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such grain shape effects can be included in the size, e.g. via equivalent circular area, this needs further evaluation, because of other possible ramifications or correlations. Thus grain shape is often interactive with local and global grain orientation, since this can influence the shape and extent of grain growth. Conversely growth of nonequiaxed grains may impart some, or enhance, local grain orientation. The phase content of the body can be important, e.g. as shown by exaggerated grain growth of α-Al2O3 or β-Al2O3 grains in α-Al2O3, of α-SiC in β- SiC, and of β-Si3N4 grains in α-Si3N4. Other body constituents can enhance or retard exaggerated grain growth and hence increase or decrease grain elongation effects. Grain shape effects by themselves and body parameters affecting them are discussed next, followed by compositional and fabrication effects on grain shape, then by effects of grain orientation.

Very limited direct assessment of grain shape on tensile strength has been made, with much of the limited data being of some uncertainty due to questions of possible impurity or additive effects, e.g. in AIN and Al2O3 bodies. Thus Sakai [260] reported an 1/3 increase in strength (from 310 to 470 MPa) of hot pressed AIN with an increase in oxygen content from 1 to 2.7 wt% (by addition of Al2O3). Part of this increased strength was attributed to reduction of AIN grain size (from Ga nearly 10 to 4 µm), but some of this was probably also due to the distinct platelet character of most grains. (Note that while Sakai’s AIN without additives had typical equiaxed grains and predominately intergranular fracture, bodies with the platelet grains showed extensive transgranular fracture.) Similarly, Komeya and Inoue [261] showed similar levels and increases of strengths of AIN-based composites with 25 w/o Y2O3 additions that resulted in similar elongated, but more fibrous and elongated, AIN grains.

Al2O3, data of Koyama et al. [135] with relatively uniform grains having aspect ratios of 2 (e.g. due to limited additives) showed very similar σ–G-1/2 behavior to that of bodies with nominally equiaxed grains (Fig. 3.15), though some uncertainty exists due to lack of comparison of G measurement techniques for varying grain shape. While alumina bodies with in situ growth of aluminabased platelet grains really fall into the category of composites (Chaps. 8–12), their results are also reasonably consistent with data for pure Al2O3 (Fig. 3.15). Similarly, strengths of the pure Al2O3 of Yasuoka et al. [262] of 430 and 660 MPa respectively for Ga 5 and 2 µm are in good agreement with Al2O3 data, as are those of bodies with 20 vol% platelet grains of lanthanum aluminate of similar average grain size as the alumina grains ( 5 µm). There is also agreement of strengths of similar data of Kim et al. [263] on Al2O3 with and without additions of Na2O and MgO to form by in situ reaction during sintering to yield several volume percent long lath, beta-alumina grains of small cross section. Strength increases, commonly 20%, but in the extreme nearly 100%, were attributed to combined effects of reduction of the matrix alumina grain size and

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formation of more and longer beta-alumina grains. Development of platelet or other exaggerated grains via other constituents is paralleled by some effects of impurities (generally with sufficient solubility or reactivity with the body phase). Note again that alkali, especially Na from Bayer processing of Al2O3, results in beta-alumina platelet grains commonly acting as fracture origins (Figure 1.3A), especially in bodies without grain growth inhibitors. Much or all of this data is at least approximately consistent with data for alumina with the same grain size as such larger beta-alumina grains. However, strengths may fall below the σ–G-1/2 relation for the pure material because of the platelet size and shape of single, and especially clustered, platelets, acting as fracture origins, possibly aided by additive or impurity effects.

The shape (and orientation) of nonequiaxed grains are important in most properties. It has been recommended that measuring of nonequiaxed grains for σ–G-1/2 evaluations can be aided by determining the size of an equivalent equiaxed grain having the same fracture surface area as the actual grain. Generally this would be essentially the same as the area of an equivalent semior full-circular flaw if an isolated grain is a possible fracture origin (e.g. for the situation depicted in Fig. 3.31A). This requires recognizing that for single larger grains at fracture origins the smaller grain dimension is the flaw size rather than the larger grain dimension (as incorrectly used in some SiC studies [96]; the larger dimension enters in determining the flaw geometry factor, Section VI.B). However, this is not necessarily pertinent to other nonmechanical properties, and it has limitations for tensile strength and especially other mechanical properties.

