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Crack stability and strength variability in alumina ceramics with rising toughness-curve behavior
Acta mater. 48 (2000) 565±578 www.elsevier.com/locate/actamat
CRACK STABILITY AND STRENGTH VARIABILITY IN ALUMINA CERAMICS WITH RISING TOUGHNESS-CURVE BEHAVIOR DESIDERIO KOVAR 1{, STEPHEN J. BENNISON 2 and MICHAEL J. READEY{ 3 1
Texas Materials Institute and Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712, USA, 2Central Research and Development, E.I. du Pont de Nemours and Co., Inc., Experimental Station, Bldg. E356/311, Wilmington, DE 19880-0356, USA and 3Glass and Electronic Ceramics, Sandia National Laboratories, Albuquerque, NM 87185-0333, USA (Received 25 May 1999; accepted 19 August 1999) AbstractÐAluminas with four distinct microstructures have been fabricated to investigate the in¯uence of grain size and grain morphology on strength variability. The four microstructures comprise two grain size scales and are characterized as either ``equiaxed'' with a narrow size distribution or ``elongate'' with a higher aspect ratio and a broader size distribution. Indentation-strength tests indicate that only the coarsegrain, elongate microstructure exhibits a strong rising toughness-curve (T-curve or R-curve). Furthermore, in situ measurements demonstrate that the coarse-grain, elongate microstructure is the only one that displays signi®cant stable crack extension from annealed indentation ¯aws free of contact-induced residual stress. Strength tests on polished specimens indicate that the highest mean strength is achieved in the ®negrain, equiaxed material with little or no T-curve. The lowest strength variability, however, is exhibited by the coarse-grain, elongate alumina and is rationalized in terms of the strong rising T-curve and its associated in¯uence on crack stability. The study suggests that maximum reliability is achieved when the T-curve is suciently strong to stabilize the propagation of natural ¯aws en route to failure. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fracture toughness; Ceramics; Mechanical properties; Microstructure; R-curve
1. INTRODUCTION
Many engineering ceramics are now designed and fabricated with microstructures that exhibit a rising fracture toughness with crack extension (T-curve or R-curve behavior) [1]. When suitably optimized, a strong T-curve has been shown to impart ``¯aw tolerance'' (i.e. a diminished sensitivity to ¯aw size) to an otherwise classically brittle ceramic [2±4]. Tcurve behavior and associated ¯aw tolerance are understandably desirable qualities for engineering materials since empirical models have been developed that predict improved reliability (i.e. reduced strength variability) as a result of T-curve behavior [5,6]. Experiments have also shown that some ceramics that exhibit T-curve behavior including zirco-
{ To whom all correspondence should be addressed. { Now at: Advanced Materials and Technology Division, Caterpillar Inc., Technical Center-E, Peoria, IL 61656-1875, USA
nia [7] and silicon nitride [8] exhibit dramatically improved reliability. However, there is also some evidence that suggests that T-curve behavior does not lead to reductions in strength variability in all ceramics [9±11]. The purpose of this study is to investigate the in¯uence of T-curve behavior on strength variability of brittle materials. Alumina ceramics are selected as a model material for this work because alumina is known to demonstrate signi®cant T-curve behavior as a result of a grain-bridging mechanism [12] and because microstructural features that give rise to T-curve behavior can be systematically altered to assess their in¯uence on crack stability and strength variability. In this study, well-characterized alumina ceramics with four distinct microstructure are processed and tested. Crack stability, strength variability and T-curve behavior are experimentally measured using natural ¯aws or indentation ¯aws to initiate failure. Conditions which lead to improved reliability are rationalized based on these experimental results.
1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 5 0 - X
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KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR 2. EXPERIMENTAL PROCEDURE
2.1. Fabrication and characterization Aluminas with four dierent microstructures were fabricated from two commercial alumina powders of varying purity: (i) 99.7%{; and (ii) 99.99%.{ Disk-shaped specimens were prepared by uniaxially pressing batches of powder in a highpurity graphite die to a pressure of 21 MPa. Density gradients in the green compacts were minimized by subsequent cold isostatic pressing to a pressure of 275 MPa. The compacts were packed in a loose powder bed of identical composition to the compacts and placed in high-purity alumina crucibles and ®red in air. All specimens were ®red at 16008C for 5 h. The ®ne-grain microstructures were removed at this point while coarse-grain microstructures were achieved by annealing at 17008C for an additional 25 h. Heating and cooling rates of 58C/min and 108C/min were used, respectively. Selected specimens were ground and polished to a one micron ®nish and thermally etched to reveal grain structures. Mean grain size, grain size distribution, and grain shape distribution for each alumina were determined from scanning electron microscopy (SEM) images with the aid of a computer-assisted image analysis system; details of this procedure are given elsewhere [9]. Densities were measured using the Archimedes method to 20.001 g/cm3 using water as the immersion medium [13]. 2.2. Mechanical properties 2.2.1. Indentation-strength tests. Biaxial ¯exure specimens (diameter 025 mm, thickness 02.5 mm) were used to measure the strength of each alumina. Specimens were prepared for testing by grinding and polishing one side of each sintered disk using successively ®ner metal-bonded diamond grinding pads and diamond paste until a mirror-like ®nish was obtained. Indentations were placed at the center of each polished disk using a Vickers diamond indentor with a dwell time of 10 s at peak forces ranging between 5 and 200 N. Contact sites were immediately covered with a drop of dry silicone oil after indentation to minimize the introduction of moisture into the indentation crack. Specimens were loaded at a constant cross-head displacement rate of 7.6 mm/min (020 000 MPa/s) using a servohydraulic testing machine according to ASTM F394 [14]. A minimum of ®ve specimens were tested at each indentation load. Strengths were calculated from thin-plate theory using peak loads and specimen dimensions. All specimens were examined after
{ A-16 SG, ALCOA Industrial Chemicals. { AKP-50, Sumitomo Chemical America Inc.
testing to ensure that fracture originated from the contact site. Only failures from indentations were included in the data pool. Short-crack T-curve behavior was deconvoluted from indentation-strength measurements following the analysis of Braun et al. [15]. This technique requires that the indentation parameter, w, and the crack geometry parameter, c, be known. These parameters are best determined by calibration using a reference material that exhibits no T-curve behavior. Once calibrated, it is assumed that these coecients do not vary with grain size or small changes in the material composition. 2.2.2. In situ crack extension. In situ crack-growth studies were performed using biaxial ¯exure testing with larger specimens (diameter 052 mm, thickness 04 mm) than those used in indentation-strength testing. These specimens were prepared in an identical manner to the smaller disks used to measure the indentation-strength response described in Section 2.2.1. Because moisture has been shown to have a profound in¯uence on both equilibrium crack size and crack con®guration [16], specimens were dried by heating to 1508C under rough vacuum overnight, transferred into a dry box under dry nitrogen (H2O content < 0.20 ppm), and indented within the dry box. Some specimens were immediately sealed into a miniature biaxial testing apparatus and transferred to an optical microscope stage where in situ tests were carried out. Other specimens were removed from the dry box after indentation and annealed in air for 4 h at 11008C to remove residual stresses associated with the indentation impression. Following annealing, these specimens were dried as before and transferred (dry) to the testing apparatus. The miniature biaxial testing apparatus consisted of a sealed chamber containing dry nitrogen (H2O content < 0.20 ppm), a load cell, and a piezoelectric driver that allowed load to be applied to the system while monitoring the surface traces of the indentation cracks in situ using an optical microscope. The design is based on an earlier model described elsewhere [15]. Crack observations were made at magni®cations from 50 to 1000 using either bright ®eld, dark ®eld, or Nomarski contrast modes. Individual crack lengths were measured to a resolution of between 0.5 and 10 mm, depending on magni®cation. In situ tests were performed by increasing the load in increments of 10±100 N until extension of one or more surface traces of the indentation cracks was apparent. Crack sizes were measured upon arrest following a speci®ed load increment. Loading followed by measurement was repeated in a stepwise fashion until specimen failure occurred. In ideal cases, four radial cracks emanated from the hardness impression corners immediately following indentation. At lower indentation loads and/or
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
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Fig. 1. Scanning electron micrographs of polished and thermally etched aluminas: (a) ®ne-grain, equiaxed; (b) coarse-grain, equiaxed; (c) ®ne-grain, elongate; and (d) coarse-grain, elongate.
when the grain size in the material was large, it was common for more than four cracks to be present. Despite such non-ideal indentation sites, failure was always preceded by the growth of two cracks on opposite sides of the indentation impression. The surface trace, c, was therefore taken as the average length of the two dominant radial cracks. 2.2.3. Strength variability. The ``natural'' strength of each material was measured using unindented, polished specimens tested in biaxial ¯exure as described in Section 2.2.1. A minimum of 24 specimens were tested for each of the four microstructures. The strength of each specimen for a given microstructure was ranked from lowest to highest and the probability of failure, Pf , for the ith specimen assigned using the following estimator: Pf
i20:5 n
1
where n is the number of specimens tested in each batch. The strength variability was assumed in all cases to be described by a two-parameter Weibull
distribution. The maximum likelihood method was used to calculate Weibull parameters and a bootstrap technique was used to estimate con®dence limits on the Weibull modulus [17]. Data are plotted on a Weibull scale to facilitate qualitative comparisons of strength variability between materials. The in¯uence of indentation ¯aws on the strength variability of the materials was also investigated. Each sample within a batch of specimens was indented at a peak contact force of 100 N and subsequently fractured in biaxial ¯exure. This contact force was selected because it is suciently large as to ensure that the indentation ¯aw was the initiation site for failure in all of the specimens tested.
