Introduction

Dynamic characteristics of rock masses are an important part of the safety and stability analysis in rock mass engineering, such as deep underground mineral mining and blasting excavation of the underground powerhouse (Xia et al. 2008; Sawicki and Kulczykowski 2014; Jalali et al. 2017; Li et al. 2018; Bednarik 2019). The rock under impact and explosion loading is in the range of high strain rate and even ultra-high strain rate (Zhang and Zhao 2014; Li et al. 2017). Compared with the medium strain rate (Dai et al. 2010) loading and quasi-static loading under the impact of an earthquake, rock materials have rate correlation and inertia effect, thus presenting distinctive mechanical features.

Relevant scholars at home and abroad have conducted a large number of studies on the dynamic characteristics of rock materials by using the SHPB system (Li et al. 2014). Ping et al. (2015) carried out the SHPB impact compression tests on sandstone specimens and discovered that their energy dissipation density is linearly proportional to the incident energy, and in a power function relationship with the average strain rate, showing a strong strain rate correlation. Liu et al. (2015a, b) and Hakalehto (1970) studied the dynamic mechanical properties of coal rock and granite, and their research results showed that the crack initiation strength has no obvious strain rate correlation, but the dynamic yield strength has a strong strain rate effect. Olsson (1991) obtained the rock constitutive model with a high strain rate by conducting the SHPB impact test on tuff, and the study indicated that the dynamic yield strength of most rock materials has obvious strain rate effect. Kimberley and Ramesh (2011) and Kim and McCarter (1998) also conducted SHPB impact tests and studied the rock failure process and damage characteristics under impact loading with the help of a high-speed camera and acoustic emission (AE) technology.

The aforementioned research is of vital importance to recognize the mechanical properties of rock materials under impact loading. However, the transient feature of the impact loading process and the limited testing techniques affect our further analysis of the dynamic fracture behavior of materials. Therefore, it is difficult to monitor the dynamic degradation process in real time and comprehensively (Li et al. 2017). Specifically, high-speed photographic technology is limited to observe the law of crack evolution on the object surface, rather than in the internal (Zhang and Zhao 2013). The acoustic measurement (Kim and McCarter 1998) and computed tomography (CT) scanning technology (Fondriest et al. 2017) are only applicable for assessing the damage of specimens before or after the tests. Moreover, in the process of impact loading, the AE signal gained is the overlay of elastic wave and impact pulse of micro-cracks (Liu et al. 2015a, b), which brings great interference to the accurate analysis of research results.

In recent years, the particle flow code (PFC) has been widely applied in the field of geotechnical engineering, which greatly expands the research scope of traditional numerical simulation (Zhang et al. 2018b). It builds the particle aggregation from mesoscopic perspectives and can reflect the highly nonlinear stress-strain relationship of the medium. Moreover, through the failure process and crack propagation of the reaction medium bonded between particles, the dynamic simulative method can trace the source of mechanical behavior to the maximum. Professors, such as Wen et al. (2019) and Cao et al. (2017), adopted particle flow to implement a numerical simulation test. Based on the mesoscopic mechanics, they analyzed uniaxial and triaxial mechanics properties of rock specimens, expounded the rules of crack propagation and evolution, and clarified the effect of mesoscopic characteristics of rocks on macro-dynamic characteristics. Zhang et al. (2018a, b) established the direct shear numerical model using the particle flow code and systematically analyzed the deformation evolution and microscopic damage under shearing.

For exploring the dynamic problems involved in phosphate rock mining engineering, the phosphate rock in Yichang is taken as the research object, and the strain rate effect and fracture fractal characteristics under impact loading are studied systematically and comprehensively. The PFC2D mesoscopic discrete element method is adopted to analyze the dynamic fracturing process of the phosphate rock in the SHPB test system, especially the real-time dynamic degradation characteristics and evolution rules of mesoscopic cracks. In the following part of this paper, a brief introduction of the materials and laboratory test methods is presented in “Materials and laboratory test methods,” including characteristics of the phosphate rock, the SHPB system, and the laboratory test plan. “Numerical model in DEM” focuses on the numerical model in DEM, which elucidates the particle flow theory and the model setup. “Results and discussion” comprehensively reports the experimental results, especially the dynamic characteristics of the phosphate rock, the size distribution characteristics of rock fragments, and meso characteristics of numerical simulation. “Conclusion” summarizes the whole study.

