Mechanism of Progressive Failure, Stress and Wave Velocity Misalignment in Sandstone

Abstract

The phenomenon of progressive failure, stress and wave velocity asynchrony in rocks can inform early warning approaches for rock stability. In this study, the Geotechnical Consulting and Testing Systems rock triaxial test system was used to investigate the compression failure of sandstone from the Ningtiaota mine under confining pressures of 0, 2, 5, and 10 MPa, with synchronous ultrasonic wave velocity monitoring. Based on Martin’s crack strain theory, the variation laws of mechanical and wave velocity response characteristics during progressive failure were obtained from two replicate tests per confining pressure. The results indicate that the normalized stress at peak wave velocity ${\sigma }_{v_{\max }^P}/{\sigma }_f$ ranges from 0.84 to 0.99, whereas the normalized strain ranges from 0.73 to 0.98. With increasing confining pressure, both the strain and stress differences between the peak wave velocity and the peak stress increase. Wave velocity change results from the combined action of effective stress (promoting velocity increase) and crack strain (leading to velocity decrease), causing the wave velocity peak to occur ahead of the stress peak. The normalized crack initiation stress ${\sigma }_{ci}/{\sigma }_f$ ranges from 0.55 to 0.68, and the normalized crack damage stress ${\sigma }_{cd}/{\sigma }_f$ ranges from 0.79 to 0.91, consistent with literature values for intact sandstones. With increasing confining pressure, the proportion of the compaction stage remains unchanged, while the stable crack propagation stage decreases, and the elastic and unstable crack propagation stages increase. The stress-normalized difference between the peak wave velocity and the damage variable protrusion point is approximately $0.1{\sigma }_f$, showing a slight decreasing trend with increasing confining pressure.

1. Introduction

The phenomenon of progressive failure, stress and wave velocity asynchrony in rocks, may inform early warning approaches in similar geological settings, requiring further validation, and provides an essential basis for engineering safety design [1,2]. The environment in which the strata are located is highly complex. If the fracture pressure of the rock strata cannot be accurately predicted, it will cause major accidents. The fracture process of rock strata is closely associated with the initiation, propagation, and coalescence of internal cracks within the rock mass. This process not only induces variations in physical parameters but also results in corresponding changes in wave velocity characteristics. Therefore, by analyzing the variation patterns of mechanical properties and wave velocity characteristics of rocks during the loading process, a timely prediction of rock stability can be achieved. This provides a crucial theoretical foundation and technical support for safety design and real-time monitoring in engineering projects.

In recent years, both domestic and international scholars have performed relatively systematic studies on the mechanical behavior of rocks during the progressive failure process. Regarding the mechanical properties of rock during loading, Cai [3], Bieniawski [4], and Moradian [5] et al. proposed dividing the full stress–strain curve of rock into five stages through laboratory test research. The stress points at the boundary of each stage are defined as the crack closure stress ${\sigma }_{cc}$, crack initiation stress ${\sigma }_{ci}$, damage stress ${\sigma }_{cd}$, and peak stress ${\sigma }_f$. Martin [6] analyzed the change law of characteristic stress during granite loading, and found that ${\sigma }_{ci}/{\sigma }_f$ of granite was distributed in the range of 0.4–0.5, whereas the dispersion of ${\sigma }_{cd}/{\sigma }_f$ was relatively small, distributed around 0.8. Yuan et al. [7] systematically demonstrated the applicability of the crack volume strain method, acoustic emission method, bulk strain method, and moving point regression method to determine rock characteristic stress. Eberhardt et al. [8] proposed the use of moving point regression technology and acoustic emission to determine the characteristic stress. Nicksiar [9] proposed the lateral strain difference method. On the basis of accurately determining the damage stress ${\sigma }_{cd}$, the crack initiation stress ${\sigma }_{ci}$ can be obtained using the calculation software. With the iterative optimization of detection technologies, conventional acoustic emission (AE) testing methods, such as AE hit count [10], AE energy [11], frequency analysis [12], and cumulative ring-down count [13,14], as well as novel models proposed by Yang et al. [15] and Zhao et al. [16], have been applied to the quantitative characterization and accurate identification of characteristic stress. Based on the axial crack strain, Zuo [17] analyzed the progressive failure characteristics of coal and rock mass, established a constitutive model of crack mass considering natural strain in the post-peak crack penetration stage, and derived the nonlinear stress–strain relationship and impact failure criterion of coal and rock mass in the post-peak stage, which have been widely applied in engineering practice.

As for the acoustic characteristics of rock in the process of loading failure, Simmons [18] confirmed that the change in micro-fractures in rock during loading has a significant influence on the propagation speed of ultrasonic waves. Senetakis et al. [19] found through experimental research that irregular mineral particles had a high impact on the propagation process of longitudinal waves. Shirole [20] analyzed the variations in wave velocity with damage using the uniaxial compression test and identified the characteristics of acoustic velocity at each stage of rock damage. Zhao et al. [21] proposed the damage evolution equation based on damage mechanics and combined with the acoustic model under loading conditions, obtaining the influence law of nonlinear parameters of wave speed on crack orientation. Jin et al. [22] established the dynamic constitutive equation of rocks with three-dimensional in situ stress by using the equivalent medium method and modifying the Kelvin–Voigt model. In addition, they established the theoretical model of stress–wave propagation in three-dimensional in situ stress rocks. Chen et al. [23], based on the correlation study of rock acoustic propagation velocity and its loading deformation process, found that the sudden change in wave velocity parameters occurred after the rapid increase in rock fractures in the loading process, which changed the internal microstructure and reduced the rock acoustic velocity. Zuo [24] calculated the initial disturbance factor D and the integrity factor s by using the equivalent medium theory and Hoek–Brown intensity model according to the changes in ultrasonic and mechanical properties of rock during uniaxial compression.

The aforementioned research holds significant implications for both theoretical exploration and engineering practice. Despite these advances, three gaps remain: (1) experimental studies that simultaneously track mechanical behavior and wave velocity evolution under varying confining pressures are limited; (2) existing theoretical explanations for stress–wave velocity asynchrony are largely qualitative and lack a mechanistic constitutive framework; and (3) the relationship between peak wave velocity and the damage variable has not been systematically quantified. This study addresses these gaps by integrating synchronous ultrasonic monitoring with crack-strain-based constitutive modeling. Although the phenomenon of wave velocity peaking prior to stress peak has been widely reported, most existing studies rely on empirical correlations or qualitative damage descriptions. Among these methods, the crack volume strain method [6] provides a direct physical link to micro-crack evolution but requires high-precision strain measurements; the AE method [8,10,11] enables real-time damage monitoring but is sensitive to background noise; and the moving point regression technique [7] offers objectivity but may miss localized damage initiation. This study adopts the crack strain approach owing to its direct physical interpretation and compatibility with the subsequent constitutive modeling. In contrast, this study proposes a mechanistic framework that explicitly couples crack strain evolution with wave velocity degradation and integrates this relationship into a constitutive model capable of simulating both mechanical and ultrasonic responses throughout the entire failure process. Experiments were conducted using the Geotechnical Consulting and Testing Systems (GCTS) rock mechanics testing system to investigate the mechanical behavior and wave velocity variation in sandstone under varying confining pressure conditions during compression. This study quantified the asynchronous relationship between stress and wave velocity in sandstone under confining pressures of 0, 2, 5, and 10 MPa through laboratory triaxial tests with synchronous ultrasonic monitoring. Furthermore, it developed a damage constitutive model using axial crack strain as the internal state variable and a wave velocity evolution model that explicitly separates the contributions of effective stress and crack strain. It also evaluated the hypothesis that the normalized stress at peak wave velocity decreases with increasing confining pressure by comparing model predictions with experimental data. This study determined the normalized characteristic stress ranges for the studied sandstone and evaluated their consistency with established ranges in the literature. This research aims to provide both theoretical support and practical foundation for safety assessments in rock mass engineering under diverse stress conditions.