3. RESULTS
3.1. Material characterization Photo-micrographs of representative polished and thermally etched specimens are shown in Figs 1(a)± (d). The densities of all materials appear high and
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KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
Fig. 2. Grain size distributions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. Grain size distributions remain approximately selfsimilar with increasing grain size for a given powder purity. However, the lower purity aluminas exhibit broader size distributions.
direct measurements con®rm that they exceed 98.5% in all four microstructures. The lower purity aluminas exhibit a relatively broad grain size distribution and many grains are tabular or elongated. In contrast, the higher purity aluminas demonstrate a narrow grain size distribution and an equiaxed grain morphology. Mean grain sizes for the ®neand coarse-grain, lower purity aluminas are 4.0 and 12.7 mm, respectively, while mean grain sizes for the higher purity aluminas are 5.0 and 10.2 mm. Grain size distributions are shown for the four materials in Figs 2(a)±(d). These distributions reveal that normal grain growth occurs in both material systems since the distributions are approximately self-similar for a given powder purity. These measurements also con®rm that the higher purity aluminas display a narrower grain size distribution compared to the lower purity aluminas. Grain shape distributions are shown in Figs 3(a)± (d). Grain shape also remains approximately selfsimilar as mean grain size is increased for both the lower and higher purity aluminas. However, grain shape is signi®cantly more elongate for the lower purity aluminas with mean aspect ratios greater than 2.0 compared to approximately 1.5 for the higher purity aluminas.
3.2. Indentation strength Indentation-strength behavior for the four aluminas is presented as Figs 4(a)±(d). The solid line in each plot is the theoretical behavior for a material with constant toughness and is characterized by a slope of ÿ1/3 [18]. The hatched box on the left side of each plot represents the range of measured strength values for polished but unindented specimens. From this data, it is apparent that the indentation-strength response for the ®negrain, equiaxed material exhibits classically brittle behavior with a similar slope to that predicted for materials with constant toughness, while the coarsegrain, equiaxed and the ®ne-grain, elongate materials exhibit slight deviations from the theoretical constant-toughness line. The coarse-grain, elongate material exhibits the largest deviation from the constant-toughness response indicating that the strength of this material exhibits the greatest degree of ¯aw tolerance. T-curves deconvoluted from indentation-strength data are shown in Fig. 5 for each of the four aluminas. The indentation-strength data for the ®ne-grain, equiaxed alumina were used to estimate the indentation and ¯aw geometry parameters, w
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569
Fig. 3. Grain aspect ratio distributions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®negrain, elongate and (d) coarse-grain, elongate aluminas. The lower purity materials demonstrate a more elongated grain shape compared with the higher purity materials.
and c. The values of w and c are estimated to be 0.100 and 1.00, respectively, which compares to previous evaluations for similar alumina ceramics of 0.071±0.082 for w and 0.77±1.26 for c [15]. The Tcurve for the ®ne-grain, equiaxed alumina gently rises and reaches a plateau toughness of approximately 3.6 MPaZm within a 250 mm extension. Tcurves for the coarse-grain, equiaxed and ®ne-grain, elongate aluminas also rise gradually but continue to increase to a toughness of approximately 5 MPaZm at a crack size of 1000 mm. The coarsegrain, elongate alumina exhibits the most pronounced T-curve behavior, starting at a lower toughness (2.8 MPaZm) and rising more steeply than the T-curves for the other materials. 3.3. In situ crack extension measurements Figure 6 plots in situ observations of crack extension from indentation ¯aws at several indentation loads for the ®ne-grain, equiaxed alumina. As applied load is increased, indentation cracks are seen to grow in an irregular, stochastic fashion jumping one or more grain facet lengths before arresting. This ``stop-start'' behavior is seen for all indentation loads and appears to be independent of crack size. At a critical size, the crack begins to accelerate and failure ensues. In the ®ne-grain, equiaxed alumina, this critical (®nal) crack size is
measured to be about 2.8 times greater then the initial indentation crack size for all indentation loads. This ratio compares favorably with the theoretical prediction that the critical crack size should be 2.6 times greater than the initial indentation ¯aw size for a material with constant fracture toughness [18]. Crack path in this material is observed to be a mixture of transgranular and intergranular. Occasionally, small microcracks are seen to open at grain boundaries ahead of the crack tip during propagation and are likely to play a role in determining crack path and the formation of grain bridges. However, microcracking is generally isolated to a few grain boundaries and there are no signs of an extended microcracked zone with associated material dilation. Figure 7 presents in situ observations of crack extension for the coarse-grain, elongate alumina at two dierent indentation loads. Crack extension is again seen to be irregular, the crack jumping at least one grain facet length during propagation. The tendency towards intergranular cracking is much greater in this material as compared to the ®negrain, equiaxed alumina leading to a more tortuous crack path and increased frequency of grain bridging. SEM photo-micrographs of the fracture surfaces for the four materials are shown in Figs 8(a)±(d). These images con®rm that crack path tortuosity
570
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
Fig. 4. Indentation-strength results for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®negrain, elongate and (d) coarse-grain, elongate aluminas. The solid line represents the theoretical behavior sf 0P ÿ1 3 for a material with constant toughness and the hatched box indicates the natural strength for polished, unindented specimens. Only the coarse-grain, elongate alumina exhibits strong ¯aw tolerance as evidenced by a signi®cant deviation from the P ÿ1/3 line.
depends on both grain size and material purity. There is a greater tendency towards intergranular cracking in the lower purity aluminas as compared to the higher purity aluminas and thus crack patch tortuosity at a given grain size is greater in the lower purity aluminas. The in¯uence of grain size on crack path is more complex, however. In the lower purity alumina, intergranular fracture is always observed, independent of grain size. As a result, the crack path tortuosity increases dramatically with increasing grain size. In the higher purity aluminas, the tendency for intergranular cracking is reduced as the grain size is increased. As a result, the crack roughness is only slightly greater in the coarse-grain high-purity alumina compared to the ®ne-grain high-purity alumina. In summary, the crack path tortuosity is greatest in the coarse-grain, elongate microstructure followed by the ®ne-grain, elongate and coarse-grain, equiaxed microstructure and then the ®ne-grain, equiaxed microstructure. 3.4. Crack stability from annealed indentations Figure 9, presents a comparison of in situ crack extension behavior in dry nitrogen from annealed indentation ¯aws for all four microstructures. Several features may be noted from these data.
First, the failure stresses in all materials are signi®cantly greater after annealing compared to asindented specimens. This is consistent with the expectation that residual contact stresses are removed by annealing. The second important feature to note from these data is that little stable crack extension is observed during loading for all materials except the coarse-grain, elongate microstructure. Cracks in the coarse-grain, elongate alumina grow to roughly two times their initial size before failure ensues. Stable crack growth in this alumina occurs in a similar ``stop±start'' fashion for both as-indented and annealed-indented specimens, the crack following an intergranular path leading to the formation of many grain bridges. 3.5. Strength variability Strength data obtained from polished specimens are plotted in Weibull form for all four materials in Fig. 10. The Weibull moduli and associated 90% con®dence limits on Weibull moduli are shown in Table 1. The Weibull moduli vary from 7.8 and 5.3 for the ®ne-grain, equiaxed and coarse-grain, equiaxed aluminas, to 9.2 and 14.8 for the ®negrain, elongate and coarse-grain, elongate aluminas. From the con®dence limits, a statistical dierence
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
571
Fig. 5. T-curves deconvoluted from indentation-strength measurements for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. The thinner solid lines represent the net driving force for fracture (sum of applied stress intensity factor and residual indentation stress intensity factor) as a function of indentation load. The broader lines represent the Tcurves and are determined by the tangency construction.
Fig. 6. In situ measured crack size as a function of applied stress and indentation load for the ®ne-grain, equiaxed alumina. Arrows indicate unstable extension. Stable crack extension is observed until the size approaches 2.6 times the initial size independent of the initial indentation force.
Fig. 7. In situ measured crack size as a function of applied stress and indentation load for the coarse-grain, elongate alumina. For both indentation loads stable crack extension greater than 2.6 times the original crack size is observed prior to failure.
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KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
Fig. 8. Scanning electron micrographs of fracture surfaces from aluminas: (a) ®ne-grain, equiaxed; (b) coarse-grain, equiaxed; (c) ®ne-grain, elongate; and (d) coarse-grain, elongate. Note that the lower purity materials exhibit a larger degree of intergranular cracking compared to the higher purity materials.
exists between the Weibull modulus for the coarsegrain, elongate alumina and all other aluminas, and between the coarse-grain, equiaxed and ®ne-grain, elongate aluminas. The low Weibull modulus of the coarse-grain, equiaxed alumina has been attributed to a very small number of abnormal grains, which did not exist in the other three materials [11]. If the data are censored to eliminate specimens that failed from this secondary ¯aw population, only the coarse-grain, elongate alumina shows a statistically dierent (increased) Weibull modulus. Strength distributions from indented specimens are plotted using a Gaussian probability scale{ in Fig. 11. The Weibull function is not used to characterize strength distributions for indented specimens since weak-link statistics are no longer appropriate
{ Note that the slope of the line graphed on a probability scale is equal to the inverse standard deviation and is thus a measure of absolute variability.
