# Failure Mechanism of Back-Break in Bench Blasting of Thin Terrane
## 2 Structural Dynamics Model of a Layered Rock Mass
### Calculation Parameters and Failure Criteria
#### 2.3.2 Explosion Load  
According to the theory of the combined action of the explosion shock wave and explosion generated gas, the retained rock mass near the blast hole was first subjected to the action of the explosion shock wave and its action time was short, at tens of subtle levels. Under the action of the explosion shock wave, the surrounding rock broke to form a broken area and consumed a lot of energy and then attenuated to a stress wave and propagated to a distance. Under the influence of the explosion gas invasion and quasi-static pressure, the rock fracture then continued to develop and the degree of fragmentation continued to increase. [PERSON] used PMMA material to study the blasting fracture law under the condition of sealing and packing. It found that 8% of crack propagation was caused by the stress wave and the remaining 92% was caused by gas pressure [27]. The development of cracks weakens the properties of rock mass material and reduces the modulus of rock mass material, as shown in Equations (6) and (7). For pre-existing cracks before blasting, its influence is reflected in the initial material properties of the rock.  
\[\frac{G_{T}}{G}=1-\frac{32}{45}\frac{(1-\
u)(5-\
u)}{2-\
u}\chi \tag{6}\]  
\[\frac{E_{T}}{E}=1-\frac{16}{45}\frac{(1-\
u^{2})(10-3\
u)}{2-\
u}\chi \tag{7}\]  
where \(E_{T}\) and \(G_{T}\) are the equivalent elastic modulus and shear modulus, \(\
u\) is Poisson's ratio and \(\chi\) is the fracture density parameter. When \(\chi\) approaches 1, \(E_{T}\) and \(G_{T}\) are zero, which indicates the complete failure of the rock mass. After a calculation with the above formula, the maximum attenuation of the elastic modulus and the shear modulus under the influence of the shock was less than 15% of the original value. According to the research results of [PERSON] [28], the failure area formed by the shock wave was within five times of the blast hole radius and the fracture area was within 30 times of the blast hole radius. Therefore, in order to reflect the impact wave on the structural failure of the remaining rock mass, it was approximately considered that the structural stiffness of the rock mass within 30 times of the blast hole radius decreased by 15%.  
According to the theory of detonation thermodynamics, the expansion process of the explosive gas in the blast hole is an isentropic adiabatic process. Under the condition of a coupling charge, when the explosive gas fills the blast hole, the blast hole wall pressure \(P_{0}\) is [29]:  
\[P_{0}=\frac{\rho C^{2}}{2(\gamma+1)} \tag{8}\]  
where, \(\rho\) is the explosive density, \(C\) is the explosive velocity and \(\gamma\) is the adiabatic index. The value of the industrial explosives such as emulsion explosive and ammonium nitrate explosive is generally 3. According to the relevant research results, a triangle function is often used to express the equivalent load curve of the explosion load. The peak value of the explosion load is \(P_{0}\) in the \(t_{1}\) period and then linearly attenuates in the \(t_{2}\) period [29; 30]. According to the actual observation results of bench blasting and related theoretical studies [20; 31], the load action time of bench blasting should be in the order of 10 ms. In this paper, the pressure rise period \(t_{1}\) of the load was 1 ms and the pressure reduction period \(t_{2}\) was 10 ms to reflect the time history of the load. In addition, according to research results [32], under reasonable plugging conditions, the amount of compression of the plugging material in the plugging section was small. Within 10% of the length of the plugging section, in order to reflect the difference between the stemming section and the charge section and considering that some of the plugging materials would be thrown out of the blast hole, the peak value of the loading action of the particle on the stemming section was reduced to 15% of the original value.