《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_brittleness12

Availableonlineatwww.sciencedirectcom ScienceDirect 98 Acta materialia ELSEVIER Acta Materialia 55(2007)5538-5548 Crack tip process zone domain switching in a soft lead zirconate titanate ceramic Jacob L. Jones, S. Maziar Motahari, Mesut Varlioglu, Ulrich Lienert Joel v bernier. mark hoffman ersan ustundas Department of Materials Science and Engineering, P.O. Box 116400, Unicersity of Florida, Gainesville, FL 32611-6400, USA Department of Materials Science and Engineering, 2220 Hooter Hall, lowa State University, Ames, IA 50011-2300, USA c Adranced Photon Source, Argonne National Laboratory, Building 401, 9700 S. Cass Avenue, Argonne, IL 60439, USA d School of Materials Science and Engineering, The University of New South Wales, NSW 2052, Australia Received ll January 2007: received in revised form 8 June 2007: accepted 8 June 2007 Available online 30 july 2007 Abstract Non-180 domain switching leads to fracture toughness enhancement in ferroelastic materials Using a high-energy synchrotron X-ray source and a two-dimensional detector in transmission geometry, non-180 domain switching and crystallographic lattice strains were measured in situ around a crack tip in a soft tetragonal lead zirconate titanate ceramic. At Ki=0.71 MPa m"and below the initiation toughness, the process zone size, spatial distribution of preferred domain orientations, and lattice strains near the crack tip are a strong function of direction within the plane of the compact tension specimen. Deviatoric stresses and strains calculated using a finite element model and projected to the same directions me sured in diffraction correlate with the measure ed spatial distributions and direction o 2007 Acta Materialia Inc. Published by Elsevier reserved Keywords: Ferroelectricity: Fracture; Ceramics: Toughness; X-ray diffraction(XRD) 1. Introduction depressions that result from the strain associated with fer roelastic switching perpendicular to the sample surface The inherent brittleness of ferroelectric ceramics is a [8, 9]. In electrically poled lead zirconate titanate(PZT) structural liability that leads to crack initiation at defects ceramics, Lupascu and co-workers showed that the change and stress concentrations such as pores and electrode and in potential energy can be mapped spatially surrounding substrate interfaces. However, non-180 domain switching the crack tip using a liquid-crystal display [2, 6]. Employing in the frontal zone and crack wake lead to a rising R-curve X-ray diffraction, Glazounov et al. [5] measured the inten behavior, or an increase in toughness with crack extension sity ratio change of certain diffraction peaks as a function [I-10]. Using various techniques, recent work has elicited of distance from the crack face. Hackemann and Pfeiffer the region in which domain switching occurs or the [7] have also demonstrated that domain orientations per- switching zone"in ferroelastic materials, the size of which pendicular to the sample surface can be measured around is related to the toughness er nhancement In BaTiO3 ceram- the crack tip using a small beam size from laboratory X ics, Nomarski interference contrast and atomic force rays in reflection geometry microscopy have been employed to measure local surface However, non-180 domain switching within the plane of the sample has yet to be reported, and it is this plane that or Tel: +1352 846 3788: fax: +1352 846 3355. exhibits a complex stress distribution contributing to the E-mail address: jones(@mse ufl.edu(J. L. Jones) toughness enhancement. The directionally dependent 1359-6454/30.00@ 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved dor: 10.1016/j. actamat. 2007.06.012

Crack tip process zone domain switching in a soft lead zirconate titanate ceramic Jacob L. Jones a,*, S. Maziar Motahari b , Mesut Varlioglu b , Ulrich Lienert c , Joel V. Bernier c , Mark Hoffman d , Ersan U¨ stu¨ndag b a Department of Materials Science and Engineering, P.O. Box 116400, University of Florida, Gainesville, FL 32611-6400, USA b Department of Materials Science and Engineering, 2220 Hoover Hall, Iowa State University, Ames, IA 50011-2300, USA c Advanced Photon Source, Argonne National Laboratory, Building 401, 9700 S. Cass Avenue, Argonne, IL 60439, USA d School of Materials Science and Engineering, The University of New South Wales, NSW 2052, Australia Received 11 January 2007; received in revised form 8 June 2007; accepted 8 June 2007 Available online 30 July 2007 Abstract Non-180 domain switching leads to fracture toughness enhancement in ferroelastic materials. Using a high-energy synchrotron X-ray source and a two-dimensional detector in transmission geometry, non-180 domain switching and crystallographic lattice strains were measured in situ around a crack tip in a soft tetragonal lead zirconate titanate ceramic. At KI = 0.71 MPa m1/2 and below the initiation toughness, the process zone size, spatial distribution of preferred domain orientations, and lattice strains near the crack tip are a strong function of direction within the plane of the compact tension specimen. Deviatoric stresses and strains calculated using a finite element model and projected to the same directions measured in diffraction correlate with the measured spatial distributions and directional dependencies. Some preferred orientations remain in the crack wake after the crack has propagated; within the crack wake, the tetrag￾onal 0 0 1 axis has a preferred orientation both perpendicular to the crack face and toward the crack front. 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ferroelectricity; Fracture; Ceramics; Toughness; X-ray diffraction (XRD) 1. Introduction The inherent brittleness of ferroelectric ceramics is a structural liability that leads to crack initiation at defects and stress concentrations such as pores and electrode and substrate interfaces. However, non-180 domain switching in the frontal zone and crack wake lead to a rising R-curve behavior, or an increase in toughness with crack extension [1–10]. Using various techniques, recent work has elicited the region in which domain switching occurs or the ‘‘switching zone’’ in ferroelastic materials, the size of which is related to the toughness enhancement. In BaTiO3 ceram￾ics, Nomarski interference contrast and atomic force microscopy have been employed to measure local surface depressions that result from the strain associated with fer￾roelastic switching perpendicular to the sample surface [8,9]. In electrically poled lead zirconate titanate (PZT) ceramics, Lupascu and co-workers showed that the change in potential energy can be mapped spatially surrounding the crack tip using a liquid-crystal display [2,6]. Employing X-ray diffraction, Glazounov et al. [5] measured the inten￾sity ratio change of certain diffraction peaks as a function of distance from the crack face. Hackemann and Pfeiffer [7] have also demonstrated that domain orientations per￾pendicular to the sample surface can be measured around the crack tip using a small beam size from laboratory X￾rays in reflection geometry. However, non-180 domain switching within the plane of the sample has yet to be reported, and it is this plane that exhibits a complex stress distribution contributing to the toughness enhancement. The directionally dependent 1359-6454/$30.00 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.06.012 * Corresponding author. Tel.: +1 352 846 3788; fax: +1 352 846 3355. E-mail address: jjones@mse.ufl.edu (J.L. Jones). www.elsevier.com/locate/actamat Acta Materialia 55 (2007) 5538–5548