when a single dominant ¯aw is introduced into a specimen and poor ®ts were obtained when data were force-®t to a Weibull distribution. The variability in strength is instead characterized using the coecient of variation, CV, for each batch of material and is calculated by taking the ratio of the standard deviation to the mean. As expected from indentation-strength measurements, the ®ne-grain, equiaxed alumina exhibits the lowest mean strength when samples are indented at a force of 100 N Table 1. Characteristic strengths (s0), Weibull modulii (m ) and 90% con®dence intervals (m5 and m95) for the Weibull modulii for the four aluminas investigated. The coarse-grain, elongate alumina exhibits signi®cantly less strength variability compared to other aluminas Material
s0
m
m5
m95
Fine-grain, equiaxed Fine-grain, elongate Coarse-grain, equiaxed Coarse-grain, elongate
507 369 346 243
7.8 5.3 9.2 14.8
6.1 4.2 7.4 12.1
10.5 7.1 11.9 19.8
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
573
Fig. 10. Weibull plot for unindented samples with polished surfaces. Note that although the ®ne-grain, equiaxed alumina exhibits the highest mean strength, the coarsegrain, elongate alumina demonstrates the highest Weibull modulus. Fig. 9. Comparison of crack extension from annealed indentations in w ®ne-grain, equiaxed, q coarse-grain, equiaxed, * ®ne-grain, elongate and Q coarse-grain, elongate aluminas. Annealing following indentation removes the residual stress ®elds associated with contact damage. Only the coarse-grain, elongate alumina demonstrates signi®cant stable extension prior to failure.
prior to testing. The coarse-grain, equiaxed and ®ne-grain, elongate aluminas have similar mean strengths, which are slightly higher than that for the ®ne-grain, equiaxed alumina. The coarse-grain, elongate material exhibits the highest mean strength. Coecients of variation in strength range from 5.7 to 6.7% for all four aluminas and indicate no statistically signi®cant dierences in strength variability. These results would seem to imply that microstructure does not in¯uence the strength variability of alumina. However, it can be seen from Fig. 12 that signi®cant dierences in indentation ¯aw size distribution between the four microstructures are evident even though all indentations are made at the same peak contact force (100 N). It is apparent that indentations in the ®ne-grain, equiaxed material are well behaved and exhibit little variability in crack length. Signi®cantly more variability in crack length and crack morphology develops in the more ¯aw-tolerant materials. These dierences in the crack morphologies have been attributed to local variations in fracture toughness which increase with grain size [9, 10, 19]. The indentation crack
sizes measured immediately following indentation are summarized in Table 2 for all four aluminas. These measurements support the qualitative observations that the ®ne-grain, equiaxed alumina exhibits the least variability in indentation crack size CV 5:0% while the coecient of variation in crack length for the other aluminas falls within the range of 8.3±11.7%. Results are consistent with previous work [19], which has shown that indentation crack sizes are sensitive to microstructure even though this variability is not re¯ected in strength variability. The relationship between variability in initial indentation crack size and microstructure is similar to that between ¯aw tolerance and microstructure: variability in crack sizes increases with grain size and grain aspect ratio. Most signi®cantly, although variability in the indentation ¯aw size increases dramatically with grain size and grain
Table 2. Comparison of indentation crack sizes P 100 N for the four aluminas investigated. Note that the variability in size increases with increasing grain size and aspect ratio Material Fine-grain, equiaxed Fine-grain, elongate Coarse-grain, equiaxed Coarse-grain, elongate
Crack size, c (mm)
CV (%)
221211 182221 181215 192222
5.0 11.7 8.3 11.6
Fig. 11. Strength distributions plotted on a probability scale for aluminas containing 100 N Vickers indentations. There is no signi®cant dierence in strength variability between the four aluminas investigated.
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Fig. 12. Optical micrographs of 100 N Vickers indentations in (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. Note that the variability in indentation crack size increases with grain size and grain aspect ratio.
aspect ratio, strength variability is not aected by changes in indentation ¯aw size distribution. 4. DISCUSSION
Our study illustrates that T-curve behavior and associated ¯aw tolerance in alumina ceramics is not controlled solely by mean grain size. For example, although the ®ne-grain, equiaxed alumina and the ®ne-grain, elongate alumina have similar mean grain sizes, the ¯aw tolerance and T-curve behaviors are much more pronounced for the more elongate grain morphology. In addition, the ®negrain, equiaxed alumina and coarse-grain, equiaxed alumina exhibit similar toughness behavior, despite signi®cant dierences in mean grain size. These results are consistent with previous work, which have shown that microstructural characteristics other than grain size such as grain shape, grain size distribution, and grain-boundary toughness have a pronounced in¯uence on crack-growth resistance in alumina ceramics. For example Cook found that T-
curve behavior was dependent on both grain size and grain-boundary chemistry since these factors in¯uence the degree of intergranular fracture [20]. Similarly, in high-purity alumina ceramics, increasing the grains size without changing the grainboundary chemistry did not lead to enhanced toughness since larger grains tended to fracture transgranularly thus reducing grain-bridge density and associated crack-tip shielding [11]. Many studies on aluminas indicate that in all but the purest materials, an amorphous phase is present at many grain boundaries following sintering [21, 22]. In addition to providing an easier path for crack propagation, the glassy phase generally results in broader grain size distributions and elongate or tabular shaped grains [23]. Such grain morphology gives rise to potentially more eective grain bridges, and associated ¯aw tolerance, and is generally observed in lower purity aluminas [3, 24]. There are a number of important features that are apparent from T-curves determined from our indentation-strength measurements. First, increases
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
in toughness at longer crack lengths come at the expense of short-crack toughness. For instance, although the ®ne-grain, equiaxed material demonstrates the lowest long-crack toughness, it also shows the highest short-crack toughness. Thus, details of the T-curve shape are tied to microstructure by grain size, grain shape, and a tendency towards intergranular cracking [20]. Materials that contain coarse, elongate grains and fracture intergranularly exhibit the steepest T-curves and the greatest increase in toughness. However, these same materials also exhibit the lowest short-crack toughness and associated strength. This ®nding is consistent with previous observations in alumina and other ceramics that derive T-curve behavior from grain bridging [3, 20, 24±26]. We now go on to consider the eect of T-curve behavior on crack stability. Stable crack extension depends on a number of factors including: initial ¯aw size (c ), magnitude and shape of the T-curve, as well as morphological details of the initiating critical ¯aw [2]. The criterion for fracture, based on linear elastic fracture mechanics, states that if the applied stress intensity, KI, exceeds the toughness, T, then crack growth occurs. However, if the Tcurve rises rapidly, stable equilibrium crack extension may precede catastrophic failure. The degree of stabilization prior to failure is governed by the rate of increase in applied stress intensity, dKI/dc, and the rate of increase of toughness, dT/dc. Stable crack extension occurs in materials with a rising toughness until: KI rT
2a
dKI dT r dc dc
2b
after which point failure occurs at an unstable equilibrium condition. For cracks emanating from natural ¯aws, the applied stress intensity is given by the familiar form: p KI cs c
3
where s is the applied stress and c is a crack geometry-dependent term. Thus, a simple graphical construction for crack stability can be made by plotting applied stress intensity vs square-root of crack size and then superimposing a T-curve on this plot (the tangency construction). However, when contact-induced residual stresses are present (e.g. as in the case of an indentation ¯aw), there is an additional local driving force for crack extension. This
{ We refer to TR as the ``eective'' toughness since residual stresses generated during indentation lead to crack extension at stress intensities much lower than the actual toughness, T.
575
driving force can be accounted for by subtracting the stress intensity due to indentation residual stress from the toughness [18] resulting in an eective toughness, TR, given by:{ TR T ÿ
wP c3 2
4
The advantage of subtracting the indentation residual stress intensity factor from the toughness rather than adding it to the applied stress intensity factor (as is usually done) is that a conventional graphical tangency construction can still be performed. This allows direct comparison between crack stability in the case of either natural or contact-induced ¯aws. T-curves determined from the indentationstrength data along with tangency constructions are shown for the four aluminas in Figs 13(a)±(d) for both indentation ¯aws and natural ¯aws. The tangency construction predicts that indentation cracks should grow in a stable manner until the crack length reaches 2.59 times the initial indentation crack size for materials of constant toughness [18]. Crack extension beyond this length is an indication that the material exhibits rising T-curve behavior. All four aluminas studied are predicted to exhibit stable equilibrium crack extension from indentation ¯aws based on the tangency constructions shown in Figs 13(a)±(d). For instance, indentation ¯aws in the ®ne-grain, equiaxed alumina should grow to 2.8 times their initial sizes before failure. In the ®negrain, elongate, coarse-grain, equiaxed, and coarsegrain, elongate materials, all with stronger T-curve behavior, additional stabilization is predicted up to approximately 3.2 times the initial ¯aw size. We see that the predicted degree of stable crack extension from the tangency constructions in Fig. 13 compares favorably to that observed and presented in Figs 6 and 7. Although all of the aluminas exhibit some Tcurve behavior, the coarse-grain, elongate alumina clearly exhibits the strongest T-curve. The tangency construction indicates that this is the only material that has a suciently steep T-curve to result in stable crack extension from ¯aws free of contactinduced residual stresses (i.e. natural ¯aws). Recall that when indentations are annealed to remove residual stresses (see Fig. 9), signi®cant stable crack extension is only observed in the coarse-grain, elongate alumina. These results show that the nature of the ¯aw and the shape of the T-curve both play an important role in determining whether strength is sensitive to initial ¯aw size. For example, although materials that exhibit weak T-curve behavior exhibit stable crack extension from indentation ¯aws, the strength still shows some dependence on indentation ¯aw size. When the same material fails from a natural ¯aw, however, no stable crack extension is observed
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Fig. 13. T-curves and tangency constructions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. The solid lines are T-curves for natural ¯aws while the shaded lines are T-curves for indentation ¯aws P 100 N). Initial crack size is denoted by c0 and the dashed lines represent the tangency condition at instability, cf . All materials are predicted to show stable crack growth from indentation ¯aws in contrast to failure from natural ¯aws where only the coarse-grain, elongate alumina demonstrates stable equilibrium growth en route to failure.
