Journal of the American Ceramic Society--Fujita er al Vol. 87. No. 2 V(+m/ where n is a junction area ratio, defined by n- and the alumina-mullite (".(iv) The modulus is determined from the unction stiffness, weighted by the fraction of the three Using the arithmetic aver types, given by Eq.(15). Both the arithmetic mean harmonic mean are considered Using the arithmetic mean of the junction stiffness, the resulti 1+y ggregate modulus is predicted to be =XAVn+2X(I-XAl X E 1+n)/2A - XAA m+2XA(1-XA) V2(1+A 3)+=xy The models were implemented in the following way. The modulus (16) EM of the pure mullite aggregate was obtained from Eg. (7) additive terms on the right side represent the contribu calibrated according to the procedures outlined above. Based on alumina-alumina, alumina-mullite, and mullite-mullite the known elastic moduli of the particles, A-2 In comparing Eq respectively. Analogously. the modulus prediction ob- (15)to the experimental data, the remaining unknown variable, n was used as a fitting parameter. The results of this fitting are on the harmonic mean of the junction stiffness is: shown in Fig. 6(a). The procedure yields n-3+ I for all aging E+2x0-x0(2+=8 times. This value is qualitatively consistent with the expectation that the alumina-containing junctions sinter more rapidly than those with only mullite, and hence their junctions are larger. A (17) predictions with the experimental me pare the toughness model similar procedure was used to ce A preliminary assessment of these results has been made the value of n inferred from the modulus measurements was used arough comparisons with numerical simulations based on the in Eq(18). The remaining unknown variable, y, was used for discrete element method. The nature of the simulations is essen- fitting(Fig. 6(b)). The procedure yields a junction toughness ratio tially the same as those used for determining toughness of particle y= 10=0.3. This result suggests that the nd that the increase chesses of the ggregates, with the exception that the aggregate now consists of two particle types with elastic properties given by those of alumina in aggregate toughness with alumina content is mainly caused by nd mullite (A-2). In the present simulations, all junctions are the increase in the average junction area. The quality of the fits to assumed to be of the same size(n= 1). Figure 7 shows the results both the modulus and the toughness data for all three aging times for two sets of simulations, for junction radii a/R= 0.053 and appears satisfactory, especially in light of the scatter in the 0.313. The results are presented in terms of a modulus parameter experimental data. Furthermore, it is recognized that the scatter hat varies from O for X0 to I for Xa= I. Also shown are the precludes determination of the fitting parameters n and y with predictions from the analytical models(Eqs. (16)and(17)).Over Can eels hoe ange of XA, the predictions from the two analytical entration of the stiffer particles approaches zero(XA-0 in VI. Implications for Crack Deflection his case), the arithmetic mean method vields more accurate esults. By contrast, in the limit where Xa- l, the harmonic mean Both the experimental results and the model predictions have method is more accurate. For the range of compositions considered een used to calculate the variation in the crack deflection preferred and is used in subsequent calculations mean method is parameter, S. with aging time, assuming the mixtures to be An analogous analytical model has been developed for the The results are plotted on Fig. 8. Over the entire range of toughness of a mixed aggregate. Following Eq (9). the contribu- parameter values considered in this study, the crack deflection tion from each junction type is assumed to be proportional to condition is predicted to be satisfied: that is, 2>1.The T(alR) weighted by the corresponding number fraction (Eq (15) experimental results are also plotted on Fig. I, demonstrating that The junction toughnesses are denoted TAA, TMM and T AM: the alumina to mullite toughness ratio is defined as y TAA/TMM 1.0 闪 100M/OA OM/10A DEM:aR=0.053,2=6.15 80M/20A o8“0“ DEM: a/R=0313z=68 ∠70M30A Arithmetic mean(Eqn. 16 60M/40A Harmonic mean(Eqn. 17) 0.6 DMOA 0.4 口9oM10A 80M20A O 70M/0A ◆60M40A 02 Aging time, t (h) Alumina content, x f aging time and composition on the crack deflection fiber properties Tr= 15 J/m"and E,=260 Numerical simulations and analytical results( Egs. (16) and lines are editions. Extrapolations of the predictions erical monosized alumina I yield the mes, Ie, at which crack penetration is ullite particles (A=2 and n= 1) occur