1o991 C 2007 The American Ceramic Society urna R Curves and Crack-Stability Map: Application to Ce-TZP/Al2O3 aur nd Raymond A. Cutler* Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112-0560 The concept of a crack-stability map is developed by considering unstable. The crack extension is stable if the following the interaction between the crack-driving force and the risi rack-growth resistance of a toughened ceramic. The map plots normalized transition crack length as function of the ratio of the dka(c dKRe) ck-initiation fracture toughness and the plateau toughness to delineate regimes of stable and unstable crack growth. The plot is used to analyze R curves and fracture stresses of a transfor On the other hand the crack extension is unstable when the mation-toughened Ce-TZP/Al2O3. It is shown that the fracture following condition is met stress and the small scatter measured for this ceramic are con- sistent with its R-curve behavior, which enables stable growth of surface cracks from flaws(pores and second-phase particles ), dKR(c) leading to a flaw-insensitive ceramic. The transition from stable to unstable crack extension or rom unstable to stable extension will occur at a crack length. c, that simultaneously satisfies the following equations RANSFORMATION-TOUGHENED ceramics, polycrystalline ce- ramics with coarse or elongated grains, and whisker-and Ka(e")=KR(c) fiber-reinforced ceramics exhibit rising crack-growth resistance or R curves. Typically, R curves are represented as plots of stress intensity versus crack length. R curves can be measured dK dKp using fracture-mechanics test specimens with large cracks, bend pecimens with surface cracks produced by indentation, or bend The prediction of the fracture strength of a ceramic that inclusions, etc.)on the surface. Of these three types of measure- exhibits an R-curve behavior is based on the solution of eq (4) ments. the one based on natural surface cracks is the most useful This is done most conveniently using an analytical function that and pertinent for predicting the fracture strengths and the describes the crack-length dependence of the crack-growth re- sistance, KR(). Theoretical models of toughening, for example, reported by marshall and Swain.3 for Mg-PSZ, Steinbrech and transformation toughening, or toughening due to crack-bridg- Schmenkel for coarse-grained alumina. and Ramachandran ing ligaments, provide closed-form analytical solutions only for et al. for Ce-TZP/Al O These studies have revealed two com the steady-state or peak fracture toughness corresponding mon characteristics associated with r curves for surface crack steady-state transformation or bridging zones. The rising part the r curve, essential for strength prediction, is typically given as the crack-growth resistance with crack growth as compared with a numerical result. This is inconvenient for analysis of strengths the corresponding measurements on large fracture-mechanics and strength bilit test specimens. These are the characteristics that enable one to The purpose of this paper is to examine the implications of lake such measurements during a bend test Rcurves on crack stability and strengths of toughened ceramics In ceramics that exhibit R-curve behavior. cracks ex- in general and that of a transformation-toughened Ce-TZP/ d when the applied crack-driving force is equal to AlO3 in particular. For this purpose, the following empirical rack-growth resistance. In terms of stress-intensity factors, this equation was chosen to fit the r curves for surface cracks criterion for crack extension is given by the following KR(c)=Ks-(Koo-Ko)exp( Ka(c)=KR(c) where c is the semi-axis of a semi-elliptical surface crack mea- K a(c) is the applied stress intensity for a crack of length ured on the surface, Ko is the steady-state fracture toughness KR(c)is the crack-growth resistance. While K(c) is a exhibited by the ceramic at large crack sizes, Ko is the crack- on of the applied load, crack size, shape and specimen/ gness.andλ ing geometry, kr(c) depends on the pertinent-toughening arameter that determines the increase in fracture toughnes mechanism and the development of the process zone during with crack