and strength is strongly dependent on initial ¯aw size. It is only for materials that exhibit very strong T-curve behavior that stable crack extension and ¯aw tolerance is observed from both natural and contact-induced ¯aws. These results are consistent with a recent theoretical model which concluded that stable crack growth was not predicted for an alumina that exhibited modest T-curve behavior when failure ensued from a natural ¯aw free of residual stresses [27]. We now turn our attention to the question of strength variability. Weibull plots presented in Fig. 10 show that strength variability decreases only for aluminas with very strong ¯aw tolerance. Similar observations were made by Ting et al. for a variety of alumina ceramics [28]. The highest Weibull modulus was displayed in aluminas with broadest grain size distribution and an elongate grain shape while the lowest Weibull modulus was displayed by an alumina with equiaxed grain morphology and narrow grain size distribution. Hirosaki et al. and Homan and Petzow both found similar results for silicon nitride [29, 30], another material toughened by a grain-bridging mechanism. Ting et al. and
Hirosaki et al. argued that the decreased strength variability in these systems was due to a narrowing in the critical ¯aw size distribution. They suggested there is a decreased probability of encountering a large strength-controlling defect in materials with narrow grain size distribution. Since large defects occur in only a few specimens, there is greater scatter in strength for such materials. They went on to argue that materials with heterogeneous microstructures contain a more uniform dispersion of large ¯aws, thus increasing the probability of ®nding large defects in all specimens and reducing strength variability. A contrary argument was presented by Homan and Petzow who suggested that T-curve behavior is responsible for the reduced strength variability observed in grain-bridging ceramics rather than a narrowing in ¯aw size distribution. However, little experimental evidence was oered to support these assertions. One potential method of distinguishing whether T-curve behavior or changes in critical ¯aw size distribution are responsible for the changes in strength variability involves quantifying the critical ¯aw population. This is a dicult task as the failure ori-
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
gin often cannot be determined unambiguously from post mortem fractography particularly in coarse-grained, heterogeneous microstructures. Furthermore, when a failure origin can be clearly identi®ed, it may not be possible to accurately assess the in¯uence of the critical ¯aw size population due to uncertainties in specifying the driving force for fracture. Even in the case of an ideal Grith-like ¯aw, knowledge of ¯aw geometry and local stress ®elds is needed together with ¯aw size to accurately specify the applied stress intensity factor that resulted in failure. A possible solution to this problem is to control failure using well-characterized critical ¯aws using indentation ¯aws [18]. The strength distributions collected for aluminas containing 100 N indentations shown in Fig. 11 reveal that there is essentially little dierence in strength variability of aluminas with vastly dierent microstructures. However, measurements of indentation crack sizes revealed that distributions in crack sizes also vary greatly depending on microstructure. Indentation cracks in the coarse-grain, elongate alumina display more than twice the size variability than the ®ne-grain, equiaxed alumina, yet no dierence in strength variability is observed. This suggests that a narrowing of the critical ¯aw size distribution by itself cannot account for the reduction in strength variability observed in the coarse-grain, elongate material. Further evidence that the critical ¯aw size distribution cannot be the only factor controlling strength in grain-bridging materials is provided by in situ crack-growth measurements on specimens containing annealed indentation ¯aws. These tests conclusively demonstrate that even when a ¯aw is not subject to strong residual stresses associated with contact damage, stable crack extension is observed in aluminas with strong T-curve behavior whereas no signi®cant crack stability is observed in aluminas with less pronounced T-curve behavior. While the in¯uence of ¯aw size distribution cannot be discounted fully, this observation suggests that the stabilizing in¯uence of a strong T-curve must also play an important role in reducing strength variability in our aluminas. From a practical viewpoint these results have several important implications with respect to materials selection when reliability is crucial. Ultimate component reliability is attained when strength becomes insensitive to both natural and processing ¯aws. For example, while it is true that ®ne-grain, highpurity aluminas demonstrate high mean strength, they exhibit poor strength variability and high susceptibility to contact-induced damage. Microstructural coarsening in high-purity aluminas does not signi®cantly increase ¯aw tolerance because grain bridging is limited by a tendency towards intergranular fracture. As a result, coarsegrain, equiaxed aluminas demonstrate inferior strength variability and a moderate susceptibility to
577
contact-induced ¯aws. Despite the presence of a broader grain size distribution, elongate grain shape, and decreased grain-boundary strength, the ®ne-grain, elongate alumina exhibits similar properties to the ®ne-grain, equiaxed alumina since the ecacy of grain bridging is scale dependent. It is only when elongate grains and weak grain boundaries are combined with a larger grain size that signi®cant grain bridging occurs. T-curve behavior in this large-grain, elongate microstructure is strong enough to stabilize both natural and contactinduced ¯aws. As a result, this material exhibits low strength variability when failure is initiated from natural ¯aws and strong ¯aw tolerance to post-processing, contact-induced ¯aws. It should be noted that improved reliability comes at the expense of reduced mean strength. Such a trade-o is a feature of materials in which the shielding mechanism responsible for strong T-curve behavior results in a low short-crack toughness.
5. CONCLUSIONS
Strength, toughness and reliability of four alumina ceramics with dierent microstructural scale and morphology have been investigated. A combination of large, elongate grains and intergranular fracture results in signi®cant grain bridging in the coarse-grain, lower purity alumina. As a result, this material exhibits the greatest ¯aw tolerance to indentation ¯aws and signi®cant rising T-curve behavior. In situ crack propagation experiments in a dry ambient atmosphere con®rm that this is the only material of the four examined that exhibits a signi®cant degree of stable crack extension from ¯aws free of contact-induced residual stress. Reliability tests to probe strength variability when failure is controlled by natural ¯aws demonstrate that the coarse-grain, elongate alumina exhibits less strength variability compared to all of the other materials that were tested. Strength tests on indented specimens (controlled ¯aws) reveal that there is no statistical dierence in strength variability between materials at a given indentation load. However, examination of the indentation cracks show that variability in crack sizes increases dramatically as grain size or grain aspect ratio is increased. Based on crack-stability measurements and strength-variability measurements from indented specimens, the reduced strength variability observed for polished specimens with the coarsegrained, elongate microstructure cannot be accounted for simply by changes in the critical ¯aw population. The study suggests that maximum reliability is achieved when the T-curve is suciently strong to stabilize the propagation of natural ¯aws en route to failure. Improved reliability is attained at the cost of mean strength when the shielding mechanism responsible for strong T-curve behavior
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results in a trade-o between strength and toughness. AcknowledgementsÐThe authors would like to acknowledge Drs Brian Lawn, Edwin Fuller, and Craig Carter for their many helpful discussions. In addition, we would like to thank Ms Terry Deis and Mr Carl Lovejoy for their assistance in the preparation of specimens used in this study. Principal funding for this research was provided by the Air Force Oce of Scienti®c Research, Contract Number F49620-92-J-0034. REFERENCES 1. Becher, P. F., J. Am. Ceram. Soc., 1991, 74, 255. 2. Bennison, S. J. and Lawn, B. R., J. Mater. Sci., 1989, 24, 3169. 3. Chantikul, P., Bennison, S. J. and Lawn, B. R., J. Am. Ceram. Soc., 1990, 73, 2419. 4. Lawn, B. R., Padture, N. P., Braun, L. M. and Bennison, S. J., J. Am. Ceram. Soc., 1993, 76, 2235. 5. Kendall, K., McN. Alford, N., Tan, S. R. and Birchall, J. D., J. Mater. Res., 1986, 1, 120. 6. Cook, R. F. and Clarke, D. R., Acta metall., 1988, 36, 555. 7. Readey, M. J., McNamara, P. D., McCallen, C. L. and Lawn, B. R., J. Mater. Sci., 1993, 28, 6748. 8. Li, C. W. and Yamanis, J., Ceram. Engng Sci. Proc., 1989, 10, 623. 9. Kovar, D. and Readey, M. J., J. Am. Ceram. Soc., 1994, 77, 1928. 10. Curtin, W. A., J. Am. Ceram. Soc., 1995, 78, 1313. 11. Kovar, D. and Readey, M. J., J. Am. Ceram. Soc., 1996, 79, 305. 12. Bennison, S. J. and Lawn, B. R., Acta metall., 1989, 37, 2659. 13. Fazio, P. C., et al. (ed.), ASTM Standard C-20-92. American Society for Testing and Materials, 1992.
14. Fazio, P. C., et al. (ed.), ASTM Standard F-394-78. American Society for Testing and Materials, 1978. 15. Braun, L. M., Bennison, S. J. and Lawn, B. R., J. Am. Ceram. Soc., 1992, 75, 3049. 16. Cook, R. F. and Pharr, G. M., J. Am. Ceram. Soc., 1990, 74, 787. 17. Johnson, C. A. and Tucker, W. T., Ceramics and Glasses. ASM International, Metals Park, OH, 1992. 18. Lawn, B. R., Fracture of Brittle Solids. Cambridge University Press, Cambridge, 1993. 19. Cook, R. F., Liniger, E. G., Steinbrech, R. W. and Deuerler, F., J. Am. Ceram. Soc., 1994, 77, 303. 20. Cook, R. F., Acta metall. mater., 1990, 38, 1083. 21. Bateman, C. A., Bennison, S. J. and Harmer, M. P., J. Am. Ceram. Soc., 1989, 72, 1241. 22. Handwerker, C. A., Morris, P. A. and Coble, R. L., J. Am. Ceram. Soc., 1989, 72, 130. 23. O'Donnell, H. L., Readey, M. J. and Kovar, D., J. Am. Ceram. Soc., 1995, 78, 849. 24. Cook, R. F., Freiman, S. W. and Baker, T. L., Mater. Sci. Engng, 1986, 77, 199. 25. Padture, N. P., Bennison, S. J. and Chan, H. M., J. Am. Ceram. Soc., 1993, 76, 2312. 26. Padture, N. P., Runyan, J. L., Bennison, S. J., Braun, L. M. and Lawn, B. R., J. Am. Ceram. Soc., 1993, 76, 2241. 27. Zimmermann, A., Homan, M., Flinn, B. D., Bordia, R. K., Chuang, T.-J., Fuller, E. R. J. and RoÈdel, J., J. Am. Ceram. Soc., 1998, 81, 2449. 28. Ting, J.-M., Lin, R. Y. and Ko, Y.-H., Am. Ceram. Soc. Bull., 1991, 70, 1167. 29. Homan, M. J. and Petzow, G., in Materials Research Society Proceedings, ed. I.-W. Chen, P. Becher, M. Mitomo, G. Petzow and T. S. Yen. MRS, Pittsburgh, PA, 1992. 30. Hirosaki, N., Akimune, Y. and Mitomo, M., J. Am. Ceram. Soc., 1993, 76, 1892.
