Progressive Damage Behaviours of Triaxially Confined Rocks under Multiple Dynamic Loads

The dynamic triaxial compression tests were conducted using a triaxial Hopkinson bar apparatus (Fig. 4). It consists of a servo-control hydraulic loading system for quasi-static triaxial confining pressures (σ_x-static, σ_y-static, σ_z-static) and a dynamic loading system to apply an additional dynamic stress wave on the testing specimen. The apparatus allows the determination of dynamic behaviours of geo-materials under different confinement conditions at high strain rates. Details of the design, operation and calibration of the triaxial Hopkinson bar system are given in the reference (Liu et al. 2019). The experimental setup and loading paths for dynamic repetitive triaxial compression are depicted in the Fig. 5, the rock specimen is initially subjected to a desired triaxial pre-stress state (σ_x-static, σ_y-static, σ_z-static) or (σ₁, σ₂, σ₃), and then followed by a dynamic load. As shown in Fig. 5.a, in the test, specimen is sandwiched between the incident bar (2.5 m in length), transmission bar (2 m in length) along X direction and four output bars (2 m in length) in Y and Z directions. The loading path of rock before impact is shown in Fig. 5b. The quasi-static confinements are achieved independently by the three hydraulic cylinders installed at the end of three pairs of steel square bars in X, Y and Z loading directions, respectively. Dynamic loads are applied by launching a striker (Φ 40 mm, 500 mm in length) in the gas gun impacting on the incident bar, which generates an incident wave towards the specimen. At the location of specimen assembly, the incident wave (σ_In.) is divided into a reflected wave (σ_Re.) propagating back into the incident bar, and a transmitted wave (σ_Tr.) into the transmission bar. The Poisson’s effect induces the lateral expansion of specimen during the impact, thus compressive pulses (σ_y1, σ_y2, σ_z1 and σ_z2) were generated and transmitted through the Y and Z output bars during the dynamic triaxial compression (Fig. 5c). It is worthy to mention that, the compression bars serving as output bars are not only applying confining stresses, but also capturing the variation of confining stresses induced by the Poisson’s effect and fracturing of the rock tested.

All square bars are made of high strength maraging 42CrMo steel with a Young’s modulus of 210 GPa, density of 7850 kg/m³, P wave velocity of 5200 m/s. The cross-section of square bars is 50 × 50 mm². Strain gauges (FLA-6-11) are diametrically mounted on the surfaces of the incident, transmission bars and output bars in Y and Z directions. The test signals are recorded at a sampling rate of 1 MHz/s by strain gauges through Wheatstone bridges and a differential amplifier (SDY-2107A dynamic strain meter). The impact velocity of striker is measured by a laser device right before the impact. Red copper disc with a dimension of 15 mm × 1.5 mm (diameter × thickness) is served as the pulse shaper, which produces a well-repeatability ramped incident wave to reduce high frequency oscillations and wave dispersion effect. The friction between the bar-specimen interfaces are lubricated by petrolatum. The assumptions and data interpretation method for dynamic stress (σ_x, σ_y and σ_z), strain (ε_x, ε_y and ε_z), strain rate (\(\dot{\varepsilon }_{x}\), \(\dot{\varepsilon }_{y}\) and \(\dot{\varepsilon }_{z}\)), and volume strain (ε_v) in dynamic triaxial compression tests are provided in previous studies (Liu and Zhang 2019; Liu et al. 2019; Xu et al. 2020). 

Once the triaxial pre-stress is applied, the strain energy W₀ stored in the specimen is calculated by:where ρ, E and m are the density, elastic modulus and mass of testing specimen, respectively; σ_i is the principal pre-stress applied on the rock.

Based on the one-dimensional elastic wave propagation theory, the elastic strain energies of the incident, reflected and transmission waves within the bars along the X direction are calculated by Lundberg (1976):

The output strain energies of induced stress waves along the Y and Z directions are expressed as:where E_b is the elasitc modulus of the bars; A_b is the cross-sectional area of the bars; C_b is the longitudinal wave velocity of the bars; σ is the stress wave in the bars; t₀ is the duration of stress wave. W_in, W_re, W_tr, W_y1, W_y2, W_z1 and W_z2 correspond to energies carried by the incident, reflected, transmitted waves and output stress waves in Y and Z bars, respectively.

The energy absorbed by the specimen W_s is given by:

The energy absorption per unit volume of rock specimen is calculated by:where V_s is the volume of rock specimen.

