Experimental Investigation of the Static and Dynamic Compression Characteristics of Limestone Based on Its Initial Damage

In the current work a new equation for initial damage assessment of limestone based on plane strain theory is proposed. Detailed investigations of the static and dynamic characteristics of limestone with different initial damage degree, using longitudinal wave speed, and static-dynamic compression tests are performed. This study investigated the static and dynamic characteristics of limestone with different initial damage degree, using longitudinal wave speed, and static-dynamic compression tests. Experimental results show that the degree of initial damage decreases with increasing longitudinal wave speed, which reaches the minimum when the longitudinal wave speed is approximately 6000 m/s, and the smaller the longitudinal wave velocity, the greater the degree of initial damage. The static and dynamic compressive strengths of limestone increase with the longitudinal wave velocity and strain rate, but the elastic modulus and Poisson’s ratio do not change significantly. Finally, based on the experimental results, the definitions of damage threshold value and strain softening are proposed, which further verify the influence of strain rate and initial damage on rock compression characteristics. The present study sheds light on the importance of initial damage for the mechanical state of rock in underground engineering.


Introduction
Natural rock contains microcracks, micropores, and other defects. The initial stress release and redistribution during the construction of underground structures lead to the further development of original microcracks and micropores in rock. As a result, the strength and stability of surrounding rock are reduced [1,2]. Therefore, a good understanding of the mechanical characteristics of rock mass with different initial damage levels is essential for stability analysis of the space underground [3].
The surrounding rock masses of the underground engineering structures, such as tunnel, champers, deep foundations, and structures [4][5][6][7], will have a certain degree of rupture and damage due to the release of the in-situ stress [8,9]. Kachanov first defined the damage variable by using the ratio of the damaged area of the material section to the initial non-damaged area, opening the way for the development of damage mechanics [10][11][12]. Over the past decades, comprehensive studies on the measurement means of rock damage have been conducted using various laboratory tests. Many scholars have since chosen parameters that are related to the rock mechanics in order to characterize the initial damage, such as the number, spacing, direction of microcracks [13][14][15][16], crack volumetric where D is the longitudinal wave velocity of the rock, v is the initial wave velocity (m/s), and v' is the faulted condition wave velocity (m/s). The non-destructive state of rock under different geographical conditions has no uniform standard. Therefore, the non-destructive state of the research rock should be analyzed according to the formation conditions; in order to better understand the non-Appl. Sci. 2021, 11, 7643 3 of 23 destructive state, a quantitative definition of non-destructive state wave speed determined from the drilling condition of rock is necessary. The non-destructive initial wave velocity is defined as the maximum value of the test results in relation to the statistical analysis and is given by Zhang [46]. The statistical formula for the longitudinal wave velocity is applied here and is defined as: where v i 0 is the longitudinal wave velocity of the rock in the i times experiment (m/s), and V 0 is the longitudinal wave velocity (m/s).
Based on the initial damage formula proposed in the above analysis, the rock sample's longitudinal wave velocity test is conducted. Moreover, the mechanical characteristics of limestone with different initial degrees of damage under the static and dynamic compression test are tested.

Tested Materials
The rock samples were taken from the limestone of the tiger ave formation in the Devonian system, which comes from five sampling depth locations. After drilling the core, the limestone core was cut and polished using a core wire cutting device. According to the Geotechnical Test Code, the rock samples' machining accuracies are as follows: the allowable deviation of the unevenness of two ends is ±0.05 mm, the allowable deviation of the height diameter is ±0.3 mm, and the allowable deviation of the vertical axis of the end face is 0.25. The rock samples of the same drilling depth were made into the abovementioned specifications, and the standard cylinder specimens with a height*diameter of 100 mm × 50 mm are made and labeled. The main mineral compositions and physical properties of the limestone are shown in Tables 1 and 2.  Figure 1 shows the non-metallic ultrasonic instrument RS-ST02C. The experimental set-up includes a waveform generator, two piezoelectric transducers with a maximum resonant frequency of 1 MHz mounted on the sample holder, and a USB interface connected to a computer. To ensure the conduction of the longitudinal wave ability of the coupling medium, vaseline was selected as the coupling agent, which is applied to the top and bottom of the limestone specimen. After installing the connection, checks were performed to ensure the sensitivity of the P-wave velocity measurement. Five sampling depths A, B, C, D, E were selected; the measurement results of longitudinal wave velocities of 12 standard samples at each depth are shown in Table 3. The initial damage degrees of the longitudinal wave velocity for five different drilling locations are shown in Figure 2. Figure 2 shows that:

