3556 Journal of the American Ceramic Society--Kaur et al crack-driving force plots, Ka(c), at these stresses( relation to the R curve. At ci and ci, both th force and the crack resistance and their derivati to c are equal as required by the conditions of Eq (4) V. Fracture Strengths of Ce-TZP/AlzO3 The critical stresses for extension of surface cracks in Ce-TZP/ AlO3 were calculated using the following equation Fig. 2. A surface crack initiated at a pore in Ce-TZP/AlO3 with attendant transformation zones surrounding the crack tips. o.()=AR() Y(e)ve (16) In Eq(16), KR(c is the R-curve function defined by eq (5) and Y(c)is the stress-intensity coefficient defined in Eq.( 8).A second-phase particles while incrementally loading beams in log-log plot of the critical stress, o c), versus c for Ce-TZP/ ace cr AlO3, calculated using the fitted R curve in Fig 3, is shown as a that was deliberately introduced into the ceramic during pro- dashed line in Fig 4. The critical stress decreases monotonically essing. It is interesting to note that the surface crack did not for crack lengths up to i= 28.7 um. This is the initial unstable develop a transformation zone in the early stage of crack crack-growth regime For ci=28.7 um c<c5=172.3 um, growth. Despite the presence of the pores, a significant fraction the critical stress increases monotonically. This is the stable of the surface cracks(0.5)initiated at second-phase particles crack-growth regime. For c>c=172.3 um, the critical stress ( CeMnAlyOjg) as extensions of the cleavage cracks in the p again decreases monotonically. This is the unstable crack ticles. There was no detectable difference in the r curves mea- growth regime at large crack lengths sured for cracks initiated at the two types of flaws. Further, The solid line in Fig. 4 represents the fracture strengths of measurements on six specimens showed little variation. Ce-TZP/Al2O3 for different initial lengths of surface cracks, c Figure 3 shows an example of an R curve measured for a There are three regimes of interest For crack lengths in the range surface crack in a Ce-TZP/Al2O3 ceramic initiated at a pore. c< co, fracture strength, or(c), is identical to the critical stress, Stable crack growth was measured in the range 50 um<c<175 o c). For co c<c, the fracture strength is constant at 392 m. The crack-growth resistance values (O in Fig 3)were cal MPa For c> ca, the fracture strength again decreases monoton- culated using Eq(6)with measured values of c, the maximum ically and is equal to the crack-extension stress. Note in Fig 4 that bending stress applied in each stage of incremental loading, and co is defined such that o(co)=o(c?). In the crack length range, the stress-intensity coefficient, Y(c), calculated using Eq( 8). The co<c<c, there is always stable growth of an initial crack to a solid line in the figure represents a"best fit "of Eq. (5)obtained ength, c,, at which point there is unstable fracture. This results in a with a curve-fitting program. The R-curve parameters, Ko=1. 11 constant fracture strength, o(co)=o(c)=392 MPa over a MPa-m 2, K=8.90 MPa. m, and 2=156.2 um,were wide range of crack lengths from co= 10.0 um to:= 172.3 um. obtained from the best fit" by minimizing the variance. The predicted plateau in the fracture strength, 392 MPa in Figure I indicates that for the normalized toughness para- Fig. 4, can be compared with the maximum bending stress neter, Ko/Ko=0.125 and 2=156.2 Hm, corresponding to pplied in incremental loading during r-curve measurements Ce-TZP/AlO3. crack growth should be unstable for initial crack before unstable fracture. For the r curve shown in Fig 3. the lengths, c< cf=28.7 um, it should be stable in the crack final applied stress was 394 MPa, a value close to the predicted length range, G< c<c=172.3 um, and it should be un- plateau. Analyses of R curves and strength predictions, similar stable for c>172.3 um. The range of stable to those shown in Figs. 3 and 4, were carried out for six differen neasured during incremental loading of Ce-TZH sets of measurements of R curves made by Ramachandran et al.s 175 um, is consistent with the prediction of Fig. I For each R curve, the parameters Ko, Ko, and A; the transition he critical stresses corresponding to stable crack extension at crack lengths, co, ci, and c; and the plateau fracture strengtH and unstable crack extension at c were calculated using eq Fig. 4)were calculated. Figure 5 shows a cumulative distribu (4)to be 353 and 392 MPa, respectively. Figure 3 shows the 1000 R-Cure Data 6 2 Fig 3. R curve(data p d solid line) and crack Fig 4. Critical stress for crack ex n(oc) and fracture stress (dashed lines) plots illustrating crack growth at the point of tangency (or(e)for surface cracks in Ce-TZP/Al20second-phase particles while incrementally loading beams in bending. Figure 2 shows a surface crack initiated at a pore that was deliberately introduced into the