In this study, the dynamic triaxial compression tests on the granite and marble were conducted under a quasi-static true triaxial stress state (σ_x-static, σ_y-static, σ_z-static) of (30, 20, 10) MPa and repetitive dynamic impacts with a striker velocity of 27 m/s until failure. The sandstone is initial confined at the same pre-stress state and then followed by successive dynamic impacts with a striker velocity of 17 m/s. The duration and amplitude of incident waves were kept approximately the same using pulse shapers and controlling the gas pressure in the gas gun. During the dynamic triaxial repetitive impacts, the progressive damage characterization of rock was performed after each impact loading. Microscopic damage was characterized by P and S wave velocity, synchrotron X-ray μCT and thin section microphotography.

The ultrasonic compressive and shear wave (P and S wave) velocities of rock specimens are measured before and after each impact to quantify the induced damages. The ultrasonic measurement setup consists of four basic components: a waveform generator, two piezoelectric transducers, a signal preamplifier and a data storage/digitizer (Fig. 6a). A rock specimen is sandwiched between two piezometers, with the Olympus Couplant B-Glycerin used at the specimen-piezometer interfaces to improve the contact for efficient sound transmission. A spike-shaped excitation pulse with a duration of 200 ns, produced by the PXI-5412 waveform generator, is sent to the transmitter as the input signal for specimen and is synchronously recorded by the digital storage system. Once the ultrasonic wave travelled through the specimen, it is recognised by the receiver as the output signal and is further amplified by an Olympus preamplifier (Model 5660B). Finally, the amplified signal is collected by a digital storage and digitizer with a sample rate at f_s = 10 MHz to achieve a high enough accuracy (Brantut et al. 2014). This sampling frequency f_s meets the condition of Nyquist-Shannon (f_s \(\ge\) f_max, where f_max is the maximum frequency contained in the spectrum of analogical signal). The P/S wave transducers used are Olympus Model V103/V153 with a working frequency of 1 MHz.

The travel time is generally determined by the time difference of transmitting and receiving moments (Aben et al. 2016). Typical waveforms are shown in Fig. 6b, and the wave velocity V is calculated as follows:where L_s is the length of testing rock, T_Input and T_Output are time moments of input and output signals, respectively. Both P and S wave velocities of cubic specimen are measured in the X, Y and Z directions for qualifying the anisotropic damages.

The mechanical failure of rock is generally associated with complicated progressive failure, as characterized by initiation, propagation, and coalescence of micro-cracks (Bobet and Einstein 1998; Eberhardt et al. 1999). Over the last decade, X-ray μCT has been widely employed to nondestructively identify micro cracks in geo-materials (Cnudde and Boone 2013; Desrues et al. 2010; Johns et al. 1993; Nasseri et al. 2011; Vicente et al. 2017; Zhu et al. 2018). In this study, the damaged specimen after each individual compression was scanned by X-ray μCT using a monochromatic beam available at Australian Synchrotron’s Imaging and Medical Beamline (IMBL) (Fig. 7), which provides non-destructive three-dimensional images of fracture network. Damaged rock specimen was digitally scanned with a spatial voxel size of 18.3 × 18.3 × 18.3 μm³, a monochromatic beam of 80 keV, an exposure time of 0.5 s and a specimen-detector distance of 50 cm. Details of X-ray μCT scanning and reconstruction parameters are provided in Table 2. Individual image stacks of the four corner parts of a specimen were stitched together using the Fiji software to reconstruct the whole images in post-processing. Reconstruction of the tomographic data was carried out using the XLI/XTRACT software on the Australian Synchrotron’s ASCI supercomputer, which allows a large number of image sets to be reconstructed quickly in a batch mode. All μCT images obtained were further processed using the Avizo 9.6 software on the MASSIVE cluster and the MATLAB for 2D analysis. The complexity and interconnectedness of the fracture network, including crack density, direction, surface area, volume and connectivity, are accurately measured for further analysis.

