J.L Jones et al 1 Acta Materialia 55(2007)5538-5548 5547 propagation. Moreover, the in-plane directionality has implications for modeling of ferroelastic toughening behav ior. Whereas most early models have related the switching zone to toughening behavior, the increased in-plane esolution of the experimental data presented here increases the dimensionality of the analysis and demon- strates that invariant measures of the process zone are insufficient. Micromechanical models that determine switching of every possible orientation (i.e. n)and at every spatial point relative to the crack position(X, y have the intrinsic information necessary to report a directional dependence. Such theoretical results can directly compare the calculated and measured degrees of switching. Other 0.0020406081.01.21.4161.8 models may incorporate some directional dependence though the degree of switching is not directly modeled. Fig. 10.( Color) Preference for 002 domain orientations in the out-of- mployed here, where domain switching is modeled indi- soaes is twntsit stres aindtenesit aston ei s and 6 Pack face he stime on are illustrated as a bolded line at y=l0. The contour the development of measurement techniques to character- lines steps of 0.025 mrd. The out-of-plane domain orientations ize the in-plane domain switching and strain directionality, were ed from in-plane domain orientations using Eq (8) improvement of theoretical models in this dimension will ignificantly advance the understanding of the dependence of these results to macroscopic properties such as enhanced reported by Hackemann and Pfeiffer [7]. A direct compar- Initiation toughness due to frontal zone switching and son is not appropriate given the different loading condi toughness enhancement with crack extension The measured in-plane switching can also be used to cal- were obtained after crack propagation with the stress inten- culate the complementary out-of-plane switching because ity factor reduced by 50%, whereas the results in Fig. 10 domain switching in orthogonal orientations is conserva- are prior to crack propagation and represent a bulk aver- domain orientations parallel to one direction correspond We iterate that one significant advancement in the work to decreases in the other variant orientations. This is best presented here compared to techniques used in the past is understood by considering the schematic in Fig 9. The that the method of X-ray transmission allows measurement of scattering vectors oriented within the plane of the sam n=450 schematics correspond with a third domain orien- ple. For example, in the earlier X-ray reflection work of pled and only scattering vectors oriented normal to the other words, a tetragonal(100) is directed out of the page face are measured, yielding information similar to that preference for the 002 domain orientation perpendicular presented in Fig. 10. However, the complex directional to the specimen plane(parallel to the Z-direction)is there- dependence of the stresses within the plane of the sample fore estimated (Fig. 2)leads to unique domain orientation distributions within this plane(Fig. 5). Nor is this directional 3-0g2(n)-02(7+90°) dence measurable the recently developed LCD (8) method by Kounga Jiwa et al. (2) because it is the change in total electrostatic potential that is measured, which is Eq.(8)averages over all in-plane directions within the also directionally invariant. The in-plane behaviors of var range 00</<75 because the 90<n<165 data are iously oriented domains measured in this work are more incorporated through the foo2(n+ 90) term and the suitable for comparison and validation of constitutive frac- 800<1<345data were included in the earlier antipodal ture models [4] because they provide more information averaging used to generate the data representing the range than is available from techniques yielding directionally 00<n<165 invariant measures. The importance of directionality exhib Using Eq ( 8)and the results from Fig. 5, the preference ited here in ferroelastic materials is also significant in the for the 002 domains oriented out of the sample surface, variant selection and phase transformation behavior of f oo2(Z), is shown in Fig. 10. The foo?(Z) values at the other materials including zirconia-containing ceramics.In crack tip in Fig. 10 are less than 1.00 mrd and therefore such structural ceramics, this approach shows promise describe a preference for c-axes to switch into the specimen for characterizing both the degree of ferroelastic switching plane during loading. This Z-direction process zone size and phase variant selection of the monoclinic and tetrago- correlates within an order of magnitude to that previously nal phases [33].propagation. Moreover, the in-plane directionality has implications for modeling of ferroelastic toughening behav￾ior. Whereas most early models have related the switching zone size to toughening behavior, the increased in-plane resolution of the experimental data presented here increases the dimensionality of the analysis and demon￾strates that invariant measures of the process zone are insufficient. Micromechanical