Mechanical Properties of Ceramics and Composites

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more time-consuming and costly to prepare. Thin sections are also more time consuming and costly to prepare but typically readily define size and shape aspects of the microstructure and can often give substantial orientation and some crystallographic information. Fracture surfaces also have the advantage of revealing the grain structure of fracture, which can be critical for revealing effects of isolated larger grains or grain clusters, especially of thin platelet grains. Replicas of fracture are sometimes used in OM and SEM to increase the amount of reflected light or contrast definition of the microstructure.

Microscopies typically give photo or screen images from which measurements can be made by various, especially stereological, techniques. A comprehensive discussion of these techniques is a large and complex topic beyond the scope of this book, since there are variations of, limitations of, and complications to many of these techniques. The reader is referred to other sources on the subject [50–63]. The goal here is instead to note basic techniques and limitations and suggest basic and practical approaches to provide guidance, and to stimulate further development of the techniques.

At the lower, basic end of a potential hierarchy of needs is a simple but important nominal average grain or particle size value. An approximate value can often be obtained by measuring a few representative grains or particles, and a more accurate value from the commonly use linear intercept and related techniques. These give G=αl, where l is the average intercept length for grains along a random sample line and α = a constant (commonly 1.5) to account for the fact that neither the sampling lines nor the plane on which they are taken cut grains at their true diameters. However, both theory and experiment show α values ranging from <1 to >2, due to only partially understood dependencies on grain shape and size distributions [64, 65] and possibly on whether the surface is polished or fractured [66, 67] and the degree of intervs. trans-granular fracture. (See Refs. 68 and 69 for other limited microstructural evaluation from fracture surfaces, and Ref. 70 for characterization of intergranular and transgranular fracture.) It is thus important to give measurement specifics, including what value is used to obtain the “true” grain size, since there is no single conversion value. Many investigators simply use α = 1 but often do not state this nor give enough other information.

Next, consider several aspects of linear intercept and related measurements starting with the reliability and repeatability of determining an average G value based on detailed comparative round robin tests [71, 72]. These showed about a 10% scatter among 25 international laboratories on an ideal computergenerated grain structure, based on counting at least 100 linear (or circular) intersections (and is estimated to be only cut in about half by going to 1000 intersections). Scatter increased to 25% for a “nice” (96% sintered alumina) microstructure (equiaxed grains of relatively uniform size with clearly marked grain boundaries). The increase was attributed to factors such as differences in

Grain and Particle Effects on Ceramic Properties

31

polishing and etching techniques and their effectiveness. Analyses of a more complicated but not uncommon (sintered 99% alumina) microstructure resulted in an doubled scatter of results (and somewhat higher scatter still for similar measurements of the limited porosity). Thus measurements of grain sizes are likely to vary by 25–50% or more, and uncertainties in conversions to a “true three-dimensional size” can double or triple this variation or uncertainty.

Consider now four issues of linear intercept (and related circle and grid) measurements. These are, in order of decreasing development and increasing seriousness, (1) that it is limited in giving values reflecting the range of grain size, e.g. the standard deviation (since analysis is based on a single-size, equiaxed grain), (2) that these are limitations in handling nonequiaxed grains or particles, e.g. of tabular or rod-shaped grains or particles, (3) there being no precise way to relate an average G or D value with an individual, e.g., maximum value, i.e. Gm or Dm (e.g. there is no way accurately to relate the measurement of a single grain diameter on a sample surface to random grain chords or linear intercepts), and

(4) a single G or D value may often not be sufficient, i.e. there can be a need for different grain or particle size values to reflect differing impacts of the grain or particle structure itself, or of its variation in size, shape, or orientation. Though progress has been made allowing estimates of the grain size distribution, it still assumes uniform, e.g. spherical or tetrakaidecahedral, grains [50–56, 64–66]. Exact accounting for a mixture of grain sizes can be made by computations for mixtures of a few groups of different size grains, with each group consisting of uniform size spherical grains [50]. Methods for a broader range of grain sizes have been presented [73].