Petrographic studies based on thin section microscopic observations provide insights into mineral compositions, rock textures and impact induced cracks. The locations of thin sections are well selected, and the rocks are cut, epoxy solidified and milled in the X, Y and Z direcitons, conresponding to maximum, intermediate and minimum principal stresses, respectively. The standard thickness of a thin section is ~ 30 μm, which is adequate for optical observations (Freire-Lista et al. 2016; Homand et al. 2000). Two polarising microscopies used for microscopic examination are Nikon Optiphot2 and Zeiss Axio. The magnifications of the two microscopies are X5, X10, X20 and X40. Both microscopies can work in plane-polarised and cross polarised light modes and are equipped with real time cameras. The observation normally starts with X5 magnification and then tune in higher magnification to observe more details such as micro-cracks. A coordinates adjusting system is mounted on the rotating stage for a precision control of the section position. The mineral composition of the rocks, intergranular and trans-granular macro/micro cracks are distinguished and characterised. Besides, the rose diagram represents the micro-crack orientation and length along specific direction for each plane, which also provides the quantitative analysis for anisotropic damage.

Thin sections were produced from the initial and impacted rock specimens, parallel and perpendicular to the impact direction, respectively. Ideally, evolution of micro-cracks should be studied using a single specimen subjected to successive impact loadings. For rock dynamic tests, a technique for coordinated microscopic examination in a single specimen is not yet available, thus micro-crack evolution was observed in a set of identical specimens cored from the same source block to minimize differences between individual specimens (Tapponnier and Brace 1976). To characterise the spatial evolution of micro-cracks, quantitative microstructural analysis was performed over a relatively large portion of a crack section when possible to obtain representative results (Moore and Lockner 1995). For each thin section, 20 subregions of 1.1 × 0.88 mm located in a thin section were analyzed. Thus, 40 subregions were observed in each dierection of tested specimen, with micro-cracks identified and characterized in the subregions and their lengths measured (Griffiths et al. 2017; Wu et al. 2000).

Ultrasonic testing as a non-destructive tool is one of the most widely used method for rock characterization and examination of damage (Lama and Vutukuri 1978; Rummel and Vanheerden 1978). The ultrasonic wave velocity, amplitude attenuation and their amplitude-frequency spectra are affected by the material’s physical and mechanical properties that are strongly related to the microstructural characteristics (Aydin 2013; Chaki et al. 2008). To qualify the progressive damage of rock under dynamic repetitive triaxial compression, the ultrasonic velocities of P and S waves, ultrasonic pulse shape, and amplitude-frequency spectra of the received ultrasonic waves were examined in the X, Y and Z directions, respectively.

During the impact, the rock is fatigued with induced micro-cracks, and the damage value D of rock is qualified by the attenuation coefficient of wave velocity:where D_i,j is the damage value, i and j are the specified impact time and measurement direction, respectively; N is the total impact time.

The wave velocities were measured before and after each impact under the triaxial confinement, and the results are plotted in Fig. 14. As expected, there is a negative correlation between the wave velocities and impact times, as shown in Fig. 14a, both of P and S wave velocities decreased with increasing impact times. The generation and propagation of micro-cracks degrade the rock specimen and thus lead to a macroscopic decrease of wave velocity. Besides, the reduction of P wave velocity of granite in the Z direction is much larger than that of X and Y directions. The reason is that micro-cracks propagated along the impact X direction (maximum principal stress) and perpendicular to the Z direction (minimum principal stress), and the specimen had the largest deformation in the Z direction. It is worthy to note that, wave velocities decrease considerably after the 7th impact, indicating that micro-cracks propagated and coalesced pervasively across the fatigued specimen. The damage values of granite characterized by wave velocities are presented in Fig. 14b, where one can see that the damage values exhibit a positive relation with increasing impact times. The damages in the Z direction are higher than that of Y and Z directions, and they all increase sharply after 7th impact. These results agree with dynamic stress-strain evolutions in the Fig. 9a at Sect. 3.2. For marble and sandstone, the P and S wave velocities decrease while the damage values increase with impact times. However, it seems to be difficult to identify the turning point that the rocks fail completely, suggesting a different fatigue and damage mechanism with that of granite.

For crystallized materials like rock, the distribution of induced microstructure has a significant influence on the propagation and scattering of ultrasonic waves. Rocks with different microstructural are considered as unique frequency-selecting filters (Aydin 2013; Li and Ma 2009; Li et al. 2010; Santos et al. 2010). Therefore, by transforming the travelled signals from time domain f(t) to the frequency domain f(ω), the microstructure evolution of rock is observed.

