# Development and Validation of Universal 3D Blast Fragmentation Model
## 2 Fragmentation Model Fundamentals  
### Fragmentation Mechanism  
Explosive charge detonation creates a pressure wave in the surrounding rock that propagates around the blasthole and is transmitted from one \"particle\" to another. Bonds between particles are stressed in tension and break accordingly, while due to the circular/cylindrical shape of the pressure wave front, a radial pattern of fractures is formed (Figure 1). The forming of each fragment requires three fracture sets to be generated in mutually perpendicular positions. Hence, the length and density (spacing) of fractures in each set will determine its size. The foundation of the model presented herein is the rock fracturing theory [22], which provides constitutive relationships between rock and explosive properties for estimating the length and density of blast-induced fractures. The tensile nature of radial fractures is investigated numerically [23] and in practical identification [24].  
The length of radial tension fractures that fall in a specific density zone is calculated according to the following relationship [22]:  
\[r_{cn}=\frac{P_{h}\cdot r_{h}}{k\cdot\sigma_{l}\cdot n}\]  
Figure 1: Schematic illustration of radial tension fractures formation [22].  
where  
\(r_{cn}\)--crack zone radius;  
\(P_{h}\)--borehole pressure;  
\(r_{h}\)--borehole radius;  
\(k=\frac{(1-\
u)}{(1+\
u)(1-2\
u)}\);  
\(\
u\)--Poisson's ratio;  
\(\sigma_{l}\)--tensile strength;  
\(n\)--number of fractures in the zone.  
For a blasthole with radius \(r_{h}=0.051\) m and pressure in granite with a tensile strength of \(\sigma_{t}=14\) MPa and Poisson's ratio \(\
u=0.25\), the cracking zones are as presented in Figure 2.  
The high energy of the pressure wave is transferred from particle to particle, and at a certain point, this load will reach boundary particles at the free surface (Figure 3). Due to highly different densities between air and rock medium, these particles have high acceleration towards void space and no rock particles to transmit the load. If the load is high enough, those particles form a tension fracture subparallel to the free surface and can practically be \"torn\" from the rest of the rock medium.  
Figure 3: Pressure wave reaching the free surface [22].  
Figure 2: Radial tension fracture length and density around the blasthole [22].  
During the pressure wave propagation, each particle moves forward, transmits the load, and tends to return to its initial state. However, due to the plastic nature of rock medium, particles will return to a position different from their origin in relation to the elastic property of the rock medium or the recoverability of strain energy (Figure 4).  
This phenomenon relationship was developed to estimate distances between tension fractures subparallel with the free surface [22]. First, tension fracture is formed at a distance:  
\[b=\frac{B}{n}=\frac{B^{2}\cdot k\cdot\sigma_{t}}{P_{h}\cdot r_{h}\cdot l_{sr}}\]  
where  
\(B\)--burden of an explosive charge (\(B=\frac{0.17\cdot P_{h}\cdot r_{h}}{k\cdot\sigma_{t}}\));  
\(I_{sr}\)--index of strain energy recoverability;  
\(I_{sr}=\frac{E_{t}}{E_{t}}\)  
\(E_{r}=\int_{\epsilon_{p}}^{E_{t}}f_{1}(e)de\)  
\(E_{t}=\int_{0}^{\epsilon_{p}}f_{2}(e)de\)  
\(E_{r}\)--recoverable strain energy;  
\(E_{t}\)--total strain energy (recoverable + absorbed).  
The following tension crack forms at the distance \(b_{1}\) that is smaller than distance \(b\) since the tensile strain is larger, so distance \(b_{2}\) is smaller than \(b_{1}\) and so on (Figure 5).  
Figure 4: Absorbed and recovered strain energy [22].  
Figure 5: Formation of tension cracks subparallel with the free surface [22].  
The radial tension fractures, after reaching the free surface, for a rock wedge that is being subjected to the loads and a third set of blast-induced fractures, are formed. This third set is perpendicular to the free surface or the face of the excavation (Figure 6).