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 tensiononly 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