Materials and laboratory test methods

Characteristics of phosphate rock

The research object is the phosphate rock, and the failure section of the specimen is examined through a JSM-5510LV scanning electron microscope. The specimens are rich in such chemical elements as P (phosphorus), Ca (calcium), and Si (silicon), and their structure is relatively dense and uniform with low porosity.

As a comparative study of the dynamic test, a static loading test has been carried out through the hydraulic testing machine to obtain such basic mechanical parameters of the phosphate rock as uniaxial strength and elasticity modulus (shown in Table 1). In the uniaxial compression laboratory test presenting in Fig. 1, when specimens reach the peak stress, the axial stress drops rapidly, and the rock fails all of a sudden, accompanied by a crisp sound, with obvious characteristics of brittleness.

Table 1 Static physical and mechanical parameters of the phosphate rock
Full size table
Fig. 1
figure 1

Uniaxial compression test of the phosphate rock

Full size image

SHPB system and test plan

For the SHPB system, the diameter of the incident bar and the transmission bar is 0.05 m, and the length is 1.5 m. The diameter of the striker and the absorbing bar is also 0.05 mm and the length is 0.2 and 0.3 m, respectively. All the components are made of 48CrMoA steel with an elastic modulus of 210 GPa and Poisson’s ratio of 0.28. In order to reduce the dispersion effect and achieve constant strain rate loading, based on the preliminary test, the soft rubber with a thickness of 1.5 mm is used as a pulse shaper.

Due to the difference in wave impedance between rock specimens and the pressure bars (incident bar and transmission bar), part of the incident wave is reflected back to the incident bar to form the reflected wave, while the other parts enter the transmission bar to form the transmitted wave. The strain gauge sensors on the incident bar and the transmitted bar monitor the corresponding signals.

Three groups of impact compression tests (І~III) are designed. The impact pressure is designed as 0.050 MPa, 0.060 MPa, and 0.070 MPa, respectively. To ensure the accuracy of laboratory tests, each group is tested three times. It should be ensured that the impact striker in the launching chamber is in the same position during each test and keeps the same distance with the incident bar, to obtain the same impact velocity under the same pressure.

Numerical model in DEM

Particle flow theory

The discrete element of PFC sets rock, soil, and other material media as an aggregate composed of particle units. In the calculation, force-displacement law and Newton’s laws of motion are applied alternatively: through the force-displacement law, the contact force between such entities as particles can be updated, and through the Newton’s laws of motion, can the location of particles to particles and particles to borders, until the unbalanced force and unbalanced moment of all particles in the model meet the design requirements of precision. This method, which has been improved repeatedly through the finite difference theory, can automatically reflect the highly nonlinear stress-strain relationship, yield strength, failure process, and crack development of the medium. This particle flow contact model adopts the Linear Parallel Bond Model, which has been widely used, and the normal stress on the bond is calculated as follows (Potyondy and Cundall 2004; Liu et al. 2019):

$$ \overline{\sigma}=\frac{F_{\mathrm{n}}}{A}+\beta \frac{\left\Vert \left.{M}_{\mathrm{b}}\right\Vert R\right.}{I} $$
(1)

The tangential stress on the bond:

$$ \overline{\tau}=\frac{F_{\mathrm{s}}}{A}+\left\{\begin{array}{l}0\ (2D)\\ {}\beta \frac{\left\Vert \left.{M}_{\mathrm{s}}\right\Vert R\right.}{I}\ (3D)\end{array}\right. $$
(2)

where \( \overline{\sigma} \) refers to the real-time normal stress; Fn, the real-time normal force on the contact; A, the cross-sectional area of the contact; β, the contributing factor of the moment; Mb, the bending moment on the bond; R, the radius of the bond; I, the moment of inertia of the contact cross section on the bond; \( \overline{\tau} \), the real-time tangential stress; Fs, the real-time tangential force on the contact; Ms, the torque on the bond; and J, the polar moment of inertia of the contact cross-sectional area on the bond.

The particle bonding area can endure certain strength. When normal stress exceeds the bonding strength, tensile failure is generated. When the tangential stress exceeds the bearing strength, shear failure is generated.