growth. Equation (5) was chosen for two reasons. crack growth. The crack extension defined by Eq. (I)can be First. it has been shown to fit r curves measured for surface cracks on a number of toughened ceramics including whisker formation-toughened Ce-TZP/AlO3. Second, the empirical parameters describing the R curve closely parallel those appear- ing in theoretical models of toughening . The paper first Manuscript No. 23020. Received April 2, 2007; approved June 22, 2007. examines the influence of the R curve on crack stability by This paper is based on research supported by the U.S. Department of Energy under developing a crack-stability map. The methodology is then Member. American Ceramic Socie extended to predict the strength and strength variability of Author to whom correspondence should be addressed. e-mail: d shetty(a utah. edu transforma e-TZP/ALO3 3554
R Curves and Crack-Stability Map: Application to Ce-TZP/Al2O3 Sarbjit Kaur, Dinesh K. Shetty,* ,w and Raymond A. Cutler* Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112-0560 The concept of a crack-stability map is developed by considering the interaction between the crack-driving force and the rising crack-growth resistance of a toughened ceramic. The map plots normalized transition crack length as function of the ratio of the crack-initiation fracture toughness and the plateau toughness to delineate regimes of stable and unstable crack growth. The plot is used to analyze R curves and fracture stresses of a transformation-toughened Ce-TZP/Al2O3. It is shown that the fracture stress and the small scatter measured for this ceramic are consistent with its R-curve behavior, which enables stable growth of surface cracks from flaws (pores and second-phase particles), leading to a flaw-insensitive ceramic. I. Introduction TRANSFORMATION-TOUGHENED ceramics, polycrystalline ceramics with coarse or elongated grains, and whisker- and fiber-reinforced ceramics exhibit rising crack-growth resistance or R curves.1 Typically, R curves are represented as plots of stress intensity versus crack length. R curves can be measured using fracture-mechanics test specimens with large cracks, bend specimens with surface cracks produced by indentation, or bend specimens with surface cracks initiated at natural flaws (pores, inclusions, etc.) on the surface. Of these three types of measurements, the one based on natural surface cracks is the most useful and pertinent for predicting the fracture strengths and the strength variability of ceramics. Such measurements have been reported by Marshall and Swain2,3 for Mg-PSZ, Steinbrech and Schmenkel4 for coarse-grained alumina, and Ramachandran et al. 5 for Ce-TZP/Al2O3. These studies have revealed two common characteristics associated with R curves for surface cracks: (a) a low crack-initiation toughness and (b) a steep increase in the crack-growth resistance with crack growth as compared with the corresponding measurements on large fracture-mechanics test specimens. These are the characteristics that enable one to make such measurements during a bend test. In ceramics that exhibit R-curve behavior, cracks extend when the applied crack-driving force is equal to the crack-growth resistance. In terms of stress-intensity factors, this criterion for crack extension is given by the following equation: KaðcÞ ¼ KRðcÞ (1) where Ka(c) is the applied stress intensity for a crack of length, c, and KR(c) is the crack-growth resistance. While Ka(c) is a function of the applied load, crack size, shape, and specimen/ loading geometry, KR(c) depends on the pertinent-toughening mechanism and the development of the process zone during crack growth. The crack extension defined by Eq. (1) can be stable or unstable. The crack extension is stable if the following condition is satisfied: dKaðcÞ dc dKRðcÞ dc (3) The transition from stable to unstable crack extension or from unstable to stable extension will occur at a crack length, c , that simultaneously satisfies the following equations: Kaðc Þ ¼ KRðc Þ dKa dc ðc Þ ¼ dKR dc ðc Þ (4) The prediction of the fracture strength of a ceramic that exhibits an R-curve behavior is based on the solution of Eq. (4). This is done most