# Damage Characteristics and Constitutive Model of Deep Rock under Frequent Impact Disturbances in the Process of Unloading High Static Stress

Authors: Chun Wang; Shuai-fei Zhan; Mei-zhi Xie; Cheng Wang; Lu-ping Cheng; Zu-qiang Xiong
Journal: Complexity
 (2020)

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## Abstract

During the mining of deep mineral resources, the rock is in the complex mechanical environment of high crustal stress and blasting excavation unloading, and the improved SHPB test system is used to carry out frequent impact disturbance test of deep rock in the process of unloading high static stress. Firstly, the general characteristics of dynamic stress-strain curve envelope of the whole process trend can be divided into four stages: the stable development stage of micro cracks, the non-stable development stage of micro cracks, the fatigue damage stage, and the fatigue failure stage. Then, the damage variables of the rock are defined by continuous factors, strain equivalence principle and statistical damage theory, which are based on the whole deformation characteristics of rock during the test. And the derived damage variable equation of rock in the damage process is proved to be reasonable, and the damage constitutive model of rock under frequent disturbances in the process of unloading high static stress is established by the combined model method. Finally, the rationality of constitutive equation is verified by test data, which shows that the dynamic stress-strain curve envelope is in good consistency with the theoretical curve of constitutive equation.


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## Body

## 1. Introduction

With the rapid development of human economy, various resources are consumed increasingly. While the shallow resources tend to run out, the exploitation of deep resources is urgently needed, especially the deep mineral resources. However, the mining of deep mineral resources is inevitable to make deep rock mass in the complex mechanical environment of high static stress and frequent dynamic disturbances [1]. Furthermore, the static mechanical theory is inadequate to explain some complex deformation characteristics and failure phenomenon of deep rock, so it is urgent to explore suitable theory and method for explaining the mechanical characteristics.In order to study the mechanical characteristics of deep rock, scholars in the field of rock mechanics have carried out a large number of experiments, involving the damage characteristics and constitutive models of deep rocks under static and dynamic loads. The research on test equipment and data monitoring technology is also paid much attention. The continuous improvement of equipment and data monitoring technology provide a basic guarantee for the study of the mechanical characteristics of deep rocks. For example, Mobayen and Tchier [2–4] have come up with novel robust adaptive second-order sliding mode tracking control technique and sliding mode disturbance observer control technique, and a nonsingular fast terminal sliding-mode stabilizer was developed. All these laid a foundation for the improvement of dynamic load test technology.The research on damage characteristics of deep rock mainly refers to damage variables, damage process, damage mechanics and so on. From the perspective of energy, the laws of energy dissipation in the process of rock deformation and failure are studied by some scholars, which can reflect the damage process effectively [5–7]. Therefore, the method to describe damage variables of rock is defined according to the law of energy dissipation, and the theoretical formula of damage variables is deduced to estimate the damage degree of rock under specific mechanical conditions [8–10]. Based on the mechanical parameters of rocks, damage variables are also defined by elastic modulus method, maximum strain method and residual strength method, and the corresponding formulas of damage variables are inferred, but none of them can reflect the initial damage degree of rocks under frequent disturbances high static stress absolutely [11–14]. As to damage mechanism, partial scholars have also discussed the influence of temperature, stress environment, and humidity on rock failure characteristics. It is found that high temperature induces germination and expansion of micro-cracks within rock and weakens the strength of rocks [15, 16]. When the stress environment is different, the direction and form of crack germination or expansion in rock internal are different, so the ultimate failure modes are different too [17, 18]. Moreover, rock with higher water content is more vulnerable to damage [19, 20].As regards with the constitutive model of deep rock, it involves the research of influencing factors, model establishment, equation deduction and so on. Because rock is the product of geological process, its characteristics affected mechanical properties are discontinuous, anisotropic inhomogeneous etc. For example, the rock strength under impact load is much greater than that under static stress [21–23], and the rock changes from brittleness to toughness under high confining pressure [24–27]. Therefore, it can be see that there are many complex factors which affect the rock constitutive model, and a large number of studies have been carried out by scholars. Considering the interaction of temperature, residual strength and mechanical conditions, a thermo-mechanical coupled damage model [28, 29] and thermal damage model [30, 31] are established, providing theoretical reference for the excavation and drilling of deep rock mass engineering. Due to the influence of water content and the pH of water on rock mechanical properties, it is found that the degree of chemical damage caused by water reflects the internal damage of rock to some extent, so a damage constitutive model of fractured rock mass under chemical water-rock environment is established [32, 33]. Scholars have also discussed the constitutive model of rock under high stress and dynamic impact disturbances, such as the non-linear damage creep constitutive model of high stress soft rock, established by the non-linear damage creep characteristics [34, 35]. Rock with high stress has the characteristics of transition from brittleness to ductility, and the damage constitutive model under high stress is established [36]. Considering the dual action of temperature and dynamic coupling, the rock mechanical behavior is analyzed to study rock constitutive model under this condition [37, 38].In summary, a lot of studies have been carried out on rock damage characteristics and constitutive models in different mechanical environments, and the results obtained are used to guide the corresponding engineering practice, for example, Liu and Dai [39] proposes a damage constitutive model describing the deformation and strength characteristics of fractured rock mass under cyclic uniaxial compression. Liu et al. [40] establish a damage constitutive model describing rock mechanical behavior under cyclic loading, based on the law of energy dissipation. The research on damage characteristics and constitutive model of deep rocks under the action of frequent impact loads is insufficient, especially the study on the damage characteristics and constitutive model of deep rock under frequent impact disturbances in the process of unloading high static stress. During the deep rock engineering construction, the ore body and surrounding rock are under the condition of high in-situ stress, especially the mining engineering. The mechanical properties of deep rock will change under the influence of frequent impact disturbance caused by blasting excavation. At the same time, the equilibrium state of the original rock stress is broken during the blasting excavation operation. Therefore, during the formation of the new stress equilibrium state, the ore and surrounding rock are in the process of unloading. Based on the above mentioned analysis, the environment of the engineering rock mass can be summarized as follows: the deep rocks are subjected to frequent impact disturbance in the process of high-stress unloading. Therefore, an experimental study on the damage characteristics and constitutive model of deep rock based on the actual engineering environment which was carried out can provide a theoretical reference for the rock breaking and support in deep rock mass engineering construction.To solve these problems, experimental research is carried out on some new ideas. Such as, the high static load applied in the axial direction of the rock sample is used to simulate the high in-situ stress of deep rock, the excavation unloading process of rock mass engineering is simulated by unloading high static load at different rates, and the impact loads frequently applied up the rock sample axis is used to simulate the impact disturbance caused by multiple blasting. At the same time, the growth model of biological population introduced to analyze the microcrack propagation in rock is the first time. In this way, the damage characteristics of rock can be quantitatively analyzed, and the dynamic constitutive relationship of deep rock can be established under the corresponding conditions. Finally, the purpose of guiding deep rock engineering construction is realized.


## 2. High Static Load Unloading Frequent Impact Disturbance Test

### 2.1. Test Specimen and Equipment

#### 2.1.1. Test Specimen

The test core was taken from the surrounding rock near mining roadway in the depth of 900m underground of Dongguashan Copper Mine, and it was skarn with dense structure and good homogeneous by analysis. To ensure that the non-parallelism and non-perpendicularity of specimens were less than 0.02 mm, the two ends of them were carefully polished. Specimens were processed into two kinds of sizes (50  mm × 100  mm, 50  mm × 50  mm), with a high diameter ratio of 2 : 1, which were used to measure the uniaxial compressive strength. In addition, the specimens with a high diameter ratio of 1 : 1 were used to study the frequent disturbances test of high static stress unloaded.


#### 2.1.2. Test Equipment

Both the uniaxial compressive test and high static stress unloaded test were completed in Central South University, and the former was studied by Instron1346 type electro-hydraulic servo testing machine, while the latter relied on the improved SHPB test system (Structure diagram as shown in Figure1 [41]) for research. As shown in Figure 1, the improved SHPB test system is capable of axial loading and unloading. Manual hydraulic pump is used for loading, and the loading time can be recorded with a stopwatch to control the loading rate. The hydraulic valve needs to be opened slowly when the pressure is unloaded, moreover, the unloaded value and time are recorded to control the unloaded speed.Figure 1
Structure diagram of the test load: (1) Pressure loading unit; (2) Buffer bar; (3) Support; (4) Strain gauge A2; (5) Transmission bar; (6) Rock specimen; (7) Strain gauge A1; (8) Incident bar; (9) Heterotype impact hammer; (10) Launch cavity; (11) Gas gun.


### 2.2. Test Scheme

For simulating high static stress state, the basic principle is that pre-added axial pressure is set as close as possible to the uniaxial compressive strength, and the axial pressure is unloaded at different rates to simulate the process of rock excavation. In order to ensure that the specimens can withstand multiple cyclic impacts, as soon as the axial pressure is reduced to 50% of pre-added high axial pressure, the impact load of 0.5 MPa is applied to specimens for simulating small disturbance. Before the impact load, the rock sample is first subjected to axial static load. The loading rate first was 0.5 MPa/s, and then changed to 0.1 MPa/s. In order to prevent the rock sample from being damaged because the loading rate is too fast, the loading rate is adjusted to 0.1 MPa/s when the later axial pressure of 5 MPa is applied. At the same time, in order to ensure the same impact load rate in the test, the heterotype impact hammer were placed at the same position in launch cavity before each impact. Then when the impact pressure is fixed, the impact load can be kept constant, which was applied to the impact end of the incident bar. Test scheme is shown in Tables1 and 2, and the average uniaxial compressive stress of skarn is 126.63 MPa according to the data of Table 1.Table 1
Test scheme and results of deep skarn under uniaxial compression.
NumberHeight (mm)Diameter (mm)Density(g/cm3Longitudinal wave velocity (m/s)Loading rate (mm/s)Peak load (kN)Uniaxial compressive strength (MPa)XK1100.0250.063.7855670.03239.08121.53XK299.3649.864.2658980.03384.10196.82XK399.2449.564.0653240.03255.80132.67XK498.9649.983.8451260.03176.9390.23XK5100.0250.023.7252970.03252.89128.76XK699.0449.943.9451350.03220.17112.46XK7100.0850.043.9050790.03204.31103.94Table 2
Test scheme of high static stress unloaded and frequent disturbances.
NumberHeight (mm)Diameter (mm)Density (g/cm3)Longitudinal wave velocity (m/s)Pre-axial pressure (MPa)Axial pressure on impact (MPa)Unloading rate of axial pressure (MPa/s)Impact pressure (MPa)XK1-150.3453.603.1044676532.50.50.5XK1-250.2252.983.0945497537.5XK1-349.8552.883.3346748542.5XK1-449.8553.703.3944969547.5XK2-149.4753.083.5643106532.51.00.5XK2-249.8152.873.2246257537.5XK2-350.0553.723.5346138542.5XK2-450.0753.143.5444289547.5XK3-149.7453.153.1945636532.51.50.5XK3-250.0753.693.5241467537.5XK3-349.4153.203.3045678542.5XK3-449.6753.343.5143669547.5XK4-150.3053.362.9447436532.52.00.5XK4-250.3053.243.2145717537.5XK4-349.7153.262.8542278542.5XK4-449.5953.313.5342529547.5