The central hypothesis of this study is that the asynchronous evolution of stress and wave velocity in sandstone under triaxial compression is governed by the competition between two opposing mechanisms: effective stress, which promotes an increase in wave velocity by closing micro-cracks, and crack strain, which promotes a decrease in wave velocity by generating and propagating micro-cracks. This competition leads to a predictable relationship in which the peak wave velocity occurs before the peak stress, and the normalized stress at peak wave velocity decreases as confining pressure increases. The flowchart of this study is illustrated in Figure 1.

2. Asynchronous Response Characteristics of Stress and Wave Velocity

2.1. Test Material Preparation

The test specimens used in this study were obtained from the Ningtiaota mine, located in the northwest region of China, specifically in Shaanxi Province (see Figure 2a). The mine’s designed annual production capacity is 20 million tons. The coal-bearing strata are primarily composed of the Middle Jurassic Zhiluo Formation (J2z) and Yan’an Formation (J2y). The minable coal seams include Nos. 2-2, 3-1, 4-2, 4-3, and 5-2. The mining area has a relatively simple structural configuration, with strata generally occurring in a near-horizontal attitude. The test specimens were collected from the overlying strata of the 2-2 coal seam, with burial depths ranging from 100 m to 200 m. From the top downward, the primary lithologies comprised the following: fine sandstone, siltstone 1, mudstone, siltstone 2, medium sandstone, and fine-silt interbedded sandstone of the Zhiluo Formation; followed by fine-silt interbedded sandstone and medium sandstone of the Yan’an Formation; and finally, fine sandstone of the Yan’an Formation. The core sample had a diameter of 90 mm. Following the rock mechanics testing procedures recommended by the International Society for Rock Mechanics (ISRM), it was processed into standard-sized specimens measuring φ50 mm × 100 mm, with 13–15 specimens prepared per group [25,26]. To minimize experimental errors, the end portions were ground to ensure flatness control within 0.05 mm, followed by a drying treatment (see Figure 2b). After the specimens were thoroughly dried, the circumferential longitudinal wave velocities of each specimen were measured using a circumferential velocity anisotropy (CVA) detection system. Specimens exhibiting a wave velocity variation of less than 5% within each layer were selected for further analysis [26].

All specimens were collected from a single mining area to minimize geological variability and enable a systematic investigation of the asynchronous stress–wave velocity behavior under controlled conditions. Although the present findings are specific to this geological setting, the proposed mechanistic framework is expected to be applicable to similar sedimentary rock formations.

The basic physical parameters of the prepared specimens are summarized in

Table 1. Average density and average wave speed of the sampled rocks.

Formation	Lithology	Range of Rock Depth (m)	Mean Density of Rock (kg/m3)	Average Wave Speed (m/s)
Zhiluo	Fine sandstone	109.81–117.34	2144.3 ± 87.2	2369.9 ± 102.4
	Siltstone 1	120.31–126.53	2279.9 ± 65.3	2621.5 ± 118.7
	Mudstone	130.16–141.45	2386.0 ± 54.8	2947.0 ± 135.2
	Siltstone 2	143.73–149.61	2348.3 ± 71.2	2965.2 ± 121.5
	Sandstone	149.74–158.58	2246.9 ± 68.5	2432.0 ± 109.3
	Fine and siltstone interbed	159.02–165.29	2387.8 ± 59.4	3356.6 ± 148.3
Yan’an	Fine and siltstone interbed	167.57–172.67	2407.2 ± 62.1	3513.3 ± 156.8
	Sandstone	180.39–185.65	2155.6 ± 73.4	2499.9 ± 112.6
	Fine sandstone	187.79–198.72	2363.2 ± 61.7	3412.8 ± 139.5

Note: Values are presented as mean ± standard deviation; wave velocity variation ≤ 5% within each layer.

. Across the sampled lithologies, the mean density ranges from 2144 to 2407 kg/m³, and the average wave velocity ranges from 2369 to 3513 m/s. The interbedded fine siltstone of the Yan’an Formation exhibits the highest wave velocity (3513 m/s), whereas the fine sandstone of the Zhiluo Formation exhibits the lowest (2369 m/s). This variability reflects differences in mineral composition, grain size, and cementation, and provides a basis for understanding the range of mechanical and acoustic responses in subsequent analyses.

2.2. The Basic Physical Properties of Sandstones

To analyze the mineral compositions of sandstone specimens at varying depths, this study employed a Bruker D8 Advance X-ray diffractometer for sample detection. The analytical results are presented in Figure 3. The adiabatic quantitative analysis method was used to process the experimental results, obtaining the mineral composition data of rocks at various depths. The lithology and burial depth of the rock lead to a relatively high degree of variability in the mineral composition. Nevertheless, as shown in Figure 3, the primary mineral constituents are quartz, feldspar, and kaolinite, which collectively account for approximately 85% of the total mineral content, with the remaining components present in minor proportions. The rock formation components are relatively homogeneous, indicating that the rock was primarily formed through sedimentary processes. No apparent signs of rock mass intrusion were detected internally, and the overall structure exhibited a high degree of stability. As the burial depth increases, the quartz content gradually decreases, whereas the contents of kaolinite and feldspar show an increasing trend. Through lithological comparative analysis, we determined that medium sandstone exhibits the highest feldspar content, with relatively similar concentrations of quartz and kaolinite. In contrast, fine sandstone, siltstone, and fine-silt interbedded sandstone display relatively higher quartz content, while the levels of feldspar and kaolinite remain comparatively stable. Quartz and feldspar constitute the primary structural framework of the rock, while other minerals fill the interstitial spaces. The framework structure of medium sandstone remains relatively stable across varying burial depths. In contrast, the framework content of fine sandstone, siltstone, and fine-silt interbedded sandstone gradually decreases with increasing burial depth. This trend can be primarily attributed to the increasing in situ stress associated with greater depth.

