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increased the machined strengths by 100 MPa (Weibull moduli were similar for all three surface conditions at 17). Earlier studies of Rice [17] showed strengths of machined samples of dense Al2O3, MgAl2O4, and ZrO2 (+ 11.2% Y2O3) decreasing upon annealing at 1400, 1600, and 1670°C (in general G increased only at 1670°C).

With limited, or no, extrinsic sources of failure, e.g. pores, or surface irregularities, the presence of (intrinsic) grain boundary grooves on as-fired surfaces also limits strengths. Coble [23] has theoretically shown that such grooves act as failure causing flaws. Such grooves are expected to be less severe for finer G bodies than for coarser ones due to the relatively lower temperatures, shorter times, or both used in obtaining finer G, thus introducing a G dependence of such grooves. Coble indicates that the equivalent flaws will generally be related to G and vary in the range of G/15 to G. Support for the deleterious effects of grain boundary grooving and its correlation with G is provided by the general fiber results as well as Simpson’s [156] fiber failures [3] and increasing the σ of FP fibers 25% by smoothing their surface with a SiO2 coating [159, 160].

Chemical finishing of polycrystalline materials has received very limited study, mainly because of frequent adverse affects of porosity, impurities, and varying grain orientation, often resulting in rough or irregular, hence weaker, surfaces. Evans and Davidge [72] showed a small (<10%) strength increase for their chemically polished UO2 (P 0.03) over the temperature range compared (to 1200°C), with this increase being greater for their 25 µm vs. 8 µm G body. Similarly, Gruszka et al.’s [124] chemical finishing of 99.5% Al2O3 electronic substrates (G 0.9 µm, P 3%) showed both some σ increases and decreases vs. as-fired or diamond lapped surfaces (both giving σ 400 MPa). They showed an approximate inverse σ-surface roughness correlation, i.e. a maximum σ of 550 MPa from use of molten borax with among the lowest surface roughness (2.9 µin.) similar to that for the as-fired and diamond lapped surfaces (respectively 2.6 and 2.0 µin), with σ 410 MPa. This supports the implication from glass, single crystal, and more specifically fiber, processing that achieving very smooth surfaces of dense, fine grain polycrystalline bodies can give substantially higher σ. Strengths of 2–5 GPa reported for small CVD SiC specimens (P = 0) with very smooth as-deposited surfaces (apparently due in part to the extremely fine G 0.01–0.1 µm) clearly show that such surfaces can yield high strengths [111, 112]. However, abrasion even by only light sanding with SiC abrasive paper readily dropped these very high strengths to normal levels observed for conventional SiC bodies, consistent with the resulting flaw sizes observed (which also indicate lower KIC e.g. by 20%). Such σ decreases with abrasion were greater than those observed for c-axis sapphire filaments [167].

An important question is what other flaw sources and populations can give similar σ–G-1/2 behavior as for specimens whose strength is primarily determined by surface flaws in machined or as-fired surfaces. Clearly, handling, impact, etc.

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of as-fired or deposited surfaces result in flaws that are generally similar to machining flaws, thus expected to give similar σ – G-1/2 behavior. Two other primary flaw sources are compositional variations (including impurities) and porosity, with heterogeneities of these often being particularly important sources of failure. Impurities and other compositional variations often have their main influence via effects on grain size, especially that local to a serious compositional fluctuation (e.g. Figure 1.5), in which case similar σ – G-1/2 trends are likely to result (but require identification of effects of such local G changes, typically by fractography). On the other hand, impurity particles of random size, location, or both are likely to be simply a source of scatter for most mechanisms envisioned, e.g. microcracking, thus obscuring or eliminating normal σ – G-1/2 behavior. The other major source of failure is pores or pore clusters. Rice [6, 144, 273] has shown that machined alumina samples with fine porosity fail from machining flaws similar to those found in dense bodies and thus should show the expected σ – G-1/2 behavior (at least over the ranges in which G can change without significantly changing the amount or character of the porosity). Recently, Zimmermann et al. [278] showed that large (artificially introduced) isolated, uniform spherical pores that were fracture origins in otherwise dense alumina samples showed strengths decreasing as G increased from 0.8 to 9.2 µm, following the same trend with G as for dense bodies without the large pores, but at about 30% lower strengths for failure from the isolated large pores. Zimmermann and Rödel [279] discussed this trend, indicating that it probably reflected localized cracks, rather than the postulated circumferential cracks around the pores, as also suggested by Rice [6, 280]. This thus indicates that some larger pore failures can give similar σ – G-1/2 behavior as for dense bodies, but this probably requires that the pore size, shape, spatial distribution, and surrounding microstructure be fairly uniform to limit data scatter, which could obscure or obliterate any consistent trend as a function of G. Another issue that needs to be addressed to assess the extent of such pore-grain failure is demonstrating such pore-grain failure relations over a larger G range, especially into the larger G regime and where spacing of larger pores is variable, which typically occurs and greatly complicates pore induced fracture [6, 280]. The generalization of the two-branch σ–G-1/2 behavior proposed by Zimmermann and Rödel [279] is a stimulus for further study but appears limited by the above-noted variations of individual mechanisms, as well as the complications of various combinations of mechanisms, e.g. of pores, second phases, and larger grains.