### 2.3. Test Results

According to the differences of preadded axial pressures and unloading rate, the test was divided into 16 groups. Each specimen was subjected to multiple impact disturbances, and the corresponding dynamic stress-strain curves can be obtained after each impact disturbances. To study the overall deformation characteristics of specimen, the envelope of the dynamic stress-strain curve of each specimen under multiple impact disturbances was selected for analysis. In the test, a corresponding dynamic stress-strain curve can be drawn based on the test data obtained from each impact. Some feature data can be selected to form a new set of data, such as the peak data and all data before the peak on the stress-strain curve obtained at the first impact, the data corresponding to the residual strength stage after the peak at the last impact, and the peak data on the dynamic stress-strain curve corresponding to the intermediate impacts. The stress-strain curve drawn with the new data combination is the envelope curve of the dynamic stress-strain curve. Figure2 shows some representative dynamic stress-strain curves, in which the numbers represent the disturbance times. Figure 3 lists the envelopes of dynamic stress-strain curves at the unloading rate of 0.5 MPa/s and the pre-added axial pressure of 85 MPa, respectively. In Figure 3(a), the numbers represent pre-added axial pressure value, and the numbers in Figure 3(b) show unloading rate of axial pressure.Figure 2
Dynamic stress - strain curve and envelope curve of skarn under the common action of high static stress unloading and impact disturbance (axial load 95 MPa and unloading rate 0.5 MPa/s).Figure 3
Envelope of typical dynamic stress-strain curve. (a) The axial unloading rate is 0.5 MPa/s. (b) The pre-added axial pressure is 85 MPa.
(a)(b)It can be seen from Figure2 that, the initial stages of dynamic stress-strain curves are approximately linear, the straight-line segment is gradually shortened with the increasing impact disturbance times, and the speed entering nonlinear stage is getting faster. It indicates that the specimen appears to have elastic deformation firstly and then plastic deformation under high static stress unloaded and frequent disturbances, and the elastic properties in specimen are weakened through frequent disturbances. Because the micro cracks in specimen are closed almost under pre-added high axial pressure, there is no compaction stage, and the specimen enters the elastic stage directly. Moreover, the pre-added axial pressure approached the uniaxial compressive strength extremely, which leads to damage in specimen, in the meantime, the damage is gradually aggravated by loading, unloading and frequent disturbances. Therefore, the elastic deformation stage becomes shortened and the plastic deformation stage enlarged gradually on dynamic stress-strain curve. Figure 2 also shows that the trends of envelope of the dynamic stress-strain curves are consistent with that of dynamic stress-strain curve at each impact, which shows that the whole process trends of specimens under same axial pressure, same unloading rate, and same impact strength can be reflected by variation characteristics of envelope in some conditions.Figure3(a) shows that the peak stress decreases corresponding to the envelope with the increase of pre-added axial pressure, when the unloading rate is constant. For example, when the preloaded axial pressures were 65 MPa, 75 MPa, 85 MPa and 95 MPa, the corresponding pressures to the peak stress of the envelope of the dynamic stress-strain curve were 115.08 MPa, 110.58 MPa, 104.25 MPa and 90.38 MPa, respectively. Upon further analysis, the peak stress reduction ratio of the corresponding envelope curve is 3.91%, 5.50% and 12.05%, respectively, when the preloaded axial pressure increased from 65 MPa to 75 MPa, from 75 MPa to 85 MPa and from 85 MPa to 95 MPa. Therefore, the internal damage is aggravated by pre-added high axial pressure, namely, the higher the axial pressure is, the weaker the rock resisted to external disturbance will be. In a word, the number of impact disturbances that the specimens can withstand reduces.Figure3(b) shows that the change trends of envelopes remains constant basically when the axial pressure is the same, and it goes through a straight line segment first and then enters a non-linear development stage, indicating that the dynamic change trend of rock is not affected by the unloading rate. When the unloading rate is 0.5 MPa/s, 1.0 MPa/s, 1.5 MPa/s and 2.0 MPa/s, the corresponding dynamic peak stress is 104.25 MPa, 103.85 MPa, 103.38 MPa and 103.15 MPa, respectively. The range between dynamic peak stresses varies from 0.2% to 0.5%, showing a small decreasing trend, and it can be concluded that the allowable dynamic stress decreases with the rising unloading rate. When the axial pressure is certain, the larger the unloading rate, the shorter the time required for unloading to same axial pressure. The time of specimen affected by the unloading process of high axial pressure is shorter before each impact. At the same time, the time for micro-cracks within specimen to germinate and expand will be shorter, resulting in a smaller internal damage accumulation. In addition, when the specimen is in the process of axial pressure and unloading, it has the effect of delaying the force about external impact. When the unloading rate is higher, the effect of delaying force is more obvious, leading to a decline in effective impact stress and internal fatigue damage on specimens. That is, the higher the unloading rate, the more conducive to stability of rock.The general characteristics of the envelope of dynamic stress-strain curve can be subdivided into four stages, such as stable development of micro-cracks, unstable development of micro-cracks, fatigue damage, and fatigue failure, as shown in Figure4.Figure 4
Envelope of typical dynamic stress-strain curve.In the stable development stage of microcracks (OA section), the curve develops in a straight line and the corresponding deformation modulus is the largest, indicating that specimen is in the elastic deformation stage at the beginning. At this stage, the microcracks in specimen are constant or stable, which reflects that the specimen has the maximum impact resistance, and it is more difficult to produce greater deformation.In the unstable propagation stage of microcracks (section AB), the curve develops in a non-linear way and slows down; meanwhile, the corresponding deformation modulus gradually decreases, indicating that the specimen is in the plastic deformation stage. The internal micro cracks of specimen show instantaneous propagation and transfixion under frequent disturbance, leading to macroscopic failure. However, each dynamic disturbance is small, so a single impact fails to make the micro cracks in specimen through instantly, but makes it in an unstable development.In the fatigue damage stage (BC segment), this section of the curve shows a downward trend, and the downward trend is relatively slow. At this stage, the rock has a large amount of compression deformation, but the dynamic stress reduction is relatively small, which shows a certain ductility. Because the elastic energy stored in specimen is relatively large, it is able to counteract part of the frequent disturbance. However, the damage degree in the specimen is still aggravated with the increasing impact numbers, and the elastic energy stored in the specimen decreases gradually, which shows a slow downward trend in the envelope curve.In the fatigue failure stage (CD segment), the brittleness of the specimen is enhanced. In general, there is no macroscopic failure of the specimen under impact stress, and the leading factor causing macroscopic failure is the pre-added high axial pressure.


## 2.1. Test Specimen and Equipment

### 2.1.1. Test Specimen

The test core was taken from the surrounding rock near mining roadway in the depth of 900m underground of Dongguashan Copper Mine, and it was skarn with dense structure and good homogeneous by analysis. To ensure that the non-parallelism and non-perpendicularity of specimens were less than 0.02 mm, the two ends of them were carefully polished. Specimens were processed into two kinds of sizes (50  mm × 100  mm, 50  mm × 50  mm), with a high diameter ratio of 2 : 1, which were used to measure the uniaxial compressive strength. In addition, the specimens with a high diameter ratio of 1 : 1 were used to study the frequent disturbances test of high static stress unloaded.


### 2.1.2. Test Equipment

Both the uniaxial compressive test and high static stress unloaded test were completed in Central South University, and the former was studied by Instron1346 type electro-hydraulic servo testing machine, while the latter relied on the improved SHPB test system (Structure diagram as shown in Figure1 [41]) for research. As shown in Figure 1, the improved SHPB test system is capable of axial loading and unloading. Manual hydraulic pump is used for loading, and the loading time can be recorded with a stopwatch to control the loading rate. The hydraulic valve needs to be opened slowly when the pressure is unloaded, moreover, the unloaded value and time are recorded to control the unloaded speed.Figure 1
Structure diagram of the test load: (1) Pressure loading unit; (2) Buffer bar; (3) Support; (4) Strain gauge A2; (5) Transmission bar; (6) Rock specimen; (7) Strain gauge A1; (8) Incident bar; (9) Heterotype impact hammer; (10) Launch cavity; (11) Gas gun.


## 2.1.1. Test Specimen

The test core was taken from the surrounding rock near mining roadway in the depth of 900m underground of Dongguashan Copper Mine, and it was skarn with dense structure and good homogeneous by analysis. To ensure that the non-parallelism and non-perpendicularity of specimens were less than 0.02 mm, the two ends of them were carefully polished. Specimens were processed into two kinds of sizes (50  mm × 100  mm, 50  mm × 50  mm), with a high diameter ratio of 2 : 1, which were used to measure the uniaxial compressive strength. In addition, the specimens with a high diameter ratio of 1 : 1 were used to study the frequent disturbances test of high static stress unloaded.


## 2.1.2. Test Equipment

Both the uniaxial compressive test and high static stress unloaded test were completed in Central South University, and the former was studied by Instron1346 type electro-hydraulic servo testing machine, while the latter relied on the improved SHPB test system (Structure diagram as shown in Figure1 [41]) for research. As shown in Figure 1, the improved SHPB test system is capable of axial loading and unloading. Manual hydraulic pump is used for loading, and the loading time can be recorded with a stopwatch to control the loading rate. The hydraulic valve needs to be opened slowly when the pressure is unloaded, moreover, the unloaded value and time are recorded to control the unloaded speed.Figure 1
Structure diagram of the test load: (1) Pressure loading unit; (2) Buffer bar; (3) Support; (4) Strain gauge A2; (5) Transmission bar; (6) Rock specimen; (7) Strain gauge A1; (8) Incident bar; (9) Heterotype impact hammer; (10) Launch cavity; (11) Gas gun.


