J.L Jones et al 1 Acta Materialia 55(2007)5538-5548 5541 split Pearson VIl profile shape function [19]and an optimi- that described above for the strain data. The standard devi- zation routine within MATLAB(Ver. 7.0.4, The Math- ation of the calculated foo2 values for 21 different positions Works, Inc ) An example of measured intensities and the away from the crack tip is 0.02 mrd. Using this descriptor, and 200 reflections are shown in Fig. 4. With this given sample direction, foo2= l mrd means there are equal approach, the intensities and positions of all measured number of c-axes as a- and b-axes in the given sample direc- aks describe preferred orientation and lattice strains in tion, and oo2=3 mrd means there are no a-or b-axes in directions within the plane of the specimen the given sample direction. In ferroelastic ceramic materi- For extraction of lattice strains, an unstressed image far als, the measured results typically fall in the range from the crack was first analyzed, from which the specimen- 0.5<f002 1.5 mrd [26] to-detector distance. detector orientation calibrant-to-sam ple distance, beam center position, and unstressed PZT lat- 3. Results tice parameters(ao, Co)were refined using the known CeO lattice constant(ac=5.411 A)and X-ray wavelength On The stress intensity factor was first slowly increased to subsequent images from stressed regions of the sample, only KI=0.71 MPa m", nearly equal to the initiation tough the calibrant-to-sample distance and the beam center posi- ness of this material [21]. The interchange in intensity tion were refined. The hk/ lattice strains were determined between the pseudo-cubic 002 peaks is shown in Fig. 4 from the distortion of the respective Debye-Scherrer rings for a position near the crack tip Using Eq (6), preference relative to the unstressed positions using Eq. (4). The for the 002 pole (ooz) is calculated at this stress intensity unstressed PZT lattice parameters(ao, Co) are used to calcu- factor and is presented in Fig. 5( filled contours)as a func- late dhk using tion of position relative to the crack tip(X, n and direc tion within the plane of the specimen (n). For example, h+k 1 (5 the n=0o map describes the preference for 002 domain orientations perpendicular to the crack face or parallel to Values of chk/ were averaged for symmetrically equivalent the Y-direction. Tensile stresses near the crack tip increase the preference for these domain orientations, the magni (antipodal) n angles within each image. In other words, tude of which decreases with increasing distance from the for the same X, y position, strains at n angles differing crack tip. The spatial distributions of the 002 orientation noe(n), the intal position(x, n) and in-plane azimuthal preference changes with increasing and ellipsoidal regions 80° were averaged (e.g.n=45°andn=225) angle(n), the integrated intensities of the pseudo-cubic 002 of highest intensity (mrd)are generally parallel to n.This peaks with the background subtracted(I002, I2oo)were uti- in agreement with the spatial stress distributions presented in Fig. 2 zed to calculate the degree of preference for the 002 pole At the same stress intensity factor of K1=0.71 MPa/ using a multiple of a random distribution(mrd)[26] the lattice strains of all measured peaks(E101, 6110, 6111, E002, 8200, 8112, and E211) were extracted using Eqs. (4)and(5) f/mrd=3 +2·(l20/ (6) Distributions of hk/ lattice strains that contain both an a and c lattice parameter component (i.e. E101, 6111, 8112, where /hEi is the integrated intensity of the hkl peak and E211) are similar in shape and are a strong function of from the pattern obtained far from the crack(5 mm in this angle(n) and distance from the crack tip. Fig. 6 (filled con- experiment). Values of foo2 were averaged for symmetri- tours)shows the spatial distribution and angular dependence cally equivalent n angles within each image, equivalent to of &11l. In contrast, lattice strains of hkl peaks containing 2000 854 204.254.304.354.40445 Fig. 4. Azimuthally integrated diffracted intensities(+) from near the crack tip(x=0.8 mm, Y=1.0 mm)during loading with a stress intensity factor of K=0.71 MPa m"at (a)n=0 and(b)n=90%. A profile shape function(-)based on two split-Pearson VII functions to model each 002 and 200 peal (--)is seen to fit the diffraction data well.split Pearson VII profile shape function [19] and an optimi￾zation routine within MATLAB (Ver. 7.0.4, The Math￾Works, Inc.). An example of measured intensities and the calculated component peaks of the tetragonal PZT 0 0 2 and 2 0 0 reflections are shown in Fig. 4. With