# Numerical Simulation Study of Cavity Formation in Soil under Blast Load
## 3 Numerical Simulation Results and Discussion  
The numerical simulation was performed using Ansys 2020 R1 Autodyn 2D hydrocode via a Compaction EOS Linear model. It is a relevant model since it is based on pressure and density dependence, and it has been proven by experimental data.  
Also, the model is described in the Euler system (2D) with fixed boundaries that ensure the continuity of the material and prevent the reflection of the stress wave at the boundaries.  
For a Compaction EOS linear model, the response of materials is represented through a plastic compaction path defined as a piecewise linear function of pressure versus density, and the elastic unloading/reloading path is defined via a piecewise function of sound speed versus density. Both, the compaction and unloading/reloading paths are indicated by tabular data of compaction pressure versus density and tabular data of sound speed against density.  
A fixed compaction path derived from static compression data was applied. This ignores the pressure enhancement due to the energy absorption. The elastic bulk stiffness of the material is defined as a piecewise linear curve of sound speed (c) versus density (\(\rho_{0}\)) and is given by following equation:  
\[K=\rho_{0}\;c^{2} \tag{1}\]  
The level of compaction in the material is given by following equation:  
\[\alpha=\rho_{s}/\rho_{0} \tag{2}\]  
Initially, \(\rho_{0}\) will be equal to the value defined in the density property of the material. Material property \(\rho_{s}\) is the solid zero pressure density of the material and corresponds to the fully compacted material density. For a porous material, the initial density will be less than the solid density; hence, the value of \(\alpha\) will be greater than 1.0. As compaction takes place, \(\alpha\) will reduce to a value of 1.0 for the fully compacted state [13].  
The isentropic expansion of the detonation products was modeled by using the Jones-Wilkins-Lee (JWL) equation of state (EOS) given by the following equation [14]:  
\[P=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega }{R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E}{V} \tag{3}\]  
where \(P\) is the pressure; \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are the constants; \(V\) is the specific volume of detonation product over the specific volume of undetonated explosive; and \(E\) is the specific (volume) theoretical detonation energy.  
The JWL (Jones-Wilkins-Lee) constants for the ANFO explosive are determined using thermochemical EXPL05 V7.01.01 [13], applying the partial equilibrium detonation model, and they are given in Table 3. It should be noted that, according to the model, the JWL constants change as the VOD changes. Thus, for example, the constants will have different values for the same explosive at different charge diameters, which is related to the non-ideal behavior.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Type of Explosive** & **Density** & **Velocity of Detonation** & **Theoretical Heat of Detonation** \\
**(Trade Name)** & **(g/cm\({}^{3}\))** & **(Measured)** & **(Literature and Producer Declaration)** \\  & & **(m/s)** & **(KJ/kg)** \\ \hline ANFO (Paleace) & 0.90 & 2000 & 3597 \\ Ammonium nitrate powder & & & \\ explosive & 1.05 & 4000 & 4276 \\ (Permonex) & & & \\ Watergel (Riegel) & 1.25 & 4576 & 4591 \\ \hline \end{tabular}
\end{table}
Table 2: Parameters of the studied explosives.