Guidelines_for_Fracture_Mirrors_Quinn

Guidelines_for_Fracture_Mirrors_Quinn - 1, July 7,2006...

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Unformatted text preview: 1, July 7,2006 GUIDELINES FOR MEASURING FRACTURE MIRRORS Conference on Fractography of Glasses and George D. Quinn Ceramics, . . Roch ter, NY, Jul 942. 2006 National Institute of Standards and Technology es y Ceramics Division Stop 8529 Gaithersburg, MD 20899 ABSTRACT Fracture mirror size analysis is a powerful diagnostic tool for glass and ceramic fractures. There is considerable variability in published fracture mirror constants, however, especially for ceramics. Mirror—mist or mist~hackle boundaries are not sharp, well—defined transitions and they are subject to interpretation This paper presents some guidelines for mirror boundary inter— pretation and analysis that are from a new NIST Guide to Practice for Fractography. Some recommendations are based on common sense, others are based on solid technical grounds, and some are based upon the experience of other experts and the author. The guidelines are intended to help students, novices, engineers, and materials scientists use fracture mirror analysis to solve practical problems. INTRODUCTION Fracture mirrors are relatively smooth regions surrounding the fracture origin. In glasses and very fine~grained ceramics, the mirror region is very smooth and highly reflective, hence the name “fracture mirror.” In polycrystalline ceramics or composites, the qualifier “relatively” must be used, since there is an inherent roughness from the microstructure even in the area immediately surrounding the origin. Mirrors are circular in ideal loading conditions such as tensir n specimens with internal origins, or near~semicircular for surface origins in tensile specimens or small mirrors in bending as shown in Figures 1 and 2. Fracture mirrors only form in moderate— to high—strength specimens. Weak specimens do not have fracture mirror boundaries since the crack does not achieve sufficient velocity within the confines of the specimen. Fracture mirrors often are elongated or distorted due to stress gradients or geometrical effects. ‘ Fracture mirrors bring one’s attention to an origin, but also give information about the magnitude of the stresses that caused fracture and their distribution. One of the earliest observations in this regard was by Brodmann in 1894:1 “The radially measured size of these mirrors in general was larger, the smaller was the strength.” The relationship between strength and mirror size was gradually refined so that in the 19505 and 1960s, many authors measured mirror sizes and correlated them with stresses in accordance with the relationship12’3’4’5’ 6189 O‘RmzA (1) where e was stress, R was the mirror radius, and m and A were constants. The data was frequently plotted on raphs of log stress versus log radius in order to check for the goodness of fit of equation 1. I was recognized fairly early that m was 1/2 for annealed glasses, and equation I was rewritten as: 0R%:A (2) Figure l Fused silica rod broken in flexure (122 MPa). The origin is a surface flaw located at the bottom of the specimen (a) where the stress was a maximum The mirror—mist boundary is small relative to the cross section size {shown in the insert in (a)} and is approximated by a circle in (b). Close examination of the fracture surface in the Vicinity of the flaw (not shown) showed that fracture started from the deepest part of the flaw. The sizes and arcs were judged while Viewing through a compound optical microscope and then marked on the photographed image. (Some subtle detail in the mist may be lost during printing) The mist~hackle boundary is slightly elongated towards the top. (c) is an SEM image of the same mirror and at the same magnification as (a) and (b). The mist is indistinct in the SEM image. (d) is a composite of two SEM images showing the transition from mirror to mist to hackle. The location of the boundaries as assessed by optical microscopy are marked by dashed lines on the SEM images. Ab A0 Ai ch Figure 2 Schematic of a fracture mirror centered on a surface flaw. The various mirror boundaries and their corresponding fracture mirror constants (A) are marked. The initial flaw at the origin may grow in size from a, to aC at which point the critical fracture toughness, K10, was reached. m .0001 .0002 .0005 001 .002 ‘1' r—i‘ r "Y v H v v i l l «ifi r—r'T—v ~1 '1 _ “f T r ' r ' ROD DtAM tea—INCHES 3 (I) 9— r A Q 9'35 w 200 Q U U.C‘+ , g v 0.39 ' ——’ 0 0,75 s g A [.00 MP8 (9 L50 _ E ° 100 o e. i 0 v.85)“ 3:1 gggeg‘gggfi” " 3 0V .ufijifi‘. . ~ [-— 8 803.3 £619. . A it) .gAC’MA-‘a M .l-SO m o ' 0 LL. :. i j 2 r w j 30 (I) “J l E I i I l _L_.L A l4 Jul-LLJ‘LL I | w .002 .004 .006 .01 0:5 .02 .04 .06 .08 .|0 .|5 MiRROR SIZE. INCHES Figure 3 Fracture stress (adjusted for the origin location) versus mirror size for a borosilicate glass from Kerper and Scuderi.8 This is the original figure from their 1966 paper and it shows the older system of units then in use with a few SI units added on the top and right side. They fitted the data with eq. 2 for 259 rods broken in the three—point bending and obtained a mirror constant of 2.08 MPan (1,891 psi/in”). The rod diameters varied almost by a factor of 10, the mirror sizes by a factor of 50, and the stresses almost by a factor of 10. In this form, A is the “mirror constant” with units of stress intensity (MPan or l<si\/in) and is considered by many to be a material property. It is possible to discern separate mist and hackle regions and the apparent boundaries between them in glasses and each has a corresponding mirror constant A. The most common notation is to refer to the mirror—mist boundary as the inner mirror boundary and its mirror constant is designated A1. The mist—hackle boundary is referred to as the outer mirror and its mirror constant is designated A0. The mirror— mist boundary is usually not perceivable in polycrystalline ceramics and only the mirror~hackle boundary is measured. v This simple relationship has tremendous practical value. It is not necessary to have a—priori information about the how the part was loaded. The smaller the mirror, the larger is the stress at the origin site. A small mirror is proof that the part was strong and had a small strength—limiting flaw. Conversely, large mirrors mean the failure stress was lower and implies a large flaw. In some instances, a part may be so weak that the mirror size is larger than the part cross—section and hence the mirror markings are not visible. That, in and of itself, is valuable information. The existence of a mirror boundary implies that the part was stressed to a moderate or high level. Alternately, very strong parts (e. g., pristine optical fibers) may have such small mirrors that they are not measurable with optical microscopes. The mirror may not even be found due to excessive fragmentation. Errol Shand of the Corning Glass works was a leader in the early analysis work3'5 His 1959 paper presented an early argument for the 1/2 power in equations 1 and 2 based on stress concentrations at the tip of a sharp crack. This paper incorporated early elements of what would become known later as fracture mechanics. Kerper and Scuderi6’7’8 at National Bureau of Standards in Washington performed meticulous experiments on hundreds of glass laths and rods and showed conclusive evidence that equation 2 was applied over a broad range of specimen and mirror sizes as shown in Figure 3. Levengood2, Shands, and later Kirchner10 ‘redited Leighton Orr of the Pittsburg Plate Glass company for equation 2. Orr did not publish his findings until 1972 when he retired,9 but presented equation 2 in empirical form at a Glass Division meeting of the American Ceramic Society in 1955. H So although many associate equation 2 with Johnson and Holloway12 in 1966, the relationship was already in use for over 10 years. Their primary contribution was to use an energy analysis to give some theoretical underpinnings to equation 2. Henry and James Kirchner and James Conwaym’m later presented compelling evidence that a fracture mechanics criterion based on a critical stress intensity (Kl) gives the best fit to data and predicts the exact shape of mirrors in various stress fields. They argued that a more fundamental material parameter might be K18, the