this approach, the intensities and positions of all measured peaks describe preferred orientation and lattice strains in all directions within the plane of the specimen. For extraction of lattice strains, an unstressed image far from the crack was first analyzed, from which the specimen￾to-detector distance, detector orientation, calibrant-to-sam￾ple distance, beam center position, and unstressed PZT lat￾tice parameters (ao, co) were refined using the known CeO2 lattice constant (ac = 5.411 A˚ ) and X-ray wavelength. On subsequent images from stressed regions of the sample, only the calibrant-to-sample distance and the beam center posi￾tion were refined. The {hkl} lattice strains were determined from the distortion of the respective Debye–Scherrer rings relative to the unstressed positions using Eq. (4). The unstressed PZT lattice parameters (ao, co) are used to calcu￾late d hkl using d hkl ¼ h2 þ k2 a2 o þ l 2 c2 o  1=2 ð5Þ Values of ehkl were averaged for symmetrically equivalent (antipodal) g angles within each image. In other words, for the same X, Y position, strains at g angles differing by 180 were averaged (e.g. g = 45 and g = 225). For each spatial position (X, Y) and in-plane azimuthal angle (g), the integrated intensities of the pseudo-cubic 0 0 2 peaks with the background subtracted (I002,I200) were uti￾lized to calculate the degree of preference for the 0 0 2 pole using a multiple of a random distribution (mrd) [26]: f002½mrd ¼ 3  I 002=I unpoled 002 I 002=I unpoled 002 þ 2  ðI 200=I unpoled 200 Þ ð6Þ where I unpoled hkl is the integrated intensity of the hkl peak from the pattern obtained far from the crack (5 mm in this experiment). Values of f002 were averaged for symmetri￾cally equivalent g angles within each image, equivalent to that described above for the strain data. The standard devi￾ation of the calculated f002 values for 21 different positions away from the crack tip is 0.02 mrd. Using this descriptor, f002 = 0 mrd means that there are no c-axes oriented in the given sample direction, f002 = 1 mrd means there are equal number of c-axes as a- and b-axes in the given sample direc￾tion, and f002 = 3 mrd means there are no a- or b-axes in the given sample direction. In ferroelastic ceramic materi￾als, the measured results typically fall in the range 0.5 < f002 < 1.5 mrd [26]. 3. Results The stress intensity factor was first slowly increased to KI = 0.71 MPa m1/2, nearly equal to the initiation tough￾ness of this material [21]. The interchange in intensity between the pseudo-cubic 0 0 2 peaks is shown in Fig. 4 for a position near the crack tip. Using Eq. (6), preference for the 0 0 2 pole (f002) is calculated at this stress intensity factor and is presented in Fig. 5 (filled contours) as a func￾tion of position relative to the crack tip (X, Y) and direc￾tion within the plane of the specimen (g). For example, the g = 0 map describes the preference for 0 0 2 domain orientations perpendicular to the crack face or parallel to the Y-direction. Tensile stresses near the crack tip increase the preference for these domain orientations, the magni￾tude of which decreases with increasing distance from the crack tip. The spatial distributions of the 0 0 2 orientation preference changes with increasing g and ellipsoidal regions of highest intensity (mrd) are generally parallel to g. This is in agreement with the spatial stress distributions presented in Fig. 2. At the same stress intensity factor of KI = 0.71 MPa m1/2, the lattice strains of all measured peaks (e101,e110,e111,e002, e200, e112, and e211) were extracted using Eqs. (4) and (5). Distributions of hkl lattice strains that contain both an a and c lattice parameter component (i.e. e101, e111, e112, and e211) are similar in shape and are a strong function of angle (g) and distance from the crack tip. Fig. 6 (filled con￾tours) shows the spatial distribution and angular dependence of e111. In contrast, lattice strains of hkl peaks containing 4.15 4.20 4.25 4.30 4.35 4.40 4.45 0 500 1000 1500 2000 2500 (002) (200) Intensity [counts] 2θ [degrees] (b) η=90o 4.15 4.20 4.25 4.30 4.35 4.40 4.45 0 500 1000 1500 2000 2500 (200) Intensity [counts] 2θ [degrees] (a) η=0o (002) Fig. 4. Azimuthally integrated diffracted intensities (+) from near the crack tip (X = 0.8 mm, Y = 1.0 mm) during loading with a stress intensity factor of KI = 0.71 MPa m1/2 at (a) g = 0 and (b) g = 90. A profile shape function (—) based on two split-Pearson VII functions to model each 0 0 2 and 2 0 0 peak (- - -) is seen to fit the diffraction data well. J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548 5541