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

Abstract

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.

1. 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 strain [17], X-ray, computed tomographic scan number [18,19], ratio of initial compression and yield strength [20,21], the acoustic emission signals [22,23,24,25], and the rock’s acoustic wave speed [26,27,28,29,30,31]. Among them, the direct methods are preferred for measuring the P-wave velocities of rocks, and the test process produces no secondary damage to the rock test sample. Hence, the P-wave velocity test provides an accurate estimation of the total material damage. Acoustic wave speed has been widely applied in rock damage evaluation research in recent years due to its ability to accurately represent rock deformation and the damage information.

Acoustic wave technology is the procedure of longitudinal wave reception after propagation in a crack-containing rock. The longitudinal wave velocity was calculated using the transmission travel time of an acoustic pulse along the axial direction of the samples [29]. The internal damage behaviors in the compression fracture tests have been introduced and are of great concern [25,32,33,34]. Aydan et al. [35], Mansouri et al. [36], Fahimifar et al. [37] and Kachanov et al. [38] observed that the crack development and rock strength were influenced by damage development in rock creep tests. Gatelier et al. [39] conducted a rock failure pattern under irreversible strain and damage accumulation in a uniaxial experiment of porous sandstone. Heo et al. [40] measured the elastic wave velocities of granite and marble under triaxial compression tests and determined the location of damaged cracks. Ghazvinian et al. [41] developed a damage threshold of brittle rock which describes the crack damage of brittle rock. Doan et al. [42] compared pre-damaged and intact granite results under a high strain rate loading test. Bayram [43] constructed model parameters for the strength damage of granite after freezing and thawing. Overall, compared to the accumulated damage of compression tests, the studies on the initial damage of brittle rock are insufficient. In addition, the relevant experimental data regarding initial damage are quite scarce, especially those in terms of high strain rate.

These above investigations have provided insights into the compression static and dynamic characteristics of limestone with different initial levels of damage. Hence, in this paper a series of compression tests were conducted on block type limestone by employing a large-scale rock multi-function shear device subjected to 2 × 10⁻⁴ s⁻¹ strain rates and a Hopkinson rod device subjected to a 100 s⁻¹ strain rate. The influences of the initial damage on the stress–strain curve, compression strength, modulus of elasticity, Poisson’s ratio, initial damage threshold value, and strain softening characteristics were evaluated and discussed. The presentation of the results in this study is helpful to better understanding the mechanical properties in order to develop a proper constitutive model of limestone with initial damage.

2. The Definition of Initial Damage

The indicators that are defined in rock damage mechanic variables macroscopically have multiple parameters such as density, resistance, elastic modulus, tensile strength, yield stress formula, elongation, ultrasonic velocity, and acoustic emission [23]. The determination methods of tensile strength, yield stress and elongation cause secondary damage to rock, and the damage degree measured by density and resistance is different from the fact. The method of elastic modulus and longitudinal wave velocity to represent initial damage is similar, and the ultrasonic testing methods use the longitudinal waveform to obtain the internal crack information of the rock [44]; these belong to non-destructive testing technologies and have little effect on the material’s original state. The damage variable is defined as [45]:

$$D=1-{\left(\frac{v^{\prime }}{v}\right)}^2$$

(1)

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-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:

$$V_0\ge \max \left(v_0^1,v_0^2,\dots v_0^i\right)$$

(2)

where $v_0^i$ is the longitudinal wave velocity of the rock in the i times experiment (m/s), and V₀ 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.

3. Materials and Methods

3.1. 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 above-mentioned 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

Table 1. Main mineral compositions of limestone samples.

Mineral Composition		Percentage Weight (%)
Component	calcite	92
	dolomite	3
	irony	2
	clay	2
	others	1

and

Table 2. Basic physical and mechanical properties of limestone.

Property	Average	Range
Initial density (g/cm3)	2.76	2.68–2.81
Natural moisture content (%)	1.9	1.1–2.7
Cohesive force (MPa)	27	21–35
Frictional angle (°)	33	28–37
Poisson’s ratio	0.27	0.25–0.31

.

3.2. Rock Ultrasonic Testing

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. Vertical wave velocities of five limestone specimen borehole locations.

