## 3 Response of the Geomaterial to Cyclic Torsional Loading  
The pile group is placed into a homogenous isotropic granular geomaterial medium. The application of cyclic torsional loading on the pile group is resisted by the flow-controlled geomaterial. The stress-strain response of geomaterial is highly nonlinear in all phases of loading. The hyperbolic stress-strain curve ([PERSON] and [PERSON] [30]) is expressed as follows:  
\[\frac{G_{s}}{G_{max}}=1-f_{s}\bigg{(}\frac{\tau}{\tau_{max}}\bigg{)}^{g_{s}} \tag{11}\]  
where \(G_{s}\) and \(G_{max}\) are the initial and maximum shear modulus of geomaterial, respectively; \(\tau\) and \(\tau_{max}\) are the current and maximum shear stress, respectively; \(f_{s}\) and \(g_{s}\) are the curve fitting parameters. The parameter \(f_{s}\) controls the magnitude or the extent of degradation whereas the parameter \(g_{s}\) controls the rate of degradation and the curvature of the curve. A two-way torsional load applied at the centre of the pile group varies sinusoidally with time. The torsional and lateral resistances concurrently resist a portion of the applied torsional load (Kong [12]). Therefore, the initial modulus of subgrade reaction for the lateral load, \(k_{h}\), and torsional load, \(k_{\Theta}\), using the nonlinear \(p-y\) and \(\tau-\theta\) curves are expressed as follows:  
\[k_{h}=\frac{P}{y}\ ;\ k_{\Theta}=\frac{2T_{s}}{\pi D^{2}L} \tag{12}\]  
where \(P\) and \(T_{s}\) are the lateral and torsional geomaterial reactions per unit length of the pile (\(L\)), respectively. In the cyclic torsional analysis of pile groups, the geomaterial degradation has been captured by modifying the geomaterial spring stiffness (lateral translational spring and a torsional spring) after the completion of a number (N) of cycles. [PERSON] and [PERSON] [31] model calculates the deterioration of the geomaterial reaction modulus, \(k_{h}\), due to cyclic lateral loading and is expressed as follows:  
\[k_{hN}=k_{h1}N^{-\alpha} \tag{13}\]  
where \(k_{hN}\) takes the value of \(k_{h}\) at the \(N\)th cycle of load; \(k_{h1}\) is the value of the geomaterial lateral reaction modulus for the first cycle of load, and \(\alpha\) is the degradation parameter for geomaterial lateral modulus. Similarly, the deterioration of the geomaterial shear modulus, \(k_{\theta}\), due to cyclic torsional loading is expressed as follows:  
\[k_{\theta N}=k_{\theta 1}N^{-\beta} \tag{14}\]  
where \(k_{\theta N}\) takes the value of \(k_{\theta}\) at the \(N\)th cycle of load; \(k_{\theta 1}\) is the value of the geomaterial torsional reaction modulus for the first cycle of load; and \(\beta\) is the degradation parameter for geomaterial torsional modulus. The degradation parameters for geomaterial lateral reaction modulus, \(\alpha\) and geomaterial torsional modulus, \(\beta\), are expressed as follows:  
\[\alpha=\frac{E_{c}}{E_{s}};\beta=\frac{G_{c}}{G_{s}} \tag{15}\]  
where \(E_{c}\) and \(E_{s}\) are the geomaterial modulus for cyclic and static loading, respectively; \(G_{c}\) and \(G_{s}\) are the geomaterial shear modulus for cyclic and static loading, respectively. The degradation is insignificant unless pile-geomaterial slippage takes place at the interface ([PERSON] [32]). As the angle of the pile twist increases, the mobilized shear stress at the pile shaft will reach the limiting value, and a slip will occur between the pile and the geomaterial. The limiting shaft friction \(\tau_{u}\) ([PERSON] and [PERSON] [11]) can be expressed as follows:  
\[\tau_{u}=A_{t}z^{t} \tag{16}\]  
where \(A_{t}\) is the constant that determines the magnitude of the shaft friction and \(t\) is the corresponding non-homogeneity factor. A correlation between \(\theta_{i}\) and \(\tau\) for an initial no-slip condition ([PERSON] [7]) is expressed as follows:  
\[\theta_{i}=\frac{\tau}{2G_{s}} \tag{17}\]  
where \(\theta_{i}\) is the angle of twist for \(i\)th element; \(\tau\) is the pile-geomaterial interfacial shear stress; and \(G_{s}\) is the initial geomaterial modulus.