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conductivity are also affected, but strengths are often not significantly reduced. Major sources of these complications in graphite are effects of grain and pore parameters, e.g. microcracks originating and branching from, or terminating on, pores have been reported [102, 103].

Consider now the case of introducing, or generating, more microcracks in conjunction with propagation of a macrocrack. While the originally predicted occurrence of microcrack formation in two lobes ahead of the crack tip significantly above and below the macrocrack plane [44] is contrary to experimental results, and attention has now been focused on wake region bridging, the issue of some microcrack formation at or closely ahead of the macrocrack is not settled. Ultrasonic probing around a stressed macrocrack by Swanson [49, 50] in both granite and alumina samples showed microcracking being confined mainly in or near the wake region of the macrocrack, mainly or exclusively associated with bridging. However, while Hoagland et al.’s [48] results (Figure 4.5 of Ref. 5) showed much wake microcracking, they also clearly showed considerable microcracking ahead of and around the macrocrack, much less in spatial extent and location than first proposed, but more than indicated by Swanson. The extent to which the sandstone porosity was a factor in this broader spatial distribution of microcracks is unknown, but Hoagland et al.’s and other results [102–105] showed that even quite limited porosity can play a significant role in some microcracking in conjunction with a macrocrack, indicating complex behavior that is not fully understood. Additionally, even where most toughening occurs via wake bridging, it appears that microcracks may initiate at or slightly ahead of the macrocrack tip as precursors for bridges, so bridging evidence is also probable evidence for microcracking. Acoustic emission, which is probably more sensitive but is not widely used, is an indication of this (Sec. III.G). There may also be contributions of other mechanisms such as crack deflection, branching, and tortuosity (see Sec. IV.B of Chap. 8), depending on material and microstructural parameters as discussed for KIC–G dependence of materials such as Al2Ti2O5 with extensive microcracking [104]. Lawn [106] theoretically considered crack tip microcracking, concluding that there was a limited opportunity for microcracking ahead of the crack tip, but noted that this might be expanded by second phase particles. This may imply some effects of porosity, though he did not address this or extreme anisotropy, e.g. of graphite.

The first of five additional observations is Peck et al.’s [107] qualitative assessment of crack propagation and fracture energy of rocks in their and other studies. While they showed that porosity (and weak interface bonding) gave lower fracture energies, as expected, higher fracture energies and greater distances of crack propagation for steady state fracture occurred when there was a preexisting network of interconnecting microcracks (or when rock texture provided multiple, incipient fracture surfaces); but the highest fracture energies were found in crystalline rocks without interconnected microcracks. Second, sig-

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nificant maxima of fracture toughness versus grain size (Figures 2.16 and 2.17), initially attributed to microcracking effects from TEA or from crystal structure transformation (e.g. BaTiO3), have since been implied to be due to bridging. Third, tests of bodies in which the microstructural stresses arise from phase transformations that occur close to room temperature, e.g. BaTiO3, offer opportunities to verify the role of microstructural stresses by testing above and below the transformation temperature. The fourth and related point is that in piezoelectric and ferroelectric materials, electric fields can increase the grain boundary stresses and microcracking (Sec. III.I). Fifth, as noted above and later, acoustic emission can be an important tool for detecting and evaluating microcracking.

D.Grain Size Dependence of Fracture Toughness of Cubic Ceramics

Though many studies of fracture toughness have been made with little or no grain size characterization, there is considerable data on the grain size dependence of fracture toughness. While there are uncertainties of test parameters such as crack size effects, important data trends are seen from four previous reviews [11, 60, 108, 109], especially the two most recent. This section addresses toughness for cubic materials, particularly overall trends, and notes variations and some of the known or indicated test differences and limitations. The following section treats single-crystal toughnesses and the transition between them and polycrystalline values for both cubic and noncubic ceramics. Section F addresses the G dependence of noncubic ceramics, Sec. G, crack wake effects such as bridging (which occur in cubic materials, and more often in noncubic ones) and Sec. H, effects of preferred grain orientation, again mainly in noncubic ceramics.

