Numerical study on fractal characteristic of blast-induced cracks in deep rock mass

The quantitative representation on the characteristics of blasting-induced cracks in rock mass is of great significance to guide the fine blasting design in practice. However, the quantitative relation between blasting-induced cracks of rock mass and in-situ stresses is currently lacking. To provide the reference for the fine blasting design to excavate rock mass with different in-situ stresses, the present study adopts the fractal theory to quantitatively represent the characteristics of blasting-induced cracks in rock mass. The effects of in-situ stress levels and lateral coefficients on the modes and fractal dimensions of blasting-induced cracks are investigated by numerical simulation combined with image processing. The results show that with the increase of in-situ stresses and lateral coefficients, the fractal dimensions of blast-induced cracks gradually decrease. In addition, quantitative relations between the fractal dimensions of blasting-induced cracks and in-situ stresses are established to provide the reference for the design of rock blasting.


Introduction
The demands on mineral resources increasingly grow and significantly promotes the development of underground mining toward the deep earth [1].However, the high in-situ stresses in deep earth greatly hinders the connection of the blast-induced cracks between different blasting boreholes, thereby easy to cause poor fragmentation of excavated rock [2,3].Therefore, it is essential to investigate the effects of in-situ stresses on the blasting-induced cracks to give the reference for the blasting design in deep mining and improve the mining efficiency in deep rock mass.
Many studies [4][5][6][7] have examined the actions of in-situ stresses on the initiation, propagation and coalescence of blasting-induced cracks.For instance, Xie et al. [8] numerically investigated the effects of in-situ stresses/confining stresses on the forming results of cracks induced by cut blasting in deep rock mass.The results showed that the lengths of blasting-induced cracks decreased with increasing hydrostatic stress.It is noted that previous studies mainly focused on the qualitative discussion on the effects of in-situ stresses on blasting-induced cracks of rock mass.Very limited study has provided quantitative relation between in-situ stresses and blasting-induced crack levels.
To quantitatively represent the levels of blasting-induced cracks, the fractal theory has been widely used in existing studies [9].For example, Yang et al. [10,11] calculated the fractal dimensions of blasting-induced cracks with varying the charge weight and eccentric charge blasting.The results indicated the fractal dimensions well characterized the levels of blasting-induced cracks.In addition, Ding et al. [12,13] experimentally investigated the effects of decoupled coefficients, coupled media, and uniaxial stress on the fractal dimensions of blast-induced cracks.However, the uniaxial stress cannot indicate the real in-situ stress states of deep rock mass.Currently, the effects of in-situ stresses on the characteristics of blasting-induced cracks has not been well investigated.The quantitative relation between them is still lacking.
In this study, The fractal characteristics of blast-induced cracks under different in-situ stresses are investigated using LS-DYNA.Firstly, the numerical model of rock mass with a single borehole is established and is calibrated based on the results of a small-scale blasting test.Then, the modes of blast-induced cracks under different confining stresses (i.e., in-situ stresses) and lateral stress coefficients are examined.The fractal dimensions of blasting-induced cracks of rock mass are quantitively compared.Finally, a quantitative relation between fractal dimensions of the blast-induced cracks and in-situ stresses is established.

Material model and validation
In LS-DYNA, many material models can be used to simulate the behavior of rock mass subjected to blasting loading.Amongst, the RHT model is popular one [3,8], which is used in this study.The details of RHT model can refer to Borrvall and Riedel [14].In this study, a small-scale blasting test of cylinder rock subjected to internal blasting loads conducted by Banadaki [15] is utilized to calibrate the model.The rock mass employed in the test is a cylindrical granite sample with a diameter of 144 mm.The PETN explosive with a diameter of 1.65 mm is filled at the center of the granite sample.A two-dimensional numerical model with the same size as the granite sample in the test is established in this study.The parameters of the RHT model to simulate the granite are given in Table 1.*EOS_JWL and *MAT_HIGH_EXPLOSIVE_BUREN are respectively used as the equation of state and material models of the PETN explosive.Its density, velocity of detonation and bursting pressure are determined as 1320 kg•m -3 , 6690 m•s -1 and 16 GPa, respectively.*EOS_JWL parameters of the explosive, i.e., A=586 GPa, B=21.6 GPa, R1=5.81,R2=1.77, ω=0.282,E0=7.38 GPa are mainly determined based on Banadaki [15].Figure 1(a) and (b) respectively show the measured and processed results of blastinduced cracks modes.Figure 1(c) shows the simulated results.It can be found that the distributions of simulated blast-induced cracks have good agreements with the experimental results.Therefore, it is concluded that the RHT model can accurately predict the blasting-induced response of rock mass.[15], (b) processed test result [15] and (c) simulated cracks.

Numerical model
With the calibrated model, a two-dimensional numerical model of rock mass with a single borehole is established in this section, as shown in Figure 2. The size of the numerical model is 5 m in width and 5 m in height.According to the specification of drilling machine, the diameter of the borehole located in the centre of the model is 42 mm.The explosive filled in the borehole has a diameter of 32 mm.The solid elements with gradually increased sizes from 2 mm to 20 mm are used to mesh the numerical model.The fluid-structure coupling algorithm is adopted to control the interaction of explosive, air and rock.The partial rock domain overlap with air domain to ensure the blasting loads to pass into rock domain.Non-reflective boundary conditions are applied to the model boundaries using the keyword *BOUNDARY_NON_REFLECTING.A quasi-static method [16] is utilized to apply in-situ stresses to the numerical model.The application time of the pre-stress is determined to be 80 ms, and the explosive is detonated at 90 ms.