2.3. Testing Apparatus and Test Plan

The loading equipment utilized in this experiment is the GCTS RTR-1000 Rock Mechanics Test System (see Figure 4a), which is housed in the State Key Laboratory of Coal Resources and Safe Mining at China University of Mining and Technology (Beijing) [24,27]. During the loading process, the axial and radial deformations of each specimen were continuously monitored in real time using a linear variable differential transformer extensometer. The confining pressures of 0, 2, 5, and 10 MPa were selected to represent the stress conditions corresponding to burial depths of approximately 100–200 m at the Ningtiaota mine, based on the average overburden density and depth estimates. According to the testing procedure recommended by the ISRM, a rock failure test was conducted on the ninth sandstone layer, which overlies the 2-2 coal seam. A strain loading rate of 0.02% per minute was applied, and owing to the limited availability of intact specimens from the same lithological layer, each test condition was repeated twice. When the specimen was fractured, the indenter of the loading system automatically retracted. During the loading process, the piezoelectric ceramic sheet within the indenter was employed to measure the axial ultrasonic velocity at both ends of the sandstone specimens across various depths (see Figure 4c). The ultrasonic frequency of 20 MHz was selected based on the sensor specifications of the GCTS system and the specimen dimensions. This frequency provides adequate penetration through 100 mm-long specimens while maintaining sufficient sensitivity to detect velocity variations associated with micro-crack evolution. Higher frequencies would lead to excessive signal attenuation, whereas lower frequencies would compromise spatial resolution. To minimize energy attenuation caused by ultrasonic wave propagation at the interface, a coupling agent was utilized to enhance interfacial coupling.

Owing to the limited availability of intact, homogeneous specimens meeting the selection criteria from the same lithological layer, two specimens were tested per confining pressure condition. Although this sample size limits statistical power, the consistency of the observed trends across confining pressures supports the reliability of the main findings.

2.4. Typical Experimental Results of Asynchronous Stress and Wave Velocity

Figure 5 illustrates the relationship between stress, P-wave velocity, and strain during the loading process of medium sandstone from the Yan’an formation under varying confining pressure conditions. As illustrated in Figure 5, the deformation and failure characteristics of sandstone remain largely consistent under confining pressures of 0, 2, 5, and 10 MPa. As the confining pressure increases, the elastic modulus increases correspondingly. The peak stress of sandstone increases with increasing confining pressure, and its post-peak residual strength also increases with increasing confining pressure. Under confining pressure, the initiation and propagation of micro-cracks are further suppressed. As the magnitude of confining pressure increases, its restraining effect on rock mass deformation becomes more pronounced. This phenomenon enhances the sandstone’s resistance to deformation, thereby increasing both its peak strength and residual strength.

Under varying confining pressure conditions, the wave velocity generally exhibits consistent variation trends, following a distinct pattern characterized by an “initial concave upward increase—linear increase—convex upward increase—oscillatory fluctuations—subsequent decline.” As the axial pressure increases, the internal cracks within the specimen gradually close, resulting in a corresponding increase in P-wave velocity $v_P$ With further elevation of the axial pressure, the existing internal cracks continue to close without the formation of new cracks. Consequently, the P-wave velocities $\mathrm{e}\mathrm{x}\mathrm{h}\mathrm{i}\mathrm{b}\mathrm{i}\mathrm{t}$ an approximately linear upward trend, and the rate of change in wave velocity stabilizes. As the axial pressure continues to increase, crack closure and propagation gradually reach a state of dynamic equilibrium. The wave velocity exhibits fluctuating variations. Although the wave velocity continues to rise, the rate of change progressively decreases. When the stress approaches its peak value ${\sigma }_f$, most of the maximum P-wave velocities $v_{P,\max }$ are observed during this stage. The wave velocity initially fluctuates upward, culminating in a peak, followed by a decline. This behavior can be attributed to the initiation, propagation, and coalescence of internal cracks. When a macroscopic crack forms and propagates, the stress enters the post-peak phase. At this stage, the P-wave velocities $v_P$ decrease rapidly. Some rocks experience a complete loss of load-bearing capacity owing to fracturing, resulting in a decline in wave velocity to 0 m/s. As confining pressure increases, the initial wave velocity gradually rises, although the rate of increase diminishes progressively. This behavior is closely associated with the confining pressure in suppressing the initiation and propagation of micro-cracks within the rock matrix.

Notably, during the loading process of sandstone, the occurrence of the wave velocity peak precedes the stress peak. This has resulted in a spatiotemporal mismatch between the peaks of their distribution patterns. As illustrated in the figure, the strain corresponding to the peak wave velocity is lower than that associated with the peak stress. Similarly, the stress at the peak wave velocity is less than the peak stress. Moreover, as the confining pressure increases, the difference between these two strains becomes more pronounced, and the disparity between the respective stress values also increases.

It is noteworthy that during the loading process of sandstone, the peak wave velocity occurs earlier than the peak stress, leading to a temporal and spatial mismatch between these two parameters. As illustrated in Figure 5, the strain ${\epsilon }_{v_{\max }}$ associated with the peak wave velocity is lower than that corresponding to the peak stress ${\epsilon }_f$; similarly, the stress ${\sigma }_{v_{\max }}$ associated with the peak wave velocity is lower than the peak stress ${\sigma }_f$. Additionally, with the increase in confining pressure, the strain difference gradually widens, and the stress difference correspondingly increases.

Owing to differences in confining pressure, the peak strain and peak stress of sandstone during the compression process vary, rendering it difficult to directly compare the stress and strain values corresponding to the peak wave velocity. Therefore, based on experimental data regarding stress and wave velocity during the sandstone compression process, the relevant stress and strain parameters were normalized. The ratio of stress ${\sigma }_{v_{\max }^P}$, which corresponds to the peak wave velocity under varying confining pressures, to the peak stress ${\sigma }_f$, as well as the ratio of strain ${\epsilon }_{v_{\max }^P}$, corresponding to the peak wave velocity under different confining pressures, to the peak strain ${\epsilon }_f$, are calculated and presented in Figure 6. As illustrated in the figure, the parameter ${\sigma }_{v_{\max }^P}/{\sigma }_f$ is predominantly distributed between 0.84 and 0.99. The normalized stress at peak wave velocity ${\sigma }_{v_{\max }^P}/{\sigma }_f$ is 0.97 ± 0.02 at 0 MPa, decreasing to 0.87 ± 0.03 at 10 MPa. With increasing confining pressure, the parameter ${\sigma }_{v_{\max }^P}/{\sigma }_f$ exhibits a slight decrease. The normalized value ${\epsilon }_{v_{\max }^P}/{\epsilon }_f$ of the peak strain corresponding to wave velocity in the comparison is primarily within the range of 0.73 to 0.98. In contrast to ${\sigma }_{v_{\max }^P}/{\sigma }_f$, ${\epsilon }_{v_{\max }^P}/{\epsilon }_f$ exhibits a clear decreasing trend as the confining pressure increases. This phenomenon is associated with the initiation of internal cracks within the rock. As axial pressure increases, internal cracking begins to occur in the sandstone. Although such micro-cracking does not result in structural instability or macroscopic failure of the sandstone, it can reduce wave velocity. Consequently, this explains the occurrence where the parameter ${\sigma }_{v_{\max }^P}/{\sigma }_f$ is less than 1. Confining pressure can suppress the propagation of micro-cracks. In other words, as the confining pressure increases, the development and coalescence of micro-cracks occur at a slower rate. Consequently, the value of ${\sigma }_{v_{\max }^P}/{\sigma }_f$ decreases slightly with increasing confining pressure.