Of broader concern for tensile strength are common complications such as grain orientation, and character relative to the matrix grain structure other than size and shape. Other issues are connection with other flaws, e.g. surface flaws (Fig. 3.31), pores (Figure 1.4B), or combinations of these. However, a major issue is grain and local compositional variations or more than one grain with or without local compositional variations acting as fracture origins. While probably fairly common in several materials, this is common in origins from larger grains in situ toughened Si3N4. There, large rod or platelet shaped grains singly, or more commonly two or more in combination with excess additive phase, are dominant fracture origins as grain sizes and the level of toughness increases. While both the number of grains in a cluster and the extent of excess additive phase, as well as the location of these relative to flexure tensile surfaces, impact the resultant effective flaw size (Fig. 3.31), there has been little evaluation. Again such evaluation is of uncertain relevance to basic σ–G-1/2 behavior, e.g. for projection of strengths to significantly larger or finer grain sizes, since such combinations often represent a significantly different flaw population. However, the above questions are relevant for understanding strength behavior of such bodies to control better their microstructure and properties.

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FIGURE 3.31 Schematic of some possible idealized flaws for fracture initiation from one or more larger grains. (A) A single larger tabular or rod grain near the surface with an associated machining flaw. Arrows indicate subsequent crack propagation.(B) and (C) Possible effects of two crossed grains, which clearly increase the effective flaw sizes (dashed lines), with the extent of this probably depending on the extent of associated excess additive phase (i.e. less of these indicated by inner dashed lines). (D) Three grains, increasing probable flaw size (dashed line) some.

D.Grain Orientation Effects

Preferred crystallographic orientation of grains can significantly effect σ. Chandler and colleagues [169, 170] showed higher σ due to preferred orientation in extruded, sintered BeO from UOX-derived powders having a significant fraction of needle shaped BeO grains (Chap. 2, Sec. III.H). The degree of preferred orientation increased with decreasing specimen cross section (i.e. with increasing area reduction in extrusion) and with subsequent grain growth (i.e. 50% orientation at G = 20 µm, reaching 80% orientation at G = 80–100 µm). Several investigators have shown similar preferred orientation from green body extrusion of Al2O3 and resultant increases in σ for fracture perpendicular to the extrusion axis (Chap. 2, Sec. III.H). McNamee and Morrell [127] subsequently corroborated the earlier favorable effects of extrusion on strengths of Al2O3 reported by Hanney and Morrell [126] and concluded that the preferred crystallographic orienta-

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tion of the Al2O3 grains was the major factor. Salem and Shannon [264] have also recently shown a KIC anisotropy for extruded Al2O3, e.g. 2.5 MPa·m1/2 for the crack plane, and propagation direction perpendicular to the extrusion direction to 4.1 MPa·m1/2 for the crack propagation direction and plane being respectively perpendicular and parallel to the extrusion direction (with an intermediate value for the third basic orientation). Similarly, preferred orientation obtained by hot pressing or hot working (press forging) Al2O3 has been shown to effect strengths significantly [11]. Preferred crystallographic orientation of MgO by hot extrusion results in increased strength attributed to reductions in the ease of crack nucleation via blocked slip bands at grain boundaries due to reduced misorientation between the grains [48, 50, 52]. Other studies of preferred orientation corroborate and extend these effects [265–267]. These prevalent effects of orientation raise questions of adequately interpreting data for specimens from materials and processes that may yield (generally unexamined) preferred orientation, e.g. in Al203 studies of Charles [149] and especially in studies of small extruded Al2O3 and TiO2 rods of Alford et al. [27] (Figs. 3.16, 3.20), where effects could be greater because of the small rod size.

Dykins’s [268] study of the tensile strength of ice from –27 to –4°C (i.e. from 90 to 98% of the melting point) showed marked anisotropy of strength relative to the axis of the freezing direction. Strengths for stressing normal to the freezing direction averaged 39 ± 5% of those for stressing parallel with the freezing direction, with little effect of grain size. Strength, even this close to melting, also commonly followed a G-1/2 dependence (Figure 6.13).