Modes of blast-induced cracks under different in-situ stresses
Figure 3 shows the modes of blast-induced cracks under confining stress of 0 MPa, 10 MPa, 20 MPa, 30 MPa and 40 MPa.It can be seen from that the confining stress significantly affects the length and distribution of the blast-induced cracks.With the increase of confining stress, the lengths of blastinginduced cracks decrease in vertical and horizontal directions.In addition, the lengths of the blastinduced cracks in the two directions are similar in the same case.It may be because the circumferential stress around the borehole is isotropic under equal bidirectional stress.The results indicate that the increasing the confining stress is disadvantageous to expand blasting-induced cracks.The modes of blast-induced cracks of rock mass under same vertical in-situ stress (i.e., 20 MPa) but different lateral stress coefficients (i.e., 0.25, 0.5, 1.0, 1.5, and 2.0) are shown in Figure 4.It can be seen that the constraint action of in-situ stresses on the levels of blasting-induced cracks is enhanced as the lateral stress coefficient increases from 0.25 to 2.0, i.e., increased horizontal in-situ stress.In addition, the blasting-induced cracks are easier to propagate along the direction of the maximum principal stress.It is because that the constraints of in-situ stresses on the blasting-induced cracks are smallest along the direction of maximum principal stress when compared other directions.

Fractal characteristic of blast-induced cracks under different in-situ stresses
To quantify the blasting-induced cracks, the box-counting fractal dimension approach is further introduced to describe the cracks.Simulated blast-induced cracks in Sections 3.1 and 3.2 are binarized by image processing and the results are shown in Figure 5.To improve the accuracy of fractal dimensions, the resolution of binary image in Figure 5 is enhanced to 6181px × 6181px.The box-counting fractal dimension can be calculated as where D is the box-counting fractal dimension, δk is the side length of the square box, N(δk) is the minimum number of square boxes covering blasting-induced cracks.Figure 6  where Cclamp is the constraint coefficient, D0 is the fractal dimension of blast-induced cracks without in -situ stress, Dstress is the fractal dimension of blast-induced cracks with given in-situ stress.The constraint coefficients under different confining stresses and lateral stress coefficients are calculated using Eq. ( 2) and the results are shown in Figure 6(c) and (d).It can be found that from Figure 6(c) and (d) the freezing effect for the blast-induced cracks becomes more obvious as the increase of in-situ stress.The quantitative relations between in-situ stresses and fractal dimensions of blasting-induced cracks and the defined constraint coefficient can give the reference for the fine blasting design to excavate the deep rock mass.

Conclusion
The present study investigates the modes and fractal dimension characteristics of blasting-induced cracks in rock mass.The effects of in-situ stress levels and lateral stress coefficients on the characteristics of blasting-induced cracks are examined.The main conclusions are summarized as follows.
(1) Blasting-induced crack levels of rock mass reduce with the increase of confining stresses.Cracks are prone to develop towards the direction of maximum in-situ stresses.(2) The quantitative relations between the fractal dimensions of blast-induced cracks and the insitu stresses are established, which can be used to guide blasting design in practice.

Figure 2 .
Figure 2. Configurations of the numerical model.

Figure 3 .
Figure 3. Modes of blast-induced cracks under different confining stresses.The modes of blast-induced cracks of rock mass under same vertical in-situ stress (i.e., 20 MPa) but different lateral stress coefficients (i.e., 0.25, 0.5, 1.0, 1.5, and 2.0) are shown in Figure4.It can be seen that the constraint action of in-situ stresses on the levels of blasting-induced cracks is enhanced as the lateral stress coefficient increases from 0.25 to 2.0, i.e., increased horizontal in-situ stress.In addition, the blasting-induced cracks are easier to propagate along the direction of the maximum principal stress.It is because that the constraints of in-situ stresses on the blasting-induced cracks are smallest along the direction of maximum principal stress when compared other directions.

Figure 4 .
Figure 4. Modes of blast-induced cracks under different lateral stress coefficients.

Figure 5 .
Figure 5. Binary image of the blast-induced cracks under different confining stresses and lateral (a) and (b) show the fractal dimensions of blast-induced crack under different confining stresses and lateral stress coefficients.It can be seen that the fractal dimensions of the blast-induced cracks change with the variations of confining stresses and lateral stress coefficients.The fractal dimensions of blastinginduced cracks nonlinearly decrease with the increase of confining stresses and lateral stress coefficients.Meanwhile, the quantitative relation between the fractal dimensions of blasting-induced cracks and in-situ stresses are established as given in Figure 6(a) and (b).

Figure 6 .
Figure 6.Fractal dimensions under different (a) confining stresses and (b) lateral stress coefficients, constraint coefficient under different (c) confining stresses and (d) lateral stress coefficients.To quantify the constraint of in-situ stress on the blast-induced cracks, the constraint coefficient is defined as

Table 1 .
The parameters of RHT model for granite.