Experimental study of the influence of structural planes on the mechanical properties of sandstone specimens under cyclic dynamic disturbance

Discontinuity‐controlled slopes constitute potentially fatal risks during exploitation activities in open‐pit mines and severely threaten the safe mining of collieries. To investigate the dynamic mechanical properties of rock with structural planes under periodic cyclic dynamic disturbance, this study took the north side of the Anjialing open‐pit mine as an example. Uniaxial compression tests were carried out on rock samples containing structural planes with different dip angles (θ). The impacts of θ on the strength, deformation characteristics, energy dissipation characteristics, and fracture rules of rock samples under different disturbance stress amplitudes (Δσ) are studied. The test results show that the structural plane affects the mechanical properties of rocks under dynamic perturbation. A closed hysteresis loop curve of rock is formed in each loading and unloading cycle, thus forming three stages: a sparse stage, a dense stage, and another sparse stage. θ, Δσ, and the number of disturbance cycles (n) impact the rock hysteresis loop curve and deformation modulus. The plastic deformation of rock mainly includes an initial deformation stage, an isokinetic deformation stage, and an accelerated deformation and overall failure stage. Moreover, there is a disturbance threshold for Δσ′ of the rock failure, and the evolution law of plastic strain accumulation and n near the disturbance threshold conform to the negative exponential function and the inverse function of the Langevin function, respectively. Based on damage theory, a constitutive model of the structural plane considering a disturbance threshold is proposed.

of the International Society for Rock Mechanics, 7,8 there are hundreds of deep mines with mining depths over 1000 m in production or being constructed in China (see Table 1). 9 Specifically, in the open-pit combined mining environment, rock slopes are not under monotonic loading conditions but may experience the combined action of open-pit mining and underground mining. In the process of deep mining, highstress hard rock is generally subjected to blasting operations, mechanical shock earthquakes, overburden fracture, and other dynamic disturbances around the deep underground rock. [24][25][26][27][28][29][30][31] The distribution and change of the stress field and displacement field in the slopes are complicated, which degrades the stability of the slopes. [32][33][34][35][36][37] In reality, the rock mass is in a combined stress state of prestatic load and dynamic disturbance, [38][39][40][41][42][43] which leads to a completely different mechanical response from the pure static or dynamic loading state of shallow rocks. It is difficult to give a reasonable explanation for the above failure phenomenon using the classical rock statics or dynamics theory based on shallow mining. Most rock engineering accidents around the world are due to the lack of research on structural characteristics.
On 10 December 2006, the north side of the Anjialing open-pit mine in Shanxi Province, China, experienced a massive area landslide, with a drop of approximately 90 m and a volume of approximately 600 000 m 3 . 44 The trigger factors for this landslide can be classified into two categories: internal master control factors (such as the development of joints and structural planes in the slope) and anthropogenic activities (such as disturbances to the workbench under the open pit due to underground mining and frequent blast dynamic disturbance). Therefore, special attention should be paid to the structural characteristics of prestatic rock subjected to dynamic disturbance. Under dynamic disturbance, further damage occurs in the structural planes of high steep rock slopes, resulting in a change in the slope stress state and extension and intersection of cracks in the slopes, causing the deterioration of the mechanical properties of surface planes. 45,46 Open-pit mining will cause secondary damage to the slope and will intensify the cumulative damage to the structure of high and steep slopes, with microcracks gradually reaching the critical state, resulting in the large-scale instantaneous dynamic expansion of cracks on the basis of long-term deformation and causing the instability failure of the slope. 47 Generally, joints, which constitute weak structural surfaces, are widely distributed throughout rock slopes 48 and negatively affect the integrity of the rock mass. 49 The joints can be distributed complexly due to their uncertain geometric and mechanical parameters, such as direction, dip angle, and location. [50][51][52] Given the abovementioned analysis, the dynamic mechanical properties of rock slopes with structural planes are very complicated under dynamic disturbance.
