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factors such as elastic anisotropy (EA) and thermal expansion anisotropy (TEA). Rice, with others, also considered microplasticity (including twinning as a possibility in some materials) but subsequently broadened consideration to include extrinsic factors such as surface finish, preferred orientation, and pore and grain characteristics and heterogeneities (e.g. of larger grains).

As the range of materials, microstructures, and testing broadened, four factors ended consideration of microplasticity as a broad mechanism controlling the G dependence of ceramic tensile strengths. First, the intercept strength levels for finer G branches were typically lower, e.g. by an order of magnitude, than the expected stresses for activating microplasticity in many materials, e.g. Al2O3 (one of the key materials in Carniglia’s review). Second, while local stresses from EA and TEA can be substantial, they were of uncertain levels and spatial extents to make up the difference between the strength intercepts and the yield stresses. Further, they are only pertinent to some materials and vary substantially from material to material. Third, TEM examination of sharp cracks in single crystal and polycrystalline Al2O3 at room temperatures showed no evidence of associated slip [14]. While the possibility of crack-twin relations was not addressed, twinning is pertinent to only a limited and uncertain (more likely noncubic) set of materials. Fourth and most fundamental was subsequent evaluation, with fractography being a key tool, showing that typically both branches result from flaw failure [2, 3, 11–13].

Thus it will be shown that the finer and larger σ – G-1/2 branches typically arise from changes in the flaw to grain size ratio, with strength levels of both finer and larger G branches being determined by flaw sizes (Fig. 3.1). However, there can be two important variations of the normal flaw controlled σ – G-1/2 behavior. First, some room temperature failure is determined by microplasticity in a limited number of ceramics and related materials (and becomes somewhat more common and extensive as temperature increases, Chap. 6). However, where this occurs, the large grain branch is the regime of this behavior, not the finer grain branch as proposed by Carniglia. Further, as the stress for microplasticity increases, e.g. as G decreases, there is often a switch to flaw failure when preexisting flaw dimensions are those of the grains, which is identical to the finer grain flaw failure (Fig. 3.1). The other deviation from the now-established normal two-branch flaw controlled strength behavior occurs with substantial microcracking.

Two aspects of σ – G behavior should be stressed. First, the above twobranch σ – G-1/2 behavior is due first to changing c/G ratios and the resultant grain boundary or single crystal polycrystalline toughness transition, and secondarily, at least for machining flaws, to impacts of grain parameters, especially size, and local hardness, elastic moduli, and toughnesses on flaws that are introduced and determine strength. This is contrary to the focus of most mechanical property studies on toughness, as measured by large cracks, as the mechanism of

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controlling σ and its microstructural dependence. Second, as comparison of this and the previous chapter shows, much of the G dependence of strength is inconsistent with that of large crack toughness, especially at larger G, since much of the larger crack toughening is precluded by the above process.

Beyond the above historical summary of the grain dependence of failure mechanisms, this review first addresses such mechanisms in more detail and then provides an extensive summary of σ – G-1/2 data, mechanisms, and parameters, updating and extending previous reviews [2, 3, 11, 13]. This starts with materials exhibiting some known or expected microplastic control of strength, often with a transition to flaw failure. Next, σ – G-1/2 behavior for cubic and then noncubic materials, where strength is generally controlled by preexisting flaws, is presented. In both cases oxide materials are reviewed first. Then pertinent microstructural effects are addressed, including grain shape and orientation as well as moderate levels of impurities and especially additives (bodies having higher levels of second phases are addressed later as composites, Chaps. 8–12). The basic microstructural question of which G to use, specifically the average or the maximum, is also addressed, showing that while tensile failure is a weak link process suggesting use of the maximum G, since it is associated with lower strengths, this is a serious oversimplification. Blind application of this (i.e. without other input, especially fractography) leads to errors and distortions, since grains by themselves are not flaws. At the other extreme, behavior of ceramics with nanoscale grains is discussed, including neglected but often significant effects of second phases commonly left from frequent unusually low processing temperatures used to obtain bodies with very fine grains. Surface finish effects, i.e. of machining flaws and effects versus as-fired surfaces, and test issues such as specimen size and shape along with loading and environmental effects and their interactions, are discussed. Then more detailed evaluation of the σ – G-1/2 model is given, followed by a comparison of strength-toughness-grain size behavior and discusson of mechanical reliability, primarily via Weibull moduli.

