# Constitutive Analysis of the Anisotropic Flow Behavior of Commercially Pure Titanium
## 4 Discussion  
A constitutive approach was employed to quantify the contribution of microstructural factors to the plastic deformation behavior in order to provide deeper insight into the CP-Ti anisotropy. The flow stress at a compressive strain of 0.2 (\(\sigma_{0.2}\)) is composed of the stress component related to the dynamic Hall-Petch strengthening (\(\Delta\sigma_{H\cdot P}\)), dislocation pile-up (\(\Delta\sigma_{disk}\)), and texture hardening (\(\Delta\sigma_{tx}\)), as well as YS (\(\sigma_{y}\)), expressed as:  
\[\sigma_{0.2}=\sigma_{y}+\Delta\sigma_{H\cdot P}+\Delta\sigma_{disk}+\Delta \sigma_{tx}. \tag{2}\]  
The dynamic Hall-Petch strengthening is determined by the reduction in the effective grain size, as follows:  
\[\Delta\sigma_{H\cdot P}=k_{y}(d_{f}{{}^{-0.5}}-d_{i}{{}^{-0.5}}), \tag{3}\]  
where \(k_{y}\) (= 0.53 MN\(\cdot\)m\({}^{-1.5}\), [14]) is a material coefficient, \(d_{i}\) is the initial grain size, and \(d_{f}\) is the effective grain size determined by high-angle boundaries with a misorientation angle higher than 15\({}^{\circ}\). The active occurrence of mechanical twinning stimulates a significant grain refinement and resultant dynamic Hall-Petch strengthening. Regarding the investigated loading directions, RD compression gave rise to the highest \(\Delta\sigma_{H\cdot P}\) value (83 MPa), followed by ND compression (69 MPa) and TD compression (30 MPa). The markedly low \(\Delta\sigma_{H\cdot P}\) value of the TD specimen is rationalized by the suppressed twinning. For further analysis, the 20%-compressed specimens were subsequently annealed at 773 K for 12 h; this condition was set to induce the recovery while suppressing the static recrystallization as much as possible. In other words, the dislocations generated by the compression were annihilated without changing granular and textural structures. Such a heat treatment enabled us to investigate the influence of dislocation hardening (i.e., \(\Delta\sigma_{disk}\) values) by excluding the other stress components. The measured \(\Delta\sigma_{disk}\) values demonstrate a clear tendency. RD and ND compression made comparable values (69 MPa and 74 MPa), whereas TD compression showed a remarkably larger dislocation hardening (114 MPa). Finally, the texture hardening of each sample was evaluated from the strain accommodated by mechanical twinning (\(\varepsilon_{tw}\)) at a compressive strain of 0.2, as follows [14]:  
\[\varepsilon_{tw}=\varepsilon_{tw}/\surd{2}, \tag{4}\]  
where \(s\) is twin shear (0.217 for [10; 11; 12] and 0.176 for [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22], [14]) and \(f_{tw}\) is the area fraction of mechanical twins. RD and ND specimens exhibited higher \(\varepsilon_{tw}\) values due to their higher fraction of twinned regions in comparison to the TD sample. It is thus inferred that slip was the primary deformation mechanism in the latter, providing a good explanation for the aforementioned high contribution of dislocation hardening. Meanwhile, the dominant twinning system in the RD compression (i.e., [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]) possessed the lower twin shear than the [10; 11; 12] twinning system. Figure 5: EBSD image quality map with marked twin boundaries at a compressive strain of 0.2 along (**a**) RD, (**b**) ND, and (**c**) TD. Red and blue lines indicate [10; 11; 12] and [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22] twin boundaries, respectively. This rationalizes the comparable dislocation hardening between the RD and ND specimens in spite of the 10% higher fraction of twinned region in the former. Activation stress for slip is determined as the ratio of CRSS (\(\tau_{CRSS}\)) and SF in a single crystal [15]. The activation stress of polycrystalline metal (\(\sigma_{a}\)) can be calculated by expanding this model as follows:  
\[\sigma_{a}=\Sigma(\tau_{CRSS}/m)n_{\text{SF}}, \tag{5}\]  
where \(n_{\text{SF}}\) is the number fraction of SF value.