## 2.2. Test Scheme

For simulating high static stress state, the basic principle is that pre-added axial pressure is set as close as possible to the uniaxial compressive strength, and the axial pressure is unloaded at different rates to simulate the process of rock excavation. In order to ensure that the specimens can withstand multiple cyclic impacts, as soon as the axial pressure is reduced to 50% of pre-added high axial pressure, the impact load of 0.5 MPa is applied to specimens for simulating small disturbance. Before the impact load, the rock sample is first subjected to axial static load. The loading rate first was 0.5 MPa/s, and then changed to 0.1 MPa/s. In order to prevent the rock sample from being damaged because the loading rate is too fast, the loading rate is adjusted to 0.1 MPa/s when the later axial pressure of 5 MPa is applied. At the same time, in order to ensure the same impact load rate in the test, the heterotype impact hammer were placed at the same position in launch cavity before each impact. Then when the impact pressure is fixed, the impact load can be kept constant, which was applied to the impact end of the incident bar. Test scheme is shown in Tables1 and 2, and the average uniaxial compressive stress of skarn is 126.63 MPa according to the data of Table 1.Table 1
Test scheme and results of deep skarn under uniaxial compression.
NumberHeight (mm)Diameter (mm)Density(g/cm3Longitudinal wave velocity (m/s)Loading rate (mm/s)Peak load (kN)Uniaxial compressive strength (MPa)XK1100.0250.063.7855670.03239.08121.53XK299.3649.864.2658980.03384.10196.82XK399.2449.564.0653240.03255.80132.67XK498.9649.983.8451260.03176.9390.23XK5100.0250.023.7252970.03252.89128.76XK699.0449.943.9451350.03220.17112.46XK7100.0850.043.9050790.03204.31103.94Table 2
Test scheme of high static stress unloaded and frequent disturbances.
NumberHeight (mm)Diameter (mm)Density (g/cm3)Longitudinal wave velocity (m/s)Pre-axial pressure (MPa)Axial pressure on impact (MPa)Unloading rate of axial pressure (MPa/s)Impact pressure (MPa)XK1-150.3453.603.1044676532.50.50.5XK1-250.2252.983.0945497537.5XK1-349.8552.883.3346748542.5XK1-449.8553.703.3944969547.5XK2-149.4753.083.5643106532.51.00.5XK2-249.8152.873.2246257537.5XK2-350.0553.723.5346138542.5XK2-450.0753.143.5444289547.5XK3-149.7453.153.1945636532.51.50.5XK3-250.0753.693.5241467537.5XK3-349.4153.203.3045678542.5XK3-449.6753.343.5143669547.5XK4-150.3053.362.9447436532.52.00.5XK4-250.3053.243.2145717537.5XK4-349.7153.262.8542278542.5XK4-449.5953.313.5342529547.5


## 2.3. Test Results

According to the differences of preadded axial pressures and unloading rate, the test was divided into 16 groups. Each specimen was subjected to multiple impact disturbances, and the corresponding dynamic stress-strain curves can be obtained after each impact disturbances. To study the overall deformation characteristics of specimen, the envelope of the dynamic stress-strain curve of each specimen under multiple impact disturbances was selected for analysis. In the test, a corresponding dynamic stress-strain curve can be drawn based on the test data obtained from each impact. Some feature data can be selected to form a new set of data, such as the peak data and all data before the peak on the stress-strain curve obtained at the first impact, the data corresponding to the residual strength stage after the peak at the last impact, and the peak data on the dynamic stress-strain curve corresponding to the intermediate impacts. The stress-strain curve drawn with the new data combination is the envelope curve of the dynamic stress-strain curve. Figure2 shows some representative dynamic stress-strain curves, in which the numbers represent the disturbance times. Figure 3 lists the envelopes of dynamic stress-strain curves at the unloading rate of 0.5 MPa/s and the pre-added axial pressure of 85 MPa, respectively. In Figure 3(a), the numbers represent pre-added axial pressure value, and the numbers in Figure 3(b) show unloading rate of axial pressure.Figure 2
Dynamic stress - strain curve and envelope curve of skarn under the common action of high static stress unloading and impact disturbance (axial load 95 MPa and unloading rate 0.5 MPa/s).Figure 3
Envelope of typical dynamic stress-strain curve. (a) The axial unloading rate is 0.5 MPa/s. (b) The pre-added axial pressure is 85 MPa.
(a)(b)It can be seen from Figure2 that, the initial stages of dynamic stress-strain curves are approximately linear, the straight-line segment is gradually shortened with the increasing impact disturbance times, and the speed entering nonlinear stage is getting faster. It indicates that the specimen appears to have elastic deformation firstly and then plastic deformation under high static stress unloaded and frequent disturbances, and the elastic properties in specimen are weakened through frequent disturbances. Because the micro cracks in specimen are closed almost under pre-added high axial pressure, there is no compaction stage, and the specimen enters the elastic stage directly. Moreover, the pre-added axial pressure approached the uniaxial compressive strength extremely, which leads to damage in specimen, in the meantime, the damage is gradually aggravated by loading, unloading and frequent disturbances. Therefore, the elastic deformation stage becomes shortened and the plastic deformation stage enlarged gradually on dynamic stress-strain curve. Figure 2 also shows that the trends of envelope of the dynamic stress-strain curves are consistent with that of dynamic stress-strain curve at each impact, which shows that the whole process trends of specimens under same axial pressure, same unloading rate, and same impact strength can be reflected by variation characteristics of envelope in some conditions.Figure3(a) shows that the peak stress decreases corresponding to the envelope with the increase of pre-added axial pressure, when the unloading rate is constant. For example, when the preloaded axial pressures were 65 MPa, 75 MPa, 85 MPa and 95 MPa, the corresponding pressures to the peak stress of the envelope of the dynamic stress-strain curve were 115.08 MPa, 110.58 MPa, 104.25 MPa and 90.38 MPa, respectively. Upon further analysis, the peak stress reduction ratio of the corresponding envelope curve is 3.91%, 5.50% and 12.05%, respectively, when the preloaded axial pressure increased from 65 MPa to 75 MPa, from 75 MPa to 85 MPa and from 85 MPa to 95 MPa. Therefore, the internal damage is aggravated by pre-added high axial pressure, namely, the higher the axial pressure is, the weaker the rock resisted to external disturbance will be. In a word, the number of impact disturbances that the specimens can withstand reduces.Figure3(b) shows that the change trends of envelopes remains constant basically when the axial pressure is the same, and it goes through a straight line segment first and then enters a non-linear development stage, indicating that the dynamic change trend of rock is not affected by the unloading rate. When the unloading rate is 0.5 MPa/s, 1.0 MPa/s, 1.5 MPa/s and 2.0 MPa/s, the corresponding dynamic peak stress is 104.25 MPa, 103.85 MPa, 103.38 MPa and 103.15 MPa, respectively. The range between dynamic peak stresses varies from 0.2% to 0.5%, showing a small decreasing trend, and it can be concluded that the allowable dynamic stress decreases with the rising unloading rate. When the axial pressure is certain, the larger the unloading rate, the shorter the time required for unloading to same axial pressure. The time of specimen affected by the unloading process of high axial pressure is shorter before each impact. At the same time, the time for micro-cracks within specimen to germinate and expand will be shorter, resulting in a smaller internal damage accumulation. In addition, when the specimen is in the process of axial pressure and unloading, it has the effect of delaying the force about external impact. When the unloading rate is higher, the effect of delaying force is more obvious, leading to a decline in effective impact stress and internal fatigue damage on specimens. That is, the higher the unloading rate, the more conducive to stability of rock.The general characteristics of the envelope of dynamic stress-strain curve can be subdivided into four stages, such as stable development of micro-cracks, unstable development of micro-cracks, fatigue damage, and fatigue failure, as shown in Figure4.Figure 4
Envelope of typical dynamic stress-strain curve.In the stable development stage of microcracks (OA section), the curve develops in a straight line and the corresponding deformation modulus is the largest, indicating that specimen is in the elastic deformation stage at the beginning. At this stage, the microcracks in specimen are constant or stable, which reflects that the specimen has the maximum impact resistance, and it is more difficult to produce greater deformation.In the unstable propagation stage of microcracks (section AB), the curve develops in a non-linear way and slows down; meanwhile, the corresponding deformation modulus gradually decreases, indicating that the specimen is in the plastic deformation stage. The internal micro cracks of specimen show instantaneous propagation and transfixion under frequent disturbance, leading to macroscopic failure. However, each dynamic disturbance is small, so a single impact fails to make the micro cracks in specimen through instantly, but makes it in an unstable development.In the fatigue damage stage (BC segment), this section of the curve shows a downward trend, and the downward trend is relatively slow. At this stage, the rock has a large amount of compression deformation, but the dynamic stress reduction is relatively small, which shows a certain ductility. Because the elastic energy stored in specimen is relatively large, it is able to counteract part of the frequent disturbance. However, the damage degree in the specimen is still aggravated with the increasing impact numbers, and the elastic energy stored in the specimen decreases gradually, which shows a slow downward trend in the envelope curve.In the fatigue failure stage (CD segment), the brittleness of the specimen is enhanced. In general, there is no macroscopic failure of the specimen under impact stress, and the leading factor causing macroscopic failure is the pre-added high axial pressure.


## 3. Analysis of Damage Characteristics

### 3.1. Damage Variable

In this test, the damage inside the specimen is intensified with each impact, so the reasonable definition of damage variable is helpful to analyze and reflect damage characteristics in the whole test process. Due to the complex internal structure of rock, the rock microelement is assumed to be continuous and random, and the damage variables are defined by continuous factor, strain equivalence principle, or statistical damage theory. Meanwhile, their weighted averages are also used for analysis.


