Numerical Simulation Study on Blasting-induced Damage Characteristics of Frozen Rock

Rock masses in cold regions are exposed to a negative temperature environment for extended periods, causing significant changes in their mechanical properties. This inevitably results in substantial differences in the blasting characteristics compared to rocks at normal temperatures. To reveal the influence of different temperatures on the blasting characteristics of frozen rocks, this paper utilized a finite element method-based LS-DYNA software for numerical simulation of single borehole blasting at freezing temperatures of -5°C, -10°C, -20°C, and -30°C. The results showed that under negative temperature conditions, the degree of rock blasting-induced damage gradually decreased as the rock temperature decreased. As the temperature decreased from -5°C to -30°C, the volume of rock blasting-induced damage decreased by approximately 14%. Additionally, this paper has further investigated the effects of ignition locations, free surface, and borehole burdens on the blasting-induced damage characteristics of frozen rock. It was found that the volume of the blast cavity is the largest in reverse detonation. While forward detonation is more favorable for damaging the top rock and helps to improve the rock fragmentation at the orifice. Increasing the free surface can significantly improve the effect of blasting-induced damage. There is an optimal borehole burden distance, which can fully utilize the explosive energy and break the rock efficiently. Finally, based on multivariate nonlinear regression analysis, an empirical formula of the reduction coefficient of the specific charge for frozen rock mass is developed, which can provide a theoretical basis for designing and optimizing rock engineering blasting in alpine and cold regions.


Introduction
Abundant metal mineral resources can be found in China's western regions.These resources are essential to China's future metal mineral reserves and supply security.However, the cold climate of these regions causes significant exertion.Extremely low temperatures cause the mineral rock to freeze, creating numerous hurdles for blasting and mining operations.These challenges range from large blast block sizes, inefficient use of explosives, and higher explosive unit consumption, all of which hinder the extraction of high-quality and environment-friendly mineral resources in alpine and cold regions [1].Therefore, it is crucial to explore the blasting properties of frozen rock to enhance the efficiency of blasting and excavation techniques for rock engineering projects in frigid regions.
The characteristics of rock blasting are intricately linked to the rock's physical and mechanical properties.To study the blasting traits of frozen rock, a firm grasp of the rock's physical and mechanical properties under negative temperature conditions is necessary.Scientists across the globe have conducted extensive research in this field.For instance, Xu et al. [2] conducted uniaxial and triaxial compression tests on dry and saturated sandstones and shales at low temperatures.The results revealed that the freezing strength of the rocks was significantly impacted by water content.The uniaxial compressive strength, angle of internal friction, and modulus of elasticity increased as the temperature decreased.Fu et al. [3] conducted uniaxial compression tests on slate in various negative temperature situations using a universal testing device and a low-temperature numerically controlled test chamber.The results revealed that the uniaxial compressive strength of the slate increased exponentially as the freezing temperature decreased.Moreover, the peak strain of the slate decreased, and the rock became more brittle as the freezing temperature dropped.Yang et al. [4] found that freezing can greatly increase the strength of rocks through triaxial compression tests on coal rocks and sandstones.He also discovered that as the temperature drops, both types of rocks become less elastic and more fragile, with sandstone being more vulnerable to temperature changes.
In recent years, there has been a growing interest in the kinetic properties of frozen rocks.Wen et al. [5] employed a low-temperature compensation device with a large-diameter separated Hopkinson compression bar (LT-SHPB) system to execute dynamic compression tests on sandstones at different temperature conditions.The results indicated that the dynamic compressive strength of the rocks increased as the temperature decreased and the strain rate increased.A study by Yang et al. [6] involved dynamic mechanical tests on three different rock types at negative temperatures utilizing a split Hopkinson compression rod system.The research showed that low temperature had the most significant effect on the properties of red sandstone, followed by marble, while granite was least susceptible to temperature effects.Kodama J et al. [7] concluded that the water-ice phase transition within the rock mass pores and the coefficient of thermal expansion of the rock (cold shrinkage rate) at low temperatures are critical factors affecting the physical and mechanical characteristics of rocks in alpine and cold regions.
In alpine and cold regions, research on rock blasting primarily focused on engineering practices to improve blasting construction efficiency in alpine and high-altitude regions by improving blasting technology and the construction process.Theoretical studies on stress wave propagation and blasting vibrations in frigid rock are relatively sparse.For instance, Dong et al. [8] presented a quantitative model that explains how blast vibration propagates in alpine environments.His research showed that when the rock layer's freezing depth increases, the peak vibration velocity's decay rate slows down, which worsens the effects on rock slopes.Notably, there is a negative relationship between the freezing depth and the blast vibration's decay coefficient.Hu et al. [9] demonstrated a 40% increase in the longitudinal wave velocity of frozen rock through indoor experiments.The above-mentioned literature unveiled that limited attention has been paid to studying blast damage and fragmentation characteristics of rock bodies in alpine and cold regions.Inquiring into the blasting-induced fragmentation volume of frozen rock is essential for improving the blasting effectiveness of frozen rock masses, maximizing utilization of blasting energy in cold region rock formations, and subsequently improving rock breaking efficiency while minimizing blasting costs.
To investigate the effect of negative temperature on the blasting features of frozen rocks, this study created a comprehensive three-dimensional numerical model of single-hole blasting in frozen rock using LS-DYNA.The study investigated the blast response and damage characteristics of frozen rocks under different low-temperature conditions (−5℃, −10℃, −20℃, and −30℃), explored the effects of ignition locations, free surface, and borehole burden on the blast effect of low-temperature frozen rocks, and quantitatively evaluated the blast effect of frozen rocks under different factors.Finally, a formula for the reduction coefficient of specific charge for frozen rock mass was derived through multivariate nonlinear regression analysis.This empirical formula can serve as a valuable reference for the precise design of rock engineering blasting in alpine and cold regions.

