## 2.3.1. Simulation method

An equivalent circuit of granite fragmentation using high voltage pulse discharge is shown in Fig. 3. Capacitor energy storage Marx generator are mostly used in .High voltage pulsed discharge technologies. The capacitor *C* is used to store energy from power supply. The switches *K* can be gas discharge switches, mechanical switches, or high voltage solid-state switches. The circuit resistance *R* consists of connecting wires and gas discharge switches. The circuit inductance *L* includes inductance on gas discharge switches, connecting wire and discharge channels formed between the electrodes. The stray capacitance *C*S is the sum of the stray capacitances of all components in the generator. The stray inductance *L*1 is the sum of the stray inductances of all components in the generator. The resistance of discharge channel *R*dc is resistance of granite when a breakdown is formed between two electrodes. Water acts as an insulting medium between the electrodes. When gas discharge switch is closed, the capacitor energy storage Marx generator produces a high voltage pulse to impact the granite and forms a main discharge channel. The energy from the capacitor is repeatedly injected into the main discharge channel. The energy causes the increase of pressure and radial size in the channel, and forms a mechanical stress wave, resulting in the breakage of the granite.

The loop resistance *R* mainly consists of the connecting wire resistance and gas discharge switch resistance in the generator. When the whole loop is turned on (*t* = 0), the resistance *R* decreases exponentially, and it can be expressed as:

$$R={R_{{\text{lm}}}}+\left( {{R_0} - {R_{{\text{lm}}}}} \right){e^{ - {t \mathord{\left/ {\vphantom {t {{t_R}}}} \right. \kern-0pt} {{t_R}}}}}$$

1

Where *R*0 is the resistance at *t* = 0 In the high-voltage pulse discharge crushing process; *R*lm is the limit minimal value of the resistance; Value of the *R*lm for the usually used 6–10 stages Marx generators with the gas discharge switches is equal to 1–3Ω [23]; *t*R is the resistance decrease characteristic time.

In accordance with Kirchhoff’s law, when the *K* is closed, the sum of the voltage drops on all components in the entire loop is 0, which can be expressed as:

$${U_C}+{U_R}+{U_L}+U{}_{{{L_1}}}+{U_{{\text{dc}}}}=0$$

2

Where *U*C is the voltage on the *C* of the Marx generator; *U*R is the voltage drop on the *R*; *U*L is the voltage drop on the *L*; *U*L1 is the voltage drop on the *L*1; *U*dc is the voltage drop on gap between the two electrodes. Similarly, the entire loop current also has a certain relationship, and it can be expressed as:

$$I={I_{{\text{dc}}}}+{I_{{C_{\text{S}}}}}={I_{{\text{dc}}}}+{C_{\text{S}}}\frac{{d\left( {{U_{{\text{dc}}}}+{U_{{L_1}}}} \right)}}{{dt}}$$

3

Where *I* the current flowing through the *C* of the Marx generator, that is, the *I* is the sum of the current on all stray capacitors and the current *I*dc on the gap between the two electrodes.

Combining equations (1)-(3), the voltage drops *U*dc(*t*) and current *I*dc(*t*) on the gap between the two electrodes can be calculated in the process of broken granite by high voltage pulse discharge.

The same side needle-pin discharge electrode structure is used in granite crushing, the strong electric field acts on the inside of the rock between the two electrodes to form discharge channels, which makes the rock change from a non-conductive state to a conductive state. The growth of discharge channel is usually described by the step probability function of local electric field. In the expansion direction *n* of the discharge channel, if the projection value *E**n* of the local electric field formed exceeds its critical electric field strength *E*c, the growth probability density of the discharge channel *ω**n* is proportional to the square of the *E**n*, and can be expressed as:

$${\omega _n}=\alpha \vartheta \left( {{E_n} - {E_{\text{c}}}} \right)E_{n}^{2}$$

4

Where *α* is the growth rate coefficient of discharge channel; *θ*(*x*) is the step function (when *x* > 0, *θ* (*x*) = 1; When *x* ≤ 0, *θ*(*x*) = 0); The square dependence of the discharge channel growth probability on electric field intensity is caused by the formation of conductive phase by electric field energy consumption.

The Gauss's theorem for calculation of the distribution of electric potential *φ* in the process of granite fragmentation by high voltage pulse is used:

$$\nabla \left( { - \varepsilon \cdot \nabla \varphi } \right)={{{\rho _{{\text{tv}}}}} \mathord{\left/ {\vphantom {{{\rho _{{\text{tv}}}}} {{\varepsilon _0}}}} \right. \kern-0pt} {{\varepsilon _0}}}$$

5

Where *ε*0 and *ε* are the absolute and relative permittivity of the granite, respectively; *ρ*tv is the total volume density of the free charges in the forming discharge channels and granite.

The change of the volume charge density *ρ*V in the granite is calculated from the charge conservation law:

$$~\frac{{\partial {\rho _V}}}{{\partial t}}=\nabla \left( {\sigma \nabla \varphi } \right)$$

6

Where *σ* is the specific conductivity of the granite.

