Mechanical Behavior Characteristics of Sandstone and Constitutive Models of Energy Damage Under Different Strain Rates

Abstract

To explore the influence of mine roof on the damage and failure of sandstone surrounding rock under different pressure rates, mechanical experiments with different strain rates were carried out on sandstone rock samples. The strength, deformation, failure, energy and damage characteristics of rock samples with different strain rates were also discussed. The research results show that with the increases in the strain rate, peak stress, and elastic modulus show a monotonically increasing trend, while the peak strain decreases in the reverse direction. At a low strain rate, the proportion of the mass fraction of complete rock blocks in the rock sample is relatively high, and the shape integrity is good, while rock samples with a high strain rate retain more small-sized fragmented rock blocks. This indicates that under high-rate loading, the bifurcation phenomenon of secondary cracks is obvious. The rock samples undergo a failure form dominated by small-sized fragments, with severe damage to the rock samples and significant fractal characteristics of the fragments. At the initial stage of loading, the primary fractures close, and the rock samples mainly dissipate energy in the forms of frictional slip and mineral fragmentation. In the middle stage of loading, the residual fractures are compacted, and the dissipative strain energy keeps increasing continuously. In the later stage of loading, secondary cracks accelerate their expansion, and elastic strain energy is released sharply, eventually leading to brittle failure of the rock sample. Under a low strain rate, secondary cracks slowly expand along the clay–quartz interface and cause intergranular failure of the rock sample. However, a high strain rate inhibits the stress relaxation of the clay, forces the energy to transfer to the quartz crystal, promotes the penetration of secondary cracks through the quartz crystal, and triggers transgranular failure. A constitutive model based on energy damage was further constructed, which can accurately characterize the nonlinear hardening characteristics and strength-deformation laws of rock samples with different strain rates. The evolution process of its energy damage can be divided into the unchanged stage, the slow growth stage, and the accelerated growth stage. The characteristics of this stage reveal the sudden change mechanism from the dissipation of elastic strain energy of rock samples to the unstable propagation of secondary cracks, clarify the cumulative influence of strain rate on damage, and provide a theoretical basis for the dynamic assessment of surrounding rock damage and disaster early warning when the mine roof comes under pressure.

1. Introduction

The rate of pressure coming from the mine roof is a key factor affecting the stability of the surrounding rock. In actual mining, the advancement speed of the working face, the dynamic adjustment of mining stress, and the disturbance of adjacent working faces may all lead to changes in the rate of roof pressure [1]. Especially in deep mining or under complex geological conditions, the non-uniformity and suddenness of roof compression increase sharply, which is prone to induce disasters such as roof failure, rib spalling, and rockburst [2,3]. Therefore, conducting research on the stability of surrounding rock under different pressure rates has significant practical significance and broad application prospects.

To explore the influence of strain rate on the mechanical properties of rocks, scholars at home and abroad have conducted a large number of studies on the strength, deformation, and energy evolution characteristics of rocks and achieved remarkable results [4,5,6,7,8]. Komurlu [9] found that the loading rate can change mechanical properties of rock. Eberhardt et al. [10] believe that the increase in damage will weaken the strength of the rock. Ma et al. [11] found through sandstone mechanics experiments that an increase in the loading rate would increase the failure strength and elastic modulus of the rock samples. At low rates, microcracks propagate along the weak surface, while at high rates, severe damage is caused. Similar laws were verified in the prefabricated fractured shale experiments of Suo et al. [12], where the peak load decreased with the maximum displacement as the rate increased. Wu et al. [13]’s research shows that the loading rate mainly regulates the morphology of the fracture zone. An increase in the loading rate significantly extends the length of the fracture zone, but the shape tends to narrow. Zhao et al. [14] revealed from the energy perspective that the total strain energy, elastic strain energy, and dissipated strain energy of brittle granite all increase with the increase in the loading rate. However, an increase in the loading rate will intensify the degree of damage and failure of the rock samples. The cyclic loading experiments of Zhang et al. [15] confirmed that a high loading rate would enhance the stress memory effect of rocks and inhibit the development of acoustic emission activities. Wei et al. [16] studied the deformation and failure characteristics of shale under different loading rates. The study found that a high loading rate would lead to severe rock damage, but the cracks were relatively simple. However, a low loading rate will lead to complex crack damage, that is, a low loading rate is conducive to the formation of more broken rock blocks in the shale matrix. In terms of sandstone experiments, Zhang et al. [17] found that the difference in loading rates directly led to the differentiation of failure modes, and the degree of damage increased with the increase in the loading rates. Gao et al. [18] conducted mechanical tests on rocks at different loading and unloading rates with the aid of a true triaxial loading device. The research finds that with the increase in the unloading rate of the horizontal principal stress, the total elastic strain energy at failure gradually decreases, while the total dissipated energy gradually increases. For special materials, Li and Liu [19] found that the mass loss rate of high-water-content materials decreased with the increase in the loading speed, and showed significant rate sensitivity. Wang et al. [20]’s research on prefabricated fractured rocks shows that their strength is always lower than that of intact rock masses, but both the peak strength and elastic modulus increase positively with the increase in the loading rate, and based on this, a constitutive model is established.

