DENTAL MATERIALS 24(2008)289-298 where AFcH is the chemical free energy change, AUe the strain increase in Ms temperatures with increasing grain size [26] energy change, AUs the surface energy change, and AU the Local residual stresses will also scale with grain size, provid- interaction term due to stress application. The surface free ing an additional or altemative explanation for the Ms tracking energy change is considered negligible and the chemical free with grain size [26] energy term can be stated as a function of transformation Transformation involves the development of a transforma entropy change Asm and temperature [26] tion zone first associated with the crack tip and later becoming a crack wake feature The size of this zone and features of the △FcH=△stm(To-T) (4) zone microstructure(grain size and microcracking, in particu- lar) control toughening. Following the analysis of Evans 2, the where T is the test temperature and To the transformation toughening increment due to transformation is a function of temperature for an unconstrained t-phase particle. Tetrago- the transformation zone width h, the local elastic modulus of nal particles embedded within a matrix will be stabilized by transformed material e, the volume fraction of transformable Constrained particles, that are metastable, will transform at is Poisson's ratio e ational strain e as given below(where v both matrix constraint as well as the strain energy term AUe. phase f, and the dilat a temperature Ms<To (with Ms being the martensitic start temperature discussed earlier). Dopants such as Y203 act to √h decrease both the Ms and To temperatures. This decrease in △Kc=0.2 22a-可 Ms below To creates an additional free energy change term essentially capturing matrix and dopant stabilization, that is The simplest, most commonly"understood"toughening equal to As -m To-Ms)[26]. Thus, without an applied stress, mechanism concept involves crack-tip shielding(from the the free energy change for transformation can be written applied stress)by the compressive dilatational stress asso- terms of the test(m)and Ms temperatures as ciated with transformation. In fact, it is now thought that purely dilatant transformation within an included angle AFo=ASmI(To-T)-(To-Ms)]=Ast-m(Ms-T)(5) 120 ahead of the crack tip(hydrostatic component in Fig. 2) leads to a decrease in toughness[9. Other toughening factors reflecting that AFo=0 as T=Ms, initiating transformation have been advanced related to the microstructure of the trans- At test(or service)temperatures above Ms, an applied stress formed material that act both locally near the crack tip and Un can reduce AFo to zero and trigger thet+m transformation in the crack wake. For example, the crack tip is embedded in a microcracked material having an elastic modulus that dif- fers from the bulk ceramic, creating an additional crack tip (6) shielding term. As cracks grow to lengths of approximately h-5 to h.10 the toughness reaches a plateau [16] determined Returning to more familiar and useful concepts involving by crack wake and crack tip toughening mechanisms (29,30) tensile stress and strain is achieved by defining the interactie to be elucidated below in the discussion of R-curve behavior term UI in terms of applied stress aa and the transformation dilatational strain g[26] 3 Formed transformation zone △U1=aE Now the critical applied stress for transformation o becomes[26] 3. Asymptote =0.22.. a-V I+ISVD=ED (8)△K Two practical implications come from this analys △a born-out by experimental evidence. First the stress required for transformation increases with temperature above Ms and 2. Partially developed zone second at a given test or service temperature(above Ms)this critical applied stress will increase as Ms is decreased [26]. The kinetics of transformation are governed by nucleation, with the probability of nucleation enhanced by local resid ual stresses as well as applied stress[ 29. Sources of local hydrostatic component residual stresses have been described above(e.g 8-phase/t phase lattice mismatch in PSZ and t-phase a anisotropy in TZP). Recall from above that for PSZ the stress required for △a/h the t- m transformation decreases from 470 to 70 MPa with Fig. 2- Schematic views of transformation zone and 8-phase formation during aging [18]. Since the probability of a toughness increment(AKe)development with crack potent nuclei existing within a grain should scale with grain extension, recreated with permission per description of volume, nucleation considerations also partially explain the Evans [2] and Evans and Heuer294 dental materials 