## 4 Numerical Results
### Quantization of Dilation Angle  
[PERSON] [29] analyzes the angle of dilation in the different shear conditions in Figure 11, assuming that the rigid zones lie outside the shear zone and the peak friction angle \(\phi_{p}^{\prime}\) on the rupture surface consists of two parts [30,31]:  
\[\phi_{p}^{\prime}=\phi_{crit}^{\prime}+\psi \tag{11}\]  
where \(\phi_{p}^{\prime}=\) friction angle to the peak strength; \(\phi_{crit}^{\prime}\) = critical state friction angle for constant volume (CV) shearing, which is \(31^{\circ}\) in triaxial test [32]; \(\psi=\) the angle of dilation.  
Combined with the effect of the effective normal stress and the relative density on state-dependent dilatancy, a relative dilatancy index \(I_{R}\) of the form is presented  
\[I_{R}=I_{D}\big{(}10-lnp^{\prime}\big{)}-1 \tag{12}\]  
where \(I_{D}\) is the relative density; \(p^{\prime}\) is the effective mean normal stress. Then substitute the relative dilatancy index for the angle of dilation in Equation (12) as  
\[\psi=\phi_{p}^{\prime}-\phi_{crit}^{\prime}=3\big{[}I_{D}\big{(}10-lnp^{\prime }\big{)}-1\big{]} \tag{13}\]  
which presents a comprehensive review of the experimental data and suggests an empirical relation for the estimation of the peak angle of shearing resistance.  
To compare the change of dilation angle, this research extracts the friction angle contour of the compression side sand before and after scouring, as shown in Figures 12 and 13.  
Figure 11: The saw blades model of dilatancy by [PERSON] (1986).  
Figure 10: The effective stress contour in the compression side in (**a**) no scour condition (**b**) local scour condition.  
With the soil extruding during the pile displacement, the soil range of the friction angle increases. The area of dilatancy develops towards surrounding and underlying soils. Compared with Figures 12a and 13a, after scouring the peak friction angle is larger, which also appears when the pile head displacement augments. The dilatancy behavior of interface soil is enhanced after scouring. To quantify variation in normal stress of pile-soil interfaces when they are subjected to loading and stress change, the friction angle of remaining interface sand surface depth during pile head displacement is extracted as shown in Figure 14.  
After local scouring, the friction angle reaches its peak at nearly 0.1 \(D\) pile head displacement. The corresponding friction angle is 31\({}^{\circ}\) for original soil in CV triaxial test. Moreover, the dilation angle by simulation is about 8\({}^{\circ}\), and the result is similar to the calculation in Equation (12) as 9.7\({}^{\circ}\). The feasibility of Equation (13) in evaluating dilatancy is proved. The friction angle increases in no scour condition undoubtedly can be explained by the excessive increase in the effective normal stress caused by the extrusion of the surrounding soil. In the post soil extrusion, the curve of friction angle drop in local scour condition can be explained by the failure of sand structure and the stress relief. Besides, the carry limit of soil surface during pile extrusion is also reduced.  
Figure 12: The friction angle contour at the pile head displacement of (a) 0.1 \(D\); (b) 0.2 \(D\) in local scour condition.  
Figure 13: The friction angle contour at the pile head displacement of (a) 0.1 \(D\); (b) 0.2 \(D\) in local scour condition.