B.Mechanisms, Parameters, and Analysis

Consider now more specific aspects of the σ – G -1/2 brittle fracture model (Fig. 3.1), whose development and interpretation arose mainly from fractographic study of machining flaws revealing their variation relative to G [2, 3, 11–13]. This showed that, in relatively dense machined ceramics, machining flaws formed in the surface region were the dominant or exclusive source of failure. Such flaw failure in finer G bodies clearly disproved Carniglia’s model of microplastic control of the finer G branches. To a first approximation, the depth of such flaws for a given material and machining condition has no dependence on grain size, i.e. did not change significantly in size with G (nor between most ceramics). (However, as discussed later, there is some second-order variation of c

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with material and microstructural parameters.) Thus failure of machined finer grain bodies was due to machining flaws whose size, c, was > G/2 (since flaw size is measured by a radius and grain size by a diameter). Typically two flaw populations are generated in machining, one parallel and the other perpendicular to the direction of the abrasive particle motion, with the depths of both being similar, but the former being substantially elongated and the former being half penny cracks [12, 13, 15–22]. This difference in flaw shape translates to two levels of strength of the finer G branches as a function of stress direction relative to the machining direction, again supporting the above model and concepts.

Other flaw sources may have similar or different trends with G. An important flaw population that is produced in the body and can change, e.g. shift, σ – G -1/2 behavior (Fig. 3.1) is that of microcracks. Surface flaws from contact damage or particle impact/erosion may often be similar or proportionate in size to machining flaws, but this needs to be established. Some processing flaws, e.g. some larger pores, may fit this trend, and some flaws may not, e.g. other pores or cracks from thermal stresses (Chap. 6). However, even in such cases, failure to account for flaws causing failure and basing analysis only on toughness can be misleading, e.g. as shown by evaluation of effects of preferred orientation on strength of hot pressed Si3N4 (Chap. 2, Sec. III.H).

Another important surface finish condition that often gives similar trends as machining as a function of G, at least over the normal G range, is failure of samples with as-fired surfaces, e.g. as for some ceramic parts and tests, especially for fibers [2]. This is consistent with Coble’s evaluation showing grain boundary grooves from firing (or annealing) acting as sharp flaws whose size is a reasonable fraction of the grain size, e.g. over the range of G/15 to G [23]. Thus at finer G such flaws are < G, in part due to the fine G and generally less severe grooving from less severe firing conditions, which means that several connected grooves along different boundaries act as the failure causing flaws. As G increases, grooving also generally increases, and fewer connected grooves are needed to form a failure causing flaw, reaching a point where failure may occur from a groove along part or all of a single grain, i.e. analogous to the above machining flaw case, as is discussed next.

Typical flaw populations have c > G at finer G, and c does not increase substantially as G increases, so c must become , then progressively, < G/2 (1/2 arises since c is measured as a radius and G a diameter) progressing from the finer G branch, its intersection with the larger G branch when the flaws encompass individual surface grain, then the larger G branch where the initial flaw size is < G/2. However, while such flaw–grain relations define much of the behavior, there are other issues. The first of three sets of issues are which G to use, especially an average, Ga, or the maximum, Gm, as used by some [24–27], and whether there is any G dependence in the finer G branches, i.e. whether the slope of the finer G branch = 0 as claimed by those proposing the use of Gm. It

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will be shown that the finer G slope of 0, while possibly having pertinence in some limited cases, is not the general case, since there is generally some to substantial decrease in strength as G increases along finer G branches. Second is that while polycrystalline fracture toughness values are appropriate at finer G, a transition to single crystal (Figure 2.15) or grain boundary toughnesses must occur as the intersection of the large and finer G branches is approached or reached [28]. The slope of the larger G branch is later shown to be between polycrystalline and single crystal toughness values. The third issue is contributions of intrinsic effects, such as slow crack growth, EA and TEA, and extrinsic effects, e.g. of secondary phases. Several, or all, of these may affect both branches and their intersection(s), e.g. SCG due to differing effects of grain size and of interversus transgranular fracture, but they are also probably important in a less recognizd but clearly established occurrence, namely large G strengths lower than those of the lowest strength crystal orientation with comparable surface finish. Thus as G increases so failure can progressively only occur from fewer and fewer individual grains or grain boundaries, strengths can decrease below those of single crystals with comparable surface finish due to effects of TEA and EA stresses, as well as residual pores or phase, especially at grain boundaries [2, 3, 13]. Such large G effects are likely to depend substantially on specimen size and shape as well as the nature of the stress, e.g. true tension versus flexure, since these will affect the limited statistics of grain structure impacting failure.

