Radial tension crack formation under explosive load

If we take a look at the curve of ideal and non-ideal detonation  in p-t (pressure - time) diagram (Figure 1) it can be concluded that pressure at the borehole wall is applied instantly, and then it lasts for certain period of time. These are all properties of impact load. This load induces pressure wave that propagates cylindrically around the cylindrical explosive charge (Figure 2). Impact load is applied at the borehole wall in the zone of the reaction which is followed with its retaining for certain period of time.


Figure 1 p-t diagram for ideal and non-ideal detonation [1]


Schematic illustration of radial tension fractures formation

Figure 2 Schematic illustration of radial tension fractures formation


At the distance $r_{cn}$ from the borehole compressive stress of the rock in the radial direction is:


$$ \sigma _{rc}=P_{h}\frac{r_{h}}{r_{cn}}  \label{eqn:1} \tag{1}$$

 

Where:
   $\sigma_{rc} - $radial compressive stress, 
   $P_{h} - $ borehole pressure, 
   $r_{h} - $  borehole radius, 
   $r_{cn} - $ crack zone radius. 

 

 On the other side:

$$ \sigma_{rc}=M \cdot e_{r}  \label{eqn:2} \tag{2}$$

 

 $$ M=E\cdot \frac{(1-\nu )}{(1+\nu )(1-2\nu )}  \label{eqn:3} \tag{3}$$

Where:

   $M - $ pressure wave modulus [2]
   $e_{r} - $ radial strain 
   $E - $ Young`s modulus of rock 
   $\nu - $Poisson's ratio

 

With expression:

$$ k=\frac{(1-\nu )}{(1+\nu )(1-2\nu )}  \label{eqn:4} \tag{4}$$

Radial compressive stress from Equation \ref{eqn:2} can be expressed as:

 $$ \sigma_{rc}=E\cdot k\cdot e_{r} \tag{5}$$

Then, radial strain is:

 $$ e_{r}=\frac{\sigma_{rc}}{k \cdot E} \tag{6}$$

Considering the Equation \ref{eqn:1} it becomes:

 $$ e_{r}=\frac{P_{h}\cdot r_{h}}{E\cdot k\cdot r_{cn}} \tag{7}$$

If we focus on rock particles on cylindrical surface at the distance $r_{cn}$ from the borehole, their perimeter before pressure wave reaches is:

 $$ O_{r_{cn}}=2\pi r_{cn} \tag{8}$$

After pressure wave reaches these particles they are moved to a new positions  in similar cylindrical form with radius $(r_{cn}+\Delta r_{cn})$. In this case perimeter is increased:

 $$ O_{(r_{cn}+\Delta r_{cn})}=2\pi (r_{cn}+\Delta r_{cn}) \tag{9}$$

Respectively:

$$ O_{(r_{cn}+\Delta r_{cn})}=2\pi (r_{cn}+e_{r}r_{cn}) \tag{10}$$

Therefore, in front of pressure wave (in direction of its propagation) rock is under compressive load, and under tensile load in perpendicular direction with strain:

$$ e_{l}= \frac{O_{(r_{cn}+\Delta r_{cn})}-O_{r_{cn}}}{O_{r_{cn}}}=e_{r} \tag{11}$$

Respectively:

$$ e_{l}=\frac{P_{h}\cdot r_{h}}{E\cdot k\cdot r_{cn}} \tag{12}$$

Strain that will form one radial tension crack at distance $r_{cn}$ is:

$$ e_{t}=\frac{\sigma_{t}}{E} \tag{13}$$

Where:

   $e_{t} - $tensile strain,
   $\sigma_{t} - $ tensile strength,
   $E - $ Young's modulus of rock.

Number of radial tension cracks at the distance $r_{cn}$ is:

$$ n=\frac{e_{l}}{e_{t}} \tag{14}$$

Respectively:

$$ n=\frac{P_{h} \cdot r_{h}}{k \cdot \sigma_{t} \cdot r_{cn}} \tag{15}$$

Therefore:

$$ r_{cn}=\frac{P_{h} \cdot r_{h}}{k \cdot \sigma_{t} \cdot n} \tag{16}$$

 

Complete paper is available at [3].

 

References

[1] CUNNINGHAM, C. (2006) Blasthole Pressure: What it really means and how we should use it. In: Proceedings of the annual conference on explosives and blasting technique, 32, pp. 255.

[2] MAVKO, G., MUKERJI, T. and DVORKIN, J. (2009) The rock physics handbook: Tools for seismic analysis of porous media. Cambridge university press.

[3] Torbica, S., & Lapčević, V. (2018). Rock fracturing mechanisms by blasting. Podzemni Radovi, (32), 15-31. https://doi.org/10.5937/PodRad1832015T

 


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