The grain size dependence of fracture energy and toughness of cubic materials is generally simpler and apparently less prone to wide variations in comparison to noncubic materials, but it is still subject to variations. The overall trend for cubic ceramics is for limited dependence on grain size over the G ranges generally covered (e.g. a few to 100 µm, Figs. 2.12–2.14), but some, typically modest, deviations from this are common and possibly universal. Thus most MgO studies do not indicate a G dependence (Fig. 2.12), but this may reflect the more limited G ranges, and numbers of different grain sizes covered; an earlier compilation of data from 1 to 200 µm [110] indicated a possible limited maximum of fracture energy at intermediate G (e.g. 10–40 µm). (The low value at 1 µm reflects in part residual additive effects, since fine-grain MgO as hot pressed with 1–2 wt% LiF had 1/2 the fracture energy as of material with similar grain sizes annealed to remove residual LiF or MgO hot pressed without LiF; both of the latter show substantially more transgranular fracture than material as-hot pressed with LiF [110, l21]). MgO with limited porosity, e.g. < 1 to 4% porosity, showed KIC increasing 3-fold from values for dense bodies, with much of

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FIGURE 2.12 Fracture toughness (KIC) versus average grain size (G) at 22°C for various investigations of MgO (upper portion), Y2O3 (middle), and ZnS (lower portion). Note (1) no clear G dependence of KIC for MgO with 0% porosity (P), but for MgO with <0.1 to >1% porosity (solid circles), KIC clearly increased above P = 0 levels with increasing G as P decreases, shifting to a more intragranular porosity [110, 111]; (2) some increases in toughness levels of MgO for 10% additions of different (partly to fully soluble) oxides in MgO [112–115]; (3) similar maxima trends, but different values for them for two of the three Y2O3 studies [116–118];

(4) the lower value (from indentation tests) for single crystals of Y2O3 [115]; values for {100} fracture of MgO crystals are typically 1 MPa·m1/2 [118]; and (5) a clear maximum for the CVD ZnS [120]. (After Rice [11], published with the permission of the Journal of Materials Science.)

this limited porosity transitioning from interto intragranular locations as G increased from 10 to 100 µm, which may reflect possible general effects either of pore location or of slip associated with pores [11], consistent with dislocation activity accompanying MgO fracture at modest temperatures (Sec. III.B). Similarly, at first glance, no grain size dependence is shown for the fracture energy of dense FeO being 7.5 J/m2 over the G range 10–90 µm [122], but there is a probable 4–8% decrease as G increases. Two data points of Evans and Davidge [123] for UO2 with G 8 and 30 µm had fracture energies of respectively 8 ±1

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FIGURE 2.13 Fracture toughness versus average grain size at 22°C for various indicated investigators for spinel and a ferrite. Upper portion: MgAl2O4 data of Stuart et al. [124] and Rice et al. [109], all made without additives, except for a low value at G = 100 µm for a body made with LiF additions. Note (1) the good agreement of the two studies showing no clear intrinsic dependence on grain size and (2) the value for fracture on either {100} or {110} single crystal planes G = ∞. Lower portion: a soft (cubic) NiZn ferrite [125] showing some decrease in toughness as G increases. (After Rice [11], published with the permission of the Journal of Materials Science.)

and 6 ±1 J/m2. Though some of this decrease may reflect differences in the limited porosity in the two bodies, some of it may be intrinsic.

Several data sets for cubic materials, commonly with no measurable porosity and no densification aids, showed modest, but probably statistically significant, toughness maxima at intermediate grain sizes. Thus all data sets for Y2O3 [116–118] (though at substantially different grain sizes of 1 and 20 µm) showed higher toughness at intermediate G as did CVD ZnS [120] at 1 µm, and various BaTiO3 bodies indicate twice the finer and larger grain fracture energy of 2.4 J/m2 at 40 µm [126–128] at 150°C where they are cubic. (A more definitive maximum is shown in the noncubic state at this same G and corroborated

FIGURE

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by strength trends [129]. Data over a more limited G range ( 2–40 µm) for a NiZn ferrite showed a decrease as G increased [125]. A significantly lower value for large G MgAl2O4 made with LiF addition (which is apparently more difficult to remove from MgAl2O4 than MgO) is reflected in the 100% intergranular fracture versus the 100% transgranular fracture of material without LiF additions [11, 110]). Equally or more definitive is Niihara’s data for CVD β SiC giving a maximum at 1–2 µm [130], and Kodama and Miyoshi’s data for HIPed polycarbosilane-derived ultrafine grain powder showed a pronounced maximum at G 0.6 µm [131] (Fig. 2.14). An earlier compilation of SiC data [109], though complicated by reflecting more noncubic, α, SiC (discussed below), also indicated a possible decrease at larger grain sizes.