3. Research on the Mechanical Mechanism of Asynchrony Between Stress and Wave Velocity

Building on the experimental observations of stress–wave velocity asynchrony presented in Section 2, this section develops a theoretical framework to explain the underlying mechanisms. The framework comprises three components: a damage constitutive model based on crack strain, a wave velocity evolution model incorporating competing mechanisms, and an analytical derivation of the condition for peak wave velocity occurrence.

3.1. Damage Constitutive Model Based on Crack Strain

The failure process of a rock mass is the result of the progressive accumulation of internal micro-cracks. Therefore, the essence of material damage can be attributed to the initiation, propagation, and coalescence of these micro-cracks within the rock mass. The evolution of such cracks results in irreversible plastic deformation of the rock. Therefore, crack strain is utilized to characterize the degree of damage and is adopted as the distribution variable for the microelement strength of coal and rock masses.

The use of axial crack strain as a damage indicator is supported by extensive experimental evidence. Eberhardt et al. [8] demonstrated, using AE monitoring, that the onset of significant axial crack strain coincides with the transition from stable to unstable crack propagation. Recently, Wang et al. [14] employed micro-CT scanning to directly visualize micro-crack evolution during compression, confirming that axial crack strain correlates with the volumetric density of micro-cracks. These findings justify the selection of axial crack strain as the primary state variable for damage characterization in our model.

Based on the crack strain calculation method proposed by Martin [6], this study quantitatively presented the evolution of internal cracks in rock during the compression process, thereby providing a criterion for identifying the characteristic stress stages of rock.

The calculation formula for crack strain is expressed as follows:

$$\left\{\begin{array}{l}{\epsilon }_1^c={\epsilon }_1-{\displaystyle \frac{1}{E}}\left({\sigma }_1-2\mu {\sigma }_3\right)={\epsilon }_1-{\displaystyle \frac{1}{E}}\left({\sigma }_{sd}+\left(1-2\mu \right){\sigma }_3\right)\\ {\epsilon }_2^c={\epsilon }_3^c={\epsilon }_3-{\displaystyle \frac{1}{E}}\left[\left(1-\mu \right){\sigma }_3-\mu {\sigma }_1\right]={\epsilon }_3-{\displaystyle \frac{1}{E}}\left[\left(1-2\mu \right){\sigma }_3-\mu {\sigma }_{sd}\right]\end{array}\right.,$$

(1)

Among these, ${\epsilon }_1^c$ denotes the axial crack strain; ${\epsilon }_2^c$ and ${\epsilon }_3^c$ represent the circumferential crack strains; ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ denote the three principal stresses acting on the rock in space; E refers to the elastic modulus, μ refers to Poisson’s ratio; ${\epsilon }_1$ represents the initial axial strain; and ${\epsilon }_2$ and ${\epsilon }_3$ correspond to the initial circumferential strains. We can observe from Equation (1) that the axial crack strain initially experiences a phase of slow growth, subsequently stabilizes, and finally undergoes rapid expansion; in contrast, the circumferential crack strain first decreases and then gradually increases. As illustrated in Figure 5, a one-to-one correspondence exists between the axial crack strain and the axial strain. In contrast, the relationship between the circumferential crack strain and the axial strain is nonunique. Furthermore, the damage process in rock is irreversible. Therefore, selecting the axial crack strain as the variable for describing the microelement strength distribution of coal and rock mass is more appropriate for establishing a constitutive model of rock damage.

The Weibull distribution is widely adopted in rock damage mechanics due to its flexibility in describing the statistical distribution of micro-defect strengths [28,29]. Assuming that the strength of the rock mass microelement follows a Weibull distribution, its probability density function is expressed as follows:

$$p\left({\epsilon }_1^c\right)={\displaystyle \frac{m}{{\epsilon }_0}}{\left({\displaystyle \frac{{\epsilon }_1^c}{{\epsilon }_0}}\right)}^{m-1}\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^c}{{\epsilon }_0}}\right)}^m\right],$$

(2)

In the formula, $p\left({\epsilon }_1^c\right)$ represents the random distribution function of the strength of rock mass microelements, whereas ${\epsilon }_0$ and m denote the Weibull distribution parameters that can be determined based on experimental data.

Within any infinitesimal interval [${\epsilon }_1$, ${\epsilon }_1+d{\epsilon }_1$], the failure of the rock mass evolves as a continuous process. Consequently, under the crack strain ${\epsilon }_1^c$ induced by an arbitrary load, the damage variable D of the rock mass can be formulated as follows:

$$D=1-\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^c}{{\epsilon }_0}}\right)}^m\right],$$

(3)

During the triaxial compression test, the deviator stress ${\sigma }_{sd}$ is defined by the difference between the axial stress ${\sigma }_1$ and the confining pressures ${\sigma }_2$ or ${\sigma }_3$, as expressed in the following relationship:

$$\left\{\begin{array}{l}{\sigma }_{sd}={\sigma }_1-{\sigma }_3\\ {\sigma }_2={\sigma }_3\end{array}\right.,$$

(4)

During the triaxial compression test, under a given load condition that results in crack strain ${\epsilon }_1^c$, the damage variable D of the rock mass can be expressed as follows:

$$D=1-\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1-\frac{{\sigma }_{sd}+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right],$$

(5)

Based on Lemaitre’s strain equivalence principle, the stress–strain behavior of rock under triaxial compression can be formulated as follows:

$${\sigma }_{sd}=E{\epsilon }_1\left(1-D\right)+\left[{\sigma }_{sd}^r-\left(2\mu -1\right){\sigma }_3\right]D+\left(2\mu -1\right){\sigma }_3,$$

(6)

Among these parameters, ${\sigma }_{sd}^r$ denotes the residual deviatoric stress. By substituting Equation (5) into Equation (6), the rock mass damage constitutive model that incorporates crack strain can be derived, as follows:

$$\begin{array}{ll}{\sigma }_{sd}& =E{\epsilon }_1\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1-\frac{{\sigma }_{sd}+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]\\ & +\left[{\sigma }_{sd}^r-\left(2\mu -1\right){\sigma }_3\right]\left\{1-\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1-\frac{{\sigma }_{sd}+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]\right\}+\left(2\mu -1\right){\sigma }_3\end{array},$$

(7)

According to the research cited in Ref. [28], selecting the peak strength point is a widely adopted method for determining distribution parameters in the analysis of rock stress–strain curves. This approach is based on the following characteristic: at the peak stress point (${\epsilon }_1^p,{\sigma }_{sd}^p$), the rock achieves its maximum strength, and the slope of the stress–strain curve at this point is zero.

$$\begin{array}{ll}{\sigma }_{sd}^p& =E{\epsilon }_1^p\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]+\\ & \left[{\sigma }_{sd}^r-\left(2\mu -1\right){\sigma }_3\right]\left\{1-\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]\right\}+\left(2\mu -1\right){\sigma }_3\end{array},$$

(8)

$$\begin{array}{ll}{\displaystyle \frac{\partial {\sigma }_{sd}^p}{\partial {\epsilon }_1^c}}=& E\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]-\left[E{\epsilon }_1^p-{\sigma }_{sd}^r+\left(2\mu -1\right){\sigma }_3\right]\exp \left[-{\left({\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^m\right]\cdot \\ & {\displaystyle \frac{m}{{\epsilon }_0}}{\left({\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\epsilon }_0}}\right)}^{m-1}\left(1-{\displaystyle \frac{1}{E}}{\displaystyle \frac{\partial {\sigma }_{sd}^p}{\partial {\epsilon }_1^p}}\right)\\ & \begin{array}{c}=0\end{array}\end{array},$$

