# Study on Shock Initiation Randomness of Energetic Materials on a Macroscopic Scale
## Problem Analysis
### BRVs of the Problem  
Since the randomness of the shock initiation of energetic materials is caused by the randomness of the detonation characteristics of energetic materials and the randomness of shock load conditions, the BRVs of SIREM should also be divided into the BRVs on the detonation characteristics of energetic materials and the BRVs on shock load conditions. The following is the identification of these two kinds of BRVs. 1. BRVs on detonation characteristics of energetic materials  
In the numerical models of two-phase reactive flow of the shock initiation problem of energetic materials, the equations of reaction rate, showing the shock initiation reaction rate of energetic materials, generally adopt the Lee-Tarver equation of reaction rate, and the equations of state of the unreacted energetic materials of the condensed phase and the reaction products of the gas phase generally adopt the JWL equation of state. See the two in Equations (1) and (2) [30; 31; 32]. \[\frac{d\lambda}{dt}=I(1-\lambda)^{b}(\mu-a)^{x}+G_{1}(1-\lambda)^{c}\lambda^{d}p^{y }+G_{2}(1-\lambda)^{c}\lambda^{g}p^{z} \tag{1}\]  
where \(\lambda\) is the fraction of an energetic material that has reacted, \(t\) is the time,\(\frac{d\lambda}{dt}\) is the reaction rate, \(\mu\) is the current compression degree, \(a\) is the critical compression degree, and \(I\), \(b\), \(x\), \(G_{1}\), \(c\), \(d\), \(y\), \(G_{2}\), \(e\), \(g\), and \(z\) are the parameters. \[p=A\bigg{(}1-\frac{w}{R_{1}V}\bigg{)}\exp(-R_{1}V)+B\bigg{(}1-\frac{w}{R_{2}V} \bigg{)}\exp(-R_{2}V)+\frac{wE}{V} \tag{2}\]  
where \(p\) is the pressure, \(V\) is the relative volume, \(E\) is the energy in per unit volume, and \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(w\) are the parameters. The five parameters of unreacted energetic materials and reaction products are respectively marked as \(A_{c}\), \(B_{c}\), \(R_{c1}\), \(R_{c2}\), \(w_{c}\), and \(A_{g}\), \(B_{g}\), \(R_{g1}\), \(R_{g2}\), \(w_{g}\) in order to distinguish between the two types. The literature [32] has pointed out that the three terms in the Lee-Tarver equation of reaction rate are the ignition term, growth term, and completion term. For the parameters of these three terms, \(a\) controls the critical compression degrees of energetic materials, and when \(\mu\) is less than \(a\), energetic materials cannot be initiated; \(I\), \(G_{1}\), and \(G_{2}\) are the coefficients of these three terms; \(b\), \(c\), \(d\), \(e\), and \(g\) are the burnup indices of these three terms; \(y\) and \(z\) are the pressure indices of the growth term and completion term; \(I\) and \(x\) control the number of hot spots; \(b\) and \(c\) control the combustion of inward spherical particles; \(G_{1}\) and \(d\) control the reaction growth duration at the early stage of hot spots; and \(G_{2}\) and \(z\) control the reaction rate at high pressure. For the parameters of the JWL equation of state, \(A_{c}\), \(B_{c}\), \(A_{g}\), and \(B_{g}\) are the parameters of the linear relationship between the detonation pressure and the isentropic expansion pressure, and \(R_{c1}\), \(R_{c2}\), \(w_{c}\), \(R_{g1}\), \(R_{g2}\), and \(w_{g}\) are the parameters related to energetic materials. Therefore, the parameters of these two equations can fully describe the detonation characteristics of energetic materials, and the effect of the changes of the factors, such as the density, temperature, and humidity, on the detonation characteristics of energetic materials can also be expressed by the numerical changes of these parameters. In addition, the parameters of the Lee-Tarver equation of reaction rate are fitted by the shock initiation tests and numerical simulations [33; 34; 35], and the parameters of the JWL equation of state are fitted by the cylinder tests and numerical simulations [36; 37; 38]. The literature [5; 22; 23; 24; 25] suggests that such parametric measurement tests and corresponding numerical simulations generally have random errors, and so do the parameters obtained.