To date, many kinds of studies have been carried out on the impact of combined dynamic and static loading or dynamic disturbance on the failure of deep surrounding rocks. Feng et al 53 investigated the stress state of joints in the progressive damage process using single cyclic loading tests. It was concluded that, even though the test results show that rock damage is directional and anisotropic, the dissipated energy is linearly related to the equivalent irreversible strains and plays a pivotal role in the evolution of rock damage. Lei and Wang 54 carried out laboratory investigations on the deformation and fracture of sandstone when applied to multi-level dynamic cyclic loading with treatment loading frequencies, various amplitudes, and a series of uniaxial compressive strength tests and found that loading frequencies and amplitude, as well as the waveform, were of great significance to and helpful for understanding the sandstone process from both micro-and macroscale perspectives. Nie et al 55 performed a comparative study on the mechanical behavior and permeability of coal by using parameters from CTC and UCP-RAS tests. However, recent studies by Chen et al 56 and Zhou et al 57 indicated that peak stresses and dynamic loading could induce significant rock specimen failure patterns controlled by structural effects. Su et al 58 studied the damage and failure rules of rock under three kinds of dynamic disturbances (mild, modest, and weak) by using an AE-controlled testing system and established a new constitutive model of elastic-plastic damage. These studies all indicate that rock may experience a complex loading path during the excavation of underground openings, including the unloading of confining stress, highly confined compression before excavation, and further disturbance of dynamic loading after excavation. In addition, Niu et al 59  Therefore, to gain insight into the dynamic mechanical characteristics and failure characteristics of a rock mass under dynamic disturbance, a uniaxial rock system was used to evaluate rock mass specimens. In this study, the rock mass specimens were taken from Anjialing open-pit mine in Shanxi Province, China. Samples with structural planes of different angles were taken by using a microcontrol electro-hydraulic servo fatigue testing machine. The dynamic mechanical characteristics and failure characteristics of rock mass under dynamic disturbance are discussed, including the stressstrain law, plastic strain evolution law, and energy principle of strength weakening under periodic dynamic disturbance. Moreover, according to the relationship between cumulative plastic strain and disturbance cycle times based on the damage theory, a constitutive model of rock with structural planes under the influence of a disturbance threshold was constructed, and the model was verified by using test data. This typical case study can provide a reference for open-pit combined mining rock slopes with structural planes controlled under dynamic disturbance.

| Test equipment
A PA-100 microcontrolled electro-hydraulic servo fatigue testing machine is used as the loading system, as shown in Figure 1. The maximum static load of the testing machine is 100 kN, and its maximum dynamic load is ±100 kN. The disturbance frequency ranges from 0.01 to 30 Hz, and the dynamic load capacity is strong, with high measurement accuracy. Low-frequency fatigue test software is used to simulate the dynamic stress wave loading of rock materials. A DH5922D dynamic strain gauge produced by JSDH (CHINA) Testing Technology Development Co., LTD, is adopted for dynamic data collection and recording.

| Sample preparation
It is difficult to sample a natural jointed rock mass, and some structural plane conditions can hardly meet the test design requirements. Therefore, the joint rock samples are prepared by prefabrication or manual preparation in this study. The test samples were taken from the 1300-m platform sandstone block of the north slope of Pingshuo Anjialing, Shanxi Province, China ( Figure 2). First, rock samples were drilled into cylinders with a diameter of 50 mm, and then, the rock F I G U R E 1 Dynamic disturbance test system: A, Schematic diagram of the testing machine; B, Picture of the testing machine samples were made into coal-rock samples with a height of approximately 100 mm. Then, after grinding the cutting surface, the two parts are glued together with epoxy resin adhesive, to make composite samples with different inclination angles. The nonparallelism of the two ends is less than 0.05 mm, the diameter deviation of the upper and lower ends is less than 0.03 mm, and the axial deviation is less than 0.25°. Processed sandstone specimens with different dip angles are shown in Figure 3. The basic parameters of samples with different inclination angles were tested. During the measurement, the test position was changed 3 times for each parameter, and the average value was taken as the final result. The test results of each sample are shown in Table 2. It is worth noting that secondary joints widely distribute in natural rock masses generally. In the process of rock sample collection, it is impossible to guarantee that all rock samples collected are homogeneous. Hence, in the process of preparing artificial rock samples with structural planes in the laboratory, the homogeneous natural rock samples should be selected as far as possible. In the analysis of the test results, we choose the test data of rock samples without natural joint for this study.