J.L Jones et al 1 Acta Materialia 55(2007)5538-5548 5539 domain switching distributions around the crack tip are It is understood that ferroelastic domain switching is discussed here using the well-known Mode I elastic stress caused by the deviatoric stresses [13-15]. The projected profiles In plane-stress Mode I loading, elastic stress distri- deviatoric stress, s=(n), is given by butions as a function of radial coordinates(r, e)are given S-(n)=0(0)-syou/ as [ll] where Si is the Kronecker delta. Using Eqs.(1)3), s-(n)is 分C0s51-8 KI calculated as a function of position(X, y surrounding a (1) crack tip for an applied stress intensity factor of Ki= 0.71MPa m/ 2. Spatial distributions of s-(n)for n=00, OYy cOS 30°,60°,and90° are shown in fig.2. Given that these devi- V2r atoric stresses induce domain switching and that their spa- K 0.03 In- cos tial distributions change with angle in the plane of the sample, it is hypothesized that the domain switching behav- aZ=TyZ= Ix=0 ior in these directions of the sample will be highly corre lated with these distributions Fig. I illustrates the crack orientation in both Cartesian(X, Because the techniques used in prior crack tip switching D and radial coordinate systems zone measurements have no in-plane directional resolution, The present work is motivated by a desire to better we present a new approach by which the directional depen- understand the directionality of domain switching near a dence described above is resolved. High-energy X-rays can mechanically loaded crack tip. To this end, the behavior penetrate through several millimeters of most materials and of local crystallographic orientations are examined at an therefore provide a powerful X-ray transmission technique array of points spanning the switching zone. A useful quan- by which to characterize in-plane behavior. When position tity for illustrating the in-plane directional variations as a sensitive area detectors are employed, high-energy synchro- function of position relative to the crack tip(as well as pro- tron X-rays enable the capture of the entire ring of scatter- viding clear comparisons between experimentally measured ing vectors associated with each Debye-Scherrer cone for a and modeled data) is the stress projection single sample position [16-18 a-(n=n(n)n(n)oy Because the Bragg angles for most lower-order reflec- (2) tions in these materials are typically 5o or less for high- where n(u) is a unit vector with an in-plane direction with energy X-rays (1<0.25 A), the cones of scattering vectors respect to the sample coordinate system shown in Fig. 1. lie nearly parallel to the x-Y plane of the sample.There- The scalar a-(n)represents a normal stress, i.e. the compo- fore, all scattering vectors for each reflecting plane hk/ nent of the traction vector acting on a surface with normal are treated as lying in the X-y plane of the sample in this n in the direction n. As a consequence of the measuring analysis. In this geometry, the normal lattice strains can convention used(see Fig. I inset), o-n)is equivalent to then be extracted from the experimental data using tem by a clockwise rotation of n degrees about Z[12]. Note +(n) dh- nkL the a ry component transformed to a new coordinate sys d (4) where dhk/ and diki are the measured mean crystallographic described by a()=a=(n+1800) lattice spacing for strained and unstrained crystallographic to the (hk/ po measured strains may, in turn, be related analytically to an average projection of the underlying strain tensors in the same crystallographic domain in a manner analogous to In this work, we combine high-energy synchrotron X- ray difraction with a two-dimensional detector to map both the preferred orientation induced by ferroelastic domain switching and thk/ lattice strains in the plane of act tension specimen at st approaching and exceeding the initiation toughness. The directionally dependent in-plane domain switching beha ior in a soft PZT ceramic is thereby resolved and discussed in the context of the complex stress state at the crack tip Fig. I. Schematic o position(x,y wi ometry Parameters r and 0 define a physical 2. Experimental procedure the crack. Rotation angle n from the Y-axis corresponds to the direction n at each individual X, Y position Differences in the of normal and shear strains in differen A soft Nb-doped Pb(Zro.52Ti048)O3(PZT) ceramic with tip are illustrated a composition near the morphotropic phase boundary

domain switching distributions around the crack tip are discussed here using the well-known Mode I elastic stress profiles. In plane-stress Mode I loading, elastic stress distri￾butions as a function of radial coordinates (r, h) are given as [11] rXX ¼ KI ffiffiffiffiffiffiffi 2pr p cos h 2 1 sin h 2 sin 3h 2  ð1Þ rYY ¼ KI ffiffiffiffiffiffiffi 2pr p cos h 2 1 þ sin h 2 sin 3h 2  sXY ¼ KI ffiffiffiffiffiffiffi 2pr p cos h 2 sin h 2 cos 3h 2 rZZ ¼ sYZ ¼ sXZ ¼ 0 Fig. 1 illustrates the crack orientation in both Cartesian (X, Y) and radial coordinate systems. The present work is motivated by a desire to better understand the directionality of domain switching near a mechanically loaded crack tip. To this end, the behavior of local crystallographic orientations are examined at an array of points spanning the switching zone. A useful quan￾tity for illustrating the in-plane directional variations as a function of position relative to the crack tip (as well as pro￾viding clear comparisons between experimentally measured and modeled data) is the stress projection rn *ðgÞ ¼ niðgÞnjðgÞrij; ð2Þ where n *ðgÞ is a unit vector with an in-plane direction with respect to the sample coordinate system shown in Fig. 1. The scalar rn *ðgÞ represents a normal stress, i.e. the compo￾nent of the traction vector acting on a surface with normal n * in the direction n *. As a consequence of the measuring convention used (see Fig. 1 inset), rn *ðgÞ is equivalent to the rYY component transformed to a new coordinate sys￾tem by a clockwise rotation of g degrees about Z * [12]. Note that Eq. (2) also implies an in-plane antipodal symmetry described by rn *ðgÞ ¼ rn *ðg þ 180Þ. It is understood that ferroelastic domain switching is caused by the deviatoric stresses [13–15]. The projected deviatoric stress, s n *ðgÞ, is given by s n *ðgÞ ¼ rn *ðgÞ dijrij=3 ð3Þ where dij is the Kronecker delta. Using Eqs. (1)–(3), s n *ðgÞ is calculated as a function of position (X, Y) surrounding a crack tip for an applied stress intensity factor of KI = 0.71MPa m1/2. Spatial distributions of s n *ðgÞ for g = 0, 30, 60, and 90 are shown in Fig. 2. Given that these devi￾atoric stresses induce domain switching and that their spa￾tial distributions change with angle in the plane of the sample, it is hypothesized that the domain switching behav￾ior in these directions of the sample will be highly corre￾lated with these distributions. Because the techniques used in prior crack tip switching zone measurements have no in-plane directional resolution, we present a new approach by which the directional depen￾dence described above is resolved. High-energy X-rays can penetrate through several millimeters of most materials and therefore provide a powerful X-ray transmission technique by which to characterize in-plane behavior. When position sensitive area detectors are employed, high-energy synchro￾tron X-rays enable the capture of the entire ring of scatter￾ing vectors associated with each Debye–Scherrer cone for a single sample position [16–18]. Because the Bragg angles for most lower-order reflec￾tions in these materials are typically 5 or less for high￾energy X-rays (k < 0.25 A˚ ), the cones of scattering vectors lie nearly parallel to the X–Y plane of the sample. There￾fore, all scattering vectors for each reflecting plane {hkl} are treated as lying in the X–Y plane of the sample in this analysis. In this geometry, the normal lattice strains can then be extracted from the experimental data using ehklðn *Þ ¼ dhkl d hkl d hkl ; ð4Þ where dhkl and d hkl are the measured mean crystallographic lattice spacing for strained and unstrained crystallographic orientations such that n * is parallel to the {hkl} pole. These measured strains may, in turn, be related analytically to an average projection of the underlying strain tensors in the same crystallographic domain in a manner analogous to Eq. (2). In this work, we combine high-energy synchrotron X￾ray diffraction with a two-dimensional detector to map both the preferred orientation induced by ferroelastic domain switching and {hkl} lattice strains in the plane of a compact tension specimen at stress intensity factors approaching and exceeding the initiation toughness. The directionally dependent in-plane domain switching behav￾ior in a soft PZT ceramic is thereby resolved and discussed in the context of the complex stress state at the crack tip. 2. Experimental procedure A soft Nb-doped Pb(Zr0.52Ti0.48)O3 (PZT) ceramic with a composition near the morphotropic phase boundary Fig. 1. Schematic of crack geometry. Parameters r and h define a physical position (X, Y) with respect to the crack. Rotation angle g from the Y-axis corresponds to the projection direction n * at each individual X, Y position. Differences in the directions of normal and shear strains in different regions relative to the crack tip are illustrated. J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548 5539