stress intensity factor at branching, since equation 2 does not take into account the free surface, geometry factors, and non—uniform stress gradients over the crack surface. Their work is considered again later in this paper. Equation 2 may be used in practice to estimate the stress at an origin. The mirror radius is measured from a fracture surface, the mirror constant found from a data table, and the origin stress calculated. A splendid example of this approach was shown by Morrell et al.15 for hip balls at the last fractography conference in 2000. The calculated stress is the net tensile stress acting on the flaw and the region around the flaw. It may include several stress sources including mechanical, thermal, and residual stresses. The author has compiled hundreds of fracture mirror constants for many materials in an American Society for Testing and Materials Standard16 and more recently in anew National Institute of Standards and Technology (NIST) Guide to Practice for Fractography.17 It became apparent that the reported values for identical materials (even for model materials such as fused silica or soda lime silica) varied by at least 20% (the range), and often much more. For some ceramics, the A values varied by a factor of two. This directly affects accuracy and precision of stress estimates using equation 2. Some of the scatter is due to metrology and judgment issues, but much of it is due to genuine material-to—material variability. Thus, one should not expect all aluminas to have the same mirror constant since there are major differences in the microstructures. The literature review also revealed that many authors were reluctant to show their mirrors with labeled photographs. In many instances authors did not document how they had measured their mirrors or critical details were missing. For example, many failed to report whether they corrected stresses for location or how they regressed the data. Confidence bounds on the A estimates were rarely reported. There is a strong subjective element to estimating the location of the mirror boundary even for glasses. Virtually everyone who has written about fracture mirrors has come to this conclusion. Johnson and Holloway12 wrote in 1966: “The position assigned to the boundary between mirror and mist zones depends upon the illumination and the magnification at which the fracture is examined, even within the range of the optical microscope. However, under given conditions a reproducible position for the boundary can be assigned.” Mecholsky and Freiman18 said in 1979: “While one might think initially that the measurements of a mirror boundary using a microscope is quite a qualitative operation and would vary from observer to observer; in fact, experiments performed over a number of years by a large number of investigators have shown that the values of mirror constants obtained in different laboratories are quite nearly tr e same.” They listed some values for a few glasses and ceramics that did in fact vary as much as 20 % to 30 % (range). For example, the soda—lime glass values varied by 23 %. Since many of the early mirror measurements were made while viewing through the optical microscope, it is safe to say that the first perception of a mirror boundary was where the surface roughness was a fraction of the wavelength of light (0.39 pm — 0.80 pm), Threshold levels of detectability have been estimated to be as small as a few tens of nanometers to as much as 0.25 pm. The judgment issue is exacerbated for ceramics, where intrinsic roughness due to the microstructure contributes to interpretation difficulties. Despite these problems, it cannot be denied that fracture mirror measurements are a powerful diagnostic tool for quantitative analysis. The accuracy and precision of the stress estimate is quite variable and depends on the material, the experience of the fractographer, the microscopy and illumination used, and component geometry and stress gradient effects. Stress estimates may be accurate to within 3 20 % with glasses if they form ideal fracture mirrors. The guidelines presented in this paper are intended to bring some long overdue consistency to fracture mirror analysis and improve the precision of this technique. The next two sections address two fundamental questions: 1. Is there really a boundary that can be measured? 2. Is there a theoretical basis for the fracture mirror constant, A? IS THERE A BOUNDARY? The word “boundary” must be used with some caution eVen with glasses. It is now clear that there probably is not a distinct transition point on the fracture surface corresponding to a mirror boundary. The higher the magnification the smaller the mirror seems to be, since fine detail that was washed out or not resolvable at lower power can be discerned at higher power. The mirror— mist boundary in glasses probably corresponds to surface roughness features that are of the order of0.l um to 0.2 pm, which is a fraction of the wavelength of light and at the threshold of observable features with an optical microscope. Mirror size estimates can vary depending upon the type of microscope used, the illumination, the objective power of the lens, and the judgment criteria of the fractographer. Careful electron microscopy and atomic force microscopy have shown that the formation of the mirror is a gradual progression of very—localized crack path deviations from the main plane. Superb transmission electron microscope images of various regions in a mirror have been shown by Beauchamp. ‘9’20 Poncelet21 showed comparable images in 1958, but Golz may have been the earliest with extraordinary electron microscope photos published in 1943.22 Starting well within the mirror, rounded ridges form that are elongated in the direction of crack propagation. These gradually coarsen in amplitude. At some point, they develop slight hackle steps where over running and under running fingerlike crack segments link. This point, where nano scale hackle lines form, could be an important transition point. Close examination of Beauchamp’s Figure 2 of ref. 19 shows they are not preSent inside the mirror, but they are just visible in his Figure 3 near the mirror—mist boundary The explanation below closely follows Fre’chette’s23 general discussion as well as Beauchamp’s19 analysis As the crack accelerates away from the origin, micro portions of the crack front begin to twist slightly or tilt up and down out of the main fracture plane. These local deviations occur as consequence of the stress field in front of a fast crack having maxima that re out ofplane, unlike the case for a static or slowly moving crack.“ The momentary tilting or twisting does not persist for very long since crack plane deviations are restricted by the energetic cost of creating additional crack surface. The slight tilt or twist variations in crack plane quickly rejoin the main propagating crack plane. These tiny local crack perturbations exist well within the mirror region, but are too small to be optically discernable. As the crack advances, they eventually become large enough to be discernable with the optical microscope as the “mist” zone surrounding the origin. The mist has a slight frosty appearance such as when water condenses on a reflecting mirror. As the crack continues to advance, the local perturbations increase and begin to oscillate and form larger tongue-like segments that may deviate from the average fracture plane to the degree that micro steps are generated running parallel to the direction of crack propagation. Some have described the out-of-plane perturbations as “fingers.” The perturbations gradually coarsen such that the tongue—like elements can over cut other portions of the crack front thereby generating large “velocity hackle” lines that run parallel to the direction of crack propagation. By this time the crack is running at a velocity that is a high fraction of the terminal velocity. Beauchamplg’19 pointed out that many of these features are similar in character from within the mirror out to the hackle zone. They differ only in scale. The transitions between the regions are gradual and are usually described as not abrupt. One intriguing observation by Beauchamp19 was that tilted or twisted hackle segments that relink with the main crack plane may generate elastic pulses and Wallner lines that could trigger additional