Progress has also been made on conversion of measurements of nonequiaxed grains or particles, e.g. of tabular or rod-shaped grains or particles and their shape factors on a plane (typically polished) surface to their true three-di- mensional character. An exact relation has been derived and validated assuming identical cylindrical grains as an extreme of elongated grains [74]. More recently, using similar idealizing assumptions based on spheroids of uniform size and shape [75], a shape factor, R, has been recommended, defined as

R = (A )-1Σ	A f	i	(1)
T i	i

where Ai is the area on the sampling plane of the individual grain, f the corresponding grain aspect ratio, and AT the total area of all grains on the surface being evaluated. Handling of bior multimodal distributions of such nonequiaxed grains is recommended via a rule of mixtures based on the areal weighting of each population of elongated grains or particles. However, as noted below for analysis of fracture behavior, specific dimensions are still important. Further, while the above-outlined procedures are of significant help, they involve uncertainties, which can be significantly compounded by factors such as shape, size, and orientation distributions in a single grain structure.

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Consider now the comparison of average values of grain or particle sizes in a given body when there are grains or particles present representing other size (or shape) populations. There are at least three cases of concern. The first is bior multimodal populations (e.g. Fig. 1.1). This presents the least problem, especially when the differing populations are clear and present in sufficient quantities so that normal techniques can be applied to each distinct population. The second, and particularly serious, case is when there are only a limited number of substantially larger (or differently shaped, or both) clustered or individual grains, especially as fracture origins (e.g. Figures 1.2–1.5). Accurate comparison of an average size from linear intercept and related measurements with the size of one or a few larger grains or particles on a fracture surface is not possible, though it can be addressed by the use of two-dimensional sizes as discussed below. A similar problem with essentially the same solution is that commonly found in bodies with some, and especially most, or all large grains. The problem arises since the large grains at the surface, though commonly substantially truncated by machining (e.g. Figures 1.2 and 1.3C) are still often the fracture origin, but the body average grain size does not reflect their, often substantially, reduced dimension.

However, even with a normal monomodal population there are issues of which average size is appropriate. While a linear average size, i.e. weighted by the first power of G or D for each grain or particle size respectively, is commonly used, since it is the simplest to obtain, it is often not the most physically meaningful [5]. Such a linear average gives a high weighting to small gains and a low weighting to larger ones [76], which is opposite to important trends for some key mechanical properties. Thus an average based on the volume of the grains or particles, hence weighted by G3 and thus substantially by the presence of larger grains or particles, may often be more appropriate for some property comparisons. Examples of this are where mass distribution or volume absorption (e.g. of radiation), or diffusion in composite, or more commonly in single-phase, bodies are important. More pertinent to this book is where properties are related to the surface areas of the grains or particles. Thus where diffusion, conduction, or fracture along grain boundary surfaces is pertinent, an average weighted by G2, which gives more, but not extreme, emphasis to larger grains or particles may be appropriate. Similar, and of somewhat greater interest in this book, are cases where properties or behavior depends on the cross-sectional area of grains or particles. Key examples are transgranular fracture in crack propagation tests and especially tensile and compressive failures, and hardness, wear, and erosion resistance evaluations. Electrical and thermal conduction, especially in composites, may often fall into this latter category. For example, weighting based on grain area, i.e. G = [∑iGi3][∑iGi2]-1, as opposed to grain diameter, i.e. G = (1/n) [∑iGi], increases the impact of larger grains on the average (e.g., in measuring diameters of 30 grains on a commercial lamp envelope Al2O3, the area-weighted and normal Ga were respectively 51 and 29 µm) [3, 5]. An alternant, direct weighting method for obtain-

Grain and Particle Effects on Ceramic Properties

33

ing a “composite” grain size was suggested by Goyette et al. [77], who noted that it was important in correlating the microstructure of various commercial alumina bodies and their response to single-point diamond machining.

The other aspect of which G or D value to use is whether it should be a twoor a threedimensional value [5]. Some properties and behavior are better correlated with a three-dimensional grain or particle size, e.g. elastic properties (of composites) and electrical and thermal conductivities. However, since fracture is an area-dependent process, it is more realistic to make measurements of grain or particle dimensions actually exposed on the fracture surfaces. Similarly, the area intercepted by larger grains on wear or erosion surfaces is more important than the three-dimensional sizes. Converting a linear intercept measurement to an average surface grain diameter, Gs, might be done using Gs ({1 + α}/2) l (i.e. assuming that half of the correction, α - 1, is due to the randomness of the sampling plane cutting the grains and half to the randomness of the linear intercept itself), but it is uncertain in both the form and the actual α value. Overall, it appears better to measure actual grain or particle diameters exposed on the fracture surface, e.g. selected by using random lines as in the linear intercept method. Such measurements would be directly related to measuring individual, e.g. the largest, grains on fracture. Having actual grain or particle diameters on a fracture surface allows the calculation of an average grain size (G) or particle size based on various weightings. Such two-dimensional values should be more pertinent to bulk fracture properties as well as wear and erosion phenomena.

