The tested rock material was obtained from an open pit slope located at the Jiama Copper Polymetallic Mine in Tibet, China, as shown in Figures 1A, B. The stepped fissures can be clearly observed and the information of rock structure is obtained by using 3D laser detection technology in Figures 1C, D. According to the method recommended by ISRM (Ulusay, 2015), the rock mass obtained from the open pit slope was prepared into a cylinder with a diameter (D) of 50 mm and a height (H) of 100 mm. Both ends of the specimens were polished to ensure that the non-uniformity error is less than 0.05 mm and the parallelism is less than 0.1 mm. Two parallel fissures were prefabricated on the specimens using a precision carving machine as shown in Figure 2A. The angle between the fissure and the horizontal plane was set to be 10°, 30°, 50°, and 70°, respectively. For all the prepared rock specimens, the angle and length of the rock bridge between the two fissures were the same. Typical marble specimens with different fissure angle used in this work are shown in Figure 2B.

All mechanical tests were conducted in an electro-hydraulic servo-controlled rock testing system RDST 1000, which can apply the dynamic loading frequencies in the range of 0–10 Hz. During rock deformation, the axial strain and lateral strain were continuously measure using grating sensors installed at the longitudinal and circumferential of rock specimens. The information of axial stress, axial strain, and lateral strain were recorded by computer at the same loading sampling frequency. The following gives the detailed experimental procedures.

(1) Static loading stage: Loading to 15 MPa using a constant loading rate which the loading rate was 0.06 mm/min (1.0 × 10⁻⁵ s⁻¹).

According to the loading path, it can be found that different flaw angles greatly affect the duration of the test. The test duration increases significantly with the increase of the fissure angles. The fatigue strength of the rocks with fissure angles of 10°, 30°, 50°, and 70° are 55.21 MPa, 70.32 MPa, 80.24 MPa, and 110.55 MPa, respectively. The rock specimens are subjected to different stages of fatigue loading until destruction, going through 9, 12, 14, and 20 loading stages, respectively. The total number of loading cycles for the four sets of specimens varied due to the different fissure angles, which are 244, 334, 398, and 582, respectively. The number of cyclic loading increases with the increase of the fissure angles, indicating that the fissure angles have a significant effect on the strength and strain of the rock.

According to the energy dissipation and release theory proposed by Xie et al. (2011), the process of compressive deformation and damage of rocks contains energy input, accumulation of releasable elastic strain energy, and continuous energy dissipated and release. The whole process is very complex. Assuming that in an energy-independent closed environment, the rock unit is subjected to external work to produce damage strain, and according to the law of conservation of energy and transformation, the following relationship exists between each strain energy: E n = E n e + E n d (1) Where: E n is the total energy; E n e is the elastic energy; E n d is the dissipated energy used for rock damage as well as irreversible strain to drive crack initiation, extension and coalescence.

According to the studies of Solecki and Conant (2003), Formula 1 can be converted in terms of energy density, as shown in Formula 2: ∫ U d V = ∫ U e d V + ∫ U d d V (2) Where: V is the volume of the rock, U is the total strain energy density, U _e is the elastic energy density, U _d is the dissipated energy density.

The total energy, elastic energy, and dissipated energy are calculated as follows: U = ∫ ε 1 ε max σ d ε = ∑ i = 1 n 1 2 σ i + σ i + 1 ε i + 1 − ε i (3) U e = ∫ ε 1 ε max σ d ε = ∑ i = 1 n 1 2 σ i + σ i + 1 ε i + 1 e − ε i e (4) U d = U − U e = ∫ ε 1 ε max σ d ε − ∫ ε 2 ε max σ d ε (5)

The calculation process of total energy (U), elastic energy (U _e) and dissipated energy (U _d) are shown in Figure 4A. The area of the ABCD region under the loading curve represents the total energy (Wang et al., 2021). The area of the EFDC region under the unloading curve represents the elastic energy released during the unloading process. The dissipated energy is the difference between the total energy and the elastic energy area.

In cyclic loading tests, the study of fatigue damage variables for accumulative damage evolution is essential for the prediction of fatigue instability. The damage variables can be defined in terms of different physical-mechanical parameters such as elastic modulus, acoustic emission, density, damping, dissipated energy, etc. Xiao et al. (2010) used six common definition methods to obtain damage variables, revealing the advantages and disadvantages of these methods. In this work, the dissipated energy is used to define the fatigue damage variable, as shown in Formula (6): D F L = ∑ i = 1 N i U i d / ∑ i = 1 N f U i d (6) Where: D F L is the damage variable caused by fatigue loading, N i is the stage number, N f is the number of stages in the failure of the rock, U i d is the dissipated energy of the i-th stage.