(9)

The values of the variable m and ${\epsilon }_0$ can be determined by simultaneously solving Equations (8) and (9).

$$m={\displaystyle \frac{E{\epsilon }_1^p-{\sigma }_{sd}^p+\left(2\mu -1\right){\sigma }_3}{\left[E{\epsilon }_1^p-{\sigma }_{sd}^r+\left(2\mu -1\right){\sigma }_3\right]\cdot \ln \frac{E{\epsilon }_1^p-{\sigma }_{sd}^p+\left(2\mu -1\right){\sigma }_3}{{\sigma }_{sd}^p-{\sigma }_{sd}^r}}},$$

(10)

$${\epsilon }_0={\displaystyle \frac{{\epsilon }_1^p-\frac{{\sigma }_{sd}^p+\left(1-2\mu \right){\sigma }_3}{E}}{{\left[\ln \frac{E{\epsilon }_1^p-{\sigma }_{sd}^p+\left(2\mu -1\right){\sigma }_3}{{\sigma }_{sd}^p-{\sigma }_{sd}^r}\right]}^{\frac{1}{m}}}},$$

(11)

Equations (10) and (11) represent the distribution parameters of the desired damage constitutive model. By substituting these equations into Equation (7), the damage constitutive model for rock mass based on crack strain can be derived.

3.2. Wave Velocity Evolution Model Based on Crack Strain

During the progressive failure process of rocks, a significant correlation exists between variations in wave velocity and micro-crack density [29]. The micro-crack density d serves as a critical parameter for characterizing the evolution of internal cracking in rocks. According to damage-mechanics theory, the rock mass can be idealized as a homogeneous continuum with an effective modulus, in which discontinuities such as pores and fractures are assumed to be uniformly distributed. Under this theoretical framework, the micro-crack density d can be equated to the fracture coefficient L_s, and can be determined in relation to the rock mass integrity coefficient K_v. The rock mass integrity coefficient, K_v, is a parameter employed to characterize the extent of joint and fracture development within the rock mass. It is typically defined as the ratio of longitudinal wave velocities measured in the rock mass to those obtained from an intact rock block (nonporous bedrock).

$$L_s=1-K_v=\frac{{\stackrel{—}{v_p}}^2-v_P^2}{{\stackrel{—}{v_p}}^2}=1-{\left(\frac{v_p}{\stackrel{—}{v_p}}\right)}^2=d,$$

(12)

Among these, $v_P$ denotes the longitudinal wave velocity of the rock mass, whereas $\overline{v_P}$ refers to the longitudinal wave velocity of the rock block, which can be calculated based on the mineral composition of the rock.

Research indicates that micro-crack density is influenced by multiple factors, primarily attributed to the effects of effective stress and crack strain. Specifically, variations in effective stress are the dominant factor contributing to the reduction in micro-crack density. From the perspective of wave velocity analysis, an increase in effective stress corresponds to a rise in wave velocity and a simultaneous decrease in micro-crack density. Conversely, an increase in crack strain results in higher micro-crack density, which in turn reduces wave velocity.

There exists

$$d=d^{+}+d^{-},$$

(13)

Among them, $d^{+}$ denotes the increase in micro-crack density, whereas $d^{-}$ denotes the decrease in micro-crack density.

As indicated by the above-mentioned analysis of crack density, the decrease in micro-crack density is primarily attributed to the influence of effective stress ${\sigma }_e$. The negative exponential form is chosen based on the analogy with stress-dependent permeability in fractured rocks [30], reflecting the rapid reduction in micro-crack density as effective stress compresses existing cracks. Based on the observation, we can infer that the effect of effective stress on micro-crack density within the rock also conforms to a similar negative exponential pattern. This relationship can be expressed explicitly as follows:

$$\begin{array}{ll}{d}^{-}& =A\exp \left[-(B{\sigma }_e)\right]+d_0^{-}\\ & =A\exp \left[-B({\sigma }_{sd}+3{\sigma }_3)\right]+d_0^{-}\end{array}$$

(14)

Among them, A, B, and $d_0^{-}$ are all fitted parameters. During data selection, prioritizing the wave velocity change data of the material before it enters the elastic stage is recommended.

The negative exponential form of Equation (14) is motivated by two considerations. First, previous studies on stress-dependent permeability in fractured rocks [30] have demonstrated that effective stress reduces pore and fracture apertures exponentially. Second, the effective stress–wave velocity data from the initial loading stage (before crack initiation) exhibit a concave upward trend that is described by a negative exponential function. The parameters A and B scale the magnitude of the wave velocity increase, whereas α controls the rate at which micro-crack density decreases with effective stress.

The increase in micro-crack density $d^{+}$ is primarily influenced by the strain induced by crack propagation. The power-law form captures the accelerating micro-crack growth observed experimentally as crack strain increases, consistent with fracture mechanics-based damage evolution models [30]. The growth amount $d^{+}$ of micro-crack density increases as the crack dilatation strain increases, and this evolution process can be mathematically expressed by the following equation:

$$d^{+}=1-\exp \left[-{({\displaystyle \frac{{\epsilon }_{cv}^{+}}{C}})}^F\right],$$

(15)

Among them, C and F are fitting parameters, whereas ${\epsilon }_{cv}^{+}$ represents the crack expansion strain, defined as ${\epsilon }_{cv}^{+}=2{\epsilon }_3^c$.

The power-law form of Equation (15) is adopted based on fracture mechanics theory, which suggests that micro-crack growth follows a power-law relationship with strain as cracks propagate from stable to unstable stages [31]. The parameter C scales the overall contribution of crack strain to micro-crack density increase, while F controls the rate of acceleration as crack strain develops. This form captures the experimentally observed behavior in which wave velocity from crack strain remains relatively stable in the initial stages and accelerates near peak stress.

Sensitivity analysis was performed by perturbing each parameter by ±10% and observing the change in model predictions. The model is most sensitive to α and F, with a 10% change resulting in approximately 8–12% change in peak wave velocity prediction. The parameters A, and B exhibited moderate sensitivity, whereas C exhibited lower sensitivity. These sensitivities are consistent with the physical roles of each parameter.

It can be observed from Equations (14) and (15) that the wave velocity is related to the strain and stress associated with the circumferential crack in the following manner:

$$v_P=\overline{v_P}\sqrt{1-\left\{A\exp \left[-B({\sigma }_{sd}+3{\sigma }_3)\right]+d_0^{-}\right\}-\left\{1-\exp \left[-{({\displaystyle \frac{2{\epsilon }_c^3}{C}})}^F\right]\right\}},$$

(16)

By integrating Equation (7) with Equation (16), a wave velocity model based on crack strain can be formulated.

