## 3 Modelling of the Detonation Process of Mines or IEDs and the Propagation of the Shock Wave  
Military vehicles should comply with special requirements in the field of protection against mines and IEDs. The NATO documents and the documents of institutes cooperating with NATO constitute the basis for determining requirements regarding the protection levels of the crew members of military vehicles against the shock wave of the explosion of mines and they also present the test methodology (e.g., a size of an explosive or its location towards a vehicle) which should be applied to every vehicle's construction. There are a few known methods which enable the numerical description of the detonation phenomenon and its effects on the construction. The most famous functions for the majority of packages intended for engineering calculations are the following: CONWEP, Arbitrary Lagrangian-Eulerian (ALE) and Smooth Particle Hydrodynamics (SPH) [30]. Each of them has a different way of preparing the model, the so-called numerical cost, and the type of output data which can be obtain through those functions. In order to estimate the maximum values of pressure at the front of the incident wave, the dependencies specified by [PERSON], [PERSON], [PERSON] and [PERSON] have been used and applied in the LS-DYNA system [31] and they have been implemented in the LOAD_BLAST_ENHANCED procedure. The variable in all models is constituted by a reduced distance. It is defined as:  
\[Z=\frac{R}{\sqrt[3]{M_{\rm INT}}} \tag{1}\]  
Figure 4: Physics of a mine explosion under a vehicle. Own study based on [25]. where:  
R--distance between the observation point and the explosion point,  
M\({}_{\rm TNT}\)--TNT mass equivalent for the considered explosion. In the case of the dependency specified by [PERSON], the maximum value of overpressure \(p_{\rm s}\) is described with the following dependency (the value in bars, \(Z\) in [m\(\cdot\)kg\({}^{-1/3}\)]):  
\[p_{\rm s}=\frac{0.975}{Z}+\frac{1.455}{Z^{2}}+\frac{5.85}{Z^{3}}-0.019 \tag{2}\]  
when: \(0.1<p_{\rm s}<10\) bar. The dependency specified by [PERSON] is presented by the following formula (the value in bars, for \(1<Z<10\)):  
\[p_{\rm s}=\frac{0.662}{Z}+\frac{4.05}{Z^{2}}+\frac{3.288}{Z^{3}} \tag{3}\]  
[PERSON] and [PERSON] provide the following dependency in order to calculate the maximum overpressure (the value bars, for \(1<Z<15\)):  
\[p_{\rm s}=\frac{0.754}{Z}+\frac{2.457}{Z^{2}}+\frac{6.5}{Z^{3}} \tag{4}\]  
In the case of a risk of a mine explosion, the detonation of an explosive located on the ground is more interesting than a free explosion in the air. The value of pressure in such a case is higher because it is strengthened as a result of interactions with the ground. In the case of an explosion on the perfectly rigid ground, the maximum value of pressure would correspond to the pressure generated for an explosive of twice the mass. However, due to the ground susceptibility, this value is smaller. The value of pressure at the front of the wave can be calculated using the following formula [32]:  
\[p_{\rm s}=exp\Big{(}A+B\mathrm{a}\cdot ln(Z)+\mathrm{Ca}\cdot ln(Z)^{2}+D \mathrm{a}\cdot ln(Z)^{3}+E\mathrm{a}\cdot ln(Z)^{4}+F\mathrm{a}\cdot ln(Z)^{5 }+G\mathrm{a}\cdot ln(Z)^{6}\Big{)} \tag{5}\]  
where: A-G--equation coefficients dependent on Z. for Z in the range from 2.9 to 23.8 m/kg1/3 have the following values: A = 7.5938, B = -0.0523, C = 0.4098, D = 0.0261, E = -0.0127, F = G = 0. The LOAD_BLAST_ENHANCED procedure implemented in the LS-DYNA system has been developed on the basis of the papers of [PERSON] and [PERSON] from 1997 [30]. The scope of use of this procedure includes:  
* For the explosion of a spherical load in the air--2.7 up to 750 load radiuses,
* For the explosion of a hemispheric load in the proximity of the ground--2.6 up to 600 load radiuses. For that reason, it is assumed that the scope of use includes the distance of the obstacle from the centre of the load of approx. 3 to 600 times bigger than the radius of the explosive.