models that determine switching of every possible orientation (i.e. g) and at every spatial point relative to the crack position (X, Y) have the intrinsic information necessary to report a directional dependence. Such theoretical results can directly compare the calculated and measured degrees of switching. Other models may incorporate some directional dependence though the degree of switching is not directly modeled. One example of this is a finite element model, such as that employed here, where domain switching is modeled indi￾rectly through a nonlinear stress–strain behavior. Given the development of measurement techniques to character￾ize the in-plane domain switching and strain directionality, improvement of theoretical models in this dimension will significantly advance the understanding of the dependence of these results to macroscopic properties such as enhanced initiation toughness due to frontal zone switching and toughness enhancement with crack extension. The measured in-plane switching can also be used to cal￾culate the complementary out-of-plane switching because domain switching in orthogonal orientations is conserva￾tive. That is, increases in the volume fraction of 0 0 2 domain orientations parallel to one direction correspond to decreases in the other variant orientations. This is best understood by considering the schematic in Fig. 9. The two in-plane domain orientations for both the g = 0 and g = 45 schematics correspond with a third domain orien￾tation in which the 2 0 0 planes are indistinguishable. In other words, a tetragonal Æ100æ is directed out of the page and parallel for both g = 0 and g = 45. The degree of preference for the 0 0 2 domain orientation perpendicular to the specimen plane (parallel to the Z-direction) is there￾fore estimated as f002ðZÞ ¼ 1 6 X 75 g¼0;15;... ½3 f002ðgÞ f002ðg þ 90 Þ: ð8Þ Eq. (8) averages over all in-plane directions within the range 0 < g < 75 because the 90 < g < 165 data are incorporated through the f002(g + 90) term and the 180 < g < 345 data were included in the earlier antipodal averaging used to generate the data representing the range 0 < g<165. Using Eq. (8) and the results from Fig. 5, the preference for the 0 0 2 domains oriented out of the sample surface, f002(Z), is shown in Fig. 10. The f002 (Z) values at the crack tip in Fig. 10 are less than 1.00 mrd and therefore describe a preference for c-axes to switch into the specimen plane during loading. This Z-direction process zone size correlates within an order of magnitude to that previously reported by Hackemann and Pfeiffer [7]. A direct compar￾ison is not appropriate given the different loading condi￾tions. Specifically, the results of Hackemann and Pfeiffer were obtained after crack propagation with the stress inten￾sity factor reduced by 50%, whereas the results in Fig. 10 are prior to crack propagation and represent a bulk aver￾age through the thickness of the sample. We iterate that one significant advancement in the work presented here compared to techniques used in the past is that the method of X-ray transmission allows measurement of scattering vectors oriented within the plane of the sam￾ple. For example, in the earlier X-ray reflection work of Hackemann and Pfeiffer [7], only the sample surface is sam￾pled and only scattering vectors oriented normal to the sur￾face are measured, yielding information similar to that presented in Fig. 10. However, the complex directional dependence of the stresses within the plane of the sample (Fig. 2) leads to unique domain orientation distributions within this plane (Fig. 5). Nor is this directional depen￾dence measurable using the recently developed LCD method by Kounga Njiwa et al. [2], because it is the change in total electrostatic potential that is measured, which is also directionally invariant. The in-plane behaviors of var￾iously oriented domains measured in this work are more suitable for comparison and validation of constitutive frac￾ture models [4] because they provide more information than is available from techniques yielding directionally invariant measures. The importance of directionality exhib￾ited here in ferroelastic materials is also significant in the variant selection and phase transformation behavior of other materials including zirconia-containing ceramics. In such structural ceramics, this approach shows promise for characterizing both the degree of ferroelastic switching and phase variant selection of the monoclinic and tetrago￾nal phases [33]. 1.00 1.05 0.95 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Y [mm] Z X [mm] Fig. 10. (Color) Preference for 0 0 2 domain orientations in the out-of￾plane direction (f002(Z)) as a function of spatial position (X, Y) during loading with a stress intensity factor of KI = 0.71 MPa m1/2 (the same stress intensity factor and region as in Figs. 5 and 6). Crack face position and orientation are illustrated as a bolded line at Y = 1.0. The contour lines vary in steps of 0.025 mrd. The out-of-plane domain orientations were calculated from in-plane domain orientations using Eq. (8). J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548 5547