Model setup

On the basis of accurately measuring the stress wave, in the SHPB particle flow model (shown in Fig. 2), the incident bar and the transmission bar are designed with a length of 1.0 m and a diameter of 0.05 m. In order to improve the transmission capacity of the stress wave and reduce the dispersion effect, particles in the incident bar and transmitted bar are set as regularly arranged ones with the same size.

Fig. 2
figure 2

SHPB numerical model

Full size image

The meso-mechanical parameters of the phosphorite model are adjusted by the “trial-and-error method” (Hsieh et al. 2008; Chong et al. 2016; Shi et al. 2019), based on the quasi-static uniaxial compression test. When the meso-parameters in Table 2 are used, the macro-mechanical parameters obtained by the PFC numerical calculation are in good agreement with the results in the laboratory mechanics tests (Fig. 3a).

Table 2 Micro-parameters of the PFC model
Full size table
Fig. 3
figure 3

Numerical model reliability verification. a Comparison of stress-strain curves gained by laboratory experiments and numerical calculation. b Dynamic stress equilibrium monitoring results

Full size image

The bonding strength between particles in the numerical model is set sufficiently high because there is almost no damage to the incident and transmission bars during the dynamic impact test. According to the results of laboratory experiments, the numerical simulation is carried out by applying a half-sine stress pulse at the end of the incident bar. Based on the three-wave method, the magnitude of the unbalanced force is calculated by stress wave overlay. In Fig. 3b, such mathematical signs σIn, σRe, and σTr refer to the incident wave, reflected wave, and transmitted wave, respectively. The results show that the superposition of the reflected and incident waves is basically coincident with the transmitted wave. With the increase of loading, the difference between the two fluctuates over a smaller range, which verifies the stress equilibrium condition effectively.

Results and discussion

Mechanical features

In the stress-strain curve of the phosphate rock laboratory impact compression tests shown in Fig. 4a, the dynamic failure process can be divided into an initial compaction stage, an elastic stage, a plastic deformation stage, and a post-peak failure stage. As specimens are relatively complete and dense, the compaction stage is not obvious. Under different loading strain rates, elastic stages of the stress-strain curve are coincident with each other, and the slope difference is not significant, which indicates that the dynamic elasticity modulus is not so sensitive to the strain rate. When the loading strain rate is larger, more micro-cracks are activated, due to the restriction by the crack propagation velocity and stress releasing area. However, the original micro-cracks and new cracks do not propagate further, showing obvious deformation hysteresis, which is enhanced with the increase of strain rate. As a result, the peak stress increases from 86.7 to 137.8 MPa (shown in Table 3). The ratio of dynamic compressive strength to the quasi-static compressive strength of specimens is defined as a dynamic increase factor (DIF) (Ma et al. 2017). When the strain rate increases from 25.20 to 47.7 s−1, the DIF increases from 1.62 to 2.57, and a strong strain rate effect has been demonstrated, which is in great consistency with the existing research results (Li et al. 2017) (shown in Fig. 4b).

Fig. 4
figure 4

Mechanical features. a Stress-strain curve of phosphate rock. b Sensitivity rules of dynamic increase factor

Full size image
Table 3 Impact compression test results of phosphate rock
Full size table

Size distribution characteristics of fragments

The characteristics of specimen fragments are of great significance to correctly understand the rock failure degree, fracturing damage, and energy dissipation and releasing, which is an effective method to study the fracture mechanism under the impact. Figure 5 shows the fragments of phosphate rock specimen IM1, IIM1, and IIIM1, respectively. In order to quantify the index of fragment distribution efficiently, the Split-desktop digital image method has been used to conduct statistics on the size distribution. When the loading strain rate is 25.20 s−1, due to the small impact energy, specimens have a low failure degree with few and relatively large fragments, which are mostly in the shape of lumps or strips. When the loading strain rate reaches 32.03 s−1, the fragment size decreases, but the number of smaller fragments increases significantly, indicating that the number of cracks increases significantly, and the failure degree of specimens tends to be severe. When the loading strain rate increases to 47.70 s−1, the impact energy is much larger, and the formation of a large number of micro-cracks directly leads to the crushing of specimens, with substantially smaller fragment size and more uniform size distribution.