J. L Jones et al. Acta Materialia 55(2007)5538-5548 Fig. 2. Spatial distributions(X, Y of the projected deviatoric stress, s-, in Mode l, plane-stress geometry at a stress intensity factor of KI=0.71 MPam Projected deviatoric stresses are shown perpendicular to the crack face(=0), parallel to the crack propagation direction (n=90), and two intermediate sitions(=30% and =60). The crack face is illustrated in each figure as a heavy line at Y=1.0 (K350, Piezo Technologies, Indianapolis, IN, USA)was significantly increases when the irradiated area encom- used in this experiment. This material composition has passes the crack. Diffraction data were collected with a been used in earlier work and is believed to contain only 50 x 50 um- beam rastered in two dimensions around the the tetragonal phase [19, 20]. Compact tension(CT) speci- crack in steps of 100 um, mapping a total area approxi mens were obtained in the dimensions 50 x 48 x 1.5 mm mately 2 x2 mm-. At each position, the beam shutter was vith a machined notch 23 mm in length parallel to the opened for 10 S, exposing scattered X-rays onto a two- 50 mm dimension. Similar to the crack geometry described dimensional digital image plate detector(MAR 345. in Ref [21], a chevron notch was cut at a 45 angle to the 150 um pixel size, Mar USA, Inc. positioned 1634 mm major surface of the sample at the end of the 23 mm long behind the center of the sample. The diffraction geometry notch using a sharp razor blade with 6 um diamond paste is illustrated in Fig 3 as an abrasive. a precrack was initiated on the major sur- The detector was centered on the transmitted X-ray face opposite the opening of the chevron notch(on the side beam such that complete Debye-Scherrer rings were col- of higher stress)using a 500 g Vickers indentation. The pre- lected(see Fig. 3 for a typical pattern). For each diffraction crack was propagated by employing the controlled crack measurement at each spatial position, the collection of growth apparatus described below. After the precrack grains sampled were contained within a 50 50x was propagated through the chevron notch and was visibly 1500 um'matchstick'diffracting volume. This high present under an optical microscope on both major sur- energy transmission geometry and the resulting low Bragg faces, the specimen was thermally annealed for 2 h at angles allow diffracting vectors oriented to within 3 of the 600C. A custom built apparatus was used to apply a con- specimen plane to be sampled. Therefore, for all practical trolled stress intensity factor. The compact tension speci- purposes, the sampled diffraction vectors are considered mens were loaded in Mode I, perpendicular to the crack to lie within the specimen plane g a piezoelectric actuator. The load was recorded Using Fit2D(Ver. 12.077)[24, the diffracted image in-line with the actuator and sample using a l kN load cell. were"caked"(rebinned within polar coordinates) within The crack tip position and crack length were identified and 15 wide azimuth sectors to obtain integrated diffracted measured prior to loading with an optical microscope. The intensity as a function of 20 [25]. The tetragonal PZT peaks crack length, load, and sample geometry were all used to 101, 110, 111, 002, 200, 1 12, and 21 l and the cubic calculate the applied stress intensity factor (K1) via stan- ceria peaks 111, 200, 220, and 3 1 l were then fit using a dard formulations [22, 23 While the sample was under constant load, high-energy synchrotron X-rays (80.8 keV, wavelength iN0.1535 A) from beamline 1-ID-c at the advanced photon source (APS) were used to measure in-plane domain switching and lattice strains surrounding the crack tip. Ceria CeO2) powder was suspended in Vaseline and spread on 0 the exit-beam side of the sample over the switching zone strain and preferred orientation [18]. The crack tip was incident Xrays i9 This serves as a internal standard in that it provides an azi- muthally uniform powder pattern that is ideally free of relocated with respect to the diffraction geometry by using 20 x 20 um- beam and measuring the transmitted inten sity as a function of sample position (x, y). Because of Fig 3 Schematic of the diffraction geometry and a typical two-dimen- the crack opening displacement, the transmitted intensity sional detector image showing the Debye-Scherrer rings

