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 3554R 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 transfor￾mation-toughened Ce-TZP/Al2O3. It is shown that the fracture 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), leading to a flaw-insensitive ceramic. I. Introduction TRANSFORMATION-TOUGHENED ceramics, polycrystalline ce￾ramics 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 measure￾ments, 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 com￾mon 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 ex￾tend 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 (2) On the other hand, the crack extension is unstable when the following condition is met: 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 re￾sistance, KR(c). Theoretical models of toughening, for example, transformation toughening,6 or toughening due to crack-bridg￾ing 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 mea￾sured on the surface, KN is the steady-state fracture toughness exhibited by the ceramic at large crack sizes, K0 is the crack￾initiation 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 whisker￾reinforced alumina,8 self-reinforced silicon nitride,8 and trans￾formation-toughened Ce-TZP/Al2O3. 5 Second, the empirical parameters describing the R curve closely parallel those appear￾ing 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