conveniently using an analytical function that describes the crack-length dependence of the crack-growth resistance, KR(c). Theoretical models of toughening, for example, transformation toughening,6 or toughening due to crack-bridging ligaments,7 provide closed-form analytical solutions only for the steady-state or peak fracture toughness corresponding to steady-state transformation or bridging zones. The rising part of the R curve, essential for strength prediction, is typically given as a numerical result. This is inconvenient for analysis of strengths and strength variability. The purpose of this paper is to examine the implications of R curves on crack stability and strengths of toughened ceramics in general and that of a transformation-toughened Ce-TZP/ Al2O3 in particular. For this purpose, the following empirical equation was chosen to fit the R curves for surface cracks: KRðcÞ ¼ K1 ðK1 K0Þ exp c l (5) where c is the semi-axis of a semi-elliptical surface crack measured on the surface, KN is the steady-state fracture toughness exhibited by the ceramic at large crack sizes, K0 is the crackinitiation fracture toughness, and l is a crack size scaling parameter that determines the increase in fracture toughness with crack growth. Equation (5) was chosen for two reasons. First, it has been shown to fit R curves measured for surface cracks on a number of toughened ceramics including whiskerreinforced alumina,8 self-reinforced silicon nitride,8 and transformation-toughened Ce-TZP/Al2O3. 5 Second, the empirical parameters describing the R curve closely parallel those appearing in theoretical models of toughening.6,7 The paper first examines the influence of the R curve on crack stability by developing a crack-stability map. The methodology is then extended to predict the strength and strength variability of a transformation-toughened Ce-TZP/Al2O3. D. Marshall—contributing editor This paper is based on research supported by the U.S. Department of Energy under Contract No. DE-FG02-87ER45312 at the University of Utah. *Member, American Ceramic Society. w Author to whom correspondence should be addressed. e-mail: d.shetty@utah.edu Manuscript No. 23020. Received April 2, 2007; approved June 22, 2007. Journal J. Am. Ceram. Soc., 90 [11] 3554–3558 (2007) DOI: 10.1111/j.1551-2916.2007.01940.x r 2007 The American Ceramic Society 3554
November 2007 R Curves and Crack-Stability Map 3555 IL. Theoretical Analysis Y(,=)=0.2139+ 1.0770 (13) The applied crack-driving force for a surface crack, expressed in rms of the stress intensity due to far-field loading, is given by the following equation 1.0770 Ka(c)=OvC (6 (14) where Y, a stress-intensity coefficient, is a function of the surface crack shape, relative crack depth, and the mode of loading, ar Newman9.10 have calculated values of y for surface cracks of The solid line in Fig. I plots the solutions of Eq.(12),i.e semi-elliptical shape( semi-major axis, c, and semi-minor axis, a) for different crack penetrations(a/w, where w is the beam thick and a specific value of the parameter, (/w)=0.0568, pertinent ness)in bending by finite-element analysis(FEA). Further, ob- to the Ce-TZP/AL,O, ceramic discussed in the following section servations of crack fronts on fracture surfaces indicate that the he resulting plot maps two regimes: a regime of stable crack aspect ratio of the crack decreases linearly with stable growth rowth and a regime of unstable crack growth with two impor through the beam thickness tant implications. For a toughness ratio, Ko/Koo>0.197, crack growth is always unstable for any initial crack length. For ratios c. The dotted-dashed vertical line in Fig. I corre- Raju and Newman's FEA results, used in conjunction with sponds to the transformation-toughened Ce-TZP/AlO Eq. (7), lead to the following equation for y in terms of a single ceramic with Ko=8.90 MPa. m, Ko=1. 11 MPa - m2.203 variable ca Ko/Koo=0.125. For these toughness parameters, the crack-sta- bility map defines(ci/2)=0.184 and(c5/)=1.103 Y(c)=0.2139+1.0770 In the limiting case, w>>7, the stress-intensity coefficient, Y, (8) is independent of the crack length and Eq(12)reduces to the Tollowing fo The substitution of Eqs.