Rock Ultrasonic Testing
1. The rock samples' longitudinal wave velocities are similar at the same drilling locations, and there is a small longitudinal wave velocity change of rock samples at each drilling position, except point D. 2. With an increasing drilling depth, the longitudinal wave velocity increases, and the maximum value is approximately 6000 m/s; the smaller the drilling depth, the greater the degree of damage disturbance, and the smaller the wave velocity. 3. The initial damage varies non-linearly with respect to the longitudinal wave velocity of the five drilling locations.
In order to quantify the extent of nonlinearity in the initial damage behavior, an exponential relationship is applied here and expressed as: where D is the initial damage value, and ν is the longitudinal wave velocity of the five drilling locations.  To ensure the conduction of the longitudinal wave ability of the coupling medium, vaseline was selected as the coupling agent, which is applied to the top and bottom of the limestone specimen. After installing the connection, checks were performed to ensure the sensitivity of the P-wave velocity measurement. Five sampling depths A, B, C, D, E were selected; the measurement results of longitudinal wave velocities of 12 standard samples at each depth are shown in Table 3. The initial damage degrees of the longitudinal wave velocity for five different drilling locations are shown in Figure 2. Figure 2 shows that: The rock samples' longitudinal wave velocities are similar at the same drilling locations, and there is a small longitudinal wave velocity change of rock samples at each drilling position, except point D.

2.
With an increasing drilling depth, the longitudinal wave velocity increases, and the maximum value is approximately 6000 m/s; the smaller the drilling depth, the greater the degree of damage disturbance, and the smaller the wave velocity.

3.
The initial damage varies non-linearly with respect to the longitudinal wave velocity of the five drilling locations.
In order to quantify the extent of nonlinearity in the initial damage behavior, an exponential relationship is applied here and expressed as: where D is the initial damage value, and ν is the longitudinal wave velocity of the five drilling locations.  Table 4. In order to carry out the compression test at high and low strain rates, four representative initial damage moduli were selected, which are 0.754, 0.625, 0.357, and 0.043.

Static Compression Test at Low Strain Rates
In this study, the experimental equipment is a TAW-2000 microcomputer controlled electro-hydraulic servo rock multi-function testing machine belonging to an all-digital control electro-hydraulic servo loading system. The overall appearance of the equipment is shown in Figure 3. Moreover, the instrument controls the loading confining pressure and strain rate. The axial strain of the specimen is monitored via an extensometer during loading. Both stress and strain are automatically recorded and saved by the data acquisition system.   Table 4. In order to carry out the compression test at high and low strain rates, four representative initial damage moduli were selected, which are 0.754, 0.625, 0.357, and 0.043. In this study, the experimental equipment is a TAW-2000 microcomputer controlled electro-hydraulic servo rock multi-function testing machine belonging to an all-digital control electro-hydraulic servo loading system. The overall appearance of the equipment is shown in Figure 3. Moreover, the instrument controls the loading confining pressure and strain rate. The axial strain of the specimen is monitored via an extensometer during loading. Both stress and strain are automatically recorded and saved by the data acquisition system. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 24 (a) (b)  Table 5 shows the experimental scheme of the static compression test. According to the limit value of the strain threshold for static and dynamic compression loading, the strain rate ε < 5.4 × 10 −4 s −1 belongs to the static loading [47]. In this study, the strain-controlled loading scheme is adopted with a strain rate of 2 × 10 −6 s −1 for all static compression tests. Moreover, the constant normal stresses of 5 Mpa, 10 Mpa, and 20 Mpa are applied to represent the confining pressures encountered in the underground rock excavation application for the triaxial compression tests.

Dynamic Compression Test of High Strain Rates
The dynamic compression instrument of limestone adopts the Hopkinson pressure bar (SHPB) device with variable sections. The measuring principle is shown in Figure 4. For this measuring system, the strain rate range is 1 − 10 3 s −1 , the dynamic impact load is 500 MPa, and both the diameters of the incident and transmission rod are 50 mm. The equal bar diameter specimen carries out the impact loading, and the special-shaped punch hits the input rod, resulting in a semi-sine stress wave type to achieve constant strain rate loading.  Table 5 shows the experimental scheme of the static compression test. According to the limit value of the strain threshold for static and dynamic compression loading, the strain rate ε < 5.4 × 10 −4 s −1 belongs to the static loading [47]. In this study, the strain-controlled loading scheme is adopted with a strain rate of 2 × 10 −6 s −1 for all static compression tests. Moreover, the constant normal stresses of 5 Mpa, 10 Mpa, and 20 Mpa are applied to represent the confining pressures encountered in the underground rock excavation application for the triaxial compression tests.

Dynamic Compression Test of High Strain Rates
The dynamic compression instrument of limestone adopts the Hopkinson pressure bar (SHPB) device with variable sections. The measuring principle is shown in Figure 4. For this measuring system, the strain rate range is 1-10 3 s −1 , the dynamic impact load is 500 MPa, and both the diameters of the incident and transmission rod are 50 mm. The equal bar diameter specimen carries out the impact loading, and the special-shaped punch hits the input rod, resulting in a semi-sine stress wave type to achieve constant strain rate loading.  Based on the one-dimensional stress wave theory and the stress uniformity hypothesis, the specimen's stress, strain, and strain rate during the impact process were calculated. The calculation principle is shown as [48]: where E is the elastic modulus of the bar; A and AS are the cross-sectional area of the bar and specimen, respectively; C0 is the vertical wave velocity; ls is the specimen length; εI, εR, and εT are the incident, reflection, and transmission strain of the samples, respectively; σs, εs and ' are the stress, strain, and strain rate of the specimen, respectively. The experimental scheme of the SHPB test of limestone is shown in Table 6.