(K350, Piezo Technologies, Indianapolis, IN, USA) was used in this experiment. This material composition has been used in earlier work and is believed to contain only the tetragonal phase [19,20]. Compact tension (CT) speci￾mens were obtained in the dimensions 50 · 48 · 1.5 mm3 with a machined notch 23 mm in length parallel to the 50 mm dimension. Similar to the crack geometry described in Ref. [21], a chevron notch was cut at a 45 angle to the major surface of the sample at the end of the 23 mm long notch using a sharp razor blade with 6 lm diamond paste as an abrasive. A precrack was initiated on the major sur￾face opposite the opening of the chevron notch (on the side of higher stress) using a 500 g Vickers indentation. The pre￾crack was propagated by employing the controlled crack growth apparatus described below. After the precrack was propagated through the chevron notch and was visibly present under an optical microscope on both major sur￾faces, the specimen was thermally annealed for 2 h at 600 C. A custom built apparatus was used to apply a con￾trolled stress intensity factor. The compact tension speci￾mens were loaded in Mode I, perpendicular to the crack face, using a piezoelectric actuator. The load was recorded in-line with the actuator and sample using a 1 kN load cell. The crack tip position and crack length were identified and measured prior to loading with an optical microscope. The crack length, load, and sample geometry were all used to calculate the applied stress intensity factor (KI) via stan￾dard formulations [22,23]. While the sample was under constant load, high-energy synchrotron X-rays (80.8 keV, wavelength k  0.1535 A˚ ) from beamline 1-ID-C at the Advanced Photon Source (APS) were used to measure in-plane domain switching and lattice strains surrounding the crack tip. Ceria (CeO2) powder was suspended in Vaseline and spread on the exit-beam side of the sample over the switching zone. This serves as a internal standard in that it provides an azi￾muthally uniform powder pattern that is ideally free of strain and preferred orientation [18]. The crack tip was relocated with respect to the diffraction geometry by using a 20 · 20 lm2 beam and measuring the transmitted inten￾sity as a function of sample position (X, Y). Because of the crack opening displacement, the transmitted intensity significantly increases when the irradiated area encom￾passes the crack. Diffraction data were collected with a 50 · 50 lm2 beam rastered in two dimensions around the crack in steps of 100 lm, mapping a total area approxi￾mately 2 · 2 mm2 . At each position, the beam shutter was opened for 10 s, exposing scattered X-rays onto a two￾dimensional digital image plate detector (MAR 345, 150 lm pixel size, Mar USA, Inc.) positioned 1634 mm behind the center of the sample. The diffraction geometry is illustrated in Fig. 3. The detector was centered on the transmitted X-ray beam such that complete Debye–Scherrer rings were col￾lected (see Fig. 3 for a typical pattern). For each diffraction measurement at each spatial position, the collection of grains sampled were contained within a 50 · 50 · 1500 lm3 ‘‘matchstick’’ diffracting volume. This high￾energy transmission geometry and the resulting low Bragg angles allow diffracting vectors oriented to within 3 of the specimen plane to be sampled. Therefore, for all practical purposes, the sampled diffraction vectors are considered to lie within the specimen plane. Using Fit2D (Ver. 12.077) [24], the diffracted images were ‘‘caked’’ (rebinned within polar coordinates) within 15 wide azimuth sectors to obtain integrated diffracted intensity as a function of 2h [25]. The tetragonal PZT peaks 1 0 1, 1 1 0, 1 1 1, 0 0 2, 2 0 0, 1 1 2, and 2 1 1 and the cubic ceria peaks 1 1 1, 2 0 0, 2 2 0, and 3 1 1 were then fit using a 0 MPa 2 MPa 4 MPa 4 MPa 6 MPa 2 MPa 8 MPa 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 X [mm] Y [mm] 0 MPa 2 MPa 2 MPa 4 MPa 4 MPa 6 MPa 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 X [mm] Y [mm] 2 MPa 0 MPa 0 MPa 2 MPa 2 MPa 4 MPa 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 X [mm] Y [mm] 2 MPa 4 MPa 6 MPa 6 MPa 4 MPa 0 MPa 8 MPa 10 MPa 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 X [mm] Y [mm] 0o 30o 60o 90o Fig. 2. Spatial distributions (X, Y) of the projected deviatoric stress, s n *, in Mode I, plane-stress geometry at a stress intensity factor of KI = 0.71 MPa m1/2. Projected deviatoric stresses are shown perpendicular to the crack face (g = 0), parallel to the crack propagation direction (g = 90), and two intermediate positions (g = 30 and g = 60). The crack face is illustrated in each figure as a heavy line at Y = 1.0. in situ compact tension specimen η η = 0° incident X-rays Y X Fig. 3. Schematic of the diffraction geometry and a typical two-dimen￾sional detector image showing the Debye–Scherrer rings. 5540 J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548

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

J. L Jones et al. I Acta Materialia 55(2007)5538-5548 420 00.20406 a20 a2a4o6082o2 060810 0204060810121,4161.8 000204060.810121416 000.2040608101.2141.61.8 0608101.2141618 00204060.8101214161.8 D00.20.40608101.2141.618 上g 5.( Color) Filled contours(in steps of 0.05 mrd) represent domain switching (oo2)in the crack front during loading with a stress intensity factor of 0.71 MPa m"as a function of spatial position(X, Y) and in-plane azimuth angle, n. Crack face position and orientation are illustrated in each map as a bolded line at Y=1.0. Overlaid contour lines designate constant deviatoric stress(0. 4, 8,.. MPa), projected as a function of rotation angle n(s-),as predicted by the finite element model. The position identified by an arrow in the n=0 map is discussed in the text

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 X [mm] Y [mm] 0.90 0.95 1.00 1.05 1.10 1.15 6 0 6 12 18 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 0o 6 0 6 12 18 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 15o 6 0 6 12 18 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 30o 6 0 12 6 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 45o 6 6 0 12 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 60o 6 6 0 0 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 75o 6 6 0 0 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 90o 6 6 0 0 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 105o 6 6 12 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 120o 0 6 6 12 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 135o 0 6 6 12 18 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 150o 0 6 6 12 18 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 165o X [mm] X [mm] Y [mm] Y [mm] Y [mm] Y [mm] X [mm] Fig. 5. (Color) Filled contours (in steps of 0.05 mrd) represent domain switching (f002) in the crack front during loading with a stress intensity factor of KI= 0.71 MPa m1/2 as a function of spatial position (X, Y) and in-plane azimuth angle, g. Crack face position and orientation are illustrated in each map as a bolded line at Y = 1.0. Overlaid contour lines designate constant deviatoric stress (0, 4, 8,... MPa), projected as a function of rotation angle g ðs n *Þ, as predicted by the finite element model. The position identified by an arrow in the g = 0 map is discussed in the text. 5542 J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548

J.L Jones et al 1 Acta Materialia 55(2007)5538-5548 543 20 0.2 02 0002040.6081012141.618 000.2040.608101.2 1.61 000204060.81012141618 20 1.2 1.0 20406081.012141.618 D00.20406081.012141.618 D00.2040.60810 141618 D00 0.60810 141.618 000.2040.608101.2141.618 000204060.81012141618 0002040.6081012141.618 000.2040.608101.2141.6 0D0.204060810121416 IMm X[mm] IMm Fig. 6.( Color) Filled contours(in steps of l x 10)represent the measured I ll lattice strain(2111) in the crack front actor of Ki=0.71 MPa mas a function of spatial position (X, y and in-plane azimuth angle, n Crack face pe osi ionagd loading with a stress intensity tation are illustrated in ch map as a bolded line at Y=1.0. Overlaid contour lines, plotted at every 2 x 10, represent the projected deviat (E-) calculated by the finite

