November 2007 R Curves and Crack-Stability Map 3557 tion plot of the plateau fracture strengths calculated from the r crack in bending(Eq(6)). the solution of Eq.(4) defines the curves and compares them with the fracture stresses measured in transition crack length, c". a plot of the normalized transition incremental loading. In each case, the R-curve analysis accu- crack length, e/, versus the normalized toughness parameter. rately predicted the measured fracture stress. The solid line fitted Ko/ Ko, defines a map with regions of crack stability and crack through the measured fracture stresses is the following two- instability. The resulting crack-stability map has two important parameter Weibull distribution function implications. First, for a toughness ratio, Ko/Koo>0.197, there is no solution for c. This means that a surface crack always (IT) Ko>0. 197. This explains why R-curve measurements for nat extends unstably irrespective of its initial length when K ural surface cracks by mor ng their stable growth are In Eq(17), F is the cumulative probability of fracture, m is reported so infrequently and limited to a few ceramics.When the Weibull modulus, and ge is the characteristic strength. The such measurements are reported they show steeply rising R estimates of the parameters, obtained by the maximum like- curves at small crack lengths consistent with the above predic- lihood method, were m=50.7 and oe=412.7 MPa. The high tion For a toughness ratio, Ko/Kn<0.197, c"has two solutions value of the Weibull modulus is a reflection of the high ch and cs. An initial crack of size, ci< c< c, undergoes stable reproducibility of the R curves, even for cracks initiated at extension to c, before it becomes unstable. This is the regime in different types of flaws in Ce-TZP/Al2O3, and the small varia- Fig 4 where the critical stress for extension of a crack increases bility in the instability crack length(c). Figure 5 also plots the Figure 4 also indicates that for R curves satisfying the condition, fracture stresses of Ce-TZP/Al2O3 measured in fast fracture Ko/Ko<0. 197, the critical stress at ci is a local minimum. A (O 100 MPa/s)tests. These tests gave a characteristic consequence of this local minimum is an expanded plateau strength,e=456.2 MPa, a value 10.5% higher than the extending from co to c where the fracture strength is constant corresponding value assessed in the incremental loading(R and independent of the crack size. It should be noted that crack curve)tests. This difference is believed to be due to moisture induced subcritical crack growth encountered in the incremental arrest, and further stable growth to c2. All cracks, c< co, will loading tests. The crack lengths measured in the incremental undergo only an unstable extension and the fracture strength is loading tests included a subcritical growth component in again dependent on the initial crack size. Figure 4 indicates that addition to the stable growth due to the rising crack-growth co= 10 um for Ce-TZP/ALO3. Because flaws in this material, resistance. Therefore, instability crack lengths(c)) were over res, and second-phase particles were larger than this size, it estimated or the r curves were underestimated due to subcritical crack growth. Alcala and Angladanoted a similar trend with a llikely that strength behavior corresponding to this small crack Y-TZP ceramic. R curves measured with single-edge precracked ASe range was ever observed. The fact that the smallest crack size corded during R-curve measurements in this material was beam tests showed dependence on both the rate of loading and about 30 um suggests that they were likely nucleated, unstably mode of loading. The R curve measured at K= 1.0 MPa. m/ extended, and arrested during one of the steps in incremental s-I was higher than the one measured at 0.1 MPa. m/2 loading. It is recognized here that initial cracks in the range co Further, R curves measured in stepped loading and static c< i that grow unstably may overshoot the arrest loading were lower than those measured under constant stres between ci and c, sufficiently to proceed directly to the unstabl sing rates. These results suggest that R curves should ideally be branch at c >c due to kinetic effects. Such effects should be measured under loading conditions identical to those used in most prominent for initial cracks slightly larger than co, a situ strength measurements ation not encountered in Ce-TZP/AlO The agreement noted in Fig. 5 between the fracture stresses asured in the incremental loading tests(o)and the plateau V. Discussion The crack-stability d in this