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 thegeometry, 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