Fig. 5
figure 5

Fragmentation distribution characteristics under different loading strain rates

Full size image

Fractal geometry mainly describes the statistical self-similarity characteristics of rock specimens during the failure process through such two parameters as the fractal dimension and the scale-free interval, which is defined as (Xia et al. 2014):

$$ D=-\underset{\varepsilon \to 0}{\lim}\frac{\lg N\left(\overline{\varepsilon}\right)}{\lg \overline{\varepsilon}} $$
(3)

where D refers to the fractal dimension; \( \overline{\varepsilon} \), the scale; and \( N\left(\overline{\varepsilon}\right) \), the measured result on the scale \( \overline{\varepsilon} \).

The fractal dimension is calculated by means of size-quantity statistics. The equivalent fragment size can be calculated as follows:

$$ {R}_{\mathrm{eq}}=\sqrt{\mathrm{ab}} $$
(4)

where a and b are the length and width of fragments, respectively.

$$ N={N}_0{\left({R}_{\mathrm{eq}}/{R}_{\mathrm{eq}\mathrm{max}}\right)}^{\hbox{-} D} $$
(5)

In this equation, N is the number of fragments whose equivalent grain size is greater than Req, and N0 is the number of fragments with maximum equivalent grain size (Reqmax).

Based on fractal geometry, the fractal dimension gained through the size distribution statistics of phosphate rock fragments is calculated. Figure 6 is the relation diagram between parameter lg(Reqmax/Req) and lgN, in which the slope of the fitting line is the fractal dimension. With the increase of strain rate, the fractal dimension (D) shows an upward trend, which is 1.756, 1.808, and 1.916, respectively. Namely, the fragment size has obvious fractal characteristics. The increase of the loading strain rate causes the rock to consume more energy, so the damage intensifies, leading to the formation of more and smaller fragments. Therefore, the failure degree of rock specimens can be quantitatively and comprehensively evaluated by the fractal dimension (D).

Fig. 6
figure 6

The relation diagram between parameters lg (Reqmax/Req) and lgN

Full size image

Meso characteristics in numerical simulation

Based on laboratory experiments, the amplitude value of incident stress wave is designed as 50 MPa, 75 MPa, 100 MPa, 125 MPa, and 150 MPa, and the cycle is 300 μs, with five kinds of working conditions of SHPB numerical simulation tests. The corresponding specimen number is IS1–IS5.

Crack feature analysis

The specimen IS3 under 100-MPa impact loading is taken as an example (in Fig. 7). It can be found that in the impact loading experiments, the number of cracks (both the total cracks, tensile cracks, and shear cracks) shows an “S”-type accumulation on the whole, which is consistent with the staged growth in the early research (Zhou et al. 2017). It experiences four phases from no cracks, to crack initiation, to the rapid growth of cracks, and to a gradual reduction in the growth rate, presenting obvious regularity.

Fig. 7
figure 7

Crack development and evolution characteristics of specimen IS3

Full size image

Accordingly, crack propagation in the specimens can be divided into the elastic crack-free stage, crack initiation stage, rapid crack growth stage, and slow crack development stage. Whether it is the minor damage of specimens under small impact loading, or the crushing of specimens under high impact loading, the number of tensile cracks always plays a dominant role. Figure 7 shows the cumulative number curve of cracks and their spatial distribution in the four stages, respectively.

Elastic crack-free stage (OA)

During the time between 276 and 370 μs, the specimen is under elastic deformation with no cracks generated inside, and the stress-strain curve almost shows a linear growth.

Crack initiation stage (AB)

At 370 μs, cracks emerge, and the number increases, with the crack growth curve presenting a concave nonlinear feature. The crack initiation strength is 52.1 MPa, which is 64.88% of the dynamic compressive strength of the complete specimen. At this stage, crack distribution is random, and the number of shear cracks is approximately equal to that of tensile cracks.

Rapid crack growth stage (BC)

At 408 μs, when the peak stress approaches, the number of cracks shows a linear and rapid growth, with a more dramatic rise in the number of tensile cracks than that of shear cracks. Due to the sudden surge in crack density, the interaction and interference between cracks tend to be obvious, and cracks continue to develop and condense. At this time, due to the increase of mesoscopic crack density, the nonlinear characteristics of rock mechanical properties have been demonstrated distinctly.

