ovember 2006 Oxide Fiber Composites simulations of fracture using the dEm (Sidebar A). The esults of the simulations are well described by Aging time 口1000hw=10±0.3 where I is the junction toughness. (The scaling with a is a 日 consequence of the dependence of toughness on junction area. Combining Eqs. (4)and(7) yields the corresponding time de- Although this sensitivity is greater than that of the modulus, the magnitude of the effect is small when surface diffusion is the ative sintering mechanism An assessment of these models is made through comparison with measurements on pure mullite compacts(Fig. 14(b)) Consistent er-law scalings of modulus and toughness with time are obtained for aging times up to 10 h. Furthermore extrapolation, the modulus and the tough dicted to increase by only 10% and 20%, respectively, for an 口 ional 1o h components). Such extrapolations are used in estimating long 日 term durability of oxide CFCCs, demonstrated below 口日口●。8 (2) Two-Phase Particle Networks The preceding models for junction growth and property changes in monophase aggregates are extended to two-phase particle mixtures(such as those comprising pore CFCCS). For a generic mixture of A and B particles, three 10 unction types are present: A-A, A-B, and B-B. The aggregate Alumina co properties are obtained by averaging the junction properties Fig 15. Effects of composition and aging time on(a) Youngs modulus weighted by the number fraction of the associated junction type d (b)toughness. The solid lines represents model predictions(Eqs. (10) To facilitate tractable solutions, the two particle types are as- nd (II)). Adapted from Fujita et al. sumed to be the same size and arranged randomly in the mix ture. A statistical analysis yields the junction fractions, f An analogous model for the toughness of a mixed aggregate is obtained when the contribution from each junction type is fAA=XA (9a) assumed to be proportional to T (a/r)and the toughnesses of the different types of junctions are then weighted by their re- spective number fractions Eq (9). The result where XA is the number fraction of A particles. +(1-XA)2 (11) 6 Young's modulus of the two-phase mixture is modeled fol- wing an approach similar to that used for monophase sys- tems, with appropriate modifications to reflect differences in where TAA, TBB, and TAB are the junction toughnesses: Y=TAA/ junction characteristics. () The description of junction stiffness IMM; and TAB is taken to be the average of TAa and TBe gtal is modified to account for the moduli of the particles on either The models are assessed by comparison with er de of the junction. In the hertzian limit, the stiffness of dis for (Fig. 15).- The unknown parameters are and y. Fitting the a=EAEB-49(i) The area of the A-B junction is assumed to be modulus measurements yields a junction area ratio n3+I the average of the areas of the A-a and B-B junctions. This consistent with the expectation that the alumina-containing result ressed in non-dimensional form as aab/aBB junctions should sinter more rapidly than those with only mu (1 +n)/2 where n is a junction area ratio, defined by ite. Then, upon fitting the toughness measurements, the inferred m=(aAAaBB).(ini) The modulus is determined from the arith- toughness ratio is p=1.0+0.3. The implications are twofold: () metic mean of junction stiffness, weighted by the respective the toughnesses of the three junction types are similar to one number fractions, given by eq(9). The result is another, and (ii the increase in aggregate toughness with alu mina content is due largely to the increase in the average junc- tion area En=X入+2X(-x-2 (3) Precursor-Derived Two-Phase Networks +(1-XA)2 When the binder phase is produced by a precursor route, the g p Upon comparing with numerical simulations, this poses, the topology is represented by one of two limiting ideal to be accurate in the domain in which the sinterable phase com- zations. In both, the major phase is treated as a contiguou prises <40%o of the total. etwork of uniform particles, radius R, and with averagesimulations of fracture using the DEM (Sidebar A).49 The results of the simulations are well described by G Gj ¼ 12 a R 2 (7) where Gj is the junction toughness. (The scaling with a2 is a consequence of the dependence of toughness on junction area.) Combining Eqs. (4) and (7) yields the corresponding time de￾pendence G Gj ¼ 12 t tR