24 (2008) 289–298 where FCH is the chemical free energy change, Ue the strain energy change, US the surface energy change, and UI the interaction term due to stress application. The surface free energy change is considered negligible and the chemical free energy term can be stated as a function of transformation entropy change St→m and temperature [26]: FCH = St→m(TO − T) (4) where T is the test temperature and TO the transformation temperature for an unconstrained t-phase particle. Tetrago￾nal particles embedded within a matrix will be stabilized by both matrix constraint as well as the strain energy term Ue. Constrained particles, that are metastable, will transform at a temperature Ms < TO (with Ms being the martensitic start temperature discussed earlier). Dopants such as Y2O3 act to decrease both the Ms and TO temperatures. This decrease in Ms below TO creates an additional free energy change term essentially capturing matrix and dopant stabilization, that is equal to St→m (TO − Ms) [26]. Thus, without an applied stress, the free energy change for transformation can be written in terms of the test (T) and Ms temperatures as: FO = St→m[(TO − T) − (TO − Ms)] = St→m(Ms − T) (5) reflecting that FO = 0 as T = Ms, initiating transformation. At test (or service) temperatures above Ms, an applied stress UI can reduce FO to zero and trigger the t→m transformation [26]: FO = St→m(Ms − T) − UI (6) Returning to more familiar and useful concepts involving tensile stress and strain is achieved by defining the interaction term UI in terms of applied stress a and the transformation dilatational strain εT [26]: UI = aεT (7) Now the critical applied stress for transformation T c becomes [26]: a = T c = St→m(Ms − T) εT (8) Two practical implications come from this analysis, both born-out by experimental evidence. First the stress required for transformation increases with temperature above Ms and second at a given test or service temperature (above Ms) this critical applied stress will increase as Ms is decreased [26]. The kinetics of transformation are governed by nucleation, with the probability of nucleation enhanced by local resid￾ual stresses as well as applied stress [29]. Sources of local residual stresses have been described above (e.g. ı-phase/t￾phase lattice mismatch in PSZ and t-phase anisoptropy in TZP). Recall from above that for PSZ the stress required for the t→m transformation decreases from 470 to 70 MPa with ı-phase formation during aging [18]. Since the probability of a potent nuclei existing within a grain should scale with grain volume, nucleation considerations also partially explain the increase in Ms temperatures with increasing grain size [26]. Local residual stresses will also scale with grain size, provid￾ing an additional or alternative explanation for the Ms tracking with grain size [26]. Transformation involves the development of a transforma￾tion zone first associated with the crack tip and later becoming a crack wake feature. The size of this zone and features of the zone microstructure (grain size and microcracking, in particu￾lar) control toughening. Following the analysis of Evans [2], the toughening increment due to transformation is a function of the transformation zone width h, the local elastic modulus of transformed material E, the volume fraction of transformable phase f, and the dilatational strain εT ij , as given below (where  is Poisson’s ratio) [2]: KC = 0.22EεT ijf √ h (1 − ) (9) The simplest, most commonly “understood” toughening mechanism concept involves crack-tip shielding (from the applied stress) by the compressive dilatational stress asso￾ciated with transformation. In fact, it is now thought that purely dilatant transformation within an included angle of 120◦ ahead of the crack tip (hydrostatic component in Fig. 2) leads to a decrease in toughness [9]. Other toughening factors have been advanced related to the microstructure of the trans￾formed material that act both locally near the crack tip and in the crack wake. For example, the crack tip is embedded in a microcracked material having an elastic modulus that dif￾fers from the bulk ceramic, creating an additional crack tip shielding term. As cracks grow to lengths of approximately h·5 to h·10 the toughness reaches a plateau [16] determined by crack wake and crack tip toughening mechanisms [29,30] to be elucidated below in the discussion of R-curve behavior; Fig. 2 – Schematic views of transformation zone and toughness increment (Kc) development with crack extension, recreated with permission per description of Evans [2] and Evans and Heuer [5].