Crack Conditions at the Failure Stage
The constant normal strain rates of 2 × 10 −6 s −1 , 2 × 10 −5 s −1 and 2 × 10 −4 s −1 are applied to represent typical static compression. Four groups of uniaxial compressions are performed as D = 0.043, 0.357, 0.625, and 0.754. The uniaxial compression is terminated once the axial strain remains constant. A comparison diagram before and after the uniaxial compression test of limestone is shown in Figure 5. Figure 5 shows the original damage crack propagation of the uniaxial compression test with D = 0.625. For viewing convenience, only two original damaged cracks were identified. It can be seen from Figure 5 that: 1. The transverse cracks expand to varying degrees for crack 1 and crack 2, the degree of circumferential crack propagation is large, and the fracture surface is formed when entering the yield failure stage. Based on the one-dimensional stress wave theory and the stress uniformity hypothesis, the specimen's stress, strain, and strain rate during the impact process were calculated. The calculation principle is shown as [48]: where E is the elastic modulus of the bar; A and A S are the cross-sectional area of the bar and specimen, respectively; C 0 is the vertical wave velocity; ls is the specimen length; ε I , ε R , and ε T are the incident, reflection, and transmission strain of the samples, respectively; σ s , ε s and ε s are the stress, strain, and strain rate of the specimen, respectively. The experimental scheme of the SHPB test of limestone is shown in Table 6.

Crack Conditions at the Failure Stage
The constant normal strain rates of 2 × 10 −6 s −1 , 2 × 10 −5 s −1 and 2 × 10 −4 s −1 are applied to represent typical static compression. Four groups of uniaxial compressions are performed as D = 0.043, 0.357, 0.625, and 0.754. The uniaxial compression is terminated once the axial strain remains constant. A comparison diagram before and after the uniaxial compression test of limestone is shown in Figure 5.
2. New macroscopic cracks in the middle of the specimen are produced, indicating that the internal microcrack aggregation in the initial compression phase eventually exhibits macroscopic cracks, as shown in crack 3. This behavior is similar to the uniaxial compression loading test reported by Fakhimi [49], in which a progressive increase in the initial damage occurs with continued uniaxial compression.

The Uniaxial Compression Test with Low Strain Rates
Uniaxial compression stress-strain curves in three low strain rates are also represented in Figure 6. It can be seen that: 1. A rapid decline after the maximum peak stress was observed, corresponding to the failure state. The sharp decline of limestone strength after the failure state is due to the macroscopic cracks' expansion and the strain softening of the residual strength.
2. An interesting phenomenon is that the stress growth curve shows several peak spikes after the line elastic phase. Moreover, for curves 1, 2, and 3, the compression phase curve is approximated, while the linear elastic phase curve grows sharply as the strain rate decreases slowly. Similar behavior was observed by Hemami [50] for different kinds of rock under compression loading and the phenomenon was given a name by Perez [51]. 3. The difference in the maximum stress is due to the load strain rate and the initial damage. For D = 0.043, the difference in the maximum uniaxial compression becomes more apparent than that of D = 0.357, 0.625, and 0.754. The value of the maximum compression stress in the load strain rate = 2 × 10 −4 s −1 is 176 Mpa, which is 11.4% and 15.0% higher than that in = 2 × 10 −5 s −1 and = 2 × 10 −6 s −1 , respectively. The stress strength of the limestone depends upon the degree of development of crack pores in the rock. A greater initial damage means the cracks inside the rock are more complicated, leading to a destruction in the compression test.  The transverse cracks expand to varying degrees for crack 1 and crack 2, the degree of circumferential crack propagation is large, and the fracture surface is formed when entering the yield failure stage.

2.
New macroscopic cracks in the middle of the specimen are produced, indicating that the internal microcrack aggregation in the initial compression phase eventually exhibits macroscopic cracks, as shown in crack 3. This behavior is similar to the uniaxial compression loading test reported by Fakhimi [49], in which a progressive increase in the initial damage occurs with continued uniaxial compression.

The Uniaxial Compression Test with Low Strain Rates
Uniaxial compression stress-strain curves in three low strain rates are also represented in Figure 6. It can be seen that:

1.
A rapid decline after the maximum peak stress was observed, corresponding to the failure state. The sharp decline of limestone strength after the failure state is due to the macroscopic cracks' expansion and the strain softening of the residual strength.

2.
An interesting phenomenon is that the stress growth curve shows several peak spikes after the line elastic phase. Moreover, for curves 1, 2, and 3, the compression phase curve is approximated, while the linear elastic phase curve grows sharply as the strain rate decreases slowly. Similar behavior was observed by Hemami [50] for different kinds of rock under compression loading and the phenomenon was given a name by Perez [51].