#### 3.1.1. Damage VariableD1 Based on Continuous Factor and Strain Equivalence Principle

There is a compact structure and good uniformity of test specimen, and it is assumed isotropous in the test. Base on strain equivalence principle, it is considered that the deformation caused by nominal stress acting on the damage material is equal to that caused by effective stress acting on virtual nondestructive material. The schematic diagram of strain equivalence principle for rock microelement is shown in Figure5.Figure 5
The schematic diagram of strain equivalence principle for rock microelement.In summary, the ratio of damage volume to total volume of rock microelement is defined to damage variableD1, and the formula is as follows:(1)D=1−S˜S,0≤D≤1.By combining continuous factor, strain equivalence principle and Hooke's law, the relationship between stress and strain can be obtained from Figure5:(2)ε=σE˜=σ˜E.Formula (3) is calculated from formulae (1) and (2):(3)D1=1−S˜S=1−σσ˜=1−E˜E.


#### 3.1.2. Damage VariableD2 Based on Statistical Damage Theory

Assuming that the defects of rock microelement are independent, random, and the distribution of them conforms to Weibull's distribution. Based on statistical damage theory, Tang [42]. Calculated the damage variable of rock, and the formula is as follows:(4)D2=1−εαm+1exp−εαm,ε≥0.In formula (4), ε, α, and m represents strain of rock damage mass, rock integrity, the and relevant parameters of rock’s properties and shapes, respectively.


#### 3.1.3. Damage VariableD under High Static Stress Unloaded and Frequent Disturbances

According to the test results, the damage variableD is rational when the proportion of D1 and D2 is 50% respectively, so D is defined as the weighted average among them:(5)D=D1+D22=121−E˜E+1−εαm+1exp−εαm,ε≥0.


### 3.2. Damage Evolution Equation

#### 3.2.1. Equation Analysis of Damage VariableD1

As the continuous factors and equivalent strain are applied to the definition of damage variableD1, the following assumptions must be satisfied. Firstly, the rock material is composed of damage and lossless microelements. Secondly, the volume of them is equal, and the lossless microelements are able to irreversibly transform into damage microelements instantaneously. Finally, the damage only occurs along the axial direction but not the laterally, and the lossless microelements obey Hooke’s law.Combined with equations (1) and (3), the equation of damage variable can be expressed as:(6)D1=NS0MS0=NM.In formula (6), N is the total number of damaged microelements, M is the sum of the number of nondestructive microelements and damaged microelements, that is, the total number of microelements in the specimen, and S0 is the volume of a single element.Assuming that the growth process of biological species group is used to simulate the damage evolution process of rock, the survival environment of species group is considered as rock external load. Moreover, the time in the species group growth model corresponds to the strain of rock under the external load, and the environmental capacity of species group corresponds to the total number of rock microelements, so the growth rate of the number of damaged microelements can be expressed as follows:(7)dNdε=rN1−NM.In formula (7), γ and ε is natural growth rate and the relevant strain of rock, respectively.Then, formula (7) is solved by the separation of variables in differential equation, and expression (8) is obtained:(8)N=M1+Ce−rε,C=M−N0N0.In formula (8), N0 is the initial quantity of rock damage microelements.The differential formula (9) of damage variable D1 is deduced by formula (6) and formula (7):(9)dD1dε=1MdNdε=rNM1−NM.Substitute formula (8) into formula (7), and formula (10) of damage variable is solved out through the method of separating variables:(10)D1=11+eβ−rε,β=lnMN0−1.In formula (10), β represents the initial damage degree of rock material.


#### 3.2.2. Equation Analysis of Damage VariableD2

According to the statistical damage theory, the distribution of microelement defects in rocks is independent and random, and it satisfies Possion distribution. Therefore, the probability of defects occurrence can be expressed by Possion law within the interval of lengthl, and the expression of probability function is shown in formula (11) as follows:(11)Pkl=λlkk!e−λl.In formula (11), λ is the mathematical expectation of microelement defects per unit length, k is the number of microelement defects, and P(k/l) is the probability function of k defects appearing in the interval with length l.Suppose that the probability of a cell body defect on the length of Δl is P1(Δl), and the probability of no element body defect on the length of l is P(l). Formulae (12) and (13) can be inferred as follows:(12)P1Δl=λΔl⋅e−λΔl,(13)Pl+Δl=Pl−PlPΔl.Formula (14) can be deduced from formulae (12) and (13):(14)Pl+Δl−PlΔl=−λPl⋅e−λΔl.Whenl = 0, P(l) = 1, and the value of Δl tends to zero, so the limit of both sides in formula (14) can be obtained:(15)Pl=e−λl.Therefore, the probability functionϕ(l) for the defect of no less than one microelement in the interval of length l can be derived:(16)ϕl=1−Pl.The probability density functionϕ(l) of microelement can be deduced from formula (16):(17)φl=dϕdl=λe−λl.Assuming that the damage microelement in the specimen loses its bearing capacity and rock strain is infinite when damaged, the expression of damage variableD2 are represented as follows:(18)D2=∫0εlφldl∫0∞lφldl=1−λε+1e−λε.While considering the defects of line, plane and body, the completeness was considered asα=1/λ [43]. Therefore, formula (18) can be rewritten as formula (19).(19)D2=1−εαm+1exp−εαm.In formula (19), ε is the strain of rock damage mass, and m represents the parameters related to materials and shapes of rock.


#### 3.2.3. Definition of Damage VariableD

Based on the above mentioned equation analysis, the damage variableD of rock under high static stress unloaded and frequent disturbances can be defined as formula (20):(20)D=121+11+eβ−rε−εαm+1exp−εαm,β=lnMN0−1,ε≥0.


#### 3.2.4. Determination of Damage Evolution Equation Parameters

If the damage variableD is to be analyzed and calculated, the parameters of β, r, m and α must be determined firstly. To determine the values of β and r, the equation of D1 needs to be analyzed, while the parameters of m and α can be determined by analyzing the equation of D2.(1) Determination of Parameters β and r. Firstly, the value of damage variable D1 corresponding to each strain of the envelope is calculated by formula (3), then E (the elastic modulus of non-destructive materials) and E˜ (the elastic modulus of damaged materials) in formula (3) are calculated by Figure 6.Figure 6
The schematic diagram ofE and E˜ with the determining method.Formula (10) is deduced, and formula (21) is ordered:(21)Y=ln1D1−1=β−rε.Calculate theY value corresponding to each strain in the dynamic stress-strain envelope and then analyze the fitting linear formula (Y-ε linear formula). Finally, by fitting the linear formula (21) with the logistic model, and values of β and r are able to be determined.(2) Determination of Parameters m and α. Statistical damage theory holds that α (rock integrity) is a function of ε [42], and α is inversely proportional to ε under high static stress unloaded and frequent disturbances as follows:(22)α=Cε,Cis a proportionality constant.Assuming that the equationε/C = 1/α1 holds, where α1 is a parameter related to rock properties and shapes, and substitute it into formula (22), it is found that the values of α and α1 is equal. Thus, formula (19) can be rewritten to formula (23) as follows:(23)D2=1−εα1m+1exp−εα1m.It is supposed that the damaged microelement loses its bearing capacity and the non-damaged microelement meets Hooke’s law, therefore, the relationship between rock stress and strain under high static stress unloaded and frequent disturbances is shown in formula (24) as follows:(24)σ=Eε1−D2.Substitute formula (23) into formula (24), and formula (25) is given as follows:(25)σ=Eεεα1m+1exp−εα1m.Furthermore, ifσ∗=σ/Eα1 and ε∗=ε/α1 are founded, formula (25) can be transformed into dimensionless form, such as formula (26):(26)σ∗=ε∗ε∗m+1exp−ε∗m.According to formula (26), the dynamic stress-strain theoretical curves of rock with different m values are drawn in Figure 7.Figure 7
The schematic diagram of dynamic stress-strain curves with one-dimensional dimensionless.By analyzing the similarity of the curves in Figure7 and the envelope of the measured dynamic stress-strain curves, the value of m can be estimated. After the values of m is determined, the maximum value of σ∗ is obtained by formula (26), and the parameter α is calculated by formula (27), which is deduced by Tang [42].(27)α=α1=PEσ∗max.In formula (27), P is the peak stress in the envelope of dynamic stress-strain curves.


### 3.3. Damage Evolution Law

To verify the definition of damage variableD in formula (5), the values of β, r, m and α are substituted into this formula, so the corresponding dynamic damage variable D is calculated by combining the corresponding strain in the envelope of dynamic stress-strain curves. By comparing the damage degree in specimens and analyzing the characteristics of dynamic variable-strain theory curve, it is conclude that the defined damage variable is reasonable. Figure 8 lists the theoretical relationship of two typical damage variable-strain curves, and Figure 9 shows the damage states of the corresponding two groups of rocks during the impact process.Figure 8
The damage variable-strain curves of rock under high static stress unloaded and frequent disturbances. (a) The axial pressure is 85 MPa. (b) The unloading rate is 0.5 MPa/s.
(a)(b)Figure 9
The damage states of rocks during the impact process. (a) The unloading rate was 0.5 MPa/s and it is the seventh impact. (b) The preloaded axial pressure is 85 MPa and it is the 10th impact.
(a)(b)It can be concluded from Figure8 that(1)
When the value of dynamic strain is 0, the value of damage variableD is greater than 0, so the defined damage reflect the initial damage degree of rock under high static stress.(2)
There exist four stages of fitted damage variable-strain curve, that is moderate increase, fast increase, slowly increase and tend to be stable. The whole curve presents S-shape, and the value of damage variableD is between 0 and 1. It can effectively correspond to four development stages of rock in the test, which is steady development of micro cracks, rapid propagation of micro cracks, fatigue damage and fatigue failure.(3)
When the unloading rate is constant, the greater the axial pressure is, the faster the value of damage variableD tends to be 1, reflecting that pre-high axial pressure accelerates the process of rock damage and failure.(4)
When the pre-loading axial pressure is constant, the smaller the unloading rate is, the faster the damage variableD tends to be 1, which shows that high unloading rate is beneficial to improving the rock’ ability to withstand frequent disturbances.As shown in Figure9(a), when the unloading rate and impact times maintain constant, the greater the pre-added axial pressure is, the more serious the rock damage is. When the pre-added axial pressure and impact times keep invariant, the smaller the unloading rate of axial pressure is during the impact process, the more serious the rock damage will be, as shown in Figure 9(b). The above mentioned two phenomena indicate that preadded high axial pressure weakens the ability of rock to resist external impact load, and high unloading rate enhances the ability of rock to withstand frequent disturbances, proving the rationality of defined damage variable.


