# Constitutive Analysis of the Anisotropic Flow Behavior of Commercially Pure Titanium
## 3 Results  
The initial microstructure of the rolled CP-Ti sheet was investigated through EBSD analysis (Figure 2). The inverse pole figure map showed an equiaxed grain structure with an average grain size of 55 \(\upmu\)m. In addition, the split basal texture was confirmed in the initial microstructure; most basal poles were tilted by approximately 20-40\({}^{\circ}\) from ND towards TD, while maintaining the perpendicularity to RD. This is the typical characteristic of rolled CP-Ti sheet, as reported earlier [6]. Figure 1: Schematic illustration of cylindrical specimens for quasi-static uniaxial compression test. They were machined along rolling direction (RD), normal direction (ND), and transverse direction (TD), respectively. The uniaxial compression along RD, ND, and TD gave rise to two obvious differences in the flow behavior of CP-Ti (Figure 3). First, the alloys exhibited different YS values depending on the compressive direction: 144 MPa for RD, 202 MPa for ND, and 182 MPa for TD. Such an anisotropic behavior originated from the difference in the critical resolved shear stress (CRSS) and the Schmid factor (SF) for favorable slip systems [11]. This is further discussed in the following paragraph with the calculated SF values for the present case. The flow stresses measured at a compressive strain of 0.2 were comparable in spite of the lowest YS for the RD specimen. Second, an anisotropic strain-hardening behavior was revealed in Figure 3b. RD and ND compressions resulted in the three-stage hardening [12]; the hardening rate decreased in Stage A, increased due to the activation of twinning in Stage B, and then decreased again after the twinning saturation in Stage C. The specimen compressed along ND, however, showed significantly less increase in hardening rate in Stage B compared to the RD specimen. This left the plateau region in Stage B of the ND specimen. Meanwhile, the TD compression led to a monotonic reduction in the strain-hardening rate. The comparable flow curves after a compressive strain of 0.2, as shown in Figure 3a, stemmed from the highest strain-hardening rate of RD. SF values (\(m\)) determine the quantitative relation between the applied normal stress (\(\sigma\)) and resolved shear stress on a given slip plane (\(\tau\)), expressed as follows [11]:  
\[\tau=m\sigma=\cos(\phi)\cos(\lambda)\sigma, \tag{1}\]  
Figure 3: Compressive deformation behavior of CP-Ti (commercially pure titanium) depending on the loading direction: (**a**) flow stress curves starting from the yield point and (**b**) strain-hardening rates. Figure 2: EBSD (electron backscatter diffraction) investigation for the initial microstructure: (**a**) inverse pole figure map from the 3-dimensional view and (**b**) pole figures with respect to (0001), (11–20), and (10–10). where \(\phi\) and \(\lambda\) are an angle of the axial force vector and shear force vector, respectively, with respect to the normal vector of the given plane. Slip is initiated when the resolved shear stress exceeds CRSS for each slip plane. Accordingly, a high SF value makes slip liable to occur, thereby decreasing YS. The SF values for each compressive direction were calculated for prismatic <\(a\)> slip (Figure 4), because it is the most favorable slip system in CP-Ti with the rolling texture at room temperature [13]. In the initial state (i.e., before the compression), RD presented the highest SF value on average, thereby rationalizing its lowest YS, as shown in Figure 3a. Similarly, the highest YS of the ND specimen is understood in light of the lowest SF value on average. Such results are further supported by consistent conclusions made in a previous work [13] that the basal slip is the most favorable slip system in ND compression. EBSD analysis suggested the types and fractions of twinning for the compressed CP-Ti alloys (Figure 5). Twin boundaries were detected on the basis of a tolerance misorientation of 5\({}^{\circ}\).