Model Building and Calibration
To study the blasting properties of frozen rock at low temperatures, this section used LS-DYNA to construct a 3D numerical model for single-hole blasting and calibrate the constructed numerical model with results from indoor rock blast tests.

Numerical Modeling of Single-hole Blasting in Frozen Rock
The single-hole blasting numerical model for rock (refer to Fig. 1) outlined in this paper had dimensions of 4.0 m × 2.2 m × 4.0 m.The borehole was 1.0 m long with radial-coupled charging and plugging lengths of 0.6 m and 0.4 m, and a diameter of 42 mm.The propagation of blast waves and their interaction with the rock was simulated via the arbitrary Lagrangian-Eulerian (ALE) method.Furthermore, the explosives, air, and rock were meshed using 3D solid elements.The cell size for the explosives and the surrounding rock and air was determined to be approximately 0.8 mm through grid cell validation.gradual increases in cell size were employed for both rock and air cells at a distance from the explosives, with a maximum size of about 100 mm to enhance computational efficiency.To analyze the explosive properties of frozen rock at various temperatures, the temperature effect of the RHT model was simulated by varying specific model parameters.The values of the parameters are listed in section 1.2.To study the influence of the number of free surfaces, we constructed numerical models featuring the top surface combined with different quantities (0, 1, 2, and 3) of lateral free surfaces.In addition, we set non-reflecting boundaries on all model surfaces except for the free surfaces to eliminate the impact of boundary reflection on the blast stress waves.Additionally, experimental models were created to investigate the effects of different borehole burden distances on single-hole blasting with a free lateral surface at a temperature of −30°C.The distances tested were 1.0 m, 1.1 m, 1.2 m, 1.3 m, 1.4 m, and 1.5 m, respectively.

Material Modeling of Rock.
The Riedel-Hiermaier-Thoma (RHT) model has been widely used to simulate the response of brittle materials, such as rocks, under the effects of blast impact and penetration, and it is also used in this paper as a material model for frozen rocks.To precisely define the RHT material model, 38 parameters must be accurately established.
To investigate the blasting characteristics of frozen rocks under different negative temperatures, the paper chose four temperatures (−5 ℃, −10 ℃, −20 ℃, and −30 ℃) as research factors.The basic parameters of the RHT model were determined through indoor tests and empirical equations for frozen rocks at various low temperatures [10].The other parameters, which are not temperature sensitive and have less influence on the rock properties, are taken from reference [11].Tables 1 and 2 detail the RHT model's parameters for frozen rocks at different negative temperatures.
where  is the relative volume of the explosive product,  is the pressure of the blasted product and,  stands for the explosive energy per unit volume (specific internal energy), with the initial value of  0 ., ,  1 ,  2 , and  are constants linked to explosives, and their exact values are given in Table 3.
In addition, this paper used the *Mat_Null model for the air material and the *Linear_Polynomial equation of state, which is expressed as follows: where μ is the volume parameter, and  1 ~5 are the material constants.Table 4 shows the values for each parameter.