To sum up, previous studies have mostly focused on the analysis of rock mechanical properties under static loading and have not yet revealed the dynamic influence law of strain rate on the energy storage–energy dissipation conversion mechanism and damage accumulation of rock samples. Moreover, there are the following deficiencies in model construction. The existing damage models are difficult to accurately characterize the nonlinear hardening characteristics in the compaction stage, resulting in a deviation in the prediction of the early deformation response of rock samples. Meanwhile, the existing constitutive model studies have failed to clarify the abrupt mechanism of the unstable propagation of secondary cracks from the perspective of energy dissipation. Therefore, in this paper, sandstone is taken as the research object, and displacement loading experiments with different strain rates were carried out on sandstone. The variation laws of peak stress, elastic modulus, and peak strain of rock samples under different strain rates are discussed in detail, and the strength and deformation evolution characteristics of rock samples were quantitatively characterized. Meanwhile, the conversion relationship between the elastic strain energy and the dissipated strain energy of the rock sample was analyzed from the energy perspective, and the energy storage characteristics and energy dissipation characteristics of the rock sample were clarified. Furthermore, by combining the failure mode of rock samples and the characteristics of energy dissipation, the failure mechanism of rock samples under different strain rates was further revealed. Finally, the damage variable was defined based on the cumulative dissipative strain energy, and the constitutive model of energy damage was constructed. The rationality of the model was judged based on R² and RMSE. This model can not only accurately characterize the nonlinear hardening characteristics of the rock sample during the compaction stage under different strain rates, but it can also respond well to the strength and deformation laws of the rock sample. Meanwhile, the evolution characteristics of the energy damage variables based on this model reveal the sudden change mechanism from the dissipation of elastic strain energy of rock samples to the unstable propagation of secondary cracks, clarify the influence of strain rate on damage accumulation, and provide a theoretical basis for the dynamic assessment of surrounding rock damage and disaster early warning when the mine roof is compressed.

2. Experiment Preparation

The sandstone rock samples in this article were taken from a certain mine in China. The selected large rock samples have good homogeneity and no obvious cracks. Core sampling and grinding were carried out on the selected rock samples to prepare multiple groups of cylindrical rock samples with diameters of 50 mm and heights of 100 mm. The processing accuracy of rock samples is strictly carried out in accordance with the standard for test methods of engineering rock mass (GBT50266-2013) [21]. To ensure the uniformity of the prepared rock samples, ultrasonic tests, weight tests, height tests, and diameter tests were conducted on the rock samples. The specific test data are shown in

Table 1. Mechanical parameters of rock samples.

