DENTAL MATERIALS 24(2008)289-298 elements of this concept are presented schematically in Fig. 2. ximated as[11] More recent theoretical modeling places much more empha sis on shear versus dilatational stresses for both activating GRiffith+△K*(△C the transformation and leading to an increase up to fourfold Y(co+△c)05 in transformation zone height[9). Perhaps the most compre- hensive review of the current state of theoretical work is found where Acf is the stable crack extension prior to failure, co the in Hannink et al. [91 initial flaw size, and Y a flaw geometry constant. Acf is further Experimentally the size of the transformation zone is found defined as [11] to be a function of test temperature and grain size; increasing [25]. Zone size as a function of temperature(relative to M. Ac=2 as t drops toward Ms and increasing as grain size increases has been visually demonstrated by Becher and Swain for a (r)05/(coag)os Y-TZP[26]. Zone size plays a critical role in the toughness with d being the transformation zone size described above increment achieved as a result of the microstructural changes Adding to this complexity is the finding that many occurring with transformation Microcracking transformation the highest toughness ceramics(especially high Mg-PSz zones have been visualized by Lutz et al. for a 20% Al203 Y- demonstrate non-linear, non-elastic yielding prior to fail- TZP in a dramatic series of photomicrographs [31]. Zone sizes ure [30]. Deformation mechanisms include both reversible have been measured by several techniques including opti- and irreversible components, with the reversible associated cal and transmission electron microscopy, X-ray diffraction with transformation and the irreversible with microcracking and Raman spectrometry [32]. Reported zone sizes include: [30, 35]. Marshall and James [35] provided both photographic 2um to 70 um in Mg-PSZ having Kic values between 4 and (Nomarski interference)and x-ray diffraction data to convinc- ues varying from 5 to 10MPam2[32 e gly establish the presence of a reversible transformation around 200 MPa in a 9mol% Mg-PSZ. This reversibility raises an issue regarding the concept of nucleation being a R-curve behavior: description, definition key feature of the transformation in this material and implications same material was found to exhibit a non-linear stress-strain curve with irreversible strain at failure being as large as the elastic strain [30] as do many transformation-toughened An increase in the resistance to crack growth (i.e. toughness) zirconia ceramics having high Kic values(36). Both the stress- during crack extension has been termed R-curve behavior. R- induced transformation along with microcracking prior to curve behavior was originally noted for large-graine failure account for the nonlinear yielding Two major implica having thermal expansion anisotropy and the increasing tions arise for highly toughened ceramics: (1)actual stresses resistance to crack extension with crack length was mainly at failure can be much lower(up to 40%)than calculated by attributed to mechanisms active in the crack wake that linear elastic analysis and(2) these materials are remarkably shielded the crack tip from the applied stress intensity 33, 34. damage tolerant, e.g. strengths of Vickers-indented speci In transforming ceramics additional toughening mechanisms mens (up to 1000N) can equal polished-surface specimens contribute to R-curve behavior, with the total available being (301 subdivided into three basic groups [31]: (1)crack deflection and crack branching: (2) contact shielding by wedging and bridg ing involving broken-out grains or rough crack-wake surfaces 6. Strength versus toughness and (3)stress-induced zone shielding involving transforma tion, microcracking and residual stress fields For linearly elastic brittle materials the highest strength and The KI given in Eq(1)as the Griffith failure criteria is no highest toughness occur in the same material; this is not longer"critical. This applied k is suffcient for initial flaw the case for transforming ceramics. As cracks grow from an growth but insufficient for catastrophic crack propagation initial size, transformation events create an incrementally ow growing with increasing length increasing toughness, AK, with toughness increments scal (Fig. 2).One new description of the critical stress intensity for ing proportional to h, where h is the transformation zone fracture, Kf, has been given by size in the nomenclature of Swain and Rose 36]. This scal- ing yields R-curve behavior discussed above, until h is fully K= GRiffith+△K*(△c) (10) developed and the toughness reaches a plateau. However, incremental crack extension, Ac, scales directly with h while mental crack extension Ac AK(Ac)(essentially the R-curve)is in scaling properties, vh versus h, yields a strength maxi- a function of the depth and location of the initial crack, speci- mum with increasing steady-state toughness 36]. In practical men dimensions, test conditions and methods; hence there is terms the ceramic begins to weaken(crack growth o h)even no unique R-curve for a material [31]. This implies that Kr is not though the toughness is still increasing(AK o h). Accor specifying a unique material failure criterion, or is a"material ing to Swain and Rose [36] strengths beyond this maximum property"in a sense analogous with KIc become bounded by the critical stress at the crack tip nec Keeping with the formulation of Heuer the fracture essary for the t-m transformation. This analytical concept trength of transformation-toughened ceramics can be appro- was illustrated using measured strength and toughness datadental materials 24 (2008) 289–298 295 elements of this concept are presented schematically in Fig. 2. More recent theoretical modeling places much more empha￾sis