## 3.1. Damage Variable

In this test, the damage inside the specimen is intensified with each impact, so the reasonable definition of damage variable is helpful to analyze and reflect damage characteristics in the whole test process. Due to the complex internal structure of rock, the rock microelement is assumed to be continuous and random, and the damage variables are defined by continuous factor, strain equivalence principle, or statistical damage theory. Meanwhile, their weighted averages are also used for analysis.


### 3.1.1. Damage VariableD1 Based on Continuous Factor and Strain Equivalence Principle

There is a compact structure and good uniformity of test specimen, and it is assumed isotropous in the test. Base on strain equivalence principle, it is considered that the deformation caused by nominal stress acting on the damage material is equal to that caused by effective stress acting on virtual nondestructive material. The schematic diagram of strain equivalence principle for rock microelement is shown in Figure5.Figure 5
The schematic diagram of strain equivalence principle for rock microelement.In summary, the ratio of damage volume to total volume of rock microelement is defined to damage variableD1, and the formula is as follows:(1)D=1−S˜S,0≤D≤1.By combining continuous factor, strain equivalence principle and Hooke's law, the relationship between stress and strain can be obtained from Figure5:(2)ε=σE˜=σ˜E.Formula (3) is calculated from formulae (1) and (2):(3)D1=1−S˜S=1−σσ˜=1−E˜E.


### 3.1.2. Damage VariableD2 Based on Statistical Damage Theory

Assuming that the defects of rock microelement are independent, random, and the distribution of them conforms to Weibull's distribution. Based on statistical damage theory, Tang [42]. Calculated the damage variable of rock, and the formula is as follows:(4)D2=1−εαm+1exp−εαm,ε≥0.In formula (4), ε, α, and m represents strain of rock damage mass, rock integrity, the and relevant parameters of rock’s properties and shapes, respectively.


### 3.1.3. Damage VariableD under High Static Stress Unloaded and Frequent Disturbances

According to the test results, the damage variableD is rational when the proportion of D1 and D2 is 50% respectively, so D is defined as the weighted average among them:(5)D=D1+D22=121−E˜E+1−εαm+1exp−εαm,ε≥0.


## 3.1.1. Damage VariableD1 Based on Continuous Factor and Strain Equivalence Principle

There is a compact structure and good uniformity of test specimen, and it is assumed isotropous in the test. Base on strain equivalence principle, it is considered that the deformation caused by nominal stress acting on the damage material is equal to that caused by effective stress acting on virtual nondestructive material. The schematic diagram of strain equivalence principle for rock microelement is shown in Figure5.Figure 5
The schematic diagram of strain equivalence principle for rock microelement.In summary, the ratio of damage volume to total volume of rock microelement is defined to damage variableD1, and the formula is as follows:(1)D=1−S˜S,0≤D≤1.By combining continuous factor, strain equivalence principle and Hooke's law, the relationship between stress and strain can be obtained from Figure5:(2)ε=σE˜=σ˜E.Formula (3) is calculated from formulae (1) and (2):(3)D1=1−S˜S=1−σσ˜=1−E˜E.


## 3.1.2. Damage VariableD2 Based on Statistical Damage Theory

Assuming that the defects of rock microelement are independent, random, and the distribution of them conforms to Weibull's distribution. Based on statistical damage theory, Tang [42]. Calculated the damage variable of rock, and the formula is as follows:(4)D2=1−εαm+1exp−εαm,ε≥0.In formula (4), ε, α, and m represents strain of rock damage mass, rock integrity, the and relevant parameters of rock’s properties and shapes, respectively.


## 3.1.3. Damage VariableD under High Static Stress Unloaded and Frequent Disturbances

According to the test results, the damage variableD is rational when the proportion of D1 and D2 is 50% respectively, so D is defined as the weighted average among them:(5)D=D1+D22=121−E˜E+1−εαm+1exp−εαm,ε≥0.


## 3.2. Damage Evolution Equation

### 3.2.1. Equation Analysis of Damage VariableD1

As the continuous factors and equivalent strain are applied to the definition of damage variableD1, the following assumptions must be satisfied. Firstly, the rock material is composed of damage and lossless microelements. Secondly, the volume of them is equal, and the lossless microelements are able to irreversibly transform into damage microelements instantaneously. Finally, the damage only occurs along the axial direction but not the laterally, and the lossless microelements obey Hooke’s law.Combined with equations (1) and (3), the equation of damage variable can be expressed as:(6)D1=NS0MS0=NM.In formula (6), N is the total number of damaged microelements, M is the sum of the number of nondestructive microelements and damaged microelements, that is, the total number of microelements in the specimen, and S0 is the volume of a single element.Assuming that the growth process of biological species group is used to simulate the damage evolution process of rock, the survival environment of species group is considered as rock external load. Moreover, the time in the species group growth model corresponds to the strain of rock under the external load, and the environmental capacity of species group corresponds to the total number of rock microelements, so the growth rate of the number of damaged microelements can be expressed as follows:(7)dNdε=rN1−NM.In formula (7), γ and ε is natural growth rate and the relevant strain of rock, respectively.Then, formula (7) is solved by the separation of variables in differential equation, and expression (8) is obtained:(8)N=M1+Ce−rε,C=M−N0N0.In formula (8), N0 is the initial quantity of rock damage microelements.The differential formula (9) of damage variable D1 is deduced by formula (6) and formula (7):(9)dD1dε=1MdNdε=rNM1−NM.Substitute formula (8) into formula (7), and formula (10) of damage variable is solved out through the method of separating variables:(10)D1=11+eβ−rε,β=lnMN0−1.In formula (10), β represents the initial damage degree of rock material.


### 3.2.2. Equation Analysis of Damage VariableD2

According to the statistical damage theory, the distribution of microelement defects in rocks is independent and random, and it satisfies Possion distribution. Therefore, the probability of defects occurrence can be expressed by Possion law within the interval of lengthl, and the expression of probability function is shown in formula (11) as follows:(11)Pkl=λlkk!e−λl.In formula (11), λ is the mathematical expectation of microelement defects per unit length, k is the number of microelement defects, and P(k/l) is the probability function of k defects appearing in the interval with length l.Suppose that the probability of a cell body defect on the length of Δl is P1(Δl), and the probability of no element body defect on the length of l is P(l). Formulae (12) and (13) can be inferred as follows:(12)P1Δl=λΔl⋅e−λΔl,(13)Pl+Δl=Pl−PlPΔl.Formula (14) can be deduced from formulae (12) and (13):(14)Pl+Δl−PlΔl=−λPl⋅e−λΔl.Whenl = 0, P(l) = 1, and the value of Δl tends to zero, so the limit of both sides in formula (14) can be obtained:(15)Pl=e−λl.Therefore, the probability functionϕ(l) for the defect of no less than one microelement in the interval of length l can be derived:(16)ϕl=1−Pl.The probability density functionϕ(l) of microelement can be deduced from formula (16):(17)φl=dϕdl=λe−λl.Assuming that the damage microelement in the specimen loses its bearing capacity and rock strain is infinite when damaged, the expression of damage variableD2 are represented as follows:(18)D2=∫0εlφldl∫0∞lφldl=1−λε+1e−λε.While considering the defects of line, plane and body, the completeness was considered asα=1/λ [43]. Therefore, formula (18) can be rewritten as formula (19).(19)D2=1−εαm+1exp−εαm.In formula (19), ε is the strain of rock damage mass, and m represents the parameters related to materials and shapes of rock.


### 3.2.3. Definition of Damage VariableD

Based on the above mentioned equation analysis, the damage variableD of rock under high static stress unloaded and frequent disturbances can be defined as formula (20):(20)D=121+11+eβ−rε−εαm+1exp−εαm,β=lnMN0−1,ε≥0.


### 3.2.4. Determination of Damage Evolution Equation Parameters

If the damage variableD is to be analyzed and calculated, the parameters of β, r, m and α must be determined firstly. To determine the values of β and r, the equation of D1 needs to be analyzed, while the parameters of m and α can be determined by analyzing the equation of D2.(1) Determination of Parameters β and r. Firstly, the value of damage variable D1 corresponding to each strain of the envelope is calculated by formula (3), then E (the elastic modulus of non-destructive materials) and E˜ (the elastic modulus of damaged materials) in formula (3) are calculated by Figure 6.Figure 6
The schematic diagram ofE and E˜ with the determining method.Formula (10) is deduced, and formula (21) is ordered:(21)Y=ln1D1−1=β−rε.Calculate theY value corresponding to each strain in the dynamic stress-strain envelope and then analyze the fitting linear formula (Y-ε linear formula). Finally, by fitting the linear formula (21) with the logistic model, and values of β and r are able to be determined.(2) Determination of Parameters m and α. Statistical damage theory holds that α (rock integrity) is a function of ε [42], and α is inversely proportional to ε under high static stress unloaded and frequent disturbances as follows:(22)α=Cε,Cis a proportionality constant.Assuming that the equationε/C = 1/α1 holds, where α1 is a parameter related to rock properties and shapes, and substitute it into formula (22), it is found that the values of α and α1 is equal. Thus, formula (19) can be rewritten to formula (23) as follows:(23)D2=1−εα1m+1exp−εα1m.It is supposed that the damaged microelement loses its bearing capacity and the non-damaged microelement meets Hooke’s law, therefore, the relationship between rock stress and strain under high static stress unloaded and frequent disturbances is shown in formula (24) as follows:(24)σ=Eε1−D2.Substitute formula (23) into formula (24), and formula (25) is given as follows:(25)σ=Eεεα1m+1exp−εα1m.Furthermore, ifσ∗=σ/Eα1 and ε∗=ε/α1 are founded, formula (25) can be transformed into dimensionless form, such as formula (26):(26)σ∗=ε∗ε∗m+1exp−ε∗m.According to formula (26), the dynamic stress-strain theoretical curves of rock with different m values are drawn in Figure 7.Figure 7
The schematic diagram of dynamic stress-strain curves with one-dimensional dimensionless.By analyzing the similarity of the curves in Figure7 and the envelope of the measured dynamic stress-strain curves, the value of m can be estimated. After the values of m is determined, the maximum value of σ∗ is obtained by formula (26), and the parameter α is calculated by formula (27), which is deduced by Tang [42].(27)α=α1=PEσ∗max.In formula (27), P is the peak stress in the envelope of dynamic stress-strain curves.