3.3. Theoretical Derivation of Asynchrony Between Stress and Wave Velocity

According to the aforementioned wave velocity model, variations in rock wave velocity result from the combined effects of crack strain and stress. The experimental data indicate that the value of $v_{\max }^P$ is lower than that of $\overline{v_P}$, and the wave velocity remains positive throughout. Consequently, Equation (16) can be appropriately reformulated.

$${\left({\displaystyle \frac{v_P}{\overline{v_P}}}\right)}^2=1-\left\{A\exp \left[-B({\sigma }_{sd}+3{\sigma }_3)\right]+d_0^{-}\right\}-\left\{1-\exp \left[-{({\displaystyle \frac{2{\epsilon }_c^3}{C}})}^F\right]\right\},$$

(17)

When $v_P$ reaches its maximum value $v_{\max }^P$, ${\left({\displaystyle \frac{v_P}{\overline{v_P}}}\right)}^2$ also attains its peak. Conversely, when ${\left({\displaystyle \frac{v_P}{\overline{v_P}}}\right)}^2$ achieves its maximum, $v_P=v_{\max }^P$.

By performing partial differentiation of Equation (17) with respect to ${\sigma }_{sd}$, the resulting expressions are as follows:

$${\displaystyle \frac{\partial {\left(\frac{v_P}{\overline{v_P}}\right)}^2}{\partial {\sigma }_{sd}}}=AB\exp \left[-B({\sigma }_{sd}+3{\sigma }_3)\right]-{\displaystyle \frac{2\mu F}{E}}·{\displaystyle \frac{{\left(2{\epsilon }_c^3\right)}^{F-1}}{C^F}}\exp \left[-{({\displaystyle \frac{2{\epsilon }_c^3}{C}})}^F\right],$$

(18)

Based on the extreme value theorem, Equation (19) can be derived.

$${\sigma }_{v_{\max }^P}={\displaystyle \frac{\ln \left(AB\right)-\ln (\frac{2\mu F}{E})-\ln (\frac{{\left(2{\epsilon }_c^3\right)}^{F-1}}{C^F})-{(\frac{2{\epsilon }_c^3}{C})}^F}{B}}-3{\sigma }_3,$$

(19)

Furthermore, combining this with Equation (7), the strain value corresponding to the peak wave velocity can be obtained. As expressed in Equation (19), with an increase in confining pressure, the deviatoric stress associated with the peak wave velocity decreases.

4. Comparative Analysis of Theoretical Models and Experimental Outcomes

4.1. Comparative Analysis of Mechanical Test Results and Theoretical Model Predictions

Based on the rock mass damage constitutive model that incorporates crack strain, the axial crack strain ${\epsilon }_1^c$, damage variable D, which are key parameters for characterizing the deformation and failure behavior of rock, and the corresponding axial stress values throughout the progressive failure process can be calculated by substituting experimental data into Equations (1), (5) and (7). Figure 7 illustrates the variation trends of calculated stress, axial crack strain, and damage variables of medium sandstone from the Yan’an formation as a function of axial strain under varying confining pressure conditions. As illustrated in Figure 7, under varying confining pressure conditions, the trend of the damage variable curve is generally consistent with that of the axial crack strain. Specifically, as the axial strain increases, the damage variable exhibits a pattern characterized by a slight increase in the initial stage, relative stability in the middle stage, and a rapid rise in the subsequent stage.

Figure 7 variation in deviatoric stress, axial crack strain, and damage variable with axial strain for sandstone under different confining pressures: (a) 0 MPa, (b) 2 MPa, (c) 5 MPa, (d) 10 MPa. Solid black lines represent experimental stress–strain curves, red dashed lines represent model-predicted stress using Equation (7), green dotted lines represent axial crack strain ${\epsilon }_1^c$ calculated from Equation (1), and blue dash-dot lines represent damage variable D calculated from Equation (5). Quantitative validation metrics (

Table 2. Quantitative validation metrics for the stress–strain model.

Confining Pressure (MPa)	R2	Normalized RMSE	MAPE (%)
0	0.994	0.05	3.8
2	0.990	0.04	3.2
5	0.989	0.06	4.5
10	0.972	0.07	5.1

) present R² values of 0.972–0.994, normalized RMSE of 0.04–0.07, and MAPE of 3.2–5.1%, confirming the model’s accuracy across all confining pressures.

The choice of axial crack strain as the primary damage variable is motivated by the experimental observation (Figure 7) that axial crack strain exhibits a monotonic, irreversible increase throughout the loading process, whereas circumferential crack strain shows nonunique behavior. This aligns with the micro-crack evolution patterns documented in previous studies [6,14].

During the rock loading process, the material experiences three sequential stages: microdamage, stable damage development, and accelerated damage. This progressive transition aligns with the evolution of internal micro-cracks within the rock under increasing load, as illustrated in Figure 8. During the rock loading process, as the pressure increases, the internal micro-cracks within the rock gradually close under compressive stress. However, owing to the end effect, a localized tensile stress zone develops at the tips of the micro-cracks, thereby increasing the axial strain. As the applied stress increases gradually, the closure degree at the specimen’s ends reaches its maximum. Under continued stress, the micro-cracks experience lateral deformation, and the tensile stress becomes increasingly concentrated at the tips of the micro-cracks. As the stress continues to increase, tensile cracking initiates at the tip of the micro-crack, and the transverse deformation correspondingly intensifies, leading to the crack propagating through the entire interior of the rock. Under increasing confining pressure, the position of the stable stage gradually decreases, whereas the strain proportion during this stage correspondingly increases.

In addition, it is noteworthy that under varying confining pressure conditions, the overall trend of the damage evolution process in coal rock remains largely consistent. However, certain discrepancies are observed in the values of the damage variables and axial crack strain under triaxial compression testing. When the axial stress gradually increases to the peak strength, the corresponding axial strain also increases with the rise in confining pressure. This observation further suggests that the presence of confining pressure enhances the capacity of coal and rock to resist deformation and failure, promotes the ductility of their internal structures, and thereby delays the onset of failure to some extent.

To validate the rationality and accuracy of the damage constitutive model based on axial crack strain, this study compared the established model and experimental data obtained under various confining pressure conditions (refer to Figure 7). Furthermore, the damage constitutive relationship curves for coal and rock under different confining pressures are plotted based on the computational results. As shown in Figure 7, the theoretical stress–strain curves from the damage constitutive model (Equation (7)) closely match the experimental data across all confining pressures. Quantitative validation metrics (Table 2) confirm the model’s performance: R² values ranged from 0.972 to 0.994, normalized RMSE values from 0.04 to 0.07, and MAPE values from 3.2% to 5.1%. The highest agreement was observed at 2 MPa (R² = 0.990, MAPE = 3.2%), whereas the lowest was at 10 MPa (R² = 0.972, MAPE = 5.1%), reflecting slightly increased scatter at higher confinement. Throughout the entire loading process, the axial strain of the coal and rock increases gradually with the increment of axial stress. In the elastic stage, both the model and test curves are approximately straight lines, indicating that the elastic modulus remains relatively unchanged and the mechanical properties of the coal and rock are stable. When the axial stress reaches the yield stage, both the model and test curves exhibit nonlinear variation characteristics, gradually bending toward the X-axis. The rock mass begins to exhibit strain-softening phenomena, and micro-cracks start to emerge and expand within it, interpenetrating to form macroscopic cracks and reducing bearing capacity. By comparing and analyzing the overall trends of the model and test curves, the damage constitutive model considering crack strain can accurately and reasonably describe the deformation and failure characteristics in the coal and rock stages.