Calibration of the Model
To verify the precision of the coupled explosives, air, and rock, the model was calibrated using a cylindrical rock explosion test performed by Banadaki.As shown in Figure 3, the granite cylindrical rock utilized in the experiment has a diameter of 144 mm.A hole measuring 6.45 mm is drilled at the rock's center, and 1.65 mm PETN explosives are inserted inside the borehole.This section modeled single-hole blasting of the same cylindrical rock as in that test.The models and parameters for both the explosive and air remained consistent with those utilized in section 2.2.2, with the RHT model employed as the material model for the rock.While the basic parameters were provided in the previous test, the remaining parameters were established following section 2.2.1.The basic parameters of the RHT model for granite are listed in Table 5.  c) and (d) show the crack results and simulated damage cloud for this cylindrical rock explosion test.It can be observed that the rock's damage aligns fittingly with the test cracking pattern in both magnitude and quantity when the damage threshold D is set at 0.2.Furthermore, the sizes of the blast crush zone and fissure zone determined in the test, which were based on the density of fractures, closely matched the simulation's findings.This demonstrated that the coupled explosive, air, and rock model used in this paper can accurately emulate the dynamic response of rock under blasting loads.

Effect of Temperature on Blasting-induced Damage in Frozen Rock
Based on the calibrated numerical model from the previous section, this section investigates the damage characteristics and blast response of frozen rock at different temperatures.Figure 4 shows the evolution of blasting-induced damage in rocks at −30°C.It can be seen that after the explosives were detonated at the bottom of the borehole, the blast wave propagated upwards along the borehole and simultaneously radiated into the area around the borehole, creating a stress wave that damaged the rock around the borehole, and the area of damage rapidly expanded in a spindle shape.The blast stress wave reached the top of the model's free surface at 0.6 ms, reflecting as a tensile wave that caused a rupture on the rock close to the free surface.The damage then expanded concurrently inward and outward along the rock's surface.It is worth pointing out that the damage expands more rapidly outward along the free surface than it does within the rock due to the low confinement near the free surface and the complete crushing and expansion of the rock near the free surface.Different negative temperatures under the rock blasting damage process are much the same as in Figure 4 and will not be repeated here.The damage caused by a blast to frozen rock under various negative temperature conditions is shown in Figure 5.The rock can be divided into three areas of damage caused by the explosion: the crushed area (shown in red), the densely cracked area, and the extendedly cracked area.As the temperature decreases, the crushed and densely cracked area rock gradually reduces, while the average length of the primary cracks in the extended cracked area gradually increases.This can be explained by the rock's increased strength at lower temperatures, resulting in less damage to the rock near the blast center.And because of tensile damage brought on by stress wave reflection at the free surface, the expanded cracks were concentrated close to the rock surface.Since the stiffness of the rock increased significantly, but the tensile strength of the rock grew less at lower temperatures, the stress wave was reflected more strongly at the rock surface at lower temperatures, which induced a larger zone of extended cracking.From the lower part of Figure 5, the damage profile view, it can be visually observed that the total fragmentation volume of the rock gradually decreased as the temperature decreased.The reason is that more free water in the rock's cracks condenses into ice when the temperature drops.This, in turn, increases the rock's strength and stiffness and makes it more resistant to damage.In other words, achieving the same crushing effect under identical charging conditions demands more energy in frozen rock than in rock at room temperature.
A rock damage parameter (η) was utilized to characterize the damage level of rock blasting as a damage indicator.This parameter was defined as: where   and   represent the area of the failed unit ( > 0.2) and the total area of the rock unit, respectively.Figure 6 depicts the trend of the damage parameter η along the shot hole profile and rock's top surface under four negative temperatures.Over the whole blasting process, the growth rate of rock damage can be approximately divided into two phases: the quick growth phase and the decelerated growth phase.This is because, during the initial blasting stage, the blasting pressure exceeds the rock's dynamic ultimate strength, resulting in rapid damage to the rock surrounding the blast hole.As the blast's stress wave spreads in all directions, the rock damage grows slowly because the wave weakens over time.It is also apparent from Figure 6 that during the initial phase of the explosion, the damage is greater in frozen rocks with lower temperatures at the same moment.As the explosion progresses, damage accumulation in rocks with lower temperatures slows down.This is because the velocity of waves in the rock rises as the temperature drops.The wave impedance of the frozen rock becomes greater and matches the wave impedance of the explosives better.This results in the explosive energy being more efficiently transferred into the rock at the start of the explosion.Moreover, blast stress waves propagate faster in the lower-temperature rock, resulting in faster damage growth in the lower-temperature frozen rock during the initial stages of the blast.However, lower temperatures can strengthen rock, increasing its resistance to damage, so colder-frozen rock is less damaged later in the blasting process.To quantitatively represent the total damage caused by rock blasting, the final damage cloud of the model underwent equal-spaced slicing along the borehole axis direction (50 mm spacing).Each slice's damage area was acquired and cumulatively computed in three dimensions to attain the total fragmentation volume of the rock through integration.Figure 7 exhibits the aggregate volume of blasting-induced damage caused to the frozen rock across four negative temperatures.It is discernible that in the temperature range of −5℃ to −30℃, the rock fragmentation volume reduces as the temperature decreases.The correlation can be depicted by a quadratic polynomial, which is consistent with the temperature-dependent change rule for rock strength.

