Introduction

In many underground rock engineering practices, such as tunnelling and mining, a large number of cavities (including roadway, chamber and goaf) have been left. The existence of such cavities lowers the stability of rock mass as stress concentration occurs around them, which is responsible for many hazards, such as gob collapse and water inrush (Yin et al. 2018). Therefore, it is of great significance to understand the mechanical behaviour of rock mass containing cavities for the security of underground engineering construction.

Over several decades, the mechanical behaviour and fracture evolution of the rock mass containing joints or cavities have been extensively studied (Tan et al. 2021; Li et al. 2020a; Liu et al. 2018b). For rock mass containing a single circular cavity under static uniaxial compression, three kinds of failure can be observed, namely the primary tensile cracks from the roof and floor, the spalling failure around the sides and far-field cracks (Lotidis et al. 2017; Li et al. 2017b; Carter et al. 1991, 1992; Dzik and Lajtai 1996). For non-circular cavities, similar failure patterns can also be concluded, but high stress concentration around cavity corners may result in serious failure from the corners (Zhou et al. 2017). In addition, for specimens containing a single cavity, the final instability of rock specimens under uniaxial compression is always characterized by crack interaction between the cavity and specimen boundary (Zhou et al. 2017; Li et al. 2011; Zhao et al. 2014; Sagong et al. 2011; Li et al. 2017a). It is concluded that cavity number and layout both dominate the stability of rock mass (Lin et al. 2015; Zhao et al. 2014; Yang et al. 2012; Tang et al. 2005). For example, Lin et al. (Lin et al. 2015) reported that the existence of cavities greatly degrades the strength of rock specimens, but the peak stress mainly depends on number of coalescent cavities rather than the whole cavities numbers. Yang et al. (2012) studied the mechanical parameters of sandstone containing two oval cavities and found that they have a nonlinear relationship with the ligament angle.

The above-mentioned studies mainly focused on quasi-static conditions. However, in practical engineering, dynamic disturbances such as blasting, drilling and seismic event are inevitable for underground rock structure (Cai et al. 2020c, 2020a; Zhou et al. 2020a; Tan et al. 2018; Zhang et al. 2020; Xu and Dai 2018). Those disturbances propagate inside the rock mass in the form of stress waves. Refraction, reflection and scattering will happen when the stress wave reaches borders of cavity. These will give rise to intense stress concentration around those free surfaces, which may lead to the catastrophic failure of cavities (Yi et al. 2014; Yi et al. 2016; Changyou et al. 2017; Liu et al. 2018a; Cao et al. 2021). Undoubtedly, cavities have a weakening influence on the dynamic mechanical properties of rock mass, which may be affected by a series of factors like shape, size and layouts of cavities (Li et al. 2017b). Moreover, the mechanical behaviours and failure pattern of rock specimens, especially those containing cavities, are tightly related to the loading rate (Zhou et al. 2014; Wang et al. 2018). For specimens containing a single circular cavity under dynamic loading, initial failure in the form of tensile crack appears around the roof and floor of the cavity parallel to loading direction, which is similar to what happened to those under static compression (Tao et al. 2017; Li et al. 2020c). However, it should be noted that results in different studies may be inconsistent in terms of final failure mode. For example, Li et al. (2015) found that the final failure for specimens containing a single circular cavity under split Hopkinson pressure bar (SHPB) tests was characterized by X-shape shear crack, while Tao et al. (2017) reported that the instability of specimens containing a single circular cavity was caused by tensile cracks parallel to dynamic loading direction without shear cracks observed. This difference may be resulted from various factors including cavity shape and size, width-height ratio of specimen, loading condition and so on.

Overall, the dynamic mechanical behaviour and cracking evolution for rock containing cavities have been extensively studied in the past. However, some research gaps should be noted. On the one hand, most related studies focused on circular or elliptic cavities. For cavities with other shapes like rectangular cavities which are popular in underground rock engineering, studies about their dynamic response characteristics are insufficient. On the other hand, little attention has been paid to the dynamic mechanical behaviour of twin-cavity system; further studies are expected for a better understanding of their dynamic mechanical behaviour.