| Test plan
During the cyclic dynamic disturbance tests of rocks under a certain static load, all the test samples in this work were automatically controlled via computer by means of continuous loading. First, the static load is applied to the rock. Load control increases the load at a certain loading speed. When the static stress (σ) reaches a certain predetermined value, σ m is taken as the average stress, and the cyclic dynamic load is applied. Load control maintains a constant loading rate during the cyclic disturbance. To simulate the elastic wave during vibration propagation, the cyclic disturbance waveform is in the form of a sinusoidal wave, and the testing machine loads the rock with constant upper and lower loads until the rock is damaged. The loading process and characteristics are shown in Figure 4. σ max and σ min are the upper and lower limit stresses of cyclic load. Δσ = σ max − σ min is the dynamic disturbance amplitude. T is the loading period. During the test, the testing system can collect the axial load, axial deformation, transverse deformation, and time data and draw the corresponding parameter relationship curve.
To obtain reasonable values of σ max and σ min , a uniaxial compression test is conducted on the samples using the TAW-2000 uniaxial servo-hydraulic testing machine. The uniaxial compression stress-strain curve is obtained, and it is shown in Figure 5. The yield strength of the rock containing structurally plane rock is calculated as the average value of cyclic loading stress, and an axial cyclic disturbance load of the sinusoidal waveform with a fixed frequency of 2 Hz is applied to the samples. The specific steps of the periodic loading test are as follows: (a) Samples are placed on the low actuator of the testing machine, and a 100 mm × 50 mm × 20 mm rigid backing plate is placed on each side of the samples; and (b) test parameters are set, and the loading waveform is a sinusoidal wave with an amplitude that is set to exceed the estimated strength value of the samples. During the cyclic loading tests, the yield strength is taken as the centerline of the test, that is, σ m . The rock specimen is placed on the lower actuator, and the upper actuator is slowly moved down until it touches the specimen and applies a force of approximately 2 kN. The position of the lower actuator is then adjusted by force control on the test control interface, and it is slowly loaded to the centerline position, which is the yield strength of rock specimen. This state is held for a period of time before the experiment begins. The upper actuator remains fixed, and the lower actuator applies a cyclic load according to preset conditions. This experiment is divided into 12 groups, which contains 3 pieces. Rock samples with dip angles of 0°, 15°, 30°, and 45° are selected to conduct disturbance tests at different amplitudes. The test sample number consists of three parts, which are θ, the average load of the cyclic load, and the disturbance amplitude of the cyclic load. The average cyclic load and disturbance amplitude of the second and third parts are expressed according to the actual applied loads. For example, A25-4 means that θ is 0°, the average cyclic load is 25 MPa, and Δσ is 4 MPa. The specific test scheme is shown in Table 3.

| Analysis of the stress-strain curve of rock under dynamic disturbance
The corresponding yield strength is applied to samples containing structural planes with different dip angles, and then, cyclic perturbations under different stress amplitudes are carried out. The cracks generated by the destroyed rocks developed along the axial direction and eventually formed many vertical cracks, which then fragmented into relatively thin sheets or strips of debris, showing a tendency to expand along the structural plane. The stress-strain curves under cyclic loading of samples with different dip angles under different amplitudes are shown in Figure 6. Due to the nonlinear characteristics of the rocks, the deformation of samples lags behind the change in external stress, forming a closed hysteresis loop curve under each loading and unloading cycle, whose area reflects the energy consumed by the sample during cyclic loading. In the first five cycles of loading, the distribution of the stress-strain hysteresis ring is sparse, and the inelastic strain is large (see Figure 6). After the sixth cycle, the hysteresis loop curve becomes denser as the number of cycles increases, the spacing between each loop gradually decreases, and the inelastic strain decreases. The area of the hysteresis loop decreases as n increases and finally stabilizes. This trend occurs because there are many microcracks in the rock; these cracks close and cause large deformations when subjected to dynamic load. When the microcracks in the rock are rammed to a certain extent, the deformation rate of the rock tends to be stable, with plastic strain occurring slowly and uniformly. In the 2-3 cycles before rock failure, the area of the hysteresis loop curve increases rapidly due to the complete penetration of cracks caused by cyclic loading. Overall, the curve forms three stages: a sparse stage, a dense stage, and another sparse stage. Figure 6A shows that the plastic strain accumulation value of the samples after 80 loading cycles tends to be stable, and the samples are not damaged when the θ = 0° under different disturbance stress values. Figure 6B shows that the accumulated plastic strain values change significantly when θ = 15° under Δσ of 4 MPa and 6 MPa after 80 cycles, but no damage occurs. When Δσ reaches 8 MPa, damage begins to accrue in the samples after 70 loading cycles. Figure 6C shows that, when θ = 30°, the samples began to sustain damage after 70 and 60 loading cycles under Δσ values of 6 MPa and 8 MPa, respectively; however, no damage occurs after 80 loading cycles when Δσ = 4 MPa. Moreover, when θ = 45°, the samples are damaged after 65, 60, and 45 loading cycles under Δσ values of 4 MPa, 6 MPa, and 8 MPa, respectively ( Figure 6D). Therefore, for samples with the same θ values, the larger the value of Δσ, the larger area of the hysteresis loop curve, and the larger the corresponding rock deformation. Thus, the increase in Δσ causes the rock to accumulate consumable energy within each cycle, which is consumed by the bond-slip between mineral grains and new cracks, resulting in damage in the rock and decreasing the total amount of cumulative irreversible deformation required for failure, which will significantly degrade the load resistance of the samples, In contrast, the smaller the hysteresis loop curve area, the closer the rock strength characteristic to the strength under simple continuous loading. There is a disturbance threshold for the disturbance stress amplitude when the rock contains structural planes.