Rock Samples	Loading Rate (mm/min)	Ultrasonic Wave (km/s)	Weight (g)	Height (mm)	Diameter (mm)
A1-1	1	3.02	504.53	100.32	49.89
A2-1		3.05	506.71	99.83	49.97
A3-1		2.99	503.97	100.6	49.7
A1-3	3	2.97	503.95	99.79	49.72
A2-3		3.03	504.89	99.94	49.91
A3-3		2.98	504.24	99.87	49.95
A1-5	5	3.04	505.24	100.51	49.96
A2-5		3.02	504.76	99.75	49.87
A3-5		2.97	504.18	100.48	49.97
A1-7	7	3.00	504.23	99.83	49.94
A2-7		3.04	505.58	100.52	49.86
A3-7		3.02	504.61	100.46	49.93

as follows. It can be known from Table 1 that the errors of ultrasonic waves, weight, height, and diameter of the rock samples all meet the experimental requirements. In Table 1, A1-1, A2-1 and A3-1, respectively, represent the three duplicate samples in the group with a loading rate of 1 mm/min. A1-3, A2-3, and A3-3, respectively, represent the three duplicate specimens in the group with a loading rate of 3 mm/min. A1-5, A2-5, and A3-5, respectively, represent the three duplicate specimens in the group with a loading rate of 5 mm/min. A1-7, A2-7, and A3-7, respectively, represent the three duplicate specimens in the group with a loading rate of 7 mm/min.

Displacement loading experiments with different strain rates were carried out on the rock samples that met the experimental requirements. Dai et al. [22] conducted mechanical experiments on rocks at different strain rates to study the mechanical properties and energy dissipation laws of rocks under different loading rates. The displacement loading rate of the rock samples is different, and so is the strain rate of the rock samples. Therefore, in order to study mechanical behavior characteristics of sandstone and constitutive models of energy damage under different strain rates, the displacement loading rates were 1 mm/min, 3 mm/min, 5 mm/min, and 7 mm/min, respectively. During the experiment, the stress–strain data and failure forms of the rock samples at each strain rate were recorded in real time to further compare and analyze the failure characteristics of the rock samples at different strain rates. It is declared that the high and low strain rates presented below are only within the quasi-static strain rate range of this article.

3. Experimental Result Analysis

3.1. Strength and Deformation Characteristics

Figure 1 shows the strength and deformation characteristics. The mechanical behavior of rock samples shows significant stage evolution characteristics under different strain rates. According to the morphological characteristics of the stress–strain curve, it is divided into three typical stages: the compaction stage, the elastic stage, and the plastic stage. During the compaction stage, the primary defects inside the rock sample close, and the curve presents an upward concave shape. After entering the elastic stage, the rock sample undergoes elastic deformation, and the curve is approximately a straight line. As the strain increases to the yield strain, the rock sample enters the plastic stage, and secondary cracks initiate in the stress concentration area and show unstable propagation characteristics. When the strain reaches the peak strain, the rock sample undergoes brittle failure, the stress shows a cliff-like decline characteristic, and the residual strength approaches zero, as shown in Figure 1a. In addition, the strain rate has a significant influence on the mechanical behavior of rock samples. By comparing and analyzing the evolution laws of mechanical characteristic parameters of rock samples under different strain rates, both the peak stress and the elastic modulus show a monotonically increasing trend with the increase in the strain rate, but their growth rates show an obvious decreasing characteristic. Especially at low strain rates, the sensitivity of these two mechanical characteristic parameters to the strain rate is the most significant. For example, when the loading rate increased from 1 mm/min to 3 mm/min, the growth ratios of the peak stress and the elastic modulus were the largest, increasing by 71.1% and 147.4%, respectively. On the contrary, the peak strain shows a monotonically decreasing law with the increase in the strain rate, presenting an evolution trend opposite to the peak stress and the elastic modulus, as shown in Figure 1b. This reverse correlation among parameters reveals the influence mechanism of the strain rate on the deformation capacity and bearing performance of rocks.