In view of this, SHPB tests were conducted on marble specimens containing one or two cavities under different driving pressures in the present study. The driving pressure refers to the pressure of nitrogen gas released from the gas tank to drive the striker for impact. Five main groups including intact specimens, specimens containing a single rectangular cavity and specimens containing two rectangular cavities with different layouts were considered. The dynamic loading capacity, failure process and energy characteristics were investigated for the purpose of a comprehensive understanding of the dynamic mechanical behaviours for rock mass containing cavities.

Experimental methodology

Specimen preparation

In this study, rectangular cuboid marble specimens with dimensions of 60 mm × 60 mm × 30 mm (length × height × thickness) were cut for SHPB tests. The aspect ratio was 1.0 for all specimens to minimize the stress shadow effect. All specimens processed from one single marble block whose density and uniaxial compression strength (UCS) are 2780 kg/m3 and 124.8 MPa, respectively. The ends of each specimen were polished and parallel to each other. To reduce the effect of specimen boundaries on the coalescence and failure modes of cavities, the size of each rectangle cavity imbedded inside specimens was designed to be 15.0 mm × 7.5 mm.

As shown in Fig. 1, five main groups named from G1 to G5 were prepared with each containing twelve specimens. Specimens were intact ones without cavity in group G1 and contained a single cavity in group G2. Two cavities were cut in specimens in groups G3, G4 and G5. The minimum distance between two adjacent cavities was 7.5 mm. The angle between the line connecting centroids of two cavities with respect to the horizontal direction was defined as the connecting angle θ, which was 0° in group G1, 45° in group G2 and 90° in group G3. To simulate the dynamic disturbance with different energy levels, three driving pressures were designed, i.e., 0.35 MPa, 0.40 MPa and 0.45 MPa.

Fig. 1
figure 1

Schematics of cavity configurations in the specimens. (a) G1. (b) G2. (c). G3. (d) G4. (e) G5

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Experimental apparatus

SHPB test system

To study the dynamic mechanical behaviour of specimens with larger sizes, Li (Li et al. 2005b) developed a large diameter SHPB system, in which the diameter of both incident bar and transmitted bar was 75 mm. However, large diameter equipment contributes to strong oscillation and dispersion of the incident wave, which may result in unreliable results. This negative effect caused by large diameter bars can be revealed by (Li et al. 2005a):

$${C}_{p}={C}_{0}\left[1-{\pi }^{2}{v}^{2}{\left(\frac{r}{\lambda }\right)}^{2}\right]$$
(1)

where \({C}_{p}\) indicates the velocity of harmonic components of stress wave;\({C}_{0}=\sqrt{E/\rho }\) is the wave velocity of bars; E is the elastic modulus of bars; ρ is the density of bars; v is the Poisson ratio of bars; r is the radius of bars; \(\lambda\) is the wavelength of harmonic components of stress wave.

From Eq. (1), the difference between \({C}_{p}\) and \({C}_{0}\) rises with the increase of bar radius and thus gives rise to stress wave dispersion. In order to address this problem, the half-sine loading waveform was suggested by Li, which was realized by using a cone-shaped striker (Zhao et al. 2016; Hong et al. 2009).

In this study, the 75 mm SHPB system was employed for all tests. As shown in Fig. 2, the SHPB test system consists of a cone-shaped striker, an incident bar (2.0 m in length), a transmitted bar (1.5 m in length) and an absorption bar (0.5 m in length). All elastic bars and the striker were made of high-strength Cr40 alloy whose density and elastic modulus were 7800 kg/m3 and 250 GPa, respectively. In the tests, rock specimens were sandwiched between the incident and transmitted bars. Moderate Vaseline grease was smeared on the interfaces between bars and specimen to minimize the end friction (Zhou et al. 2020b).

Fig. 2
figure 2

The schematic illustration of SHPB system with high-speed camera

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Dynamic stress equilibrium was firstly checked before tests and the result was presented in Fig. 3. \({\sigma }_{i}\), \({\sigma }_{r}\), \({\sigma }_{i+r}\) and \({\sigma }_{t}\) indicates the incident stress wave, reflected stress wave, the superposition of incident and reflected wave stress and the transmitted stress wave, respectively. It can be seen that the \({\sigma }_{i+r}\) almost coincides with \({\sigma }_{t}\) during the loading process, which means that stress equilibrium in the specimen has been achieved.