Slow crack development stage (CD)

After 468 μs, the number of cracks grows slowly, and finally, the number amounts to 3837, among which tensile cracks account for about 75.3% and shear cracks, 24.7%. Under the dominant action of tensile cracks, tensile failure emerges in the specimen and more macroscopic cracks form, which are approximately along the axis direction.

Failure mode analysis

Figure 8 shows the crack spatial distribution (DFN [discrete fracture net]) and fracture morphology of the specimen under different loading strain rates. In the fracture morphology diagram, different pieces are given different ID and are displayed in different colors. With the increase of impact loading, a large number of mesoscopic cracks are activated (Table 4), and the specimen gradually changes from local damage mode to axial splitting failure mode and to crushing failure mode.

Fig. 8
figure 8

Mesoscopic cracks and fracture morphology under different loading strain rates

Full size image
Table 4 SHPB test results of numerical simulation
Full size table

Local damage mode

Under the condition of low loading stress rate (\( \overset{.}{\varepsilon } \) = 15.2 s−1), the contact force of local particles reaches the bearing limit of the bonding force, and a small number of mesoscopic cracks are generated due to the bond failure. Although crack distribution is random, the specimen does not fail.

Splitting failure mode

When the loading stress rate is \( \overset{.}{\varepsilon } \) = 21.0 s−1, micro-cracks are rapidly activated, which develop and condense continuously, forming several main cracks which penetrate the specimen and are nearly parallel to the direction of the main stress, with obvious direction. At the same time, a certain number of multi-level cross-cracks form at the ends of the specimen, and the entire specimen is subjected to axial splitting failure, resulting in a total of 1812 cracks and 94 fragments, among which the fragments are relatively large in size and mostly in the shape of bars.

Crushing failure mode

When the loading stress rate reaches \( \overset{.}{\varepsilon } \) = 34.9 s−1, due to the limitation of crack growth rate and stress release zone, more micro-cracks are activated and join in the failure process before the main crack goes through the whole rock specimen. Due to the sudden increase in crack density, the interaction between cracks enhances, resulting in the change of propagation direction and forming a multi-layer and cross-mesh crushing mode. Finally, a total of 4366 cracks and 340 fragments are generated in the specimen, with the latter being smaller in shapes and larger in numbers.

Conclusion

The phosphate rock in Yichang is taken as the research object, and the strain rate effect and fracture fractal characteristics under impact loading are studied systematically, which is of great significance for understanding the dynamic mechanical properties of rock materials. However, a comprehensive understanding of the dynamic fracture mechanism is plagued by the transient characteristics of the impact process and the limited testing techniques. In this study, the PFC2D mesoscopic discrete element method is adopted to analyze the dynamic fracturing process of the phosphate rock by the SHPB testing system. Starting from the mesoscopic structure, we have studied the real-time dynamic degradation characteristics and evolution rules of mesoscopic cracks, such as nucleation, propagation, penetration, and interaction, which greatly makes up for the deficiency of laboratory experiments. Major findings are the following.

The failure process of phosphate rock specimens shows obvious brittle features under both static and dynamic impact loading. With the increase of strain rate, the dynamic elastic modulus shows little change, but the yield strength increases obviously, presenting a strong effect of loading strain rate. In addition, with the increase of the impact loading strain rate, the rock consumes more energy; more severe damages are generated, and smaller fragments emerge substantially. The fragment size has obvious fractal characteristics, and the failure degree of rock specimens can be quantitatively and comprehensively evaluated by the fractal dimension (D).

Moreover, based on the discrete element simulation research, we can find that the initiation and propagation of macroscopic cracks are essentially the results of the alternating action of local stress concentration and stress relaxation. With the increase of loading, the internal part of specimens reaches the bearing limit of the bonding force between particles. Thus, cracks are created, propagated, condensed, and penetrated, resulting in macroscopic failure. According to the characteristics of crack propagation, the failure process can be divided into four stages: an elastic crack-free stage, a crack initiation stage, a rapid crack growth stage, and a slow crack development stage. At the same time, mesoscopic cracks are activated in large quantities, and the interaction between cracks tends to be obvious with the increase of strain rate. Rock failure pattern changes from local damage to splitting failure and to crushing failure.