B.Test Condition Effects on Strength–Grain Size Behavior

Effects of test conditions on resulting σ–G-1/2, which entail specimen size and geometry, and test factors, e.g. uniaxial versus biaxial stress, test environment, and cyclic stressing, need to be considered. The primary effects of specimen

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size, geometry, as well as some aspects of test methodology are due to changes in the amount of materials being at substantial stress levels and hence the availability of different flaw types and sizes to cause failure. As noted earlier, use of flexure specimens of rectangular cross section raises some questions with regard to the appropriateness of using Gm, since the latter usually represents a volume flaw distribution, whereas this test configuration favors surface failure. The use of round versus rectangular cross section flexure rods is even more questionable, since this presents significantly more surface area of varying stress from which failure can occur. Binns and Popper [121] showed σ of round rods (1 cm dia.) averaging 25% higher (ranging from 20% lower, the only case of round rods being weaker than rectangular rods, to as much as 90% higher) than for 1 cm square bars of the same materials for 10 different alumina bodies. Similarly, McNamee and Morrell’s [127] 95% Al2O3 fabricated by various means (G = 3–7.5 µm, P = 0.035–0.05) showed σ of round rods and rectangular bars made by extrusion higher by 40 and 60% respectively than for bars machined from die pressed, isopressed, or slip cast plates. Further, within an extruded body σ was greater for fracture perpendicular vs. parallel with the extrusion axis. Although all three of their direct comparisons showed as-fired strengths 10% higher than for as-machined surfaces, their machining was with a somewhat coarser [180] grit transverse to the bar lengths, which gives lower σ. Fractographic determination of specific failure origins, if sucessful, is the basic solution to this problem. However, since this is often not done, and is not always practical or feasible, the next best procedure should be to attempt at least an approximate statistical evaluation of the frequency of potential failure sources, e.g. as done by McNamee and Morrell [127]. They showed the occurrence of larger G (> 70 µm) varying from 0.01/mm2 to 0.45/mm2 in their samples. Thus three point flexure (preferably with rectangular cross section bars to avoid large surface areas, at variable stress) more closely approximates average G dependence of σ.

That statistical effects continue to small size and high σ is shown by the definitive gauge section dependence of σ in tensile testing fibers [158–165]. This is consistent with the frequently low Weibull moduli in such tests, e.g. 4–6. The frequent correspondence of tensile strengths of fibers and corresponding flexure strengths of small bars (e.g. Figs. 3.17, 3.18, and 3.21) indicate similar volumes under high stress.

An important question is the effects of stress conditions, e.g. biaxial versus uniaxial stress, on σ–G-1/2 behavior, both due directly to the differences of the stress states as well as indirectly on other factors such as slow crack growth and microcracking. One of the few tests of these issues is data of Chantikul et al. [131] on biaxial strength as a function of G (Fig. 3.14). Their results cannot be distinguished from uniaxial results, indicating little or no effect of biaxial loading, but more testing of other materials and microstructures is needed. Another stress factor is repeated stressing, i.e. mechanical fatigue. Aspects of such tests

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were briefly outlined in Chapter 2, Section II.I, but these were focused on large crack behavior with very little attention to G effects. One exception to this was Lewis and Rices’ [285] tests of Lucalox Al2O3, which indicated an intrinsic fatigue mechanism due to microcracking from TEA (or EA), which would thus be G dependent.