Fig. 3
figure 3

Stress equilibrium on the two ends of the specimen

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High‑speed digital image correlation technique

In recent years, the digital image correlation (DIC) technique has been extensively applied to obtain the deformation and strain fields of rock materials subjected to external loadings (Zhou et al. 2019; Yan et al. 2020; Li et al. 2020b; Sharafisafa et al. 2020), which was also used to analyse the fracture evolution of specimen under dynamic loading in this study. To improve DIC performance for deformation and failure evolution analysis, artificial speckles were randomly fabricated on the surface of rock against a very thin layer of white paint.

During SHPB tests, the failure process of each specimen was captured by a high-speed camera in the resolution of 256 × 256 pixels at 79,166 frame/s. The open source DIC code Ncorr was then employed to calculate specimen deformation (Blaber et al. 2015). Afterwards, the processed data was interpreted to strain contour figures and displacement vector figures via self-documenting codes.

Determination of dynamic parameters

According to the one-dimensional stress wave theory, the dynamic forces on both ends of the specimen can be expressed as (Cai et al. 2020b):

$$\left\{\begin{array}{l}{F}_{1}\left(t\right)={A}_{b}{E}_{b}\left[{\varepsilon }_{i}\left(t\right)+{\varepsilon }_{r}\left(t\right)\right]\\ {F}_{2}\left(t\right)={A}_{b}{E}_{b}{\varepsilon }_{t}\left(t\right)\end{array}\right.$$
(2)

where F1 and F2 are the forces on the specimen end connecting with the incident bar and transmitted bar, respectively; ɛ is the measured strain; the subscripts i, r and t are incident, reflected and transmitted waves, respectively; Eb is the elastic modulus of the bar; Ab is the cross-section areas of the bar. When the stress equilibrium is satisfied, the dynamic loading capacity of the specimens is the peak value of \(\left[{F}_{1}\left(t\right)+{F}_{2}\left(t\right)\right]/2\).

In the dynamic SHPB test, the strain energy in elastic bars induced by stress waves can be calculated as (Li et al. 2018):

$$\left\{\begin{array}{l}{W}_{i}={A}_{b}{C}_{b}{E}_{b}\int {\varepsilon }_{i}{\left(t\right)}^{2}dt\\ {W}_{r}={A}_{b}{C}_{b}{E}_{b}\int {\varepsilon }_{r}{\left(t\right)}^{2}dt\\ {W}_{t}={A}_{b}{C}_{b}{E}_{b}\int {\varepsilon }_{t}{\left(t\right)}^{2}dt\end{array}\right.$$
(3)

where \({W}_{i}\), \({W}_{r}\) and \({W}_{t}\) are the strain energy induced by incident, reflected and transmitted bars, respectively; \({C}_{b}\) is the elastic wave speed in the bars.

Without the consideration of the energy consumed in the interfaces between the specimen and bars, the energy absorbed by the specimen, which includes the stored strain energy, the released kinetic energy and other forms of consumed energy, can be calculated as (Zhang et al. 2000; Weng et al. 2019; Yue et al. 2020):

$${W}_{s}={W}_{i}-{W}_{r}-{W}_{t}$$
(4)

Accordingly, the absorbed energy per unit volume e can be determined as:

$$e={W}_{s}/V$$
(5)

where V is the volume of a rock specimen excluding the volume of cavity.

Results and discussion

Dynamic loading capacity

The geometric and strength parameters for marble specimens whose test results were available are presented in Table 1. Specimens in the same main group have the same cavity number and cavity layout. The sub-group index 1, 2 and 3 indicates the driving pressure of 0.35 MPa, 0.40 MPa and 0.45 MPa actuating the striker bar, respectively.