Thus, in summary, cubic, in contrast to noncubic, materials generally show no G dependence of fracture energy and toughness, or modest decreases at finer, or larger grain sizes, or both. These trends are indicated despite the variations in test methods, especially between different investigators (e.g. for indentation techniques). Extreme results reported for the fine grain SiC from HIPed polycarbosilane powder are unexplained. More broadly, the mechanism(s) for the com-

2.14 Fracture toughness (KIC) (and some corresponding flexure strength) versus average grain size (G) at 22°C for α and β SiC from a previous survey [60]. The data of Kodama and Miyoshi [131] is for polycarbosilane derived SiC, Niihara [130] for CVD SiC, and Seshardi et al. [132] for sintered alpha SiC. (After Rice [11], published with the permission of the Journal of Materials Science.)

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monly indicated KIC decrease at finer G are uncertain, but extrinsic effects common to preparing many fine-grain bodies may be a factor in some, possibly many, of these results (Chap. 3, Sec. IV.A). Though also uncertain, the first of two possible, not mutually exclusive, intrinsic mechanisms for decreasing toughness at larger G is elastic anisotropy, which has similar effects to TEA in noncubic materials [65]. Unfortunately EA data is often either uncertain or nonexistent, so quantitatively testing is limited. The second intrinsic reason for decreased toughness at larger grain sizes is the transition to single crystal or grain boundary toughness, as is discussed in the next section.

E.Single Crystal/Grain Boundary-Polycrystalline Fracture Toughness Transition

This section addresses single crystal and grain boundary toughnesses for cubic, then noncubic, materials and their relation to typical polycrystalline values. Single crystal fracture toughnesses are important to understanding mechanical failure of not only single crystal but also polycrystalline bodies, e.g. for transgranular grain microcracking. Generally toughness of polycrystalline bodies with normal, e.g. machining, flaw populations must transition to those of the corresponding single crystal or grain boundary values as G increases to larger sizes, as is noted above and discussed below and in Chap. 3. Further, polycrystalline toughnesses for any grain size with increasing preferred grain orientation often begins to approach that of the corresponding single crystal orientation, provided bridging and other polycrystalline toughening mechanisms are not significant. Grain boundary fracture toughnesses are pertinent to the broader case of intergranular microcracking and are expected to be the toughness values that large grain polycrystalline bodies will approach where intergranular failure is dominant (Chap. 3, Sec. I.B).

Single crystal fracture toughnesses of greatest pertinence are those for lower toughness cleavage or preferred fracture planes, since these control most failure of single crystals and are dominant in most transgranular fracture; the latter usually with varying degrees of mixed mode failure for the one or more lower toughness crystal planes involved. Such single crystal values are invariably substantially below those of polycrystalline bodies of the same material and crystal structure and hence provide a general check on the probable validity of limited data for singleor polycrystal values, especially where one or both sets of data are limited. Table 2.1 summarizes much single crystal data and corresponding typical, usually finer grain, polycrystalline values. Again, note that single crystal values approaching or exceeding lower polycrystalline values, or lower polycrystalline values falling below single crystal values for preferred fracture surfaces, should raise suspicion about the validity of one or both values. Thus, for example, values for YAG single crystals seem high, e.g. in view

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	TABLE 2.1 Comparison of Ceramic Single Crystal and Polycrystalline Fracture
	Toughnessesa
	Material						Typical Sxl	Sxl KIC	Pxl KIC
							fracture	(MPa·m1/2)	(MPa·m1/2)	Pxl fractureb
	(A) Cubic Ceramics
	MgO						{100} cleavage	1	2	Mainly TGF
	ThO2						{111} cleavage	0.6–1	1.1	Mainly TGF
	Y2O3						{111} cleavage	<0.9	1.3	Mainly TGF
	ZrO2						Possibly {110} cleavagec	1.1–2.4	2.6	Mainly TGF
	MgAl2O4						{100} and {110} cleavage	1	2	Mainly TGF
	Y3Al5O12						{111} cleavage	2.2	1.7–3.1	Mainly TGF?
	TiC						{100} cleavage	1.2	3–4	Mainly TGF
	CaF2						{111} cleavage	0.4	1.1	Mainly TGF
	ZnSe						{110}	0.3	0.9	Mainly TGF
	Diamond						{111} cleavage	2.9–4.1	5.3–13d	Mainly TGF
	(B) Noncubic Ceramics
	Al2O3						Rhomb. & other nonbasal
							fract.	1.5–2	3.5–4	Mixed IGF & TGF
	β-Al				O e		Basal, (0001)	0.16	2.7–3.3	Mainly TGF
			2			3
	TiO2						{110}	0.8	2.5	Mainly TGF
	SiC						Probably (1010)or (1120)f	2	3–4	Mixed IGF & TGF
	Si	3	N		g		?	1–1.5	4	Mainly TGF
				4			¯
	Calcite							0.23	1	Mainly IGF?
							{1011}