Consider next effects of environmentally driven slow crack growth, i.e. beyond subcritical growth considered below. Environmentally driven crack growth, which has been evaluated primarily, if not exclusively, by uniaxial stressing, clearly increases flaw sizes as the rate of crack growth increases, stressing rate decreases, and strength increases (e.g. with decreasing G). However, intrinsic increases in SCG as G decreases (Figure 2.7) would, at least partially, if not completely, counter this trend. As G increases down the finer G branch, the resulting decreased strength will decrease the contribution of slow crack growth, hence also adding to the slope of the finer G branch. As strengths approach those of the intersection on with the larger G branch, i.e. with the grain dimensions approaching those to contain the failure causing flaw in a single grain, crack growth rates will transition from those for polycrystalline samples to those of single crystals or of grain boundaries. Besides possibly further changing crack growth rates, this may result in a change in fracture mode as discussed in Chapter 2, Section III.A.

Kirchner and Ragosta [281] calculated that small (e.g. 10 µm) flaws within single Al2O3 grains would not lead to catastrophic failure at single crystal KIC values unless G was ≥ 100 µm and loading rates high (e.g. 104 MPa/s) to limit environmentally induced slow crack growth. They concluded that cracks would otherwise be arrested as the grain boundary and failure would be determined by the polycrystalline KIC. While these calculations show that large Al2O3 grains could be failure sources at single crystal KIC values, their assumptions lead to more restricted calculated growth. They assumed that the transition from single to polycrystalline KIC values occurs at the first grain boundary rather than over a range of e.g. 2–6 grains, as indicated for materials investigated [28] (Figure 2.16). They also assumed that TEA stresses were not a factor in such slow crack growth, i.e. flaws would not grow in the absence of external mechanical stressing, which has received little direct study. However, McMahon’s [276] and Rice’s [282] Al2O3 studies show that this either does not occur or readily saturates early in the life of a specimen. That the latter may be the case is shown by observations of Hunter et al. [283, 284] that microcracking of pure HfO2 upon cooling in a sealed furnace only began occurring upon opening the furnace to air and continued for several tens of hours before saturation.

Gruver et al. [128] showed similar 96% Al2O3 origins from isolated large grains over a range of temperatures. Using 1/2 these Gm values as c for the eight large grain (of 20) fracture origins in liquid N2 gave KIC 3.9 ± 0.6 MPa·m1/2, and

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seven of 20 in the 95% (and two of 20 hot pressed) Al2O3 having large grain origins at 22°C gave KIC 3.2 ± 0.3 MPa·m1/2. While these all generally agree with the polycrystalline KIC, all gave at least one value ≥ two standard deviations below the average, suggesting that in those cases the grains were not the complete flaw, i.e., c extended into the surrounding average grain structure, so plotting σ value for those Gm values is questionable. Similarly, it cannot be ruled out that large grains giving the highest calculated KIC values may have resulted in failure before the flaw reached the full grain size, i.e. the critical flaw size was <G.

Chantikul et al.’s [131] model based on crack bridging is an interesting extension of the basic fracture mechanics-c/G model. It also predicts two branches, with branch intersections at c 2 – 3G. They showed the larger G slope decreasing as G increases, indicating a transition to single (or bi-) crystal strengths. Although it predicts a zero slope for the finer G branches, factors leading to a nonzero slope in this region could be included. Use of a grain boundary KIC of 2.75 MPa·m1/2 Al2O3 is questionable in view of KIC for many common orientations of sapphire being 2 MPa·m1/2, and grain boundary values commonly being 1/2 such values (Chapter 2, Section III.E). However, there are more fundamental issues concerning their model. It is based on TEA stresses, yet the same type of σ–G-1/2 behavior is observed for cubic materials not having any TEA stresses. Although bridging has been observed in cubic as well as noncubic materials, there are a variety of concerns regarding the applicability of bridging to normal strength behavior, as discussed earlier and later.