Table 1 Geometric parameters for valid marble specimens
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The dynamic loading capacity of specimens under different incident energy is presented in Fig. 4. Under the same driving pressure, the specimens in group G1 (intact specimens) have the maximum dynamic loading capacity. The specimens in main group G3 and G4 where specimens containing two cavities with connecting angle of 30° and 45° show the lowest dynamic loading capacity. Figure 5 further shows the dynamic loading capacity of specimens containing two cavities versus the connecting angle. Clearly, regardless of driving pressure, as the connecting angle rises, the dynamic loading capacity slightly decreases first at 45° and then dramatically increases.

Fig. 4
figure 4

Dynamic loading capacity for specimens subjected to different driving pressures. (a) pi = 0.35 MPa. (b) pi = 0.40 MPa. (c) pi = 0.45 MPa

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Fig. 5
figure 5

Dynamic loading capacity of specimens containing two cavities versus

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The fitting curves of dynamic loading capacity fd versus incident energy Wi for specimens in different main groups are presented in Fig. 6. The fitting functions between fd and Wi for all groups are list in Table 2. As illustrated in Fig. 6, for specimens in the same group, with the increase of incident energy, the dynamic loading capacity increases significantly at first and then tends to remain stable. However, the threshold of incident energy varies from groups to groups. For intact specimens in group G1, the loading capacity keeps increasing on the whole with incident energy increasing from 89.7 J to 271.9 J. For specimens containing a cavity (G2) or two cavities (G3, G4 and G5), the dependence of dynamic loading capacity on driving pressure is not obvious anymore once the incident energy exceeds about 150.0 J. This result shows that the existence of cavity obviously limits the strengthening effect of strong dynamic loading on rock specimens.

Fig. 6
figure 6

Dynamic loading capacity versus incident energy for specimens

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Table 2 The fitting functions between dynamic loading capacity fd and incident energy Wi
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Energy characteristics

It has been reported that in static mechanical tests, the fragmentation of rock is highly related to the energy input (Song et al. 2020, 2018; Wang et al. 2019). A higher e indicates that more energy will be released during rock fracturing, which usually results in the violent failure and more fragments (Zhou et al. 2017). The absorbed energy per unit volume e for specimens in different groups is shown in Fig. 7. Overall, the change trend of e is very similar to that of dynamic loading capacity (Fig. 4). For specimens in the same group, they have the lowest e when suffering the driving pressure of 0.35 MPa. Then, a considerable increase of e can be observed when the driving pressure raises to 0.40 MPa, but the change of e is less distinct when the driving pressure increases from 0.40 MPa to 0.45 MPa. Take the intact specimens (main group G1) for example, the average value of e is 0.43 mJ/mm3 under the driving pressure of 0.35 MPa, then substantially increases to 0.67 mJ/mm3 under 0.40 MPa driving pressure, but lastly remains almost the same despite the highest driving pressure of 0.45 MPa.

Fig. 7
figure 7

Absorbed energy per unit volume for specimens. (a) pi = 0.35 MPa. (b) pi = 0.40 MPa. (c) pi = 0.45 MPa

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As for the influence of cavities on absorbed energy, in call cases with different driving pressures, the intact specimens have the highest e. Although the specimens in G5 have two cavities with a connecting angle of 90°, their e is close to that of the specimens containing one cavity in G2. The absorbed energy for specimens with connecting angle 0° and 30° is the lowest. As shown in Fig. 7b, when the driving pressure is 0.40 MPa, the average value of e is the highest (0.67 mJ/mm3) for intact specimens (G1), then reduces by 22.7% to 0.44 mJ/mm3 for specimens containing a single cavity (G2), which is almost identical with that for specimens containing two cavities with connecting angle of 90° (G5), followed by G3 of 0.27 mJ/mm3, where specimens containing two cavities with connecting angle of 0°. With the connecting angle changes to 45°, specimens have the minimum value of e of 0.21 mJ/mm3. Similar energy change trend is also observed when the driving pressures are 0.35 MPa and 0.45 MPa.

For a better understanding of the influence of cavity layouts on absorbed energy for specimens containing two cavities, the variation of e with the increasing connecting angle is presented in Fig. 8. When the driving pressure is 4.5 MPa, with the increase of connecting angle from 0° to 45°, the average e declines from 0.31 mJ/mm3 to 0.21 mJ/mm3 by 31.0% and then rebounds to a peak of 0.50 mJ/mm3 at 90°. Similar change is also observed when the driving pressure is 4.0 MPa. However, when the connecting angle is increased from 0° to 45°, the fluctuation of e is much slighter when the driving pressure is 3.5 MPa, whose e only decreases by 9.8% from 0.19 mJ/mm3 to 0.17 mJ/mm3. Afterwards, e significantly rises to 0.25 mJ/mm3 by 45.6% when the connecting angle is increased to 90°.