(6)and(5)into Eq (4) leads to the foll exp Y(c)ovc=Koc -(Koo -Ko)exp The dashed line in F Similarly, the slopes of the crack-driving force and the crack a plot of Eq(15)and rep- esents the crack-stabilit growth resistance functions can b ted to obtain the fol crack-growth regime is or the limiting case relative to the more generalized presented by Eq (12). In the following section, the implications of crack stability on fracture 「Y()dY (K-K0) are Ce-TZP/ALO Equation(9)divided by Eq(10)obtains the following equa Ill. Surface Cracks and R Curves for Ce-TZP/AlO3 tion for c the critical crack size for transition from sta ble to unstable growth, or vice versa Ramachandran et al. measured R curves for Ce-TZP/AlO3 by measuring the lengths of surface cracks initiated at pores or dy (11) r Ce. TZPlAlo Inst:hetRick G:newth Equation (I1)can be transformed into the following conve- nient form Instable Crack Grwth 05 In Eq(12), the stress-intensity coefficient and its deriva ms of the normalized parameters, (c/) and (/w) resistance
II. Theoretical Analysis The applied crack-driving force for a surface crack, expressed in terms of the stress intensity due to far-field loading, is given by the following equation: KaðcÞ ¼ Ys ffiffi c p (6) where Y, a stress-intensity coefficient, is a function of the surface crack shape, relative crack depth, and the mode of loading, and s is the maximum tensile stress in bending of a beam. Raju and Newman9,10 have calculated values of Y for surface cracks of semi-elliptical shape (semi-major axis, c, and semi-minor axis, a) for different crack penetrations (a/w, where w is the beam thickness) in bending by finite-element analysis (FEA). Further, observations of crack fronts on fracture surfaces indicate that the aspect ratio of the crack decreases linearly with stable growth through the beam thickness11: a c ¼ 1 a w (7) Raju and Newman’s FEA results, used in conjunction with Eq. (7), lead to the following equation for Y in terms of a single variable c: YðcÞ ¼ 0:2139 þ 1:0770 w w þ c (8) The substitution of Eqs. (6) and (5) into Eq. (4) leads to the following equation: Yðc Þs ffiffiffiffi c p ¼ K1 ðK1 K0Þ exp c l (9) Similarly, the slopes of the crack-driving force and the crackgrowth resistance functions can be equated to obtain the following equation: s ffiffiffiffi c p YðcÞ 2c þ dY dc ðc Þ ¼ ðK1 K0Þ l exp c l (10) Equation (9) divided by Eq. (10) obtains the following equation for c , the critical crack size for transition from stable to unstable growth, or vice versa: YðcÞ YðcÞ 2c þ dY dc ðc Þ ¼ K1l ðK1 K0Þ exp c l 1 (11) Equation (11) can be transformed into the following convenient form: K0 K1 ¼ 1 exp c l 2c l Y c l ; l w Y c l ; l w þ 2c l l dY dc c l ; l w 8 >>>: 9 >>= >>; þ 1 0 BB@ 1 CCA 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 (12) In Eq. (12), the stress-intensity coefficient and its derivative have been written in terms of the normalized parameters, (c /l) and (l/w): Y c l ; l w ¼ 0:2139 þ 1:0770 1 þ c l l w (13) dY dc c l ; l w ¼ 1:0770 w 1 þ c l l w 2 (14) The solid line in Fig. 1 plots the solutions of Eq. (12), i.e., values of (c /l) for different values of the parameter, K0/KN, and a specific value of the parameter, (l/w) 5 0.0568, pertinent to the Ce-TZP/Al2O3 ceramic discussed in the following section. The resulting plot maps two regimes: a regime of stable crack growth and a regime of unstable crack growth with two important implications. For a toughness ratio, K0/KN40.197, crack growth is always unstable for any initial crack length. For ratios o0.197, crack growth is unstable for initial crack lengths, c c 2. The dotted–dashed vertical line in Fig. 1 corresponds to the transformation-toughened Ce-TZP/Al2O3 ceramic with KN 5 8.90 MPa m1/2, K0 5 1.11 MPa m1/2, and K0/KN 5 0.125. For these toughness parameters, the crack-stability map defines ðc 1=lÞ ¼ 0:184 and ðc 2=lÞ ¼ 1:103. In the limiting case, w l, the stress-intensity coefficient, Y, is independent of the crack length and Eq. (12) reduces to the following form: K0 K1 ¼ 1 exp c l 2 c l þ 1 2 6 6 4 3 7 7 5 (15) The dashed line in Fig. 1 shows a plot of Eq. (15) and represents the crack-stability map for the limiting case. The stable