3.
The difference in the maximum stress is due to the load strain rate and the initial damage. For D = 0.043, the difference in the maximum uniaxial compression becomes more apparent than that of D = 0.357, 0.625, and 0.754. The value of the maximum compression stress in the load strain rate ε = 2 × 10 −4 s −1 is 176 Mpa, which is 11.4% and 15.0% higher than that in ε = 2 × 10 −5 s −1 and ε = 2 × 10 −6 s −1 , respectively. The stress strength of the limestone depends upon the degree of development of crack pores in the rock. A greater initial damage means the cracks inside the rock are more complicated, leading to a destruction in the compression test. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 24

The SHPB Test with High Strain Rates
The SHPB stress-strain curves at different strain rates are presented in Figure 7. It can be seen from Figure 7 that: 1. The maximum stress in the three strain rates gradually decreases with a decrease in vertical wave velocities; thus, the damage variable and maximum strain increase. Furthermore, the differences in the maximum stress are due to the load strain rate. For D = 0.043, the differences in the maximum uniaxial compression becomes more obvious than for D = 0.357, 0.625, and 0.754. 2. The value of the maximum compression stress in the load strain rate ε = 100 s −1 is 207 Mpa, which is 12.4%, and 15.6% higher than that in = 10 s −1 and 1 s −1 , respectively. The strain strength of the maximum stress becomes greater for D = 0.754, and the value of the maximum strain in the load strain rate = 100 s −1 is 0.0117, which is 11.4% and 37.6% higher than that in = 10 s −1 and 1 s −1 , respectively. The stress strength of limestone depends on the development of the microcracks and micropores in rock. The larger the initial damage degree is, the more complex the microcracks in the rock are, and the rock is more prone to failure under the same external load. 3. The dynamic destructive stress at a low strain rate is less than the static destructive stress, while higher strain rates are greater than the static destructive stress. The dynamic destructive stress of the rock is much higher than the static destructive stress. The cracks are activated at low stress levels. In addition, the expansion and polymerization lead to the fracture of rock before the stress level extends to the other cracks. Enter the yield stage

The SHPB Test with High Strain Rates
The SHPB stress-strain curves at different strain rates are presented in Figure 7. It can be seen from Figure 7 that: The maximum stress in the three strain rates gradually decreases with a decrease in vertical wave velocities; thus, the damage variable and maximum strain increase. Furthermore, the differences in the maximum stress are due to the load strain rate. For D = 0.043, the differences in the maximum uniaxial compression becomes more obvious than for D = 0.357, 0.625, and 0.754.

2.
The value of the maximum compression stress in the load strain rate ε = 100 s −1 is 207 Mpa, which is 12.4%, and 15.6% higher than that in ε = 10 s −1 and 1 s −1 , respectively. The strain strength of the maximum stress becomes greater for D = 0.754, and the value of the maximum strain in the load strain rate ε = 100 s −1 is 0.0117, which is 11.4% and 37.6% higher than that in ε = 10 s −1 and 1 s −1 , respectively. The stress strength of limestone depends on the development of the microcracks and micropores in rock. The larger the initial damage degree is, the more complex the microcracks in the rock are, and the rock is more prone to failure under the same external load. 3.
The dynamic destructive stress at a low strain rate is less than the static destructive stress, while higher strain rates are greater than the static destructive stress. The dynamic destructive stress of the rock is much higher than the static destructive stress. The cracks are activated at low stress levels. In addition, the expansion and polymerization lead to the fracture of rock before the stress level extends to the other cracks. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 24

Triaxial Compression Test
Constant normal stresses of 5 MPa, 10 MPa, and 20 MPa are applied to represent typical normal stresses encountered in underground engineering construction. The test results are shown in Table 7.

Triaxial Compression Test
Constant normal stresses of 5 MPa, 10 MPa, and 20 MPa are applied to represent typical normal stresses encountered in underground engineering construction. The test results are shown in Table 7.  Figure 8 shows the typical triaxial compression stress-strain curves of limestone with different σ 3 . It can be seen from Figure 8 that: 1.
The stress-strain curves can be divided into three stages: 1-the initial compression stage, 2-the line elasticity growth stage, and 3-the yield failure stage. The greater the surrounding pressure, the greater the peak stress and the strain corresponding to the peak stress.

2.
The principal stress increases with increasing confining pressure. For σ 3

Compressive Strength and the Yield Strain
The variations of compressive strength and the yield strain with the level of initial damage are shown in Figure 9. The horizontal coordinates are the logarithm of the strain rate. It can be seen that:

Compressive Strength and the Yield Strain
The variations of compressive strength and the yield strain with the level of initial damage are shown in Figure 9. The horizontal coordinates are the logarithm of the strain rate. It can be seen that: The interfacial compressive strength varies nonlinearly with the increasing of initial damage. In order to quantify the extent of nonlinearity in the compression behavior, a nonlinear relationship is applied and expressed as: where Rc is the compressive strength; εc is the compressive strain of strength; D is the initial damage; and a, b, and c are the quotieties.