Two further points should be noted. First, for fracture from elongated grains or particles, use of a size reflecting their area on the fracture surface is typically a good approximation [78], but more accurate calculations require the actual dimensions. Some used the maximum grain or particle dimension, i.e. length [11], but fracture mechanics uses the smaller dimension of an elongated, e.g. elliptical, flaw as C (the larger dimension impacts the flaw geometry parameter), so this is inappropriate (as is the smallest grain dimension by itself, as previously suggested [41]); an intermediate grain or particle size value is appropriate. However, the aspect ratio (and orientation effects) may be important, e.g. as indicated by Hasselman’s [79] modeling of effects of elastic anisotropy of grains on mechanical properties. Second, it is important that the different, i.e. twoand three-dimensional grain or particle size values be relatable, which requires substantial further analytical and experimental evaluation.

C.Spatial Distribution and Orientation Measurements

Besides grain or particle size and shape, the spatial distributions of these parameters can also be important, especially if fracture origins were not identified. Systematic spatial variations of sizes and shapes, e.g. between the surface and the interior, e.g. from loss of additives near the surface or machining truncating

34

Chapter 1

large surface grains (Figures 1.2 and 1.3C), are easier to handle. Handling random or irregular distributions of larger grains or particles (e.g. due to variations in initial particles, additives or impurities, or porosity) indicates the need for statistical methods, especially to determine spacings of larger grains, particles, or their clusters. Stoyan and Schnabel [80] used a pair correlation approach to address this problem, characterizing the frequency of interpoint distances (e.g. between grain vertices or centers—the latter was preferred). They showed a higher correlation of strength for nearly dense Al2O3 bodies than with Ga ( 9 to 15 µm) itself. Modern stereological tools make such characterization more practical (and potentially applicable to pores and pore–grain or pore–particle associations), but again fractography is the most assured method of addressing this.

Overall, i.e. global, preferred crystallographic orientation of grains or particles clearly occurs in varying degrees as a function of forming methods and processing and material parameters, as is discussed in Chaps. 2–12. Such orientation can clearly affect properties, especially mechanical ones, in a desirable or undesirable fashion, and is often a major issue in understanding the microstructural dependence of properties, especially when its presence is not accounted for. When associated with grain or particle geometry, especially elongation, substantial orientation information can be obtained from stereological measurements. However, x- ray diffraction techniques, which range from qualitative Laue patterns, to comparisons of intensities of various x-ray lines, to complete pole figures, are typically more versatile, effective, and widely used. Pole figures are the most accurate, comprehensive, costly, and time-consuming but have been greatly aided by modern computer aided characterization, and possibly by newer, additional methods of determining grain orientations [81–83]. Also of potential importance is local grain or particle orientation, e.g. of individual elongated or platelet grains or particles, or clusters of them, especially at fracture origins. Such information can be important to determine if microcracking from thermal expansion anisotropy was a factor in fracture. In the past such information was difficult to obtain. However, modern microscopic analytical techniques are providing increasing capabilities in this area [84].

Thus, in summary, at least two different but related G or D values may often be needed, one based on cross-sectional, especially fracture, area for fracture, as well as probably wear and erosion, and one on three-dimensional size for elastic and conductive properties. Both values should be relatable, which is a challenge due to varying size, shape, and orientation not necessarily being independent of one another and their interrelations probably varying for different properties. Further, the impact of these grain parameters on properties can depend significantly on the spatial distributions of each of these variables. Thus the size of isolated larger platelet grains is likely to have limited effect on some properties such as electrical or thermal conductivities, provided they are not associated with other important microstructural complications such as accumulations of second phases or microcracking. However, even in the absence of the latter complications, they can have varying effects on fracture properties ranging from erosive particle im-