For multi-stage fatigue loading tests on rock specimens with different fissure angles, the axial and lateral stress and strain curves can be obtained as shown in Figures 5A, B. It can be found that as the test time increases, the loading curve and unloading curve no longer coincide due to the plastic deformation of the specimen’s interior, thus forming a hysteresis loop. The form of the hysteresis loop varies with the loading time, and for each fatigue loading stage before specimen damage, the hysteresis loop exhibits a change from sparse to dense. The sparse morphology of the hysteresis curve is due to the increase in stress amplitude, which causes a large plastic strain. Subsequently the formed cracks are gradually closed, resulting in the hysteresis curve becoming dense. Compared with the hysteresis curves of the previous fatigue loading stage, the final stage of fatigue loading shows violent crack development, a significant increase in dissipated energy, and a significant increase in deformation of the specimen during the compressive yielding stage. Therefore, the hysteresis curve in this stage becomes increasingly sparse until the specimen failure.

In order to study the effect of different fissure angles on marble specimens, the effect of different fissure angles on the strength and fatigue life of marble specimens are analyzed, as shown in Figure 5C. It can be visualized from the Figure 5C that the strength as well as the lifetime of the marble specimens increases gradually with the increase of the fissure angles under the fatigue loading condition.

In order to describe more intuitively the effect of fissure angles on the marble specimens fatigue responses, the loading path (Figure 3) and the stress strain curves (Figure 5) are combined to obtain the damage characteristics of the marble specimens under fatigue loading conditions, detailed mechanical parameters as shown in Table 1.

The form of the hysteresis loops at each fatigue loading stage reflects the changes in the internal microstructure of the rock, which is closely related to the crack extension behavior. Afterwards, the relationship curves of the maximum axial strain with the number of fatigue cycles for marble specimens with different fissure angles are plotted, as shown in Figure 6. It can be observed that the axial strain increases with the increase in the number of cycles and exhibits a significantly different trend at the beginning of cyclic loading than at the middle and the end. At the beginning of each loading stage, the strain increases rapidly and then tends to become stable until the last cyclic loading stage. At the same time, the sudden increase in stress amplitude also leads to larger axial deformation, and the accumulative plastic strain increases continuously with the cyclic loading stage, and the axial strain increases sharply at the last fatigue loading stage until the specimens are damaged.

In order to further analyze the relationship between lateral strain and fissure angle during the whole fatigue loading process, the relationship curves between the lateral strain and the number of fatigue cycles during the fatigue loading process of the marble specimens with different fissure angles are plotted as shown in Figure 7. It can be found that the lateral strain increases significantly faster at the end of cyclic loading relative to the earlier loading stages, and the radial strain increases suddenly at the last loading level until the damage of the rock. Meanwhile, in order to be able to better describe the effect of axial and lateral strains on the volume change of the specimen, the volumetric strain is calculated from the axial and lateral strains. The volumetric strain reflects the volume change of the rock during deformation and is also a good indicator to reveal the fracture behavior of the rock. The relationship curves between the volumetric strain and the number of fatigue cycles are shown in Figure 8. Combining Figures 7, 8, it can be found that the trends of lateral strain and volumetric strain are very similar, which indicates that the lateral strain of the rock has a greater effect on the fatigue damage of the rock.

The relationship curves between axial, lateral and volumetric strains and fatigue loading stages can be more intuitive to see that with the increasing loading stages, the axial, lateral and volumetric strains gradually become larger, and in the final stage of fatigue loading, they all increase sharply. In addition, the peak axial strain, peak lateral strain and peak volumetric strain of the marble specimens with different fissure angles under fatigue loading conditions are obtained as shown in Figure 9D. It can be found that each peak strain gradually increases with the increasing angle of the joint. Meanwhile, with the increase of cyclic loading stage, the volumetric strain changes from positive to negative, as shown in Figure 9C. The phenomenon indicates that the rock specimens underwent a process of compression followed by expansion. The results also further illustrate that the degree of damage as well as the damage form of different marble specimens of fissure angles under fatigue loading conditions are different.

The relationship between the total input energy, elastic energy, dissipated energy and the number of cycles is plotted using Eqs 3–5, as shown in Figures 10A–D. The results show that the overall trend of the energy curves is similar under the effect of static loading. During the initial cyclic loading stage, the total input energy is almost entirely converted into elastic energy and stored in the interior of the rock. At this time, the damage of the rock is smaller, so the dissipated energy is also smaller, the total energy curve and the elastic energy curve basically overlap, and the dissipated energy is almost zero.