While some of the above effects at large G have limited documentation (presented later), there is clear evidence of significant effects of extreme TEA at finer G. Thus an important flaw population that leads to a substantial variation in the σ–G-1/2 is for bodies with extensive microcracking, such as that resulting from high TEA, e.g. in MgTi2O5 and Fe2TiO5 (Figs. 3.1, 3.23). Subsequent microcracking results in much faster decreases in strength as G increases above the critical value Gs for microcracking [per Eq. (2.4)], since the effective flaw size is increasing due to more and larger microcracks as G increases, but then the decrease begins to saturate. Such effects are typically paralleled by similar initially rapid and then decreasing reductions in elastic moduli, again due to saturation of the microcracking process.

The above fractographic studies and resultant flaw model development partly overlapped with conceptual arguments based on σ–G data. Thus aspects of this model, i.e. finer G branches due to c > G/2, the higher slope large G branch, and higher single crystal strengths than for some large G bodies were schematically indicated by Emrich in his review of crystallized glasses [29]. Similarly, finer and larger G branches both being due to flaw failure with the two branches intersecting when c G/2 was proposed by Rhodes and Cannon [30]. Bradt, Tressler, and students [24–26] also subsequently discussed such a model in conjunction with their surface finish studies.

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The other basic, but less common, failure mechanism is by microplastic crack nucleation and possible slip assisted crack growth. This also has a distinct G dependence, since stress levels for it decrease with increasing slip band length, the maximum of which in a polycrystalline body is normally G, since grain boundaries are generally both sources and barriers to dislocations. This inverse relation between slip band length (hence also grain size) and stresses for crack nucleation results because blockage of a slip band at a barrier generates a back stress resisting dislocation motion from the source into the barrier. The effect of the back stress on the dislocation source decreases with increasing source–bar- rier separation, hence with G. The mathematical relation for nucleation of a crack by a slip band extending across a grain being blocked at the opposite grain boundary is the Hall–Petch relation:

σ	= σ	y	+ kG-1/2	(3.1)
f

where σf = the fracture stress (in materials where there is insufficient bulk ductility), σy = the yield stress, i.e. the tensile stress to activate the easiest activated slip system in single crystals of comparable composition, crystal perfection, and surface finish, and k a constant. Note (1) both σy and k are clearly dependent on the specific material and on the absence or degree of grain orientation relative to the stress axis, and (2) Eq. (3.1) is very similar to the Griffith equation, Eq. (2.2), when the grain and flaw sizes are the same or similar, i.e. in such cases the primary difference is σy, which was a major source of confusion, as was discussed earlier. Twin nucleation of cracks or assistance in their growth, though more restricted in the number of materials it is applicable to, is more complex, since twins (though often thin) are three-dimensional defects rather than linear ones as dislocations are. However, twin caused failure is expected to follow the same or similar G dependence as Eq. (3.1), since G again is typically the upper limit to the scale of twin lengths. Note that microplastic initiated failure can be variable, e.g. due to variable G and σy, which may increase or decrease σf as well as variable grain boundary character. Slip assisted growth of cracks (e.g. from machining) is a transition between slip initiated and preexisting flaw failure, which lowers strengths. While the latter would give some variation similar to flaw initiated failure, microplastic initiated failure should have less extreme variations.

