Mechanical Properties of Ceramics and Composites

ergy/toughness–G dependence, hardness related cracking (Chap. 4, Sec. II.D), and related phenomena at room temperature. The differing trends of EA with T for different materials prevent general guidance on its effects by extrapolating those indicated at room temperature to higher temperatures, since the extrapolations would often be different, and commonly uncertain or unknown. Further, there are other factors increasing intergranular fracture as T increases besides possible increases in EA. However, consideration of specific materials where the EA–T dependence is known is at least suggestive, especially where it can be compared with trends for other similar materials but with different EA.

Thus the high EA of ZrO2 bodies at modest T and its increases with increasing T (Fig. 7.14) suggest that it may be a factor in the transition from transto intergranular fracture at very modest T (e.g. Fig. 2.5, Chap. 6). Similarly, such a fracture mode transition at higher T in MgO has been suggested as reflecting its increasing EA with increasing T [51]. However, again other mechanisms may be involved, e.g. as indicated by decreases in E of ZrO2, as shown in Figure 6.18, e.g. this would presumably not occur due to EA, unless it was causing microcracking (and then only in tests with substantial applied stress, most likely static versus dynamic modulus measurements), but it may be due to effects of lattice defect structures formed.

B.Mechanisms, Comparison with Tensile Strength and Self-Consistency

The temperature dependence of properties provides an important opportunity to corroborate mechanisms by comparing expected with observed temperature dependences of the same and related properties, i.e. evaluating self-consistency. Thus comparison of the temperature dependence of microplastically controlled compressive strength with actual data and with the temperature dependence of hardness is of value, as is comparison of the G and T dependences of compressive and tensile strengths. While data is limited, such comparisons are suggestive, as will be summarized below.

Comparison of measured or extrapolated (i.e. to G = ∞) single crystal compressive strengths, or preferably both, supports the concept that much compressive failure, even at room temperature, is controlled by microplastic processes of a more general nature than may be involved in some tensile failure. Thus values for MgO crystals of 50–120 MPa for tensile strength and 100–400 MPa for compressive strengths at 22°C (Figs. 3.5,5.3) are consistent with higher stresses activating more deformations systems in compression. On the other hand, such values are respectively 20 and 60 MPa at 1300°C, which is at least approximately consistent with expected temperature reductions of stresses for slip. Also, where slip is the mechanism of both tensile and compressive failure, large differences in their strength–G–1/2 slopes would not be expected (at least until grain

boundary failure becomes important, as will be discussed below). Thus while there is considerable variation in the room temperature tensile data giving slopes of 4–12 MPa·cm1/2, that for compression at 22°C falls in the lower half of this range, i.e. reasonably consistent, and at 1300°C the slopes for both are 4–5 MPa·cm1/2. The even more limited data for UO2 appears consistent with such trends, i.e. G = ∞/single crystal values of 700 and 60 MPa respectively in compression and tension at 22°C and slopes for both of 2 MPa·cm1/2, and no significant change for tensile values at 1000°C, thus supporting possible microplastic control of room temperature tensile strength of high-quality UO2.

Comparison of G = ∞/single crystal values for compressive and tensile strengths of materials with established flaw initiated failure is less meaningful, since the two values reflect differing mechanisms. Similarly there are uncertainties in comparing their strength–G-1/2 slopes. However, comparison of these values for compressive strengths between different materials and temperatures, though data is again very limited, is suggestive. Thus compressive data for TiB2 giving extrapolated crystal values of only 500 MPa (Fig. 5.1) versus 3.5 GPa for sapphire (Fig. 5.1) and still 450 MPa for TiB2 at 1750°C (Fig. 7.12) all indicate that the earlier TiB2 data at 22°C is substantially low, e.g. due to parasitic testing stresses.

Finally, the comparison of Al2O3 and ice tensile and compressive strength data at higher temperatures is also suggestive, e.g. Figure 7.8 shows a common single crystal value at 1600°C. This is reasonable, since anisotropy of plastic flow generally decreases with increasing temperature, i.e. the yield stresses for more difficult to activate slip generally decrease faster than the average yield stress. Thus at higher temperatures the yield stress for the easiest and hardest slip systems will approach one another, as thus will the stresses for single crystal failure in tension versus compression, i.e. similar to ductile metals. However, the strength–G-1/2 slope for compressive failure is higher than for tensile failure (e.g. a ratio of 30). This may reflect greater ease and propensity for polycrystalline failure via grain boundary sliding mechanisms in tension than in compression, i.e. consistent with, and probably a precursor to, different types and character of creep failure in compression and tension.

