Mechanical Properties of Ceramics and Composites

properties across grain boundaries. While the number of scattering events increases as G decreases, the net ray deflection decreases, so the net effect is for more serious scattering as G increases, but this is also a function of wavelength. Scattering decreases as the grain size decreases below the wavelength, e.g. a body with intrinsic transmission and G 1 m will have limited scattering in the infrared and substantial scattering in the visible, and greater scattering in the UV. However, other effects are typically dominant in scattering, since extrinsic effects of impurities and intrinsic effects of residual porosity (often at grain boundaries) are generally much greater than intrinsic effects of refractive index differences across grain boundaries.

Dielectric breakdown, i.e. failure of an electrical insulator at some level of applied electrical field, e.g. in volts per cm, also commonly increases with increasing G [14,15], so breakdown fields are often significantly lower than for finer G, but also lower than single crystal values. However, much if not all of the G dependence is due to extrinsic effects of residual pores and second phases along grain boundaries, often increasing in extent as G increases. The role of larger grain size on dielectric behavior is also reflected in effects of larger grains on the breakdown voltage for TiO2 doped ZnO varistors. Hennings et al. [16] reported that seeding the varistor body to eliminate large exaggerated grains produced a finer, more uniform grain size giving more consistent onset of nonlinear conduction.

Clearly the significant anisotropy that can occur in the above nonmechanical properties makes grain shape and especially orientation effects important. Also, where there is substantial anisotropy, this can change substantially with temperature due to differing dependences of property values in various crystal directions on temperature.

B.Composite Ceramics

Composite ceramics intrinsically have broader property dependence on microstructure than monolithic ceramics do, as is discussed in Chapters 8–11. For example, while monolithic ceramics normally have no dependence of elastic properties on grain structure unless there is microcracking or preferred orientation (in noncubic ceramics), there is more and broader dependence in composites. Thus the dependence of elastic properties on the nature and volume fraction of the dispersed phase has been addressed in Chapter 8, and the effects of preferred orientation, possible microcracking, and contiguity of the phases (especially for fiber composites) have been discussed to limited extents. Extending the discussion to other properties of composites could be a very large task. Instead, the focus of this section will be to outline the effects of particulate parameters on three important properties that are impacted more in composites than in monoliths. Two of these, namely thermal expansion and conductivity, are particularly

pertinent to mechanical properties, especially thermal stresses and thermal shock, and are addressed first.

Consider first thermal expansion, which does not depend on grain size in monolithic ceramics, nor generally on particle size in composites, unless microcracking occurs. However, the thermal expansion of composites clearly depends on the expansions of the matrix and dispersed phase and their volume fractions, and may also be affected by preferred orientations of the matrix grains and dispersed particles if they are noncubic. Composite expansion can also depend on contiguity of the dispersed phase, especially for fibers, but the focus here is on particulate composites. The challenge in calculating composite expansion is to account for the elastic interactions of the matrix grains and the dispersed particles due to differences of thermal strains between them.

A variety of expressions for thermal expansion of various composites have been derived, e.g. as reviewed by Raghava [17] reflecting different results for different dispersed phase geometry and for the same geometry, the latter reflecting the complexity of the problem and the effects of differing assumptions. A simple equation given by Kingery [18] is

where α and k are respectively the linear thermal expansion and the bulk modulus, the subscripts C, M, and P refer to the composite, matrix, and dispersed (particulate) phase, and φ is the volume fraction dispersed phase. A simpler equation used with some success for composites of constituents without large differences in elastic properties (e.g. ceramic–metal composites) is [19]

However, this clearly does not address the occurrence of factors such as microcracking, e.g. during cooling from fabrication when αM > αP, or particles acting more as pores due to combinations of αP > αM, particle size, and limited parti- cle–matrix bonding. Accounting for effects such as varying bonding between Ni particles and an alumina matrix has been reported to have a significant effect on expansion behavior of composites [19].