Thus, in summary, the dominant failure mechanism for ceramics is from preexisting flaws, such as from as-fired and especially machined surfaces, giving the σ–G-1/2 behavior of Fig. 3.1. Other flaw populations can deviate from this, e.g. flaws from microcracking giving faster initial strength decreases at finer G and less at larger G. Another fundamental variation is with microplastic initiated or assisted failure, which typically results in cracks in, or along boundaries of, grains. The key distinction between this and typical brittle failure from preexisting flaws from a σ–G-1/2 plot is that strengths of the latter at larger G commonly

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extrapolate to σ values well below any realistic σy values, e.g. 0, while microplastic failure extrapolates to σy at G= ∞ (G-1/2 = 0). Thus it is typical for larger grain bodies having flaw failure at strengths less than those for comparably finished single crystals oriented for the lowest strength. Conversely, data reflecting microplastic caused failure have large grain strengths that extrapolate to σy and thus do not fall much below single crystal strengths. Brittle fracture initiated by microplasticity is typically in competition with failure from preexisting flaws, with slip assisted crack growth being a transition between these two mechanisms. Flaw failure is favored by higher stresses for finer G fracture and larger preexisting flaws, e.g. from harsher machining, handling, etc., or increased severity due to second phases or pores. Note that as G decreases, the stress for microplastic failure increases in a similar fashion as for flaw initiated failure increases (e.g. since both are ultimately controlled by cracks). Thus it is both logical and observed that at some finer G, competition between microplastic and flaw initated failure shifts depending on material and surface finish. However, in the absence of significant extrinsic effects, this change should also occur when the size of microplastically introduced crack is G/2, i.e. as for the flaw model (Fig. 3.1).

II.CERAMICS AND RELATED MATERIALS WITH KNOWN OR EXPECTED MICROPLASTIC INITIATED FAILURE

Some softer, less refractory ceramics and related materials often exhibit macroscopic yield (Fig. 3.2A). NaCl macroscopic yield stresses of Stokes and Li [11, 31] lie on, or slightly below, while their fracture stresses lie slightly above, that of KCl [32]. CdTe may or may not exhibit yield before fracture [33–35]. Somewhat harder, stronger materials like PbTe [36] and CsI [37], while not showing macroscopic yield, extrapolate to σ values (at G= ∞) that are probably their yield stresses. Similarly, ZnSe [38, 39] (large G) extrapolation to σ 15–30 MPa at G= ∞ (Fig. 3.2B) may indicate microplastic control of σ (e.g. by twinning). Purity, processing, and testing are all probably factors in σ differences.

Previous data [40–43] for hot pressed and sintered (commercial) BaTiO3 shows two σ–G-1/2 branches (Fig. 3.3A), with the large G data for unalloyed (and some alloyed) samples extrapolating to single crystal strengths [41]. This is taken as evidence of microplastic controlled failure given yield shortly preceding fracture of (“butterfly”) BaTiO3 crystals and associated substantial birefringence around their fractures (Fig. 3.3B). The higher σ of BaTiO3 hot pressed with 1% LiF + 2% MgO at larger G (hence higher heat treatment for more homogenization) also supports microplastic control via indicated alloying effect [41], as do other indications of microplasticity in BaTiO3 [44, 45]. Statistical analysis of much of the data shows the fine G branch having a slope > 0, both above and below the Curie temperature [42], but with higher strengths (by 50%) above the

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FIGURE 3.2 Yield or failure stress versus G-1/2 for materials of known or probable microplastic controlled failure (A) Csl [37], KCl [32], PbTe [36], and CdTe [33]; (B) ZnSe data of Roy and Natale for various hot-pressed (H) and PVD and CVD bodies from a previous survey [38, 39]. Vertical and horizontal bars represent standard deviations.

Curie temperature where it is in a cubic structure with no microstructural stresses.