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 X [mm] Y [mm] -6E-4 -5E-4 -4E-4 -3E-4 -2E-4 -1E-4 0 1E-4 2E-4 3E-4 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 0o X [mm] X [mm] Y [mm] Y [mm] Y [mm] Y [mm] 0 2E-4 0 -2E-4 4E-4 2E-4 -4E-4 6E-4 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 120o 0 0 2E-4 -2E-4 4E-4 -4E-4 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 105o -2E-4 0 2E-4 2E-4 4E-4 0 6E-4 -4E-4 4E-4 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 150o 0 2E-4 2E-4 4E-4 -2E-4 4E-4 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 165o -2E-4 0 2E-4 2E-4 4E-4 0 -4E-4 6E-4 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 135o 0 0 2E-4 -2E-4 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 90o 2E-4 0 0 -2E-4 4E-4 -4E-4 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 75o 2E-4 2E-4 0 0 -2E-4 4E-4 -4E-4 6E-4 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 60o 2E-4 0 2E-4 -2E-4 4E-4 0 -4E-4 6E-4 8E-4 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 45o 2E-4 0 2E-4 -2E-4 4E-4 0 6E-4 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 30o 2E-4 0 2E-4 4E-4 -2E-4 6E-4 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 15o X [mm] Fig. 6. (Color) Filled contours (in steps of 1 · 104 ) represent the measured 1 1 1 lattice strain (e111) in the crack front during loading with a stress intensity factor of KI = 0.71 MPa m1/2 as a function of spatial position (X, Y) and in-plane azimuth angle, g. Crack face position and orientation are illustrated in each map as a bolded line at Y = 1.0. Overlaid contour lines, plotted at every 2 · 104 , represent the projected deviatoric strain ðe n *Þ calculated by the finite element model. J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548 5543

J. L Jones et al. I Acta Materialia 55(2007)5538-5548 0.4 04 00020.406081012141618 00.20.40608101214161 0.00.2040.60.81012141618 iMm Fig. 7.( Color) Filled contours represent domain switching (ooz)at the same position as in Fig. 5 after the stress intensity KI=1.2 MPa m -and the crack propagates Filled contour lines vary in steps of 0.05 mrd with a maximum value of 1. 28 mrd Posi ntified by the I1 1 lattice strains only one a or c lattice parameter component (ie 8110, E002, crack face, Y=1.0. The shear stresses predicted by eq and Eoo exhibit more uniform lattice strains with less corre- (1)are the first indicator as to the source of this direction lation to the crack tip position. For brevity, only the mea- ality. Fig. I illustrates the directions of the normal and sured E111 lattice strains are presented here. The spatial shear stresses predicted by Eq (1)in different regions with distributions of E111 as a function of n are similar to those respect to the crack tip. The txy shear stress is opposite in of foo2 in Fig. 5, again agreeing with the stress distributions sign in the regions-90o<8<0 and 0<0<90. In con- presented in Fig. 2 trast, the Gxx and ry components are symmetric about The stress intensity factor was slowly increased until the 0=0 crack started to propagate and was then held constant at A finite element model was developed to quantitatively KI= 1. 2 MPa m"until the crack was arrested This corre- demonstrate the directional dependence of the measured lates with the earlier-observed R-curve behavior measured distributions and their magnitudes near the crack tip. The macroscopically [21]. The preferred domain orientations finite element model was constructed in ABAQUS(ver for select n angles at this increased stress intensity factor 6.3)using elastic-plastic elements with power-law harden- are shown in Fig. 7(filled contours) for the same area as ing rules. The power-law hardening stress-strain behavior described in Fig. 5. The measured 11 l lattice strains is described by (E111) are also shown in Fig. 7(overlaid contour lines) crack propagation. It is apparent from the n=0 map that EE/+9-17 The positive a111 values have moved in the direction of the crack propagated by approximately I mm, leaving a domain-switched region in the crack wake. The spatial dis- where a and o are the macroscopic stress and strain values tributions of domain orientations and lattice strains remain o is the yield stress, and o and n are hardening coefficients a strong function of n By using elastic-plastic elements, this model can incorpo- rate nonlinearity not modeled by Eq (1). Thus, while the 4. Discussion finite element model does not model domain switching directly, the model incorporates the " plasticity"and"hard One notable characteristic of the domain orientation ening " associated with this behavior. The real stress-strain nd strain distributions in Figs. 5-7 is the apparent spatial behavior of this ceramic material was incorporated within asymmetry about the crack tip for a given off-axis angle n. the model by fitting the hardening coefficients to the mea In other words, at intermediate angular ranges (e.g. sured stress-strain behavior in tension(from Ref. [2ID 300<n< 60 and 120< n< 150), domains have different using an optimization routine. Using an elastic modulus degrees of preferred orientation and strains on opposite 66 GPa, and yield stress ll MPa [21]. the extracted harden- sides of the crack face. That is, foo2 and &111 are asymmet- ing coefficients are a=0.39 and n= 1. 69. A Poisson's ratio ric about the crack face for a single off-axis direction. How- of 0. 2 was also employed. The model and experimental ever,sample symmetry about the crack face is satisfied stress-strain behaviors are compared in Fig 8. The geom- when considering the entirety of orientation space etry, loading, and boundary conditions in the finite element =45°andn=135° maps are mirror images about th model were equivalent to the experimental compact tension

