brittle materials used in microelectronic devices provide a pervasive example of the structural requirements “hidden” in many functional applications. To span a range of electrical properties—conducting, semiconducting, and insulating—a wide range of materials is required in the fabrication of microelectronic devices, e.g., copper, silicon, silica, and polyimide. This range leads to ubiquitous thermal expansion mismatch effects between the disparate materials. The inevitable consequence is the development of large tensile forces arising during thermal excursions in device fabrication and operation and the equally inevitable consequence of brittle component failure, e.g., of the silicon and silica glass components. In such cases, the device ceases to perform not because of a change in material electrical properties but because of inherent mechanical design factors. (Hence, significant attention is given in this book to brittle failure of materials used in microelectronic devices—Chapters 8–11.)

The force required to cause a structural component to fail in its load‐bearing function is the component failure force (sometimes referred to as the supportable force, Ashby 1999). The terms loaded and loading are used throughout as broad descriptors to refer to any configuration that generates applied forces on components. Loading may arise from the direct application of a force, as in an overlying mass giving rise to a gravitational weight (a “load”). Loading may arise from an imposed displacement, in which case the force arises indirectly as a reaction to the displacement, as in thermal expansion mismatch configurations. Loading may also arise from an imposed change in displacement rate, in which case the applied force arises indirectly as a consequence of deceleration of a moving mass, as in impact configurations. A loading cycle is thus composed of load–unload segments, separated by the peak force generated in the cycle.

A key aspect of brittle component design is thus to ensure that the component failure force always exceeds the operational peak force. Design with brittle components thus requires knowledge of . In turn, depends on the component dimensions and the mechanical properties of the brittle materials forming the component. This dependence and brittle material properties are considered in Section 1.1.2.

1.1.2 Materials

Attention here is focused on the linear elastic behavior in Figure 1.2a, prior to bar failure. To compare bars of different dimensions, it is convenient to define the engineering stress acting on the bar by and the engineering strain of the bar by . The linear elastic behavior of the bar expressed in terms of these quantities is thus or . The usual condition for the observation of linear elasticity can thus be restated as one in which the strain is very small, . Using an invariant combination of extensive (size dependent) parameters, , experiments show that the stress–strain expressions describe the deformation of bars formed of the same material but with differing dimensions. The implications are that and are intensive (size independent) quantities describing the loading on the bar and the deformation of the bar, respectively, and that the invariant extensive combination can be expressed as , where is an invariant intensive material property. If the bar is long or “slender” (Ashby 1999), , the quantity is termed the Young's modulus of the material. is positive. The expression for the intensive linear elastic stress–strain response of the material forming the bar is thus . The brittle failure stress or strength of the bar material is given by . Figure 1.2b shows the brittle failure behavior of the bar material in a stress–strain plot. In this example, , , and are all positive. A major focus of this book is measurement and prediction of the material brittle fracture strength. Such measurements and predictions enable determination of component failure forces for use in design with brittle materials.

Two additional aspects of elastic deformation are factors in brittle fracture strength considerations. The first is the transverse strain generated by the axial loading in Figure 1.1. The second is the work performed by the loading. Much more detailed information regarding elasticity can be found elsewhere (Mase 1970; Timoshenko and Goodier 1970; Sadd 2009). Atomic interactions within the bar lead to internal connectivity of the bar material and transverse contractions in the and dimensions associated with extension in the dimension. For the bar here, as and both dimensions are unconstrained transversely along ( indicates “approximately equal to”), the transverse strain along the length of the bar is uniform and given by . is an invariant material property termed the Poisson's ratio. is intensive and usually positive, 0.3. An elastically isotropic material is one in which the stress–strain response of the material is identical in all directions and for which and completely describe material deformation. Such a material is also elastically homogeneous, in which the deformation response is identical at all points. If the bar of Figure 1.1 is uniformly composed of an elastically isotropic material and is in equilibrium with the applied force, stress and strain are distributed homogeneously throughout the bar. If one of the transverse dimensions of the bar in Figure 1.1 is large relative to the other, say , the stress–strain relation becomes , where is termed the “plane strain modulus.” Other geometrical constraints lead to other moduli expressed as functions of and are used in this book, for example, the “biaxial modulus” and the “indentation modulus” (Mase 1970; Timoshenko and Goodier 1970; Sadd 2009). (The symbol is used for an arbitrary elastic modulus defined in context.)

