1 Introduction

It is of great importance to evaluate a blasting-induced fragmentation zone beyond the proposed excavation line of a tunnel because the unwanted damaged zone requires extra support systems for tunnel safety. However, complicated blasting process which may hinder a proper characterization of the damaged zone can be effectively represented by two loading mechanisms (e.g., Brinkman 1987). The first one is a dynamic impulsive load generating stress waves that radiate outwards immediately after detonation. This load creates a crushed annulus around blasthole with many radial cracks (fracture zone in Fig. 1) and remains in the order of micro-seconds. The second one is a gas pressure that remains for a few micro-seconds, which is relatively long time. Since the gas pressure reopens up the arrested cracks and continues to extend some cracks, it contributes to the formation of fragmentation zone (Fig. 1) induced by blasting.

Fig. 1
figure 1

Fracture process of explosive blasting (Whittaker et al. 1992)

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Although Fjellborg and Olsson (1996), Olsson et al. (2004), Mindess (1991), Rossmanith (1983) and Maji and Wang (1992) carried out experimental test to predict the blasting-induced crack propagation, they do not explain the mechanisms involved with the fragmentation process.

Recent advances in the numerical simulation tools which have powerful computational functions have made numerical analysis a most promising approach to study the fracturing processes in rock mass. Particularly, to simulate the crack propagation due to blasting, various numerical methods have been used. Zhu et al. (2007a, b), Chen and Zhao (1998) and Ma et al. (1998) conducted numerical study with AUTODYN code for simulation of rock fracturing. AUTODYN code has been applied to solve nonlinear problems based on the Lagrange computation scheme. This code can be used to simulate the rock fragmentation due to blasting. To consider the gas flow through the crack, finite difference method (FDM), which uses Lagrangian computation process, also has been adopted by Cho et al. (2004b), Goodarzi et al. (2011, 2013), and Cho et al. (2002).

A finite element method (FEM) such as ABAQUS has been used to simulate a brittle cracking in brittle rock (Saharan and Mitri 2008). However, explicit crack generation cannot be effectively presented with general FEM code. To overcome this limitation, XFEM has been used to simulate the crack propagation in solid media. Fracturing of rock mass is also studied with the powerful extended finite element method (XFEM) (Majid et al. 2015; Ren et al. 2009). Recently, advanced XFEM which can consider hydro-mechanical coupling has been introduced in various researches (Mohammadnejad and Khoei 2013; Gordeliy and Peirce 2013; Gholami et al. 2013).

To simulate the rock fragmentation, various numerical methods have been used: three-dimensional discrete element program PFC3D (Potyondy et al. 2004), combined finite element–discrete element program MBM2D (Minchinton and Lynch 1996), cross-format centered finite difference procedure (Wang et al. 2008), AUTODYN 2D (Zhu et al. 2007a, b), Johnson–Holmquist model in LS-DYNA (Ma and An 2008), ANSYS–LSDYNA (Wei et al. 2009), fully coupled gas flow-lattice model (Onederra et al. 2013), DEM–SPH simulation (Ali and Mark 2014). A summary of numerical analysis on crack propagation due to blasting is presented by Ali and Mark (2014).

Ouchterlony (1997), Ouchterlony et al. (2002) gave an equation for estimating radial crack length. Hustrulid (2010) analyzed the energy and work done by an explosive charge in a borehole to develop the extent of damage based on explosive energy. Kanchibolta et al. (1999) offered an equation to estimate the crushing zone radius. Although these works tried to provide theoretical solution for the blasting-induced damaged zone, simplification and assumption cannot be avoided due to the complexity involved with the fragmentation process. And these models are only considered for estimating the extent of the blast-damaged zone (Torbica and Lapcevic 2015). In fact, there are several studies to evaluate fragmentation zone by introducing gas-driven fractures in blasting (e.g., Nilson et al. 1985; Paine and Please 1994). To study the gas fracturing process, the deformation of the rock around blasthole and gas flow needs to be considered (Paine and Please 1994).

