## 7 Results, Discussion and Validation  
The response of footings and structures supported by pile groups depends upon the cyclic stiffness and damping generated by geomaterial-pile group interaction. The three-dimensional sectional view of the geomaterial-pile group (2,2) for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation, respectively, is shown in Figure 4a-c. Figure 4d-f show the geomaterial-pile group (2,2) sectional view for plastic strain values for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation, respectively. The peak plastic strain decreases with an increase in dilation. The peak plastic strain of the geomaterial has been computed as 43.44, 27.34, and 14.19% for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation, respectively (Table 2). The color scheme is selected to represent the intensity of plastic strain, where the green color scheme is for maximum and minimum plastic strain for 2\({}^{\circ}\) and 12\({}^{\circ}\) dilation, respectively. Figure 5 shows the variation of twist angle measured at the pile cap with the number of cycles of normalized cyclic torque for pile group (2,2) evaluated for 2\({}^{\circ}\), 6\({}^{\circ}\), and 12\({}^{\circ}\) dilation. The twist angle decreases with an increase in the number of cycles. The peak twist angle decreases with an increase in dilation. The peak twist logarithmically decreases with an increase in the number of cycles and dilation and then becomes asymptotic after 30 cycles of loading in a flow-controlled geomaterial. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cycle** & \(\mathcal{O}_{D}\) = **12\({}^{\circ}\)** & \(\mathcal{O}_{D}\) = 6\({}^{\circ}\) & \(\mathcal{O}_{D}\) = 2\({}^{\circ}\) \\ \hline
1 & 0.92 & 1.34 & 1.67 \\
10 & 7.08 & 12.74 & 18.49 \\
20 & 11.02 & 20.76 & 32.35 \\
30 & 14.19 & 27.34 & 43.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Peak values of plastic strain (%) of the geomaterial. Figure 3: 3D pile group (2,2) model with (**a**) continuum of geomaterial and boundary conditions, (**b**) the surface–surface contact between the master (pile) and slave surface (geomaterial), (**c**) 3D pile group (2,2) model with C3D8R, an eight-node linear brick hexahedral element with reduced integration, (**d**) isometric view of pile group (2,2). The relationship between the peak twist angle and the number of cycles, \(N\) is expressed in matrix form as follows:  
\[[\theta_{t}]= \begin{bmatrix}\log_{e}N&1\end{bmatrix}\begin{bmatrix}a_{t}\\ b_{t}\end{bmatrix}  \tag{25}\]  
\[\theta_{t}=a_{t}\log_{e}N+b_{t} \tag{26}\]  
\[N=e^{\frac{b_{t}-b_{t}}{a_{t}}};\;N=1;\;\theta_{t}=b_{t} \tag{27}\]  
where \(a_{t}\) and \(b_{t}\) are the logarithmic degradation parameters for the twist (Table 3), which depend upon the number of cycles, dilation angle, and plastic strain. The logarithmic parameter \(b_{t}\) is a measure of the residual angle of twist (\(\theta_{0}\)) required in excess to mobilize the torque in the pile group over and above the maximum dilation at the end of the first cycle (\(N=1\)). Figure 4: Sectional view of the geomaterial-pile group (**a–c**) Color scheme is selected to represent the magnitude of displacement, where red and blue color schemes are for maximum (7.694 \(\times\) 10\({}^{3}\)) and minimum (5.532 \(\times\) 10\({}^{3}\)) displacement respectively, (**d–f**) The geomaterial–pile group (2,2) sectional view showing plastic strain (43.44–14.19%) for 2\({}^{\circ}\) to 12\({}^{\circ}\) dilation. Figure 6 shows the variation of plastic strain (%) with the 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 plastic strain increases with an increase in the number of cycles.