hackle along the crack front. This could set off a cascade of hackle that forms a mist— hackle boundary. Attempts to make objective determinations of the mirror boundary by quantitative surface roughness characterization have been unsuccessful, since there are different scales of roughness for the various features that comprise the mirror and its apparent boundaries. A single threshold roughness value corresponding to the boundaries cannot be specified although many have tried. Duckworth et al.25 carefully studied mirror sizes in float glass using optical photographs and a conventional surface profilometer. They obtained a good correlation with optical boundary estimates when the surface roughness reached a level of 0.25 pm for the mirror- mist A; boundary, and 5 for the mist~hackle A0 boundary. Hull,26’27 WUnsche et al.,28 and Kuluwansa et al. ) used atomic force microscopy to show that the mist region in glass or brittle epoxy has a roughness of as small as a few tens of nanometers, which is much less than the wavelength of visible light. The scanned regions were quite small, however, and if the AF M scanned a small region between hackle and river line steps, the measured roughness was much less than if the latter were included. Hull pointed out that the large undulations from Wallner lines needed to be factored out when evaluating the intrinsic mist roughness. His study showed that roughness increased continuously and there were no dramatic jumps in roughness at the boundaries, but the rate of change of roughness did increase significantly at the mirror~mist boundar . Hull 6 considered the matter in some detail and pointed out that different surface roughness characterization devices such as atomic force microscopes (AFMS), mechanical profilometers, and laser optical profilometers all have different advantages, disadvantages, sensitivities and scanning zone sizes. AFMs can measure tiny regions with very high sensitivities, but may miss large hackle steps in a mist or hackle zone. These can dramatically alter the average or root mean square roughness. A mechanical stylus profilometer or laser profilometer with a l um spot size may miss the small undulations and be more sensitive to larger hackle steps on the fracture surface. Mist and hackle regions have different roughness at different scales. In summary, there are two potential transition points. The mirror~mist boundary in glass may be a transition Where nanonmicro steps form between ridges that are tens of nanometers tall. The mist~hackle transition boundary may be where a band of microhackle forms that is triggered by self—generated or external elastic pulses and Wallner lines. It is not known whether these possible transition points can be detected or measured accurately and precisely using ordinary microscopy. IS THERE A THEORETICAL BASIS FOR THE F RACTURE MIRROR CONSTANTS? The underlying principal that accounts for the micro and macro branching has been attributed to the crack reaching a critical velocity,24 a critical energy level,12 a critical stress intensity,13’l4’ 30’3 I or a critical strain intensity.32 A velocity criterion for crack branching was discounted by data shown by Congleton and Petch.31 Richter and Kerkhof, 33 Field,34 and Doll35 used ultrasonic fractography to show that cracks approached terminal velocity before the formation of the mist boundary and there was no pronounced change in velocity when the mirror formed. Congleton and Petch31 demonstrated that a stress intensity criterion controlled branching and also showed the Johnson and Holloway energy criterion12 was related to the stress intensity criterion. It is difficult to distinguish between the energy, stress intensity, or strain intensity criteria since they are all related for isotropic materials.36 The two Kirchners and Conway!“4 showed that a fracture mechanics criterion based on a critical stress intensity gave a better fit to mirror shapes in various stress fields than a simple application of equation 2. A crucial observation was that the stress intensity model fully accounts for the small inward facing cusp at specimen surfaces whereas the other models do not. They argued that a more fundamental material parameter might be KIB, the stress intensity factor at branching, rather than the As in equation 2 since the latter do not take into account the free surface, geometry factors, and non-uniform stress gradients over the crack surface. There is validity to their argument, but in practice the ease of use of equation 2 has led to its widespread utilization. Evaluation of a KIB is more difficult and involves an unusual curve fitting exercise on a 0 versus l/\/R graph.l4 One other consideration is that their stress intensity shape factors (the geometry terms, Y) for various surface cracks were based on solutions for static cracks, whereas fracture mirrors are formed by cracks moving at or near terminal velocity. Nevertheless, the size and shape predictions based on A or K113 match very closely for the limiting cases of small mirrors in tension specimens. This is also true for small Semicircular mirrors centered on surface flaws in strong flexure specimens. So, at least for some special mirror cases, A should be directly related to a more fundamental parameter K18. Mecholsky et al.37 have shown that A can be correlated to the critical fracture toughness, K10. The relationship depends upon the mirror to flaw size ratios, which differ for glasses and ceramics. KIC and K113 may in turn be related, but little work has been done to establish this relationship. ch pertains to the resistance to crack propagation of a static flaw in the vicinity of the origin, but Km pertains to a rapidly moving crack that has had ample opportunity to interact with more of the microstructure and be affected by possible Rucurves. THE GUIDELINES The boundary criteria for glasses and ceramics adopted in the new NIST Guide17 are as follows: The mirror—mist boundary in glasses is the periphery where one can discern the onset of mist. This boundary corresponds to A, the inner mirror constant. The mist~hackle boundary in glasses is the periphery where one can discern the onset of systematic hackle. This boundary corresponds to A0, the outer mirror constant. The mirror~hackle boundary in polycrystalline ceramics is the periphery where one can discern the onset of systematic new hackle and there is an obvious roughness change relative to that inside the mirror region. This boundary corresponds to A0, the outer mirror constant. Ignore premature hackle and/or isolated steps from microstructural irregularities in the mirror or irregularities at the origin. These are very general criteria and they rely upon perception, but they furnish guidance for the practical steps which follow in the guidelines. The microstructure of most polycrystalline ceramics obscures the fine details of fracture mirrors. Mist is usually not recognizable. In coarse—grained or porous ceramics, it may be impossible to identify a mirror boundary. in very weak materials, the mirror may be larger than the specimen or component and the boundaries will not be present. Sometimes coarse microstructural elements or flaws in a mirror may trigger early hackle lines that will generate wake hackle within the mirror, prior to the onset of generalized velocity hackle that forms the mirror boundary. The mode of crack propagation, whether it is trans- or intergranular, also affects the mirror markings. Mirrors may be easier to see in materials that start out with transgranular fracture but then develop increasing fractions of intergranular fracture. On the other hand, it may be more difficult to define mirrors when the fracture mode is intergranular from the origin right through the entire mirror. For these reasons it is appropriate to add the qualifier “relatively” as in “relatively smooth” when describing the mirror region since there is an inherent roughness from the microstructure even in the area immediately surrounding the origin. If the mirror is being measured for a component failure analysis, and if the mirror constant A is known, steps 1 through 9 below are followed and the stress at the origin is calculated in accordanCe with equation 2. Note that this origin stress may or may not necessarily be the maximum stress in the part. If the fracture mirror constant A is being evaluated by means of testing laboratory specimens and the origin stresses are known, then additional steps 10 through 12 may be used. _ Examples of how to judge the boundaries are shown in Figures 1, 4 and 5. Low~power images are also shown to provide an overview. The boundaries were assessed while looking through an optical microscope and the digital images were then marked. The twelve guidelines are presented below with clarification notes and details. Space limitation precludes a full treatment in this summary paper, but the NIST Guide includes copious notes, examples, and marked illustrations. 