Bodies having both preferred grain orientation and pronounced grain elongation often show greater effects on strength, since both often reflect greater crystalline anisotropy. Thus data of Virkar and Gordon [195] for beta-alumina bodies with substantial basal plane texture of the platelet grains in the hot pressing plane showed 30% higher strength in the finer grain regime for specimens tested for fracture normal, versus parallel, with this basal texture (Figure 3.21, i.e. respectively mode A and A′ versus mode C). Clearly much of this difference is due to the anisotropy of single crystal fracture energy, but it may also reflect some effect of grain elongation. Note that the latter could also affect the branch intersection some, but primarily if the grain elongations were oriented to accommodate machining flaw elongation for tests based on failure from flaws formed parallel with the abrasive particle motion.

More pronounced effects are seen in bodies with substantial uniaxial alignment of rod or needle shaped grains. Thus data of Yon et al. [269] for bodies of SbSI (a piezoelectric material of interest) with highly aligned needle grains showed anisotropies of strengths for four bodies ranging from 40 to 170% higher for stress parallel with the C (and elongation) axis of the needle shaped grains as the grain diameter increased from 0.38 to 600 µm and lengths increased from 11 to 6000 µm (Fig. 3.32). Though the much less refractory and much

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FIG. 3.32 Strength at 22°C versus the inverse square root of the grain size[G] for SbSI bodies with highly oriented needle grains. Strengths plotted using (1) the diameter of elongated grains as G for stressing parallel with the axis of elongation with resultant fracture transverse to the grain axis, and (2) either the length (1) of the elongated grains or the equivalent diameter (d, assuming an elliptical cross section of the elongated grains) for stressing normal to the grain orientation axis for fracture parallel with the grain elongation. Note that in either of the latter cases the much greater initial rates of strength decrease as the corresponding grain dimension increases. (Data from Yon et al. [269].)

softer nature of this material raise questions of the specific mechanisms, the strengths for stressing parallel with the c-axis indicate finer G branch flaw failure over most of the G range studied with a branch to either larger G flaw or microplastic failure at G (= needle diameter) of 55 µm. On the other hand, the much more rapid initial decrease in strengths for stressing normal to the needle

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axes may indicate microcrack failure, which may be present for two reasons. The first is the much larger needle dimension in the plane of fracture for this test orientation and the high anisotropy of the material with substantial, but imperfect, alignment. Second is that this is an orthorhombic material and thus has unequal thermal expansion (and other properties) not only between the c direction and a normal direction but also in the other direction normal to both the c-axis and a normal to it. A more extreme case of strength anisotropy due to orientation of elongated grains is that of Ohji et al. [270], who extensively seeded α-Si3N4 with fine rod shaped β-Si3N4 grains that were oriented by tape casting. Resultant dense sintered bodies had strengths of 1100 MPa for stressing normal to the resultant elongated grains and just over half of this, 650 MPa, for stressing in the normal direction. This strngth anisotropy was despite nearly identical toughness values ( 11 MPa·m1/2) and Young’s moduli in the two directions but was similar to that of the Weibull moduli for strengths in these two directions, i.e. 26 and 46 respectively. Much more extreme is the > 7-to-1 ratio of strengths for stressing parallel versus perpendicular to the fibrous grain axis of the fine grain jadeite [271] (Table 2.3). This data, though limited to three strengths for two different jade compositions stressed parallel with (but only one stressed normal to) the fiber axis is consistent with a definite positive slope to the finer G branch of the σ–G-1/2 relation.

Further note the high strength anisotropy of CVD graphite (Table 2.14), which roughly correlates with the anisotropies of E and K, but much less so for the orientation giving very high K due to delamination normal to the crack. The latter orientation is not the strongest, again showing large crack behavior not reflecting normal strength behavior. More generally note that while the orientation dependence of strengths often at least partly correlates with that of K (Sec. III.H), there are a variety of variations, some with opposing toughness and strength trends. Examples of such variations besides CVD graphite are oriented bodies of Al2O3, β-Al2O3, and Si3N4. In all these cases fractography has been necessary to identify possible or definite orientation dependences of failure causing flaws, which together with the anisotropy of K explained that of strength.

VI. OTHER FACTORS

A.Flaw Character—Surface Finish Effects

The typically prominent role that surface finish plays in the tensile strength of ceramics is reflected in their resultant σ–G-1/2 behavior, with the surface finishing and grain size dependence of tensile strength each providing insight into the role and character of the other. This is especially true for the dominant surface finishing methods of machining that have been most extensively investigated with regard to the range of parameters, materials, and grain sizes and has been

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supplemented by successful fractography. Thus machining effects, which were the basis for the σ–G-1/2 model (Fig. 3.1) for flaw failure, are discussed first and most extensively, but with some discussion of the more limited data on effects for as-deposited or as-fired or annealed and chemically polished surfaces.

