3318 Journal of the American Ceramic Society--Zok 150 the same value of a. Operationally, this estimate is obtained by replacing Ep in Eq (13)with the volume-weighted average of the moduli of the two solid phases, yielding the result a=0112k E=04(1+14x2)[EPUP+EB(1-UB where I-Po+vB and EB is the modulus of the binder. Upon combining Eqs.(13H (15), the estimated modulus of the two-phase system becomes =0.B=1 E E=0.4(1+14x (1-Po)+(EB/Ep)VB (1-Po)+VB measurements predictions An analogous approach yields estimates of toughn the properties of the two phases are the same the results from DEM simulations(Sidebar A)are well described by =12a2 When adapted to the general case in which the properties of the two phases differ, the toughne ome 2VB+2 o=0.1,B=12 3(1-po) Comparisons between the model predictions and the exper- imental measurements for the mullite-alumina system are plot 20 ted in Fig. 16. When the parameters are selected to be b= 1.2 Alumina concentration, VA (% and ao/R=O1, the model provides a good fit to the modulus data. In contrast, when they are taken as B= l and ao R=O, the nd mode modulus of the pure mullite is erroneously predicted to be 0 and odulus and(b) toughness. The the rate of increase with volume fraction is underestimated. For network, strengthened by precursor-derived alumina( from Fujita et al. are l modeling the toughness, with Ti the only unknown parameter junction radius, ao. In one limit, the binder phase is a porous Upon fitting the data( Fig. 16), the junction toughness is inferred homogeneous continuum occupying all available space in the to be li=4 J/m*: only slightly greater than that of pure mullite interstices of the particle network. This idealization is applicable (i=3J/m) to the as-processed aluminosilicate matrices, wherein the silica glass exists as a contiguous nanoporous phase within the alu- (4) Implications for Crack Deflection mina particle network. In the other, the binder is modeled Once calibrated, the preceding models are coupled with the This is the preferred representation for the mullite- tion, as manifest in the parameter 2. The results are po lla coating, thickness h, on the particle surfaces(inset of analysis in Section Ill to assess the propensity for crack defle nalysis Nextel720 fibers In this case, crack deflection is predicted over From geometry, the normalized net junction radius a is given the entire range of compositions(0%40% alumina) and aging times(to 1000 h). As a complementary representation, a subset of these results is plotted in Fig. 12(a)(assuming @= 1). Here, 2β-VB+3 again, the property combinations lie within the crack deflection 2=R+h~V3(1-Po) ( 12) domain. Moreover, the interface sliding stress in these systems is low, typically <10 MPa (Sidebar B), re-affirming that mullite-alumina mixtures are good candidates for use in oxide composites The critical aging time le for crack penetration is obtained by Co=Mo/(R+h); and B is a non-dimensional parameter that ac- extrapolating the predictions in Fig. 17(a)to 2=0=1.It counts for preferential binder accumulation at the particle junc nges from 4000 h for mixtures of 60% mullite 40% alumina tions(= l for uniform coatings). For the case in which the roperties of the two phases are the same, computer to 60000 h for pure mullite, the latter being comparable with the targeted service lives of CFCC components. With knowledge of ions based on the DEM of Youngs modulus are accur- the activation energy of the sintering mechanism, the model can escribed by the empirical equation be readily extended to other temperatures A similar assessment is made of the mullite particle networks =0.4x(1+14x strengthened by precursor-derived alumina, again assuming the C-inforcements to be Nextel 720 fibers. The results are plotted n Fig. 17(b). Here 2 decreases with increasing alumina con where Ep is the modulus of the solid particles. More genera entration, VA, and eventually falls below the critical value. when the properties of the two phases differ, the modulus of the oRl, at VA9%. This point is expected to mark the onset of wo-phase aggregate is estimated from the weighted average of crack penetration into the fibers and a significant loss in damage the moduli of the two monophase aggregates, evaluated atjunction radius, ao. In one limit, the binder phase is a porous homogeneous continuum occupying all available space in the interstices of the particle network. This idealization is applicable to the as-processed aluminosilicate matrices, wherein the silica glass exists as a contiguous nanoporous phase within the alu￾mina particle