Turning to thermal conductivity, there is an even greater number of models and more diversity of them than for expansion, reflecting a greater challenge (e.g. Ref. 20 for an earlier survey of models and their fits to data). However, besides the complications of solving the basic problem of thermal conductivity through a densely consolidated (i.e. pore-free) body with dispersed particles of different conductivity than the matrix and having various sizes, shapes, etc., there are three serious complications that can occur. The first, for which there is some theoretical and experimental understanding, is percolation of at least one of the phases, i.e. the onset of that phase forming continuous paths through the body along the axis parallel to which conductivity is of interest. This is impor-

tant because, as this occurs for a phase of substantially higher conductivity than the other, typically the matrix, phase with which it is mixed, the thermal conductivity increases rapidly. The combined uncertainties in predicting the approach, onset, and extensive percolation combined with incorporating such effects in the prediction of thermal conductivity are a challenging task. The second complication, for which there is less information, is that of having an interfacial phase between many or all of the dispersed particles and the matrix, particularly when such phases have much lower conductivity than the dispersed phase, especially when the latter has high conductivity. The third complication, which is widely overlooked, is that the conductivity of the dispersed phase is not certain, especially for refractory ceramic particulates. This arises because the process of making the particulates can result in factors such as limited impurities or poor crystalline perfection that can substantially reduce their thermal conductivity, as shown by work of Slack on bulk materials (e.g. Refs. 21 and 22). Further, there is no way to ascertain whether the conductivity of the particles is at all consistent with accepted values for the bulk material, except via evaluation of data against models. However, the latter presents significant challenges, since lower than expected composite thermal conductivity, which would result from poorer particulate conductivity, can also result for less particulate percolation or interfacial phases of lower conductivity, or both, either of which can be challenging to ascertain accurately.

Focusing on particle size dependence, note that models not addressing percolation or interfacial layers predict no dependence on dispersed particle size. On the other hand, percolation generally occurs at lower volume fraction particles as the particle size decreases, thus giving increased conductivity with finer particles of good conductivity. However, interfacial phases of lower conductivity tend to reduce body conductivity less with larger particles, since there are fewer larger particles for a given volume fraction of particles and hence fewer interfacial layers and less total area covered by such layers. Thus, for example, the net particulate surface area for a fixed volume fraction of spherical particles varies inversely with the particle diameter, e.g. doubling the particle size gives only half the net particle surface area. Thus such layers can give an opposite dependence on particle size (if layer thickness is independent of G); and if both percolation and interfacial layers are present with similar effects, thermal conductivity behavior similar to that predicted by models neglecting both effects can result.

Consider now some limited experimental data on thermal conductivity of composites. Hasselman et al. [23] showed that thermal conductivity of composites of 15 v/o diamond particles in a cordierite matrix increased as the particle size increased, but that the benefits of larger particle size diminished as temperature increased (Fig. 12.1). (This reduction with temperature is similar to reduced grain size effects on conductivity of monolithic ceramics as temperature increases.) They applied a model including interfacial effects to their data, which

FIGURE 12.1 Thermal conductivity versus particle size for a composite of 15 v/o diamond particles in a cordierite matrix at various temperatures. Note the decrease in thermal conductivity with decreasing particle size and the decrease in both thermal conductivity and its particle size dependence as temperature increases. Values of the matrix alone are shown at zero particle size. (From Ref. 23.)

was also applied to earlier data for Al2O3-SiC particulate composites, showing similar particle size and temperature dependence of composite conductivity (which was typically less than for the matrix alone) [24]. Chen et al. [25] also showed that the thermal conductivity of composites with 10–60 v/o Cu particles in an epoxy matrix showed no significant difference in the increase in conductivity with volume fraction between 11 and 100 m dia particles. However, composites with 7 m Al2O3 showed modest, but statistically significant, higher conductivity despite Cu having a conductivity 10-fold higher than Al2O3, i.e. suggesting that other effects such as interfacial effects (e.g. with the Cu) were probably operative.

Finally, note that Neilsen’s model [26] is commonly used, e.g. by those in the important field of developing organic, commonly rubber, matrix composites with dielectric ceramic fillers (e.g. MgO, AlN, and more commonly Al2O3 and especially hexagonal BN) for use as thermally conducting, electrical insulating gaskets for electronic components. This model’s use is based on both its fitting

considerable data (e.g. Ref. 20) and its involving more microstructural dependence, and it reflects both some of the advances and the problems in the field. Thus besides the usual parameters of conductivities and volume fractions of the constituents, it includes particle shape and orientation (but not size) and maximum packing fraction, which reflects essentially a fully percolated volume dependent on particle shape and orientation. However, neither the incorporation nor the determination of these terms is rigorous, and so the important question is raised of whether the frequent fitting of data reflects real correlation with actual composite parameters or just curve fitting by using parameters in the model as adjustable parameters. Note also that conductivities predicted by this equation can become excessively large or infinite, near and above the percolation limit.