Fractography showed that to a first approximation machining flaw sizes causing failure for a given material and machining condition were independent of grain size (including single crystals, i.e. for G ∞, G-1/2 = 0) [13, 15–22, 138, 142–144], e.g. note machining flaws in Figs 3.7, 3.9, 3.10BC, 3.12, and 3.16. A central aspect of this is that the larger and finer G branches intersect when the flaw size and grain size are the same (recognizing that the former is measured as a radius and the latter as a diameter, as well as effects of varying flaw and grain factors). The first of two machining factors that corroborate this model is that more severe machining, commonly by use of coarser abrasive grits, reduces strengths across the limited, but useful, finer G ranges encompassed [24–26]. Though it has been claimed that such finer G branches have no G dependence, the data cited for this is at best insufficient, and other data clearly shows some G dependence (e.g. Figs. 3.10–3.19), as indicated theoretically below.

The second aspect of machining effects that provides even stronger support for the basic model of Fig. 3.1 is the effect of machining direction on strength. It has been extensively shown in a variety of singleand polycrystal ceramics that tensile testing a machined body as a function of the angle of the stress axis relative to the direction of motion of the abrasive particles in machining plays a major role in resultant strengths [15–22, 142–144]. Thus strengths for samples of the same body machined in a direction parallel with uniaxial stressing are commonly nearly twice the strengths of the same samples tested machined in a direction perpendicular. This has been extensively shown to be due to abrasive motion generating two flaw populations, one of elongated flaws, and the other of half penny cracks respectively parallel and normal to the abrasive motion. Both flaw populations are the same depth, and while there are some other variations in shape, the differences in flaw aspect ratios, commonly of 2 to 4 versus 1 respectively, is the predominant factor in the resultant strength difference. Thus specimens uniaxially tested in two different directions relative to the abrasive machining direction result in two different finer G strength branches. Other limited variations in flaw dimensions are indicated experimentally, consistent with limited theoretical dependence on E, H, and K per Eq. (3. 2), as is discussed later.

Recent further evaluation or earlier and additional machining direction studies show that such machining direction effects are grain size–dependent, and that this dependence significantly further corroborates the basic model [142–144]. Thus, as summarized in Fig. 3.33, the difference in strength as a function of stress versus machining direction goes to 0 at an common G value, designated GC, but increases as G either decreases or increases from this

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FIG. 3.33 Percent change in σ vs. G-1/2 for various materials for machining perpendicular to the tensile axis vs. machining parallel with it. Machining was by diamond grinding except for one set of Al2O3 tests sanded with SiC fixed abrasive. All data shows reduction in the strength anisotropy as a function of stress versus machining direction as G increases to intermediate G values and then increases again consistent with substantial strength anisotropy as a function of machining direction in single crystals. The shift to higher values at intermediate and large G as well as single crystal for MgO is attributed to surface work hardening in these ranges [49]. (Published with the permission of the Journal of the American Ceramic Society [143].)

GC value, for a given machining operation. This is completely consistent with, and strongly corroborates, the basic model that as the grain and flaw dimensions approach one another, the grain dimensions begin to constrain, hence become, the flaw dimensions. However, this is more significant for elongated flaws, since their greater length versus depth becomes incompatible sooner with grain dimensions as these approach each other than does the depth of the flaws. (Also, flaw depths are probably a more basic result of the flaw generation process, e.g. as indicated by depths of both flaw populations being similar.) As G decreases below

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GC for a given machining operation, flaws become progressively larger than the grain size and thus progressively less constrained in shape by individual grain boundaries. Conversely, for G > GC, flaw dimensions are progressively < the G values, so there is progressively less effect of the boundaries of the single grain in which the flaw formed on the flaw shape.