1. Use an optical microscope whenever possible. A compound optical microscope with bright—field viewing is best for glasses. A stereo optical microscope is best for ceramics. A scanning electron microscope may be used if optical microscopy is not feasible. Differential interference contrast (DlC, also known as Nomarski) mode viewing with a research compound microscope is not recommended by this Guide. It is not suitable for rough ceramic fracture surfaces. It also creates complications with glass fracture surfaces. There is no question that DlC mode viewing can discern very subtle mist features in glasses, but the threshold of mist detectability is highly dependent upon how the polarizing sliders are positioned. Hence, DIC measured radii are quite variable. DlC red radii can be substantially smaller than those obtained with conventional viewing modes. It also must be borne in mind that not all users have access to interference contrast microscopes. Dark—field illumination may be used with glasses but dark—field images may lose a little resolution with glasses and radii may be slightly larger as a result. It is very effective with highly~reflective mirror surfaces in single crystals. Confocal optical microscopes and optical interferometers have been occasionally USed to examine fracture mirrors, but the author is unaware of any systematic study to correlate apparent mirror sizes to those measured with conventional optical microscopes. A resolution limit of about 0.25 pm may be a problem with the confocal microscope. Scanning electron microscope images of mirrors are not recommended for glasses since the mirror—mist boundary is usually indiscernible. SEM images often appear flat and do not have adequate contrast to see the fine mist detail at the ordinary magnifications used to frame the whole mirror. SEM images may be used with very small mirrors that would be difficult to see with optical microscopy, e. g., high— strength optical fibers. Scanning electron microscope images may be used for ceramics if necessary, but contrast and shadowing should be enhanced. 2. The fracture surface should be perpendicular to the microscope optical path or camera. This simple and fairly obvious requirement is intended to avoid the foreshortening that can occur if the specimen is tilted. A small amount of tilting is acceptable in order to get a favorable reflection in a glass piece. The requirement poses a small problem if the mirrors are examined with stereo binocular microscopes. These have two different tilted optical paths. If viewing (b) Figure 4 Silicon carbide tension strength specimen (371 MPa) with a mirror centered on a compositional inhomogeneity flaw. Note how clear the mirror is in the low power images in (a,b). The mirror boundary (arrows in c) is where systematic new hackle forms and there is an obvious roughness difference compared to the roughness inside the mirror region. Figure 5 Silicon nitride bend bar with a Knoop surface crack in a silicon nitride (449 MPa). The mirror is incomplete into the stress gradient, but the mirror sides can be used to construct boundary arcs in (c). Radii are measured in the direction of constant stress along the bottom. with both eyes in a stereo microscope, the specimen should be flat and facing directly upwards. The observer’s brain will interpret the image as though he is facing it directly. Alternatively, if a camera is mounted on one light path of the stereo microscope and it is used to capture or display the mirror, then the specimen should be tilted so that the camera axis is normal to the fracture surface. For example, tilt the specimen to the right if the camera is attached to the right optical path. 3. Optimize the illumination to accentuate topographical detail. The mist and hackle features should be accentuated. Glasses may either be illuminated from directly down onto a fracture surface or by grazing angle, vicinal illumination. Ceramics should not be directly illuminated since the light will reduce contrast, especially in translucent or transparent materials. Ceramics should be illuminated with vicinal illumination. Thin coatings may be applied to translucent or transparent ceramics as needed. Stereo microscopes are strongly preferred for ceramics. Vicinal illumination is less convenient with compound light microscopes, but the observer should experiment with whatever illumination options are available to accentuate subtle surface roughness and topography features. 4. Use a magnificationsuch that the fracture mirror area occupies about 75 % to 90 % of the width of the field .of View for glasses, and approximately 33 0/0 to 67 0/0 of the width of the field of View for ceramics. Observers usually mark the mirror boundaries closer to the origin at greater magnifications than they would at lower magnifications. This is because mist or micro hackle markings are easier to see at distances closer to the origin at high magnification. This is particularly the case with glasses. Conversely, at very low magnifications, much detail is lost and observers typically overestimate the mirror size. Fracture mirrors are reasonably easy 0 see in glasses and magnifications should be used such that they nearly fill the field of view. Mirror interpretation is more problematic with polycrystalline ceramics. Excessive magnification often leads to confusion as to where the boundary is located. Even though a mirror may be obvious at low or moderate magnification, at higher magnification it may be impossible to judge a boundary. It is more practical to view the mirror region and the natural microstructural roughness therein compared to the hackle roughness in the regions outside the mirror. “Stepping back” and using the 33 % to 67 % rule should help an observer better detect the topography differences. Images recorded at these magnifications are also more convincing when shown to other fractographers or engineers. Supplemental lower magnification images may also be made to aid interpretation such as shown in Figures 4 and 5. The images should not be more than 5 times lower magnification, otherwise it is difficult to correlate features in one image to another. The guide has additional information on how to interpret ceramics with microstructures that make judgment difficult. Some self—reinforced silicon nitrides (Figure 4) and yttria~stabilized tetragonal zirconia polycrystals (Y T ZP) are difficult (Figure 6). The mirror regions are somewhat bumpy in the self—reinforced silicon nitride. The zirconia has intrinsic micro hackle lines well within the mirror. The mirror boundary was judged to be the point where systematic radiating new hackle commenced and there is an obvious roughness change relative to the inside— mirror region. The word systematic requires elaboration. Mirror hackle lines are created after the radiating crack reaches terminal velocity. Premature, isolated hackle can in some instances be generated well within a mirror, however, but it should be disregarded. Wake hackle from an isolated obstacle inside the mirror (such as a large grain or agglomerate) can trigger early “premature” hackle lines. Steps in scratches or grinding flaws can trigger hackle lines that emanate from the origin itself. 