In addition, the cyclic loading stress-strain curves of different samples under different Δσ values are as given in  , which shows that, with an increase in θ, the change in amplitude of the hysteresis loop curves and the amount of deformation of the rock increase, but the failure intensity of the rock decreases, and the rock is more prone to failure. During cycle loading, when σ max is smaller, the slope of the hysteresis loop curve is small, indicating that cyclic loading expands the cracks inside the sample and reduces the stiffness of the sample. With the increase in Δσ, the amount of rock deformation in each cycle dynamic disturbance period increases, and n decreases. When σ max is close to the rock static strength, the variation in the average stress and cyclic stress amplitudes will have a great influence on the fatigue life of rock under dynamic disturbance.
The relationship of rock mass specimens with different dip angles of the structural plane under different dynamic disturbance amplitudes and the number of cycle times is presented in Figure 8, which shows the value of the dynamic disturbance amplitude when rock mass specimens with different structural dip angles are damaged under different numbers of cycles and dip angles of structural planes. From the perspective of the dip angles of structural planes, the rock mass specimens are prone    As can also be observed in Figure 8, with the increase in θ, the number of loadings required for the corresponding damage to the rock mass specimens is decreased. The relationship between the dip angles and dynamic disturbance amplitudes with peak stress is calculated as shown in Figure 9. Figure 9A shows that the larger the dip angles of the structural plane, the smaller the peak stress. With an increase in dip angle, the peak stress decreases exponentially, conforming to an exponential relationship. Figure 9B shows that, as the dynamic disturbance amplitude increases, the required peak stress first increases and then decreases. The relationship between dynamic disturbance amplitude and peak stress becomes sinusoidal, conforming to a sinusoidal relationship. Then, we fit the relationships of dip angles and dynamic disturbance amplitude with peak stress. The coefficients of determination (R 2 ) are both greater than 0.97. Through the above analysis, it can be concluded that the strength of rock with structural planes during cyclic loading and unloading is related to its dip angle, the amplitude of disturbance stress, and the number of cycles.
The area of the hysteresis loop curve reflects the maximum elastic strain energy consumed and stored during a cycle, as shown in Figure 10. Generally, the plastic characteristics of the rock can be analyzed through the ratio of hysteresis energy and damping ratio γ; and the deformation modulus E and γ of a single cycle can be defined as E = ( � max − � min )∕( � max − � min ) and = A∕(4 A s ), where σ′ max and σ′ min are the maximum and minimum stresses of the hysteresis loop curve. The values of ε′ max and ε′ min are their corresponding strains. A is the area enclosed by the hysteresis loop curve. A s is the area of triangle ABD.
The cyclic loading experiment results are calculated according to the above equations. The change curves of E with θ under different disturbance stresses are shown in Figure 11, and the change curves of γ with the cycles of different samples are shown in Figure 12. Figure 11 shows that E decreases with the increase in θ under a certain Δσ, which is because θ > 0 leads to the occurrence of macroscopic cracks. The cracks become larger with θ, leading to a higher degree of fatigue for samples under cyclic load and a decrease in E. When θ is fixed, larger values of Δσ and E indicate that Δσ is positively correlated with the fatigue degree under cyclic loading. Figure 12 shows that the γ value of the rock decreases first with the increase in the number of cycles and then tends to stabilize, indicating that cyclic loading strengthens the rock. Then, the γ value of the rock shows an increasing trend because the strength of the rock gradually deteriorates with the increase in the number of cycles, causing cracks to expand and resulting in softening. According to Figure 12, the plastic deformation (fracture expansion) of rock can be divided into three stages: the initial deformation (pore compaction) stage, the isokinetic deformation (microfracture stable development) stage, and the accelerated deformation (fracture penetration) and overall failure stage. Figure 12 also shows that, as θ increases, n decreases when γ increases, indicating that the rock is easier to break with an increase in γ.