CRACK STABILITY AND STRENGTH VARIABILITY IN ALUMINA CERAMICS WITH RISING TOUGHNESS-CURVE BEHAVIOR DESIDERIO KOVAR 1{, STEPHEN J. BENNISON 2 and MICHAEL J. READEY{ 3 1
Texas Materials Institute and Department of Mechanical Engineering, University of Texas at Austin, Austin, TX 78712, USA, 2Central Research and Development, E.I. du Pont de Nemours and Co., Inc., Experimental Station, Bldg. E356/311, Wilmington, DE 19880-0356, USA and 3Glass and Electronic Ceramics, Sandia National Laboratories, Albuquerque, NM 87185-0333, USA (Received 25 May 1999; accepted 19 August 1999) AbstractÐAluminas with four distinct microstructures have been fabricated to investigate the in¯uence of grain size and grain morphology on strength variability. The four microstructures comprise two grain size scales and are characterized as either ``equiaxed'' with a narrow size distribution or ``elongate'' with a higher aspect ratio and a broader size distribution. Indentation-strength tests indicate that only the coarsegrain, elongate microstructure exhibits a strong rising toughness-curve (T-curve or R-curve). Furthermore, in situ measurements demonstrate that the coarse-grain, elongate microstructure is the only one that displays signi®cant stable crack extension from annealed indentation ¯aws free of contact-induced residual stress. Strength tests on polished specimens indicate that the highest mean strength is achieved in the ®negrain, equiaxed material with little or no T-curve. The lowest strength variability, however, is exhibited by the coarse-grain, elongate alumina and is rationalized in terms of the strong rising T-curve and its associated in¯uence on crack stability. The study suggests that maximum reliability is achieved when the T-curve is suciently strong to stabilize the propagation of natural ¯aws en route to failure. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fracture toughness; Ceramics; Mechanical properties; Microstructure; R-curve
1. INTRODUCTION
Many engineering ceramics are now designed and fabricated with microstructures that exhibit a rising fracture toughness with crack extension (T-curve or R-curve behavior) [1]. When suitably optimized, a strong T-curve has been shown to impart ``¯aw tolerance'' (i.e. a diminished sensitivity to ¯aw size) to an otherwise classically brittle ceramic [2±4]. Tcurve behavior and associated ¯aw tolerance are understandably desirable qualities for engineering materials since empirical models have been developed that predict improved reliability (i.e. reduced strength variability) as a result of T-curve behavior [5,6]. Experiments have also shown that some ceramics that exhibit T-curve behavior including zirco-
{ To whom all correspondence should be addressed. { Now at: Advanced Materials and Technology Division, Caterpillar Inc., Technical Center-E, Peoria, IL 61656-1875, USA
nia [7] and silicon nitride [8] exhibit dramatically improved reliability. However, there is also some evidence that suggests that T-curve behavior does not lead to reductions in strength variability in all ceramics [9±11]. The purpose of this study is to investigate the in¯uence of T-curve behavior on strength variability of brittle materials. Alumina ceramics are selected as a model material for this work because alumina is known to demonstrate signi®cant T-curve behavior as a result of a grain-bridging mechanism [12] and because microstructural features that give rise to T-curve behavior can be systematically altered to assess their in¯uence on crack stability and strength variability. In this study, well-characterized alumina ceramics with four distinct microstructure are processed and tested. Crack stability, strength variability and T-curve behavior are experimentally measured using natural ¯aws or indentation ¯aws to initiate failure. Conditions which lead to improved reliability are rationalized based on these experimental results.
1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 5 0 - X
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2.1. Fabrication and characterization Aluminas with four dierent microstructures were fabricated from two commercial alumina powders of varying purity: (i) 99.7%{; and (ii) 99.99%.{ Disk-shaped specimens were prepared by uniaxially pressing batches of powder in a highpurity graphite die to a pressure of 21 MPa. Density gradients in the green compacts were minimized by subsequent cold isostatic pressing to a pressure of 275 MPa. The compacts were packed in a loose powder bed of identical composition to the compacts and placed in high-purity alumina crucibles and ®red in air. All specimens were ®red at 16008C for 5 h. The ®ne-grain microstructures were removed at this point while coarse-grain microstructures were achieved by annealing at 17008C for an additional 25 h. Heating and cooling rates of 58C/min and 108C/min were used, respectively. Selected specimens were ground and polished to a one micron ®nish and thermally etched to reveal grain structures. Mean grain size, grain size distribution, and grain shape distribution for each alumina were determined from scanning electron microscopy (SEM) images with the aid of a computer-assisted image analysis system; details of this procedure are given elsewhere [9]. Densities were measured using the Archimedes method to 20.001 g/cm3 using water as the immersion medium [13]. 2.2. Mechanical properties 2.2.1. Indentation-strength tests. Biaxial ¯exure specimens (diameter 025 mm, thickness 02.5 mm) were used to measure the strength of each alumina. Specimens were prepared for testing by grinding and polishing one side of each sintered disk using successively ®ner metal-bonded diamond grinding pads and diamond paste until a mirror-like ®nish was obtained. Indentations were placed at the center of each polished disk using a Vickers diamond indentor with a dwell time of 10 s at peak forces ranging between 5 and 200 N. Contact sites were immediately covered with a drop of dry silicone oil after indentation to minimize the introduction of moisture into the indentation crack. Specimens were loaded at a constant cross-head displacement rate of 7.6 mm/min (020 000 MPa/s) using a servohydraulic testing machine according to ASTM F394 [14]. A minimum of ®ve specimens were tested at each indentation load. Strengths were calculated from thin-plate theory using peak loads and specimen dimensions. All specimens were examined after
{ A-16 SG, ALCOA Industrial Chemicals. { AKP-50, Sumitomo Chemical America Inc.
testing to ensure that fracture originated from the contact site. Only failures from indentations were included in the data pool. Short-crack T-curve behavior was deconvoluted from indentation-strength measurements following the analysis of Braun et al. [15]. This technique requires that the indentation parameter, w, and the crack geometry parameter, c, be known. These parameters are best determined by calibration using a reference material that exhibits no T-curve behavior. Once calibrated, it is assumed that these coecients do not vary with grain size or small changes in the material composition. 2.2.2. In situ crack extension. In situ crack-growth studies were performed using biaxial ¯exure testing with larger specimens (diameter 052 mm, thickness 04 mm) than those used in indentation-strength testing. These specimens were prepared in an identical manner to the smaller disks used to measure the indentation-strength response described in Section 2.2.1. Because moisture has been shown to have a profound in¯uence on both equilibrium crack size and crack con®guration [16], specimens were dried by heating to 1508C under rough vacuum overnight, transferred into a dry box under dry nitrogen (H2O content < 0.20 ppm), and indented within the dry box. Some specimens were immediately sealed into a miniature biaxial testing apparatus and transferred to an optical microscope stage where in situ tests were carried out. Other specimens were removed from the dry box after indentation and annealed in air for 4 h at 11008C to remove residual stresses associated with the indentation impression. Following annealing, these specimens were dried as before and transferred (dry) to the testing apparatus. The miniature biaxial testing apparatus consisted of a sealed chamber containing dry nitrogen (H2O content < 0.20 ppm), a load cell, and a piezoelectric driver that allowed load to be applied to the system while monitoring the surface traces of the indentation cracks in situ using an optical microscope. The design is based on an earlier model described elsewhere [15]. Crack observations were made at magni®cations from 50 to 1000 using either bright ®eld, dark ®eld, or Nomarski contrast modes. Individual crack lengths were measured to a resolution of between 0.5 and 10 mm, depending on magni®cation. In situ tests were performed by increasing the load in increments of 10±100 N until extension of one or more surface traces of the indentation cracks was apparent. Crack sizes were measured upon arrest following a speci®ed load increment. Loading followed by measurement was repeated in a stepwise fashion until specimen failure occurred. In ideal cases, four radial cracks emanated from the hardness impression corners immediately following indentation. At lower indentation loads and/or
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
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Fig. 1. Scanning electron micrographs of polished and thermally etched aluminas: (a) ®ne-grain, equiaxed; (b) coarse-grain, equiaxed; (c) ®ne-grain, elongate; and (d) coarse-grain, elongate.
when the grain size in the material was large, it was common for more than four cracks to be present. Despite such non-ideal indentation sites, failure was always preceded by the growth of two cracks on opposite sides of the indentation impression. The surface trace, c, was therefore taken as the average length of the two dominant radial cracks. 2.2.3. Strength variability. The ``natural'' strength of each material was measured using unindented, polished specimens tested in biaxial ¯exure as described in Section 2.2.1. A minimum of 24 specimens were tested for each of the four microstructures. The strength of each specimen for a given microstructure was ranked from lowest to highest and the probability of failure, Pf , for the ith specimen assigned using the following estimator: Pf
i20:5 n
1
where n is the number of specimens tested in each batch. The strength variability was assumed in all cases to be described by a two-parameter Weibull
distribution. The maximum likelihood method was used to calculate Weibull parameters and a bootstrap technique was used to estimate con®dence limits on the Weibull modulus [17]. Data are plotted on a Weibull scale to facilitate qualitative comparisons of strength variability between materials. The in¯uence of indentation ¯aws on the strength variability of the materials was also investigated. Each sample within a batch of specimens was indented at a peak contact force of 100 N and subsequently fractured in biaxial ¯exure. This contact force was selected because it is suciently large as to ensure that the indentation ¯aw was the initiation site for failure in all of the specimens tested.