Modulus of Elasticity
The relationship between modulus of elasticity and the degree of initial damage at different strain rates is shown in Figure 10. It can be seen that: The elastic modulus is between 21 and 23 Gpa, and the value of the maximum elastic modulus in D = 0.043 is 68 Gpa, which is 28.3%, 78.9%, and 195.7% higher than that in D = 0.0357, 0.625, and 0.754, respectively. Compared to the degree of damage, the strain rate has little effect on the elastic modulus. In order to quantify the extent of nonlinearity in limestone compression behavior, a nonlinear relationship is applied and expressed as: where E is the modulus of elasticity; D is the initial damage; and a, b, and c are the quotieties. The interfacial compressive strength varies nonlinearly with the increasing of initial damage. In order to quantify the extent of nonlinearity in the compression behavior, a nonlinear relationship is applied and expressed as: ε c = e (a+b * D+c * D 2 ) where R c is the compressive strength; ε c is the compressive strain of strength; D is the initial damage; and a, b, and c are the quotieties.

Modulus of Elasticity
The relationship between modulus of elasticity and the degree of initial damage at different strain rates is shown in Figure 10. It can be seen that:

Poisson's Ratio
The relationship between Poisson's ratio and the degree of initial damage at different strain rates is shown in Figure 11. It can be seen that: 1. The Poisson's ratio decreases nonlinearly with respect to strain rate increases. The elastic modulus is between 21 and 23 Gpa, and the value of the maximum elastic modulus in D = 0.043 is 68 Gpa, which is 28.3%, 78.9%, and 195.7% higher than that in D = 0.0357, 0.625, and 0.754, respectively. Compared to the degree of damage, the strain rate has little effect on the elastic modulus. In order to quantify the extent of nonlinearity in limestone compression behavior, a nonlinear relationship is applied and expressed as: where E is the modulus of elasticity; D is the initial damage; and a, b, and c are the quotieties.

Poisson's Ratio
The relationship between Poisson's ratio and the degree of initial damage at different strain rates is shown in Figure 11. It can be seen that: 1.
The Poisson's ratio decreases nonlinearly with respect to strain rate increases.

2.
The Poisson's ratio is between 0.26 and 0.3, and the value of the maximum elastic modulus in D = 0.043 is 68 Gpa, which is 28.3%, 78.9%, and 195.7% higher than that in D = 0.357, 0.625, and 0.754, respectively. In addition, compared with the degree of the initial damage, the strain rate has little effect on the elastic modulus. In order to quantify the extent of the initial damage in the limestone compression behavior, a nonlinear relationship is applied and expressed as: where µ is the Poisson's ratio; D is the initial damage; and a, b, and c are the quotieties.

Poisson's Ratio
The relationship between Poisson's ratio and the degree of initial damage at different strain rates is shown in Figure 11. It can be seen that: 1. The Poisson's ratio decreases nonlinearly with respect to strain rate increases. 2. The Poisson's ratio is between 0.26 and 0.3, and the value of the maximum elastic modulus in D = 0.043 is 68 Gpa, which is 28.3%, 78.9%, and 195.7% higher than that in D = 0.357, 0.625, and 0.754, respectively. In addition, compared with the degree of the initial damage, the strain rate has little effect on the elastic modulus. In order to quantify the extent of the initial damage in the limestone compression behavior, a nonlinear relationship is applied and expressed as: where μ is the Poisson's ratio; D is the initial damage; and a, b, and c are the quotieties.

Comparison of the Two Initial Damage Determination Methods: Cyclic Loading-Unloading and Longitudinal Wave Speed
Hamza [53] has taken the strain energy and elastic modulus in the loading and unloading stage as the entry point of the initial damage. The definition of the initial damage is a change law of the elastic modulus, and the expression is shown as: However, the definition of the initial damage does not consider the effects of plasticity during the loading-unloading process. The influence of the irreversible plastic strain and the total strain is added to the equation. Moreover, the irreversible plastic deformation of the elastic-plastic material in one-dimensional condition is considered. The expression of the new initial damage variable is shown as: where ε is the total strain; ε p is the irreversible plastic strain after unloading; E is the injury elastic modulus; and E is the initial elastic modulus. The stress-strain curves of four initial injuries under loading-unloading conditions are shown in Figure 12. It can be seen from Figure 12 that:

1.
The constant normal three, four, five, and six loading and unloading cycles were developed for four of the initial damaged rock samples. Herein, the difference between starting and ending strain values for the last loading is plastic strain ε p , and the stress peak point strain in the yield phase is the total strain ε which were circled in the drawings 1-4. The final parameters and initial damage values are shown in Table 8. The initial damage calculated by the loading and unloading method is greater than that represented by the longitudinal wave velocity. The reason for this phenomenon is the plastic strain after unloading and is the total value of the original accumulated damage during loading, leading to a greater damage than that of the initial damage in the compression process.