In this paper, the simple analytical method is presented to evaluate the fragmentation zone induced by gas pressure during blasting in rock. The fragmentation zone is characterized by analyzing crack propagation from the blasthole.

2 Evaluation of Crack Propagation from the Blasthole

2.1 Basic Assumptions

To evaluate crack propagation in rock, a model of blasthole with radial cracks is introduced in an infinite elastic plane (Fig. 2). Two-dimensional plane is applied since the diameter of blasthole is far smaller than its length and there is no free surface.

Fig. 2
figure 2

Pressurized blasthole and propagating radial cracks. a Uniform pressure inside blasthole and cracks, and b uniform pressure inside blasthole and linear pressure inside cracks

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The typical number of symmetric cracks of equal length around blasthole is chosen in an elastic rock and is assumed to propagate radially from the blasthole by gas pressure. It was experimentally investigated that the number of main cracks that could be formed ranges from 3 to 8 (Garnsworthy 1990) even though a huge number of cracks would be generated after detonation. For the simplicity, the number of major cracks for model is assumed to be four (N = 4) and eight (N = 8), which are symmetric.

In Fig. 2, the gas pressure is applied to the blasthole and is penetrated into the cracks. p A and p B are gas pressures applied to the blasthole and cracks, respectively. It is assumed that the distribution of pressure along blasthole is uniform. In the case of pressure inside cracks, two types of pressure distribution, uniform and linear pressure distributions, are considered for the analysis. Uniform pressure distribution gives upper bound estimation of calculating crack length (Fig. 2a). Due to the fast crack propagation during blasting and narrow opening of the crack, however, the actual pressure inside cracks will not be distributed uniformly so that linear pressure distribution is also considered for the analysis. In case of linear pressure distribution, p B is the pressure at the crack mouth (Fig. 2b).

Simply, two conditions only are considered to describe the crack propagation in this paper. The first condition is the criterion for crack propagation. It states that the stress intensity factor (SIF) for mode I, K I, needs to be greater than the fracture toughness K IC, that is, K I > K IC. The second condition indicates that the total mass of gas in the blasthole and cracks need to be conserved. That is, the mass change, ΔM, needs to be equal to zero. Based on these two conditions, the final fragmented area is characterized by analyzing the probable length of the crack propagation at given conditions. It may be reasonable to ignore the effect of gravity because the explosive pressure is expected to highly exceed the in situ stress.

2.2 Formulation

The total mass of gas during the crack propagation remains constant as following:

$$M_{\text{blasthole}} + M_{\text{crack}} = M_{\text{i}}$$
(1)

where M blasthole and M crack are the mass of gas in the blasthole and the crack, respectively. And M i is the mass of gas in the blasthole at the moment of crack initiation when most of the gas is assumed to be generated. Each mass of the gas in Eq. (1) can be expressed as follows:

$$M_{\text{i}} = V_{\text{i}} \rho_{\text{gi}}$$
(2)
$$M_{\text{blasthole}} = V_{\text{blasthole}} \rho_{\text{g}}$$
(3)
$$M_{\text{crack}} = V_{\text{crack}} \rho_{\text{g}}$$
(4)

where V i, V blasthole, and V crack are volume of the blasthole at the moment of crack initiation, volume of the blasthole, and volume of the crack by gas pressure, respectively. The density of the gas, ρ g, during the blasting follows adiabatic process. Then

$$\rho_{\text{g}} = \left( {\frac{p}{{p_{\text{i}} }}} \right)^{{\frac{1}{\gamma }}} \cdot \rho_{\text{gi}}$$
(5)

where γ is the adiabatic exponent ranging from 1.2 to 3 in blasting (e.g., Paine and Please 1994; Persson et al. 1994), ρ gi and p i are the density of the gas and pressure at the moment of crack initiation, respectively. p i is reasonably assumed to be peak pressure since the most of the gas is generated at the moment of crack initiation.