VI. SUMMARY AND CONCLUSIONS

There is very little data on the G dependence of H as a function of T, leaving much of such dependence to be implied by H–G relations at 22°C and their extrapolation from H–T data for a single polycrystalline body (often of at best uncertain G) and for single crystals. There is reasonable data for the latter providing a data base to estimate limits of H anisotropy as a function of T for some important ceramics. Tests of both singleand polycrystals showed lower H due to adsorbed H2O, and resultant reduced increases in H as T is increased to a

few hundred degrees, which is a widely neglected factor in the T dependence of H. A more significant change in the H–T trends occurred in a medium G, dense Al2O3, namely an H minimum at 200–300°C (Fig. 7.6A), which appears to correlate, with similar minima of increasing severity in respectively compressive and tensile (Figs. 6.12,7.6A) strengths. Beyond this, there are often changes in, e.g. inflections in, H–T relations observed in both singleand polycrystals as T increases, indicating changes in plastic flow, with differences between singleand polycrystals indicating differences in mechanisms that may also be dependent on G. Both H measurements on different crystal planes and of H anisotropy on individual planes as a function of T show that H anisotropy does not change in a simple similar pattern as a function of T. Data on ice shows substantial H anisotropy to ≥ 0.98 Tm. The very limited data on indent cracking shows this continuing in a medium G, dense Al2O3 to at least 1000°C, with increased intergranular fracture, as is common for other fracture as a function of T. These observations and other property correlations, especially with compressive strength, indicate mainly gradual if any changes in H–G relations as T changes, except where phase transformations occur, which can cause sharp, often large, H changes. However, while some G dependence of H is thus expected at higher T similar to that at lower T, the overall reduction of H with increasing T implies decreases differentiation of H as a function of G as T increases.

In contrast to H data, there is almost no single crystal data, but there is reasonable data on the G dependence of σC as a function of T, showing a normal Petch-type dependence to substantial T, e.g. > 0.96 Tm in ice and > 0.65 Tm in TiB2. While compressive failure by macroscopic fracture generally ceases as macroscopic plastic deformation begins, there can often be continued, but diminishing, macrocracking as T increases. Correlation between H and σC continues as a function of T, including the occurrence of a strain-rate-dependent (Fig. 7.6A, 7.13) σC minimum at 400°C that appears to correlate with one for H (Fig. 7.6A) and for tensile strength (Fig. 6.12). However, the H–σC correlation, i.e. the H/σC ratio, is more complex, e.g. involving a changing constraint factor, as shown by direct H and σC measurements on the same bodies as a function of T (Fig. 7.6). Extrapolations of tensile and compressive strengths to G–1/2 = 0 (i.e. implied or actual single crystal values) often become similar or the same as T further increases. This also indicates basic changes that probably underlie changing H/σC ratios, since much higher compressive versus tensile strengths at lower T are attributed to σC being driven by highly constrained local plastic deformation, while tensile strength is predominantly controlled by brittle failure from the most severe flaw. Thus while the H/σC ratio probably changes due to differing measurements, e.g. in terms of strain rates and environmental sensitivities, and changes in the extent and character of plastic flow, there is some useful, but variable, relation.

There is very little data on high-temperature erosion or wear, let alone on

their G dependence. However, what limited data does exist clearly indicates that complex changes are probable in such properties as a function of T and possibly also G as T increases. Increased bonding, reaction, and deformation are all factors in these changes.

Besides limited or no data on the grain size dependence of these properties, there is much less, if any, data on effects of grain shape and orientation (though single crystal data does indicate the limits of the latter for H). Further, the properties of this and previous chapters are probably often affected by other properties and factors, e.g. TEA, EA, and plastic anisotropy, but limited data and probable complex interactions cloud such interactions, which appear to be an important area for further research. However, the effects of at least some of these other factors are also apparently complicated by their dependence on grain shape and orientation. Note that while TEA (which exists only in noncubic materials) decreases with increasing T, EA may increase substantially, decrease, or change little, depending on the material.

Finally, the first of two overall observations is that the transition from a Hall–Petch dependence with strengths increasing as G decreases to creep and related deformation with strengths decreasing with decreasing G as T increases is often pushed to high values of relative Tm for many materials at normal or higher strain rates. Studies of both creep and superplasticity support such changes, but many more details are needed. These opposite dependences on G pose serious challenges for processing, designing, and using ceramics at high temperatures. Second, the increased, often dominant, role of grain boundary phases, even in very limited amounts, also poses important challenges. These phases also pose serious challenges for processing, designing, and using ceramics at high temperatures.