2.
The effects of two different initial damages on the compressive strength and destructive strain of limestone are illustrated. For ε = 2 × 10 −6 s −1 , the compressive strengths are 127. 43 Figure 13 shows the different definitions of two initial damage methods. It can be seen from Figure 13 that: 1. The difference of the corresponding compressive strengths of Dv and DE decreases as the degree of initial damage increases. When D enters the one initial damage D0, which is the initial damage boundary point, the two curves have an equal compres-   Figure 13 shows the different definitions of two initial damage methods. It can be seen from Figure 13 that: The difference of the corresponding compressive strengths of D v and D E decreases as the degree of initial damage increases. When D enters the one initial damage D 0 , which is the initial damage boundary point, the two curves have an equal compressive strength of R 0 . In addition, the compressive strength obtained by the longitudinal wave velocity is larger than that obtained by the loading-unloading test.

2.
The yield-stage strain values of the four initial damage values in Figure 13a obtained by the longitudinal wave velocity method are on average 2% larger than those obtained from the loading-unloading method.

3.
The initial damage obtained by the loading-unloading method is lower than that obtained by the longitudinal wave velocity method. Based on the data analysis, two engineering practice suggestions for limestone have been given: (1) When the initial damage is less than the D 0 , the rock compressive strength value obtained by the loading-unloading method is relatively conservative. The results of the rock stability analysis obtained from this can be better combined with the stability analysis of underground excavation structures. (2) Conversely, when the initial damage is greater than D 0 , the compressive strength obtained by the longitudinal wave velocity method can be applied to the engineering practice. Due to the rock's inelastic material, rock strain cannot fully recover after loading and unloading. Furthermore, the plastic strain and initial damage accumulate accordingly. Hence, the degree of initial damage is larger than that obtained by the longitudinal wave velocity method, DE > Dv. The compressive strength and destructive strain of DE and Dv are shown in Table 9. The compressive strength varies nonlinearly with respect to the initial damage, and an exponential relationship is applied and expressed as: where Rc is the compressive strength; εc is the compressive strain of strength; ε is the loading strain rate; D is the initial damage; and a, b, and c are the quotieties. Due to the rock's inelastic material, rock strain cannot fully recover after loading and unloading. Furthermore, the plastic strain and initial damage accumulate accordingly. Hence, the degree of initial damage is larger than that obtained by the longitudinal wave velocity method, D E > D v . The compressive strength and destructive strain of D E and D v are shown in Table 9. The compressive strength varies nonlinearly with respect to the initial damage, and an exponential relationship is applied and expressed as: where R c is the compressive strength; ε c is the compressive strain of strength; ε is the loading strain rate; D is the initial damage; and a, b, and c are the quotieties.

Initial Damage Threshold Value of Different Strain Rates
The damage threshold of a non-destructive rock is the minimum damage variable when macroform cracks appear in the rock. Figure 14 shows the analysis of the relationship between the damage variables and compressive strength. It can be seen from Figure 14 that: 1.
The peak compressive strength remains stable when the initial damage is minor. When the initial damage value reaches a particular value, the corresponding compressive strength begins to decrease dramatically, and the initial damage value reaches an inflection point, which is the initial damage threshold value, as shown by a red dotted line. The initial damage threshold values of the two loading strain rates (ε = 100 s −1 , 2 × 10 −4 s −1 ) are D 1 thr and D 2 thr , and the values of D 1 thr and D 2 thr are 0.2 and 0.18, respectively.

2.
When the damage variable meets 0 < D < D thr , the microcracks of the limestone expand stably, and the stress-strain curve is OB, which is shown in Figure 8. Furthermore, the compression peak stress point C is entered as the loading stress level increases. Moreover, when D > D thr , the damage deterioration of the rock enters the macroscopic crack formation stage, which belongs to an unstable damage development stage. 2. When the damage variable meets 0 < D < Dthr, the microcracks of the limestone expand stably, and the stress-strain curve is OB, which is shown in Figure 8. Furthermore, the compression peak stress point C is entered as the loading stress level increases. Moreover, when D > Dthr, the damage deterioration of the rock enters the macroscopic crack formation stage, which belongs to an unstable damage development stage.
In conclusion, when D < Dthr, the microcracks inside the rock determine the degree of initial damage, and its compressive strength is almost unchanged. However, for D > Dthr, the macroscopic form crack determines the initial degree of damage of the rock; thus, decreasing the extent of the rock's compressive strength and the initial degree of damage increases. Hence, the rock's main stage of damage deterioration is the unstable expansion stage of the macroform cracks.

Strain Softening Characteristics of Limestone
The strain softening property of rock is the decline rate of the curve after the peak stress point slows down with increasing initial damage. To quantitatively analyze the strain softening degree of rock after the peak point of rock with different degrees of initial damage, the strain softening modulus is proposed as: In conclusion, when D < D thr , the microcracks inside the rock determine the degree of initial damage, and its compressive strength is almost unchanged. However, for D > D thr , the macroscopic form crack determines the initial degree of damage of the rock; thus, decreasing the extent of the rock's compressive strength and the initial degree of damage increases. Hence, the rock's main stage of damage deterioration is the unstable expansion stage of the macroform cracks.