## 3.2.1. Equation Analysis of Damage VariableD1

As the continuous factors and equivalent strain are applied to the definition of damage variableD1, the following assumptions must be satisfied. Firstly, the rock material is composed of damage and lossless microelements. Secondly, the volume of them is equal, and the lossless microelements are able to irreversibly transform into damage microelements instantaneously. Finally, the damage only occurs along the axial direction but not the laterally, and the lossless microelements obey Hooke’s law.Combined with equations (1) and (3), the equation of damage variable can be expressed as:(6)D1=NS0MS0=NM.In formula (6), N is the total number of damaged microelements, M is the sum of the number of nondestructive microelements and damaged microelements, that is, the total number of microelements in the specimen, and S0 is the volume of a single element.Assuming that the growth process of biological species group is used to simulate the damage evolution process of rock, the survival environment of species group is considered as rock external load. Moreover, the time in the species group growth model corresponds to the strain of rock under the external load, and the environmental capacity of species group corresponds to the total number of rock microelements, so the growth rate of the number of damaged microelements can be expressed as follows:(7)dNdε=rN1−NM.In formula (7), γ and ε is natural growth rate and the relevant strain of rock, respectively.Then, formula (7) is solved by the separation of variables in differential equation, and expression (8) is obtained:(8)N=M1+Ce−rε,C=M−N0N0.In formula (8), N0 is the initial quantity of rock damage microelements.The differential formula (9) of damage variable D1 is deduced by formula (6) and formula (7):(9)dD1dε=1MdNdε=rNM1−NM.Substitute formula (8) into formula (7), and formula (10) of damage variable is solved out through the method of separating variables:(10)D1=11+eβ−rε,β=lnMN0−1.In formula (10), β represents the initial damage degree of rock material.


## 3.2.2. Equation Analysis of Damage VariableD2

According to the statistical damage theory, the distribution of microelement defects in rocks is independent and random, and it satisfies Possion distribution. Therefore, the probability of defects occurrence can be expressed by Possion law within the interval of lengthl, and the expression of probability function is shown in formula (11) as follows:(11)Pkl=λlkk!e−λl.In formula (11), λ is the mathematical expectation of microelement defects per unit length, k is the number of microelement defects, and P(k/l) is the probability function of k defects appearing in the interval with length l.Suppose that the probability of a cell body defect on the length of Δl is P1(Δl), and the probability of no element body defect on the length of l is P(l). Formulae (12) and (13) can be inferred as follows:(12)P1Δl=λΔl⋅e−λΔl,(13)Pl+Δl=Pl−PlPΔl.Formula (14) can be deduced from formulae (12) and (13):(14)Pl+Δl−PlΔl=−λPl⋅e−λΔl.Whenl = 0, P(l) = 1, and the value of Δl tends to zero, so the limit of both sides in formula (14) can be obtained:(15)Pl=e−λl.Therefore, the probability functionϕ(l) for the defect of no less than one microelement in the interval of length l can be derived:(16)ϕl=1−Pl.The probability density functionϕ(l) of microelement can be deduced from formula (16):(17)φl=dϕdl=λe−λl.Assuming that the damage microelement in the specimen loses its bearing capacity and rock strain is infinite when damaged, the expression of damage variableD2 are represented as follows:(18)D2=∫0εlφldl∫0∞lφldl=1−λε+1e−λε.While considering the defects of line, plane and body, the completeness was considered asα=1/λ [43]. Therefore, formula (18) can be rewritten as formula (19).(19)D2=1−εαm+1exp−εαm.In formula (19), ε is the strain of rock damage mass, and m represents the parameters related to materials and shapes of rock.


## 3.2.3. Definition of Damage VariableD

Based on the above mentioned equation analysis, the damage variableD of rock under high static stress unloaded and frequent disturbances can be defined as formula (20):(20)D=121+11+eβ−rε−εαm+1exp−εαm,β=lnMN0−1,ε≥0.


## 3.2.4. Determination of Damage Evolution Equation Parameters

If the damage variableD is to be analyzed and calculated, the parameters of β, r, m and α must be determined firstly. To determine the values of β and r, the equation of D1 needs to be analyzed, while the parameters of m and α can be determined by analyzing the equation of D2.(1) Determination of Parameters β and r. Firstly, the value of damage variable D1 corresponding to each strain of the envelope is calculated by formula (3), then E (the elastic modulus of non-destructive materials) and E˜ (the elastic modulus of damaged materials) in formula (3) are calculated by Figure 6.Figure 6
The schematic diagram ofE and E˜ with the determining method.Formula (10) is deduced, and formula (21) is ordered:(21)Y=ln1D1−1=β−rε.Calculate theY value corresponding to each strain in the dynamic stress-strain envelope and then analyze the fitting linear formula (Y-ε linear formula). Finally, by fitting the linear formula (21) with the logistic model, and values of β and r are able to be determined.(2) Determination of Parameters m and α. Statistical damage theory holds that α (rock integrity) is a function of ε [42], and α is inversely proportional to ε under high static stress unloaded and frequent disturbances as follows:(22)α=Cε,Cis a proportionality constant.Assuming that the equationε/C = 1/α1 holds, where α1 is a parameter related to rock properties and shapes, and substitute it into formula (22), it is found that the values of α and α1 is equal. Thus, formula (19) can be rewritten to formula (23) as follows:(23)D2=1−εα1m+1exp−εα1m.It is supposed that the damaged microelement loses its bearing capacity and the non-damaged microelement meets Hooke’s law, therefore, the relationship between rock stress and strain under high static stress unloaded and frequent disturbances is shown in formula (24) as follows:(24)σ=Eε1−D2.Substitute formula (23) into formula (24), and formula (25) is given as follows:(25)σ=Eεεα1m+1exp−εα1m.Furthermore, ifσ∗=σ/Eα1 and ε∗=ε/α1 are founded, formula (25) can be transformed into dimensionless form, such as formula (26):(26)σ∗=ε∗ε∗m+1exp−ε∗m.According to formula (26), the dynamic stress-strain theoretical curves of rock with different m values are drawn in Figure 7.Figure 7
The schematic diagram of dynamic stress-strain curves with one-dimensional dimensionless.By analyzing the similarity of the curves in Figure7 and the envelope of the measured dynamic stress-strain curves, the value of m can be estimated. After the values of m is determined, the maximum value of σ∗ is obtained by formula (26), and the parameter α is calculated by formula (27), which is deduced by Tang [42].(27)α=α1=PEσ∗max.In formula (27), P is the peak stress in the envelope of dynamic stress-strain curves.


## 3.3. Damage Evolution Law

To verify the definition of damage variableD in formula (5), the values of β, r, m and α are substituted into this formula, so the corresponding dynamic damage variable D is calculated by combining the corresponding strain in the envelope of dynamic stress-strain curves. By comparing the damage degree in specimens and analyzing the characteristics of dynamic variable-strain theory curve, it is conclude that the defined damage variable is reasonable. Figure 8 lists the theoretical relationship of two typical damage variable-strain curves, and Figure 9 shows the damage states of the corresponding two groups of rocks during the impact process.Figure 8
The damage variable-strain curves of rock under high static stress unloaded and frequent disturbances. (a) The axial pressure is 85 MPa. (b) The unloading rate is 0.5 MPa/s.
(a)(b)Figure 9
The damage states of rocks during the impact process. (a) The unloading rate was 0.5 MPa/s and it is the seventh impact. (b) The preloaded axial pressure is 85 MPa and it is the 10th impact.
(a)(b)It can be concluded from Figure8 that(1)
When the value of dynamic strain is 0, the value of damage variableD is greater than 0, so the defined damage reflect the initial damage degree of rock under high static stress.(2)
There exist four stages of fitted damage variable-strain curve, that is moderate increase, fast increase, slowly increase and tend to be stable. The whole curve presents S-shape, and the value of damage variableD is between 0 and 1. It can effectively correspond to four development stages of rock in the test, which is steady development of micro cracks, rapid propagation of micro cracks, fatigue damage and fatigue failure.(3)
When the unloading rate is constant, the greater the axial pressure is, the faster the value of damage variableD tends to be 1, reflecting that pre-high axial pressure accelerates the process of rock damage and failure.(4)
When the pre-loading axial pressure is constant, the smaller the unloading rate is, the faster the damage variableD tends to be 1, which shows that high unloading rate is beneficial to improving the rock’ ability to withstand frequent disturbances.As shown in Figure9(a), when the unloading rate and impact times maintain constant, the greater the pre-added axial pressure is, the more serious the rock damage is. When the pre-added axial pressure and impact times keep invariant, the smaller the unloading rate of axial pressure is during the impact process, the more serious the rock damage will be, as shown in Figure 9(b). The above mentioned two phenomena indicate that preadded high axial pressure weakens the ability of rock to resist external impact load, and high unloading rate enhances the ability of rock to withstand frequent disturbances, proving the rationality of defined damage variable.


## 4. Constitutive Model

### 4.1. Fundamental Assumption

Based on analysis results of the envelope of rock’s dynamic stress-strain curve in this test, the damage evolution law during the impact process is combined to establish the constitutive model under high static stress unloaded and frequent disturbances. However, the establishment of the constitutive model needs to meet certain assumptions, as follows:(1)
The constitutive relationship is not affected by inertia effect at constant train rate [44].(2)
Because the rock element is characterized by elastic-viscous and statistical damage, it is designed as a Maxwell body (composed by an elastic element and a viscous element) which is in parallel with damage bodyDa1 firstly, and then in series with damage body Da2. The mechanical model of rock unit assemblage is shown in Figure 10.(3)
According to the stress state, the constitutive relation ofσ and ε is expressed as follows [45]:(28)σ=Eε1−Dε≥0.In formula (28), σ and ε are the stress and strain of damage body, respectively.(4)
Because of non-destructive properties, the constitutive relation of viscous element is expressed as follows [46]:(29)σ2=ηdε2dt,In formula (29), σ2 and ε2 are the stress and strain of viscous component respectively, η is viscosity coefficient, and t is the time corresponding to strain.(5)
The stress-strain relationship of rock element before damage conforms to the linear differential equation, and it is approximately considered that the principle of strain superposition is still valid [47].(6)
Though the viscoelasticity constitutive equation and the principle of strain equivalence, the influence of damaged rock mass, under high static stress unloaded and frequent disturbances, on constitutive relationship can be predicted [48].Figure 10
The mechanical model of rock unit assemblage.