only one a or c lattice parameter component (i.e. e110, e002, and e200) exhibit more uniform lattice strains with less corre￾lation to the crack tip position. For brevity, only the mea￾sured e111 lattice strains are presented here. The spatial distributions of e111 as a function of g are similar to those of f002 in Fig. 5, again agreeing with the stress distributions presented in Fig. 2. The stress intensity factor was slowly increased until the crack started to propagate and was then held constant at KI = 1.2 MPa m1/2 until the crack was arrested. This corre￾lates with the earlier-observed R-curve behavior measured macroscopically [21]. The preferred domain orientations for select g angles at this increased stress intensity factor are shown in Fig. 7 (filled contours) for the same area as described in Fig. 5. The measured 1 1 1 lattice strains (e111) are also shown in Fig. 7 (overlaid contour lines). The positive e111 values have moved in the direction of crack propagation. It is apparent from the g = 0 map that the crack propagated by approximately 1 mm, leaving a domain-switched region in the crack wake. The spatial dis￾tributions of domain orientations and lattice strains remain a strong function of g. 4. Discussion One notable characteristic of the domain orientation and strain distributions in Figs. 5–7 is the apparent spatial asymmetry about the crack tip for a given off-axis angle g. In other words, at intermediate angular ranges (e.g. 30 < g < 60 and 120 < g < 150), domains have different degrees of preferred orientation and strains on opposite sides of the crack face. That is, f002 and e111 are asymmet￾ric about the crack face for a single off-axis direction. How￾ever, sample symmetry about the crack face is satisfied when considering the entirety of orientation space – the g = 45 and g = 135 maps are mirror images about the crack face, Y = 1.0. The shear stresses predicted by Eq. (1) are the first indicator as to the source of this direction￾ality. Fig. 1 illustrates the directions of the normal and shear stresses predicted by Eq. (1) in different regions with respect to the crack tip. The sXY shear stress is opposite in sign in the regions 90 < h < 0 and 0 < h < 90. In con￾trast, the rXX and rYY components are symmetric about h = 0. A finite element model was developed to quantitatively demonstrate the directional dependence of the measured distributions and their magnitudes near the crack tip. The finite element model was constructed in ABAQUS (ver. 6.3) using elastic–plastic elements with power-law harden￾ing rules. The power-law hardening stress–strain behavior is described by e ¼ r E 1 þ a r ro  n1 " # ð7Þ where e and r are the macroscopic stress and strain values, ro is the yield stress, and a and n are hardening coefficients. By using elastic–plastic elements, this model can incorpo￾rate nonlinearity not modeled by Eq. (1). Thus, while the finite element model does not model domain switching directly, the model incorporates the ‘‘plasticity’’ and ‘‘hard￾ening’’ associated with this behavior. The real stress–strain behavior of this ceramic material was incorporated within the model by fitting the hardening coefficients to the mea￾sured stress–strain behavior in tension (from Ref. [21]) using an optimization routine. Using an elastic modulus 66 GPa, and yield stress 11 MPa [21], the extracted harden￾ing coefficients are a = 0.39 and n = 1.69. A Poisson’s ratio of 0.2 was also employed. The model and experimental stress–strain behaviors are compared in Fig. 8. The geom￾etry, loading, and boundary conditions in the finite element model were equivalent to the experimental compact tension 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 X [mm] Y [mm] 0.90 0.95 1.00 1.05 1.10 1.15 0 1E-4 1E-4 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 0o 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 90o 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 45o Y [mm] X [mm] X [mm] X [mm] Fig. 7. (Color) Filled contours represent domain switching (f002) at the same position as in Fig. 5 after the stress intensity factor is increased to KI = 1.2 MPa m1/2 and the crack propagates. Filled contour lines vary in steps of 0.05 mrd with a maximum value of 1.28 mrd. Position identified by the arrow in g = 0 is discussed in the text. Overlaid contour lines at 0, 1 · 104 , 2 · 104 , and 3 · 104 represent the corresponding measured 1 1 1 lattice strains. 5544 J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548

J. L. Jones et al. Acta Materialia 55(2007)5538-5548 545 of the crack tip, which also correlates with the measured preference for 002 domain orientations in these regions Part of the good correlation between the calculate stresses and the measured domain orientation distributions is due to the fact that ferroelastic switching in polycryst line soft pzt occurs as a smooth function of stress. In other words, domain switching does not occur at a discrete critical (or "yield")stress but instead is continuous with Finite element model power-Law applied stress. This is evidenced by the stress-strain behav- ior shown in Fig. 8. Recent constitutive models of fracture Strain [ behavior have begun incorporating incremental switching criteria in place of discrete criteria [4] and the empirical Fig.8. Experimental stress-strain behavior of PZT in tension(from Ref. results presented here further support this approach [21]compared with the stress-strain behavior employed within the finite The agreement between the deviatoric stresses and the referred domain orientations is better in regions of lower stress. In spatial regions and sample directions with geometry, including the location of the crack tip relative to increased domain orientations, the deviatoric stress con- the specimen and the applied load required to generate a tours deviate from the domain switching results. Possible stress intensity factor of KI=0.71 MPa m/2 sources of discrepancy between the measurements and After comparing a plane-stress and plane strain state, model in the regions of increased domain switching are: the former was employed within the finite element model (1)strain-hardening behavior in regions of higher stress because the resulting stress and strain distributions better which are not considered in the model (e. g, stiffening due correlate with the experimental results than the stresses cal- to domain switching saturation) and (2) thickness effects culated using a plane strain condition. This is supported by that may be obscured through thickness-averaging(thick the fact that the process zone size is on the order of the ness effects are discussed further in Section 4.2).Nonethe sample thickness. This effect has also been observed in ear- less, it is apparent from Fig. 5 that domain switching lier work [2, 7, 27] near the crack tip on the 100 um to millimeter size scale The stress components, Oij, were extracted from the is sufficiently approximated as proportional to the pro- finite element model as a function of position surrounding jected deviatoric stress the crack tip for the first applied stress intensity factor of KI=0.71 MPa m"2. These stress components were then 4.2. Strain used to calculate the projected deviatoric stress, s=(n) using Eqs. (2)and(3). The resulting spatial distributions The lattice strains of several hkl planes are measured in of projected deviatoric stress parallel to n are shown as the diffraction geometry. The &111 strains are reasonably overlaid contour lines in Fig. 5 with the domain orientation representative of the behavior of the polycrystal. Differ measurements. The deviatoric model elastic strains e(n) ences between a1 11 and other hkl-type strains can be attrib projected on n were determined using an analogous formu- uted to a complex interaction between domain switching lation for strains to that of Eqs. (2)and(3). These projected and anisotropy in the intrinsic elastic stiffness [20, 28-30] model strains are shown as overlaid contour lines in Fig. 6 Fig. 6 demonstrates good correlation between the shape with the measured 1l l lattice strains, Elm(n). of the measured E111 distributions and the projected devia- Sections 4.1 and 4.2 describe in more detail the correla- toric strain(E-)predicted by the finite element model. The tion between the domain switching and lattice strain mea- strain magnitudes are also comparable; the contour line surements and the deviatoric stress and strain values corresponding to zero strain in the model nearly follows extracted from the finite element model the measured 8111=-l x 10 contour line. Recently, the 11 1 lattice strain was shown to correlate with the mac- 4.1. Stress and domain switching roscopic polycrystalline strain in some tetragonal ferroelas tic ceramics during mechanical loading [30] and after Fig. 5 demonstrates that the spatial distributions and unloading [28, 29]. However, an exact correlation between lative magnitudes of the projected deviatoric stress corre- these values is not expected here because the experimental late well with the domain orientations in the corresponding results only describe the 1 11 lattice strain, whereas the directions and positions relative to the crack tip. At n model describes the averaged polycrystalline strain preferred domain orientations in the frontal zone occur at Some discrepancy exists between the shape of the model stresses below 6 MPa and generally increase with increas- strain distributions and the measured eu values in regions ing stress. The individual stress profiles are a strong func- of high strain(i.e. at locations very near the crack tip). For tion of n, correlating with the measured preferred domain example, in the n=0o orientation, the measured E111 orientations. Additionally, the stress behind the crack tip strains drop in intensity near the crack tip, the region of at intermediate angles (e.g. n=30)is higher than ahead highest model strain. These are also regions where the