5. Measure the mirror size while viewing the fracture surface with an optical microscope whenever possible. Mirrors should be evaluated while looking in a microscope. Use either calibrated reticules in the eyepieces or traversing stages with micrometer-positioning heads. If a digital camera and high—resolution computer monitor are available, the image can be viewed both through the microscope and measured on the digital monitor provided that it has been calibrated. (a) (b) 4-—-----—> ....._...., 2R 200 um Figure 6 A fine— grained 3 mol % yttria—stabilized tetragonal zirconia (BY—TZP) polycrystal (Ref. 38). The mirror is difficult to mark in this material. (a) shows the uncoated fracture surface of a 2.8 mm thick flexural strength specimen (486 MPa) with Vicinal illumination. (b) shows an interpretation for a mirror-hackle boundary where systematic new hackle is detected (small white arrows) as compared to the roughness inside the mirror. The marked circle is elongated somewhat into the depth due to the stress gradient. The radius was 345 um. Interpretation from two—dimensional photos alone only should only be done as a last resort. Measurements from photos may be necessary for very small specimens or very strong specimens with tiny mirrors, such as in fibers or microelectro—mechanical system (MEMS) devices. Scanning electron microscope images may be used. Again, the fractographer should take an overall framing photograph or image shot in accordance with the 75 % to 90 % rule for glasses and 33 % to 67 % rule for ceramics. Higher and lower magnification images may be used to help aid in interpretation. Mirror size measurements from photographs are usually less accurate or precise. They frequently overestimate mirror sizes unless conditions are carefully optimized to accentuate contrast and topographic detail. Two—dimensional photographic renditions of a three—dimensional fracture surface usually lose much of the topographic detail discernable by the eye with a compound optical or stereo microscope. Video cameras should not be used to capture mirror images since they lack adequate resolution. Mirror size measurements made on computer monitor screens are also subject to inaccuracies, also because they are two~dimensional renditions of a three—dimensional fracture surface. Nevertheless, higharesolution cameras and monitors are beginning to match the capabilities and accuracy of an observer peering through the optical microscope. 6. Measure radii in directions of approximately constant stress whenever possible. Measurements should be taken from the center of the mirror region, but some judgment may be necessary. A common practice is to make a judgment whether a mirror is indeed approximately semicircular or circular. If it is, then multiple radii measurements may be made in different directions and averaged to obtain the mirror size estimate. The center of the mirror may not necessarily be the center of the flaw at the origin. Careful inspection of tiny localized fracture surface markings (Wallner lines and micro hackle lines may reveal that fracture started one spot on a flaw periphery. For example, fracture from grinding or impact surface cracks glass often starts from the deepest point of the flaw and not at the specimen outer surface. Figure I shows an example. Large pores often trigger unstable fracture from one side. If an exact mirror center cannot be determined, measure a mirror diameter and halve the measurement. This is commonly done for semicircular mirrors centered on irregular surface-located flaws whereby the mirror center may be difficult to judge. Circular embedded mirrors are easiest to interpret such as in Figure 4. Small semicircular mirrors on the surface of a part, such as in a bend bar or a flexurally loaded plate, are also not too difficult to interpret. The mirror relationship holds up remarkably well in glass optical fibers tested in tension for mirror radii almost as large as the fiber diameter.39 The mirror radius should simply be measured from the origin to the mirror— mist or mist-hackle boundary on the opposite side of the fiber, R; as shown in Figure 7. Mirror shapes are commonly affected by stress gradients in a plate or a beam. Mirror radii are elongated in the direction of decreasing stress. Examples are shown in Figure 8. In such cases, measure the mirror radius along the tensile surface where the stress is constant. Do not measure the mirror radii into the gradient. Even with this precaution, there is considerable evidence that the data begins to depart from the stress—mirror size curves and the relationship in equation 2 when the sizes approach the cross section thickness, For mirrors radii larger than the plate thickness, mirrors are larger than they would otherwise be in uniform tension. A trend for mirrors to elongate the opposite way, along the external surface of a specimen, was detected by the author in recent work on the fractographic analysis of grinding flaws in structural ceramics.40’4l Long surface cracks often caused mirrors to have perceptible deviations from a semicircular shape as shown in Figure 9. Figure 7 Mirrors surrounding surface origins in rods or fibers loaded in direct tension. (After Ref. 39) Measure both the mirror—mist radius and mist—hackle radii into the depth. Figure 8 Elongated mirrors in bending stress fields. If the mirror is small relative to the part size then the mirror may be semicircular as shown in (a): Weaker parts have larger mirrors that flare into the interior and are incomplete as shown in (b) and (c). Measure the mirror size (R1 or 2Ri for the mirror—mist in the illustrations here) in the direction of constant stress. Ceramic N 53% Will! Jll V W x \ \ , ,/ (b) E . 5"\[email protected]/ a u) d J/f/w/ . Figure 9 Grinding cracks and scratches can cause mirror elongations along the surface, even in bend bars with stress gradients. (a) shows a schematic of such a mirror with the mist—hackle boundary marked in glass, and (b) shows a comparable image in a polycrystalline ceramic. It has some intrinsic microstructural roughness inside the mirror and the mirror—hackle boundary is marked. Use an average radius: Ravg = {(R1 + R2 + Rd}/3. In some cases, it may be difficult to measure mirrors in directions of constant stress. The two sides of a mirror may have unequal lengths since the stresses are different on either side of the mirror. Figure 10 shows examples from the author’s research of round rods broken in flexure.40’41 Many origins were not at the rod bottom where the stresses were a maximum, but part way up the side of the specimen. The specimen orientation was easily determined from obServation of the cantilever curl. The maximum tensile stress on the bottom of the specimen (b) (C) Figure 10 Fracture mirrors in two rods tested in flexure. The maximum tensile stress is at bottom center. Fractures started at flaws part way up the sides of the rods causing the mirrors to have unequal radii. One rod (a,b,c) was sufficiently strong that a nearly semicircular mirror formed, albeit with unequal radii due to the stress gradient. Use R = Rh if the origin and mirror center is distinct. Otherwise use Ravg = (R1 + R2 + Rd) / 3 if the mirror is nearly semicircular. Use Ravg = (R1 + R2) / 2 if the mirror is elongated into the interior and Rd is large. (def) show a weaker glass rod. Use R = Rh if the origin and mirror center are distinct, otherwise use Ravg = (R + R2) / 2. Truncate surface cusps. \ was on the rod directly opposite the cantilever curl. The mirror radii had obviously different lengths due to the stress gradient. A radius in the direction of constant stress, Rh, should be measured as shown in Figure 10, if the mirror is centered on a well—defined origin site. If there is any doubt, then an average radius may be computed. Use Ravg = (R1 + R2 + Rd) / 3 if the mirror is nearly semicircular. Use Ravg = (R1 + R2) / 2 if the mirror is elongated into the interior and Rd is large or is incomplete. For origins located in the interior of a rod broken in flexure, only use the radii in the direction of constant stress. There is one important detail about mirror sizes that warrants discussion. Mirrors located on a specimen external surface have small cusps at the intersection with the outer surface as shown in Figure 1]. Cusps are often detected in glass mirrors, but they are rarely if ever discerned in polycrystalline ceramic mirrors. The small cusp is a consequence of fracture mechanics. A small element of material near the tip of a crack at the specimen exterior surface experiences greater stress intensity than a similar element buried in the interior whereby neighboring elements can “share the load.” The slightly—greater stress intensity at the surface triggers the mirror markings a bit sooner than for interior elements. The convention adopted in the NIST Guidelines is to truncate the cusp. Extend