a Sxl = single crystal, Pxl = polycrystalline, KIC = fracture toughness (polycrystalline fracture toughness generally in the absence of significant microcracking, wake, etc. effects)

b Pxl fracture: typical or representative fracture mode, with TGF = transgranular fracture and IGF = intergranular fracture.

c While {111} cleavage may be expected due to the fluorite structure, cleavage is often on {110} but may be variable, depending on stabilization, composition, and stoichiometry, e.g. as indicated by indent crack complexities as reported by Pajares et al. [134].

d Higher polycrystalline toughness is for metal bonded, sintered diamond [142].

e In such extreme cases of a single very low toughness plane, other fracture planes are more important, fracture normal to the basal plane gives KIC of 2 MPa·m1/2 [144].

f Respectively firstand second-order prismatic cleavage (per private communication with P. T. B. Shaffer).

g CVD Si3N4, i.e. without densification aids and resultant grain boundary phases. Source: Refs. 4, 11, 39, 40, 60, 76, 124, 133–150. See also Chap. 6, Figs. 1, 5, and 6.

of Pardavi-Horváth’s [151] values of two other garnets (GGG and CaGeGG) respectively of 1.2 and 0.8 MPa·m1/2 for (111) fracture.

There is typically substantial anisotropy of fracture toughness in single crystals of various crystal structures. However, except for materials of extreme crystal anisotropy, i.e. noncubic materials with very platy structures such as mi-

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cas, beta-aluminas, graphite, and hexagonal BN, there is no clear difference in the anisotropy of single crystal toughnesses between cubic and noncubic materials. Many crystals, whether of cubic or noncubic structures, have one significantly preferred fracture surface, usually a cleavage plane, with {100} cleavage in many NaCl structure materials being an example, e.g. in CaO and MgO, while showing very limited cleavage on {110} planes, does cleave on {110} planes only under conditions that are not fully understood. This is not surprising in view of the elastic anisotropy of crystals and the relation of elastic moduli and toughness. (Thus the comment by some that cubic ZrO2 has unusually anisotropic toughness is partly true as well as misleading, since like its elastic moduli, it is fairly anisotropic.) A good test for the primacy of a cleavage or fracture surface is that cracks propagating on other planes will often branch onto the primary, i.e. the lowest toughness, plane, but the reverse crack branching, i.e. from the primary to other planes, occurs only occasionally, if at all. Stoichiometric MgAl2O4 is unusual in having two cleavage planes of very similar toughnesses [73–76, 124] (Table 2.1). (MnZn ferrite approaches this similarity; see the note at the end of this chapter.) Table 2.2 lists a few other crystals with two or three sets of cleavage planes with similar toughnesses. See also Chap. 6, Sec. IV for additional single crystal toughness data.

Turning to the transition from singleto polycrystal fracture toughnesses, this is seen as a natural consequence of grain sizes increasing to approach and subsequently become larger than the strength controlling flaw. Thus fracture (energy and) toughness must decrease toward single crystal values for transgranular fracture origins given the commonly significantly lower single crystal fracture toughness for preferred single crystals fracture planes, e.g. by factors of 2–3 (Table 2.1, Fig. 2.15). Some mapping of such polycrystalline–single crystal transitions has been by DCB tests of bodies of increasing G and especially from fractographic evaluations of larger grain, transgranular, polycrystalline fracture origins [15, 70] (Fig. 2.15). For example, lower than expected DCB toughness

TABLE 2.2 Fracture Toughnesses for Different Preferred Fracture/Cleavage Planes of Some Crystals with More Than One Such Planea

Crystal		Structure	Fracture/cleavage plane- K		IC	(MPa·m1/2)
Si		Cubica	(111) 0.8	(110), (100) 0.9–0.95
SiO2		Quartz	¯	¯
	O b		{1120} 0.85–0.95	{1011} 0.85–1.0
KalSi		Triclinic	{110} 0.31	{001}, {010} 0.39
3	8		¯	¯
	c	Hexagonal
Be3Al2(SiO3)6			{1010} 0.2	{1120} 0.25

a Diamond structure. b Microline feldspar.

c beryl, which with green coloration, e.g. from Cr doping, is the precious gem emerald. Source: Refs. 149, 151–155.

