## 7 Results, Discussion and Validation
The variation in the peak shear stress with the number of cycles for pile group (2,2) evaluated for \(2^{\circ}\), \(6^{\circ}\), and \(12^{\circ}\) dilation, respectively, has been shown in Figure 7.  
Figure 5: Variation of twist angle with number of cycles for pile group (2,2) evaluated for \(2^{\circ}\), \(6^{\circ}\), and \(12^{\circ}\) dilation, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Dilation**, \(\mathcal{O}_{D}\) & \(\mathbf{D}_{f}^{\circ}\) & \(\mathbf{D}_{f}^{\circ}\) & \(\mathbf{D}_{f}\) & \(\mathbf{a_{t}}\) & \(\mathbf{b_{t}}\) & \(\mathbf{a_{s}}\) & \(\mathbf{b_{s}}\) \\ \hline \(2^{\circ}\) & \(-0.019\) & \(2.006\) & \(0.1445\) & \(-0.007\) & \(0.0357\) & \(-1.079\) & \(8.289\) \\ \(6^{\circ}\) & \(-0.0156\) & \(1.3342\) & \(0.6356\) & \(-0.003\) & \(0.0244\) & \(-1.298\) & \(8.246\) \\ \(12^{\circ}\) & \(-0.009\) & \(0.7087\) & \(0.6871\) & \(-0.002\) & \(0.0189\) & \(-1.364\) & \(8.118\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Flow-controlled cyclic degradation parameters. Figure 6: Variation of plastic strain with number of cycles for pile group (2,2) evaluated for \(2^{\circ}\), \(6^{\circ}\), and \(12^{\circ}\) dilation with enlarged views magnifying the convexities of plastic strain. The relationship between the peak values of plastic strain and the number of cycles has been expressed in matrix form as follows:  
\[\left[\varepsilon_{P}\right]= \begin{bmatrix}N^{2}&N&1\end{bmatrix}\begin{bmatrix} D_{f}^{\prime\prime}\\ D_{f}^{\prime}\\ D_{f}^{\prime}\end{bmatrix}  \tag{28}\]  
where \(D_{f}^{\prime\prime}\) is the power law cyclic degradation parameter, which reduces the magnitude of plastic strain at a decrement rate for increasing dilation angle; \(D_{f}^{\prime}\) represents the linear cyclic degradation parameter, which decreases with the increasing dilation angle; and \(D_{f}\) represents the constant cyclic degradation parameter for a cyclic strain, which increases with the dilation angle (Figure 8a and Table 3). The peak shear stress for pile group (2,2) and a single pile for normalized cyclic torque is found in the range of (5-8.2) kPa at dilations of 2\({}^{\circ}\) to 12\({}^{\circ}\), respectively, in the present analysis. The peak shear stress for the pile group (2,2) logarithmically decreases and then asymptotically converges after 60 cycles of loading in a flow-controlled geomaterial, which, in turn, is a function of the number of cycles and dilation (Figure 7). The relationship between the peak shear stress and the number of cycles, \(N\) is expressed in matrix form as follows:  
\[\left[\tau_{p}\right]= \begin{bmatrix}\log_{e}N&1\end{bmatrix}\begin{bmatrix} a_{s}\\ b_{s}\end{bmatrix}  \tag{29}\]  
\[\tau_{p}=a_{s}\log_{e}N+b_{s} \tag{30}\]  
\[N=e^{\frac{\tau_{p}-b_{s}}{a_{s}}};\;N=1;\;\tau_{p}=b_{s} \tag{31}\]  
where, \(a_{s}\) and \(b_{s}\) are the logarithmic degradation parameters for shear stress. The experimental data of [PERSON] and [PERSON] (2001) [22] have the values of \(a_{s}\)= \(-\)0.782 and \(b_{s}\) = 2.9964, and the same for the present analysis for a single pile are \(-\)1.438 and 6.3183, respectively. Figure 8b shows the variation of a set of logarithmic degradation parameters with the angle of dilation (Table 3). Figure 7: Variation of peak shear stress with the number of cycles in a flow-controlled geomaterial. The peak shear stress for pile group (2,2) and a single pile for normalized cyclic torsional load is captured at dilations of 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\), respectively, in the present analysis. Figure 8: (**a**) Variation of cyclic degradation parameter for pile group (2,2) evaluated for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation; (**b**) variation in logarithmic cyclic degradation parameter for pile group (2,2) evaluated for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation.