Study on Infrasonic Signal Characteristics and Energy Characterization of Damage and Failure in Red Sandstone Under Uniaxial Cyclic Loading and Unloading Conditions

Abstract

The instability and collapse of surrounding rock in mine goaf areas often lead to the destabilization of geological structures, surface subsidence, and mining safety accidents. To investigate the evolutionary mechanisms and precursor characteristics of rock instability and failure processes, uniaxial loading and cyclic loading–unloading tests were conducted on red sandstone using a rock mechanics loading system. These experiments aimed to explore the mechanical behavior of the rock and the development process of internal fractures. The characteristics of infrasonic signals generated during red sandstone fracturing and the laws governing damage evolution were analyzed with an infrasonic acquisition system. The research results indicate that the infrasonic signal activity generated by rock under loading conditions can be characterized by three distinct stages, namely the relative stability period, the active period, and the pre-failure precursor period. Prior to peak strength, a substantial number of infrasonic signals are generated in rocks with significant activity; this characteristic is independent of the loading path but dependent on the stress magnitude. The variation in cumulative infrasonic energy reflects the accumulation of damage in rock specimens during the loading process, and as damage accumulates, the stress–strain curve exhibits hysteresis effects and nonlinear increases, accompanied by a rapid rise in infrasonic energy. By analyzing the characteristics of infrasonic parameters and characterizing the damage and its evolutionary features in red sandstone based on infrasonic energy, the internal crack damage evolution process in rocks can be effectively characterized. This approach provides theoretical foundations and technical support for early warning and monitoring prior to rock failure.

1. Introduction

In the process of underground mining, ore rock is easily affected by the disturbance of external loads such as mining equipment and rock drilling and blasting [1,2], and at the same time, the changes in crustal stress have different degrees of influence on the joints, fissures, and other structural plane in the rock body, which poses a certain degree of risk to the stability of the mines and mining safety [3,4]. The frequency of an infrasonic wave is 0.01~20 Hz [5,6]. It has the characteristics of low frequency, long wavelength, long propagation distance, and low attenuation and does not need coupling contact when monitoring rock stability [7]. It is suitable for use in complex terrain environments, such as mines. Therefore, it is of great significance to study the evolution mechanism and precursor characteristics of the rock damage destruction process by using infrasound technology for the early warning and analysis of rock damage.

In recent years, a large number of scholars have used infrasound technology to study the infrasonic signal characteristics of coal and rock stability monitoring and the deformation damage process. Zhu et al. conducted uniaxial compression tests on six kinds of rocks, detected the low-frequency infrasonic signals generated in the process of micro-fracturing, and analyzed the characteristics of infrasonic signals in the process of rock fracturing [8]. Xu et al. conducted compression failure tests on sandstone by using laboratory tests, collected infrasonic signals during rock deformation and failure, and analyzed the energy characteristics of abnormal signals [9]. Wei et al. analyzed the corresponding relationship between infrasonic signal and loading stress during the uniaxial compression of coal and rock. Before the instability and failure of coal and rock occurred, significant infrasonic signals were generated [10,11]. Jia et al. analyzed the variation characteristics of sound energy in coal samples during uniaxial loading, and with the increase in strength, the proportion of low-frequency sound waves gradually increased [12,13,14]. Zhao et al. used infrasound methods to study the mechanical characteristics of cemented tailings backfill and rock damage processes at different curing ages [15,16,17]. Valerie L. Zimmer et al. deployed seismic and infrasound sensors on cliffs for the in situ detection and localization of rockfalls at a distance [18]. Shan et al. conducted infrasound experiments to study the damage process of granite under dynamic uniaxial compression [19]. Zhang et al. investigated the relationship between infrasonic signals characteristics and loading rate during the uniaxial compression damaging of marble [20]. Most scholars analyzed the infrasonic parameter characteristics of coal, rock, and backfill during the damage process under different loading conditions, which is best for promoting the application of infrasound technology in rock engineering; however, few studies have been carried out on the characterization of the damage evolution of red sandstone based on infrasonic energy under uniaxial loading and cyclic loading and unloading.

This study investigates the infrasonic signal characteristics and damage evolution mechanisms of red sandstone under loading conditions, and key parameter characteristics such as mechanical characteristics, infrasonic amplitude changes, infrasonic energy rate, and cumulative energy during uniaxial loading and cyclic loading and the unloading of red sandstone are analyzed. On the basis of infrasonic energy being used to characterize rock damage, the damage evolution law of red sandstone is further studied, which provides theoretical support and technical guarantee for the stability assessment and instability warning of rock engineering.

2. Materials and Methods

2.1. Specimen Preparation

Red sandstone is widely distributed in the mines of the Gannan region, China. And its loading produces better acoustic signal response characteristics [21]. In view of this, this paper conducts research on red sandstone in the deep stope of a lead–zinc mine in the Gannan region, using diamond core drilling to minimize mechanical disturbance, and according to the test procedure of the International Society for Rock Mechanics (ISRM), 10 standard cylindrical specimens with specifications of Ø50 mm × 100 mm were obtained. The specimens numbered A-1, A-2, and A-3 were subjected to the uniaxial loading test, while the specimens numbered from A-4 to A-10 were subjected to the cyclic loading and unloading test in order to carry out a comparative analysis of the deformation and failure characteristics of red sandstone and their infrasonic activity patterns and damage. The specimen sizes are shown in

Table 1. Specimen size.

Specimen No.	Diameter/mm	Height/mm
A-1	49.56	100.12
A-2	49.84	100.08
A-3	49.62	100.02
A-4	49.58	99.94
A-5	49.88	100.06
A-6	49.64	99.98
A-7	49.72	100.08
A-8	49.76	100.04
A-9	49.68	99.84
A-10	49.82	100.06

.

2.2. Mechanical Test

The loading equipment utilized the RMT-150C rock mechanics test system, which is manufactured by the Wuhan Rocksoil Mechanics Institute of the Chinese Academy of Sciences, Wuhan, China, the uniaxial loading test utilized the displacement control, and the loading rate was 0.005 mm/s. The cyclic loading and unloading utilized load control, the loading rate was 0.5 kN/s, and 5 cycles of loading and unloading were performed, with stress endpoints of 25 kN, 50 kN, 75 kN, 100 kN, and 125 kN at each level of the cycle. In order to provide consistency in the mechanical data, at each stage the samples were loaded to the upper limit, then unloaded to 5 kN, and in the last stage, the specimens were loaded until they were damaged.

2.3. Infrasound Test

The infrasound acquisition equipment consists of the CASI-MDT-2011 Network Transmitter and matching infrasound sensors with a sampling frequency of 640 Hz. The infrasound acquisition system was placed 0.5 m away from the test platform, the relevant parameters were set, and other low-frequency noise interferences in the test environment were avoided insofar as possible. The test was carried out at night when noise was very low; doors, windows, air conditioners, etc., were closed before the test; people were prohibited from walking, talking, etc., during the test; and the mechanical loading system and infrasound acquisition system were run synchronously until the specimen was destroyed. The test procedure is shown in Figure 1.

3. Results

3.1. Mechanical Data Analysis

Uniaxial compression tests were conducted on specimens A-1 to A-3, with an average uniaxial compressive strength of about 87.097 MPa, and cyclic loading and unloading tests were conducted on specimens A-4 to A-10. Of these, A-5 specimen failed the test due to a data acquisition error, and the rest of the specimens had obvious stress–strain hysteresis loops with an average cyclic loading and unloading with a compressive strength of 84.862 MPa, as shown in

Table 2. Peak stress and peak strain of each specimen.

Specimen No.	Peak Stress/MPa	Peak Strain	Loading Method
A-1	105.152	0.006451	Uniaxial loading
A-2	73.257	0.006788	Uniaxial loading
A-3	82.883	0.005471	Uniaxial loading
A-4	84.502	0.005134	Cyclic loading and unloading
A-6	98.172	0.006098	Cyclic loading and unloading
A-7	75.223	0.009260	Cyclic loading and unloading
A-8	95.147	0.005187	Cyclic loading and unloading
A-9	77.423	0.009532	Cyclic loading and unloading
A-10	78.707	0.009524	Cyclic loading and unloading

and Figure 2, Figure 3 and Figure 4.

The loading deformation modulus E₁ is defined as the ratio of the stress difference between the end and the beginning of the loading phase to the corresponding strain difference during the same loading interval. The unloading deformation modulus E₂ is defined as the absolute value of the ratio between the stress difference at the end and beginning phases of unloading and its corresponding strain difference. The calculation formula is shown in Equation (1), and the results are shown in Table 3 and Table 4, Figure 5 and Figure 6.

$$E(i)={\displaystyle \frac{{\sigma }_{\max }(i)-{\sigma }_{\min }(i)}{{\epsilon }_{\max }(i)-{\epsilon }_{\min }(i)}}$$

(1)

• $i$—Number of cycles;
• $E(i)$—Deformation modulus per cycle of loading and unloading, GPa;
• ${\sigma }_{\max }(i)$, ${\sigma }_{\min }(i)$—Maximum and minimum stresses per cycle of loading and unloading, GPa;
• ${\epsilon }_{\max }(i)$, ${\epsilon }_{\min }(i)$—Load and unload maximum and minimum strains per cycle.

Figure 5. Stress–strain curves under cyclic loading and unloading.

Figure 5. Stress–strain curves under cyclic loading and unloading.

Figure 6. Loading deformation modulus–number of cycles.

Figure 6. Loading deformation modulus–number of cycles.

Table 3. Calculation results of deformation modulus at each loading stage.

Number of Times Loaded	Deformation Modulus E1 at Different Loading Stages/GPa
	A-4	A-6	A-7	A-8	A-9	A-10
1	12.70	11.46	5.64	13.99	6.16	5.74
2	19.07	19.98	9.92	21.88	10.07	9.27
3	20.28	21.71	11.75	22.76	11.73	11.03
4	21.16	22.05	12.89	23.35	12.79	12.47
5	20.04	22.70	13.48	24.01	13.32	13.32
6	20.25	19.05	12.31	22.13	11.94	12.41

Table 3. Calculation results of deformation modulus at each loading stage.

Number of Times Loaded	Deformation Modulus E1 at Different Loading Stages/GPa
	A-4	A-6	A-7	A-8	A-9	A-10
1	12.70	11.46	5.64	13.99	6.16	5.74
2	19.07	19.98	9.92	21.88	10.07	9.27
3	20.28	21.71	11.75	22.76	11.73	11.03
4	21.16	22.05	12.89	23.35	12.79	12.47
5	20.04	22.70	13.48	24.01	13.32	13.32
6	20.25	19.05	12.31	22.13	11.94	12.41

Table 4. Calculation results of deformation modulus at each unloading stage.

Number of Times Unloaded	Deformation Modulus E2 at Different Unloading Stages/GPa
	A-4	A-6	A-7	A-8	A-9	A-10
1	20.70	23.66	12.78	23.81	12.71	12.84
2	21.84	22.84	13.16	25.07	13.32	12.71
3	21.77	23.81	14.45	25.13	14.40	13.68
4	22.70	24.08	15.50	25.26	15.38	14.94
5	22.93	24.30	16.34	25.94	16.09	15.94

Table 4. Calculation results of deformation modulus at each unloading stage.

Number of Times Unloaded	Deformation Modulus E2 at Different Unloading Stages/GPa
	A-4	A-6	A-7	A-8	A-9	A-10
1	20.70	23.66	12.78	23.81	12.71	12.84
2	21.84	22.84	13.16	25.07	13.32	12.71
3	21.77	23.81	14.45	25.13	14.40	13.68
4	22.70	24.08	15.50	25.26	15.38	14.94
5	22.93	24.30	16.34	25.94	16.09	15.94

As shown in Figure 5 and Figure 6, the first cycle of loading and unloading E₁ of the red sandstone is the smallest, and its stress–strain diagram exhibits the smallest stress–strain slope for the first cycle of loading; the second cycle E₁ has the largest increment, indicating that the mechanical properties of the rock mass are significantly enhanced after the previous loading. Each cycle E₁ increases with the number of cycles, except for the high-stress phase where a decrease occurs. The decrease in the loaded deformation modulus of specimen A-4 at the fifth loading was due to the rock rupture sound of specimen A-4 when it was loaded at the upper stress of the fifth cycle. Specimen A-4 showed a greater degree of internal damage during the fifth cycle, so the loaded deformation modulus of specimen A-4 showed a decrease in the fifth cycle. The loading deformation modulus E₁ of the remaining specimen exhibited a decline during the sixth loading cycle, indicating that the progressive deterioration of rock mass damage occurred with increasing cycle number and upper stress limits, ultimately leading to structural failure. From the perspective of mechanical parameter analysis, this phenomenon corresponds to the gradual slowing of growth rate in modulus E₁ followed by eventual decrease, which can serve as a critical failure characteristic indicator in rock mechanics. The unloading deformation modulus E₂ shows a gradual increase with the number of cycles, indicating the enhanced elastic recovery capacity of the rock during unloading.

3.2. Infrasonic Time Domain Characteristics

Waveform analysis provides a comprehensive characterization of signal features in a complete picture, image and continuity. The background infrasonic signals and the test infrasonic signals are decomposed and reconstructed by the wavelet analysis and are combined with the time–mechanical parameters. The infrasonic time domain characteristics of the rock under cyclic loading and unloading are investigated and compared with the infrasonic time domain characteristics of uniaxial compression. Figure 7 shows the infrasonic time domain characteristics of the rock specimen A-1 under uniaxial compression. The horizontal axis shows the loading time (s), the left-hand side vertical axis shows the infrasonic amplitude (V), and the right-hand side vertical axis shows the axial stress (GPa). Figure 7 shows the infrasonic time domain feature of rock specimens under uniaxial compression, and the peak strength of rock specimen A-1 is 105.152 MPa. The variation in infrasonic amplitude before peak strength is significant. The axial stress at point A is 81.02 MPa (about 85% of the peak strength), and the infrasonic amplitude is 0.0047 V. The infrasonic amplitude during the elastic deformation stage of rock specimen. The infrasonic wave amplitude is low and stable, and the internal fracture in the rock is low. After point A, the variation in infrasonic amplitude is more significant, and when the rock specimen reaches the point of plastic deformation onset, the infrasonic amplitude gradually increases, indicating that microfractures begin to appear within the rock. At points B and C, the axial stresses were 99.69 MPa and 102.49 MPa, and the infrasonic amplitudes were 0.0071 V and 0.0051 V. Although the infrasonic amplitude at point C was not the highest, the infrasonic signals with higher amplitude were the most intensive, and the infrasonic amplitude reached a peak of 0.008 V when approaching peak strength, and the rock specimen undergoes rapid fracture propagation and coalescence.

As shown in Figure 8, the peak strength of rock specimen A-4 is 84.502 MPa. In the low-stress stage of the first and second cycles, infrasonic amplitude demonstrates relatively stable variation characteristics, which is attributed to the fact that the rock is in the compaction stage. During this phase, pre-existing microcracks and pores within the rock gradually close, thereby resulting in nonlinear stress–strain behavior. In the initial loading stage, pre-existing microcracks and pores in the rock gradually close, leading to nonlinear stress–strain behavior, and the internal microfractures and micropores are compressed, resulting in a small amount of nascent fracture. With the increase in axial stress, the infrasonic amplitude shows a more obvious increase in the third and fourth cycle stages, such as at the upper limit stress of the third cycle at point A (437.2 s), the infrasonic amplitude is 0.006722 V, which is obviously higher than the previous infrasonic amplitude, and with the generation of new fractures, the infrasonic amplitude also changes accordingly. The amplitude of the infrasonic waves changes most significantly when entering the high-stress stage, especially in the fifth cycle stage, such as at the amplitude of 0.01142 V at point B (1125 s) of the fifth cycle loading stage. The amplitude of the infrasonic waves at point C (1280 s) after the fifth cycle of the upper limit stress reaches a maximum value of 0.0117 V. Point D of the fifth cyclic unloading stage (1387 s), with an amplitude of 0.008382 V, indicates that the internal fracture of the rock rapidly propagation, converges, and penetrates during the high-stress stage, and the mechanical properties deteriorate sharply. After point D and before peak strength, the infrasonic amplitude exhibits a state of low-magnitude stability with minimal fluctuation.

As shown in Figure 9, the rock specimen underwent five cycles of cyclic loading and unloading, with progressive stress increments during the loading phases and decrements during the unloading phases, with each cycle forming a closed hysteresis loop. The peak stress of the rock specimen gradually increased with each loading, the peak strength of the rock specimen under cyclic loading and unloading was 98.127 MPa, and the internal structure of the rock specimen gradually accumulated damage. During the low-stress phase of the first and second cycles, the infrasonic amplitude exhibited relatively stable variations with minimal fluctuations, consistently remaining below 0.004 V. During the initial loading cycle, the relatively low infrasonic amplitude indicates fewer internal fractures and higher structural integrity of the rock specimen, which is attributed to the compaction characteristics inherent in the initial stage of rock deformation. With the increasing stress, the amplitude of infrasonic waves changes significantly, and the amplitude of infrasonic waves increases rapidly when the peak stress is reached in each loading, such as the infrasonic amplitudes of 0.00672 V and 0.01037 V at point A (691 s) and point B (886 s), respectively, of the fourth cycle, and the amplitude change is significantly enhanced, indicating that the internal fracture in the rock specimen is initiated and starts to expand. During the unloading phase, the amplitude of infrasonic waves may decrease, but they do not fully return to the initial level, indicating the accumulation of damage within the rock specimen. In the high-stress stage, the amplitude of infrasonic waves changes most significantly: the peak of infrasonic amplitude gradually increases, the fracture inside the rock specimen gradually propagates and accumulates, and the damage of the rock specimens is further aggravated by each loading cycle. For example, at point C of the upper stress of the fifth cycle (1231 s), the amplitude of the infrasonic waves changes significantly, with an amplitude of 0.008604 V, indicating that the internal fracture of the rock crack propagates and coalesces rapidly. At points D and E of the sixth cyclic loading stage, the amplitudes of infrasonic waves were 0.00358V and 0.00604V, respectively, and while the changes in amplitude were more pronounced, the number of infrasonic signals that showed significant changes was significantly reduced. After point E and before peak strength, the rock specimen approaches critical failure state while exhibiting infrasonic amplitude stabilization with low-magnitude fluctuations.

3.3. Characteristics of Infrasonic Energy Rate

Energy rate, as an important characteristic parameter of signal processing, provides a better representation of the energy and strength of the signal. Infrasonic energy rate is the energy of infrasonic waves passing through a unit area per unit time, and the energy rate is not affected by the threshold value. The energy corresponding to a certain event is the square of the amplitude; the rock specimens change under the action of the load, the infrasonic amplitude is changing constantly, and the infrasonic waves energy rate is changed accordingly. Figure 10 shows the infrasonic energy rate characteristics of the rock specimens under uniaxial compression, and there is a significant change in the infrasonic energy rate before peak strength. The axial stress at point A is 81.02 MPa (about 85% of the peak strength), and the infrasonic energy rate is 0.0027 V². After point A, the infrasonic energy rate starts to increase significantly, with infrasonic energy rates of 0.005 V² and 0.0068 V² at points B and C, respectively, where the infrasonic energy rate reaches a maximum at point C. When peak strength is reached, the infrasonic energy rate is also more significant at 0.005 V². Prior to reaching peak strength, the rock specimen undergoes rapid fracture propagation and coalescence, accompanied by significant infrasonic signal variation.

As shown in Figure 11, rock specimen A-4 exhibits relatively low and stable infrasonic energy rates during the low-stress phase, which can be attributed to its compaction stage with the minimal generation of new fractures. As stress increases, the infrasonic energy rate begins to show more obvious changes, and at the point A of the upper stress of the third cycle, the infrasonic energy rate is 0.00879 V², which is obviously increased, indicating that the internal fracture of the rock changes significantly with the increase in the stress. The variation in infrasonic energy rate is most obvious in the high-stress stage, such as at point B of the fifth cyclic loading stage, where the infrasonic energy rate is 0.0245 V², which reaches the maximum value. The infrasonic energy rates at point C and D of the fifth cycle are 0.0165 V² and 0.0108 V², respectively, which are also relatively high, but gradually show a decreasing trend. In the high-stress phase, the internal fracture and microfracture structures of the rock propagation and coalescence dramatically, releasing a large amount of energy, and the infrasonic energy rate varies significantly. The infrasonic energy rate values are lower after the D point and before peak strength.

As shown in Figure 12, the infrasonic energy rate of rock specimen A-6 is relatively low during the low-stress stage, which is determined by the characteristics of the compaction phase. With the increase in stress, the infrasonic energy rate appears to increase significantly in the fourth cycle, such as the infrasonic energy rate of 0.0064 V² and 0.0107 V² at points A and B, respectively, where the infrasonic energy rate reaches the maximum value at point B, which indicates that there is a large change in the internal fracture of the rock. In the high-stress phase, the infrasonic energy rate also undergoes a very pronounced change. For example, the infrasonic energy rate at point C of the upper stress in the fifth cyclic phase is 0.0085 V², and the higher infrasonic energy rate occurs most intensively. The infrasonic energy rates of 0.00282 V² and 0.00582 V² at points D and E, respectively, in the sixth cyclic loading stage are also relatively high, indicating that the structures such as cracks and fractures inside the rock propagation and coalescence dramatically in the high-stress stage. The infrasonic energy rate values are similarly low after the E point and before peak strength.

3.4. Cumulative Infrasonic Energy Characteristics

The variation in infrasonic cumulative energy reflects the cumulative process of damage in the loading process of rock specimens. In order to better reflect the change characteristics of rocks in the loading action of infrasound, infrasonic cumulative energy characterization is further carried out after the cumulative infrasonic energy rate is added. Figure 13 shows the infrasonic cumulative energy characteristics of the rock specimens under uniaxial compression, and the infrasonic cumulative energy before peak strength shows a significant dramatic increase. In the initial stage, the stress and strain of the rock specimens show a linear relationship, showing elastic deformation, and there is basically no damage or fracturing inside the rock specimens; the infrasonic cumulative energy grows slowly, and the infrasonic cumulative energy at point A is 0.1156 V². After point A, the growth rate of infrasonic cumulative energy increases significantly, and microfractures and damage began to appear inside the rock specimen. By point B, the infrasonic cumulative energy is 0.1375 V², and from point A to point B, the infrasonic energy increased by 0.0219 V². By point C, the infrasonic cumulative energy was 0.1658 V², and from point B to point C, the infrasonic energy increased by 0.0283 V². The infrasonic cumulative energy continues to increase, the number of fractures rapidly increases and propagates. The infrasonic cumulative energy is 0.1754 V² at point D. The infrasonic energy increases by 0.0096 V² from point C to point D. The infrasonic cumulative energy tends to stabilize or slightly increase, and the destruction of the rock specimens and the generation of fracture are essentially completed. Before peak strength, the infrasonic energy increases by a total of 0.0598 V² from point A to point D, indicating that there is a significant expansion and penetration of the internal fracture of the rock at this stage.

The cumulative energy signature of infrasonic waves under cyclic loading and unloading can be similarly categorized into three stages, as shown in Figure 14 and Figure 15. (1) The relatively stable period of infrasonic cumulative energy: During the low-stress period of the first and second cycles, the infrasonic cumulative energy increases slowly, and there is no obvious sudden change in infrasonic amplitude, which is due to the fact that during the compaction stage of the rock, the internal microfractures and micropores are compressed, and a small amount of new fractures is produced, which causes less damage. (2) The period of rapid growth of infrasonic cumulative energy: With the continuous strengthening of stress, the cumulative energy increases significantly, and the amplitude changes are obvious, especially when the peak stress is reached each time, the infrasonic cumulative energy increases significantly. The increment of the cumulative energy of the rock specimens near point A is also significantly increased by 0.0067 V². The abnormal fluctuation of the infrasonic amplitude is significant in this stage, and a large number of new fractures are generated or expand inside the rock and the infrasonic activity starts to be active. (3) Destruction precursor period: In the high-stress stage of the fifth and sixth cycles; infrasonic activity is very active; the cumulative energy increment is the most significant; the cumulative energy increment of the rock specimens near point C is 0.1796 V², of which the cumulative energy increment is the largest near the point C of the upper stress limit of the fifth cycle; and the cumulative energy near points D and E is not significant but the amplitude appears to have very obvious fluctuations. Fractures rapidly expand, converge, and penetrate and the mechanical properties deteriorate sharply. After point E and before peak strength, the infrasonic amplitude is relatively flat and the cumulative energy growth rate is significantly lower, but a large number of infrasonic signals are still generated.

3.5. Characterization of Damage and Its Evolution Characteristics Based on Infrasonic Energy

The deformation and destruction of rock is the result of the internal evolution of rock damage, the internal damage of rock will release energy, which not only produces high-frequency signals but also produces low-frequency signals, and these signals reflect and contain information about the damage to the rock material, so infrasonic energy was chosen to characterize the rock damage. It was assumed that the microcell strength of the rock obeys the Weibull distribution [22]. The Weibull distribution of microelement strength is given as:

$$F(\sigma )=1-\exp \left[-{({\displaystyle \frac{\sigma }{{\sigma }_0}})}^m\right]$$

(2)

where

$\sigma$ is the microelement strength, ${\sigma }_0$ is the scale parameter, and $m$ is the shape parameter.

The damage area of the rock cross-section is as follows:

$$S=S_m{\displaystyle {\int }_0^{\epsilon }\phi (x)}$$

(3)

where

$S_m$ is the cross-sectional area of the material in its undamaged state;

$\phi (x)$ is the probability distribution of the strength of the microelement.

When the strain increment of the specimen is $\Delta \epsilon$, the damage area of the specimen is obtained from Equation (3). When the damage area of the specimen is $\Delta S$,

$$\Delta S=S_m{\displaystyle {\int }_0^{\Delta \epsilon }\phi (x)dx}$$

(4)

The damage variable $D$ is defined as the ratio of the damage area $\Delta S$ to the initial cross-sectional area $S_{\mathrm{m}}$:

$$D=\Delta S/S_m={\displaystyle {\int }_0^{\Delta \epsilon }\phi (x)dx}$$

(5)

Assuming that the infrasonic waves energy produced by damaging the microcell area is $n$, then the infrasonic waves energy $N$ will be produced by damaging the $\Delta S$ area:

$$N=n\Delta S=N_m\Delta S/S_m$$

(6)

The ratio of infrasonic energy $N$ to the cumulative infrasonic energy $N_m$ at full damage:

$$N/N_m=\Delta S/S_m={\displaystyle {\int }_0^{\Delta \epsilon }\phi (x)dx}$$

(7)

The relationship between the damage variable $D$ and the infrasonic cumulative infrasonic energy is obtained by couplings (4) and (6):

$$D=N/N_m$$

(8)

It is assumed that the infrasonic cumulative infrasonic energy produced by the rock at all levels of loading and unloading phases are $N_{i+}$ and $N_{i-}$, respectively; the damage variables are $D_{i+}$ and $D_{i-}$; and the cumulative damage variables are $D_{+}$ and $D_{-}$. The cumulative infrasonic energy of the cyclic infrasonic waves per stage is $N_i$. The cumulative infrasonic energy $\Sigma N_i$ is obtained by accumulating $N_i$, as shown in

Table 5. The infrasonic energy and damage variables of rock specimen A-4.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.0224	0.0132	0.0356	0.0356	0.0142	0.0084	0.0142	0.0226
2	0.0335	0.0323	0.0658	0.1014	0.0212	0.0204	0.0438	0.0642
3	0.1121	0.1386	0.2507	0.3521	0.0709	0.0877	0.1351	0.2228
4	0.0736	0.0641	0.1377	0.4898	0.0466	0.0406	0.2694	0.3099
5	0.5702	0.4019	0.9721	1.4619	0.3608	0.2543	0.6708	0.9251
6	0.1184		0.1184	1.5804	0.0750		1

,

Table 6. The infrasonic energy and damage variables of rock specimen A-6.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.0347	0.0142	0.0489	0.0489	0.0408	0.0167	0.0408	0.0575
2	0.0269	0.0376	0.0645	0.1134	0.0316	0.0442	0.0892	0.1334
3	0.0481	0.0392	0.0873	0.2007	0.0566	0.0461	0.1900	0.2361
4	0.1210	0.1223	0.2433	0.444	0.1424	0.1439	0.3785	0.5224
5	0.1739	0.1095	0.2834	0.7274	0.2046	0.1288	0.7269	0.8558
6	0.1226		0.1226	0.85	0.1442		1

,

Table 7. The infrasonic energy and damage variables of rock specimen A-7.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.0799	0.0743	0.1542	0.1542	0.0297	0.0277	0.0297	0.0574
2	0.1259	0.1077	0.2336	0.3878	0.0469	0.0401	0.1043	0.1444
3	0.2629	0.2450	0.5079	0.8957	0.0979	0.0912	0.2422	0.3334
4	0.2785	0.2415	0.5200	1.4157	0.1037	0.0899	0.4371	0.5270
5	0.4208	0.2856	0.7064	2.1221	0.1567	0.1063	0.6837	0.7900
6	0.5641		0.5641	2.6862	0.2100		1

,

Table 8. The infrasonic energy and damage variables of rock specimen A-8.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.0803	0.0319	0.1122	0.1122	0.0147	0.0058	0.0147	0.0206
2	0.1095	0.1207	0.2302	0.3424	0.0201	0.0221	0.0407	0.0628
3	0.2149	0.3665	0.5814	0.9238	0.0394	0.0672	0.1022	0.1694
4	0.6135	0.7850	1.3985	2.3223	0.1125	0.1439	0.2819	0.4258
5	0.9095	1.0728	1.9823	4.3046	0.1668	0.1967	0.5926	0.7893
6	1.1492		1.1492	5.4538	0.2107		1

,

Table 9. The infrasonic energy and damage variables of rock specimen A-9.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.2379	0.0799	0.3178	0.3178	0.0508	0.0171	0.0508	0.0678
2	0.1760	0.1003	0.2763	0.5941	0.0376	0.0214	0.1054	0.1268
3	0.2272	0.2658	0.493	1.0871	0.0485	0.0567	0.1753	0.2320
4	0.2395	0.3029	0.5424	1.6295	0.0511	0.0646	0.2831	0.3477
5	1.4659	0.8111	2.2770	3.9025	0.3128	0.1731	0.6606	0.8337
6	0.7794		0.7794	4.6859	0.1663		1

and

Table 10. The infrasonic energy and damage variables of rock specimen A-10.

Number of Cycles	Parameters
	Ni+	Ni−	Ni	ΣNi	Di+	Di−	D+	D−
1	0.1664	0.1356	0.3020	0.3020	0.0242	0.0197	0.0242	0.0439
2	0.2439	0.2200	0.4639	0.7659	0.0354	0.0320	0.0793	0.1113
3	0.4039	0.5237	0.9276	1.6935	0.0587	0.0761	0.1700	0.2461
4	0.6718	1.1125	1.7843	3.4778	0.0976	0.1616	0.3437	0.5053
5	0.9244	1.3398	2.2646	5.7420	0.1343	0.1947	0.6396	0.8343
6	1.1406		1.1406	6.8826	0.1657		1

and Figure 16 and Figure 17.

$$N_i=N_{\mathrm{i}+}+N_{\mathrm{i}-}$$

(9)

$$D_{i\pm }=N_{\mathrm{i}\pm }/N_{\mathrm{m}}$$

(10)

$$D={\displaystyle \sum _1^i{D}_{i-}}+{\displaystyle \sum _1^i{D}_{i+}}$$

(11)

As shown in Figure 17 and Figure 18, During the low-stress phase of the first and second cycles, the cumulative damage variable averaged 0.1072 by the end of the second cycle, and the rock specimens were less damaged. As the stress continues to increase, the damage variable and cumulative damage variable increase more significantly in the third and fourth cycles, where the cumulative damage variable averages 0.4397 after the fourth cycle; in the third and second cycle of the unloading phase of rock specimen A-4, the damage variable grows from 0.0204 to 0.0709, and the cumulative damage variable grows from 0.0642 to 0.1351. Entering the high-stress stage, in the fifth and sixth cycles, the maximum value of the damage variable occurs and the cumulative damage variable increases sharply. In the fifth cycle of the loading stage of rock specimen A-6, the damage variable grows from 0.1439 to 0.2046, and in the fourth cycle of the unloading stage, the cumulative damage variable grows from 0.5224 to 0.7269. After the fifth cycle of loading, reaching peak strength, the damage variable decreases and the rock specimen damage, although increasing, increases at a reduced rate.

4. Discussion

A comprehensive analysis of the infrasonic wave time domain characteristics of the red sandstone under uniaxial compression and cyclic loading and unloading shows that the changes in infrasonic waves amplitude reflect the development and expansion of the internal fracture of the rock, and that higher infrasonic amplitude under cyclic loading and unloading conditions generally occurs at and around the upper limit stresses of each cycle before peak strength. In the low-stress stage, the infrasonic wave amplitudes are relatively low and smooth. With the increase in stress, the infrasonic wave amplitudes change significantly, indicating the rapid expansion of the fracture. At the high-stress stage, the variation in infrasonic amplitude is most significant for both uniaxial compression and cyclic loading and unloading, with a maximum in infrasonic amplitude, a feature which is shown to be independent of the loading path and is related to the stress. Before peak strength, infrasonic waves amplitude appear in a low and flat state, and cyclic loading and unloading under this condition is more obvious and can be used as a precursor characteristics of rock rupture.