crack-growth regime is compressed for the limiting case relative to the more generalized solution represented by Eq. (12). In the following section, the implications of crack stability on fracture strength are examined for the transformation-toughened Ce-TZP/Al2O3. III. Surface Cracks and R Curves for Ce-TZP/Al2O3 Ramachandran et al. 5 measured R curves for Ce-TZP/Al2O3 by measuring the lengths of surface cracks initiated at pores or Fig. 1. Crack-stability map showing the ranges of crack length for stable and unstable growth in a ceramic exhibiting rising crack-growth resistance. November 2007 R Curves and Crack-Stability Map 3555
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 cc=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 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 co172.3 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/Al20
second-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 processing.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 particles. There was no detectable difference in the R curves measured 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 calculated 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 parameter, 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 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 2, the fracture strength again decreases monotonically 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 distribuFig. 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
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/Knc 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 accurately predicted the measured fracture stress. The solid line fitted through the measured fracture stresses is the following twoparameter 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 likelihood 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 variability 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 moistureinduced 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 overestimated 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 stressing 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 natural 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 prediction. For a toughness ratio, K0/KNo0.197, c has two solutions: c 1 and c 2. An initial crack of size, c 1 c 2 due to kinetic effects. Such effects should be most prominent for initial cracks slightly larger than c 0, a situation 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, especially for ceramics susceptible to subcritical crack growth. Obviously, 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 ceramics 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 reproducible 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 corresponding 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
3558 Journal of the American Ceramic Society--Kaur et al VoL 90. No lI VI. Conclusions D. B. Marshall and M. V Swain, Crack Resistance Curves in Magnesia- Partially-Stabilized Zirconia, J. Am. Ceram. Soc. 71. 399-404(1988). 1. The interaction between crack-driving force and crack ce Curves of surface growth resistance can be depicted on a crack-stability map that plots normalized transition crack lengths for stable and unstable CR田 handran.ly chao anddk s8C or and flaw crack growths for a toughened ceramic as a function of the ratio (1993) ensitivity of Ce-TZP/Al O3 Composite, J. Am. Ceran. Soc., 76. 961-9 of the crack-initiation toughness and the plateau tough eR M. MeMeeking and A. G. Evans, "Mechanisms of Transformation Tough- 2. For the R curve described by the empirical exponential ening in Brittle Materials. " J. Am. Ceran. Soc., 65, 242-6(1982) function, stable crack growth is encountered when Ko/ko< 0.197 R-Curve) Behavior of Toughened Alumina and Silicon Nitride, " J. Am. Ceram. of crack lengths for toughened ceramics that satisfy the condi tion Ko/K <0. 197 when they exhibit the exponential R-curve 41 Small Elliptical Surface Cracks in Finite-Thickness Plates, Eng. Fract. Mech., 11, behavior 4. All ceramics with an R-curve behavior exhibit an in- IL. S. Raju and J. C. Newman Jr,"An Empirical Stress-Intensity Factor creased Weibull modulus as compared with ceramics with flat qmi K aw thar S and M. Kur an. "Fatigue crack rowth (ri a surface A truly flaw-insensitive strength can be expected only for (1978) Flaw, " Adv. Res. Strength Fract. Mater, Int. Conf. Fract, 4th, 1977. 