It is important to note that there are three intrinsic statistical effects impacting the specific relations between flaw and grain dimensions as they approach one another, so the dimensions of the failure causing flaws are only approximately those of the grains at the branch intersections or the strength anisotropy being zero at GC. First is the variation in grain size, which will vary from specimen to specimen, with greater variations between different bodies, but some between a given set of samples, as well as within specific samples. An important one in larger G machined samples is truncation of surface grains (Figs. 1.2A&C, 3.9). Thus there will be some tendency for branch intersections and GC values to be toward the larger size of the G range, but other factors such as truncation of larger grains can partially counteract such shifts. Further, orientation of individual surface grains relative to preferred planes for both forming of flaws and their causing subsequent mechanical failure is important. Additionally, favorable orientation of two adjacent surface grains can allow flaw formation partly in both grains instead of just one even though the flaws and grain dimensions are similar. While the above grain orientation affects impact flaw formation, this is also impacted by the second factor, the specific local micromechanics of each abrasive particle–surface interaction, e.g. of the size, shape, and force on the particle forming flaws. The impact of this is clearly indicated by flaws not forming in much larger grains despite their being favorably oriented (as shown by their subsequent cleavage fracture, Fig. 3.27).

There is a general and two specific effects that need to be considered in evaluation of effects on σ – G relations. The general one is introduction of surface compressive stresses, which occurs in all materials and obviously does not override effects of machining direction. However, whether such effects change with G and possibly alter the anisotropy of strength due to machining direction with G is unknown. The first of the two specific cases is work hardening of surface grains of materials with easier activated slip such as MgO and CaO. However, this merely shifts the relative machining anisotropy curve (Fig. 3.33) higher. The second specific case is the substantial surface transformation of tetragonal to monoclinic ZrO2 from machining of TZP bodies with 20% increased strengths attributed to expected surface compression [272]. Thus while the baseline for measuring such strength increases is uncertain, study shows substantial reduction, or complete elimination, of strength anisotropy as a function of machining direction possibly for some TZP and PSZ polyand especially single crystals [273]. However, there is no evidence that these trends for toughened

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bodies change σ – G relations. In fact, the similar effect from fine G to G = ∞ reinforces similar behavior across the complete G range.

An important question is what effects other surface treatments have on σ – G relations. Abrasion, wear, and impact (e.g. from tumbling in a mill) are examples of other mechanical effects on surfaces. The nature and size of many impact flaws [16] suggests that they may follow similar trends as for machining flaws. This is also suggested by similar increases in strengths as a result of grit blasting TZP bars as for machining them [8, 274, 275].

Consider now other surface finishing effects, primarily of as-deposited or as-fired or annealed surfaces, which have received only limited investigation. Some single crystal, and especially glass, specimens can commonly be flame or chemically polished to remove surface flaws such as from machining and achieve high strengths, so there is a common impression that as-fired or annealed surfaces yield higher strengths. However, such surface treatments are commonly not applicable to, or much less effective with, polycrystalline samples due to probable intrinsic factors such as TEA and EA, and are clearly limited by a variety extrinsic surface effects. The latter arise from factors such as sintering of dust (e.g. specimen powder, including agglomerates from spray drying) on their surface as well as flashing and other surface and internal variations resulting from imperfect specimen forming and densification, all leading to stress concentrations limiting strengths. Further, machining again commonly introduces some surface compression stress which aids strengths, as directly shown by Hanney and Morrell [126], introducing some, usually moderate, uncertainty in calculating flaw sizes.

For the above reasons, as-deposited, fired, of annealed surfaces often do not give the highest strengths, but even in some, possibly many, of these cases useful σ – G-1/2 data may be obtained, but probably more scattered. Direct comparison with machined samples is valuable. Thus Hanney and Morrell [126] showed strengths of their as-fired Al2O3 bodies generally being less than that for specimens that were diamond ground. (Mechanical polishing resulted in σ roughly intermediate between these two levels.) They also showed that annealing of bars with coarse, but not fine, machining reduced σ. Similarly, Steele et al. [122] showed strengths of ground and polished Al2O3 bars being similar to each other, but greater than as-sintered or thermally etched specimens. McMahon [276] showed that Al2O3 specimens tested as-fired had lower strength, those ground perpendicular to the specimen and tensile axis intermediate strengths and those ground parallel with the specimens and tensile axis the highest strengths. Lino and Hubner [277] also showed lower as-fired vs. machined strengths of Al2O3 (P 0.02, G 4 µm), but an increase of the latter upon annealing (e.g. at 1400–1500°C). They also showed that machining increased strengths of their assintered Al2O3 bars (P 2%, G 4 µm) by 100 MPa, i.e. 50% to 300 MPa (still low for such bodies), but annealing near the original sintering conditions