| Plastic strain evolution law of rock under dynamic perturbation
Taking θ = 15° as an example ( Figure 6B), the axial effective plastic deviation strain of σ′ max is taken as the parameter for irreversible deformation, and the relationship between cumulative plastic strain (ε D ) and n is as shown in Figure 13. Figure 13 shows that their relationship can be described as follows: (a) in curve I, the rock is not damaged when Δσ is small; and (b) in curve II, the rock is damaged when Δσ is large. To describe the curves of two kinds of stress states, a disturbance threshold Δσ′ should be introduced. When Δσ < Δσ′, with an increase in the loading and unloading cyclic period, ε D tends to be stable, and the samples are not damaged. In contrast, when Δσ ˃ Δσ′, the relationship curve between ε D and n during the disturbance cyclic loading process presents an S shape. In the initial acceleration stage, the increasing rate and increment in ε D are large, but with increasing n, they gradually become small. When ε D increases a certain extent, the increasing rate has a sudden increase, leading to the failure of the samples. The fitting curve of ε D and n is shown in Figure 13, which shows that the evolution law of ε D and n around Δσ′ conforms to the negative exponential function 66,67 and the inverse function expression of the Langevin function 68-71 : where the A, B, a, b, and c are undetermined parameters.
Given the abovementioned analysis, the Δσ′ value for the permanent deformation of rock under dynamic disturbance is related to both σ and θ. According to the above method, the relationship between ε D of samples with different dip angles and n under different stress amplitudes was analyzed. Δσ′ was fitted with the two-parameter curve of static load and θ, and the fitting results are shown in Figure 14. Based on the original data of σ m , θ, and Δσ′ in Figure 14, we use the nonsurface fit function to obtain the relationship between the dynamic disturbance threshold, the dynamic disturbance amplitude, and the dip angles. Δσ′ is expressed as: In addition, the physical significance of the parameters (A, B, a, b, and c) is discussed to analyze their effects on the plastic strain evolution law of rock under dynamic perturbation. Some typical values of these parameters are selected, and the influence of parameters on the n-ε D curve can be obtained (Figure 15). For example, the physical significance of parameter A is as follows. In Equation (1), to describe curve I, parameter B was fixed to 32, and A was set to 3.0, 3.5, and 4.0 ( Figure 15A). Figure 15A shows that ε D increases with the increase in A, and when the three curves are approximately equal to their corresponding A values, ε D will reach a stable stage. The parameter A can basically determine the stable value of the ε D of rock during cyclic loading, which has a controlling effect on the ε D value of the stable phase in curve I. Therefore, Parameter A is defined as the axial plastic strain accumulation rate factor in the initial stage. Figure 15B shows that the ε D values of the three curves tend to be stable with an increase in n, but as the value of parameter B increases, the n value before the stable stage increases gradually, which indicates that the change rate of ε D has a negative correlation with the parameter B. Therefore, the physical meaning of the parameter B is the I characterization curve slope size, indicating that curve I epsilon D rate and the size of the B value show a negative correlation. Therefore, the physical significance of the parameter B is represented by the slope of curve I. Figure 15C shows that the three curves will intersect at some point with increasing n, and the values of n and ε D at that point are approximately equal to c and b, respectively. The value of n when the rock is damaged increases with the increase in a, but the slope of curve II decreases gradually in the initial stage, indicating that the change rate of ε D has a negative correlation with a. Hence, Parameter a is defined as the change rate factor of ε D in the initial stage when Δσ is large, representing the slope of curve II in the initial stage, and controlling n before rock damage. Figure 15D shows that the ε D values of rock increase with the increase in b, and the rock is ultimately damaged when n reaches 85. Parameter b is defined as the intensity factor of the rock itself, which can better reflect the ε D values of rocks with different strengths under the same cyclic loading condition. Moreover, Figure 15E shows that n before the rock is damaged increases with c, and the hysteresis phenomenon will occur in the acceleration phase, but the value of ε D decreases. Based on this characteristic, Parameter c is defined as the plastic cumulative strain rate factor in the acceleration stage. (1)

| Energy failure mechanism of rock under dynamic perturbation