To assess the model’s predictive capability beyond the calibration dataset, parameters calibrated at 2 MPa were used to predict the stress–strain response at 5 MPa. The prediction achieved R² = 0.89 and normalized RMSE = 0.11, indicating that the model captures the main trends even when parameters are not optimized for the target condition. This provides preliminary evidence of the model’s structural validity.

The model captures the full range of deformation behavior, including the linear elastic stage, the nonlinear yield stage, and the post-peak strain-softening stage. The slightly lower R² at 10 MPa (0.972) is attributed to increased micro-crack heterogeneity under high confinement, which introduces more scatter in the experimental data. Nevertheless, the normalized RMSE remains below 0.08 for all conditions, indicating that the model’s prediction error is less than 8% of the peak stress, which is considered acceptable for rock constitutive modeling [28].

4.2. Comparative Analysis of the Wave Velocity Test Results with the Theoretical Model

By substituting the experimental data into Equation (16), the variation curve of wave velocity during the progressive failure of the rock can be obtained, as shown in Figure 9.

It can be observed from the figure that the wave velocity $v_P^{+}$ caused by the effective stress shows a trend of first increasing and then stabilizing. Under different confining pressures, the final $v_P^{+}$ stabilizes within the range of 2900–3050 m/s. The wave velocity, $v_P^{-}$, caused by crack strain, shows a trend of first stabilizing and then increasing. Moreover, as the confining pressure increases, the protrusion value of the wave velocity $v_P^{-}$ increases. This causes the peak wave velocity to occur ahead of the peak stress, and the lead value increases with the increase in confining pressure. This is the same as the result discussed in Section 2.4. As the confining pressure increases, the micro-cracks inside the rock are further compressed and closed, and the increment of the wave velocity with the rise in axial pressure decreases (the increment of the wave velocity is 642 m/s at 0 MPa, 405 m/s at 2 MPa, 224 m/s at 5 MPa, and 221 m/s at 10 MPa). Therefore, as the confining pressure increases, the formation of tiny cracks also causes significant changes in wave velocity. The test data were inserted into Equations (7) and (19) to obtain the stress ${\sigma }_{v_{\max }^P}$ and strain ${\epsilon }_{v_{\max }^P}$ corresponding to the peak wave velocity under different confining pressures. The obtained results were compared to the test results, as shown in Figure 10.

The parameters A, B, and α in Equation (14) were determined by fitting the effective stress–wave velocity data obtained during the initial loading stage (prior to crack initiation) using a nonlinear least-squares method. Similarly, the parameters C and F in Equation (15) were calibrated using the crack strain–wave velocity data from the stable crack propagation stage. All fitting procedures were performed separately for each confining pressure condition. The fit quality was evaluated using the coefficient of determination (R²), with all values exceeding 0.929, indicating good agreement between the model and experimental data.

Notably, the model parameters were calibrated using the same experimental dataset presented in this study. Therefore, the comparisons shown in Figure 7, Figure 9 and Figure 10 represent model fitting rather than independent validation. A rigorous validation would require additional experimental data from different rock types or testing conditions, which is beyond the scope of the present study but represents an important direction for future work.

5. Discussion

5.1. Method for Distinguishing Characteristic Stress Stages

It can be observed from Equation (2) that the transition point of crack strain can be considered the distinguishing point of the characteristic stress stage. Figure 11 shows the progressive failure process curve of a typical specimen, which can be divided into the following stages: I-compaction stage, II-elastic stage, III-stable crack propagation stage, IV-unstable crack propagation stage, and V-post-peak stage. An unchanging stage exists for crack strain. When both the axial crack strain and the circumferential crack strain are stable around a fixed value, it indicates that with stress loading, no significant changes have occurred in the internal cracks of the rock. This stage is called the elastic stage. Before this stage is the compaction stage, and the distinguishing point is the compaction stress ${\sigma }_{cc}$. After the elastic stage, a trend exists that the strain variation amplitude of axial cracks is relatively small, whereas that of circumferential cracks increases. This stage is the stable crack propagation stage, and the distinguishing point from the elastic stage is the crack initiation stress ${\sigma }_{ci}$. When the axial crack strain increases significantly, it indicates that the bearing capacity of the specimen gradually decreases until the peak stress ${\sigma }_f$ is reached. This stage belongs to the unstable crack propagation stage and is classified as damage stress ${\sigma }_{cd}$. The post-peak stage follows the peak stress, and the final stop point is the residual stress ${\sigma }_{cr}$. The corresponding strains are the compaction strain ${\epsilon }_{cc}$, cracking strain ${\epsilon }_{ci}$, peak strain ${\epsilon }_f$, damage strain ${\epsilon }_{cd}$, and residual strain ${\epsilon }_{cr}$.

5.2. Relationship Between Peak Wave Velocity and Damage Variables

According to the theoretical derivation in Section 3, the change in wave velocity is the result of effective stress and crack strain. Among them, the change rate of effective stress decreases as the damage variable increases. Therefore, a specific relationship exists between wave velocity deformation and the damage variable.

In the mechanical constitutive model, protruding points of the damage variable exist. According to Equation (7), the axial crack strain and the damage variable D deform synchronously during the rock compression process. In other words, the strain protrusion value of the axial crack is the same as that of the damage variable D; therefore, the protrusion value of the damage variable D is related to the damage stress ${\sigma }_{cd}$. In the calculation of the wave velocity model, a protrusion value exists for the wave velocity $v_P^{-}$ that changes owing to crack strain. According to Equation (16), the wave velocity $v_P^{-}$ changed owing to the crack strain, which is related to the expansion crack ${\epsilon }_{cv}^{+}$. Therefore, the protrusion value of the wave velocity $v_P^{-}$ changed owing to the crack strain, which is associated with the crack initiation stress ${\sigma }_{ci}$. Thus, based on the results of the stress test, the variations in the cracking stress ${\sigma }_{ci}$ and the damage stress ${\sigma }_{cd}$ of sandstone under different confining pressure environments were obtained, as shown in Figure 12.

Characteristic stress is a crucial node in the progressive failure process of rocks. Numerous studies have shown [30,32] that the ${\sigma }_{ci}/{\sigma }_f$ values of most intact sandstones range from 0.525 to 0.7, and the ${\sigma }_{cd}/{\sigma }_f$ values range from 0.75 to 0.95. Overall, the ${\sigma }_{ci}/{\sigma }_f$ values and ${\sigma }_{cd}/{\sigma }_f$ values of different sandstones (Figure 9) conform to the general characteristic stress law of rocks. This agreement validates our crack-strain-based stress stage identification method. The characteristic stress of some specimens exceeded this area, and this was particularly evident in the interbedded fine siltstone of the Zhiluo Formation. This was related to the inclusion of fine coal lines within the interbedded fine siltstone of the Zhiluo Formation, which led to the ${\sigma }_{ci}/{\sigma }_f$ values of the interbedded fine siltstone of the Zhiluo Formation being higher than the general characteristic stress law of rocks.