It is assumed that the uniform pressure inside cracks is same as that of blasthole as follows:

$$p_{\text{A}} = p_{\text{B}} = p_{0}$$
(6)

In case of linear pressure distribution, above Eq. (6) is valid since the pressure at the crack mouth, p B, is also same as pressure in blasthole, p A. Substituting Eqs. (2)–(6) into Eq. (1), we obtain

$$V_{\text{blasthole}} \left( {\frac{{p_{0} }}{{p_{\text{i}} }}} \right)^{{\frac{1}{\gamma }}} + V_{\text{crack}} \left( {\frac{{p_{0} }}{{p_{\text{i}} }}} \right)^{{\frac{1}{\gamma }}} = V_{\text{i}}$$
(7)

As shown in Eq. (7), volumes of the blasthole and cracks in proposed model (Fig. 2) need to be evaluated for arbitrary gas pressure. To simply do this, an original model is decomposed into two auxiliary problems as shown in Fig. 3. In Problem A, the gas pressure is applied only inside blasthole but no pressure inside cracks. In Problem B, the gas pressure is applied only inside cracks but no pressure inside blasthole. Each problem is affected by one single parameter, p 0.

Fig. 3
figure 3

Decomposition of the original model (N = 8, uniform pressure) into two sub-problems. a Uniform pressure model and b linear pressure model

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In both problems, the SIFs of the cracks and volumes of the cracks and blasthole can be expressed as

$$K_{\text{IZ}} = p_{0} \sqrt {\pi r} k_{\text{IZ}}^{\text{crack}} \left( {l,r} \right)$$
(8)
$$V_{\text{Z}}^{\text{crack}} = \frac{{p_{0} r^{2} }}{E}f_{\text{Z}}^{\text{crack}} \left( {\nu ,l,r} \right)$$
(9)
$$V_{\text{Z}}^{\text{blasthole}} = \frac{{p_{0} r^{2} }}{E}f_{\text{Z}}^{\text{blasthole}} \left( {\nu ,l,r} \right)$$
(10)

where subscript Z is A or B, \(K_{\text{IZ}}\) is SIF of the crack in problem Z, \(V_{\text{Z}}^{\text{crack}}\) and \(V_{\text{Z}}^{\text{blasthole}}\) are volumes of cracks and blasthole in problem Z, respectively, \(k_{\text{IZ}}^{\text{crack}}\), \(f_{\text{Z}}^{\text{crack}}\), and \(f_{\text{Z}}^{\text{blasthole}}\) are dimensionless functions to be determined in plane strain condition using finite element code, FRANC2D (Wawrzynek and Ingraffea 1987), l and r are the crack propagation length and blasthole radius, respectively, and ν and E are Poisson’s ratio and elastic modulus of the rock, respectively. Also, \(k_{\text{IZ}}^{\text{crack}}\) is the function of l and r, and both \(f_{\text{Z}}^{\text{crack}}\) and \(f_{\text{Z}}^{\text{blasthole}}\) are functions of ν, l, and r.

Suppose that all three dimensionless functions with respect to the radial crack length, l, are known from the numerical calculations (refer to “Appendix”). Then, the superposition of problems A and B results in

$$K_{\text{I}} = \left( {p_{0} k_{\text{IA}}^{\text{crack}} \left( {l,r} \right) + p_{0} k_{\text{IB}}^{\text{crack}} \left( {l,r} \right)} \right)\sqrt {\pi r}$$
(11)
$$V_{\text{crack}} = \frac{{p_{0} c^{2} }}{E}f_{\text{A}}^{\text{crack}} \left( {\nu ,l,r} \right) + \frac{{p_{0} c^{2} }}{E}f_{\text{B}}^{\text{crack}} \left( {\nu ,l,r} \right)$$
(12)
$$V_{\text{blasthole}} = \frac{{p_{0} c^{2} }}{E}f_{\text{A}}^{\text{blasthole}} \left( {\nu ,l,r} \right) + \frac{{p_{0} c^{2} }}{E}f_{\text{B}}^{\text{blasthole}} \left( {\nu ,l,r} \right)$$
(13)

Our notations can be further simplified by introducing the function

$$f_{\text{Z}} = f_{\text{Z}}^{\text{crack}} \left( {\nu ,l,r} \right) + f_{\text{Z}}^{\text{blasthole}} \left( {\nu ,l,r} \right)$$
(14)

