### Tension cracks subparalel to the free surface

As it was already mentioned, strain in direction of pressure wave propagation (compression) is numerically equal to strain in the plane of wave front (tension). Looking from the borehole it is possible to differentiate 3 zones:

1. First zone where pressure load is larger than strength of the rock and where rock between radial cracks is sheared. Many authors identify this zone as crushing zone [1] ,
2. Zone where only radial tension cracks are formed due to the tension in plane of the pressure wave front and compression in direction perpendicular to the pressure wave front in elastic zone. This is zone where only tensile failure occurs.
3. In third zone, all strains are smaller than strain that cause rock failure. This is zone of elastic deformations.

Figure 1 Pressure-time history for gas pressure in the boreholes [2]

After the drop of pressure in the blasthole, after few milliseconds Figure 1, rock between pressure wave front and crushing zone, that has been under elastic deformation, is returned to its initial deformation state.
Cylindrical explosive charge, whose axis is parallel with the free surface, placed at the distance "B" from the free surface (Figure 2) is the charge with normal burden. Distance $B$ is the burden of the explosive charge and can be calculated using the expression:

$$B=r_{c4} \cdot \cos 45^{\circ}$$

Or:
$$B=\frac{0.17 \cdot P_{h} \cdot r_{h}}{k\cdot \sigma_{t}}$$

Figure 2 Normal burden of explosive charge

With the cylindrical explosive charge, with normal burden, pressure wave propagates cylindrically and forms radial tension cracks. When two radial cracks reach the free surface rock \emph{wedge} is formed (Figure 3).

Figure 3 Rock wedge formation from two radial tension cracks

Pressure wave reaches the free surface before the wedge is formed as illustrated at Figure 4. Rock particles that form the free surface have no rock medium to transfer the strain energy so they continue to move in same direction (pressure wave propagation). Next $rows$ of particles are following this motion in same manner. Distance between rock particles was decreased proportionally to compressive load, i.e. intensity of pressure wave. If rock material would be ideally elastic, rock particles would move to the equilibrium state and then continued to move for the quantity of compressive strain. This means that between two particles tension is formed instead of compression. Strain would be same, but with different sign.

Figure 4 Pressure wave reaching the free surface

Since real rock material is not ideally elastic, but plastic, only one portion of compressive energy will be recoverable and available for tension after sudden unloading (Figure 5).

Figure 6 illustrates the complete stress-strain (loading-unloading) curves for fine grained magmatic and porous sedimentary rocks. From here it is easy to notice the large difference between absorbed and recoverable strain energies for those typical rock materials. It is logical to conclude that ratio between compressive strain and tensile strain is same as ratio between total strain energy (absorbed + recoverable) and recoverable strain energy.

Figure 5 Absorbed and recovered strain energy

Figure 6 Complete stress-strain curves for a) fine grained magmatic rocks b) porous sedimentary rocks

Index of strain energy recoverability (Figure 5) can be expressed as:

$$I_{sr} =\frac{E_{r}}{E_{t}}$$
$$E_{r} =\int_{e_{p}}^{e_{t}} f_{1}(e)de$$
$$E_{t} =\int_{0}^{e_{t}} f_{2}(e)de$$

Where:
$I_{sr}$- Index of strain energy recoverability,
$E_{r}$- recoverable strain energy,
$E_{t}$- total strain energy (recoverable + absorbed).

Tensile strain in radial direction, at distance \emph{B} from the borehole, is expressed as:
$$e_{rt}=\frac{P_{h} \cdot r_{h} \cdot I_{sr}}{k \cdot E \cdot B}$$

For the formation of one tensile crack necessary strain is:
$$e_{t}=\frac{\sigma_{t}}{E}$$
At the distance \emph{B} number of formed tensile cracks is:
$$n=\frac{e_{rt}}{e_{t}}=\frac{P_{h} \cdot r_{h} \cdot I_{sr}}{k \cdot B \cdot \sigma_{t}}$$

Therefore, first rension crack subparallel with the free surface is formed at the distance \emph{b} from the free surface:
$$b=\frac{B}{n}=\frac{B^{2} \cdot k \cdot \sigma_{t}}{P_{h} \cdot r_{h} \cdot I_{sr}}$$

Figure 7 Formation of tension cracks subparallel with the free surface

Next tension crack forms at the distance $b_{1}$ that is smaller than distance $b$ since tensile strain is larger, so distance $b_{2}$ is smaller than $b_{1}$ and so on (Figure 7).
If explosive charge is placed at the distance:
$$B + b < r_{4}$$
spalling would occur since explosive charge is further than $B$ and radial cracks are not reaching the free surface and rock wedge is not separated. Anyhow, if rock has large recoverable strain energy, then $b$ is very small and that part of rock is fractured as it is already explained (Figure 8).

Figure 7 Spalling of free surface

Complete paper is available at [3].

### References

[1] B. Whittaker, R. Singh, and G. Sun, Rock fracture mechanics: principles, design, and applications, Developments in geotechnical engineering, Elsevier, 1992.

[2] S.H. Cho and K. Kaneko, Rock fragmentation control in blasting, Materials transactions 45 (2004), pp. 1722–1730.

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