2. 1361-72 those ceramics that exhibit stable crack growth due to rising J. Alcala and M. Anglada, "Indentation Precracking of Y-TZP: Implications ack-growth resistance to r-Curves and su Mat.Sc.Eg,A245,267-76 Kendall, N. MeN. Alford. S. R. Tan, and J D Toughness on Weibull Modulus of Ceramic Bend Stren 中 R. F. Cook and D. R. Clarke. "Fracture Stability. R-Curves and Strength D. Munz. "What Can We Learn from R-Curve Measurements. J. Am. Ceram. Variability. "Acta Metall. 36, 555-62(1988). ID. K Shetty and J S. Wang,""Crack Stability and Strength Distribution of -D. B. Marshall. ""Strength Characteristics of Transformation-Toughened Ceramics That Exhibit Rising Crack-Growth-Resistance (R-Curve) Behavior Zirconia, J. Am. Ceran. Soc., 69, 173-80(1986 J.Am.Cera.Soc,72.11s862(1989)
VI. Conclusions 1. The interaction between crack-driving force and crackgrowth resistance can be depicted on a crack-stability map that plots normalized transition crack lengths for stable and unstable crack growths for a toughened ceramic as a function of the ratio of the crack-initiation toughness and the plateau toughness. 2. For the R curve described by the empirical exponential function, stable crack growth is encountered when K0/KNo 0.197. 3. A flaw-insensitive fracture stress is expected over a range of crack lengths for toughened ceramics that satisfy the condition K0/KNo0.197 when they exhibit the exponential R-curve behavior. 4. All ceramics with an R-curve behavior exhibit an increased Weibull modulus as compared with ceramics with flat crack-growth resistance. 5. A truly flaw-insensitive strength can be expected only for those ceramics that exhibit stable crack growth due to rising crack-growth resistance. References 1 D. Munz, ‘‘What Can We Learn from R-Curve Measurements,’’ J. Am. Ceram. Soc., 90, 1–15 (2007). 2 D. B. Marshall, ‘‘Strength Characteristics of Transformation-Toughened Zirconia,’’ J. Am. Ceram. Soc., 69, 173–80 (1986). 3 D. B. Marshall and M. V. Swain, ‘‘Crack Resistance Curves in Magnesia– Partially-Stabilized Zirconia,’’ J. Am. Ceram. Soc., 71, 399–404 (1988). 4 R. W. Steinbrech and O. Schmenkel, ‘‘Crack-Resistance Curves of Surface Cracks in Alumina,’’ J. Am. Ceram. Soc., 71, C-271–3 (1988). 5 N. Ramachandran, L. Y. Chao, and D. K. Shetty, ‘‘R-Curve Behavior and Flaw Insensitivity of Ce-TZP/Al2O3 Composite,’’ J. Am. Ceram. Soc., 76, 961–9 (1993). 6 R. M. McMeeking and A. G. Evans, ‘‘Mechanisms of Transformation Toughening in Brittle Materials,’’ J. Am. Ceram. Soc., 65, 242–6 (1982). 7 A. G. Evans and R. M. McMeeking, ‘‘On the Toughening of Ceramics by Strong Reinforcements,’’ Acta Metall., 34, 2435–41 (1986). 8 N. Ramachandran and D. K. Shetty, ‘‘Rising Crack-Growth-Resistance (R-Curve) Behavior of Toughened Alumina and Silicon Nitride,’’ J. Am. Ceram. Soc., 74, 2634–41 (1991). 9 I. S. Raju and J. C. Newman Jr., ‘‘Stress-Intensity Factors for a Wide Range of Small Elliptical Surface Cracks in Finite-Thickness Plates,’’ Eng. Fract. Mech., 11, 817–29 (1979). 10I. S. Raju and J. C. Newman Jr., ‘‘An Empirical Stress-Intensity Factor Equation for the Surface Crack,’’ Eng. Fract. Mech., 15, 185–92 (1981). 11M. Kawahara and M. Kurihara, ‘‘Fatigue Crack Growth from a Surface Flaw,’’ Adv. Res. Strength Fract. Mater., Int. Conf. Fract., 4th, 1977, 2, 1361–72 (1978). 12J. Alcala and M. Anglada, ‘‘Indentation Precracking of Y-TZP: Implications to R-Curves and Strength,’’ Mat. Sci. Eng., A245, 267–76 (1998). 13K. Kendall, N. McN. Alford, S. R. Tan, and J. D. Birchall, ‘‘Influence of Toughness on Weibull Modulus of Ceramic Bend Strength,’’ J. Mater. Res., 1, 120–3 (1986). 14R. F. Cook and D. R. Clarke, ‘‘Fracture Stability, R-Curves and Strength Variability,’’ Acta. Metall., 36, 555–62 (1988). 15D. K. Shetty and J. S. Wang, ‘‘Crack Stability and Strength Distribution of Ceramics That Exhibit Rising Crack-Growth-Resistance (R-Curve) Behavior,’’ J. Am. Ceram. Soc., 72, 1158–62 (1989). & 3558 Journal of the American Ceramic Society—Kaur et al. Vol. 90, No. 11