And the total volume can be

$$V = V_{\text{crack}} + V_{\text{blasthole}} = \frac{{p_{0} r^{2} }}{E}f_{\text{A}} \left( {\nu ,l,r} \right) + \frac{{p_{0} r^{2} }}{E}f_{\text{B}} \left( {\nu ,l,r} \right)$$
(15)

To calculate the change of the total volume due to crack propagation, the initial volume of the pressurized blasthole should be subtracted and can be expressed as:

$$V_{\text{i}} = \frac{{p_{\text{i}} r^{2} }}{E}f_{\text{A}} \left( {\nu ,l = 0,r} \right)$$
(16)

In above Eq. (16), it is assumed that the crack length is zero (l = 0) at the moment of initiation. Then, the total volume change due to crack propagation is obtained by subtracting Eq. (16) from Eq. (15):

$$\Delta V = \frac{{p_{0} r^{2} }}{E}f_{\text{A}} \left( {\nu ,l,r} \right) + \frac{{p_{0} r^{2} }}{E}f_{\text{B}} \left( {\nu ,l,r} \right) - \frac{{p_{\text{i}} r^{2} }}{E}f_{\text{A}} \left( {\nu ,0,r} \right)$$
(17)

The change of blasthole volume, meanwhile, is defined as

$$\Delta V_{\text{blasthole}} = V_{\text{blasthole}} - V_{\text{i}}$$
(18)

Inserting Eq. (18) into Eq. (7) yields

$$\Delta V_{\text{blasthole}} + V_{\text{crack}} = \Delta V = V_{\text{i}} \left\{ {\left( {\frac{{p_{\text{i}} }}{{p_{0} }}} \right)^{{\frac{1}{\gamma }}} - 1} \right\}$$
(19)

From Eqs. (17) and  (19)

$$p_{0} f_{\text{A}} \left( {\nu ,l,r} \right) + p_{0} f_{\text{B}} \left( {\nu ,l,r} \right) - p_{\text{i}} f_{\text{A}} \left( {\nu ,0,r} \right) = f_{\text{A}} \left( {\nu ,0,r} \right)p_{\text{i}} \left\{ {\left( {\frac{{p_{0} }}{{p_{\text{i}} }}} \right)^{{\frac{1}{\gamma }}} - 1} \right\}$$
(20)

Finally, gas pressure can be derived as follows:

$$p_{0} = p_{\text{i}} \left( {\frac{{f_{\text{A}} \left( {\nu ,0,r} \right)}}{{f_{\text{A}} \left( {\nu ,l,r} \right) + f_{\text{B}} \left( {\nu ,l,r} \right)}}} \right)^{{\frac{\gamma }{\gamma + 1}}}$$
(21)

In addition, following condition should be satisfied for the crack propagation.

$$K_{\text{I}} \ge K_{\text{Ic}}$$
(22)

Therefore, using Eq. (11), finally propagated length of the crack can be determined from the following equation.

$$K_{\text{I}} = \left( {p_{0} k_{\text{IA}} \left( {l,r} \right) + p_{0} k_{\text{IB}} \left( {l,r} \right)} \right)\sqrt {\pi r} \ge K_{\text{Ic}}$$
(23)

2.3 No Gas Penetration into the Crack

Ouchterlony (1974) illustrated various pressure distribution conditions including no gas penetration into the cracks in relation to rock blasting. During the blasting, crushing annulus created by stress wave may prevent gas penetration into the cracks so that it is closely related to the blasting efficiency. To deal with this condition, only Problem A out of two auxiliary problems can be considered. Iterating same formulation procedure, equations for gas pressure and SIF can be obtained as follows:

$$p_{0} = p_{\text{i}} \left( {\frac{{f_{\text{A}} \left( {\nu ,0,r} \right)}}{{f_{\text{A}} \left( {\nu ,l,r} \right)}}} \right)^{{\frac{\gamma }{\gamma + 1}}}$$
(24)
$$K_{\text{I}} = p_{0} k_{\text{IA}} \left( {l,r} \right)\sqrt {\pi r} \ge K_{\text{Ic}}$$
(25)

Therefore, it is expected that the crack length obtained from Eq. (25) gives lower estimate while one from Eq. (23) gives upper estimate.