However, the difference in the facts e s) validates the analysi fast-fracture tests and those measured in the incremental loading a useful insight into the role of an R curve in the sta tests highlights the need to exercise caution wh akins surface cracks and their effects on fracture strengths R-curve measurements. Ideally, R curves should be measured Rcurve defined by Eq (5)and a crack-driving force for under conditions identical to those used in strength tests ally for ceramics susceptible to subcritical crack growth. Ob- sly, such measurements are difficult unless one is able to 1.00 make crack-length measurements using a high-speed camera or e R-curve Analysis a crack- measuring grid A final erest is the high Weibull modulus noted Fig. 5. It has been recognized for some time that toughened ce- ramics exhibiting R-curve behavior also exhibit a higher Weibull modulus and, therefore, higher reliabilit However. one 乐0.00 must distinguish between two cases of toughened ceramics. In one case, the toughened ceramic exhibits R-curve behavior, but the r curve does not meet the requirement, Ko/ Ko <0. 197. In this case. the fracture stress is sensitive to the initial crack size but the spread of fracture stress is not as wide as it would be for a ceramic with flat crack-growth resistance. A more desirable case of a toughened ceramic is one where Ko/K<0. 197. In this case, one can expect a range of crack lengths, co to ca, where the fracture stress is independent of the initial crack size. It is this situation that applies to Ce-TZP/AlO3. The fracture stress in 0.00 this case is determined by crack-growth resistance KR(c)and the corresponding value of c2. If R curves are highly reproduc Fracture Stress, a (MPa) ble from specimen to specimen, one should expect an invarian trength. The small variation in fracture stress seen in Fig. 5 Fig. 5. Cumulative distributions of fracture stre likely reflects small variations in the R curves and the corre- (a)assessed from analyses of R curves(.). (b)me sponding small variations in c. This material can be considere loading(O), and(c)measured in fast-fracture tests(A).tion plot of the plateau fracture strengths calculated from the R curves and compares them with the fracture stresses measured in incremental loading. In each case, the R-curve analysis accu￾rately predicted the measured fracture stress. The solid line fitted through the measured fracture stresses is the following two￾parameter Weibull distribution function: F ¼ 1 exp s sy    m (17) In Eq. (17), F is the cumulative probability of fracture, m is the Weibull modulus, and sy is the characteristic strength. The estimates of the parameters, obtained by the maximum like￾lihood method, were m 5 50.7 and sy 5 412.7 MPa. The high value of the Weibull modulus is a reflection of the high reproducibility of the R curves, even for cracks initiated at different types of flaws in Ce-TZP/Al2O3, and the small varia￾bility in the instability crack length ðc
2Þ. Figure 5 also plots the fracture stresses of Ce-TZP/Al2O3 measured in fast fracture (s_  100MPa=s) tests. These tests gave a characteristic strength, sy 5 456.2 MPa, a value 10.5% higher than the corresponding value assessed in the incremental loading (R curve) tests. This difference is believed to be due to moisture￾induced subcritical crack growth encountered in the incremental loading tests. The crack lengths measured in the incremental loading tests included a subcritical growth component in addition to the stable growth due to the rising crack-growth resistance. Therefore, instability crack lengths ðc
2Þ were over￾estimated or the R curves were underestimated due to subcritical crack growth. Alcala and Anglada12 noted a similar trend with a Y-TZP ceramic. R curves measured with single-edge precracked beam tests showed dependence on both the rate of loading and mode of loading. The R curve measured at K_ ¼ 1:0MPa m1=2 s1 was higher than the one measured at 0.1 MPa m1/2 s 1 . Further, R curves measured in stepped loading and static loading were lower than those measured under constant stres￾sing rates. These results suggest that R curves should ideally be measured under loading conditions identical to those used in strength measurements. V. Discussion The crack-stability map developed in this paper (Fig. 1) provides a useful insight into the role of an R curve in the stability of surface cracks and their effects on fracture strengths. For an R curve defined by Eq. (5) and a crack-driving force for a surface crack in bending (Eq. (6)), the solution of Eq. (4) defines the transition crack length, c