VII. EVALUATION OF THE Σ–G-1/2 MODEL PARAMETERS

A.Overall Review and Assessment

The primary failure mechanism in most ceramics at or near room temperature is brittle failure from preexisting flaws per Fig. 3.1. The first of two deviations from this is failure from a substantial density of microcracks that are preexisting, developing, or both relative to stressing to failure. These lead to rapid initial strength decreases (commonly closely paralleling those of the elastic moduli) with increasing G when Gs for substantial microcracking is reached or exceeded. Presumably strength shows normal G dependence for the preexisting (e.g. machining) flaw population at finer G (< Gs) with a transition to microcrack determined failure that depends on the G distribution and TEA, and possibly EA, specifics. The other deviation is microplastic initiation of flaws and failure in some ceramics. This becomes more marginal as stresses for microplasticity increase due to intrinsic effects at any temperature, decreasing temperatures, or inhibition of the microplasticity by impurities or additives. Higher stress for microplasticity makes this failure mechanism more dependent on better surface finish, larger G, and higher temperatures. The G = ∞ strength projection and in-

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tercept from microplastic failure is typically the stress to activate the microplasticity, which is usually the strength of single crystals oriented for easiest activation of the microplasticity determining failure of polycrystalline bodies. Such extrapolation of polycrystalline to single crystal strengths, rather than polycrystalline strengths falling well below single crystal strengths, is a basic difference respectively between microplastic and preexisting flaw failure. Increasing competition of preexisting flaw versus microplastic failure as microplastic stresses increase commonly results in a switch from microplastic to preexisting failure as strengths increase with decreasing G. The grain size of this transition should incrase as the surface finish quality decreases and the stress for the microplasticity increases, and again appears to occur when c G/2. Thus all three failure mechanisms will commonly have a finer G branch due to preexisting flaw failure.

The primary factors controlling tensile strength in brittle failure are typically the initial flaw, any slow crack growth, and the toughness controlling its propagation to failure. Conventional fracture mechanics approaches to the G dependence of strength have focused on toughness, neglecting possible variations of flaw character, i.e. size and location relative to the grain structure, and thus does not provide guidance for understanding or predicting the slopes and intersection(s) of the finer and larger G branches. The slopes of these branches are important first and foremost since this is a primary tool for estimating the benefits of reducing G and secondarily since most proposing use of Gm assumed (based on limited data) a zero slope of such branches. However, data clearly shows variable, often substantial, positive slopes with some possibility of zero or even negative slopes, all of which are consistent with expectations of varying flaw and other changes with G. A key factor in the finer G branch slopes is that while, to a first approximation, flaw size is independent of G, there are second-order variations. These are indicated for machining flaws by the following representative theoretical equation predicting effects of material properties on machining flaw sizes [286]:

	c	E 1/ 3 L						2 / 3
								(3.2)
			H			K

where E=Young’s modulus, H=hardness, L=load, and K= the appropriate toughness. For c < G the toughness should be the appropriate single crystal or grain boundary values, while that for c> G should be a polycrystalline value for small cracks, i.e. typically no R-curve or related effects.

Consider first effects of the above flaw size variations as a function of G. In a given material E typically has no G dependence, and K often has limited or no G dependence at finer G but clearly decreases for c approaching G/2. H measurably increases with decreasing G in the finer G region (Chap. 4, Sec. II), thus indicating some limited reductions in c with decreasing G. These effects increase

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strength as G decreases, giving a positive finer G slope. Different materials with different E and K values would shift flaw sizes, and thus also strengths, some in the finer G branches, with differing G dependences of K varying finer G branch slopes. Note also that the minimum in H (Chap. 4) occurs when the indent and grain sizes are similar, which may be a factor in the grain size where c G/2.

Similar trends are expected for as-fired surfaces. Thus a decrease in flaw sizes in as-fired surfaces as G decreases is also expected, since decreasing grain sizes are typically obtained by reduced temperature–time exposure of the sample, e.g. of fibers. Such reduced exposure reduces the extent of grain boundary grooving associated with finer G, and the finer G itself increases the tortuosity of the sequence of grain boundary grooves acting as a flaw, decreasing severity of the resultant flaw, often probably more than for machined flaws, indicating higher positive finer G slopes.