Fig. 8
figure 8

Absorbed energy per unit volume of specimens containing two cavities

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An interesting phenomenon worth to be mentioned is that for specimens containing two cavities with 90° connecting angle, e is very close to and even slightly higher than that of specimens containing a single cavity in some tests. Similar phenomenon is also observed in Fig. 4 in terms of loading capacity, where the loading capacity of specimens in G5 containing two holes with a 90° connecting angle are very close to those in G2 containing a single hole. This suggests that the influence of the second cavity on the impact resistance capacity and absorbing energy of rock mass is very limited. For specimens containing a single cavity during dynamic impact, stress wave results in high compressive stress concentration at both shorter side walls and reflects as tensile stress at the longer side suffering it (the right end of the specimens). When two cavities are arranged with the connecting angle of 90°, the left cavity is behind the first one along the impact direction, where stress wave has less influence. Therefore, the second cavity has much less influence on specimen’s mechanical properties. Consequently, the average absorbing energy as well as loading capacity of specimens in G1 is close to that of specimens in G2 under the same driving pressure.

Fracture evolution

Failure patterns

Zhang and Wong (2014) innovatively proposed the displacement trend lines (DTLs) method to identify the crack types of rock materials. Zhou et al. (2020b) first combined the DTL method with DIC technique to study rock fracturing process. As shown in Fig. 9, the displacement vectors around a crack can be decomposed into two components parallel to and perpendicular to the crack respectively, based on the relative movement on both sides around the crack. The crack is determined as a tensile crack when the relative movement is perpendicular to it and as a shear crack when the relative movement is parallel to it. If both relative movements are involved, it is characterized by a mixed crack.

Fig. 9
figure 9

Three crack types based on displacement trend lines around cracks: (a) tensile failure, (b) shear failure and (c) mixed tensile/shear failure

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The displacement contour for specimens in each group under different driving pressures is shown in Fig. 10, where the DTLs are plotted to determine the crack type. As seen in Fig. 10a, the failure pattern of intact specimens is dominantly characterized by the tensile crack through the specimen along loading direction. For specimens containing a single cavity (Fig. 10b), the strong compressive stress concentrates around the shorter sides parallel to loading direction and causes serious splitting failure. This phenomenon can be also observed in the areas around the outer sides of twin-cavity system. As for the coalescence patterns between two cavities, the connecting area between two horizontally aligned cavities in group G3 (Fig. 10c) is fractured via splitting pattern. When the connecting angle is 45° (G4), two cavities are coalesced with a shear crack between them. When θ increases to 90° (G5), two cavities are connected by the tensile cracks along the edges of the layer between them. It can be clearly seen that the failure patterns of specimens in the same main group are similar to each other under different driving pressures. Therefore, it can be concluded that driving pressure or incident energy has no significant effect on the failure pattern of specimens.

Fig. 10
figure 10

The displacement contour and displacement vector field for specimens under different driving pressure p (the blue and red arrows indicate relative movement on the two sides of some typical cracks)

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Generally speaking, when there is a single cavity or two cavities in the specimen, the overall instability is caused by two V-shape shear cracks from the cavity or twin-cavity system to specimen corners. However, the coalescence pattern between two cavities varies with their layout.

According to the displacement vector fields presented in Fig. 10, failure modes for specimens in different main groups are concluded and sketched in Fig. 11. Besides three typical types of cracks namely tensile crack, shear crack and mixed tensile-shear crack, surface spalling caused by stress concentration on side walls of cavities is also observed. For specimens containing one or two cavities, the final instability is characterized by cracks connecting cavity system and specimen corners. For specimens in main groups G2, G3 and G5 where the distance between cavity sides and specimen corners is farther than in group G4, the connecting cracks are characterized by shear manner. However, it seems that in group G5 where cavity sides are closer to specimen corners, the shorter connecting cracks tend to be mixed tensile-shear cracks with tensile failure involved (Fig. 11d). It is shown that cracks initiating from cavities are more likely to propagate to specimen corners through the shortest path and thus their types may change correspondingly.

