# Real-Time Behaviour of Dredged Slurry Treated by Air-Booster Vacuum Consolidation
## 8 Discussion
### Crack Initiation Pressure
The initial water content of the test soil is given in Table 3. During the vacuum consolidation process before air boosting, the water yield is recorded as 8610 g. Accordingly, the water content of the test soil at the moment of air boosting can be determined as 59.4%. Thus, \(I_{L}\ =\ 1.12\) and \(c\) = 1.21 kPa can be estimated. The [PERSON]'s modulus, \(E\), of the test soil can be estimated based on the unloading strain of the soil or by the empirical formula ([PERSON] and [PERSON], 1987):  
\[E\ =\ 100q_{u} \tag{7}\]  
from which \(E\ =242\) kPa can be obtained. It is known that vacuum consolidation will not change the total stress state of the soil. The vertical stress \(\sigma_{1}\) of the slurry at the mid-height of the booster plate, where the cracks first appear, can be estimated according to the self-weight; i.e., \(\sigma_{1}\ =\ \gamma h\ =\ 1600\frac{\mathrm{kN}}{\mathrm{m}}\times 0.4\ \mathrm{m}\ =\ 6.4\) kPa. Since the water content of the slurry soil at the end of vacuum consolidation is still higher than the liquid limit, the lateral pressure coefficient, \(K_{0}\), of the test soil is taken as \(K_{0}\ =\ 1\). Thus, the minimum principal stress \(\sigma_{3}\ =\ K_{0}\sigma_{1}\ =\ 6.4\) kPa can be determined. The values of the abovementioned parameters are summarised in Table 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Minimum principal stress (kPa) & \(\sigma_{3}\) & 6.4 \\ Vertical stress (kPa) & \(\sigma_{1}\) & 6.4 \\ Lateral pressure coefficient & \(K_{0}\) & 1 \\ Unconfined uniaxial compression strength (kPa) & \(q_{u}\) & 2.42 \\ Tensile strength & \(\sigma_{t}\) & \((\frac{1}{5}\times\frac{1}{5})\ q_{u}\) \\ Cohesion (kPa) of soil & \(c\) & 1.21 \\ Friction angle of soil & \(\varphi\) & 0 \\ Liquidity index & \(I_{L}\) & 1.12 \\ Unit weight of soil (kN/m\({}^{3}\)) & \(\gamma\) & 1600 \\ Thickness of soil (m) & h & 0.4 \\ Shear modulus of the soil (kPa) & \(G\) & 80.67 \\ Young’s modulus (kPa) & \(E\) & 242 \\ Poisson’s ratio & \(\theta\) & 0.5 \\ Expansion displacement (mm) & \(\delta\) & 1.5 \\ Cavity radius (mm) & \(a\) & 3.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Intermediate parameters. Substituting the parameters in Table 5 into Equations (1)-(5), the crack initiation pressure of the test soil can be quantified (Table 6). For Equation (4), \(p\) at \(t^{*}~{}=~{}7\) s is taken as the crack initiation pressure. From Figure 7, the expansion displacement \(\delta~{}=~{}1.5\) mm can be obtained. The initial cavity radius, \(a\), can be determined according to the area equivalence between the circular cavity section and the rectangular section of the booster plate (sectional dimensions given in Table 1). From Table 6, it is clear that Equations (2) and (3) provide the closest crack initiation pressure to that observed in our model test, i.e., 8 kPa (Figures 7 and 9). The crack initiation pressure estimated from the cavity expansion displacement is approximately 36% higher than 8 kPa. This discrepancy may be due to the fact of an overestimated shear modulus or to the shape effect of the PVD section.