Strengths of dense CaO, from hot pressing [46] or recrystallizing single crystals (via press forging or hot extruding) [47, 48] all gave single σ – G-1/2 branches, with different slopes for different bodies (Fig. 3.4). All extrapolate to σ = 20–35 MPa at G = ∞, reasonably consistent with macro yield stresses of 40–45 MPa for CaO crystals having some probable solution hardening, suggesting probable microplastic control of strength. This is reinforced by some strength increase for specimens tested with as-sanded (and more frequent internal fracture origins, Fig. 3.4B) versus as-annealed surfaces indicating surface work hardening and tests at 1100 and 1300°C showing limited strength decreases, frequently with macroscopic yield preceding fracture, but maintaining transparency [47, Chap. 6]. It is also indicated by fractography studies of recrystallized CaO single crystal samples (with 100% transgranular fracture, Fig. 3.4B-D) and

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FIGURE 3.3 BaTiO3 data indicating microplastic control of strengths at larger grain size in dense bodies. (A) strength versus G-12 for hot-pressed (with or without additions of LiF + MgO as shown), sintered (commercial, with machined surfaces), and single crystal bars. Vertical bars are standard deviations; numbers in parentheses represent the number of tests, and numbers at the bottom of the bars the % porosity. Note data for pure bodies extrapolating to single crystal strengths (which are shortly preceded by yielding). (Data after Rice, Pohanka, and colleagues [40–43].) (B) and (C) Transmitted, polarized light, lower and higher magnification micrographs of failed crystal bend bar showing substantial birefringence of respectively a coarser (B) and fine (C) scale indicative of plastic deformation (the X cut by the fracture in B) was associated with observed yielding preceding failure [41].

probable slip bands at fracture origins [46], similar to (unetched) origins in MgO (Fig. 3.6), including their internal nature indicating work hardening of surface grains from machining [48, 49].

Substantial testing of (transparent) MgO as hot pressed, annealed, or hot extruded (giving a <100> axial texture) showed primarily single σ – G-1/2 branches with significant slope increases in the respective order listed [48, 50–52] (Fig 3.5). All extrapolate to σ 70 MPa at G = ∞, less than typical single

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	B
			C

FIGURE 3.3 Continued.

crystal flexure strengths of 150 MPa, but consistent with higher purity crystal yield stresses of 75 MPa [46, 48]. Such indications of microplastic controlled strength at large G are reinforced by fractographic observations showing internal failures (attributed to work hardened surface grains) [48–52], and especially substantial identification of slip bands blocked at fracture origins on both unetched and etched fractures (Figs. 6A and B). This data is generally consistent with substantial data for hot pressed and annealed MgO [25, 53–55] and sintered MgO [56, 57], mostly with machined surfaces, which in turn reinforces the above trends and mechanisms. Thus Evans and Davidge’s (transparent) hot pressed and annealed MgO with chemically polished versus as-sawn surfaces [55] showed respectively higher and lower σ at larger G reflecting respectively microplastic and flaw failure as shown by respective intercepts at the yield stress and 0. Harrison’s dense [56], sintered MgO with various machining straddles the hot pressed and annealed curves of Rice with a trend for higher strength with finer surface finish, as well as probable branching to less G dependence of σ at finer G (e.g., at < 10 µm) due to a transition to flaw failure at finer G. Spriggs and Vasilos’ [53, 54] and Nishida et al.’s [57] sintered MgO data support this, indicating a decreased, but >0, finer G branch slope. Such a transition is also indicated by effects of additives [52], and greater reduction of machined versus annealed strengths at finer G (5–10 µm). Subsequent studies [15] clearly showed (1) sig-

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A

FIGURE 3.4 Strength and fracture of dense CaO indicating microplastic controlled strength at 22°C. (A) Strength versus G-1/2 for various bodies hot-pressed [46] (with or without LiF additions) with various annealing as well as of single crystals and bodies recrystallized from these by press forging or hot extrusion [47, 48]. Note intersection of the stress axis at 30–40 MPa in good agreement with the yield stress for CaO crystals, with a higher slope for the extruded material (tested parallel with the <100> axial texture), and similar results for bars tested with as-an- nealed or annealed and sanded surfaces. (B–D) Fracture origins of dense, recrystallized CaO test bars. In (B) note the internal fracture origin (attributed to surface work hardening) and mist and hackle starting somewhat before the crack reached the grain boundaries. (C) and (D) Fracture origins at or near the surface in surface grains. In (C) note the fracture step (at 45° to the surface) indicating a slip band, i.e. similar to MgO failures (where slip was corroborated by dislocation etching which does not exist for CaO), and the onset of mist and hackle when the crack reached the grain boundary in this smaller G body. In (D), a larger G body; note the onset of mist and hackle 1/2, way through the larger grain of origin.