Grain and Particle Effects on Ceramic Properties

35

pacts, through crack propagation such as for fracture toughness, to tensile failure. Effects of platelet grains on erosion depend on both the extent of their being in particle impact zones and their effects on erosion via local fracture in these zones, both of which depend separately and collectively on their volume fraction and their size, shape, and (local and global) orientation. However, once the critical size, shape, and orientation range for local fracture is reached for the given erosive environment, then the volume fraction dominates. Similarly, volume fraction of platelet grains is important in fracture toughness, but their orientation, shape, size, and spatial distribution, and the scale of these relative to the scale of crack propagation, can all be factors that may impact typical fracture toughness values in a different fashion than erosion. Typically more extreme is tensile failure, which is determined more by the occurrence of individual or clustered platelets near or above the critical flaw size, oriented at or near normal to the tensile axis, and in combination with another defect such as a pore or a crack (e.g. from machining the surface, or mismatch stresses). Thus it is important to recognize that grain and particle sizes and related characterization for correlation with strength typically represent a fair amount of uncertainty, e.g. a factor of 2 or more, and can thus be a factor in variations between different studies along with uncertainties and differences in the properties themselves. Poor coordination of property and microstructural measurements can exacerbate such problems.

V.COORDINATION OF PROPERTY AND MICROSTRUCTURAL MEASUREMENTS

The microstructural dependence of fracture properties, especially tensile strength, is a good example of the importance of judicious interrelation of test and microstructural evaluations. Thus, as shown extensively later, larger isolated or clustered grains or particles commonly determine the strength of a given specimen. Clearly, both the grain or particle size and the strength values used to relate strength and microstructure need to be self-consistent with each other. However, strength is very commonly based on the outer, i.e. maximum, fiber stress in flexural failure (σm, e.g. as used by all investigators coordinating strengths with maximum grain size). Such a maximum strength is, however, basically inconsistent with use of a maximum grain size (Gm) and is often more consistent with use of an average grain size (Ga), since the latter has a moderate to very high probability of being associated with σm, while Gm has a moderate to very high probability of being associated with < σm (indicating lower larger G slopes for those using Gm and σm, as is discussed later). These probabilities and the errors involved in using Ga or Gm depend on both the size and spatial distribution of grains and the stressed volume and surface, along with the extent or absence of stress gradients. Smaller volumes under high stresses more likely reflect less deviation from the Ga, i.e. the use of threeversus four-point flexure, as well as smaller specimen cross sections and corresponding shorter spans. Round flex-

36

Chapter 1

ure rods have the smallest stressed volume but a larger surface area from which surface-related flaws can be activated at variable stress. Progressing to true tensile tests (e.g. from threeto four-point bending, to hoop tension then uniaxial tension) as well as increasing specimen sizes in each test gives grater emphasis to failure from microstructural extremes and to the role of associated defects (mainly pores and cracks). Of equal or greater importance is the effect of temperature. A recent review [85] and Chaps. 6, 7, and 11 show that temperature changes of a few hundred degrees Celsius can shift the microstructural dependence of behavior in different fashions for different materials and thus can be a valuable tool in defining the mechanisms controlling behavior.

Fractography, besides being important for microstructural characterization, is also the first and most fundamental of three approaches to properly correlate fracture, especially strength, and grain or particle parameter values. This, if successful, allows both the actual G and the location of fracture initiation to be determined. With the latter the failure stress (if <σm) can be corrected for stress gradients into the sample depth. (Correction for off-center failures due to gradients along the sample length, e.g. for three-point flexure, is a separate operation from fractography.) However, as noted above, even with fractography there can still be considerable uncertainty; hence the need for other approaches. The second approach is to use the various microstructure measurements and analysis discussed in the previous section, especially in conjunction with specimen stress–volume and surface area relations noted earlier. Thus smaller specimens and stressed volumes reflect less G variation, so Ga is more reasonable, while larger specimens and more uniform stressing emphasizes effects of microstructural extremes. The third approach is to use the known property-microstructure behavior as a guide. This and the other approaches are best when done in combination with one another, e.g. for specimens known to have a range of G, the statistical fit of its σ with other data, especially for more homogeneous grain structures, at the pertinent G values can be used as a guide for the placement (or rejection) of a data point probably also aided by fractography. Lack of such combinations and comparison has been a serious shortcoming of many earlier studies, including those using Gm.