Influence of Blasting Parameters on the Effectiveness of Blasting in Frozen Rock
The study findings clearly show that the efficiency of blasting frozen rocks is significantly affected by the negative temperature.This section further analyzed the effects of ignition location, free surfaces, and borehole burdens on the efficacy of blasting in frozen rock.Details are as follows.

Effect of Ignition Location
To examine the impact of the initiation position on the blasting effect of frozen rock, this study utilized frozen rock at a temperature of −30 ℃.Three different initiation positions were considered: at the bottom of the blast hole (reverse detonation), in the middle, and at the top (forward detonation).Figure 8 shows blasting-induced damage maps of frozen rock under three ignition locations, and the rock damage distribution varies significantly.Concretely, when the detonation point is positioned at the top of a borehole, the rock near the top experiences more damage, while the rock near the bottom experiences less damage.Overall, the reverse detonation resulted in the largest fragmentation volume (3.668 m3), followed by the intermediate initiation (3.224m3), and the smallest fragmentation volume occurred from the forward detonation (3.184 m3).This phenomenon can be attributed to the fact that, under forward detonation conditions, the initial blast wave is closer to the free surface, and more of the blast energy is released towards it, resulting in less damage to the internal rock.Under reverse detonation conditions, the initial blast wave is farther from the free surface, allowing greater blast energy to destroy the surrounding rock.Therefore, it can be generalized that reverse detonation leads to optimal results for frozen rock destruction.It should be noted that the damage at the top of the borehole was more substantial under forward detonation conditions, which may contribute to reducing the proportion of large rocks.Thus, in practice, the combination of reverse detonation and forward detonation techniques can result in more effective rock damage and reduce the proportion of large rocks.

Effect of Borehole Burden
To investigate the impact of borehole burden distance, a single-hole blasting model was developed, which has two free surfaces located at the top and parallel to the axis of the borehole.The distance from the center of the hole to the side was set to 1.0 m, 1.1 m, 1.2 m, 1.3 m, 1.4 m, and 1.5 m.All other model conditions were unchanged from Section 3. The findings appear in Figure 9.If the distance of the borehole burden exceeds 1.3 m, the damaged area surrounding the hole will not be connected to the damaged area closer to the surface.This means that the rock will not be effectively destroyed and will result in a higher percentage of large rock pieces after blasting, which lowers the efficiency of blasting and increases the cost of secondary crushing.Figure 10 shows the final fragmentation volume of frozen rock at distances of different borehole burdens.The trend graph indicates that the fragmentation volume tends to increase and then decrease as the borehole burden distance increases.This is mostly because when the borehole burden distance is small, the explosive is closer to the free surface, and more energy is released prematurely into the air toward the free surface.As a result, the explosion energy is not fully utilized to destroy the rock.While the distance is too large, the explosion stress wave at the free surface of the reflected wave is smaller, resulting in less damage, so that the surface of the damage area and the internal damage area cannot pass through.Thus, an optimal distance exists to fully utilize the energy of the explosive and ensure that damaged areas inside and on the surface of the rock can be penetrated.For frozen red sandstone at −30℃, the ideal borehole burden distance is approximately 1.3 meters.