Turning now to electrical conductivity, note that many models for thermal conductivity are the same for electrical conductivity and vice versa, e.g. including Nielsen’s model [26]. However, the review by McLachlan et al. [27] is recommended as a more current assessment of the field, including more extensive discussion of incorporating percolation effects on electrical conductivity, which unfortunately still is much more phenomenological than rigorous. Electrical conductivity can be even more affected by percolation effects because of the much broader range of electrical versus thermal conductivities, so percolation of particles of a highly conductive phase in a matrix makes large increases in conductivity. For this same reason, electrical conductivity of composites can be more susceptible to corrosion or other chemical actions forming interfacial phases of more limited conductivity, hence exacerbating complications noted above for thermal conductivity. Ota et al. [28] presented data showing the onset of large increases in conductivity (i.e. large decreases in resistivity) at higher volume fraction of conductive particles as particle size increases in composites with organic matrices (often silicone rubbers for switches). While the deformation response of such composites is far greater than for ceramic matrices, the dependence of conductivity on particle volume fraction and size is relevant to many ceramics and ceramic–metal composites.

Data on the changes of either electrical or thermal conductivity as a function of microstructural parameters at elevated temperatures is very limited. Prediction from models is limited not only by their uncertainties but also by the frequently limited data on the conductivity of the constituent materials, with the former exacerbated by interfacial phases and their changes. Thus, for example, the conductivities of crystalline phases typically decrease with increasing temperature, while the thermal conductivity of glassy phases increases, which can be important in ceramic refractories, e.g. combinations of these two effects giving varying conductivities [29,30].

Electronic conductivity typically decreases as temperature increases, while that due to ionic conductivity increases, thus also giving opportunity for more variable and complex behavior of the two combined mechanisms. Again, interfa-

cial phases can be important, as indicated by the anomalous increase in ionic conductivity in composites of CaF2 or BaF2 with limited additions of Al2O3 at elevated temperatures reported by Fujitsu et al. [31]. Thus at 500°C Al2O3 additions in CaF2 increased conductivities by a maximum of nearly an order of magnitude; then they decreased with further Al2O3 addition, with the effect dependent on Al2O3 particle sizes. The maximum conductivity occurred at 5, 10, and 15 v/o Al2O3 additions with respective particle sizes of 0.06, 0.3, and 8 m that gave decreases back to the matrix conductivity at respective v/o Al2O3 additions of 20, 25, and 30 v/o Al2O3. These conductivity effects are attributed to formation of an interfacial phase, i.e. consistent with less addition of finer particles needed because of its greater surface area for such phase formation.

III.SUMMARY OF GRAIN AND PARTICLE SIZE DEPENDENCE OF MECHANICAL PROPERTIES OF MONOLITHIC AND COMPOSITE CERAMICS

A.Grain Dependence of Fracture Mode, Toughness, and Crack Propagation of Monolithic Ceramics

Microcracking occurs in noncubic ceramics spontaneously without an external applied stress, at and above a threshold grain size as a function of material properties, especially the thermal expansion anisotropy, with Eq. 2.4 being a reasonable approximation for spontaneous cracking (Fig. 2.10). While such cracking is typically intergranular and on the scale of the grains, some transgranular cracking has been observed (apparently more at larger G), as has variation of the scale of the cracking with G (Chap. 2, Secs. II.C and III.C). Besides such general trends, other trends and effects have been identified but leave a great deal to be better documented and understood. Thus such cracking also appears to occur at finer G as the applied stress increases, and at least some can also be quite dependent on the occurrence of SCG and may also occur in stressed cubic ceramics of high elastic anisotropy, usually at much larger G (e.g. on a cm grain scale, Fig. 1.7B).

