# Numerical Analysis and Experimental Test for the Development of a Small Shaped Charge
## 2 Materials and Methods
The quantities \(M\), \(Q\) and \(E\) are total mass, total momentum and total energy, respectively, of control volume \(V(t)\), bounded by surface \(S\), which moves in the fluid (gas-air) with arbitrary velocity \(\omega\) which may be zero in Eulerian coordinates or \(v\) in Lagrangian coordinates. The vector \(n\) is the outwards normal to the surface \(S\). The Johnson-Cook (JC) constitutive model was used to describe the proper dynamic behavior of the cumulative insert and the steel block affected by the insert. It is an elastic-plastic model of the material. Plastic deformation in the material is modeled using isotropic material hardening. The yield point of the material is described by the following relationship:  
\[\sigma_{Y}=\Big{(}A+B\dot{e}_{p}^{n}\Big{)}\Big{(}1+Cln\dot{\varepsilon}^{*} \Big{)}(1-T^{*m}) \tag{4}\]  
where:  
\(A\)--static yield point,  
\(\dot{\varepsilon}_{p}\)--plastic deformation,  
\(\dot{\varepsilon}^{*}\)--dimensionless strain rate, \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{0}\)  
\(\dot{\varepsilon}_{0}\)--reference strain rate,  
\(T^{*}\)--the ratio of the absolute temperature of the sample to its melting point, which is determined by the following relationship:  
\[T^{*}=\left\{ \begin{array}{ccc}0&\text{d}la&T<T_{room}\\ \frac{T-T_{room}}{T_{melt}-T_{room}}&\text{d}la&T_{room}\leq T\leq T_{melt} \\ 1&\text{d}la&T>T_{melt}\end{array} \right. \tag{5}\]  
\(T_{room}\)--temperature at which the experiment was carried out,  
\(T_{melt}\)--material melting point,  
\(n\)--parameter determining the material's susceptibility to hardening by deformation,  
\(m\)--thermal plasticization exponent,  
\(B\)--hardening constant,  
\(C\)--strain rate constant. [PERSON] and [PERSON] proposed that fracture strain typically depends on the stress triaxiality ratio, the strain rate and the temperature. The strain at fracture is given by:  
\[\varepsilon^{f}=max\Big{(}[D_{1}+D_{2}expD_{3}\sigma^{*}]\Big{[}1+D_{4}ln\dot {\varepsilon}^{*}\Big{]}[1+D_{5}T^{*}],EFMIN\Big{)} \tag{6}\]  
where \(\sigma^{*}\) is the ratio of pressure divided by effective stress:  
\[\sigma^{*}=\frac{p}{\sigma_{eff}} \tag{7}\]  
Fracture occurs when the damage parameter:  
\[D=\sum\frac{\Delta\overline{\varepsilon}^{p}}{\varepsilon^{f}} \tag{8}\]  
reaches the value of 1. The data of the copper insert material [20] with the EOS (Equation of State) model, and the steel material model [21] with the EOS data [22] from which the target was made, are presented in Table 1.  
For the JC model, it is necessary to define a polynomial equation of state describing the relationship between pressure, volume and the internal energy in a material. The linear polynomial equation of state was used for the model of the copper insert material. This equation is expressed as:  
\[P=c_{0}+c_{1}\mu+c_{2}\mu^{2}+c_{3}\mu^{3}+\left(c_{4}+c_{5}\mu+c_{6}\mu^{2} \right)E_{0} \tag{9}\]  
where: \(c_{0}\div c_{6}\), state equation parameters; \(\mu\), compression factor \(\mu=\rho/\rho_{0}\) expressed as the ratio of the actual density \(\rho\) to the original density \(\rho_{0}\); \(E_{0}\), internal energy. The polynomial equation of state in a simplified form is used to describe the gas (air) medium surrounding the explosive charge and the tested object:  
\[P=(c_{4}+c_{5}\mu)E \tag{10}\]  
where: \(\mu=\rho/\rho_{0}\), \(C_{4}\) and \(C_{5}\), equation coefficients; \(\rho\), density; \(\rho_{0}\), starting density; E, internal energy. The Gruneisen equation of state was defined for the steel material model. The equation defines the pressure in the shock-compressed material as:  
\[p=\frac{\rho_{0}C^{2}\mu\left[1+\left(1-\frac{2\eta}{2}\right)\mu-\frac{\mu}{ 2}\mu^{2}\right]}{\left[1-\left(S_{1}-1\right)\mu-S_{2}\frac{\mu^{2}}{\mu+1}- S_{3}\frac{\mu^{3}}{\left(\mu+1\right)^{2}}\right]^{2}}+(\gamma_{0}+a\mu)E \tag{11}\]