Fig. 11
figure 11

Typical failure modes for specimens in different main groups

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It should be noted that in many studies based on quasi-static compression tests, for the specimens containing a single cavity with circular or rectangular shape, small tensile cracks or strain concentration appears in the area around the middle part of the cavity sides vertical to loading direction (Zhou et al. 2017; Lotidis et al. 2017; Dzik and Lajtai 1996; Li et al. 2017b). However, in those areas, neither macro cracks nor strain concentration is observed not only in this study but also in other studies for specimens containing a single cavity with other shapes in dynamic loading tests (Li et al. 2015). Some studies have revealed that for specimens containing a single flaw subjected to dynamic loading, shear cracks rather than tensile cracks are more likely to appear around the flaw tips because of the compressive stress induced by the arrival of the stress wave (Li et al. 2017b; Li and Wong 2012). This may be also reasonable to explain what happens to specimens containing a cavity. Once the stress wave arrives the cavity boundary, intensive compressive stress concentration occurs around cavity corners where failure will be generated under high strain rate to form shear cracks. Meanwhile, the wave arriving at the middle part of the cavity sides vertical to the loading direction is still in the form of compressive stress. Before its reflection and transform to tensile stress, initial shear cracks have been generated and are keeping developing from the corners, leading to strain energy release and stress redistribution around the cavity. As a result, the tensile failure may will be suppressed.

Comparing Fig. 10c and Fig. 11c, however, we can find that for specimens containing two cavities with connecting angle of 0°, a tensile crack generates from the twin-cavity system and developed parallel to loading direction under the driving pressure of 0.45 MPa. This may be because the coalescence of the two cavities with the failure of the connecting area contributes tensile stress concentration around the damaged connecting area and thus leads to tensile failure once the stress transformed from compressive stress to tensile stress. For those under the driving pressure of 0.40 MPa, the crack-like strain concentration appears in this area although no visible tensile crack is observed. However, for specimens under driving pressure of 0.30 MPa, the reflected tensile stress wave is too weak to generate tensile failure even the twin cavities have been coalesced.

Specimen fragments collected for typical specimens in different cases after tests are shown in Fig. 12. It is clearly seen that the higher driving pressure contributes more violent failure and much more fragments. When the driving pressure reaches to 0.45 MPa, several larger fragments still remain for specimens in groups G3 and G4. Conversely, specimens are crushed into a large number of small pieces of fragments in other groups, especially in group G1.

Fig. 12
figure 12

Typical fragments for specimens in different cases

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From Fig. 7 and Fig. 12, it can be found that the fragmentation characteristics are tightly associated with the absorbed energy per unit volume e, which depends on driving pressure as well as cavities’ number and layouts. A higher e allows more fragments to be produced. When the connecting angle is 0° or 45° for specimens containing two cavities, the lower loading capacity limits the energy stored in the specimens before failure such that less fragments are generated in these specimens.

Evolution of strain field during dynamic loading

Figure 13 shows the evolution of maximum principal strain field for specimens under 4.0 MPa driving pressure. The measuring points are plotted in Fig. 11. The starting time is set as 0, which indicates the time when the high-speed camera was triggered by a transistor-logic signal generated by the strain gauge on the incident bar. From Fig. 13a, for the intact specimen, the cracks mainly propagate along the loading direction. The initial strain concentration regions nucleate around the right boundary of the specimen, namely incident boundary. At 442 μs, the strain concentration regions can be observed at the left boundary. Meanwhile, several stripe-like strain concentration regions appear through the specimen surface. At 568 μs, the specimen is cut off by a strain concentration region though its middle area, which is later turned out to be a tensile crack.