3 Results and Discussion

Two cases that have distinctly different elastic properties of the rocks for the analyses are shown in Table 1. Case 1 corresponds to the hard rock property while Case 2 corresponds to the weathered rock property. For the comparison, the same initial gas pressure, p i, is applied. The diameter of blasthole, 2r, and the adiabatic exponent, γ, are fixed as 0.045 m and 3, respectively.

Table 1 Two different elastic properties of the rocks and initial gas pressure for the analyses
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Figures 4 and  5 show the comparison of the normalized SIFs and gas pressure variations in blasthole between Case 1 and Case 2 for both uniform and linear pressure distribution inside crack. The normalized SIFs generally decrease as cracks propagate. However, the crack can only propagate if the normalized SIF (K I /K Ic) is greater than one. As expected, longer crack is obtained in Case 2 than in Case 1. For example, in case of Case 2 and uniform pressure inside crack as in Fig. 4a, obtained crack length is larger than 1 m, while in case of Case 1 with same pressure condition, obtained crack length is about to 0.98 m. Also, if the gas does not penetrate into the cracks, the obtained lengths of the cracks are much shorter than those of the gas penetration. Similar trend is obtained from the case of linear pressure distribution inside crack (Fig. 5).

Fig. 4
figure 4

Comparison of the SIFs and gas pressure between Case 1 and Case 2 for uniform pressure distribution inside cracks. a N = 4, and b N = 8

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Fig. 5
figure 5

Comparison of the SIFs and gas pressure between Case 1 and Case 2 for linear pressure distribution inside cracks. a N = 4, and b N = 8

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The normalized gas pressure inside blasthole in Figs. 4 and 5 decreases as the crack propagates since the total mass of the gas is constant during the crack propagation. Also, the gas pressure in Case 1 is generally bigger than that of the Case 2 because larger gas containment with larger deformation in Case 2 can make gas pressure lower.

Figures 6 and 7 show the comparison of the normalized SIFs and gas pressure variations between N = 4 and N = 8 for uniform and linear pressure distribution inside crack. In these results, the lengths of the crack with same pressure condition are similar. In other words, initially created cracks due to stress wave do not affect the length of the cracks because of the mechanical interaction between cracks. During the crack propagation, interaction between cracks in case of N = 8 is more significant than in case of N = 4. Crack opening by internal gas pressure is constrained by adjacent cracks so that gas pressure in case of N = 8 is a bit higher than in case of N = 4 (Figs. 6, 7). In Table 2, generated crack lengths are summarized with respect to the pressure type.

Fig. 6
figure 6

Comparison of the SIFs and gas pressure between N = 4 and N = 8 cases for uniform pressure distribution inside cracks. a Case 1, and b Case 2

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Fig. 7
figure 7

Comparison of the SIFs and gas pressure between N = 4 and N = 8 cases for linear pressure distribution inside cracks. a Case 1, and b Case 2

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Table 2 Summary of finally propagated lengths of the cracks (unit: m)
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4 Verification of Analytical Solution

4.1 Numerical Simulation on Crack Propagation Due to Pressure Inside Blasthole

Discrete element method (DEM) has been adopted to simulate the crack propagation in rock and rock mass (Ruest et al. 2006). Discrete fracture network and geological structure can be effectively handled by the DEM. Lisjak and Grasselli (2014) comprehensively summarized the simulation of fracture process in discrete rock masses. Wang and Konietzky (2009) used UDEC (Itasca 2013) to simulate the fractures in jointed rock mass due to blasting-induced stress wave. Particularly, it is found that the Voronoi joint generator is very conducive for numerical simulation of crack creation and propagation due to blasting-induced stress wave.

In this study, UDEC, two-dimensional discrete element method, is implemented to simulate the crack propagation due to blasted gas pressure induced by blasting and to validate the analytical solution presented in previous chapter. Particularly, material properties for numerical model are determined from the reference case. Furthermore, the crack length obtained from numerical analysis is compared with the analytical solution depending on the various blasthole pressure level and ground type to validate the analytical solution.