The amplitude spectra of received ultrasonic waves stemmed from rock specimen at different impact times were calculated by the application of fast Fourier transformation, which decomposes a function of time (a signal) into its constituent frequencies. The amplitude frequency data of three rocks are presented in Fig. 15, and the signals here are obtained along specimen in the Z direction. As can be seen in Fig. 15a, c, e, the frequency properties of the received ultrasonic waves changed with increasing impact times, due to the accumulative microstructure and progressive damage under repeated impacts. For all rock types, the high-frequency parts of the received ultrasonic waves decrease with increasing impact time, and the low frequency parts become dominant as the microstructure and defects generated progressively during each impact. Besides, when frequency spectrums are zoomed in between 0 and 200 kHz (Fig. 15b, d, f), it is interesting to find out that, multiple distinct frequency spectrum ranges are detected and amplitudes attenuated with increasing impact times. The damaged rock filters the high frequencies of ultrasonic waves and saves the low ones. Additionally, the scattering in the frequency and amplitude attenuation shows different patterns with rock types, this generally results from different induced damage mechanism in grain scale.

Figure 16 shows the μCT crossion images of granite under dynamic repetitive triaxial compression. The full crossion sections (a-o) are obtained from the same slice number in three directions that presented in Fig. 16p. Note here that the impact direction is along the maxinum principal stress σ₁. As shown in Fig. 16b, g, i, the initiated micro-cracks, most exhibiting in shear, were visible after the 4th impacts. The micro-cracks then propagated progressively from the corner to middle part of the specimen with increasing impact times from 4 to 10. Also, it can be seen from Fig. 16d, i and n that the shear bands exhibit different angles, e.g. 37°, 20° and 30° in XY, YX and XZ planes, respectively. Finally, after the 10th impact, the shear bands intersected each other and coalescenced across the whole volume.

On a smaller scale, it can be observed in Fig. 16q, r, t that micro-cracks propagated across the mineral grains, such as biotite grains, which is defined as transgranular cracking. Besides, crack surfaces show a stepwise pattern with a certain of roughness. The intergranular cracking (Fig. 16q) also occurred along the biotite boundary during the impact. Because the grey values of mineral grains like quartz and feldspar are very close, it is difficult to identify whether the transgranular or intergranular is prominent from these μCT images. To address this, optical microphotographs by thin section will be further disscussed in the below section.

In addition to intergranular and transgranular cracking, variable crack branching are obeserved and presented in Fig. 16s, u. Crack branching is commonly caused by the crack interaction in the form of en echelon linking or en passant linking (Blenkinsop 2007; Kranz 1979). As shown in Fig. 16s, the branching caused by en echelon interactions occured between two straight, sub-parallel micro-cracks, which are generally linked by a third straight micro-crack. However, Fig. 16u presents crack branching induced by en passant linking, which involves deviation of micro-cracks due to the interaction of micro-crack tip stress fields between each other. Specifically, crack arrest also occurred when the crack tip was stopped by biotite grains with higher elastic modulus.

Figure 16w-z present the enlarged views of damage zone of granite in the YZ plane from 4 to 10 impact times. Micro-cracks that initiated during 4th impact distributed discontinuously along a shear band within the specimen volume. After the 8th impact, with the development of damage, micro-cracks branched off along the grain boundaries and produced obvious shear bands. With increaing impact times to 10, the crack coalescenced as well as the local grain breakage appeared, leading to the pervasive damage of granite specimen.

The X-ray microtomography acquisition on rock after each static-dynamic impact shows the propgressive development of fracture network. Table 4 shows the fracture network evolution of granite under a triaxial pre-stress of (30, 20, 10) MPa and the repetitive impact velocity of 27 m/s, corresponding to the strain rate of 100-150/s. Note here that the impact direction (X direction) is along the maxinum principal stress σ₁. The 3D-reconstruction of fracture network presents that the granite damaged slightly at the first three static-dynamic coulpling loads, accompanied with a shear band generated and propagated from the corner, which results in similar dynamic stress-strain behaviors. In the 4th impact, the shear band propagated further into the specimen. Besides, several sets of micro-cracks initiated in the middle part. With increasing impact times to 7, micro-cracks propagated along the impact direction (maximun principal stress direciton σ₁) and intercted with each other. Notablely, starting from the 8th impact, a large number of micro-cracks developed fully throughout the granite, leading to a high degree of connectivity in the macroscopic fracture network. Although experienced 10th impacts, the whole specimen still kept high cohesion.