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FIGURE 2.15 Data on the single crystal to polycrystalline fracture toughnesses transition as a function of flaw to grain size ratio [15, 62, 70]. Note different scales for CVD Si3N4.

values were found to be associated with larger grains at the crack tip [15], showing the utility of fractographic examination of toughness test specimens, not just strength specimens or components. Indentation MgAl2O4 data of Sakai et al. [156] is consistent with such a transition, as is single crystal data of Chen and colleagues [152, 153] for two polycrystalline bodies, but with flaw sizes slightly smaller than G. More recent evaluation of crack tip stresses [157] may be an alternative approach to such evaluation. The level of polycrystalline fracture toughnesses versus corresponding lower single crystal values must reflect the range of single crystal values, especially the lower ones, but is impacted by two factors. The first is the multiplicity of the sets of planes with low toughnesses. A single, e.g. noncubic basal, fracture plane having only one member, while limiting polycrystalline toughness, must have less effect in limiting polycrystalline toughnesses since it allows more effect of other, higher toughness, crystal fracture planes. Increased multiplicity of lower toughness fracture planes, e.g. {100} cleavage in a cubic material such as CaO or MgO which reflects three planes probably limits polycrystalline toughness [15, 62]. Second is the resultant mixed mode combinations of different crystal planes making up transgranular polycrystalline fracture, which is probably a major factor in the greater polycrystalline versus single crystal fracture toughness.

A similar transition must occur from polycrystalline to grain boundary

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toughness values for intergranular fracture origins. Such boundary values should typically be lower than the lowest single crystal values, often substantially so, especially for the extreme case of fracture of a single grain boundary facet, as represented in fracture of bicrystals. It is commonly estimated that fracture surface energies of grain boundaries relative to crystal fracture, e.g. cleavage, energies would parallel those of surface energies, where boundary energies are often1/2 those of lower crystal surface energies for a given material. Such estimated trends are reasonably supported by the limited measurements. A classic study is the DCB fracture energy measurements of Class and Machlin [158] of KCl bicrystals grown from the melt with grain boundaries parallel with (100) surfaces, but with controlled twist angles between the two “grains.” Their results showed fracture energies decreasing rapidly as the twist angle increased from zero to 0.08 J/m2 at 5° twist, then bottoming out at 0.04–0.05 J/m2 at twist angles of 15–45°, in contrast to (100) values of 0.11 J/m2 (independent of angle of propagation on the (100) plane). This reflects an average boundary fracture energy 1/2 that of the cleavage energy as suggested by surface energy differences and translates to a decrease to a minimum boundary toughness versus that for cleavage of respectively 0.06 and 0.09 MPa·m1/2.

More recently, Tatami et al. [159] reported similar NB measurements of sapphire bicrystals made by pressure sintering crystals with a common c-axis, i.e. <1000>, normal to the boundary, which was parallel with the (0001) plane, and typically contained 0–10% porosity. (Such purely twist boundaries avoid significant boundary stresses from thermal expansion anisotropy, as shown by Mar and Scott [160] for sapphire bicrystals with twist angles about other axes.) Fracture energies at 0° twist varied from 6 to 15 J/m2 (for toughnesses of 2.7–4.3 MPa·m1/2) with higher values consistent with values for basal fracture (Fig. 6.1) and lower values with increasing residual porosity. The overall trend for fracture energies with increasing twist angle was for substantial decrease, with a few higher spikes of fracture energy at boundary angles where adjacent “grains” had coincident lattices, but the maximum of these spikes also decreased as the twist angle increased. Minimum fracture energies of ≤ 1 J/m2 occurred over most twist angles over the range of 25–45°, which corresponds to a fracture toughness of 1 MPa·m1/2. Such decreases were greater than expected from surface energies, e.g. by a factor of 2 or more.

F.Grain Size Dependence of Fracture Toughness of Noncubic Ceramics

The previous two sections have addressed the G dependence of toughness in cubic ceramics, then single crystal toughnesses, their anisotropy, and relation to typical polycrystalline values. This section addresses the G dependence of toughness of noncubic polycrystalline ceramics in the order: Al2O3 (the most