Effect of Free Surface
The free surface is one of the key factors influencing how successful blasting is.This section developed a single-hole blast model with the number of lateral free faces of 0, 1, 2, and 3, respectively, using a frozen rock at −30℃ as the study object.The models' goal is to investigate the impacts of the number and relative positions of the free faces.The distance from the center of the borehole to the lateral free surface was limited to 1.3 m.And other conditions were the same as in section 3. When there are two lateral free surfaces, the ones that are opposite and adjacent to each other are considered.Figure 11 shows the damage cloud maps of frozen rocks under different free surface numbers and distribution conditions.The results confirm that a greater number of free surfaces can significantly extend the rock's damage.Based on the statistical results in Figure 12, it can be concluded that the relative position of the free surface has little impact.And the volume of the rock damage increases by approximately 22% for each additional free surface.

Empirical Formula for the Reduction of Explosive Unit Consumption in Frozen Rock
The results of the aforementioned studies indicate that freezing has a significant influence on the blasting performance of rocks.To provide a significant reference for blasting design in high-cold regions, the study developed an empirical formula through numerical simulation results.The formula utilized multivariate nonlinear regression analysis to fit the fragmentation volume as an evaluation index, incorporating the impact of temperature, the number of free surfaces, and the distance of the borehole burden on the blasting effect. = 0.2636 •  −0.0733 •  0.4114 •   0.3604 (4) where Y is the discount factor for the unit consumption of explosives, specifically the ratio of the unit consumption of explosives at negative temperatures to that at room temperature.T is the absolute value of the negative temperature (℃).N is the number of free surfaces.And S is the borehole burden distance (m). Figure 13 compares the predicted results based on Equation (1) to the results obtained from the numerical simulation.The predicted values correspond well with the simulation results, exhibiting an R2 of 0.9505 and a residual of 0.0137.Thus, the formula can accurately predict the discount factor of explosive unit consumption under different negative temperatures within the parameter range given in this paper, which can provide a reference for the design of rock blasting in high-cold regions.4).

Conclusions
The paper investigated the blasting-induced damage characteristics of frozen rocks at different negative temperatures based on numerical simulation.It also studied how the ignition locations, free surfaces, and borehole burdens affect the blast effect.The main conclusions are as follows: 1) Due to the freezing effect on rock, blasting-induced rock damage decreases as the temperature drops.Blasting rock under sub-zero temperatures poses greater challenges compared to room temperatures.Therefore, blast designs intended for regular room temperature cannot be used for rock blasting in negative temperatures.
2) Since the point of detonation is away from the upper free surface, reverse detonation can more effectively distribute the blast energy into the rock, resulting in better blast effects.Top detonation helps to demolish the upper rock and diminishes the proportion of large rocks in the top rock following blasting.Therefore, a composite of the two is recommended for engineering blasting.
3) The efficiency of rock blasting can be greatly improved by increasing the number of free faces.Additionally, there is an optimal distance for the borehole burden that enables both effective utilization of explosive energy and efficient rock breaking.
4) The equation for the discounted explosive unit consumption of frozen rock, obtained through multivariate nonlinear regression analysis, can be used as a guideline for designing blasts at negative temperatures.

Figure 1 .
Figure 1.Diagram of model size and mesh details.

Figure 2 .
Figure 2. Diagrams of different free surfaces and borehole burdens.

Figure 6 .
Figure 6.Variation of the damage parameter (η) with time for the top surface (a) and profile (b) of frozen rock at different negative temperatures.

Figure 7 .
Figure 7. Variation of rock fragmentation volume with temperature.

Figure 8 .
Figure 8. Damage contours of frozen rock along the top surface and axial direction of the borehole under different ignition modes.(a) Reverse detonation; (b) Intermediate initiation; (c) Forward detonation.

Figure 9 .
Figure 9. Damage of frozen rock under different borehole burden distances.

Figure 10 .
Figure 10.Variation of fragmentation volume in frozen rock with borehole burden distance.

Figure 11 .
Figure 11.Damage contours of frozen rock under different free surface conditions.(a) One lateral free surface; (b) Two adjacent lateral free surfaces; (c) Two opposite lateral free surfaces; (d) Three lateral free surfaces.

Figure 12 .
Figure 12.Effect of free surface on frozen rock fragmentation volume.

Figure 13 .
Figure 13.Comparison of simulated reduction factors and predicted ones based on Eq. (4).

Table 5 .
RHT model parameters of granite.