Crytallized rocks like granite are highly heterogeneous with diffenrent minerals and grains, and also contain randomly distributed micro-cracks. When a granite is stressed, the heterogeneity leads to local concentrations of tensile and shear stresses distributed at the micro and meso scale(Denoual et al. 1997). Micro-cracks initiate locally in the material once the local stress reaches the specified strength, and they usually bend in the direction of lower yield stress or tend to arrest if suffering a grain of higher strength like quartz or lower local stress(Renard et al. 2009). During the propagation, the crack tips can be also sudjected to the random mode II loading and then change the direction, which makes the fracture surfaces with a certain roughness (Bouchaud et al. 2002). The heterogeneous deformation such as grain dislocation, crushing and rotation results in the initiation, propagation and coalesce of internal flaws, including interguanlar and transgranular cracks (Blair and Cook 1998). For the granite under static-dynamic coupling loads, there is an increase in micro-crack numbers with increasing impact times. Except for the main shear band, some micro-cracks with random directions also generate during the repetitive loads. Due to the crack intersection and coalescence, some obvious cracks branched and propagated with an angle to the main shear band are observed.

Compared to the heterogeneous granite, marble is also brittle but with homogeneous grains. Under the same loading condition as that of granite, one can see in the Table 5 that two cracks initiated on the middle part of the incident and transmission surfaces of marble, respectively. Then, they propagated towards the center part with increasing impact times. The scans taken after the 5th impacts reveal that these two cracks coalesced to form a macro-fracture whose direction is nearly parallel to the impact loads and the marble was finally split in two parts. The damage mechanism of marble is somehow different with granite, because marble has homogeneous, pure and crystalline grains. After initiation, micro-cracks underwent a straight propagation stage without surffering the high local stress concentration. Few branching crack as well as localized brittle fracture were observed.

Table 6 presents the fracture network of sandstone under the triaxial pre-stress of (30, 20, 10) MPa and the repetitive impact velocity of 17 m/s. Apparently, the damage mechanism of sedimentary sandstone is different with that of granite and marble. Sandstone containing quartz, feldspar and weak rgillaceous cement allows a larger deformation once compressed. During the dynamic triaxial impact, the sandstone was confined in a limited deformation space, rusulting in the compaction of micro-pores. The compaction process is accompanied by pore collapse, grain deformation, grain rearrangement and boundary degradetion. After the first two impacts, the compaction induced fractures were too small to be recognised by μCT. With the increasing impact times, a deformation band generated from a compaction band. Deformation bands, in general, are developed with an angle of 30.5º to the loading direction. After the 5th impact, the deformation bands coalesced to form a macro “V” shape fracture zone within the rock volume.

During the dynamic loads, the damage and fracture process of rocks are associated with energy consuming. The dynamic fracture energy is partitioned by the creation of fracture surface area, kinetic energy and other dissipative terms during the dynamic fragmentation process (Liu et al. 2020a). For triaxial static-dynamic loads tests in this study, the input energy of specimen includes static strain energy induced by the pre-stress and dynamic strain energy induced by the impact, while the output energy can be captured on the output bars. Therefore, the energy absorbed by the specimen W_s can be calculated by the Eq. (4). Besides, the crack network, generated surface area and volume of fractures after each static-dynamic load are determined by synchrotron-based μCT, as listed in Tables 4, 5 and 6.

Figure 17 shows the evolution of absorbed energy, new crack surface and crack volume of three rock types with increasing impact times. It can be observed in the Fig. 17a that the crack surface and volume increment of granite increases nonlinearly with increasing impact times, accompanied with an increasing trend of the adsorbed energy. Specially, there is a sharp rise of adsorbed energy after the 7th impact, which results in highly damaged zone and new cracks. Similar observation of the increasement of crack parameters can be obtained for brittle marble, as shown in Fig. 17b. These results are well consistent with the results of ultrasonic measurement presented in Fig. 14.

There is a different evolution trend of crack parameter and absorbed energy for sandstone, whose deformation is dominant by compaction and deformation bands. During the first three impacts, the porosity was gradually compacted and the grain was damaged, which consumed a relatively stable energy. While at the 4th impact, the deformation band of sandstone generated a macroscopic shear band, which results in a noticeable increasement of crack surface and volume, as demonstrated in Fig. 17c.