The need for fractography for other mechanical tests has also been demonstrated in crack propagation–fracture energy and toughness tests, e.g. again showing the impact of microstructural heterogeneities such as larger grains [86]. Thus tests with different crack sizes and extents of propagation should be of value, especially with fractographic examination. Similarly, evaluation of wear and erosion as a function of impacting particle sizes, velocities, and materials can be important, again especially when coupled with microstructural (i.e. often local fractographic) examination of various wear or impact sites. Again tests as a function of even limited temperature increases can be very valuable.

Finally, the importance of demonstrating isotropy of properties, instead of assuming it without substantial reason, is critical because of the frequent occurrence of some anisotropy in bodies commonly assumed to be isotropic. Such eval-

Grain and Particle Effects on Ceramic Properties

37

uations should include measurements of other properties, which are of broader importance than just the issue of isotropy. Thus elastic property measurements are important to correlate not only with other mechanical properties but also with nonmechanical ones, e.g. electrical and thermal conductivities, as well as of the latter with other mechanical properties, especially in composites. An important aspect of such intercomparison of properties is not only the specific property values but also their distribution. Thus the Weibull modulus of failure from mechanical testing is similar, but not identical, to that for failure from dielectric breakdown [87, 88], reflecting the similarities and differences of the sources of such fracture and breakdown [1]. While both failures are impacted by locally higher porosity, mechanical failure is determined more by a compact area of more, larger, or both pores, usually intergranular, often close or connected, ones, acting as much or all of the failure causing flaw. Thus the cross-sectional area of the pores parallel with the stress is not a key factor, while their area normal to the stress is, along with their close spacing. Electrical breakdown also is fostered by accumulations of pores, especially intergranular ones (often associated with larger grains, and probably boundary phases). However, pores that are most serious in electrical breakdown are those in an (often discontinuous) chain forming a failure path with the least solid material through the body. The closeness or connection of the pores and their cross-sec- tional area normal to the breakdown is not as critical, while their net area along the resultant breakdown path is. Another factor is that most mechanical tests are in flexure, and hence in a stress gradient, while most electrical breakown tests are in a uniform electrical field analogous to uniaxial tensile testing. Recognition and use of such differences in the details of the microstructural effects of varying tests and properties is thus a very valuable, but seldom used, tool to separate out specifics of the microstructural dependences of properties.

VI. SUMMARY AND CONCLUSION

Grain and particle parameters, especially but not exclusively size, play an important role in many, especially mechanical, properties. Though often less in magnitude than effects of porosity, grain and particle shapes and orientations and especially sizes can commonly vary important properties by up to a few to several fold. Since increased properties from control of grain or particle parameters are typically over and above those obtained by minimizing porosity, they are essential factors in obtaining high property levels. While their impact is greatest through primary effects, secondary effects can also be important but have received much less attention.

Microstructures range from fairly simple to very complex, offering both challenges for characterization and opportunities for achieving desired properties. However, accurate characterization of even simple microstructures can be complex and uncertain in meeting needs. For example, at least two grain and particle sizes are needed, one based on area and one on volume (i.e. respectively

38

Chapter 1

twoand three-dimensional) sizes for respectively fracture and wear and erosion behavior versus elastic and conductive properties. Interrelation of these sizes is also needed, but many current measurements reflect mainly a single size value with substantial variation and uncertainty, e.g. by factors of the order of 50 to a few hundred percent. This allows useful comparison of different studies of the same property, but these variations must be considered as a source of difference along with property variations due to differences in specimens and the tests used.

There are a diversity of tools for improved microstructural measurements that can be of considerable aid if more extensively applied, but these are not the complete answer. Some alternate or additional measurements are needed, e.g. to compare maximum with average grain or particle sizes, with the latter reflecting twoor three-dimensional, i.e. surface or volume effect, values. However, the most immediate and important needs are in first perspective on properties and second in their measurement. Thus given the diversity of microstructures, materials, and resultant properties, improved information, perspective, and understanding is needed in three areas. The primary need is for a more balanced perspective, in particular recognition of two key factors: (1) many toughness results are of uncertain or limited pertinence to much strength behavior, and (2) much of the grain and particle dependence of strengths derives from their impact on the size of flaws introduced that control failure, as extensively shown in this book. The other two needs are for better and refined observations and improved documentation via more and better microstructural characterization and more data on a broader range of bodies and especially microstructures.

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