As the number of fatigue cycles increases, damage occurs inside the rock, the accumulation of damage gradually increases, and part of the input energy is consumed to produce cracks, so the elastic energy curve no longer coincides with the total energy curve. As the number of fatigue cycles gradually increases, the dissipated energy also begins to increase, and the growth rate accelerates. It can be found that the increase of dissipated energy within the same cyclic loading stage is not obvious. When the stress amplitude increases, the dissipated energy increases suddenly. In the final fatigue loading stage, the dissipated energy increases faster and faster until the rock is damaged (Figure 10). The results show that fatigue loading causes further damage to the marble specimens, but the damage accumulation is not severe, while the increasing stress amplitude accelerates the damage deterioration of the rock. Meanwhile, the overall increasing trend in dissipated energy prior to damage of the marble specimens accelerates, indicating a continued increase in the energy required to drive crack expansion and a gradual increase in the degree of rock damage.

In order to investigate the effects of different fissure angles on the damage extension and the corresponding energy release and dissipation of marble specimens under fatigue loading conditions, the evolution process between the total energy, elastic energy, dissipated energy and the number of cycles of marble specimens with different angles of joint is analyzed and obtained as shown in Figures 11A–C. It can be visualized from the figure that the input energy is converted into elastic energy and dissipated energy, and the proportion of dissipated energy increases with the increase of cyclic loading stage. During the fatigue deformation of the marble specimens, most of the energy is released from the elastic energy, and the accumulation of damage caused by fatigue loading is increasing, which in turn leads to an increase in dissipated energy. The dissipated energy increases slowly at the initial stage of fatigue loading, but increases rapidly near the failure of the specimen, and increases sharply at the last fatigue loading stage until the specimen is damaged. The sharp increase of dissipated energy indicates the unstable propagation and coalescence mode of the crack. The dissipated energy is maximum for a fissure angle of 70° and minimum for a fissure angle of 10°. The dissipated energy for different fissure angles are 0.016 MJ/m³, 0.030 MJ/m³, 0.059 MJ/m³ and 0.115 MJ/m³, respectively.

The energy curves exhibit a stepped growth mode, and the energy increment is larger when the stress amplitude increases abruptly. The results indicate that the number of cycles of fatigue loading and the fatigue loading stage greatly affect the dissipation and release characteristics of rock energy. The relationship between the total energy, elastic energy, dissipated energy and maximum axial stress of the rock specimens are shown in Figures 12A–D. The curve fitting method is adopted to reveal the relationship between strain energy density and the upper limit of stress amplitude. The fitting results show that during each fatigue loading, the total energy, elastic energy and dissipated energy of four groups of marble specimens with different fissure angles increase with the increase of the upper limit of axial stress, and the increasing rate is gradually accelerated. In addition, a significant non-linear increasing trend is observed, and the data satisfies a power function with high correlation. The fitting results are listed in Table 2.

After investigation, it is found that there is a non-linear damage accumulation law in rocks under multi-stage fatigue cyclic loading conditions, and the previous models are not suitable for describing the damage evolution characteristics of rocks under this loading path (Figure 13). As rock damage and fracture is energy-driven, in my opinion, the energy index is better than other parameters to express damage propagation. In addition, the energy parameter contains rock stress and strain information, it is the comprehensive reflection of rock failure. According to Formula 6, a new dissipated energy-based damage accumulation model is proposed to describe the damage process of rocks by using dissipated energy to calculate the damage variables (Wang Y et al., 2020) which is shown in Formula 7. D = 1 − 1 − n / N f a b (7)

Where: D is the damage variable caused by dissipative energy, when n is 0, D is 0, when n is N_f, D is 1; N is the number of total cycles; N_f is the fatigue lifetime; a and b are the material related parameters.

According to the fitting results and the value of R², it can be seen that the accumulative damage curves show a trend of slow increasing and then fast increasing, and have obvious non-linear characteristics as well as high correlation.

Since rock crack initiation and fracture evolution are essentially energy-driven, and the different joint angle may lead to different capacities for energy storing and releasing in stepped marble, which causes differences in the damage forms and fracture morphology of the specimens. The macroscopic fracture forms of the typical marble specimens after destruction are shown in Figure 14. It can be clearly observed that the joints play a dominant and controlling role in the crack expansion path of the whole specimen. The crack extension paths of the four groups of typical marble specimens have similarities. The crack extensions of the specimens at the time of damage all start from the end of the fracture, and the development direction of the rock bridge is consistent with that of the joint. All of them develop through along the rock bridge starting fracture and are similar to the wing crack, and the crack extension direction gradually approaches 90° and gradually converges to the maximum principal stress direction. When the angles of the joint vary, there are also some differences. From Figure 14, it can be found that the cracks gradually transition from tension failure to shear failure as the angles of the joint increases. When β is 10°, in the longitudinal, the specimen shows vertical splitting cracks with some secondary cracks. When β is 30°, 50°, and 70° respectively, the upper and lower main crack tips are obviously controlled by the maximum principal stress. The angle between the main crack and the maximum principal stress is basically the same, showing a symmetric development trend. The expansion path of “X” conjugate crack is formed by the expansion from the tip of the joints.