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 boundarydomain 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