Fig. 13
figure 13

Maximum principal strain contours of specimen surfaces at different time

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As shown in Fig. 13b, when there is a single rectangular cavity in the specimen, initial strain concentration regions can be observed around the two shorter sides of the cavity parallel to loading direction. Afterwards, these strain concentration regions develop diagonally, connecting the cavity with the specimen corners. At 252 μs, obvious failure appears around the two shorter sides of the cavity as a consequence of high compressive stress concentration. Finally, four inclined crack-like high strain concentration regions fracture the specimen from the cavity to specimen corners.

For the specimen containing two horizontally aligned cavities whose longer sides vertical to the loading direction (see Fig. 13c), scattered strain concentration regions can be observed surrounding the twin-cavity system in the earlier stage. At 303 μs, three strain concentration stripes clearly appear and two of them are V-shape regions connecting the cavity system with the specimen corners. The rest generates in the connecting area between the two cavities and propagates parallel to the loading direction until reaching the left specimen boundary. The serious stress concentration in the connecting area gives rise to spalling failure in this area.

When two cavities are aligned along an inclined line (see Fig. 13d), the strain concentration regions firstly generate between the two cavities and their outer sides. Then, two V-shape regions formed in the upper and lower area outside of the twin-cavity system from the short sides to the specimen corners. At 366 μs, the splitting failure appears at the outer shorter sides of the cavities.

As for the specimens containing two vertically aligned cavities whose longer sides vertical to the loading direction (Fig. 13e), the initial strain concentrates between the two cavities. Subsequently, the strain concentration regions form in the out area of the twin-cavity system and slight splitting failure can be observed around the shorter sides of the cavities.

Overall, combining Fig. 11 with Fig. 13 shows that the stress wave propagation path as well as strain distribution are significantly changed in specimens containing one or two cavities compared with those in intact ones. The different failure modes presented are mainly dominated by the existence and configuration of cavities in the specimens, which greatly influenced the propagation path and concentration area of the stress wave. For example, for the intact specimen (Fig. 13a), failure was characterized as tensile crack initiated at the middle area of the specimen’s right side resulting from reflected tensile stress wave. For the specimen containing a single cavity (Fig. 13b), high compressive stress concentration formed on the cavity shorter sides and caused initial compressive failure. The interaction between cavity and specimen boundary gave rise to the shear cracks connecting them.

Conclusions

In this study, dynamic compressive tests were conducted on intact marble specimens and specimens containing cavities by using SHPB apparatus to investigate the effects of cavity number and layout on the mechanical behaviours and failure modes of rock specimens under dynamic loading conditions. The loading capacity and absorbed energy of specimens were obtained and compared. The failure process was recorded by a high-speed camera and then the crack evolution was analysed by means of DIC technique, based on which the failure modes were further determined using DTL method. The following conclusions can be drawn:

  1. (1)

    The embedded cavities contribute to the degradation of dynamic loading capacity of rock specimens. Generally, the intact specimens have the highest dynamic loading capacity, which is significantly reduced when there is a single cavity embedded in the specimen. For specimens containing two cavities, the strength decreases slightly with connecting angle increasing from 0° to 45° and then reaches to the highest level when connecting angle raised to 90°. In such a case, average dynamic loading capacity is very close to that of specimens containing a single cavity. Absorbed energy per unit volume e also shows a similar trend with the change of cavity number and layout.

  2. (2)

    DIC results reveal that crack evolution and failure modes are similar for specimens in the same main group with the same cavity number or cavity layout in spite of different incident energies. The influences of incident energy on crack evolution and failure modes are quite limited.

  3. (3)

    For all specimens containing one or two cavities, the final instability of them is caused by inclined cracks, which are characterized by shear cracks or mixed shear-tensile cracks, from cavities to specimen corners following the shortest path. For specimens containing one cavity, the initial strain concentration appears around the shorter side walls of the cavity and then extends to specimen corners, forming macro shear cracks. For specimens containing two cavities, initial strain concentration always appears in the connecting area between two cavities and gives rise to coalescence failure, followed by the strain concentration appeared around the outer shorter sides of the twin-cavity system, which then inclinedly propagated to the specimen corners.

  4. (4)

    The fragmentation of specimens is reflected by absorbed energy per unit volume e. A high e means more energy is absorbed and stored in the form of strain energy. Once instability occurs, it allows more energy to release and thus procedures more fragments with violent failure.