It is believed that normally only a portion of absorbed energy W_s is used to create new surface area and the rest of the energy is dissipated through other mechanism such as friction, noise, etc. Barber and Griffith (2017) investigated the energy consumed in creating new surface area and found that it likely constitutes of the order of 5%, but possibly up to 40% of absorbed energy W_s for highly fragmented specimens. For triaxial compression test, confinements significantly influence the evolutions of dissipated energy, strain energy, friction, noise, and kinetic energy. It is difficult to determine the exact fraction of each energy terms experimentally. Based on the numerical simulation of triaxial Hopkinson bar tests (Hu et al. 2020), around 20% of the absorbed energy W_s is used to create new crack surface in sandstone under dynamic triaxial compression and the kinetic energy is less than 5% of W_s. Therefore, using the data in Table 3, the absorbed energies W_s used to create new surface in sandstone under dynamic triaxial impacts are 35.6, 35.8, 38.8 and 49.5 J, with the increasing impact times from 1 to 4, and assuming that the energy percentage for new crack surface is a constant value (~ 20%) at each impact. This method could be used for the determination of crack surface energy in granite and marble.

Figures 19, 20 and 21 show the plane polarized and crossed polarized transmitted light images in the X, Y and Z directions featuring representative microstructures of granite after 10 impact times. Generally, micro-cracks are classified into three types: transgranular cracks generated across the grain; intergranular cracks propagation along grain boundaries; and transgranular-intergranular cracks, which are a combination of intergranular and transgranular cracks. In the Fig. 19a, b, it is obvious that a shear band generated acoss the specimen with an angle of 33.5° to the impact X direction, accompanied with a large number of locally highly pulverized zones. Besides, as can be observed at the enlarged view of the grain in the Fig. 19c, d, the grains are intensely damaged and crushed with many transgranular and intergranular micro-cracks following the cleavage planes of a mineral, which resulted in the columnar debris smaller than 20 μm. This grain fraturing with little or no observable grain rotation is the characteristic of pulverized texture, which is a special or unque type of brecciation (Whearty et al. 2017). The volume expansion, minor shear offset and lack of grain rotation are also recognized pervasively in the brittile minerals such as quartz and plagioclase, under the triaxial confinements and multiple impacts. The pulverized quarz seems to remain in a same extincion angle, and is consistent with the observation of pulverized granite along the San Andreas and Carlock faults (Rockwell et al. 2009). Although crushed into small particles, the original grain boundaries are easily observed. The stress induced micro-cracks throughout a grain have a perfered orientation that is nearly along the impact X direction.

The confinements and high rate loadings contribute to the brittle grain pulverization, leading to the damage generation and pulverization in the rock faults (Aben et al. 2017; Fondriest et al. 2017; Incel et al. 2019; Rodríguez-Escudero et al. 2020; Yuan et al. 2011). Whearty et al (2017) also pointed out that the pulverized texture initiated during the dynamic rupture within the confining pressture between about 1.4 and 2.4 MPa at shallow burial depths. Complete pulverized grains and seperated fragments are also observed along cataclastic rock fault under the confining pressure of about 10.8 MPa and with the depth of ~ 400 m (Morton et al. 2012; Wechsler et al. 2009).

Figure 20a, b show the microscopic observations in the XZ section, where a main pulverized shear band is recognised across different mineral grains. The shear fracture consists of intergranular and transgranular cracks, is generally closed and filled with small forming fibrous, columnar or dendritic fragments. The closeups of grain crushing within the thin shear band are shown in Fig. 20c, d. It is clear that in situ “explosion” of grain occurs after the multiple impacts, which is the result of energy rapid accumulation and dissipation, especially under dynamic cyclic loads. The remnant crystals of grains within the shear band have a heavily altered, intergranular apperance, caused by the post shear friction.

Figure 21 shows the microscopic images in the YZ plane, where it is evident that a tensile crack generated perpendicular to the minimum principal stress direction (Z direction). In this regard, the size increment of specimen in the Z direction is larger than that in other two directions, similar measurements is confirmed in the decrement of ultrasonic wave velocities, as presented in Fig. 14. The tensile crack propagated across serveral grains including quarz, biotite, feldspar as well as the orthoclase, and also shows intergranular crack apperance (Fig. 21a). Under the same loading condition, the brittle minerals exbibit different deformation and failure behaviours with that of ductile ones, resulting in the uncoordinated micro-deformations and localized breakage. As zoomed in Fig. 21c, d, the brittle quartz was crushed into smaller fragments with mode I cracks while the adjacent biotite retained relaltively intact, and the major tensile crack deviated near the kinked biotites. Although the boundaries of quarz and biotite domains are generally well identified, there is a mixing of small particles across the phase boundaries due to the offset and dilation. Besides, the stepwise patterns recognized in the X-ray microtomography images (Fig. 16z) are related to the crushing of quartz (Fig. 21c).