3. RESULTS
3.1. Material characterization Photo-micrographs of representative polished and thermally etched specimens are shown in Figs 1(a)± (d). The densities of all materials appear high and
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Fig. 2. Grain size distributions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. Grain size distributions remain approximately selfsimilar with increasing grain size for a given powder purity. However, the lower purity aluminas exhibit broader size distributions.
direct measurements con®rm that they exceed 98.5% in all four microstructures. The lower purity aluminas exhibit a relatively broad grain size distribution and many grains are tabular or elongated. In contrast, the higher purity aluminas demonstrate a narrow grain size distribution and an equiaxed grain morphology. Mean grain sizes for the ®neand coarse-grain, lower purity aluminas are 4.0 and 12.7 mm, respectively, while mean grain sizes for the higher purity aluminas are 5.0 and 10.2 mm. Grain size distributions are shown for the four materials in Figs 2(a)±(d). These distributions reveal that normal grain growth occurs in both material systems since the distributions are approximately self-similar for a given powder purity. These measurements also con®rm that the higher purity aluminas display a narrower grain size distribution compared to the lower purity aluminas. Grain shape distributions are shown in Figs 3(a)± (d). Grain shape also remains approximately selfsimilar as mean grain size is increased for both the lower and higher purity aluminas. However, grain shape is signi®cantly more elongate for the lower purity aluminas with mean aspect ratios greater than 2.0 compared to approximately 1.5 for the higher purity aluminas.
3.2. Indentation strength Indentation-strength behavior for the four aluminas is presented as Figs 4(a)±(d). The solid line in each plot is the theoretical behavior for a material with constant toughness and is characterized by a slope of ÿ1/3 [18]. The hatched box on the left side of each plot represents the range of measured strength values for polished but unindented specimens. From this data, it is apparent that the indentation-strength response for the ®negrain, equiaxed material exhibits classically brittle behavior with a similar slope to that predicted for materials with constant toughness, while the coarsegrain, equiaxed and the ®ne-grain, elongate materials exhibit slight deviations from the theoretical constant-toughness line. The coarse-grain, elongate material exhibits the largest deviation from the constant-toughness response indicating that the strength of this material exhibits the greatest degree of ¯aw tolerance. T-curves deconvoluted from indentation-strength data are shown in Fig. 5 for each of the four aluminas. The indentation-strength data for the ®ne-grain, equiaxed alumina were used to estimate the indentation and ¯aw geometry parameters, w
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
569
Fig. 3. Grain aspect ratio distributions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®negrain, elongate and (d) coarse-grain, elongate aluminas. The lower purity materials demonstrate a more elongated grain shape compared with the higher purity materials.
and c. The values of w and c are estimated to be 0.100 and 1.00, respectively, which compares to previous evaluations for similar alumina ceramics of 0.071±0.082 for w and 0.77±1.26 for c [15]. The Tcurve for the ®ne-grain, equiaxed alumina gently rises and reaches a plateau toughness of approximately 3.6 MPaZm within a 250 mm extension. Tcurves for the coarse-grain, equiaxed and ®ne-grain, elongate aluminas also rise gradually but continue to increase to a toughness of approximately 5 MPaZm at a crack size of 1000 mm. The coarsegrain, elongate alumina exhibits the most pronounced T-curve behavior, starting at a lower toughness (2.8 MPaZm) and rising more steeply than the T-curves for the other materials. 3.3. In situ crack extension measurements Figure 6 plots in situ observations of crack extension from indentation ¯aws at several indentation loads for the ®ne-grain, equiaxed alumina. As applied load is increased, indentation cracks are seen to grow in an irregular, stochastic fashion jumping one or more grain facet lengths before arresting. This ``stop-start'' behavior is seen for all indentation loads and appears to be independent of crack size. At a critical size, the crack begins to accelerate and failure ensues. In the ®ne-grain, equiaxed alumina, this critical (®nal) crack size is
measured to be about 2.8 times greater then the initial indentation crack size for all indentation loads. This ratio compares favorably with the theoretical prediction that the critical crack size should be 2.6 times greater than the initial indentation ¯aw size for a material with constant fracture toughness [18]. Crack path in this material is observed to be a mixture of transgranular and intergranular. Occasionally, small microcracks are seen to open at grain boundaries ahead of the crack tip during propagation and are likely to play a role in determining crack path and the formation of grain bridges. However, microcracking is generally isolated to a few grain boundaries and there are no signs of an extended microcracked zone with associated material dilation. Figure 7 presents in situ observations of crack extension for the coarse-grain, elongate alumina at two dierent indentation loads. Crack extension is again seen to be irregular, the crack jumping at least one grain facet length during propagation. The tendency towards intergranular cracking is much greater in this material as compared to the ®negrain, equiaxed alumina leading to a more tortuous crack path and increased frequency of grain bridging. SEM photo-micrographs of the fracture surfaces for the four materials are shown in Figs 8(a)±(d). These images con®rm that crack path tortuosity
570
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Fig. 4. Indentation-strength results for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®negrain, elongate and (d) coarse-grain, elongate aluminas. The solid line represents the theoretical behavior sf 0P ÿ1 3 for a material with constant toughness and the hatched box indicates the natural strength for polished, unindented specimens. Only the coarse-grain, elongate alumina exhibits strong ¯aw tolerance as evidenced by a signi®cant deviation from the P ÿ1/3 line.
depends on both grain size and material purity. There is a greater tendency towards intergranular cracking in the lower purity aluminas as compared to the higher purity aluminas and thus crack patch tortuosity at a given grain size is greater in the lower purity aluminas. The in¯uence of grain size on crack path is more complex, however. In the lower purity alumina, intergranular fracture is always observed, independent of grain size. As a result, the crack path tortuosity increases dramatically with increasing grain size. In the higher purity aluminas, the tendency for intergranular cracking is reduced as the grain size is increased. As a result, the crack roughness is only slightly greater in the coarse-grain high-purity alumina compared to the ®ne-grain high-purity alumina. In summary, the crack path tortuosity is greatest in the coarse-grain, elongate microstructure followed by the ®ne-grain, elongate and coarse-grain, equiaxed microstructure and then the ®ne-grain, equiaxed microstructure. 3.4. Crack stability from annealed indentations Figure 9, presents a comparison of in situ crack extension behavior in dry nitrogen from annealed indentation ¯aws for all four microstructures. Several features may be noted from these data.
First, the failure stresses in all materials are signi®cantly greater after annealing compared to asindented specimens. This is consistent with the expectation that residual contact stresses are removed by annealing. The second important feature to note from these data is that little stable crack extension is observed during loading for all materials except the coarse-grain, elongate microstructure. Cracks in the coarse-grain, elongate alumina grow to roughly two times their initial size before failure ensues. Stable crack growth in this alumina occurs in a similar ``stop±start'' fashion for both as-indented and annealed-indented specimens, the crack following an intergranular path leading to the formation of many grain bridges. 3.5. Strength variability Strength data obtained from polished specimens are plotted in Weibull form for all four materials in Fig. 10. The Weibull moduli and associated 90% con®dence limits on Weibull moduli are shown in Table 1. The Weibull moduli vary from 7.8 and 5.3 for the ®ne-grain, equiaxed and coarse-grain, equiaxed aluminas, to 9.2 and 14.8 for the ®negrain, elongate and coarse-grain, elongate aluminas. From the con®dence limits, a statistical dierence
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
571
Fig. 5. T-curves deconvoluted from indentation-strength measurements for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. The thinner solid lines represent the net driving force for fracture (sum of applied stress intensity factor and residual indentation stress intensity factor) as a function of indentation load. The broader lines represent the Tcurves and are determined by the tangency construction.
Fig. 6. In situ measured crack size as a function of applied stress and indentation load for the ®ne-grain, equiaxed alumina. Arrows indicate unstable extension. Stable crack extension is observed until the size approaches 2.6 times the initial size independent of the initial indentation force.
Fig. 7. In situ measured crack size as a function of applied stress and indentation load for the coarse-grain, elongate alumina. For both indentation loads stable crack extension greater than 2.6 times the original crack size is observed prior to failure.