The work performed by the applied force in deformation of the bar is expressed in extensive terms as (the area under the – response in Figure 1.2a). Substituting the expressions for stress and strain into the extensive work expressions gives . These expressions embody the First Law of Thermodynamics and make clear that the work performed by the applied force generates a change in the internal energy of the bar. The change can be expressed as , where the E subscript indicates internal elastic energy, and is set at . For example, internal elastic energy can thus be expressed as ; a product of a combination of intensive parameters within the first set of parentheses and a combination of extensive parameters in the second set. The other combinations of intensive parameters are similarly expressed. The combination of extensive parameters is recognized as the volume of the bar, . The implication is that the combinations of parameters within the first sets of parentheses must represent the mean elastic energy density (energy/volume) of the bar. Denoting the elastic energy density as gives , where, for example, (the area under the – response in Figure 1.2b). is extensive and is intensive, both are positive. As stress and strain are distributed homogeneously throughout the bar, elastic strain energy is also distributed homogeneously and describes the elastic energy density at all points in the bar material.

1.2 Transformations on Brittle Fracture

1.2.1 Observations

Figure 1.3 illustrates a beam loaded in three‐point bending. The applied loading consists of force imposed through an upper movable roller and directed toward two lower fixed rollers. The direction of the force and the roller axes are perpendicular to the beam axis. Under the action of the force the beam bends and the beam center exhibits relative displacement . and are taken as positive when directed as drawn. At a maximum force the beam fractures and separates and the force supported by the beam decreases to zero with negligible further increase in displacement, identical to that shown in Figure 1.2a. However, in the bending example of Figure 1.3, the stress, strain, and elastic strain energy are not homogeneously distributed throughout the beam. The maximum longitudinal stress is modified by the aspect ratio of the beam, , and occurs on the lower face directly beneath the central roller (using the same parameters as Figure 1.1). Defining gives , reflecting the strain energy density variation. Other inhomogeneously loaded systems exhibit similar variations (Mase 1970; Timoshenko and Goodier 1970; Ashby 1999; Sadd 2009). In particular, Figure 1.4 illustrates a disk, radius , loaded in biaxial flexure. The applied loading consists of force imposed through an upper movable punch directed toward lower fixed supports arranged in a circle. The direction of the force is parallel to the disk axis. Under the action of the force the disk flexes and the disk center exhibits relative displacement (not shown). is taken as positive when directed as drawn. At a maximum force , the disk fractures and separates and the force supported by the disk decreases to zero with negligible further increase in displacement, identical to that shown in Figure 1.2a. The maximum biaxial stress, , occurs on the lower face directly beneath the punch and (where indicates “varies as”). Stress analyses for bending and flexure are considered in detail elsewhere (Timoshenko and Goodier 1970).

Figure 1.3 Schematic diagram of a beam under loading by three‐point bending, leading to uniaxial flexure. On application of force the center of the beam undergoes displacement perpendicular to the beam axis. Loading in bending is common for brittle materials.

To avoid contact damage associated with gripping (Figure 1.1), brittle materials are usually mechanically tested in bending (Figure 1.3) or flexure (Figure 1.4). In addition, although the tensile bar of Figure 1.1 is a simple loading example, most brittle components also fail in bending or flexure, Figures 1.3 and 1.4. Familiar examples include: a dropped ceramic plate experiences bending on impact with a floor, between the moving upper sections of the plate and the lower stationary sections at the fixed floor; a glass window experiences flexure on loading by the wind, between the center and the fixed frame. Less familiar, more technological, examples are shown in Figures 1.5 and 1.6 and provide insight into the energy transformation that occurs on brittle fracture. Figure 1.5a shows an image of an intact semiconductor device. The metal and glass lines are visible on the underlying silicon substrate. Figure 1.5b shows an image of the two fragments from a device loaded in bending (Figure 1.3) until fracture. The fracture surfaces are visible in the center of the image. Figure 1.5c shows an image of the two fragments from Figure 1.5b reassembled into the original unloaded configuration. A faint separation is visible in the center of the image but otherwise the configuration is identical to Figure 1.5a. Figure 1.6a shows an image of a disk of translucent polycrystalline alumina () similar to that used as the bulb material in Na vapor street lamps. Figure 1.6b shows an image of three fragments from a disk loaded in biaxial flexure until fracture. Figure 1.6