4.2 Blasting-Induced Gas Pressure

Maximum blasthole pressure (P B) can be obtained from equation suggested by Morhard (1987), Clark (1987), and Nie and Olsson (2000).

$$P_{\text{B}} = \rho \left( {\frac{{{\text{VOD}}^{2} }}{8}} \right) \cdot \left( {r_{\text{c}}^{2\gamma } } \right)$$
(26)

where ρ is density of explosive (kg/m3), VOD is velocity of detonation (m/s), r c is the decoupling ratio (i.e., = explosive diameter/blasthole diameter), and γ is an adiabatic exponent. Gurit of 17 mm in diameter which is used for precise blasting for contour holes has 2000–3000 m/s of VOD, 1050 kg/m3 of density. A kimulux42 of 22 mm in diameter which is used for the stopping holes has 3000–4000 m/s of VOD and 1150 kg/m3 of density (Saharan and Mitri 2008; Song et al. 2014). Thus, the pressure inside blasthole can vary depending on the adiabatic exponent and decoupling ratio. In this study, maximum blasting pressure acting in the blasthole for crack growth ranges from 50 to 200 MPa.

Numerically, blasthole pressure profiles due to blasting can be approximated with John–Wilkinson–Lee (JWL) method (Liu 1997), pressure decay functions (Cho et al. 2003; Lima et al. 2002; Kutter 1967) and direct input of pressure–time profile (Donze et al. 1997; Valliapan et al. 1983) and so on. Although the JWL method can consider the rock-explosive interaction, it is difficult to derive reliable parameters (Liu 1997). The blasthole pressure can be expressed in the form of decay functions (Duvall 1953; Jung et al. 2001; Lima et al. 2002; Olatidoye et al. 1998; Robertson et al. 1994). Gaussian function and triangular function have been implemented to simplify the blasting pressure. However, the Gaussian function is mainly introduced to avoid numerical errors. In this study, the dynamic pressure P(t) acting on the blasthole is expressed as a transient time history function as follows (Starfield and Pugliese 1968; Histake et al. 1983):

$$P\left( t \right) = 4P_{\text{B}} \left[ {\exp \left( {\frac{ - 16338 \cdot t}{\sqrt 2 }} \right) - { \exp }\left( { - \sqrt 2 \cdot 16338 \cdot t} \right)} \right]$$
(27)

where t is the elapsed time. Figure 8 shows the transient time history of blasthole pressure applied to the numerical model. Internal pore pressure is applied in the blasthole according to transient time history function so that the pressure can penetrate and open the crack. To get a clear response, the blasthole pressure is activated after 0.01 s for the model; it is activated at the beginning of the simulation (i.e., 0 s) for the dynamic analysis.

Fig. 8
figure 8

Blasted gas pressure. Peak pressure ranges from 50 to 200 MPa

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4.3 Numerical Model

The fracturing procedure due to blasting can be divided into two phases; rapid rising detonation pressure initiates multiple cracks around the blasthole and the gas penetration into the crack which leads to crack extension (Majid et al. 2015). Basically, continuing penetration of the explosion gases is guided by initiated crack tips. Cho et al. (2004a, b), Cho and Kaneko (2004), Zhu et al. (2004, 2007a, b, and Ma and An (2008) proposed various random crack generation techniques in rock mass under blasting pressure. Particularly, growth of crack length is mainly contributed by gas pressure. In this study, to validate the analytical solution to estimate the crack length due to blasting, it is assumed that the detonation-induced stress wave creates the crack around the blasthole and gas pressure is responsible for the crack propagation.

Material properties of hard rock and weathered rock used in numerical model are identical to the analytical model. In this study, rock media are assumed as elastic solid as it is assumed in analytical model. Fracture toughness (K IC) is important parameter to be defined to control the crack propagation. In UDEC, residual Coulomb slip with residual strength is used. Sensitivity analysis is carried out to find the optimum joint property for fracture toughness. Cohesion and contact tensile strength are the most significant factors related to toughness. Cohesion and tensile strength of joint are found from trial and error (Kazerani 2011).