The comprehensive analysis of infrasonic energy rate characteristics in red sandstone under uniaxial compression and cyclic loading–unloading conditions reveals distinct patterns across stress stages. During low-stress conditions, the infrasonic energy rate remains consistently low and stable, reflecting minimal internal microcrack activity. As the stress increases to an intermediate level, significant fluctuations emerge in the energy rate, with cyclic loading–unloading amplifying these variations due to cumulative damage effects from repeated stress cycles. The most pronounced changes occur in the high-stress stage, where the energy rate peaks immediately before rock failure—a critical precursor regardless of loading paths. Notably, these characteristic variations demonstrate stress-level dependency rather than sensitivity to specific loading sequences, indicating that internal stress states predominantly govern energy release dynamics. The significant variation in infrasonic energy rate typically manifests prior to peak strength, and the rate generally exhibits a gradually decreasing trend after its initial emergence. The infrasonic energy rate remains notably low prior to peak strength, and this characteristic becomes more pronounced under cyclic loading–unloading conditions. Therefore, the emergence of substantial infrasonic signals accompanied by a sustained increase in energy rate can serve as a precursor characteristic for rock failure.

The infrasonic cumulative energy under cyclic loading and unloading conditions can be characterized as three phases: a relatively stable period of infrasonic cumulative energy, a period of rapid growth of infrasonic cumulative energy, and a period of damage precursor. Before peak strength, the cyclic loading and unloading infrasonic amplitude is relatively flat, the infrasonic cumulative energy growth rate is significantly lower than before, and a large number of infrasonic signals are still generated, but the infrasonic cumulative energy in uniaxial compression exhibits a significant and dramatic increase in the characteristics. A significant increase in infrasonic energy can be used as an early warning indicator of impending damage to the rock specimens, indicating that the internal fracture is approaching a critical state. The changes in the cumulative infrasonic energy reflect the process of damage accumulation in the rock specimens during the loading process, and the infrasonic energy increases rapidly as the damage accumulates.

Based on the analysis of infrasonic energy damage characterization, damage is generated in both the loading and unloading phases, the cumulative damage variable D increases with the increase in cycles, and the curves show an upward bending nonlinear growth pattern, indicating that the damage increases during the damage process in rock specimens. In the low-stress stage, the rock specimens have less damage, which is due to the fact that the rock specimens are in the compaction stage, the internal microfractures and micropore holes are compressed, and only a small amount of nascent fissures are produced. As the stress continues to increase, the damage variables and cumulative damage variables show a more pronounced increase, more microfractures are generated within the rock specimens, and the mechanical properties decrease. Entering the high-stress stage, the internal cracks of the rock specimens show a greater degree of expansion and penetration, the damage is intensified, and the mechanical properties deteriorate sharply. As secondary energy can better characterize red sandstone damage, the rock damage under external loading increases monotonically and irreversibly and characterizes not only the damage in the loading phase but also that in the unloading phase.

5. Conclusions

In this study, the key parametric characteristics of red sandstone, including mechanical properties, infrasonic amplitude variations, energy rates, and cumulative energy, were analyzed under both uniaxial and cyclic loading–unloading conditions. By utilizing infrasonic energy as a quantitative indicator of rock damage, the evolutionary mechanisms governing damage progression in red sandstone were comprehensively investigated, the main conclusions are as follows:

(1) The infrasonic signal activity generated by rock under loading can be characterized by three distinct stages: relative stability period, active period, and pre-failure precursor period. During the low-stress stage (relative stability period), both infrasonic amplitude and cumulative energy increment remain minimal and stable, reflecting limited microcrack activity as the rock primarily undergoes elastic deformation. As stress progresses to intermediate levels (active period), significant fluctuations emerge in these parameters due to intensified microcrack interactions and localized damage accumulation and the infrasonic amplitude and cumulative energy increment exhibit significant variations, indicating intensified infrasonic activity. At the high-stress stage (pre-failure precursor period), the characteristics of infrasonic signals undergo the most significant changes, with the infrasonic amplitude reaching its maximum value and the cumulative energy increment attaining its highest level, signaling the accelerated microcrack coalescence preceding the macroscopic failure.

(2) The significant variation in infrasonic energy rate typically manifests prior to peak strength, the infrasonic energy rate generally exhibits a gradually decreasing trend after its initial emergence. The infrasonic energy rate remains notably low prior to peak strength, and this characteristic becomes more pronounced under cyclic loading–unloading conditions. Therefore, the emergence of substantial infrasonic signals accompanied by a sustained increase in energy rate can serve as a precursor characteristic for rock failure.

(3) Rock damage characterization based on infrasonic energy shows that both loading and unloading phases generate damage. Under external loading, rock damage exhibits a monotonically increasing and irreversible trend, with the curve demonstrating an upward-curving nonlinear growth pattern. Based on infrasonic energy, it is possible to characterize damage not only during the loading phase but also during the unloading phase. Infrasonic energy can effectively reflect the evolutionary process of internal crack damage in rocks.