the semicircular (or other mirror shape) arcs as shown in Figure ll. Kirchner et al.13’14 discussed the shapes of fracture mirrors that intersect outer Figure 11 Fracture mirror in a fused silica rod (1 16 MPa). Mirrors on the surface have inward tilting cusps. These should be truncated and a mirror site measured by using the arc of the overall mirror shape. This mirror is slightly elongated into the interior due to a bending stress gradient. (Remember that some subtle mist detail may not duplicate well in printing.) surfaces and showed that the local enhancement of the stress intensity K1 accounts for the cusps. Another reason to be wary of measurements right along the surface is that surface roughness, machining damage, or other surface irregularities may trigger mist or hackle formation a bit earlier than in the interior. Others have noted that measurements taken right on an exterior surface are slightly different than those taken into the interior. Even Shand4’5 recommended that size measurements be taken 0.1 mm (.004 in) beneath the exterior surface to avoid “distortions.” Mecholsky and Freiman in 1979 recommended ignoring the cusp at the surface on the basis of fracture mechanics considerations: “In measuring the mirror—mist and mist—hackle boundaries, these should be projected to the tensile surface to compete a circular arc, since there is curvature at the surface due to free surface effects.” A dilemma occurs when mirrors are large in plates or beams broken in flexure as shown in Figure 8c. Cusps cannot be detected. In this caSe the only plausible way to measure a mirror radius is directly along the surface. In such cases, the general warning of step 7 below is applicable. Residual stresses alter mirror shapes. If the mirror is very small relative to the stress gradient, the mirror shape may remain circular or semicircular if along the surface. On the other hand, if the mirror is larger or the stress gradient is steep, then the gradient alters the mirror shape as shown in Figure 12. Figure l2a shows an annealed plate that requires an applied stress of Ga = (If to cause fracture. Figure l2b shows the case where the same plate has residual surface compression stress or = 0"C from ion exchange or thermal tempering, so that an applied stress to cause fracture is Ga = of + (5C. In other words, the applied stress must be increased to overcome the residual surface stress. Nevertheless, the net stress at the surface at the moment of fracture is o = oa — GC 2 (61’ + oc) — (50 = of, the same stress as in the annealed plate. Hence the mirror radii along the surface are unchanged compared to the annealed plate. In contrast, in the direction into the interior, tensile stresses combine with the applied tensile stress to cause the (a) ‘ . (b) (C) Interior GT, tension , . Interior (50! compressin 4:th surface GR = so compression surface 6R = GT lenSiON (53"Gf Ua=Uf+Gc Ua=Gf"GT Figure 12 Surface residual stresses also may alter a mirror shape. (5;, is the applied stress to cause fracture and of is the fracture stress in an annealed plate in tension. (a) shows a surface mirror in an annealed plate. (b) shows the mirror shape in a plate with surface compression stresses that decrease into the interior and become tensile. (c) shows a mirror in a plate with surface tensile stresses that diminish into the interior and become compressive. mirror markings to form sooner, at a shorter radius into the interior than in the annealed plate. In this example, the mirror shape is flattened to a semiellipse. Mirror radii should be measured only along the surface (or just beneath the surface if there is a cusp) in these cases. Figure 12c shows that surface residual tensile stresses have the opposite effect: mirror radii are elongated into the interior. Mirror radii again should only be measured along the surface, since againthe net stress to cause fracture is o = 6f. There are two possible paths for analysis if there are residual stresses: (a) The mirror is measured on a component. The applied stress and the residual stresses are unknown. in this instance the net stress 0 at the origin can be evaluated from R and equation 2. (b) The mirrors are collected in laboratory conditions with multiple specimens such that the apparent origin stresses oa, from applied external stresses are known. in this instance, one or more matched pairs of (3a and R are obtained. Graphical analysis shown below in step ll reveal the existence and magnitude of the residual stresses. 7. Exercise caution when mirrors are large relative to the specimen cross~sccti0n size. At some point, one can expect departures from the stress ~ mirror size relationship. The point where the departure Occurs depends upon the loading geometry and the stress state. Pronounced deviations occur once the mirror size approaches or is greater than the component thickness in plate or beam bending fractures. Experimentally measured radii are much greater than predicted by equation 2. Shand recommended that the maximum mirror size should be no more than 15 0/0 of the rod diameter for flexure tests.4 Kirchner and Conway15 warned about limitations in the fracture mechanics models for mirror radii exceeding 20 % of a rod diameter tested in flexure. On the other hand, Castilone et al.39 had success with mirrors that were almost as large as the fiber diameters for fibers tested in direct tension. Mecholsky and Freiman18 warned that systematic deviations from the mirror size relationship occur at large mirror sizes but also at very small sizes, the latter due to internal stress effects, e.g., from thermal expansion anisotropy of grains in ceramics. 8. Show at least one photo with arrows or lines marking the mirror size. 9. Report how the mirrors were measured. This simple last step is often overlooked or ignored, and the reader is left wondering exactly what had been done. The fractographer should report the microscope used, confirm that interpretation was made while looking through the microscope, whether photos had to be used, and approximately what magnifications were used. The direction the mirror radii were measured should be documented. The approximate shape of the mirrors (semicircular, circular, or elliptical) should be noted. It should also be noted whether the mirrors were an appreciable fraction of the size of the cross section or not. Lastly, and most importantly, the judgment criterion used should be reported. 10. Use the stress at the origin site. If the specimen was broken in controlled conditions where the stress distribution was known (e.g., beams, rods, or plates in flexural loadings) correct the stress for location in specimens with stress gradients. No correction is needed if a part was stressed in uniform tension. On the other hand, many parts or laboratory specimens do have stress gradients. The general principal that should be followed is that the mirror formation is guided by the stresses were in the immediate vicinity of the origin. While this may seem obvious, it is probable that some analysts in the past have erroneously used nominal stresses in a specimen or component rather than the actual stress that was acting upon the mirror region in a body. In contrast, some researchers have correlated the stress at every site along the mirror periphery with the mirror radius at that periphery site, but this complex process is not practical on a routine basis. 