While effects of slow crack growth (SCG) on spontaneous microcracking have not been studied in detail, they can clearly be a factor. There has been extensive study of SCG in ceramics due especially to the effects of water, but many microstructural factors are only partially explored. Thus it is known that SCG generally occurs intergranularly, while subsequent fast fracture typically is transgranular (e.g. Fig. 2.5), which can be a substantial aid in identifying and studying it, but there are exceptions (e.g. in some ferrites), for which there is limited understanding. Further, there is some evidence that the rate of SCG significantly decreases as G decreases (Fig. 2.8), but documentation and understanding of this factor, which could significantly impact life predictions, is

lacking, as is information on the temperature dependence of SCG. While some nonoxides may have intrinsic SCG due to water or other chemicals, some have extrinsic SCG due to oxide-based intergranular phases and fracture. Some oxides such as MgO do not exhibit SCG in single crystals but have SCG in polycrystals via intergranular fracture. Whether this is due to oxide impurities or an intrinsic effect of grain boundaries is unknown. Finally, other degradation due to water and other environmental agents can occur at room and modest temperatures. Thus MgO and especially CaO singleand polycrystals can be degraded by the expansion of hydration products formed in surface cracks and pores, and some TZP bodies can be destroyed by moisture destabilization that is accelerated at temperatures and pressures above the normal ambient (Fig. 2.9). At least some of these latter effects are dependent on G as well as chemistry.

Fracture toughness at room temperature, which has been a major focus of research on mechanical properties of ceramics for a number of years, varies from simple to complex dependence on material, microstructure, and test parameters in only partially documented and understood fashions. Roughly there are three interrelated factors that lead from simpler to more complex toughness behavior, the first of which is material character. The simplest material systems are glasses, which generally have less variation in fracture toughness for a given glass and lower values (generally ≤ 1 MPa·m1/2). However, even here, trends with other basic physical properties, Young’s modulus in this case, are only approximate and show considerable variation (Fig. 12.2). The next simplest type of material is single crystals, which have varying degrees of complexity due to general dependence of toughness on crystal orientation, the extreme of which is highly preferred cleavage, which generally gives lower toughness values (Table 2.1). The prediction of cleavage planes, and hence their multiplicity, toughness values, and interactive effects of these is limited, and the effects of frequently resulting significant mixed mode crack propagation (giving higher apparent toughness values) are often neglected. However, single crystal toughnesses, though not as extensively documented as desired, generally have more limited variations for a given crystal material and crack propagation plane in it than for many polycrystalline bodies. An important but often neglected factor that allows some checking of the self consistency of toughness data is that most fracture toughness values for a given crystal will be substantially less than those of corresponding polycrystals (Table 2.1, Fig. 2.15).

Fine grain polycrystalline materials (e.g G< a few microns) of both cubic and noncubic structures also generally have more limited variations in toughness values for a given material with different tests. Cubic materials tested over a broader range of typical grain sizes often show a tendency for a maximum of toughness, usually a modest one, at some intermediate G, with variations in both the overall level of toughness and the G of its occurrence (Figs. 2.12, 2.14). The most frequent and extreme variations of toughness occur in noncubic materials,

FIGURE 12.2 Fracture energy of glasses versus Young’s modulus at 22°C. Note that AS, BS, and LS are respectively alumino-, boro-, and lead-silicate glasses. FS-fused silica, SL = soda lime glass, LV = leached Vycor (i.e very porous before sintering, 96% silica) and GC = glassy carbon (via polymer pyrolysis) with 20% closed, spherical porosity; while glassy in appearance and behavior (e.g. being isotropic), it is really mostly nanocrystalline graphite. Dashed lines are the upper and lower bounds for the trend of the bulk of the data in the survey of Mechlosky et al. [32].

frequently as a significant maximum as a function of G in the commonly investigated G range. While there are significant variations in the level and G for the maximum, or even its occurrence, depending on material and especially test method and parameters (Figs. 2.16–2.18, most extreme toughnesses are associated with large cracks.

The above significant increases in toughness that occur most extensively in large crack tests of noncubic materials (and ceramic composites, Sec. III.B), can also occur to some extent in (mainly larger grain) cubic materials and are generally related to crack-microstructure interactions that also depend on test method and parameters. The primary mechanism of increased toughness is crack bridging in the crack wake zone commonly resulting in R-curve behavior, which develops over a range of crack propagation distances before saturating. This development clearly depends on the extent of crack propagation and hence on

the test method and parameters, but this can be complex, since it may also be a function of the initial crack size and character and aspects of the crack propagation that have been incompletely investigated. Crack bridging and its effects are enhanced by intergranular fracture along larger and elongated grains, especially if they continue to have substantial intergranular fracture, which is not commonly the case unless grain boundaries are weaker, e.g. due to boundary phases. Two other mechanisms that also have much of their effect via the crack wake zone are microcracking and transformation toughening, with the latter also sometimes involving microcracking. However, how these, especially microcracking, differ in their origins and character from other bridging effects is not well understood. For example, bridging probably involves some microcracking, but whether this initiates at or near the crack tip or in the wake zone, and the effects of more general and extensive microcracking versus that only due to the crack itself, are not well understood.