572
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
Fig. 8. Scanning electron micrographs of fracture surfaces from aluminas: (a) ®ne-grain, equiaxed; (b) coarse-grain, equiaxed; (c) ®ne-grain, elongate; and (d) coarse-grain, elongate. Note that the lower purity materials exhibit a larger degree of intergranular cracking compared to the higher purity materials.
exists between the Weibull modulus for the coarsegrain, elongate alumina and all other aluminas, and between the coarse-grain, equiaxed and ®ne-grain, elongate aluminas. The low Weibull modulus of the coarse-grain, equiaxed alumina has been attributed to a very small number of abnormal grains, which did not exist in the other three materials [11]. If the data are censored to eliminate specimens that failed from this secondary ¯aw population, only the coarse-grain, elongate alumina shows a statistically dierent (increased) Weibull modulus. Strength distributions from indented specimens are plotted using a Gaussian probability scale{ in Fig. 11. The Weibull function is not used to characterize strength distributions for indented specimens since weak-link statistics are no longer appropriate
{ Note that the slope of the line graphed on a probability scale is equal to the inverse standard deviation and is thus a measure of absolute variability.
when a single dominant ¯aw is introduced into a specimen and poor ®ts were obtained when data were force-®t to a Weibull distribution. The variability in strength is instead characterized using the coecient of variation, CV, for each batch of material and is calculated by taking the ratio of the standard deviation to the mean. As expected from indentation-strength measurements, the ®ne-grain, equiaxed alumina exhibits the lowest mean strength when samples are indented at a force of 100 N Table 1. Characteristic strengths (s0), Weibull modulii (m ) and 90% con®dence intervals (m5 and m95) for the Weibull modulii for the four aluminas investigated. The coarse-grain, elongate alumina exhibits signi®cantly less strength variability compared to other aluminas Material
s0
m
m5
m95
Fine-grain, equiaxed Fine-grain, elongate Coarse-grain, equiaxed Coarse-grain, elongate
507 369 346 243
7.8 5.3 9.2 14.8
6.1 4.2 7.4 12.1
10.5 7.1 11.9 19.8
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573
Fig. 10. Weibull plot for unindented samples with polished surfaces. Note that although the ®ne-grain, equiaxed alumina exhibits the highest mean strength, the coarsegrain, elongate alumina demonstrates the highest Weibull modulus. Fig. 9. Comparison of crack extension from annealed indentations in w ®ne-grain, equiaxed, q coarse-grain, equiaxed, * ®ne-grain, elongate and Q coarse-grain, elongate aluminas. Annealing following indentation removes the residual stress ®elds associated with contact damage. Only the coarse-grain, elongate alumina demonstrates signi®cant stable extension prior to failure.
prior to testing. The coarse-grain, equiaxed and ®ne-grain, elongate aluminas have similar mean strengths, which are slightly higher than that for the ®ne-grain, equiaxed alumina. The coarse-grain, elongate material exhibits the highest mean strength. Coecients of variation in strength range from 5.7 to 6.7% for all four aluminas and indicate no statistically signi®cant dierences in strength variability. These results would seem to imply that microstructure does not in¯uence the strength variability of alumina. However, it can be seen from Fig. 12 that signi®cant dierences in indentation ¯aw size distribution between the four microstructures are evident even though all indentations are made at the same peak contact force (100 N). It is apparent that indentations in the ®ne-grain, equiaxed material are well behaved and exhibit little variability in crack length. Signi®cantly more variability in crack length and crack morphology develops in the more ¯aw-tolerant materials. These dierences in the crack morphologies have been attributed to local variations in fracture toughness which increase with grain size [9, 10, 19]. The indentation crack
sizes measured immediately following indentation are summarized in Table 2 for all four aluminas. These measurements support the qualitative observations that the ®ne-grain, equiaxed alumina exhibits the least variability in indentation crack size CV 5:0% while the coecient of variation in crack length for the other aluminas falls within the range of 8.3±11.7%. Results are consistent with previous work [19], which has shown that indentation crack sizes are sensitive to microstructure even though this variability is not re¯ected in strength variability. The relationship between variability in initial indentation crack size and microstructure is similar to that between ¯aw tolerance and microstructure: variability in crack sizes increases with grain size and grain aspect ratio. Most signi®cantly, although variability in the indentation ¯aw size increases dramatically with grain size and grain
Table 2. Comparison of indentation crack sizes P 100 N for the four aluminas investigated. Note that the variability in size increases with increasing grain size and aspect ratio Material Fine-grain, equiaxed Fine-grain, elongate Coarse-grain, equiaxed Coarse-grain, elongate
Crack size, c (mm)
CV (%)
221211 182221 181215 192222
5.0 11.7 8.3 11.6
Fig. 11. Strength distributions plotted on a probability scale for aluminas containing 100 N Vickers indentations. There is no signi®cant dierence in strength variability between the four aluminas investigated.
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Fig. 12. Optical micrographs of 100 N Vickers indentations in (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. Note that the variability in indentation crack size increases with grain size and grain aspect ratio.
aspect ratio, strength variability is not aected by changes in indentation ¯aw size distribution. 4. DISCUSSION
Our study illustrates that T-curve behavior and associated ¯aw tolerance in alumina ceramics is not controlled solely by mean grain size. For example, although the ®ne-grain, equiaxed alumina and the ®ne-grain, elongate alumina have similar mean grain sizes, the ¯aw tolerance and T-curve behaviors are much more pronounced for the more elongate grain morphology. In addition, the ®negrain, equiaxed alumina and coarse-grain, equiaxed alumina exhibit similar toughness behavior, despite signi®cant dierences in mean grain size. These results are consistent with previous work, which have shown that microstructural characteristics other than grain size such as grain shape, grain size distribution, and grain-boundary toughness have a pronounced in¯uence on crack-growth resistance in alumina ceramics. For example Cook found that T-
curve behavior was dependent on both grain size and grain-boundary chemistry since these factors in¯uence the degree of intergranular fracture [20]. Similarly, in high-purity alumina ceramics, increasing the grains size without changing the grainboundary chemistry did not lead to enhanced toughness since larger grains tended to fracture transgranularly thus reducing grain-bridge density and associated crack-tip shielding [11]. Many studies on aluminas indicate that in all but the purest materials, an amorphous phase is present at many grain boundaries following sintering [21, 22]. In addition to providing an easier path for crack propagation, the glassy phase generally results in broader grain size distributions and elongate or tabular shaped grains [23]. Such grain morphology gives rise to potentially more eective grain bridges, and associated ¯aw tolerance, and is generally observed in lower purity aluminas [3, 24]. There are a number of important features that are apparent from T-curves determined from our indentation-strength measurements. First, increases
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
in toughness at longer crack lengths come at the expense of short-crack toughness. For instance, although the ®ne-grain, equiaxed material demonstrates the lowest long-crack toughness, it also shows the highest short-crack toughness. Thus, details of the T-curve shape are tied to microstructure by grain size, grain shape, and a tendency towards intergranular cracking [20]. Materials that contain coarse, elongate grains and fracture intergranularly exhibit the steepest T-curves and the greatest increase in toughness. However, these same materials also exhibit the lowest short-crack toughness and associated strength. This ®nding is consistent with previous observations in alumina and other ceramics that derive T-curve behavior from grain bridging [3, 20, 24±26]. We now go on to consider the eect of T-curve behavior on crack stability. Stable crack extension depends on a number of factors including: initial ¯aw size (c ), magnitude and shape of the T-curve, as well as morphological details of the initiating critical ¯aw [2]. The criterion for fracture, based on linear elastic fracture mechanics, states that if the applied stress intensity, KI, exceeds the toughness, T, then crack growth occurs. However, if the Tcurve rises rapidly, stable equilibrium crack extension may precede catastrophic failure. The degree of stabilization prior to failure is governed by the rate of increase in applied stress intensity, dKI/dc, and the rate of increase of toughness, dT/dc. Stable crack extension occurs in materials with a rising toughness until: KI rT
2a
dKI dT r dc dc
2b
after which point failure occurs at an unstable equilibrium condition. For cracks emanating from natural ¯aws, the applied stress intensity is given by the familiar form: p KI cs c
3
where s is the applied stress and c is a crack geometry-dependent term. Thus, a simple graphical construction for crack stability can be made by plotting applied stress intensity vs square-root of crack size and then superimposing a T-curve on this plot (the tangency construction). However, when contact-induced residual stresses are present (e.g. as in the case of an indentation ¯aw), there is an additional local driving force for crack extension. This
{ We refer to TR as the ``eective'' toughness since residual stresses generated during indentation lead to crack extension at stress intensities much lower than the actual toughness, T.
575
driving force can be accounted for by subtracting the stress intensity due to indentation residual stress from the toughness [18] resulting in an eective toughness, TR, given by:{ TR T ÿ
wP c3 2
4
The advantage of subtracting the indentation residual stress intensity factor from the toughness rather than adding it to the applied stress intensity factor (as is usually done) is that a conventional graphical tangency construction can still be performed. This allows direct comparison between crack stability in the case of either natural or contact-induced ¯aws. T-curves determined from the indentationstrength data along with tangency constructions are shown for the four aluminas in Figs 13(a)±(d) for both indentation ¯aws and natural ¯aws. The tangency construction predicts that indentation cracks should grow in a stable manner until the crack length reaches 2.59 times the initial indentation crack size for materials of constant toughness [18]. Crack extension beyond this length is an indication that the material exhibits rising T-curve behavior. All four aluminas studied are predicted to exhibit stable equilibrium crack extension from indentation ¯aws based on the tangency constructions shown in Figs 13(a)±(d). For instance, indentation ¯aws in the ®ne-grain, equiaxed alumina should grow to 2.8 times their initial sizes before failure. In the ®negrain, elongate, coarse-grain, equiaxed, and coarsegrain, elongate materials, all with stronger T-curve behavior, additional stabilization is predicted up to approximately 3.2 times the initial ¯aw size. We see that the predicted degree of stable crack extension from the tangency constructions in Fig. 13 compares favorably to that observed and presented in Figs 6 and 7. Although all of the aluminas exhibit some Tcurve behavior, the coarse-grain, elongate alumina clearly exhibits the strongest T-curve. The tangency construction indicates that this is the only material that has a suciently steep T-curve to result in stable crack extension from ¯aws free of contactinduced residual stresses (i.e. natural ¯aws). Recall that when indentations are annealed to remove residual stresses (see Fig. 9), signi®cant stable crack extension is only observed in the coarse-grain, elongate alumina. These results show that the nature of the ¯aw and the shape of the T-curve both play an important role in determining whether strength is sensitive to initial ¯aw size. For example, although materials that exhibit weak T-curve behavior exhibit stable crack extension from indentation ¯aws, the strength still shows some dependence on indentation ¯aw size. When the same material fails from a natural ¯aw, however, no stable crack extension is observed
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KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
Fig. 13. T-curves and tangency constructions for (a) ®ne-grain, equiaxed, (b) coarse-grain, equiaxed, (c) ®ne-grain, elongate and (d) coarse-grain, elongate aluminas. The solid lines are T-curves for natural ¯aws while the shaded lines are T-curves for indentation ¯aws P 100 N). Initial crack size is denoted by c0 and the dashed lines represent the tangency condition at instability, cf . All materials are predicted to show stable crack growth from indentation ¯aws in contrast to failure from natural ¯aws where only the coarse-grain, elongate alumina demonstrates stable equilibrium growth en route to failure.