No	ν′	Average	No	ν′	Average	No	ν′	Average	No	ν′	Average	No	ν′	Average
A1	2823	2976	B1	3230	3365	C1	3564	3675	D1	3952	4811	E1	5599	5871
A2	2846		B2	3250		C2	3602		D2	4150		E2	5645
A3	2875		B3	3298		C3	3624		D3	4221		E3	5742
A4	2902		B4	3311		C4	3649		D4	4465		E4	5778
A5	2940		B5	3326		C5	3672		D5	4723		E5	5891
A6	2961		B6	3348		C6	3688		D6	4904		E6	5911
A7	2986		B7	3369		C7	3693		D7	5021		E7	5946
A8	3010		B8	3389		C8	3702		D8	5086		E8	5979
A9	3054		B9	3426		C9	3721		D9	5126		E9	5982
A10	3097		B10	3475		C10	3740		D10	5242		E10	5990
A11	3104		B11	3480		C11	3780		D11	5367		E11	5992
A12	3114		B12	3485		C12	3797		D12	5478		E12	5997

. 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.
• 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.
• 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:

$$D=1.52-0.39{\mathrm{e}}^{\left(v/4402.95\right)}$$

(3)

where D is the initial damage value, and ν is the longitudinal wave velocity of the five drilling locations.

According to the test results, the mean longitudinal wave velocity values of the five samples (A–D) are 2976 m/s, 3365 m/s, 3675 m/s, 4811 m/s, and 5871 m/s, respectively, which corresponds to the initial damage area D = 0.754, 0.685, 0.625, 0.357, and 0.043, as is shown in

Table 4. Range and mean deviation values of the density, longitudinal wave speed and initial damage.

Depth(m)	Item	$\mathit{\rho }$	V (m/s)	D
85–85.9	Maximum	2.65	3114	0.730
	Minimum	2.63	2823	0.779
	Mean	2.635	2976	0.754
87–88.5	Maximum	2.71	3485	0.663
	Minimum	2.69	3230	0.710
	Mean	2.7	3365	0.685
88.5–89.9	Maximum	2.75	3797	0.601
	Minimum	2.71	3564	0.647
	Mean	2.74	3675	0.625
91.2–93.5	Maximum	2.79	5478	0.166
	Minimum	2.77	3952	0.566
	Mean	2.78	4811	0.357
95.2–97.9	Maximum	2.81	5997	0.009
	Minimum	2.78	5599	0.129
	Mean	2.79	5871	0.043

. 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.

3.3. Uniaxial, and Triaxial Compression Tests

3.3.1. 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.

3.3.2. Test Program

Table 5. Experimental scheme of static compression test.

No.	Test Type	D	σ3/MPa	Strain Rate/s	Criterion for End
1	Uniaxial compression	0.754	0	2 × 10−6; 2 × 10−5; 2 × 10−4	Deformation displacement
		0.625
		0.357
		0.043
2	Triaxial compression	0.754	5;10;20	2 × 10−6	Axial strain
		0.625
		0.357
		0.043

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.

3.4. 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 1–10³ 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]:

$${\sigma }_s\left(t\right)=\frac{EA}{2A_s}\left[{\epsilon }_I\left(t\right)+{\epsilon }_R\left(t\right)+{\epsilon }_T\left(t\right)\right]$$

(4)

$${\epsilon }_s\left(t\right)=\frac{C_0}{l_s}{{\displaystyle \int }}_0^t\left[{\epsilon }_I\left(t\right)-{\epsilon }_R\left(t\right)-{\epsilon }_T\left(t\right)\right]dt$$

(5)

$${\epsilon }_s^{‘}\left(t\right)=\frac{C_0}{l_s}\left[{\epsilon }_I\left(t\right)-{\epsilon }_R\left(t\right)-{\epsilon }_T\left(t\right)\right],$$

(6)

where E is the elastic modulus of the bar; A and A_S are the cross-sectional area of the bar and specimen, respectively; C₀ 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 ${\epsilon }_s^{‘}$ 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. Experimental scheme of dynamic compression test.

No.	Test Type	D	${\mathit{\epsilon }}^{\prime }$$/{\mathbf{s}}^{-1}$	Criterion for End
1	Stress–strain	0.754	1;10;100	Axial strain
2		0.625
3		0.357
4		0.043

.