11. Evaluate the Fracture Mirror Constants Once a set of matching mirror radii and fracture stresses is compiled, plot the data on linear stress versus inverse square root of mirror size as shown in Figure l3. A linear regression analysis is then performed and a mirror constant calculated from the regressed line. The older graphical representation of log stress versus log radius may be used if necessary, but is not preferred. The mirror constants should be reported as either MPan (ksh/in). Use linear regression methods to obtain A in accordance with equation 2 with a zero intercept on a graph of 6 versus \lR. In a typical strength test experiment in a laboratory, applied stress, 03, and mirror radius, R, are independently measured. It is customary to regress G on R. A is the slope of the regression line. Use some judgment in the regression analysis since fracture mirror data usually has moderate scatter. If the data do not appear to fit a trend that has a zero intercept, regress the data with a non—zero intercept as shown in Figure 13b. Again use some judgment in the interpretation, since a strict linear regression fit may produce implausible outcomes, particularly if the data is collected over a limited range of mirror sizes and stresses. Report the intercept if it deviates significantly (> 10 MPa) from zero. Investigate possible residual stresses or specimen size or shape issues if the intercept deviates significantly from zero. Consistent units should, be used with this approach. That is to say, if the stress axis is MN/m2 or MPa, then large small mirrors mirrors (a) (b) Figure 13 Plot of applied stress Ga versus l/VR. (a) shows the trend for residual stress—free parts. (b) shows it for parts with residual stresses. Compressive residual stresses move the locus up with a positive intercept of, but with the same slope. Tensile residual stresses shift the data downwards with a negative intercept (not shown). the abscissa (horizontal axis) should be 1/\/R where R is in units of meters. The mirror constant A as a slope is easily visualized. In addition, a nonzero intercept may be conveniently interpreted as an effective residual stress. If residual stresses or are present in addition to the externally applied stress, Ga, then the net stress acting on the origin site is: a 2(0‘ +0)=—’€\~ 3 net 3 r \/_R5 and: 0— :_Lo—, <4) a re An intercept below the origin corresponds to a net tensile residual stress. A positive intercept corresponds to residual compressive stress since the usual sign convention is for compressive stresses to have a negative sign. . Some caution is advised since residual stresses are often nonuniform. The estimate from the intercept is an effective residual stress, which in reality may vary in magnitude through the mirror region. Once again is it prudent to measure mirror radii in directions of constant stress. If the mirror is in a heat~strengthened or tempered piece (where stress may be constant along the surface, but change dramatically through the thickness) the mirrors should only be measured along the surface (or just underneath to avoid the cusp). Residual streSSes from an indentation or impact site are very local to the origin and may have very little effect on a mirror size. Although most researchers have felt that the regressed lines should go through the origin in annealed test pieces, there is evidence by .l. Quinn42 that a small but measurable intercept may exist in even annealed materials. The intrinsic intercept was evaluated as 10 MPa (1,500 psi) for glass, a value that interestingly concurs withOrr’s9 estimate of the minimum stress necessary for branching in glass. One popular a ternati e a ialysis method is based on plotting the data on a log stress versus log radius graph as shown in Figure 14. This method of showing the results and calculating a mirror constant was common in the older technical literature and is occasionally still found today. Graphs of this type were used when researchers were not sure whether R=t LogR=O tog R (a) (b) Figure 14 Plot of log Ga versus log R for residual stress~free parts (a) and parts with residual stress (b). Compressive residual stresses move the locus upwards, but with a different slope and intercept. Tensile residual stresses move the loci below the baseline curve (not shown). equation 2 with the \/R relationship was appropriate. Forty years of research have shown it is, so there no longer is a need to test for the trend. From equation 2: log 68 =~%IogR+|ogA (5) If stresses are in units of MN/m2 (MPa) and the mirror size is measured in meters, then the mirror constant A has units of MN/rnl'5 or MPa\/m. If the mirror size is l m, then log R = 0. Then log 6 = log A and hence, o = A. Hence, the mirror constant A corresponds to the value of stress that would create a mirror of size 1 m. Since most actual mirrors that are measured are usually much smaller than unit size, it is apparent from Figure 14 that the mirror constant (or the stress for R = l) lies somewhat beyond the range of data usually collected. Deviations from the linear relationship on the log - log plot occur when residual stresses are present but unaccounted for, or when the mirror size is large relative to the component size, or when there are stress gradients. The residual stress deviations cause the line to have a slope other than —“/2 as shown in Figure 14b. Attempts to compute the residual stresses may then be made by guessing values of the residual stresses or, replotting the data, and checking the goodness of fit of a line of slope 4/2. This is a fairly cumbersome process and the 6 versus l/\/R procedure is simpler. The two analyses put different weights on large and small mirror measurements. In one case the mirror constant is a slope ofa line, in the other it is an intercept at R = l, a rather large mirror size not likely to be realized in practice. Some of the variability in published mirror probably is due to the use of the two different curve—fitting schemes. It is also certain that some researchers have evaluated unannealed test pieces and then force fitted regression lines through the data with zero intercepts in the former scheme or lines of slope — 1/2 through the data in the latter scheme. Regression analyses on the log — log graph are more vulnerable to deviations of the data from the correct trends when mirror sizes are large. Upward deviations from the log stress lo g radius graphs have been noted in a number of studies (e.g., Shandf’5 Orr9). Regression lines chase the upward deviations and dramatically alter the estimate of the mirror constant. On the other hand, with the 6 versus l/VR graph, oversized mirror data points are closer to the origin and have less influence on the regression line and less effect upon the slope, A. Finally, regression analysis with the 6 versus lNR approach minimizes the deviations of stress 6 from the fitted line. Regression analysis for the log — log graph minimizes deviations of log 0 from the fitted line. The former is preferred from a mathematical perspective. For all the reasons above, the linear stress versus inverse square root radius approach was adopted in the NIST Guidelines. Analysis is simple and intuitive. The uncertainty of the slope can be estimated from routine analyses available in many statistical software packages. 12. Mirrors sizes should be collected over a broad range of sizes and fracture stresses if possible. Data from different specimen types and sizes may be combined. This is a fairly obvious conclusion. Superb examples are shown by Kerper and Scuderi8 for borosilicate glass rods with diameters that varied by a factor of ten and by Mecholsky and Rice43 for various sized fused silica rods, disks, and fibers. Ideally, data from many small specimens could be complemented by judicious testing of a few large specimens. Another common procedure is to anneal or fine grind/polish some specimens to obtain high strengths, but also abrade or damage others to obtain low strengths. Sometimes the mode of loading can be changed to alter the fracture stress. For example, some studies have generated mirrors with large four~point and small three-point flexure specimens. Some specimens may be tested in inert conditions and others in conditions conducive to slow crack growth. SOME FINAL THOUGHTS One is also struck by the fact that nearly all the surface—centered mirrors shown in the literature, even in the classical papers, are not exactly semicircular despite all the schematics that imply that they are. So fractographers should not be alarmed if their mirrors are not perfect. The steps in this Guide have occasionally appeared sporadically in the past. Shand recommended that stresses be corrected for the origin location,3 that radii be measured beneath the surface to avoid surface effectsf’5 that low angle vicinal illumination be used.3’5 He also warned about deviations from the trends if the mirror sizes were too large relative to the component thickness.3 Shand also said that mist could not be discerned in glass ceramics.3 Morrell et al.15 agonized over the interpretation of mirrors in Y—TZP zirconia, but settled stereo optical microscopy at a fixed magnification, with grazing incidence illumination. The specimen sides were masked to block transmitted light scatter. Matching fracture halves