The variation of the linear charge density *ρ**l* along the forming discharge channel is obtained by the continuity equation and Ohm's law:

$$\begin{gathered} \frac{{\partial {\rho _l}}}{{\partial t}}= - \frac{\partial }{{\partial l}}{I_{\text{D}}} \hfill \\ {I_{\text{D}}}=\eta {E_l} \hfill \\ \end{gathered}$$

7

Where *η* is the conductivity of the discharge channel per unit length which is determined by the product of specific conductivity of the plasma within the discharge channel and square of the discharge channel the cross-section; *l* is the coordinate along the discharge channel; *E**l* is the electric field projection along the coordinate direction of the discharge channel.

the variation of the linear conductivity during the development of discharge channel in granite is defined by the modified Rompe-Wiezel equation:

$$\frac{{\partial {\kern 1pt} \eta }}{{\partial {\kern 1pt} t}}=\upsilon \cdot \eta \cdot E_{l}^{2} - \kappa \cdot \eta$$

8

Where *υ* and *κ* are parameters of the growth and decrease rate of the discharge channel conductivity respectively. The first term on the right side of the Eq. (8) represents the increase of the conductivity due to joule energy release within the discharge channels. The second term depicts the decrease of the conductivity caused by the energy dissipation into the surrounding space.

With the help of the boundary condition, the operation characteristics of a high voltage pulse discharge breaking device are matched with the development of the discharge channel. Supposing the potential on the high voltage electrode is the voltage *U*dc between the two electrodes, and the potential on the ground electrode is 0. The total current *I*dc between the two electrodes calculates:

$$\int\limits_{{{S_{{\text{he}}}}}} { - \varepsilon {\varepsilon _0}} \frac{d}{{dt}}\left( {\overrightarrow \nabla \varphi - \sigma \overrightarrow \nabla \varphi } \right)\overrightarrow d s+\sum {\eta {E_l}} ={I_{{\text{dc}}}}$$

9

Where *S*he is the upper right side of the entire simulation region and high-voltage electrode. The first term of the integral on the right of Eq. (9) is the displacement current flowing through the surface of the high voltage electrode, and the second term on the right is the volume conduction current of the discharge channel.

The energy expression within the discharge channel per unit length is acquired by energy conservation equation between mechanical energy and electrical energy:

$$\frac{1}{{{\gamma _*}}}\frac{{d\left( {{P_{{\text{dc}}}}{S_{{\text{dc}}}}} \right)}}{{dt}}+{P_{{\text{dc}}}}\frac{{d{S_{{\text{dc}}}}}}{{dt}}=\frac{d}{{dt}}\left( {{S_{{\text{dc}}}}\sigma {E_\ell }^{2}} \right)$$

10

Where *P*dc is the pressure within the discharge channel; *γ** is the adiabatic index; *S*dc is the cross-sectional area of the discharge channel.

For homogeneous and isotropic materials, it is supposed that the discharge channel is depicted by cylindrical elements of fixed length *l*dc and raidus *r*dc(*t*), and *l*dc > > *r*dc(*t*). Meanwhile, each element is independent and its physical quantities are the same. The relationship between the stress tensor and the strain tensor can be defined by the Hooke's law:

$${\sigma _{ij}}=2\mu {e_{ij}}+\lambda \left( {\sum\limits_{k} {{e_{kk}}} } \right){\delta _{ij}}$$

11

Where *σ*ij is the stress tensor; *e*ij is the strain tensor; *δ*ij is the mechanical stress; *λ* and *µ* are respectively the Raman constants.

the dynamic change of the displacement vector \(\overrightarrow v\) is determined by the following equation

$${\rho _{{\text{tv}}}}\frac{{{d^2}\overrightarrow v }}{{d{t^2}}}=\left( {\lambda +\mu } \right)\overrightarrow \nabla \left( {\overrightarrow \nabla \cdot \overrightarrow v } \right)+\mu {\overrightarrow \nabla ^2}\overrightarrow v$$

12

In the process of granite deformation, the pressure within the discharge channel is equal to the elastic force exerted by the surrounding materials on the discharge channel. The matching between the discharge channel model and the material deformation model can be realized by the Newton's third law,

$$P=\frac{{{F_{\text{w}}}}}{{2\pi {r_{{\text{dc}}}}{l_{{\text{dc}}}}}}$$

13

Where *l*dc is the total length of the discharge channel; *F*w is the force of the surrounding materials acting on the discharge channel; *r*dc is the radius of the discharge channel.

In the simulation, the initial conditions need to be given:

$$\left\{ \begin{gathered} t=0 \hfill \\ U(t=0)={U_{\text{C}}} \hfill \\ I(t=0)=0 \hfill \\ W(t=0)=C{U_C}/2 \hfill \\ \end{gathered} \right.$$

14

The boundary condition is that electric potential *φ* of the upper left and the positive electrode of the model region is *U*D, and the electric potential *φ* of the upper right, the ground electrode and the bottom surface of the model region is 0.

On the basis of the finite difference approximation of the model equations (1) ~ (13) and the initial conditions (14), the discharge voltage and current, energy and power released into the discharge channel, conductivity, resistance, quantity, length, pressure and radius of the discharge channel are calculated by method of the explicit-implicit scheme, and it describes the process of forming a discharge channel in the granite under the action of the discharge of a high voltage pulse.

For high voltage pulse discharge granite breaking, the crushing efficiency depends on the depth of the main discharge channel into the granite. The maximum depth of the main discharge channel can be expressed as:

$${h_{\hbox{max} }}=\mathop {\hbox{max} }\limits_{{{L_{\text{M}}}}} {h_{\text{p}}}$$

14

Where *h*p is the depth of the discharge channel part; *l*M is the main discharge channel.