Turning to experimental evaluations of flaw populations, much of this must be inferred from the G dependence of strength, particularly for as-fired surfaces, since there is no independent flaw data for them. This is also partly true for machined sample, since while there is considerable flaw data for them, detailed comparisons of different materials for varying G as a function of machining parameters are limited. However, extensive studies of machining flaws, mainly by Rice and colleagues [12, 13, 15–22], show that machining flaw sizes for failure for a broad range of ceramics with typical moderate to finer grit diamond grinding do not vary widely, e.g. c generally is in the 20–50 µm range. However, materials of higher than normal machined strengths such as WC-Co, TZP, and some Si3N4 bodies (Fig. 3.34) have finer machining flaw sizes controlling failure, e.g. more commonly in the 10–20 µm range (Fig. 3.34). Thus such bodies with high toughness at small crack sizes, typically in finer G bodies with effective toughening at finer G, have smaller flaw sizes, which are less likely to be affected significantly by R-curve and other related large crack effects but may experience more benefit from surface compressive stresses than is typical for most machining.

Besides statistical variations of factors noted above, there are other important sources of differing flaw dependences on G at finer G. The first of three to be noted is preferred grain orientation, local or global, especially if it changes with G. This clearly occurs, e.g. in BeO [169, 170], where extrusion results in some preferred orientation of substantially nonequiaxed powder particles, with the orientation then increasing with grain growth, apparently due to preferential growth of oriented grains. The σ–G-1/2 for such bodies, though limited, indicates a negative finer G branch slope which is consistent with the data showing orientation increasing with G, which could counter the normal strength decrease as G increases.

A second source of changing finer G slopes is changing flaw shape as c increases to approach G/2. Strength anisotropy studies as a function of stress versus machining directions originally did not focus on effects of G on the extent of

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FIG. 3.34 SEM fractograph showing the fracture origin (arrow) in a dense hot pressed, higher strength Si3N4 from a typically smaller than normal machining flaw (shown at 10-fold higher magnification in the insert).

flaw elongation parallel with abrasive motion. However, evidence now shows that flaw elongation is reduced as the largest flaw dimension, its length in this case, approaches the dimensions of the surface grains being machined [143, 144, 273]. As G increases toward and beyond the flaw dimensions, grains first begin to constrain flaw elongation, then the smaller flaw dimension, i.e. its depth. Further G increase then progressively relaxes these constraints so that anisotropic strength as a function of machining direction increases with further G increase. This is extensively shown by studies of strength anisotropy as a function of machining and stressing directions (Fig. 3.33) and is supported by some direct fractographic evidence. The progressive reduction in flaw elongation as G increases clearly reduces slopes of finer G branches for specimens tested with the stress axis normal to the machining direction. Machining studies indicate that finer G branch slopes for such specimens are often 0 and may actually become negative as grain sizes approach the flaw depths [17].

A third possible source of changing finer G slopes is changing mechanisms involved in the failure process. An example of this is increasing contribution of

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microcracking as G increases. Thus, Tomaszewski’s data for Al2O3 [134] shows evidence of microcracking increasing as G increased via decreased E (from strength not vibrational measurements). Such increasing microcracking as G increases should increase the finer G slope as indicated by σ–G-1/2 trends for microcracking materials (Fig. 3.23) as well as similar associated decreases in E. Such microcrack contribution to the finer G slope could well be the source of the substantial finer G slopes commonly seen for Al2O3 (Figs. 3.15–3.17). Other possible combined effects are those due to effects of EA, TEA, or both. While these, especially TEA, can be important in microcracking, they may also have effects at grain sizes below those for microcracking, but at reduced failure stress levels, as is discussed for the larger G branches below.

Consider next the intersections of the finer and larger G branches. While there can be multiple finer G branches for a given material reflecting different finer G flaw populations, all will join a single larger G branch for that material when the flaw dimensions for each finer G branch equals G/2. Flaw variations in the finer G branches will lead to corresponding variations of the G range of each branch intersection. However, there are also intrinsic variations to the G of such intersections due to statistical effects on forming flaws of about the same dimensions as those of the grains as discussed earlier. Some variable contributions due to factors such as EA and TEA may also occur.

