# Shear Band Characterization of Clayey Soils with Particle Image Velocimetry
## 4 Experimental Results and Discussion
### Shear Band Characteristics: Angle and Thickness  
The shear strain was calculated with the displacement vectors from the PIV results for the four stress-strain behavior stages of the NC and OC clays. As the repeat test results were similar for each NC and OC clay, only the representative figures for NC1 and OC1 are shown in Figures 12 and 13; all experiment results are tabulated in Tables 3 and 4. Similar to the localized shear displacement, the localized shear strain did not show any particular behavior in the initial and peak stages of the four specimens. The NC clay vaguely showed a potential shear band in the peak stage (for both of NC1 and NC2). In both NC and OC clays, areas of shear strain concentration (shear band zone) formed in the softening stage and then gradually developed until the steady-state stage. As shear bands are normally characterized according to their slope and thickness [8; 17; 23; 34], the slope and thickness were also used to characterize the shear bands of the NC and OC clays in this study. Figure 11: Displacement contours for the steady-state stage of the stress–strain behavior (5.5–5.75% strain): (**a**) NC1 horizontal; (**b**) NC1 vertical; (**c**) OC1 horizontal; and (**d**) OC1 vertical. Figure 12: Increasing shear strain for NC1: (**a**) initial; (**b**) peak; (**c**) softening; and (**d**) steady state. Tables 3 and 4 indicate the slope and thickness of shear bands for all experiments (NC1, NC2, OC1, and OC2). The slope of the shear band was measured as the average angle of the contour lines around the shear band (\(\theta_{\textit{mosmard}}\) in Table 3). This was compared to previous theories on the slope of the shear band after shearing is complete. [PERSON] and [PERSON] criterion shows that failure occurs on the plane with the maximum ratio of the shear stress to the vertical confining stress. They calculated the shear band angle as follows:  
\[\theta_{\textit{MC}}=45^{\circ}+\phi^{\prime}/2 \tag{1}\]  
where \(\theta_{\textit{MC}}\) is the shear band angle calculated by [PERSON] and the Coulomb criterion and \(\phi\prime\) is the soil's internal friction angle. [PERSON] [35] experimentally confirmed that the shear band slope is more strongly related to deformation than stress and introduced the concept of the dilatation angle to propose the following shear band slope equation:  
\[\theta_{R}=45^{\circ}+\psi/2 \tag{2}\]  
where \(\theta_{R}\) is the shear band angle calculated by [PERSON] [35] and \(\psi\) is the soil's dilatation angle, which is calculated as the ratio of the volume strain to shear strain. [PERSON] et al. [36] showed that this equation \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Description** & **NC1** & **NC2** & **OC1** & **OC2** \\ \hline Softening & 4.42 mm & 5.20 mm & 3.84 mm & 4.66 mm \\ \hline Steady state & 7.56 mm & 6.34 mm & 6.56 mm & 7.01 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Thickness of shear band for each stage. Figure 13: Increasing shear strain for OC1: (**a**) initial; (**b**) peak; (**c**) softening; and (**d**) steady state. did not agree with experimental results and proposed an equation that considers both the internal friction angle and dilatation angle:  
\[\theta_{A}=45^{\circ}+(\phi^{\prime}+\psi)/4 \tag{3}\]  
where \(\theta_{A}\) is the shear band angle calculated by [PERSON] et al. [36]. Table 3 indicates that the shear band slopes of the NC1, NC2, OC1, and OC2 were 52.1\({}^{\circ}\), 49.6\({}^{\circ}\), 51.7\({}^{\circ}\), and 52.3\({}^{\circ}\), respectively, in the softening stage, which is similar to \(\theta_{R}\) that considers the dilatation angle of the soil. As noted above, the shear band starts to appear in the softening stage, and its slope continuously increases as the shearing progressed and remained fixed in the steady state. As shown in Table 3, the shear band slopes after shearing were complete (i.e., steady state) were 59.2\({}^{\circ}\) and 56.1\({}^{\circ}\) for the NC1 and NC2 and was between \(\theta_{MC}\) and \(\theta_{A}\), as reported by other researchers [18; 22; 23; 37].