geometry, including the location of the crack tip relative to the specimen and the applied load required to generate a stress intensity factor of KI = 0.71 MPa m1/2. After comparing a plane-stress and plane strain state, the former was employed within the finite element model because the resulting stress and strain distributions better correlate with the experimental results than the stresses cal￾culated using a plane strain condition. This is supported by the fact that the process zone size is on the order of the sample thickness. This effect has also been observed in ear￾lier work [2,7,27]. The stress components, rij, were extracted from the finite element model as a function of position surrounding the crack tip for the first applied stress intensity factor of KI = 0.71 MPa m1/2. These stress components were then used to calculate the projected deviatoric stress, s n *ðgÞ, using Eqs. (2) and (3). The resulting spatial distributions of projected deviatoric stress parallel to g are shown as overlaid contour lines in Fig. 5 with the domain orientation measurements. The deviatoric model elastic strains e n *ðgÞ projected on n * were determined using an analogous formu￾lation for strains to that of Eqs. (2) and (3). These projected model strains are shown as overlaid contour lines in Fig. 6 with the measured 1 1 1 lattice strains, e111ðn *Þ. Sections 4.1 and 4.2 describe in more detail the correla￾tion between the domain switching and lattice strain mea￾surements and the deviatoric stress and strain values extracted from the finite element model. 4.1. Stress and domain switching Fig. 5 demonstrates that the spatial distributions and relative magnitudes of the projected deviatoric stress corre￾late well with the domain orientations in the corresponding directions and positions relative to the crack tip. At g = 0, preferred domain orientations in the frontal zone occur at stresses below 6 MPa and generally increase with increas￾ing stress. The individual stress profiles are a strong func￾tion of g, correlating with the measured preferred domain orientations. Additionally, the stress behind the crack tip at intermediate angles (e.g. g = 30) is higher than ahead of the crack tip, which also correlates with the measured preference for 0 0 2 domain orientations in these regions. Part of the good correlation between the calculated stresses and the measured domain orientation distributions is due to the fact that ferroelastic switching in polycrystal￾line soft PZT occurs as a smooth function of stress. In other words, domain switching does not occur at a discrete critical (or ‘‘yield’’) stress but instead is continuous with applied stress. This is evidenced by the stress–strain behav￾ior shown in Fig. 8. Recent constitutive models of fracture behavior have begun incorporating incremental switching criteria in place of discrete criteria [4] and the empirical results presented here further support this approach. The agreement between the deviatoric stresses and the preferred domain orientations is better in regions of lower stress. In spatial regions and sample directions with increased domain orientations, the deviatoric stress con￾tours deviate from the domain switching results. Possible sources of discrepancy between the measurements and model in the regions of increased domain switching are: (1) strain-hardening behavior in regions of higher stress which are not considered in the model (e.g., stiffening due to domain switching saturation) and (2) thickness effects that may be obscured through thickness-averaging (thick￾ness effects are discussed further in Section 4.2). Nonethe￾less, it is apparent from Fig. 5 that domain switching near the crack tip on the 100 lm to millimeter size scale is sufficiently approximated as proportional to the pro￾jected deviatoric stress. 4.2. Strain The lattice strains of several hkl planes are measured in the diffraction geometry. The e111 strains are reasonably representative of the behavior of the polycrystal. Differ￾ences between e111 and other hkl-type strains can be attrib￾uted to a complex interaction between domain switching and anisotropy in the intrinsic elastic stiffness [20,28–30]. Fig. 6 demonstrates good correlation between the shape of the measured e111 distributions and the projected devia￾toric strain (e n *) predicted by the finite element model. The strain magnitudes are also comparable; the contour line corresponding to zero strain in the model nearly follows the measured e111 = 1 · 104 contour line. Recently, the 1 1 1 lattice strain was shown to correlate with the mac￾roscopic polycrystalline strain in some tetragonal ferroelas￾tic ceramics during mechanical loading [30] and after unloading [28,29]. However, an exact correlation between these values is not expected here because the experimental results only describe the 1 1 1 lattice strain, whereas the model describes the averaged polycrystalline strain. Some discrepancy exists between the shape of the model strain distributions and the measured e111 values in regions of high strain (i.e. at locations very near the crack tip). For example, in the g = 0 orientation, the measured e111 strains drop in intensity near the crack tip, the region of highest model strain. These are also regions where the 0 10 20 30 40 50 0.00 0.05 0.10 0.15 0.20 Measured Finite Element Model, Power-Law Hardening Approximation Strain [%] Stress [MPa] Fig. 8. Experimental stress–strain behavior of PZT in tension (from Ref. [21]) compared with the stress–strain behavior employed within the finite element model. J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548 5545

J. L Jones et al. I Acta Materialia 55(2007)5538-5548 greatest magnitude of domain switching occurs, leading to in ferroelastic ceramics, the zone size is typically defined likely changes in constitutive behavior not considered in as the region within which any domain switching occurs the model. However, we cannot preclude the possibility and is therefore limited by the experimental resolution. In that slight deviations in the position of the crack front the LCD technique, the first onset of domain switching is through the thickness of the sample can also lead to the measured, or the first deviation from linear-elastic behav reductions in measured a111 strains near the crack tip. In ior, yielding a larger process zone size than earlier tech- ther words, the behavior at the apparent crack tip posi- niques based on the measurement of remanent plastic tion in Figs. 5 and 6 may be receive a contribution from strain [2]. The X-ray approach described in this work also positions slightly behind the crack tip in addition to those measures in situ the onset of domain switching as a devia- precisely at the crack tip. Such a case might lead to the tion from foo2=1.00 mrd. Accounting for the +0.02 mrd decreases in measured &111 strain near the crack tip variability, fo02=1.05 mrd can be considered as a lower (Fig. 6) but not decreases in preferred orientation of bound estimate for the size of the switching zone. Using domains(Fig. 5), since regions of high domain orientation this definition, the n=0 domain orientation maps in Figs intensity foo2)also occur in the crack wake(Fig. 7) 5 and 7 demonstrate process zone half-heights before and after crack propagation of >500 um (frontal) and 43. General discussion 100 um(wake), respectively. The frontal zone size, mea sured in situ under an applied stress intensity factor Several additional conclusions can be drawn from the below that required for crack propagation, is comparable domain switching distributions. Figs. 5 and 7 illustrate that to that measured in Ref. [2] under similar loadin there are different degrees of domain switching reversibility conditions at different distances from the passing crack tip For exam- The domain orientation maps in Figs. 5 and 7 also dem- ple, for a constant spatial position in each n=00 map onstrate that the switching zone size is dependent on the in- X=1.0 mm, Y=0.6 mm, identified with an arrow in plane direction. After crack propagation(Fig. 7), the half- Figs. 5 and 7), the foo2 domain switching intensity height is much larger at n=45(500 um) than n=00 lecreases from 1.06 to 1.00 mrd after the crack tip passes; (100 um). In other words, further from the crack there this position experiences partial reversibility. In contrast, a are greater preferences for domain orientations oriented position near the crack tip(e.g, X=0. 8 mm, Y=1.0 mm) 45 to the crack face than the preferences of domains ori- increases in intensity from 1. 12 to 1. 25 mrd after the crack ented perpendicular to the crack face(n=0%). This direc tip passes; this position experiences a larger degree of irre- tional dependence of the zone height is illustrated in versibility. The fact that the degree of preferred orientation Fig 9 Domains oriented at 45 to the crack face tend to increases as opposed to remaining constant near the crack have their c-axes(001 direction) oriented toward the crack tip could be due to a lower applied stress intensity factor in front and exhibit a larger process zone size than those ori- ig. 5 than that required to propagate the crack and the ented perpendicular to the crack face. The directionality of resulting high stresses preceding brittle fracture. In con- switching in the crack wake demonstrates a usefulness for trast, it is clear in comparing Figs 6 and 7 that the tensile post-mortem failure studies of fractured ferroelectrics E111 lattice strains are entirely relieved after the crack axes orientations are tilted towards the direction of crack passes. Compressive lattice strains exist in the region sur rounding the crack wake, though they are of small magni Before Propagation After Propagation m吗叫 The domain orientation maps in Figs. 5 and 7 also allow measure of the size of the zone in which domain switching occurs within the plane of the sample. The importance of the height of the switching zone in toughening of ceramics is inherited from the study of phase transformation tough ening in zirconia-containing ceramics, where models of oughening enhancement utilize the total volumetric strain 45 which is a function of several parameters including the, transformation strain and the process zone height [31, 32]. The half-height of the switching zone continues to remain ntegral to current models of ferroelastic toughening(a good review is given in Ref. [4]), though it is complicating in that domain switching is incremental and never physi- cally saturates (i.e. subcritical). Therefore, a distribution ig. 9. Schematic identifying regions of strongest domain preference and of domain switching exists throughout the process zone. unique process zone half-heights(h) for two different pseudo-cubic grain In prior experimental measurements of the process zone orientations before and after crack propagation