4. Results

4.1. 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:

• 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.
• 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.

4.1.1. 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:

• 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.
• 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].
• 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 $\overline{\epsilon }$ = 2 × 10⁻⁴ s⁻¹ is 176 Mpa, which is 11.4% and 15.0% higher than that in $\overline{\epsilon }$ = 2 × 10⁻⁵ s⁻¹ and $\overline{\epsilon }$ = 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.

4.1.2. 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.
• 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 $\overline{\epsilon }$ = 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 $\overline{\epsilon }$ = 100 s⁻¹ is 0.0117, which is 11.4% and 37.6% higher than that in $\overline{\epsilon }$ = 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.
• 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.

4.2. 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 test data of different initial damages and confining pressures.

No	${\mathit{\sigma }}_3/\mathbf{MPa}$	${\mathit{\sigma }}_1/\mathbf{MPa}$	$\mathit{c}$	$\mathit{\phi }$
0.043	5	154.573	27.615	25.4
	10	171.748	32.956	35.2
	20	214.685	40.074	37.3
0.357	5	136.618	26.880	24.3
	10	156.134	29.512	34.0
	20	195.168	38.466	40.1
0.625	5	122.956	22.576	16.6
	10	136.618	26.375	25.5
	20	150.279	28.721	26.4
0.754	5	98.365	19.157	16.3
	10	122.956	22.675	16.8
	20	136.618	26.421	25.6

.

Figure 8 shows the typical triaxial compression stress–strain curves of limestone with different σ₃. It can be seen from Figure 8 that:

• 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.
• The principal stress increases with increasing confining pressure. For σ₃ = 20 Mpa, the maximum principal stress is 214.685 Mpa of D = 0.043, which increases by 10.1%, 42.9%, and 57.1% for D = 0.357, 0.625, and 0.754, respectively. Moreover, for D = 0.043, the maximum principal stress is 154.573 Mpa of σ₃ = 5 Mpa, which decreases by 10.0%, and 28.0% for σ₃ = 10 Mpa and 20 Mpa, respectively. The confinement pressure inhibits the expansion of the microcracks in the limestone; thus, this affects the growth rate of the damage, which can delay the damage and strain softening of the rock. The side reinforcement behaviors of the confinement pressure are similar to the results reported by Palchik [52].
• The slope of the line elastic phase of σ₃ = 5 Mpa is greater than that of σ₃ = 10 Mpa, 20 Mpa and D = 0.043. The slopes of the line elastic phase for σ₃ = 5 Mpa and σ₃ = 10 Mpa are approximate. Furthermore, the differences in the slopes of the line elastic phase are due to the initial damage and are diminished with increases in the confinement pressure. In addition, for D = 0.625 and 0.754, the line elasticity growth stages of three confinement pressures are approximated, further indicating that the characteristics of the confinement pressure restrained microcrack expansion into the macroscopic crack. A significant initial damage means that the large defects of the internal cracks in the limestone lead to the decreased strength and damage generation within the triaxial compression test process.

5. Discussion

5.1. 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:

$$R_{\mathrm{c}}= e^{\left(a+b\ast D+c\ast D^2\right)}$$

(7)

$${\epsilon }_{\mathrm{c}}= e^{\left(a+b\ast D+c\ast D^2\right)}$$

(8)

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.

5.2. 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:

$$E= e^{\left(a+b\ast D+c\ast D^2\right)}$$

(9)

where E is the modulus of elasticity; D is the initial damage; and a, b, and c are the quotieties.

5.3. 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:

• The Poisson’s ratio decreases nonlinearly with respect to strain rate increases.
• 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:

$$\mu = e^{\left(a+b\ast D+c\ast D^2\right)}$$

(10)

where μ is the Poisson’s ratio; D is the initial damage; and a, b, and c are the quotieties.

5.4. 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:

$$D=1-\frac{E^{\prime }}{E}$$

(11)

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:

$$D=1-\frac{\epsilon -{\epsilon }_p}{\epsilon }\left[\frac{E^{\prime }}{E}\right]$$

(12)

where ε is the total strain; ε_p is the irreversible plastic strain after unloading; $E^{\prime }$ 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:

• 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. Initial damage value determined by the loading-unloading and longitudinal wave speed.