Figure 13 shows the normalized values of the magnitudes of each stress stage and the peak wave velocity of sandstone under different confining pressures in the Yan’an Formation. The dotted lines in the figure represent the boundaries between each stress stage, and the solid lines indicate the positions of the peak wave velocity. As shown in the figure, with the increase in confining pressure, the proportion of the compaction stage remains nearly unchanged. In contrast, the proportion of the stable crack propagation stage decreases, and the proportions of the elastic stage and the unstable crack propagation stage increase. This behavior arises because higher confining pressure forces the grains into tighter contact, causing the pores between particles to be compressed or even closed. Therefore, when axial pressure continues to increase, the pore space becomes further compacted and intergranular squeezing intensifies, rendering particle slippage increasingly difficult. In addition, stress transmission between internal particles becomes more efficient. Once slippage occurs and micro-cracks initiate, these cracks become more prone to rapid propagation and coalescence. Nevertheless, owing to the strong confinement, the reduction in the rock’s load-bearing capacity is mitigated, causing the stress–strain curve to transition from a brittle response at low confining pressure to a ductile response at high confining pressure.

By comparing the peak wave velocity variation curve in the figure with the boundary line between the stable and unstable crack propagation stages, it can be observed that the two lines are approximately parallel.

In other words, after stress normalization, the stress normalization difference between the peak wave velocity and the protrusion point of the damage variable is approximately equal, and the difference is approximately 0.1 ${\sigma }_f$. This finding, derived from the model and validated experimentally, suggests that the wave velocity peak serves as a reliable precursor to the damage variable inflection point. The consistent offset between the peak wave velocity and the damage variable (Figure 13) provides a quantitative basis for using wave velocity monitoring as an early warning indicator. Under confining pressure, the difference shows a slight decreasing trend, which is also consistent with the result of Equation (16). This aligns with the findings of Shirole et al. [21] on a different sandstone. Despite differences in lithology and test conditions, our study yielded predicted normalized stresses at peak wave speeds (${\sigma }_{v_{\max }^P}/{\sigma }_f$ ≈ 0.85–0.95) that fall within the range reported in their study, offering preliminary evidence of qualitative consistency across diverse datasets.

Equation (19) predicts that the normalized stress at peak wave velocity decreases as confining pressure increases. This prediction is quantitatively consistent with the experimental data shown in Figure 6a, where the normalized stress ranges from 0.84 to 0.99 and exhibits a slight decreasing trend with confining pressure.

5.3. Comparison and Analysis of Damage–Wave Velocity Coupling Models

Existing models often define damage empirically based on modulus degradation or AE counts. In contrast, our model explicitly derives damage from axial crack strain (Equation (5)), thereby providing a clear physical interpretation. For wave velocity evolution, existing approaches typically rely on empirical fitting without a clear physical basis, whereas our model mechanistically captures the competition between effective stress (which increases wave velocity) and crack strain (which reduces wave velocity) through Equations (14)–(16). Moreover, many existing models focus only on pre-peak response, whereas our model successfully reproduces both strain softening and velocity drop in the post-peak stage (Figure 5 and Figure 9). Finally, existing studies typically employ separate models for mechanical behavior and acoustic response, whereas our work provides a single constitutive framework that simultaneously predicts both responses, validated under multiple confining pressures (Figure 7 and Figure 10). Collectively, compared to existing models, the proposed approach offers three distinctive advantages: (1) a physically transparent decomposition of wave velocity evolution into competing effective stress and crack strain mechanisms; (2) a unified constitutive framework that simultaneously predicts stress–strain and velocity–strain responses; and (3) the use of axial crack strain as a single internal variable that links micro-crack evolution to both mechanical degradation and acoustic response. However, the model’s current limitations include a lack of direct microstructural validation and the use of the same dataset for calibration and evaluation, which should be addressed in future studies.

Future work should validate the proposed model against independent datasets, including rocks from different geological origins, different lithologies, and under different loading paths. Such validation would further establish the generalizability of the mechanistic framework. Although this study presented a mechanistic explanation for the asynchronous stress–wave velocity behavior, direct microstructural validation using techniques such as AE monitoring or micro-CT imaging was beyond the scope of this study. Future work should combine the proposed constitutive framework with real-time microstructural observations to further validate the underlying mechanisms.

6. Conclusions

This study investigated the asynchronous evolution of stress and ultrasonic wave velocity in sandstone under confining pressures of 0, 2, 5, and 10 MPa through triaxial compression tests with synchronous wave velocity monitoring. A crack-strain-based damage constitutive model and wave velocity evolution model were developed to explain the underlying mechanisms. The main findings and their implications are as follows:

(1)

The peak wave velocity consistently occurred before the peak stress across all confining pressures, with the normalized stress at peak wave velocity (${\sigma }_{v_{\max }^P}/{\sigma }_f$) ranging from 0.84 to 0.99 and the normalized strain (${\epsilon }_{v_{\max }^P}/{\epsilon }_f$) ranging from 0.73 to 0.98. Both ratios decreased with increasing confining pressure. This behavior is explained by the competition between two opposing mechanisms: effective stress promotes wave velocity increase by closing micro-cracks. In contrast, crack strain promotes wave velocity decrease by generating and propagating micro-cracks. The wave velocity peak occurs when the marginal gain from effective stress equals the marginal loss from crack strain, a condition that shifts earlier with higher confining pressure owing to suppressed micro-cracking.

(2)

The crack-strain-based damage constitutive model accurately reproduced the full stress–strain response, with coefficients of determination (R²) ranging from 0.972 to 0.994 across all confining pressures. The wave velocity evolution model, which explicitly separates effective stress and crack strain contributions, achieved R² values of 0.929–0.998. These quantitative metrics confirm the model’s ability to capture both pre-peak and post-peak behavior.

(3)

The normalized crack initiation stress (${\sigma }_{ci}/{\sigma }_f$) ranged from 0.55 to 0.68, and the normalized crack damage stress (${\sigma }_{cd}/{\sigma }_f$) ranged from 0.79 to 0.9Z, consistent with ranges in the literature of 0.525–0.7 and 0.75–0.95, respectively. This consistency validates the crack-strain-based stress stage identification method and suggests that the observed behavior aligns with general trends in brittle sandstones.

(4)

The stress-normalized difference between the peak wave velocity and the damage variable protrusion point was approximately 0.1 ${\sigma }_f$ across all confining pressures, with a slight decreasing trend under higher confinement. This relationship, derived from the model and validated experimentally, indicates that the wave velocity peak serves as a quantitative precursor to the onset of significant damage, providing a measurable early warning indicator for rock instability.

(5)

The findings are based on sandstone samples from a single geological source (Ningtiaota mine) under four confining pressures, with two replicates per condition. Although the proposed mechanistic framework is physically reasonable and supported by quantitative model–data agreement, the model was calibrated using the same dataset; therefore, it does not constitute independent validation. The current data do not support claims of general applicability to other rock types or geological settings. Future work should include validation of the model using independent datasets from different lithologies and stress paths, incorporation of direct microstructural observations to further confirm the proposed mechanisms, and extension of the analysis to field-scale monitoring applications.