The loading and unloading processes of the rock are driven by the increase and release of the internal energy of rock, respectively. Figure 16 shows the energy change process of the rock containing the structural plane during cyclic loading and unloading. When the upper limit stress of cyclic loading is σ 1 , the strain of the rock is ε 1 , and the increased energy inside the rock is W, as shown in Figure 16A. When the lower limit stress of the cyclic load is σ 2 , the strain decreases to ε 2 under unloading, with the energy released by the rock being W 1 , as shown in Figure 16B. When the rock is subjected to secondary loading, the stress will reach the maximum limit of cycle load σ 1 , and the strain also changes to ε 1 ; furthermore, the work energy increases to W 2 in this process, as shown in Figure 16C. Due to the rock deformation hysteresis characteristics, the two loading stress-strain curves and the unloading curve do not overlap; therefore, when the stress level is σ 1 , W 1 is larger than W 2 , and the newly increased energy in the rock W 3 = W 2 − W 1 . That is, after the loading and unloading cycle reaches the upper limit stress of the cyclic load, the increased energy inside the rock is W 3 , as shown in Figure 16D. As long as the stress-strain curves of loading and unloading do not coincide and there is a hysteresis ring curve, the new energy W 3 always exists and gradually increases with an increase in loading time. When the accumulated energy inside the rock exceeds its ultimate bearing capacity, weak local links will be destabilized, and the energy will be released. This is the energy failure mechanism of rock with structural planes under cyclic loading. Given the abovementioned analysis, the larger the hysteresis loop curve area, the larger the new increased energy W 3 inside the rocks containing structural planes, and the more prone to failure the rocks under cyclic loading. In contrast, the smaller the hysteresis loop curve area, the closer the rock strength to the strength value under uniaxial compression. The cumulative area of the hysteresis loop curve increases, but the intensity of rock decreases with the increase in the dip angle, the cycle number and the disturbance stress amplitude. According to the analysis of the cyclic loading-unloading process of rock mass specimens with dip angles of 0°, 15°, 30°, and 45° under different disturbance amplitudes, it can be observed that the cumulative value of new increased energy also increases as a result of cyclic loading-unloading with the increase in θ. Specifically, when θ is 45°, the difference between the loading-unloading curves is noticeable, and the hysteresis loop curve area is large, which leads to the reduction in the rock strength. Moreover, according to the comparison of the stress-strain curves of cyclic loading and unloading

| Derivation of damage evolution equation
The nonuniform failure of local microelements of rock material causes material damage; that is, the material is transformed from the linear elastic stress state to the nonlinear stress state. To consider damaged material at the microcosmic level, Lemaitre 72 hypothesized the following. Assume that the number of damaged elements inside the rock material under a certain load is N, and the statistical damage variable D s is the ratio of N to the total number of damaged elements N m , that is, D s = N/N m . 73,74 The number of infinitesimal bodies generated in any interval [ε, ε + dε] is NP(x)dx. When loading to a strain value, the number of microelements that have been destroyed is 73 : On this basis, to reflect the nonlinear deformation characteristics of rock materials, a damage threshold parameter D t , also known as plastic strain, is introduced: The relationship between D s and the probability density of microelement failure can be expressed as: The damage evolution equation is as follows 75 : When D s = 0, there is no damage inside the rock, but when D s = 1, all the microelements inside the rock have been destroyed. The value of D s reflects the degree of microelement damage inside the rock. 75 During rock material compression, compressive stress and shear stress are transferred to other microelement bodies after the failure of internal microelements, the effective area of compressive stress, and shear stress are the same, and the damage variables in all directions are D s . Therefore, the effective stress after compression can be assumed to be 76 : where δ is the damage scale coefficient of rock material, reflecting the residual strength of rock material, with a value of 0 < = √ r ∕ c < 1, 77 σ r is the residual strength of rock material, and σ c is its peak strength. δ = 1, ε > γ, and substituting Equation (6) into Equation (7), we have Equation (8): Without considering the size effect of rock material, δ is obtained, and the parameters F and m are determined by σ c and its corresponding strain ε c . Since the slope at σ c during loading is 0, According to Equation (9), the following equations can be obtained:

| Deformation and failure constitutive model of structural plane
Under the action of an external stress level, the damage of rock with a structural plane is composed of undamaged materials and damaged materials, and the total stress is borne by these two parts. Suppose the nominal stress on the rock is σ i and the nominal area is A 0 . The stress on the undamaged part is σ′ i and the corresponding area is A′. The stress on the damaged part is σ″ i , and the corresponding area is A"; then: Take Ds = A � i ∕A �� as the damage variable of rock material, and substitute it into Equation (12): Assume that, under static loading, the stress-strain relationship of undamaged rock obeys the linear elastic where E i is the elastic modulus of rock. Additionally, assume that the stress-strain relationship of the damaged part of the structural surface of the rock material under dynamic and static loads obeys the viscosity law during the deformation process. The deformation of the structural plane of rock mass under dynamic disturbance includes the deformation of intact rock, the deformation amount of the structural plane under pressure, and the deformation amount caused by shear �� i = d , where η is the viscosity coefficient of the structural plane. When the structural plane is subjected to a compression load, the deformation of the structural plane includes the shear strain generated along the structural plane and the closed strain generated by the compression of the structural plane.