network. In the other, the binder is modeled as a uniform coating, thickness h, on the particle surfaces (inset of Fig. 16). This is the preferred representation for the mullite– alumina system (Fig. 13) and forms the basis for the ensuing analysis. From geometry, the normalized net junction radius a is given by40 a  a R þ h  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2b2 VB 3 1ð Þ po þ a2 o s (12) where 1po is the volume fraction of the particulate phase; VB is the volumetric concentration of the precursor-derived binder; aoao/(R1h); and b is a non-dimensional parameter that ac￾counts for preferential binder accumulation at the particle junc￾tions (b 5 1 for uniform coatings). For the case in which the elastic properties of the two phases are the same, computer simulations based on the DEM of Young’s modulus are accur￾ately described by the empirical equation E EP ¼ 0:4a 1 þ 14a3   (13) where EP is the modulus of the solid particles. More generally, when the properties of the two phases differ, the modulus of the two-phase aggregate is estimated from the weighted average of the moduli of the two monophase aggregates, evaluated at the same value of a. Operationally, this estimate is obtained by replacing EP in Eq. (13) with the volume-weighted average of the moduli of the two solid phases, yielding the result E ¼ 0:4a 1 þ 14a3  ½ EPuP þ EBð Þ 1 uB (14) where uP ¼ 1 po 1 po þ VB (15) and EB is the modulus of the binder. Upon combining Eqs. (13)– (15), the estimated modulus of the two-phase system becomes: E EP ¼ 0:4a 1 þ 14a3   ð Þþ 1 po ð Þ EB=EP VB ð Þþ 1 po VB  (16) An analogous approach yields estimates of toughness. When the properties of the two phases are the same, the results from DEM simulations (Sidebar A) are well described by G Gj ¼ 12a2 (17) When adapted to the general case in which the properties of the two phases differ, the toughness becomes G Gj ¼ 12 b2 2VB 3 1ð Þ po þ a2 o
(18) . Comparisons between the model predictions and the exper￾imental measurements for the mullite–alumina system are plot￾ted in Fig. 16. When the parameters are selected to be b 5 1.2 and ao/R 5 0.1, the model provides a good fit to the modulus data. In contrast, when they are taken as b 5 1 and ao/R 5 0, the modulus of the pure mullite is erroneously predicted to be 0 and the rate of increase with volume fraction is underestimated. For consistency, the same (former) values of b and ao/R are used for modeling the toughness, with Gj the only unknown parameter. Upon fitting the data (Fig. 16), the junction toughness is inferred to be Gj 5 4 J/m2 : only slightly greater than that of pure mullite (Gj 5 3 J/m2 ). (4) Implications for Crack Deflection Once calibrated, the preceding models are coupled with the analysis in Section III to assess the propensity for crack deflec￾tion, as manifest in the parameter S. The results are plotted in Fig. 17(a) for mullite–alumina particle mixtures combined with Nextelt 720 fibers. In this case, crack deflection is predicted over the entire range of compositions (0%–40% alumina) and aging times (to 1000 h). As a complementary representation, a subset of these results is plotted in Fig. 12(a) (assuming o 5 1). Here, again, the property combinations lie within the crack deflection domain. Moreover, the interface sliding stress in these systems is low, typically o10MPa (Sidebar B), re-affirming that mullite–alumina mixtures are good candidates for use in oxide composites. The critical aging time tc for crack penetration is obtained by extrapolating the predictions in Fig. 17(a) to S 5 o 5 1. It ranges from 4000 h for mixtures of 60% mullite–40% alumina to 60 000 h for pure mullite, the latter being comparable with the targeted service lives of CFCC components. With knowledge of the activation energy of the sintering mechanism, the model can be readily extended to other temperatures. A similar assessment is made of the mullite particle networks strengthened by precursor-derived alumina, again assuming the re-inforcements to be Nextelt 720 fibers. The results are plotted in Fig. 17(b). Here, S decreases with increasing alumina con￾centration, VA, and eventually falls below the critical value, o1, at VA9%. This point is expected to mark the onset of crack penetration into the fibers and a significant loss in damage tolerance. Fig. 16. Summary of measurements and model predictions of (a) Young’s modulus and (b) toughness. The material is made of a mullite particle network, strengthened by precursor-derived alumina. (Adapted from Fujita et al. 40). 3318 Journal of the American Ceramic Society—Zok Vol. 89, No. 11