Strain Softening Characteristics of Limestone
The strain softening property of rock is the decline rate of the curve after the peak stress point slows down with increasing initial damage. To quantitatively analyze the strain softening degree of rock after the peak point of rock with different degrees of initial damage, the strain softening modulus is proposed as: where σ c and ε c are the stress and strain at the peak stress points, respectively; σ f and ε f are the stress and strain during a complete rupture of the rock, respectively; and η is the strain softening modulus. The strain softening modulus of four degrees of initial damage are shown in Table 10.  Figure 15 shows the strain softening modulus versus the initial damage for limestone. It can be seen from Figure 15 that:

1.
The slopes of the four initial damages are −151, −130.2, −105.5, and −85.2, respectively. The decline rate of the stress peak curve slows down with an increase in the degree of initial damage. The complete rupture point and strain softening degree of rock is different under different initial damages. The strain softening degree and the brittleness of failure strain increases with an increasing of the softening modulus.

2.
With the increase in the initial damage the strain softening modulus of rock decreases, and the degree of strain softening increases. The failure ductility of the rock after peak stress is more apparent. Conversely, the smaller the initial damage degree is, the more pronounced the rock's brittle failures are.

3.
For the rocks with four initial degrees of damage, the strain softening modulus is much larger than the corresponding initial damage elastic modulus, and the descending section of the stress-strain relationship curve is steeper than the ascending section. Therefore, the results obtained from the analysis of the strain softening properties reflect that the limestone in the experiment is a brittle failure rock. In order to quantify the extent of nonlinearity in the limestone compression behavior, a nonlinear relationship is applied and expressed as: where η is the compressive strength; D is the initial damage; and a, b, and c are the quotieties.
properties reflect that the limestone in the experiment is a brittle failure rock. In order to quantify the extent of nonlinearity in the limestone compression behavior, a nonlinear relationship is applied and expressed as: where η is the compressive strength; D is the initial damage; and a, b, and c are the quotieties.

Application of Blasting Parameters Considering Initial Damage in Main Building Construction of Xi Luodu Hydropower Station
Xi Luodu Hydropower Station is located in Xi Luodu Valley in the lower Jinsha River. The main powerhouse is excavated in layers, and the integrity of surrounding rock is relatively complete-complete. The pre-splitting blasting method was used in the construction of the hydropower station to reduce the influence of blasting excavation on the rock platform. Therefore, it was necessary to carry out an on-site drilling acoustic wave test before construction, determine rock mechanics parameters, and then determine blasting excavation parameters, and carry out blasting test. The field tests are shown in Figure 16.

Application of Blasting Parameters Considering Initial Damage in Main Building Construction of Xi Luodu Hydropower Station
Xi Luodu Hydropower Station is located in Xi Luodu Valley in the lower Jinsha River. The main powerhouse is excavated in layers, and the integrity of surrounding rock is relatively complete-complete. The pre-splitting blasting method was used in the construction of the hydropower station to reduce the influence of blasting excavation on the rock platform. Therefore, it was necessary to carry out an on-site drilling acoustic wave test before construction, determine rock mechanics parameters, and then determine blasting excavation parameters, and carry out blasting test. The field tests are shown in Figure 16.

Field Measurements
The rock parameters obtained from the P-wave velocity test and rock mechanics test are shown in Table 11. The structure of the single-hole charge is shown in Figure 17. The field chart is shown in Figures 18 and 19.

Field Measurements
The rock parameters obtained from the P-wave velocity test and rock mechanics test are shown in Table 11. The structure of the single-hole charge is shown in Figure 17. The field chart is shown in Figures 18 and 19. The structure of the single-hole charge is shown in Figure 17. The field chart is shown in Figures 18 and 19.   The structure of the single-hole charge is shown in Figure 17. The field chart is shown in Figures 18 and 19.

Analysis of Single-Hole Blasting Damage Test Results
1. The P-wave velocity at test points before and after presplitting blasting is shown in Figure 20. It can be seen that, with the increase in borehole depth, the longitudinal wave velocity increases nonlinearly, which is consistent with the above conclusion. 2. The influence radius curve of single-hole blasting under different borehole longitudinal wave velocity as is shown in Figure 21a. The vibration accelerations in the vertical and horizontal directions under different longitudinal wave velocities are shown in Figure 21b. It can be seen that, with the increase in longitudinal wave velocity, the blasting influence radius and vibration acceleration decrease nonlinearly. The blasting influence radius is related to the rock strength. The larger the rock strength is, the smaller the blasting influence radius is under the same blasting condition, that is, the nonlinear relationship between rock strength and longitudinal wave velocity is proved.

1.
The P-wave velocity at test points before and after presplitting blasting is shown in Figure 20. It can be seen that, with the increase in borehole depth, the longitudinal wave velocity increases nonlinearly, which is consistent with the above conclusion.