greatest magnitude of domain switching occurs, leading to likely changes in constitutive behavior not considered in the model. However, we cannot preclude the possibility that slight deviations in the position of the crack front through the thickness of the sample can also lead to the reductions in measured e111 strains near the crack tip. In other words, the behavior at the apparent crack tip posi￾tion in Figs. 5 and 6 may be receive a contribution from positions slightly behind the crack tip in addition to those precisely at the crack tip. Such a case might lead to the decreases in measured e111 strain near the crack tip (Fig. 6) but not decreases in preferred orientation of domains (Fig. 5), since regions of high domain orientation intensity (f002) also occur in the crack wake (Fig. 7). 4.3. General discussion Several additional conclusions can be drawn from the domain switching distributions. Figs. 5 and 7 illustrate that there are different degrees of domain switching reversibility at different distances from the passing crack tip. For exam￾ple, for a constant spatial position in each g = 0 map (X = 1.0 mm, Y = 0.6 mm, identified with an arrow in Figs. 5 and 7), the f002 domain switching intensity decreases from 1.06 to 1.00 mrd after the crack tip passes; this position experiences partial reversibility. In contrast, a position near the crack tip (e.g., X = 0.8 mm, Y = 1.0 mm) increases in intensity from 1.12 to 1.25 mrd after the crack tip passes; this position experiences a larger degree of irre￾versibility. The fact that the degree of preferred orientation increases as opposed to remaining constant near the crack tip could be due to a lower applied stress intensity factor in Fig. 5 than that required to propagate the crack and the resulting high stresses preceding brittle fracture. In con￾trast, it is clear in comparing Figs. 6 and 7 that the tensile e111 lattice strains are entirely relieved after the crack passes. Compressive lattice strains exist in the region sur￾rounding the crack wake, though they are of small magni￾tude and exhibit little identifiable trend in their distributions relative to the crack orientation. Compressive lattice strains are therefore omitted from Fig. 7 for clarity. The domain orientation maps in Figs. 5 and 7 also allow a measure of the size of the zone in which domain switching occurs within the plane of the sample. The importance of the height of the switching zone in toughening of ceramics is inherited from the study of phase transformation tough￾ening in zirconia-containing ceramics, where models of toughening enhancement utilize the total volumetric strain which is a function of several parameters including the transformation strain and the process zone height [31,32]. The half-height of the switching zone continues to remain integral to current models of ferroelastic toughening (a good review is given in Ref. [4]), though it is complicating in that domain switching is incremental and never physi￾cally saturates (i.e. subcritical). Therefore, a distribution of domain switching exists throughout the process zone. In prior experimental measurements of the process zone in ferroelastic ceramics, the zone size is typically defined as the region within which any domain switching occurs and is therefore limited by the experimental resolution. In the LCD technique, the first onset of domain switching is measured, or the first deviation from linear-elastic behav￾ior, yielding a larger process zone size than earlier tech￾niques based on the measurement of remanent plastic strain [2]. The X-ray approach described in this work also measures in situ the onset of domain switching as a devia￾tion from f002 = 1.00 mrd. Accounting for the ±0.02 mrd variability, f002 = 1.05 mrd can be considered as a lower bound estimate for the size of the switching zone. Using this definition, the g = 0 domain orientation maps in Figs. 5 and 7 demonstrate process zone half-heights before and after crack propagation of >500 lm (frontal) and 100 lm (wake), respectively. The frontal zone size, mea￾sured in situ under an applied stress intensity factor just below that required for crack propagation, is comparable to that measured in Ref. [2] under similar loading conditions. The domain orientation maps in Figs. 5 and 7 also dem￾onstrate that the switching zone size is dependent on the in￾plane direction. After crack propagation (Fig. 7), the half￾height is much larger at g=45 (>500 lm) than g = 0 (100 lm). In other words, further from the crack there are greater preferences for domain orientations oriented 45 to the crack face than the preferences of domains ori￾ented perpendicular to the crack face (g = 0). This direc￾tional dependence of the zone height is illustrated in Fig. 9. Domains oriented at 45 to the crack face tend to have their c-axes (0 0 1 direction) oriented toward the crack front and exhibit a larger process zone size than those ori￾ented perpendicular to the crack face. The directionality of switching in the crack wake demonstrates a usefulness for post-mortem failure studies of fractured ferroelectrics; c￾axes orientations are tilted towards the direction of crack 45° 0° Before Propagation After Propagation h h Fig. 9. Schematic identifying regions of strongest domain preference and unique process zone half-heights (h) for two different pseudo-cubic grain orientations before and after crack propagation. 5546 J.L. Jones et al. / Acta Materialia 55 (2007) 5538–5548

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