were mounted together to aid the interpretation. The most comprehensive set of recommendations predating the new NIST guidelines were crafted by Mecholsky and Freiman.18 Six of their recommendations match steps in the guidelines: optical microscopy is preferred over scanning electron microscopy whenever possible, suitable magnifications should be used, mirror boundary arcs should be projected to the outer surface to complete a circular arc to eliminate the surface cusps, lighting should be varied to obtain optimum contrast, radii should be measured in directions of constant stress and not into gradients, and that caution should be used with data from large V mirrors relative to part thickness. CONCLUSIONS A set of guidelines for measurement of fracture mirrors and determinations of fracture mirror constants have been devised. The goal is to bring consistency to the procedures used to measure fracture mirrors. This should facilitate improved data bases and better estimates of failure stresses. Most students of the technique have concluded that consistent readings are possible, provided that defined procedures are used. The new guidelines have been prepared on the basis of a review of sixty years of literature, recommendations by experts, and the author’s own experiences. Measurement of the mirror sizes requires interpretation. The perception of the observer and the type of equipment are important factors. Although advanced microscopy and software tools hold considerable future promise, it is unlikely that a simple definitive boundary criterion such as a specific surface roughness will emerge. REFERENCES I C. Brodmann, “A Few Observations on the Strength of Glass Articles,” Nachriehlen van der Gessellschqften zu Gottingen, Mathematisch—Physikalische Klasse, l, 44 4 58 (1894). 2 W. C. Levengood, “Effect of Origin Flaw Characterization on Glass Strength,” J. Appl. Phys, 29 [5], 820 ~ 826 @958). 3 E. B. Shand, “Breaking Stress of Glass Determined from Dimensions of Fracture Mirrors,” J Am. Ceram. Soc, 42 [10], 474 — 477 (1959). 4 E. B. Shand, “Strength of Glass — The Griffith Method Revised,” ibid, 48 [l], 43 — 48 (l 965). 5 E. B. Shand, “Breaking Stresses of Glass Determined from Fracture Surfaces,” T he Glass Industry, April (1967) 190 — 194. 6 M. .J. Kerper and T. G. Scuderi, “Modulus of Rupture of Glass in Relation to Fracture Pattern,” Cer. Bull, 43 [9], 622 ~ 625 (1964). 7 M. J. Kerper and T. G. Scuderi, “Relation of Strength of Thermally Tempered Glass to Fracture Mirror Size,” ibia’, 44 [12], 953 — 955 (1965). 8 M. J. Kerper and T. G. Scuderi, “Relation of Fracture Stress to the Fracture Pattern for Glass Rods of Various Diameters,” ihia’, 45 [12], 1065 ~ 1066 (1966). 9 L. Orr, “Practical Analysis of Fractures in Glass Windows,” Materials Research and Standards, 12 [1], 21 ~ 23, 47 (1972). I O H. P. Kirchner, R. M. Gruver, and W. A. Setter, “Fracture Stress—Mirror Size Relations for Polycrystalline Ceramics, Phil. Mag, 33 [5] (1976) 775 v 780. H Levengood, Ref. 2 above, footnote page 821. 12 J. W. Johnson and D. G. Holloway, “On the Shape and Size of the Fracture Zones on Glass Fracture Surfaces,” Phil. Mag. 14, 731 ~ 743 (1966). 13 H. P. Kirchner and J. W. Kirchner, “Fracture Mechanics of Fracture Mirrors,” ibid, 62 [3- 4], 198 — 202 (1979). 14 H. P. Kirchner and J. C. Conway, Jr., “Criteria for Crack Branching in Cylindrical Rods: l, Tension; and H, Flexure,” ibia’, 70 [6], 413 - 418 and 419 — 425 (1987). 15 R. Morrell, L. Byrne, and M. Murray, “Fractography of Ceramic Femoral Heads,” pp. 253 ~ 266 in Fractography 0f Glasses and Ceramics, I V, Ceramic Transactions Vol. 122, eds, J. R. Varner and G. D. Quinn, American Ceramic Society, Westerville, OH, 2001. 16 ASTM C 1322—96, “Standard Practice for Fractography and Characterization of Fracture Origins in Advanced Ceramics” Annual Book of Standards, Vol. 15.01 , ASTM Int, West Conshohocken, PA. 1996. 17 G. D. Quinn, “Guide to F ractography of Ceramics and Glasses,” NIST Special Publication 960—16, Gaithersburg, MD, 2006. 18 J. J. Mecholsky and S. W. Freiman, “Determination of Fracture Mechanics Parameters Through Fractographic Analysis of Ceramics,” pp. 136 —150 in Fracture Mechanics Applied t0 Brittle Materials, ASTM STP 678, S. W. Freiman, ed, ASTM, Int, West Conshohocken, PA, 1979. s 19E. K. Beauchamp, “Mechanisms ofHackle Formation and Crack Branching,” pp. 409 — 446 in Fractography 0f Glasses and Ceramics 1H, eds. J. Varner V. D. Fre’chette, and G. D. Quinn, Ceramic Transactions, Vol. 64, American Ceramic Society, Westerville, OH, 1996. 20 E. K. Beauchamp, “Fracture Branching and Dicing in Stressed Glass,” Sandia Laboratories Research Report, SC—RR—70—766, Jan. 1971. 2‘ E. F. Poncelet, “The Markings on Fracture Surfaces,” J. Soc. Glass Technol, 42, 279T ~ 288T (1958). 22 E. G'o‘lz, “Ubermikroskopische F einstructuren an Glassbruchflachen,” (Ultramicroscopic Fine Structures on Glass Fracture Surfaces), Zeitschrifl Physik, 120, 773 4 777 (1943). 23 V. D. Frechette, Failure Analysis of Brittle Materials, Advances in Ceramics, Vol. 28, American Ceramic Society, Westerville, OH, 1990. 24 E. H. Yoffe, “The Moving Griffith Crack,” Phil. Mag, 42, 739 — 750 (1951). 25 W. H. Duckworth, D. K. Shetty, and A. R. Rosenfeld, “Influence of Stress Gradients on the Relationship Between Fracture Stress and Mirror Size for Float Glass,” Glass Technology, 24 [S], 263 — 273 (1983). 26 D. Hull, Chapter 5, in Fractograpby, Observing, Measuring and Interpreting Fracture Surface Topography, Cambridge Univ. Press, Cambridge, 1999. 27 D. Hull, “Influence of Stress Intensity and Crack Speed on Fracture Surface Topography: Mirror to Mist Transition,” J Mat. Sci, 31, 1829 — 1841 (1996). 28 C. Wunsche, Radlein, and G. H. Frischat, “Morphology of Silica and Borosilicate Glass Fracture Surfaces by Atomic Force Microscopy,” Glastecb. Ber. Glass Sci. Technol, 72 [2], 4 (1999). 29 D. M. Kuluwansa, L. C. Jensen, S. C. Langford, and .1. T. Dickinson, “Scanning Tunneling Microscope Observations of the Mirror Region of Silicate Glass Fracture Surfaces,” J. Mater. Res, 9 [2] (1994) 476 ~ 485. 30 A. B. .1. Clark and G. R. Irwin, “Crack—Propagation Behaviors,” Exptl, Mech, 6, 321 — 330 (1966). 3‘ J. Congleton and N. i. Fetch, “Crack—Branching,” Phil. Mag, 16, 749 — 760 (1967). 32 H. P. Kirchner, “Brittleness Dependence of Crack Branching in Ceramics,” J. Am. Ceram. Soc, 69 [4], 339 — 342 (1986). 33 H. G. Richter and F. Kerkhof, “Stress Wave Fractography,” pp. 75 — 109 in Fractograpby ofGlass, eds, R. C. Bradt and R. E. Tressler, Plenum, NY, 1994. 34 I. E. Field, “Brittle Fracture: Its Study and Application,” Contemp. Phys, 12, 1 — 31 (1971). 35 W. Doll, “Investigations ofthe Crack Branching Energy,” Int. J. Fract, 1 1, 184 — 186 (1975). r 36 Y. L. Tsai and .I. .I. Mecholsky, .Ir., “Fracture Mechanics Description of F raetur Mirrors Formation in Single Crystals,” Int. J. Fract, 57, 167 — 182 (1992). 37 I. .I. Mecholsky, .112, S. W. Freiman, and R. W. Rice, “Fracture Surface Analysis of Ceramics,”J. Mat. Sci, 11, 1310 ~ 1319 (1976). 38 G. D. Quinn, .1. Eichler, U. Eisele, and I. Rodel, “Fracture Mirrors in a Nanoscale 3Y— TZP,” J. Amer. Ceram. Soc, 87 [3] (2004) 513 — 516. 39 R. J. Castilone, G. S. Glaesemann, and T. A. Hanson, “Relationship Between Mirror Dimensions and Failure Stress for Optical Fibers,” pp. 11 - 20 in Optical Fiber and Fiber Component .Meclzanical Reliability and Testing 11, eds, M. I. Matthewson and C. R. Kurkjian, Proc. SPIE, 4639 (2002). 40 G. D. Quinn, L. K, Ives, and S. Jahanmir, “On the Nature of Machining Cracks in Ground Ceramics,” Machining Science and Technology, 9, 169 — 210 (2005). 4‘ G. D. Quinn, L. K. Ives, S. Jahanmir, and P. Koshy, “Fractographic Analysis of Machining Cracks in Silicon Nitride Rods and Bars,” pp. 343 w 365 in F ractograpby of Glasses and Ceramics 1V, , 2001. 42 I. B. Quinn, “Extrapolation of Fracture Mirror and Crack—Branch Sizes to Large Dimensions in Biaxial Strength Tests of Glass,” J. Am. Ceram. Soc, 82 [8], 2126 — 2132 (1999). 43 I. I. Mecholsky, Jr. and R. W. Rice, “Fractographic Analysis of Biaxial Failure in Ceramics,” pp. 185 — 193 in Fractograpby ofCeramic-and Metal Failures, eds. I. .I. Mecholsky, Jr. and S. R. Powell, ASTM STP 827 American Society for Testing and Materials, Westerville, OH, 1984. i) ...
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