Evidence for crack wake effects and bridging as an important component of large-scale crack propagation is well established. Thus R-curve effects showing toughness increasing with crack propagation are extensively demonstrated experimentally, and their effect in increasing toughness is demonstrated by reduced toughness when the wake area is removed until further crack propagation develops a new wake zone. Bridging is also seen by microscopic examination of crack wake regions where they intersect specimen surfaces (Figs. 2.4, 8.7, 8.15). However, wake and bridging observations have been restricted almost exclusively to large cracks propagated at limited but unknown velocities and then arrested and examined primarily at or near the intersection of the crack and the, typically machined, specimen surface. Contrary to many earlier and some current assumptions, crack bridging and other wake and R-curve effects commonly have limited or no effect on normal strengths of higher strength ceramics. Reasons for this limited relation to strength behavior are discussed below in conjunction with a discussion of the effects of grain parameters on tensile strength; here issues of the nature of wake and bridging observations, themselves pertinent to their applicability to strength, are outlined.

The above observations leave critical questions of crack size effects, low velocities and the arrested aspect of the crack, and possible effects of surface machining flaws and stresses on surface bridging observations. Thus machining flaws commonly in the range of a few tens of microns, with limited increase as G increases, control most strength behavior, raising serious questions of the effects of the orders of magnitude smaller size of such machining flaws from cracks of most toughness tests. The limited observations of crack wake bridging of specimens with as-fired surfaces show much less bridging, indicating that much but not all crack bridging is due to machining effects. The low-velocity-arrested crack aspect of bridging observations raises serious issues of what effect this has on bridging, e.g. whether bridges are sustained as cracks accelerate to, hence as-

suring significant effects on, catastrophic failure. As noted earlier and in the next section, composite tests raise serious questions of crack velocity on crack propagation, as do tests of MgAl2O4 crystals. The latter show basic changes in crack propagation with crack velocity, i.e. macroscopic crack propagation shifting from a zigzag pattern on {100} planes at low velocities to flat fracture on {110} planes in (Fig. 2.7) at high velocities.

Another question regarding wake bridging effects on strength is fracture mode behavior. Though it is often not given much scrutiny, it can be an important factor indicative of failure mechanisms. Typically toughness and strength tests at or near room temperature show mostly or exclusively transgranular fracture in dense ceramics (Fig. 2.5), with four exceptions or variations. First, fracture generally transitions to intergranular fracture at finer G, e.g. below 1 to a few micron grain sizes, which may often be at least partly extrinsic due to residual grain boundary impurities from use of finer powders and lower processing temperatures to obtain fine G. Second, there is, in at least some cases, a transition back to intergranular fracture in cubic materials at very large G (e.g. mm to cm scale) and in noncubic materials at much more modest G. Third, residues of some densification aids, e.g. LiF in MgO or MgAl2O4, often greatly enhance intergranular fracture (as does increased temperature, as is discussed later). Fourth, environmentally driven slow crack growth (SCG), which occurs in many oxides and some nonoxides (some intrinsically and some extrinsically often due to oxide additives) generally occurs by intergranular fracture. However, additive residues have also been observed to enhance grain bridging of cracks while they also lower strength, again raising questions of the impact of bridging on strength, i.e. in this case why a grain boundary phase that weakens the boundaries but enhances grain bridging of cracks still lowers body strength. Similarly, and more broadly, SCG, by mainly intergranular fracture, raises the question of whether this is a significant factor in the surface observations of crack bridging by grains or clusters or fragments of them.

Consider effects of grain shape and orientation and of temperature on fracture toughness. Elongated grains increase bridging effects, mainly with intergranular fracture, and may often correlate with preferred grain orientation because elongated grains often are more prone to orientation in forming precesses. Preferred grain orientation also directly affects fracture toughness, but the extent and nature of the effects are intimately related to material factors, especially preferred cleavage planes, and grain size and shape and their effects, e.g. on fracture mode. Thus oriented grains, especially larger ones and particularly elongated ones, will impact toughness to the extent that crack propagation is parallel or perpendicular to the grain orientation texture and the extent of resultant transversus intergranular fracture.

Very little data on toughness at intermediate temperatures exists, so directions of microstructural effects are uncertain (but composite studies in the next