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 tetragonal 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 ceramics, Nomarski interference contrast and atomic force microscopy have been employed to measure local surface depressions that result from the strain associated with ferroelastic 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 intensity 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 perpendicular to the sample surface can be measured around the crack tip using a small beam size from laboratory Xrays 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 distributions 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 quantity for illustrating the in-plane directional variations as a function of position relative to the crack tip (as well as providing 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 component 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 system 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 deviatoric stresses induce domain switching and that their spatial distributions change with angle in the plane of the sample, it is hypothesized that the domain switching behavior in these directions of the sample will be highly correlated 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 dependence 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 synchrotron X-rays enable the capture of the entire ring of scattering vectors associated with each Debye–Scherrer cone for a single sample position [16–18]. Because the Bragg angles for most lower-order reflections in these materials are typically 5 or less for highenergy X-rays (k < 0.25 A˚ ), the cones of scattering vectors lie nearly parallel to the X–Y plane of the sample. Therefore, 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 Xray 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 behavior 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) specimens 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 surface opposite the opening of the chevron notch (on the side of higher stress) using a 500 g Vickers indentation. The precrack 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 surfaces, the specimen was thermally annealed for 2 h at 600 C. A custom built apparatus was used to apply a controlled stress intensity factor. The compact tension specimens 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 standard 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 azimuthally 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 intensity as a function of sample position (X, Y). Because of the crack opening displacement, the transmitted intensity significantly increases when the irradiated area encompasses 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 approximately 2 · 2 mm2 . At each position, the beam shutter was opened for 10 s, exposing scattered X-rays onto a twodimensional 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 collected (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 highenergy 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-dimensional 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 optimization routine within MATLAB (Ver. 7.0.4, The MathWorks, 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 specimento-detector distance, detector orientation, calibrant-to-sample distance, beam center position, and unstressed PZT lattice 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 position 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 calculate 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 utilized 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 symmetrically equivalent g angles within each image, equivalent to that described above for the strain data. The standard deviation 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 direction, and f002 = 3 mrd means there are no a- or b-axes in the given sample direction. In ferroelastic ceramic materials, 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 toughness 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 function of position relative to the crack tip (X, Y) and direction 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 magnitude 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 contours) 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