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earlier work on composites where strengths were studied but toughness measurements were not developed or in common use. This strength data provides insight into other important factors such as the separation of dispersed particles (λ), consideration of which has been largely neglected in the focus on crack bridging and related R-curve effects with large cracks. It will be shown that considerable strength data varies as λ-1/2, and that this dependence is related to the grain size (G) dependence of strengths of monolithic ceramics. The similarities between monolithic and composite ceramics are also shown by strength variations, e.g. Weibull moduli of composites also often being very similar to those of monolithic ceramics.

II.THEORETICAL BACKGROUND

In principle all the models and concepts considered for crack propagation and toughness are applicable to strength. However, as noted above and in Chapters 2 and 3, there is the important issue of the pertinence of mechanisms whose effectiveness is most significant at large crack sizes, not at the much smaller cracks typically controlling strengths. Serious attention to this issue has been largely absent, so effects can often only be inferred. Thus the implication that crack bridging controls strengths needs further examination in view of its significant dependence on crack size (as well as other uncertainties, Chapters 2, 3, 8). Other mechanisms must also be examined in terms of their crack size dependence, as well as in terms of dependence on volume fraction of dispersed phase. In all cases the central issue is typically the crack size relative to the strength controlling microstructure. Thus the effects of stresses in and around dispersed particles, which has not been found to be a major factor in fracture toughness, may be more important when flaws are on the scale of the dispersed particles and where the volume fraction of dispersed phase is modest. At higher volume fractions crack interactions with the particles may be driven more by their number (and size) than their type or level of stress (unless leading to microcracking, discussed below). Clearly possible line tension effects depend directly on the scale of the crack versus that of the second phase spatial and size distributions, diminishing with increasing second phase content (via their effect on resultant spacing). Similarly, crack deflection presumably becomes less effective beyond some levels of second phase size and content. While this is also likely to be true for fiber pullout, probably at higher fiber volume fractions, even then some systems may perform well at very high levels of fibers (e.g. as shown by ropes on the one hand and implied by natural fibrous materials such as jades, Fig. 8.15D).

Another basic factor that must be considered in strength is that in addition to (or instead of) the mechanisms impacting fracture toughness, there can be effects of composite microstructure on failure causing flaws. A model was proposed by Hasselman and Fulrath [1] for ceramic composites where flaw

similar to, and possibly impacted by, other effects of composites on machining flaws, is a basic mechanism that can occur independently of other machining flaw effects.

A key flaw factor that needs to be considered is machining flaw generation, where flaw character and size are functions of the local material elastic moduli, toughness and hardness [e.g. via Eq. (3.2)], and machining conditions [2,3]. This is different from such flaws being specifically constrained by the spacing between particles, as was discussed earlier, and can occur independently of, or in addition to, such flaw constraint effects. Subsequent evidence is shown in this chapter that this plays a major role in strengths of composite, as with monolithic, ceramics. More attention to this and other flaw factors is needed, e.g. residual surface compressive stresses generated by machining many ceramics, and possibly somewhat greater in materials with ZrO2 transformation toughening, which can be important. Another factor is generation of flaws by microcracking and possible linking of microcracks with one another or with machining flaws, again with size and spacing of such cracks relative to the flaw size controlling strength being important.

III.PARTICLE PARAMETER EFFECTS ON TENSILE STRENGTHS OF CERAMIC COMPOSITES

A.Composites with Glass Matrices

1.A Synthetic Glass Matrix Composites

Consider first synthetic composites made by consolidating glass matrices around dispersed crystalline particles, beginning with Binns’ [4] earlier, substantial study of such composites with up to 40 v/o of zircon or alumina particles (of 10, 45, or 180 m dia.) in glass matrices of varying thermal expansions. Both sets of composites with φ=0.2 showed a maximum in E when the thermal expansion of the glass and dispersed particles were matched, progressively decreasing as the expansion difference increased, probably somewhat greater when the glass expansion exceeded that of the dispersed phase versus when it was lower than that of the dispersed phase by the same increment (Chap. 8, Sec. III). Thus the Young’s modulus (E) versus glass expansion curves for the two sets of composites were very similar except that the maximum E for the zircon composite was 20% lower than that with Al2O3 particles and shifted to lower expansion by the alumina–zircon expansion difference. There was no effect of particle size on E with either type of particle, except for substantially greater decreases for the coarsest Al2O3 as the glass expansion increasingly exceeded that of the Al2O3.

Biaxial flexure strengths of these composites with the coarsest particles also showed a maximum when the expansions of the glass and the Al2O3 or ZrO2 dispersed phase were matched. The decreases in strengths with increasing ex-

pansion mismatch followed those of E, again with some probable greater decrease as the glass expansion exceeded that of the particles. All composites had increasing strengths as φ increased, but at differing rates, levels, or both. Composites of Al2O3 particles with matching matrix expansion and intermediate or finer particles had 20% higher strengths than those with the coarsest particles. However, as the glass expansion progressively exceeded that of the Al2O3 particulates, composites with the finest particles reached higher strength levels and a maximum, while composites with intermediate size particles, which had essentially the same strengths as with finer particles when the glass expansion was ≤ that of Al2O3, had maximum strengths when the expansion coefficients were the same. At higher glass expansions strengths fell sooner and to a greater degree than with the finer particles, e.g. to 75% that of the finer particle composite. In contrast, composites with 20 v/o of the coarsest Al2O3 particles had a maximum strength 2/3 that of composites with intermediate size Al2O3 particles when the expansions were the same, and strengths decreased more relative to composites with intermediate and especially finer Al2O3 particles as expansion differences increased in either direction, but again with greater decreases with glass expansion exceeding that of Al2O3. Composites with zircon particles had much less increase in strengths as φ increased with intermediate or coarser particles than with fine particles, so the net result was strengths of the latter 2 times those of the former at φ= 0.4 (and ≥or 70% those of composites with respectively intermediate or coarser Al2O3 particles). Thus this more comprehensive work shows substantial parallels between Young’s modulus and strength behavior with similar effects based on the sign, and especially the magnitude, of the particle–matrix expansion difference, as well as effects of particle size (and composition, and hence probably E in the greater decreases in E of composites with larger Al2O3, but not zircon, particles). Analysis of this data shows a systematic dependence of strength on particle size in conjunction with other key parameters such as φ, consistent with behavior of other similar composites, as will be discussed below (Figs. 9.2A, B).

Hasselman and Fulrath’s [1] strength data for synthetic composites of Al2O3 particles of differing sizes and volume fractions in a glass matrix of nearly the same thermal expansion showed the bilinear behavior of Fig. 1 when plotted versus the mean free spacing between particles, λ, per Fullman’s equation,

leading to their model for the strength of composites of discrete ceramic particles in a ceramic matrix. Again, their rationale for this plot and behavior was that at larger λ values machining flaws controlling strengths would be < λ, so that λ would have no effect on strengths as shown, but as λ decreased it would reach a point where λ= the flaw size, beyond which decreasing λ would then constrain the flaw size to λ and hence increase strength in proportion to λ–1/2. Both the