A key factor in both the intersection of the finer and larger G branches and in larger G branch slopes is the extent of subcritical crack growth whether due to environmental effects or intrinsic growth. As previously noted, Kirchner and Ragosta [281] calculated that small (e.g. 10 µm) flaws within single Al2O3 grains undergoing environmentally driven SCG could lead to catastrophic failure at single crystal KIC for G ≥ 100 µm and loading rates high (e.g. 104 MPa/s) instead of arresting at the grain boundary. However, this must be an upper limit, since they assumed that the singleto polycrystal KIC transition occurs at the first grain boundary rather than over multiple, e.g. 2–6, grains, indicated experimentally for most ceramics investigated [28] (Figure 2.16) and that TEA stresses were not a factor in such slow crack growth, which is contrary to some results [283, 284]. More fundamentally, there is direct experimental support for fracture occurring from flaws whose initial size is smaller than the grain in which they are located at or before their reaching the boundaries of this grain. This is shown by cases of fracture occurring from a single larger grain, with calculations frequently showing that the failure causing flaw size was smaller than the grain dimensions. More specific demonstration of this is given by results reported for MgAl2O4, and especially Y2O3, [2, 28] showing that failure frequently became catastrophic before the flaw reached the first grain boundary. Further, some origins from larger grains show single crystal mist-hackle features [2, 3, 12, 13, 21] within individual grains for failure from preexisting flaws (e.g. Figs. 3.7, 3.12) as well as slip nucleated failure (Fig. 3.4 B–D). Since mist boundary to flaw size ratios for

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single crystals are at least 3 to 1, and probably closer to those for glasses and polycrystals ( 10 to 1), these observations set severe limits on the extent of subcritical crack growth. Thus fracture had to be critical before the flaw boundary reached the grain boundaries of the large grains in which failure initialed.

Consider in more detail the slopes of the larger G branches, where it should again be noted that the slopes of these branches on σ–G-1/2 plots need to be multiplied by (2)-1/2 0.71 to obtain K since G is measured by a diameter and flaws by a radius. The K values from these slopes should be between those for easier grain boundary or single crystal fracture at one extreme and the polycrystalline values for a given material at the other extreme. Singh et al. [287] considered the issue of intragrain flaws first propagating at the single crystal KIC then possibly arresting at the grain boundary without environmental effects. The first of three regions identified was the very large G region in which growth of the initial flaw cannot be arrested, and hence failure is determined by the initial flaw size and the appropriate single crystal KIC, with no dependence on G. The transition from the larger G to the finer G branch occurs when c G/2 if there is a step function change between single and polycrystalline KIC values, and when c = 3G for a more gradual KIC transition. For the G range between these two extremes, i.e. the larger G region, they concluded that σ would be controlled by the polycrystalline KIC and c>G/2 but with the specific c depending upon the type of KIC transition. Evans [288] subsequently showed, based on a dimensional analysis, that the large G region could exhibit an intrinsic G-1/2 dependence (independent of the original c, but with a slope intermediate between the single and polycrystalline KIC values). The conditions cited for this were that the stress–crack length relationship have a maximum, and polycrystalline KIC have no (or limited) G dependance. Subsequently Virkar et al. [289] combined and refined these two analyses, noting the need for better definition of the local KIC values e.g. their dependence on c. The more complex case of crack propagation and arrest along grain boundaries has not been considered, with or without environmental effects.

The portion of the basic model for which there is little data is the transition from the larger G branch to single crystal strengths. Substantial data exists showing extension of strengths of both cubic and noncubic materials to and well below strengths of single crystals of the same material with the same machining finish (Figs. 3.10–3.13, 3.16–3.18, 3.20, 3.21, 3.24). Thus there must be some transition from these larger G strengths to those of the weaker single crystal orientations as sketched in Fig. 3.1. Besides the clear implications of this transition by strengths of larger G and single crystals, there is some limited data in this area. Thus there are a few strengths for (mainly fusion cast, optical grade) CaF2 with G in the range of a few hundred to 1000 µm that are similar or somewhat < those for comparably finished single crystal specimens [59]. Similarly Gentilman’s [92] strengths of specimens from fusion-cast, transparent 2 Al2O3·1MgO with large (2–5 mm) grains tested with grain boundaries normal to the tensile