2.
The influence radius curve of single-hole blasting under different borehole longitudinal wave velocity as is shown in Figure 21a. The vibration accelerations in the vertical and horizontal directions under different longitudinal wave velocities are shown in Figure 21b. It can be seen that, with the increase in longitudinal wave velocity, the blasting influence radius and vibration acceleration decrease nonlinearly. The blasting influence radius is related to the rock strength. The larger the rock strength is, the smaller the blasting influence radius is under the same blasting condition, that is, the nonlinear relationship between rock strength and longitudinal wave velocity is proved. vertical and horizontal directions under different longitudinal wave velocities are shown in Figure 21b. It can be seen that, with the increase in longitudinal wave velocity, the blasting influence radius and vibration acceleration decrease nonlinearly. The blasting influence radius is related to the rock strength. The larger the rock strength is, the smaller the blasting influence radius is under the same blasting condition, that is, the nonlinear relationship between rock strength and longitudinal wave velocity is proved.

Conclusions
In this study, in order to evaluate the static and dynamic mechanical properties of limestone, longitudinal wave velocity tests were conducted. The influence of the initial damage on the compression strength, failure strain, modulus of elasticity, and Poisson's ratio is illustrated. Based on the analysis and interpretation of experimental results, the conclusions are summarized: 1. With the increasing degree of longitudinal wave speed, the initial damage decreases nonlinearly and remains constant when entering 6000 m/s, indicating the correlation between the initial damage degree of limestone and the location depth. 2. The compression strength and modulus of elasticity decrease nonlinearly with increasing initial damage and increase nonlinearly with an increasing strain rate. Conversely, the failure strain increases nonlinearly with increasing initial damage and strain rate. The Poisson's ratio changes slightly with the increase in initial damage and strain rates. Moreover, an empirical formula for describing the nonlinear relationship between initial damage and strength is presented and verified by an engineering example. The threshold of the damage variable of the limestone under the initial damage state is proposed, indicating that the initial damage value corresponds to the stress peak during the compression deformation of the nondestructive limestone. The initial strain threshold at a high strain rate of ε = 100 s −1 is approximately D = 0.2, which is 0.02 higher than that at a low strain rate of ε = 2 × 10 −4 s −1 . When the threshold of initial damage of limestone is less than 0.18, the compressive strength of the limestone is found to meet the engineering requirements. 3. As the initial damage increases, the strain softening modulus of the limestone decreases; thus, the fragility of the limestone decreases after the peak stress. With different initial degrees of damage of the rock, the strain softening modulus is much

Conclusions
In this study, in order to evaluate the static and dynamic mechanical properties of limestone, longitudinal wave velocity tests were conducted. The influence of the initial damage on the compression strength, failure strain, modulus of elasticity, and Poisson's ratio is illustrated. Based on the analysis and interpretation of experimental results, the conclusions are summarized:

1.
With the increasing degree of longitudinal wave speed, the initial damage decreases nonlinearly and remains constant when entering 6000 m/s, indicating the correlation between the initial damage degree of limestone and the location depth.

2.
The compression strength and modulus of elasticity decrease nonlinearly with increasing initial damage and increase nonlinearly with an increasing strain rate. Conversely, the failure strain increases nonlinearly with increasing initial damage and strain rate. The Poisson's ratio changes slightly with the increase in initial damage and strain rates. Moreover, an empirical formula for describing the nonlinear relationship between initial damage and strength is presented and verified by an engineering example. The threshold of the damage variable of the limestone under the initial damage state is proposed, indicating that the initial damage value corresponds to the stress peak during the compression deformation of the nondestructive limestone. The initial strain threshold at a high strain rate of ε = 100 s −1 is approximately D 1 thr = 0.2, which is 0.02 higher than that at a low strain rate of ε = 2 × 10 −4 s −1 . When the threshold of initial damage of limestone is less than 0.18, the compressive strength of the limestone is found to meet the engineering requirements.

3.
As the initial damage increases, the strain softening modulus of the limestone decreases; thus, the fragility of the limestone decreases after the peak stress. With different initial degrees of damage of the rock, the strain softening modulus is much larger than the corresponding initial damage of the elastic modulus. The descending section of the stress-strain curve is steeper than the rising section, so the conclusions obtained by the strain softening analysis reflect the brittle characteristics of the rock. 4.
The initial damage modulus defined based on the longitudinal wave velocity can be applied to understand the compression strength properties of rock, and it also has a certain reference value for the study of damage constitutive theory. It should be noted that while the definition of initial damage proposed in this paper has provided useful insights for the construction of the rock damage constitutive model, the determination of damage degree requires accurate analysis and evaluation on the number, spacing, angle and other factors of cracks. The above conclusions are only valid for the qualitative analysis of initial damage, or simple semi-quantitative analysis, and without sufficient field verification. Further numerical model analysis is suggested to determine a more perfect initial damage model to reasonably determine the long-term damage characteristics of rock for engineering applications.