Blasted gas pressure has been changed from 50 to 200 MPa for weathered rock and hard rock mass. Transient time history gas pressure is applied in the blasthole as a non-wet pore pressure. 20 m by 20 m rock media with zero velocity at boundary is make it possible to simulate the infinite boundary. As it is reported by Wang and Konietzky (2009), Voronoi tessellation is very useful for simulation of crack propagation. Thus, Voronoi tessellation generator is used to create randomly sized polygonal blocks along the crack paths as it is shown in Fig. 9. Joint area model with Coulomb slip failure is used for joint and joint properties are tabulated in Table 3. When joint shear or tensile strength is exceeded, fracture can be generated along the pre-defined four crack paths.

Fig. 9
figure 9

DEM computation model for simulation of blasting

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Table 3 Rock joint properties used for numerical analyses
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4.4 Comparisons

Crack lengths in case of uniform pressure inside cracks are measured at four directions (A, B, C and D) as it is presented in Fig. 9, respectively. Due to the random Voronoi tessellation process, the length of crack is slightly different depending on the direction. Crack lengths are obtained from analytical solution depending on the gas pressure and rock condition, and plotted as a dotted line in Fig. 10. And crack length obtained from numerical analysis is plotted as symbols with respect to direction. Figure 10a presents the crack length depending on the gas pressure for Case 1. Crack length increases linearly depending on the pressure. Although the crack length is slightly varied with respect to directions, the crack lengths obtained from numerical analysis show good agreement with analytical solution.

Fig. 10
figure 10

Comparison of crack length between numerical analysis and analytical solution a Case 1 b Case 2

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For Case 2 shown in Fig. 10b, crack length obtained from analytical solution increases linearly according to pressure inside blasthole. Definitely, crack length obtained from Case 2 is greater than that obtained from Case 1 both numerical analysis and analytical solution. Analytical solution does not converge when the pressure exceed 150 MPa due to the numerical instability. Within the range of gas pressure (i.e., 50–150 MPa), crack length obtained from analytical solution is very close to numerical result. It implies that the analytical solution can predict the crack length induced by blasting.

Although the analytical solution is derived from simplified assumptions, the crack length obtained from analytical solution is very close to the crack length obtained from dynamic numerical analysis regardless gas pressure and ground condition. Moreover, when the gas penetrates into the crack, this simplified analytical solution can be used to estimate the fragmented zone around the blasthole. In summary, the accuracy of analytical solution is properly validated with the numerical analysis in this study.

5 Conclusions

This paper presents the simple method to predict the fragmentation zone induced by gas pressure during blasting in rock. The fragmentation zone is characterized by analyzing crack propagation from the blasthole. To do this, a model of the blasthole with four (N = 4) and eight radial cracks (N = 8) of equal length in an infinite elastic plane is considered. In this model, the crack propagation is simulated by using two conditions only, the crack propagation criterion and the mass conservation of the gas.

Generally, the SIF of the crack decreases as crack propagates from the blasthole so that the finally propagated crack length can be determined. As expected, fragmentation zone in weathered rock is wider than that in hard rock because the length of crack obtained in weathered rock is longer than that in hard rock. In addition, gas penetration into the cracks significantly affects the extension of fragmentation zone in rock so that this factor is closely related to the blasting efficiency.

Gas pressure inside blasthole also continues to decrease during crack propagation since the total mass of the gas is assumed to be constant. As crack propagates, deformed blasthole and cracks are repeatedly filled with gas resulting in pressure decrease. In addition, the gas pressure in weathered rock is generally less than that of the hard rock since the crack opening that can contain gas in the weathered rock is much more than in the hard rock. The gas pressure in case of no gas penetration into the cracks is larger than that in case of the gas penetration.

The number of cracks around blasthole has a little effect on the fragmentation formation because of the mechanical interaction between cracks. The mechanical interaction between cracks during crack propagation hinders their openings by internal gas pressure.