2=n (8) Although this sensitivity is greater than that of the modulus, the magnitude of the effect is small when surface diffusion is the operative sintering mechanism. An assessment of these models is made through comparison with measurements on pure mullite compacts (Fig. 14(b)).42 Consistent power-law scalings of modulus and toughness with time are obtained for aging times up to 103 h. Furthermore, upon extrapolation, the modulus and the toughness are pre￾dicted to increase by only 10% and 20%, respectively, for an additional 104 h of exposure at 12001C (typical of turbine engine components). Such extrapolations are used in estimating long￾term durability of oxide CFCCs, demonstrated below. (2) Two-Phase Particle Networks The preceding models for junction growth and property changes in monophase aggregates are extended to two-phase particle mixtures (such as those comprising porous matrices in CFCCs).42 For a generic mixture of A and B particles, three junction types are present: A–A, A–B, and B–B. The aggregate properties are obtained by averaging the junction properties, weighted by the number fraction of the associated junction type. To facilitate tractable solutions, the two particle types are as￾sumed to be the same size and arranged randomly in the mix￾ture. A statistical analysis yields the junction fractions, f fAA ¼ X2 A (9a) fBB ¼ ð Þ 1 XA 2 (9b) fAB ¼ 2XAð Þ 1 XA (9c) where XA is the number fraction of A particles. Young’s modulus of the two-phase mixture is modeled fol￾lowing an approach similar to that used for monophase sys￾tems,46 with appropriate modifications to reflect differences in junction characteristics. (i) The description of junction stiffness is modified to account for the moduli of the particles on either side of the junction. In the Hertzian limit, the stiffness of dis￾similar particle junctions is proportional to 2l/(11l) where lEA/EB. 49 (ii) The area of the A–B junction is assumed to be the average of the areas of the A–A and B–B junctions. This result is expressed in non-dimensional form as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aAB=aBB ¼ ð1 þ ZÞ=2 p where Z is a junction area ratio, defined by Z(aAA/aBB) 2 . (iii) The modulus is determined from the arith￾metic mean of junction stiffness, weighted by the respective number fractions, given by Eq. (9). The result is: E EB ¼ X2 Al ffiffiffi Z p þ 2XAð Þ 1 XA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 þ Z 2 r 2l 1 þ l
þ ð Þ 1 XA 2 (10) Upon comparing with numerical simulations, this result is found to be accurate in the domain in which the sinterable phase com￾prises r40% of the total. An analogous model for the toughness of a mixed aggregate is obtained when the contribution from each junction type is assumed to be proportional to Gj(a/R) 2 and the toughnesses of the different types of junctions are then weighted by their re￾spective number fractions Eq. (9). The result is G GM ¼ X2 ACZ þ 2XAð Þ 1 XA 1 þ Z 2
1 þ C 2
þ ð Þ 1 XA 2 (11) where GAA, GBB, and GAB are the junction toughnesses; CGAA/ GMM; and GAB is taken to be the average of GAA and GBB. The models are assessed by comparison with experimental measurements on the mullite/alumina system, for which l 5 2 (Fig. 15).42 The unknown parameters are and C. Fitting the modulus measurements yields a junction area ratio Z371: consistent with the expectation that the alumina-containing junctions should sinter more rapidly than those with only mul￾lite. Then, upon fitting the toughness measurements, the inferred toughness ratio is C 5 1.070.3. The implications are twofold: (i) the toughnesses of the three junction types are similar to one another, and (ii) the increase in aggregate toughness with alu￾mina content is due largely to the increase in the average junc￾tion area. (3) Precursor-Derived Two-Phase Networks When the binder phase is produced by a precursor route, the resulting topology is markedly different. For modeling pur￾poses, the topology is represented by one of two limiting ideal￾izations. In both, the major phase is treated as a contiguous network of uniform particles, radius R, and with average Fig. 15. Effects of composition and aging time on (a) Young’s modulus and (b) toughness. The solid lines represents model predictions (Eqs. (10) and (11)). Adapted from Fujita et al. 42 November 2006 Oxide Fiber Composites 3317