### 4.2. Establishment of Constitutive Model

In the mechanical model of rock element, the relationship between stress and strain of the viscoelasticity body and damage body is as follows:(30)σ=σa1+σ2=σa1+σ1=σa2,ε=εa1+εa2,εa1=ε1+ε2,σ1=σ2.In formula (30), σ and ε are the stress, strain of the combined model respectively, σa1 and εa1 are the stress and strain of damage body Da1 respectively, σa2 and εa2 are the stress and strain of damage body Da2 respectively, σ1 and ε1 are the stress and strain of elastic component respectively, σ2 and ε2 are the stress and strain of viscous component respectively.The constitutive equations of Hooke body, viscous body and damaged body are substituted into formula (30), so the constitutive equation of mechanical model of rock assemblage is deduced:(31)ηEa21−D+Ea11−D+E1σ˙+E1Ea21−D+E1Ea11−Dσ=ηE1Ea21−D+Ea1Ea21−D2ε˙+E1Ea1Ea21−D2ε.In formula (31), E1, Ea1 and Ea2 are elasticity modulus of elastic element, damage body Da1 and damage body Da2 respectively.The elastic modulusE before damage should be replaced by efficient elastic modulus E(1 − D) due to the principle of strain equivalence [49], but the initial damage characteristics are not considered firstly for the better solution of formula (31), that is, the elastic modulus Ea1 and Ea2 are adopted to replace Ea1(1 − D) and Ea2(1 − D) in formula (31), respectively. Therefore, the constitutive equation of combined model is obtained:(32)ηEa2+Ea1+E1σ˙+E1Ea2+E1Ea1σ=ηE1Ea2+Ea1Ea2ε˙+E1Ea1Ea2ε.Because the specimen is subjected to high static pressure, ift = 0, the initial conditions ε(0) = ε0 and σ(0) = F are established. Therefore, formula (33) is obtained by Laplace transformation:(33)σt+t0=Ea2E1+Ea1E1+Ea1+Ea2εt+t0−E12Ea22ηE1+Ea1+Ea22∫0tετ+t0e−E1Ea1+E1Ea2/ηE1+Ea1+Ea2t+t0−τdτ.In formula (33), t0 is the loading time of static pressure, and when t0 = 0, both ε(t0 = 0) = 0 and σ(t0 = 0) = 0 are true.Ifε(t + t0) = ε0 + εr(t0) = ε0 + ct, where c is constant strain rate and invariable, formula (33) is able to transform into formula (34):(34)σt+t0=Ea2E1+Ea1E1+Ea1+Ea2ε0+εrt−ε0E1Ea22Ea1+Ea2E1+Ea1+Ea2eAt0−eAεrt/c+t0+ηcEa22Ea1+Ea22eAt0−eAεrt/c+t0−E1Ea1+E1Ea2εrtηcE1+Ea1+Ea2eAt0,where:(35)A=−E1Ea1+Ea2ηE1+Ea1+Ea2.In formula (34), εr(t) is the strain under the interaction of static stress and impact disturbances, ε0 is the strain produced by static stress F.Due to the principle of strain equivalence [43], Ea1 and Ea2 in formula (34) are replaced by Ea1(1 − D) and Ea2(1 − D), and the constitutive equation is calculated as follows:(36)σt+t0=Ea21−DE1+Ea11−DE1+Ea11−D+Ea21−Dε0+εrt−ε0E1Ea221−D21−DEa1+Ea2E1+Ea11−D+Ea21−Dekt0−ekεrt/c+t0+ηcEa221−D21−D2Ea1+Ea22ekt0−ekεrt/c+t0−1−DE1Ea1+E1Ea2εrtηcE1+Ea11−D+Ea21−Dekt0,where:(37)D=Dt+t0=1−ε0+εrtαm+1exp−ε0+εrtαm,k=−E11−DEa1+Ea2ηE1+Ea11−D+Ea21−D.


### 4.3. Verification of Test Results

For calculating the constitutive equation of rock under high static stress unloaded and frequent disturbances, the values ofE1, Ea1, Ea2, m, α, η, β, and r need to be determined by analyzing the measured data and carrying out the trial calculation. In the constitutive equation, the strain εr(t), the measured value of static loading time t0 and the constant strain rate c are test data, where εr(t) is the strain corresponding to the envelope of dynamic stress-strain curve, and t0 is the average time consumed by pre-added static load before multiple impacts.According to the test data and the constitutive equation of rock, the following results are obtained: the value ofE1 is similar to deformation modulus of the initial stage of stress-strain curve under first impact in the experimental process, so the former can be substituted by the latter. The value of Ea1 is able to express by the dynamic deformation modulus Ed, and Ed defined as the weighted average of secant modulus, secant modulus of type II and deformation modulus of loading section in the envelope, which is used to reflect the compressive deformation characteristics in the loading stage of rock. Moreover, there is a proportional relationship between Ea2 and Ed, that is h = Ea2/Ed, where h is dynamic expansion factor. In the test, the action time of impact load is short and the combined dynamic and static resultant force of rock increases rapidly, and the corresponding strain decreases or remains unchanged, which expresses a resilient phenomenon on dynamic stress-strain curve, therefore, the calculated instantaneous dynamic deformation modulus suddenly changes. As a consequence, the ratio h (h ≥ 1) of transient dynamic deformation modulus Ea2 and dynamic deformation modulus Ed after mutation is defined as dynamic expansion coefficient, and the value of h can be calculated by the measured data. The four parameters of β, r, m and α are the relative parameters in the damage evolution equation, he viscous coefficient η is deduced according to the fourth basic assumption, and its value range is 500∼1000 GPa·s generally.By analyzing the test data, two groups of typical envelope data of dynamic stress-strain curves are selected for trial calculation, as shown in Table3.Table 3
The experimental parameters and theoretical fitting parameters.
Unloading rate (MPa·s−1)F (MPa)E1 (GPa)Ea1 (GPa)Ea2 (GPa)h (GPa·s)mα (10−3)c (10−3 s−1)βrt0 (s)Dynamic expansion coefficient (h)0.565184.66135.11358.048001.210.9120.053.0522161302.650.575208.03165.89432.968002.080.8323.392.8721221502.610.585148.21124.60505.888001.481.0720.671.1815151704.060.595217.70150.52514.778000.820.5019.851.6718651903.42185144.50125.13472.978001.541.1122.541.8718351703.781.585151.56119.56280.988001.381.0521.672.1016571702.35285225.99152.87502.968001.230.6818.981.1312211703.29The parameters in Table3 are substituted into rock constitutive (36), and the corresponding theoretical dynamic stress-strain curve is fitted and compared with the experimental envelope for analysis, as shown in Figures 11 and 12.Figure 11
The comparison diagram of theoretical and experimental envelope curves of stress-strain under the axial unloading rate of 0.5 MPa/s. (a) Preload axial pressure 65 MPa. (b) Preload axial pressure 75 MPa. (c) Preload axial pressure 85 MPa. (d) Preload axial pressure 95 MPa.
(a)(b)(c)(d)Figure 12
The comparison diagram of theoretical and experimental envelope curves of stress-strain under the pre-added axial compression of 85 MPa (a) axial pressure unloading rate is 0.5 MPa/s. (b) axial pressure unloading rate is 1.0 MPa/s. (c) axial pressure unloading rate is 1.5 MPa/s, and (d) axial pressure unloading rate is 2.0 MPa/s.
(a)(b)(c)(d)As shown in Figures11 and 12, when the axial unloading rate or pre-added axial compression remain unchanged, there is a good consistency between theoretical dynamic stress-strain curve and experimental dynamic stress-strain envelope curve. Therefore, the constitutive equation accurately predicts the relationship between dynamic stress and strain of rock in the whole process under high static stress unloaded and frequent disturbances, and indirectly reflects the dynamic deformation characteristics of rock. Moreover, the yield and maximum stress of rock are able to predict effectively by the constitutive equation, as well as the rock deformation corresponding to yield stress, and all these provide theoretical basis for disaster prevention of deep rock excavation engineering.


## 4.1. Fundamental Assumption

Based on analysis results of the envelope of rock’s dynamic stress-strain curve in this test, the damage evolution law during the impact process is combined to establish the constitutive model under high static stress unloaded and frequent disturbances. However, the establishment of the constitutive model needs to meet certain assumptions, as follows:(1)
The constitutive relationship is not affected by inertia effect at constant train rate [44].(2)
Because the rock element is characterized by elastic-viscous and statistical damage, it is designed as a Maxwell body (composed by an elastic element and a viscous element) which is in parallel with damage bodyDa1 firstly, and then in series with damage body Da2. The mechanical model of rock unit assemblage is shown in Figure 10.(3)
According to the stress state, the constitutive relation ofσ and ε is expressed as follows [45]:(28)σ=Eε1−Dε≥0.In formula (28), σ and ε are the stress and strain of damage body, respectively.(4)
Because of non-destructive properties, the constitutive relation of viscous element is expressed as follows [46]:(29)σ2=ηdε2dt,In formula (29), σ2 and ε2 are the stress and strain of viscous component respectively, η is viscosity coefficient, and t is the time corresponding to strain.(5)
The stress-strain relationship of rock element before damage conforms to the linear differential equation, and it is approximately considered that the principle of strain superposition is still valid [47].(6)
Though the viscoelasticity constitutive equation and the principle of strain equivalence, the influence of damaged rock mass, under high static stress unloaded and frequent disturbances, on constitutive relationship can be predicted [48].Figure 10
The mechanical model of rock unit assemblage.