. A plot of the normalized transition crack length, c
/l, versus the normalized toughness parameter, K0/KN, defines a map with regions of crack stability and crack instability. The resulting crack-stability map has two important implications. First, for a toughness ratio, K0/KN40.197, there is no solution for c
. This means that a surface crack always extends unstably irrespective of its initial length when K0/ KN40.197. This explains why R-curve measurements for nat￾ural surface cracks by monitoring their stable growth are reported so infrequently and limited to a few ceramics. When such measurements are reported, they show steeply rising R curves at small crack lengths consistent with the above predic￾tion. For a toughness ratio, K0/KNo0.197, c
has two solutions: c
1 and c
2. An initial crack of size, c
1 < c < c
2, undergoes stable extension to c
2 before it becomes unstable. This is the regime in Fig. 4 where the critical stress for extension of a crack increases. Figure 4 also indicates that for R curves satisfying the condition, K0/KNo0.197, the critical stress at c
1 is a local minimum. A consequence of this local minimum is an expanded plateau extending from c
0 to c
2 where the fracture strength is constant and independent of the crack size. It should be noted that cracks in the range c
0 < c < c
1 will undergo unstable extension, arrest, and further stable growth to c
2. All cracks, c < c
0, will undergo only an unstable extension and the fracture strength is again dependent on the initial crack size. Figure 4 indicates that c
0 ¼ 10 mm for Ce-TZP/Al2O3. Because flaws in this material, pores, and second-phase particles were larger than this size, it is unlikely that strength behavior corresponding to this small crack size range was ever observed. The fact that the smallest crack size recorded during R-curve measurements in this material was about 30 mm suggests that they were likely nucleated, unstably extended, and arrested during one of the steps in incremental loading. It is recognized here that initial cracks in the range c
0 < c < c
1 that grow unstably may overshoot the arrest position between c
1 and c
2 sufficiently to proceed directly to the unstable branch at c > c
2 due to kinetic effects. Such effects should be most prominent for initial cracks slightly larger than c
0, a situ￾ation not encountered in Ce-TZP/Al2O3. The agreement noted in Fig. 5 between the fracture stresses measured in the incremental loading tests (J) and the plateau stresses calculated from the R curves () validates the analysis. However, the difference in the fracture stresses measured in the fast-fracture tests and those measured in the incremental loading tests highlights the need to exercise caution while making R-curve measurements. Ideally, R curves should be measured under conditions identical to those used in strength tests, espe￾cially for ceramics susceptible to subcritical crack growth. Ob￾viously, such measurements are difficult unless one is able to make crack-length measurements using a high-speed camera or a crack-measuring grid. A final point of interest is the high Weibull modulus noted in Fig. 5. It has been recognized for some time that toughened ce￾ramics exhibiting R-curve behavior also exhibit a higher Weibull modulus and, therefore, higher reliability.13–15 However, one must distinguish between two cases of toughened ceramics. In one case, the toughened ceramic exhibits R-curve behavior, but the R curve does not meet the requirement, K0/KNo0.197. In this case, the fracture stress is sensitive to the initial crack size, but the spread of fracture stress is not as wide as it would be for a ceramic with flat crack-growth resistance. A more desirable case of a toughened ceramic is one where K0/KNo0.197. In this case, one can expect a range of crack lengths, c
0 to c
2, where the fracture stress is independent of the initial crack size. It is this situation that applies to Ce-TZP/Al2O3. The fracture stress in this case is determined by crack-growth resistance KRðc
2Þ and the corresponding value of c
2. If R curves are highly reproduc￾ible from specimen to specimen, one should expect an invariant strength. The small variation in fracture stress seen in Fig. 5 likely reflects small variations in the R curves and the corre￾sponding small variations in c
2. This material can be considered truly flaw insensitive. Fig. 5. Cumulative distributions of fracture stresses of Ce-TZP/Al2O3: (a) assessed from analyses of R curves (), (b) measured in incremental loading (J), and (c) measured in fast-fracture tests (D). November 2007 R Curves and Crack-Stability Map 3557