# Double-Borehole Superimposed Effect of a New Non-Explosive Directional Rock-Breaking Method
## 2 Methodology
### The Numerical Model Calculation Principle of IESF  
#### 2.3.1 The Numerical Calculation Theory of IESF  
In order to characterize the uneven distribution of rock composition and structure in different directions, the Weibull distribution function [41] was used to describe the mesoscopic unit mechanical parameters of rock.  
\[\varphi(s,m)=\frac{m}{s_{0}}(\frac{s}{s_{0}})^{m-1}\exp\left[-(\frac{s}{s_{0} })^{m}\right] \tag{19}\]  
where \(s\) is the mesoscopic unit mechanical parameters, including modulus of elasticity, Poisson's ratio, etc. \(s_{0}\) is statistical averages of the mesoscopic unit mechanical parameters. \(m\) is used to characterize the distribution pattern of mesoscopic unit mechanical parameters and is taken as 5.  
When the maximum principal stress of the meso-element reaches its tensile strength, the maximum tensile stress criterion is used as the damage threshold criterion, which can be expressed as  
\[F_{1}\equiv\sigma_{1}-\sigma_{t}=0 \tag{20}\]  
where \(\sigma_{1}\) is the maximum principal stress. \(\sigma_{t}\) is the uniaxial tensile strength.  
For the damage of meso-element under uniaxial compression or shear stress, the Mohr-Coulomb criterion is adopted as the damage threshold criterion, which can be expressed as  
\[F_{2}\equiv\sigma_{1}-\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}-\sigma_{c}=0 \tag{21}\]  
where \(\sigma_{3}\) is the minimum principal stress. \(\varphi\) is the internal friction angle. \(\sigma_{c}\) is uniaxial compressive strength.  
The elastic modulus of rock mass can be expressed as  
\[E^{\prime}=(1-D)E^{\prime} \tag{22}\]  
where \(E^{\prime}\) and \(E^{\prime\prime}\) are the elastic modulus of the rock before and after damage, respectively. \(D\) is the damage variable, which can be expressed as [42; 43]  
\[D=\left\{\begin{array}{ll}0&F_{1}<0,F_{2}<0\\ 1-\left|\varepsilon_{t}/\varepsilon_{1}\right|^{n}&F_{1}=0,dF_{1}/dt>0\\ 1-\left|\varepsilon_{c}/\varepsilon_{3}\right|^{n}&F_{2}=0,dF_{2}/dt>0\end{array}\right. \tag{23}\]  
where \(\varepsilon_{t}\) and \(\varepsilon_{c}\) are the maximum tensile principal strain and maximum compressive principal strain corresponding to tensile damage and shear damage of the element, respectively. \(\varepsilon_{t}\) and \(\varepsilon_{c}\) are the first principal strain and the third principal strain, respectively. \(n\) is a coefficient of unit damage evolution, and is 2.  
When IESF breaks the rock, the high energy gas pressure [44] generated by IESF is  
\[P_{g}=P_{g0}\frac{m}{s_{0}}\bigg{(}\frac{s^{\prime}}{s^{\prime}_{0}}\bigg{)}^{ m-1}\exp\bigg{(}-\bigg{(}\frac{s^{\prime}}{s^{\prime}_{0}}\bigg{)}^{m}\bigg{)} \tag{24}\]  
where \(P_{g}\) is the gas pressure generated by IESF. \(P_{g0}\) is the constant related to the peak pressure. \(s^{\prime}\) is the loading step. \(s^{\prime}_{0}\) is the constant related to the loading step. \(m\) is the shape parameter of homogeneity.  
When an explosive is used to break rock, the rock mass is subjected to the action of both stress wave and blast-induced gas. The loading equation of the stress wave is [45]  
\[P_{d}=P_{0}\exp\bigg{(}-g_{a}\frac{t}{t_{0}}\bigg{)}\sin\bigg{(}\frac{4\pi}{1 +t/t_{0}}\bigg{)} \tag{25}\]  
where \(P_{d}\) is the stress under the action of stress wave. \(P_{0}\) is the peak stress, which can be calculated by \(140e^{6}Q^{3/2}\); \(Q\) is the amount of explosive. \(g_{a}\) is the explosive attenuation rate. \(t_{0}\) is the loading time. The loading equation of blast-induced gas is shown in Equation (24).