## 4.2. Establishment of Constitutive Model

In the mechanical model of rock element, the relationship between stress and strain of the viscoelasticity body and damage body is as follows:(30)σ=σa1+σ2=σa1+σ1=σa2,ε=εa1+εa2,εa1=ε1+ε2,σ1=σ2.In formula (30), σ and ε are the stress, strain of the combined model respectively, σa1 and εa1 are the stress and strain of damage body Da1 respectively, σa2 and εa2 are the stress and strain of damage body Da2 respectively, σ1 and ε1 are the stress and strain of elastic component respectively, σ2 and ε2 are the stress and strain of viscous component respectively.The constitutive equations of Hooke body, viscous body and damaged body are substituted into formula (30), so the constitutive equation of mechanical model of rock assemblage is deduced:(31)ηEa21−D+Ea11−D+E1σ˙+E1Ea21−D+E1Ea11−Dσ=ηE1Ea21−D+Ea1Ea21−D2ε˙+E1Ea1Ea21−D2ε.In formula (31), E1, Ea1 and Ea2 are elasticity modulus of elastic element, damage body Da1 and damage body Da2 respectively.The elastic modulusE before damage should be replaced by efficient elastic modulus E(1 − D) due to the principle of strain equivalence [49], but the initial damage characteristics are not considered firstly for the better solution of formula (31), that is, the elastic modulus Ea1 and Ea2 are adopted to replace Ea1(1 − D) and Ea2(1 − D) in formula (31), respectively. Therefore, the constitutive equation of combined model is obtained:(32)ηEa2+Ea1+E1σ˙+E1Ea2+E1Ea1σ=ηE1Ea2+Ea1Ea2ε˙+E1Ea1Ea2ε.Because the specimen is subjected to high static pressure, ift = 0, the initial conditions ε(0) = ε0 and σ(0) = F are established. Therefore, formula (33) is obtained by Laplace transformation:(33)σt+t0=Ea2E1+Ea1E1+Ea1+Ea2εt+t0−E12Ea22ηE1+Ea1+Ea22∫0tετ+t0e−E1Ea1+E1Ea2/ηE1+Ea1+Ea2t+t0−τdτ.In formula (33), t0 is the loading time of static pressure, and when t0 = 0, both ε(t0 = 0) = 0 and σ(t0 = 0) = 0 are true.Ifε(t + t0) = ε0 + εr(t0) = ε0 + ct, where c is constant strain rate and invariable, formula (33) is able to transform into formula (34):(34)σt+t0=Ea2E1+Ea1E1+Ea1+Ea2ε0+εrt−ε0E1Ea22Ea1+Ea2E1+Ea1+Ea2eAt0−eAεrt/c+t0+ηcEa22Ea1+Ea22eAt0−eAεrt/c+t0−E1Ea1+E1Ea2εrtηcE1+Ea1+Ea2eAt0,where:(35)A=−E1Ea1+Ea2ηE1+Ea1+Ea2.In formula (34), εr(t) is the strain under the interaction of static stress and impact disturbances, ε0 is the strain produced by static stress F.Due to the principle of strain equivalence [43], Ea1 and Ea2 in formula (34) are replaced by Ea1(1 − D) and Ea2(1 − D), and the constitutive equation is calculated as follows:(36)σt+t0=Ea21−DE1+Ea11−DE1+Ea11−D+Ea21−Dε0+εrt−ε0E1Ea221−D21−DEa1+Ea2E1+Ea11−D+Ea21−Dekt0−ekεrt/c+t0+ηcEa221−D21−D2Ea1+Ea22ekt0−ekεrt/c+t0−1−DE1Ea1+E1Ea2εrtηcE1+Ea11−D+Ea21−Dekt0,where:(37)D=Dt+t0=1−ε0+εrtαm+1exp−ε0+εrtαm,k=−E11−DEa1+Ea2ηE1+Ea11−D+Ea21−D.


## 4.3. Verification of Test Results

For calculating the constitutive equation of rock under high static stress unloaded and frequent disturbances, the values ofE1, Ea1, Ea2, m, α, η, β, and r need to be determined by analyzing the measured data and carrying out the trial calculation. In the constitutive equation, the strain εr(t), the measured value of static loading time t0 and the constant strain rate c are test data, where εr(t) is the strain corresponding to the envelope of dynamic stress-strain curve, and t0 is the average time consumed by pre-added static load before multiple impacts.According to the test data and the constitutive equation of rock, the following results are obtained: the value ofE1 is similar to deformation modulus of the initial stage of stress-strain curve under first impact in the experimental process, so the former can be substituted by the latter. The value of Ea1 is able to express by the dynamic deformation modulus Ed, and Ed defined as the weighted average of secant modulus, secant modulus of type II and deformation modulus of loading section in the envelope, which is used to reflect the compressive deformation characteristics in the loading stage of rock. Moreover, there is a proportional relationship between Ea2 and Ed, that is h = Ea2/Ed, where h is dynamic expansion factor. In the test, the action time of impact load is short and the combined dynamic and static resultant force of rock increases rapidly, and the corresponding strain decreases or remains unchanged, which expresses a resilient phenomenon on dynamic stress-strain curve, therefore, the calculated instantaneous dynamic deformation modulus suddenly changes. As a consequence, the ratio h (h ≥ 1) of transient dynamic deformation modulus Ea2 and dynamic deformation modulus Ed after mutation is defined as dynamic expansion coefficient, and the value of h can be calculated by the measured data. The four parameters of β, r, m and α are the relative parameters in the damage evolution equation, he viscous coefficient η is deduced according to the fourth basic assumption, and its value range is 500∼1000 GPa·s generally.By analyzing the test data, two groups of typical envelope data of dynamic stress-strain curves are selected for trial calculation, as shown in Table3.Table 3
The experimental parameters and theoretical fitting parameters.
Unloading rate (MPa·s−1)F (MPa)E1 (GPa)Ea1 (GPa)Ea2 (GPa)h (GPa·s)mα (10−3)c (10−3 s−1)βrt0 (s)Dynamic expansion coefficient (h)0.565184.66135.11358.048001.210.9120.053.0522161302.650.575208.03165.89432.968002.080.8323.392.8721221502.610.585148.21124.60505.888001.481.0720.671.1815151704.060.595217.70150.52514.778000.820.5019.851.6718651903.42185144.50125.13472.978001.541.1122.541.8718351703.781.585151.56119.56280.988001.381.0521.672.1016571702.35285225.99152.87502.968001.230.6818.981.1312211703.29The parameters in Table3 are substituted into rock constitutive (36), and the corresponding theoretical dynamic stress-strain curve is fitted and compared with the experimental envelope for analysis, as shown in Figures 11 and 12.Figure 11
The comparison diagram of theoretical and experimental envelope curves of stress-strain under the axial unloading rate of 0.5 MPa/s. (a) Preload axial pressure 65 MPa. (b) Preload axial pressure 75 MPa. (c) Preload axial pressure 85 MPa. (d) Preload axial pressure 95 MPa.
(a)(b)(c)(d)Figure 12
The comparison diagram of theoretical and experimental envelope curves of stress-strain under the pre-added axial compression of 85 MPa (a) axial pressure unloading rate is 0.5 MPa/s. (b) axial pressure unloading rate is 1.0 MPa/s. (c) axial pressure unloading rate is 1.5 MPa/s, and (d) axial pressure unloading rate is 2.0 MPa/s.
(a)(b)(c)(d)As shown in Figures11 and 12, when the axial unloading rate or pre-added axial compression remain unchanged, there is a good consistency between theoretical dynamic stress-strain curve and experimental dynamic stress-strain envelope curve. Therefore, the constitutive equation accurately predicts the relationship between dynamic stress and strain of rock in the whole process under high static stress unloaded and frequent disturbances, and indirectly reflects the dynamic deformation characteristics of rock. Moreover, the yield and maximum stress of rock are able to predict effectively by the constitutive equation, as well as the rock deformation corresponding to yield stress, and all these provide theoretical basis for disaster prevention of deep rock excavation engineering.


## 5. Conclusion

Deep rock mass is in the mechanical environment of high static stress dropped load and frequent impact disturbance during excavation, and the study on damage characteristics and constitutive model of deep rock is carried out, and the main conclusions are as follows:(1)
The envelopes established by dynamic stress-strain curves can reflect the whole variation trend of rock under frequent disturbances in the process of unloading high static stress, as well as the stages of stable development, non-stable expansion, fatigue damage and fatigue failure of rock.(2)
Based on continuous factor, strain equivalence principle and statistical damage theory, the damage variables of rock under frequent disturbances in the process of unloading high static stress are defined, and the methods to determine the parameters of damage variables are deduced. Meanwhile, damage variables defined is proved to be reasonable by experimental data and failure history of rock.(3)
According to the change characteristics of the envelopes of dynamic stress-strain curves, the rock constitutive model under frequent disturbances in the process of unloading high static stress is established with certain assumptions, and the corresponding constitutive equation is deduced. Finally, the experimental curve and theoretical curve of envelope are compared, and it is found that they have good consistency.


### 5.1. Deficiencies and Suggestions

In the process of studying the damage characteristics and constitutive model of deep rock under frequent impact disturbances in the process of unloading high static stress, it is found that there are some shortcomings, such as unloading rate control is difficult, the lithology is simple, constitutive equations involve many parameters, and the research results have not been verified by deep rock engineering. Therefore, the following aspects need to be further improved or deepened in the future.(1)
Improve the unloading rate control equipment and reduce the unloading rate control difficulty. At the same time, the damage characteristics and constitutive model of deep rocks need to be further discussed by expanding rock types and increasing the influence of factors such as high temperature.(2)
In order to improve the application of constitutive equation, it is necessary to carry out further research on optimizing the relationship between various parameters and simplifying the expression of constitutive equation.(3)
Based on the specific mechanical characteristics of ore bodies and surrounding rocks in deep rock mass engineering, the quantitative relationship between damage characteristics and constitutive models of deep and deep rocks should be further explored.


## 5.1. Deficiencies and Suggestions

In the process of studying the damage characteristics and constitutive model of deep rock under frequent impact disturbances in the process of unloading high static stress, it is found that there are some shortcomings, such as unloading rate control is difficult, the lithology is simple, constitutive equations involve many parameters, and the research results have not been verified by deep rock engineering. Therefore, the following aspects need to be further improved or deepened in the future.(1)
Improve the unloading rate control equipment and reduce the unloading rate control difficulty. At the same time, the damage characteristics and constitutive model of deep rocks need to be further discussed by expanding rock types and increasing the influence of factors such as high temperature.(2)
In order to improve the application of constitutive equation, it is necessary to carry out further research on optimizing the relationship between various parameters and simplifying the expression of constitutive equation.(3)
Based on the specific mechanical characteristics of ore bodies and surrounding rocks in deep rock mass engineering, the quantitative relationship between damage characteristics and constitutive models of deep and deep rocks should be further explored.



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*Source: 2706091-2020-05-30.xml*