and strength is strongly dependent on initial ¯aw size. It is only for materials that exhibit very strong T-curve behavior that stable crack extension and ¯aw tolerance is observed from both natural and contact-induced ¯aws. These results are consistent with a recent theoretical model which concluded that stable crack growth was not predicted for an alumina that exhibited modest T-curve behavior when failure ensued from a natural ¯aw free of residual stresses [27]. We now turn our attention to the question of strength variability. Weibull plots presented in Fig. 10 show that strength variability decreases only for aluminas with very strong ¯aw tolerance. Similar observations were made by Ting et al. for a variety of alumina ceramics [28]. The highest Weibull modulus was displayed in aluminas with broadest grain size distribution and an elongate grain shape while the lowest Weibull modulus was displayed by an alumina with equiaxed grain morphology and narrow grain size distribution. Hirosaki et al. and Homan and Petzow both found similar results for silicon nitride [29, 30], another material toughened by a grain-bridging mechanism. Ting et al. and
Hirosaki et al. argued that the decreased strength variability in these systems was due to a narrowing in the critical ¯aw size distribution. They suggested there is a decreased probability of encountering a large strength-controlling defect in materials with narrow grain size distribution. Since large defects occur in only a few specimens, there is greater scatter in strength for such materials. They went on to argue that materials with heterogeneous microstructures contain a more uniform dispersion of large ¯aws, thus increasing the probability of ®nding large defects in all specimens and reducing strength variability. A contrary argument was presented by Homan and Petzow who suggested that T-curve behavior is responsible for the reduced strength variability observed in grain-bridging ceramics rather than a narrowing in ¯aw size distribution. However, little experimental evidence was oered to support these assertions. One potential method of distinguishing whether T-curve behavior or changes in critical ¯aw size distribution are responsible for the changes in strength variability involves quantifying the critical ¯aw population. This is a dicult task as the failure ori-
KOVAR et al.: TOUGHNESS-CURVE BEHAVIOR
gin often cannot be determined unambiguously from post mortem fractography particularly in coarse-grained, heterogeneous microstructures. Furthermore, when a failure origin can be clearly identi®ed, it may not be possible to accurately assess the in¯uence of the critical ¯aw size population due to uncertainties in specifying the driving force for fracture. Even in the case of an ideal Grith-like ¯aw, knowledge of ¯aw geometry and local stress ®elds is needed together with ¯aw size to accurately specify the applied stress intensity factor that resulted in failure. A possible solution to this problem is to control failure using well-characterized critical ¯aws using indentation ¯aws [18]. The strength distributions collected for aluminas containing 100 N indentations shown in Fig. 11 reveal that there is essentially little dierence in strength variability of aluminas with vastly dierent microstructures. However, measurements of indentation crack sizes revealed that distributions in crack sizes also vary greatly depending on microstructure. Indentation cracks in the coarse-grain, elongate alumina display more than twice the size variability than the ®ne-grain, equiaxed alumina, yet no dierence in strength variability is observed. This suggests that a narrowing of the critical ¯aw size distribution by itself cannot account for the reduction in strength variability observed in the coarse-grain, elongate material. Further evidence that the critical ¯aw size distribution cannot be the only factor controlling strength in grain-bridging materials is provided by in situ crack-growth measurements on specimens containing annealed indentation ¯aws. These tests conclusively demonstrate that even when a ¯aw is not subject to strong residual stresses associated with contact damage, stable crack extension is observed in aluminas with strong T-curve behavior whereas no signi®cant crack stability is observed in aluminas with less pronounced T-curve behavior. While the in¯uence of ¯aw size distribution cannot be discounted fully, this observation suggests that the stabilizing in¯uence of a strong T-curve must also play an important role in reducing strength variability in our aluminas. From a practical viewpoint these results have several important implications with respect to materials selection when reliability is crucial. Ultimate component reliability is attained when strength becomes insensitive to both natural and processing ¯aws. For example, while it is true that ®ne-grain, highpurity aluminas demonstrate high mean strength, they exhibit poor strength variability and high susceptibility to contact-induced damage. Microstructural coarsening in high-purity aluminas does not signi®cantly increase ¯aw tolerance because grain bridging is limited by a tendency towards intergranular fracture. As a result, coarsegrain, equiaxed aluminas demonstrate inferior strength variability and a moderate susceptibility to
577
contact-induced ¯aws. Despite the presence of a broader grain size distribution, elongate grain shape, and decreased grain-boundary strength, the ®ne-grain, elongate alumina exhibits similar properties to the ®ne-grain, equiaxed alumina since the ecacy of grain bridging is scale dependent. It is only when elongate grains and weak grain boundaries are combined with a larger grain size that signi®cant grain bridging occurs. T-curve behavior in this large-grain, elongate microstructure is strong enough to stabilize both natural and contactinduced ¯aws. As a result, this material exhibits low strength variability when failure is initiated from natural ¯aws and strong ¯aw tolerance to post-processing, contact-induced ¯aws. It should be noted that improved reliability comes at the expense of reduced mean strength. Such a trade-o is a feature of materials in which the shielding mechanism responsible for strong T-curve behavior results in a low short-crack toughness.
5. CONCLUSIONS
Strength, toughness and reliability of four alumina ceramics with dierent microstructural scale and morphology have been investigated. A combination of large, elongate grains and intergranular fracture results in signi®cant grain bridging in the coarse-grain, lower purity alumina. As a result, this material exhibits the greatest ¯aw tolerance to indentation ¯aws and signi®cant rising T-curve behavior. In situ crack propagation experiments in a dry ambient atmosphere con®rm that this is the only material of the four examined that exhibits a signi®cant degree of stable crack extension from ¯aws free of contact-induced residual stress. Reliability tests to probe strength variability when failure is controlled by natural ¯aws demonstrate that the coarse-grain, elongate alumina exhibits less strength variability compared to all of the other materials that were tested. Strength tests on indented specimens (controlled ¯aws) reveal that there is no statistical dierence in strength variability between materials at a given indentation load. However, examination of the indentation cracks show that variability in crack sizes increases dramatically as grain size or grain aspect ratio is increased. Based on crack-stability measurements and strength-variability measurements from indented specimens, the reduced strength variability observed for polished specimens with the coarsegrained, elongate microstructure cannot be accounted for simply by changes in the critical ¯aw population. The study suggests that maximum reliability is achieved when the T-curve is suciently strong to stabilize the propagation of natural ¯aws en route to failure. Improved reliability is attained at the cost of mean strength when the shielding mechanism responsible for strong T-curve behavior
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results in a trade-o between strength and toughness. AcknowledgementsÐThe authors would like to acknowledge Drs Brian Lawn, Edwin Fuller, and Craig Carter for their many helpful discussions. In addition, we would like to thank Ms Terry Deis and Mr Carl Lovejoy for their assistance in the preparation of specimens used in this study. Principal funding for this research was provided by the Air Force Oce of Scienti®c Research, Contract Number F49620-92-J-0034. REFERENCES 1. Becher, P. F., J. Am. Ceram. Soc., 1991, 74, 255. 2. Bennison, S. J. and Lawn, B. R., J. Mater. Sci., 1989, 24, 3169. 3. Chantikul, P., Bennison, S. J. and Lawn, B. R., J. Am. Ceram. Soc., 1990, 73, 2419. 4. Lawn, B. R., Padture, N. P., Braun, L. M. and Bennison, S. J., J. Am. Ceram. Soc., 1993, 76, 2235. 5. Kendall, K., McN. Alford, N., Tan, S. R. and Birchall, J. D., J. Mater. Res., 1986, 1, 120. 6. Cook, R. F. and Clarke, D. R., Acta metall., 1988, 36, 555. 7. Readey, M. J., McNamara, P. D., McCallen, C. L. and Lawn, B. R., J. Mater. Sci., 1993, 28, 6748. 8. Li, C. W. and Yamanis, J., Ceram. Engng Sci. Proc., 1989, 10, 623. 9. Kovar, D. and Readey, M. J., J. Am. Ceram. Soc., 1994, 77, 1928. 10. Curtin, W. A., J. Am. Ceram. Soc., 1995, 78, 1313. 11. Kovar, D. and Readey, M. J., J. Am. Ceram. Soc., 1996, 79, 305. 12. Bennison, S. J. and Lawn, B. R., Acta metall., 1989, 37, 2659. 13. Fazio, P. C., et al. (ed.), ASTM Standard C-20-92. American Society for Testing and Materials, 1992.
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