## 4 Application Example
### Case for Blast Loading
#### 4.1.2 Numerical Result  
The results of the numerical modelling of the multi-layer pavement under blast loading, with the incorporation of the above-mentioned material models, are summarized and compared with the blast test results. In the numerical results, the fringe level in the damage contour is the value for the scaled damage indicator \(\delta\), which is defined to describe the damage level of the material [14; 20; 23]. A scaled damage indicator \(\delta\) is related to the effective plastic strain \(\lambda\) in the material: (i) at the yield surface, \(\lambda\) = 0, leading to \(\delta\) = 0; (ii) at the maximum strength surface, \(\lambda\) = \(\lambda_{m}\), leading to \(\delta\) = 1; and (iii) at the residual strength surface, \(\lambda\) = \(\lambda_{r}\) >> \(\lambda_{m}\), leading to \(\delta\) = 1.99 \(\approx\) 2. Thus, the \(\delta\) value moving from 0 to 1 to 2 indicates that the failure surface migrates from the yield surface to the maximum strength surface and to the residual strength surface, respectively, as the material being stressed. In this study, when the residual strength of material reduces to 20% of its peak strength, the material seems to suffer severe failure. The plastic strain corresponded to that residual strength used to calculate the delta value. Both the laboratory and field test for AC material indicated that when \(\delta\) was greater than 1.8, the material would be severely damaged. The damage situation for the multi-layer pavement slab in field blast test is shown in Figure 11. Figure 11a shows that the blast pressure destroyed the upper half of the AC layer above the GST reinforcement. It is also noted that only the centre of the GST piece was burned off during the blast event. Figure 11b shows the resulting damage on the HSC layer after removing the top layer of asphalt. From this figure, it could be seen that the crater was very shallow and did not punch through the whole layer, and a crater of around 700 mm in diameter and a depth of 10 mm was formed on the HSC layer. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Contact type & TIEBREAK \\ Friction for static & 0.71 \\ Friction for dynamic & 0.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters for the interface simulation. Figure 11: Damage of the multi-layer pavement after blast loading: (**a**) damage pattern of the AC layer in field blast test; (**b**) damage pattern of the HSC layer in field blast test. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Units** & **Value** \\ \hline Density & \(\rho\) & kg/m\({}^{3}\) & 2100 \\ Shear modulus & \(G\) & MPa & 13.8 \\ Poisson’s ratio & \(\
u\) & - & 0.3 \\ Cohesion & \(c\) & kPa & 62 \\ Friction angle & \(\phi\) & \({}^{\circ}\) & 26 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Material properties of the soil mass. The damage contour for the AC layer in numerical model is shown in Figure 12a. Comparing Figures 11a and 12a, it is observed that the damage pattern in the numerical model is symmetrical, while that in the field measurement was skewed. This is because the bomb in the field was not placed at the centre of the slab, and one side of the AC layer was more severely damaged than the other. Shear cracking near the anchor point was observed in the numerical model, which was similar to the experimental observations in the field test. It could be concluded that the basic failure pattern given by the numerical model agrees well with the results obtained from the field-testing. Figure 12b shows the damage pattern for the HSC layer. Comparing Figure 11b with Figure 12b, the damage pattern for HSC is very consistent between field measurement and numerical results. The diameter of the crater was about 750 mm in the numerical model, which is very close to that of the blast test result. As shown in Figure 12b, shear cracks are also observed near the anchor points.