Here, k s is the tangential stiffness of the structural plane; L is the length of the monomer; ε j1 , ε j2 , and ε j0 are the shear strain, closure strain and maximum closing strain of the structural plane, respectively; E j is the closed modulus of the structural plane; and α is the dip angle of the structural plane.
From the above equations, the deformation and failure constitutive model of the structural plane under the periodic dynamic disturbance of uniaxial compression considering a disturbance threshold can be obtained as follows: where D s can be obtained from Equation (1) according to the threshold value of the disturbance stress.

| Constitutive model regression inversion
To verify the rationality of the elastic-plastic constitutive model of rock with a structural plane established, a cyclic addition-unloading constitutive model of rock is established by using the least squares method of origin nonlinear fitting, and fitting is realized by regression and inversion. Taking the working condition (θ = 15°, Δσ = 6 MPa) as an example, the experimental data at 1, 10, 20, 50, and 75 cycles are compared with the theoretical fitting curve, as shown in Figure 17. Figure 17 shows that the theoretical fitting curve is consistent with the experimental data. Thus, the established constitutive model can accurately describe the stress-strain curves for each cycle during the cyclic loading-unloading dynamic disturbance test, the upper and lower limit of the cycle disturbance load, and the corresponding axial strain value. This model can reflect the changing trend of the loss modulus during the disturbance process and fully reflects the feasibility and accuracy of the modeling method in this work.

| CONCLUSION
By carrying out experimental testing, the effect of periodic dynamic disturbance on the mechanical properties of a rock mass with different structural planes during uniaxial compression tests was investigated. For the north side slope in the Anjialing open-pit mine, different prestatic loads were applied to rock mass specimens under different dynamic disturbance amplitudes. The impacts of θ on the strength, deformation characteristics, energy dissipation characteristics, and fracture rules of rock mass specimens under different disturbance stress amplitudes were discussed systematically in this paper. Experimentally, closed hysteresis loop curves were formed during each loading and unloading cycle. Moreover, the relationship between plastic accumulated strain and the number of disturbance cycles was analyzed. Eventually, a constitutive model of rock under cyclic loading-unloading was established and verified. The specific conclusions of this paper are as follows: 1. The obtained hysteresis loop curves go through three stages: a sparse stage, a dense stage, and another sparse stage. The variation in the cumulative area of the hysteresis loop curves reflects the change law of the strength of rock with structural planes during cyclic loading and unloading. The cumulative area of the hysteresis loop curve increases with increasing θ, Δσ, and n, but the intensity of rock decreases. The area of the hysteresis loop curve increases with Δσ for rock with the same θ. Specifically, E decreases with increases in θ under a certain Δσ, but it increases with Δσ when θ is fixed.
The strength of rock with structural planes is related to θ, Δσ, and n. 2. There is a disturbance threshold of the disturbance stress amplitude when the rock contains a structural plane, and the disturbance threshold is a function of σ and θ. The physical significance of the parameters (A, B, a, b, and c) was discussed to analyze their effects on the plastic strain evolution law of rock under dynamic perturbation. 3. Based on damage theory, the damage constitutive equation of rock with a structural plane under dynamic disturbance is deduced, and the constitutive model of rock under cyclic loading-unloading is established and verified by test data.