  No	Parameter				DE	Dv
  	ε	εp	E’	E
  1	0.0042	-4	31.98	33.45	0.044	0.043
  2	0.0045	0.96 -4	17.67	27.65	0.361	0.357
  3	0.0052	1.12 -4	7.52	20.21	0.628	0.625
  4	0.0061	1.62 -4	4.81	19.96	0.759	0.754

  . 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.
• The effects of two different initial damages on the compressive strength and destructive strain of limestone are illustrated. For $\overline{\epsilon }$ = 2 × 10⁻⁶ s⁻¹, the compressive strengths are 127.43 Mpa, 97.69 Mpa, 94.89 Mpa, and 82.56 Mpa for the initial damage D = 0.044, 0.361, 0.628, and 0.459, respectively. Additionally, the compressive strengths are 130.56 Mpa, 106.37 Mpa, 95.51 Mpa, and 81.23 MPa for the initial damages of 0.044, 0.361, 0.628, and 0.459, respectively.

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 D₀, which is the initial damage boundary point, the two curves have an equal compressive strength of R₀. In addition, the compressive strength obtained by the longitudinal wave velocity is larger than that obtained by the loading-unloading test.
• 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.
• 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 D₀, 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 D₀, 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, D_E > D_v. The compressive strength and destructive strain of D_E and D_v are shown in

Table 9. Compressive strength and destructive strain of D_E and D_v.

No		Rc (mpa)	ε
Dv	0.043	130.56	0.0043
	0.357	106.37	0.0045
	0.625	95.51	0.0052
	0.754	81.23	0.0064
DE	0.044	127.43	0.0042
	0.361	97.69	0.0044
	0.628	94.89	0.0051
	0.759	82.56	0.0061

. The compressive strength varies nonlinearly with respect to the initial damage, and an exponential relationship is applied and expressed as:

$${\epsilon }_{\mathrm{c}}= e^{\left(a+b\ast D+c\ast D^2\right)}$$

(13)

$$R_{\mathrm{c}}= e^{\left(a+b\ast D+c\ast D^2\right)}$$

(14)

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.

5.5. 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:

• 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 ($\overline{\epsilon }$ = 100 s⁻¹, 2 × 10⁻⁴ s⁻¹) are $D_{thr}^1$ and $D_{thr}^2$, and the values of $D_{thr}^1$ and $D_{thr}^2$ are 0.2 and 0.18, respectively.
• 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.

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.

5.6. 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:

$$\eta =\frac{{\sigma }_c-{\sigma }_f}{{\epsilon }_f-{\epsilon }_c}$$

(15)

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. The strain softening modulus of four degrees of initial damage.

ε/s−1	D	η	ε/s−1	D	η
2 × 10−6	0.043	37.102	1	0.034	36.224
	0.357	31.178		0.357	30.884
	0.625	18.853		0.625	16.744
	0.754	15.376		0.754	14.148
2 × 10−4	0.043	28.822	100	0.043	29.825
	0.357	24.816		0.357	25.922
	0.625	15.551		0.625	15.824
	0.754	11.400		0.754	12.413

.

Figure 15 shows the strain softening modulus versus the initial damage for limestone. It can be seen from Figure 15 that:

• 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.
• 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.
• 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:

$$\eta =e^{\left(a+b\ast D+c\ast D^2\right)}$$

(16)

where η is the compressive strength; D is the initial damage; and a, b, and c are the quotieties.

5.7. 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.

5.7.1. Field Measurements

The rock parameters obtained from the P-wave velocity test and rock mechanics test are shown in

Table 11. The parameters of single-hole blasting test parameters.

NO	D/ms−1	Charge Diameter/mm	Single-Hole Charge/kg	Single-Hole Diameter/mm	Single-Hole Depth/m
T1	1115, 1279, 1492, 1824, 2116, 2452, 2661, 3152	32	0.2	90	8.4
T2	3297, 3587, 3928, 4116, 4258, 4359, 4447, 4530

.

The structure of the single-hole charge is shown in Figure 17. The field chart is shown in Figure 18 and Figure 19.

5.7.2. Analysis of Single-Hole Blasting Damage Test Results

• 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.
• 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.

6. 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:

• 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.
• 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 ${\mathrm{D}}_{thr}^1$ = 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.
• 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.
• 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.