NOTE ADDED IN PROOFS

Since completion of this chapter Palko et al. [54] have reported that the elastic anisotropy of yttria is 1% at room temperature and does not change measurably up to limits of their testing of 1200°C. This is consistent with the suggestion that the absence of a hardness minimum for yttria (Fig. 4.6) and high transgranular fracture in crack propagation tests reflect low EA.

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and thermal conductivity, these properties can be dependent on the particle parameters of the dispersed phase in ceramic composites. Again, as noted in Chapter 1, the term particle, while often used in the specific sense for composites of a matrix with dispersed (single crystal or polycrystalline) particles, is also used to include platelet, whisker, and (mainly short) fibers in composites, which are also addressed, though the latter only a very limited amount for comparative purposes.

It should be noted that it is only feasible to review selected aspects of papers in this extensive area of research. The goal is to provide a substantial summary of much of the pertinent data, and suitable background, on the mechanical behavior of ceramic composites with a focus on microstructural control of, or impact on, mechanical properties. This focus is intended to aid understanding of such composites and their design and processing from both scientific and engineering standpoints. Further note that it is important that material of this chapter on crack propagation be compared to that particularly of Chapter 9 on tensile strength, and secondarily to that of Chapters 10 and 11 on other mechanical properties and elevated temperature behavior. While this was also the case for monolithic ceramics in Chapters 2–5, it is even more important here due to less extensive microstructural evaluation of both crack propagation and strength behavior and especially of both on the same composite.

II.THEORETICAL AND CONCEPTUAL BACKGROUND

A.Elastic Properties

Since elastic properties of the two or more phases in a composite are seldom identical, and may in fact be significantly different, the elastic properties of such composites are very much a function of their composition and secondly of their microstructure. Elastic properties of dense monolithic ceramics, though having their complexities, are overall simpler since there is only one phase and there is no dependence on grain size [unless this correlates with microcracking, e.g. per Eq. (2.4)]. An important complexity of the elastic properties of monolithic ceramics shared by composites is that the elastic properties of both depend on the degree of preferred orientation of the phases involved, but composites are more complex in having two or more phases as opposed to one whose orientation must be considered. Further, elastic properties of composites can depend on the shapes of the matrix and second phase particles over and above effects of these shapes on preferred orientations of each phase. Elastic properties of composites may also depend on grain and particle sizes, e.g. as both such sizes and shapes may affect contiguity of one or more of the phases, which may affect elastic properties of the composite.

There has been substantial development of models for the elastic properties

of composites because of both the importance and the complexity of the subject, but a detailed review of this development is not conducted here. While there is no single expression or family of expressions adequately predicting elastic properties of all ceramic composites of interest, only key points, reviews, and useful expressions will be summarized here for two reasons. First, first-order predictions of elastic properties available are generally adequate relative to the other uncertainties and needs to understand microstructural dependences of the mechanical properties of composites of primary interest here. Second, neither the models nor the composite characterization are sufficiently detailed and accurate to address fully many of the more detailed aspects of the elastic behavior of ceramic composites.

Consider first-order predictions of elastic moduli, especially Young’s modulus, of ceramic composites based on bounding techniques. These yield upper and lower limits for the elastic properties based on the assumptions made, which, while broad, are often for simplified idealized systems. Thus the simplest and widest bounds are obtained from a model based on slabs of two isotropic materials, which for stressing parallel with the plane of the slabs gives a rule of mixtures upper bound for Young’s modulus (EUC):

where φ = the volume fraction second (e.g. particulate) phase, EP = the Young’s modulus of the second (e.g. particulate) phase, and EM= the Young’s modulus of the other (e.g. matrix) phase. (This equation is commonly a good approximation for the modulus of fiber composites in their linear elastic region for stressing parallel with the fibers [1].) The lower bound (ELC) from such a model is obtained when the parallel slabs are stressed normal to their planes, giving

These expressions, especially Eq. (8.1), are often suitable for a first-order estimate of many composites, especially for those with constituents whose moduli do not differ substantially, e.g. by a few fold or less (see Fig. 8.11).

Models for tighter bounds have been derived, with that of Hashin and Shtrickman [2] being well known. More recently Ravichandran [3] has presented a model based on an idealized composite structure of a uniform simple dispersion of identical isotropic cubic particles in a dense surrounding isotropic matrix (i.e. so only a single unit cell needs to be considered). He obtained for the upper (EUC) and lower (ELC) bounds respectively