on shear versus dilatational stresses for both activating the transformation and leading to an increase up to fourfold in transformation zone height [9]. Perhaps the most compre￾hensive review of the current state of theoretical work is found in Hannink et al. [9]. Experimentally the size of the transformation zone is found to be a function of test temperature and grain size; increasing as T drops toward Ms and increasing as grain size increases [25]. Zone size as a function of temperature (relative to Ms) has been visually demonstrated by Becher and Swain for a Y-TZP [26]. Zone size plays a critical role in the toughness increment achieved as a result of the microstructural changes occurring with transformation. Microcracking transformation zones have been visualized by Lutz et al. for a 20% Al2O3 Y￾TZP in a dramatic series of photomicrographs [31]. Zone sizes have been measured by several techniques including opti￾cal and transmission electron microscopy, X-ray diffraction and Raman spectrometry [32]. Reported zone sizes include: 0.2m to 70m in Mg-PSZ having KIC values between 4 and 14 MPa m1/2 and 0.8–4.6m for Y-TZP (2–4mol%) with KIC val￾ues varying from 5 to 10 MPam1/2 [32]. 5. R-curve behavior: description, definition and implications An increase in the resistance to crack growth (i.e. toughness) during crack extension has been termed R-curve behavior. R￾curve behavior was originally noted for large-grained ceramics having thermal expansion anisotropy and the increasing resistance to crack extension with crack length was mainly attributed to mechanisms active in the crack wake that shielded the crack tip from the applied stress intensity [33,34]. In transforming ceramics additional toughening mechanisms contribute to R-curve behavior, with the total available being subdivided into three basic groups [31]: (1) crack deflection and crack branching; (2) contact shielding by wedging and bridg￾ing involving broken-out grains or rough crack-wake surfaces and (3) stress-induced zone shielding involving transforma￾tion, microcracking and residual stress fields. The KI given in Eq. (1) as the Griffith failure criteria is no longer “critical”. This applied K is sufficient for initial flaw growth but insufficient for catastrophic crack propagation since crack resistance is now growing with increasing length (Fig. 2). One new description of the critical stress intensity for fracture, Kf, has been given by Heuer as [11]: Kf = KGriffith + K ∗ (c) (10) where K* is the toughening increment as a function of incre￾mental crack extension c. K*(c) (essentially the R-curve) is a function of the depth and location of the initial crack, speci￾men dimensions, test conditions and methods; hence there is no unique R-curve for a material [31]. This implies that Kf is not specifying a unique material failure criterion, or is a “material property” in a sense analogous with KIC. Keeping with the formulation of Heuer the fracture strength of transformation-toughened ceramics can be appro￾ximated as [11]: f = [KGriffith + K ∗ (cf)] Y(cO + cf) 0.5 (11) where cf is the stable crack extension prior to failure, cO the initial flaw size, and Y a flaw geometry constant. cf is further defined as [11]: cf = 2 () 0.5 (cOd) 0.5 (12) with d being the transformation zone size described above. Adding to this complexity is the finding that many of the highest toughness ceramics (especially high Mg-PSZ) demonstrate non-linear, non-elastic yielding prior to fail￾ure [30]. Deformation mechanisms include both reversible and irreversible components, with the reversible associated with transformation and the irreversible with microcracking [30,35]. Marshall and James [35] provided both photographic (Nomarski interference) and X-ray diffraction data to convinc￾ingly establish the presence of a reversible transformation at around 200 MPa in a 9mol% Mg-PSZ. This reversibility raises an issue regarding the concept of nucleation being a key feature of the transformation in this material [35]. This same material was found to exhibit a non-linear stress-strain curve with irreversible strain at failure being as large as the elastic strain [30] as do many transformation-toughened zirconia ceramics having high KIC values [36]. Both the stress￾induced transformation along with microcracking prior to failure account for the nonlinear yielding. Two major implica￾tions arise for highly toughened ceramics: (1) actual stresses at failure can be much lower (up to 40%) than calculated by linear elastic analysis and (2) these materials are remarkably damage tolerant, e.g. strengths of Vickers-indented speci￾mens (up to 1000 N) can equal polished-surface specimens [30]. 6. Strength versus toughness For linearly elastic brittle materials the highest strength and highest toughness occur in the same material; this is not the case for transforming ceramics. As cracks grow from an initial size, transformation events create an incrementally increasing toughness, K, with toughness increments scal￾ing proportional to √h, where h is the transformation zone size in the nomenclature of Swain and Rose [36]. This scal￾ing yields R-curve behavior discussed above, until h is fully developed and the toughness reaches a plateau. However, incremental crack extension, c, scales directly with h while R-curve toughness becomes developed [36]. This difference in scaling properties, √h versus h, yields a strength maxi￾mum with increasing steady-state toughness [36]. In practical terms the ceramic begins to weaken (crack growth ∝ h) even though the toughness is still increasing (K ∝ √h). Accord￾ing to Swain and Rose [36] strengths beyond this maximum become bounded by the critical stress at the crack tip nec￾essary for the t→m transformation. This analytical concept was illustrated using measured strength and toughness data