Based on the above observation, thin section photomicrographs of the crack phenomenon thus indicate that: (1) with an inclined angle to impact direction, shear bands generate across the different grains with a large number of transgranular cracks and often filled with small dendritic fragments; (2) tensile cracks appear to propagate perpendicular to the minimum principal stress direction, and transgranular cracks predominate and dissect large grains, forming fibrous or columnar fragments that are often bounded by cleavage planes. (3) the mechanical property mismatch between different mineral phases leads to the uncoordinated micro-deformations and localised grain breakage, and the grain crushing results from the coupling effects of confining pressure and stress waves.

Micro-crack orientation is another important paramteter to characterise the anisotropy of fracture networks, which is highly associated with the stress field (Moore and Lockner 1995). To measure the micro-crack orientation of granite after dynamic triaxial compression, the Directionality plugin of FIJi was used, which exploits the Local Gradients orientation method (Liu 1991; Schindelin et al. 2012). The original inputs are the crossed polarized transmitted light images (e.g. Figs. 19b, 20b, 21b) from three orthogonal directions of the granite specimen, which comprise abundant information of micro-cracks. For instance, the crossed polarized image Fig. 22a is firstly binarized and then extracted the cracks, and is shown in Fig. 22c. After the processing, the micro-cracks are well identified and presented in Fig. 22b, d. Based on that, the crack orientation is finally computed and shown in Fig. 22e, note here that 0° is the East direction, and the orientation is usually counterclockwise. The amount indicates the portion of cracks in spcified orientation to the total. Moreover, using the FractalDimension plugin on the binarized thin section image, the fractal dimension of micro-cracks are computed by the box counting method. The box sizes are chosen from 640 to 6 with a reduction rate of 1.2 and 1 translations of each box. The relation between the box size and box counts is linear fitted in the bi-logarithm coordinate and thus the slope of the best fitting line is the value of fractal dimension D, as shown in Fig. 22f.

The rose diagrams of micro-crack directionality of granite in the X, Y and Z directions are shown respectively in Fig. 18, where the induced micro-cracks by dynamic triaxial impact exhibit different peak orientations, in other words, the anisotropic micro-cracking. The peak orientation angle A_peak of the micro-cracks in XY, YZ and XZ planes are 6°, 14° and 133°, respectively. In the view of mechanism, the orientation distributions of micro-cracks depend on the stress state that the rock experieced during the impact. Micro-cracks appear to be preferentially parallel to the impact X direction (maximum principal stress σ₁) and perpendicular to the Z direction (minimum principal stress σ₃).

The micro-crack desity ρ_crack, defined as the total trace lenth devided by the observed area, is calculated from the same images as for orientaition analysis. Using the AnalyzeSkeleton plugin-in Fiji, after despeckling for branches corresponding to single pixel, the crack densities ignoring the original are calculated and given in the Fig. 18. Crack densities in the XY, YZ and XZ planes are 1.55, 0.28 and 2.06 mm/mm², respectively. Generally, the ratio of minimum to maxinum crack density of unstressed granite, reported by Takemura and Oda (2005), is closed to 0.64-0.70. However, under dynamic triaxial compresion, the ratio of maxinum to minimum crack density of impacted granite is 7.4, which suggests that the crack densities in three directions also show an obvious anisotropic distribution. The crack density ρ_crack varies considerably with the observed planes and the stress states, which has been confirmed in the field observation (Whearty et al. 2017). Besides, the anisotropy of crack density is highly associated with the ultrasonic wave velocity and attenuation that discussed in Sect. 3.5.

The fractal dimension is another parameter to describe the crack complexity, related to, for example, micro-crack modes and rock micro-structures. The fractal dimension D of thin sections in three directions are calculated as 1.753, 1.758 and 1.661 and listed in Fig. 18, respectively. It seems that the fractal values of induced micro-cracks in triaxial confined rock are different in three directions, and this indicates that the effect of confinements on fractal dimensions of rock is apparent. Fractal dimension D of rock is generally governed by crack propagation and branching behaviour (Heping 1989), during which the dynamic stress intensity factors and crack velocities are greatly dependent on the confining pressure.