ceramic during pro￾cessing.5 It is interesting to note that the surface crack did not develop a transformation zone in the early stage of crack growth. Despite the presence of the pores, a significant fraction of the surface cracks (B0.5) initiated at second-phase particles (CeMnAl11O19) as extensions of the cleavage cracks in the par￾ticles. There was no detectable difference in the R curves mea￾sured for cracks initiated at the two types of flaws. Further, measurements on six specimens showed little variation.5 Figure 3 shows an example of an R curve measured for a surface crack in a Ce-TZP/Al2O3 ceramic initiated at a pore. Stable crack growth was measured in the range 50 mmoco175 mm. The crack-growth resistance values (J in Fig. 3) were cal￾culated using Eq. (6) with measured values of c, the maximum bending stress applied in each stage of incremental loading, and the stress-intensity coefficient, Y(c), calculated using Eq. (8). The solid line in the figure represents a ‘‘best fit’’ of Eq. (5) obtained with a curve-fitting program. The R-curve parameters, K0 5 1.11 MPa m1/2, KN 5 8.90 MPa m1/2, and l 5 156.2 mm, were obtained from the ‘‘best fit’’ by minimizing the variance. Figure 1 indicates that for the normalized toughness para￾meter, K0/KN 5 0.125 and l 5 156.2 mm, corresponding to Ce-TZP/Al2O3, crack growth should be unstable for initial crack lengths, c < c
1 ¼ 28:7 mm, it should be stable in the crack length range, c
1 < c < c
2 ¼ 172:3 mm, and it should be un￾stable for c4172.3 mm. The range of stable crack lengths measured during incremental loading of Ce-TZP/Al2O3, 50– 175 mm, is consistent with the prediction of Fig. 1. The critical stresses corresponding to stable crack extension at c
1and unstable crack extension at c
2 were calculated using Eq. (4) to be 353 and 392 MPa, respectively. Figure 3 shows the crack-driving force plots, Ka(c), at these stresses (dashed lines) in relation to the R curve. At c
1 and c
2, both the crack-driving force and the crack resistance and their derivatives with respect to c are equal as required by the conditions of Eq. (4). IV. Fracture Strengths of Ce-TZP/Al2O3 The critical stresses for extension of surface cracks in Ce-TZP/ Al2O3 were calculated using the following equation: scðcÞ ¼ KRðcÞ YðcÞ ffiffi c p (16) In Eq. (16), KR(c) is the R-curve function defined by Eq. (5) and Y(c) is the stress-intensity coefficient defined in Eq. (8). A log–log plot of the critical stress, sc(c), versus c for Ce-TZP/ Al2O3, calculated using the fitted R curve in Fig. 3, is shown as a dashed line in Fig. 4. The critical stress decreases monotonically for crack lengths up to c
1 ¼ 28:7 mm. This is the initial unstable crack-growth regime. For c
1 ¼ 28:7 mm < c < c
2 ¼ 172:3 mm, the critical stress increases monotonically. This is the stable crack-growth regime. For c > c
2 ¼ 172:3 mm, the critical stress again decreases monotonically. This is the unstable crack growth regime at large crack lengths. The solid line in Fig. 4 represents the fracture strengths of Ce-TZP/Al2O3 for different initial lengths of surface cracks, c. There are three regimes of interest. For crack lengths in the range c < c
0, fracture strength, sf (c), is identical to the critical stress, sc(c). For c
0 < c < c
2, the fracture strength is constant at 392 MPa. For c > c
2, the fracture strength again decreases monoton￾ically and is equal to the crack-extension stress. Note in Fig. 4 that c
0 is defined such that scðc
0Þ ¼ scðc
2Þ. In the crack length range, c
0 < c < c
2, there is always stable growth of an initial crack to a length, c
2, at which point there is unstable fracture. This results in a constant fracture strength, sfðc
0Þ ¼ sfðc
2Þ ¼ 392 MPa over a wide range of crack lengths from c
0 ¼ 10:0 mm to c
2 ¼ 172:3 mm. The predicted plateau in the fracture strength, 392 MPa in Fig. 4, can be compared with the maximum bending stress applied in incremental loading during R-curve measurements before unstable fracture. For the R curve shown in Fig. 3, the final applied stress was 394 MPa, a value close to the predicted plateau. Analyses of R curves and strength predictions, similar to those shown in Figs. 3 and 4, were carried out for six different sets of measurements of R curves made by Ramachandran et al. 5 For each R curve, the parameters KN, K0, and l; the transition crack lengths, c
0, c
1, and c
2; and the plateau fracture strength (Fig. 4) were calculated. Figure 5 shows a cumulative distribu￾Fig. 2. A surface crack initiated at a pore in Ce-TZP/Al2O3 with attendant transformation zones surrounding the crack tips. Fig. 3. R curve (data points and solid line) and crack-driving force (dashed lines) plots illustrating crack growth at the point of tangency. Fig. 4. Critical stress for crack extension (sc(c)) and fracture stress (sf (c)) for surface cracks in Ce-TZP/Al2O3. 3556 Journal of the American Ceramic Society—Kaur et al. Vol. 90, No. 11