3.2. Failure Characteristics

Figure 2 shows the damage characteristics. Under each strain rate, due to the end effect, the secondary cracks inside the rock sample first crack in the central region. With the increase in the loading stress, secondary cracks extend along the internal weak surface of the rock sample to the upper end face. According to this study [23], under high stress conditions, secondary cracks accelerate their propagation and penetration to form shear main cracks. Under uniaxial compression, the propagation path of the main crack shows nonlinear characteristics, and there is a significant correlation between its geometric morphology and the strain rate. When the stress reaches the peak stress, the elastic strain accumulated inside the rock sample can be released sharply, and the rock sample undergoes shear slip failure along the main crack path, as shown in Figure 2. The particle size distribution of the destroyed rock blocks varies under different strain rates. Rock samples with low strain rates mainly consist of more large-sized complete rock blocks, while rock samples with high strain rates retain more small-sized fragmented rock blocks. The difference in this failure mode stems from the variation of secondary crack propagation characteristics. The main reason for this is as follows. When the strain rate is low, the propagation rate of secondary cracks exceeds the threshold of the internal defect response of the rock sample. The main cracks rapidly penetrate along a single main path but fail to promote the formation of bifurcated secondary cracks. On the contrary, high-rate loading provides sufficient elastic strain energy for the bifurcation evolution of secondary cracks, which is conducive to the formation of complex reticular secondary cracks and promotes the rupture of rock samples along multi-path crack surfaces.

3.3. Energy Characteristics

3.3.1. Energy Calculation

Under different strain rates, the essence of rock sample failure is the mutual transformation among various forms of strain energy [24]. The energy conservation relationship of its deformation process can be expressed as [25]. Therefore, the total input strain energy can be further expressed as

$$U=U^d+U^e$$

(1)

Under uniaxial stress loading, the total input strain energy, elastic strain energy, and dissipated strain energy are further expressed as [26]

$$U={\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon$$

(2)

$$U^e={\displaystyle \frac{{\sigma }^2}{2E}}$$

(3)

$$U^d={\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon -{\displaystyle \frac{{\sigma }^2}{2E}}$$

(4)

where σ is the stress; ε is the strain; E is the elastic modulus.

3.3.2. Energy Conversion Characteristics

Figure 3 shows the relationship between the strain energy and strain. At the initial stage of loading, the primary fractures inside the rock sample undergo progressive closure under the influence of axial stress. During this period, the dissipated strain energy gradually increases with the increase in the strain. The total strain energy input from the outside is mainly converted into dissipated strain energy, and the energy dissipation is mainly manifested as frictional sliding at the fracture surface and the fragmentation of mineral particles. After entering the middle stage of loading, there are still residual fractures that have not been completely closed inside the rock sample. Under the action of stress, these fissures continuously generate frictional effects and lead to the dissipation strain energy maintaining an increasing trend. However, with the improvement of the bearing capacity of rock samples, the energy conversion mechanism has changed. The total strain energy input from the outside is mainly converted into elastic strain energy and is mainly used for the elastic deformation of rock samples. The change of this energy conversion characteristic marks that the rock sample begins to have a mechanical response dominated by elastic deformation, and its conversion mechanism can be explained as a significant enhancement in the ability of mineral crystals to store elastic strain energy. As the strain increases to the later stage of loading, the secondary cracks inside the rock sample accelerate their bifurcation and expansion, resulting in a further increase in the dissipated strain energy. However, the total strain energy input from the outside is still mainly converted into elastic strain energy. When the strain reaches the peak strain, the elastic strain can be released sharply, resulting in macroscopic brittle failure of the rock sample. This indicates that the elastic strain energy stored in the mineral crystals of the rock sample forms a high stress concentration at the tip of the secondary crack. When the secondary crack propagation reaches the critical length, the stored elastic strain energy is dissipated in the forms of accelerated propagation of the secondary crack, clastic kinetic energy, and acoustic emission.

Figure 4 shows the relationship between the strain energy and the loading rate. With the increase in the strain rate, the total strain energy, elastic strain energy, and dissipated strain energy all show a monotonically increasing trend. Among them, the growth ratio of the dissipated strain energy is the largest, followed by the total strain energy, and the elastic strain energy is the smallest, as shown in Figure 4. The main reasons for this are as follows. The high-rate loading inhibits the stress relaxation process at the tip of the secondary crack, prompts the bifurcated propagation mode of the secondary crack to shift from quasi-static to dynamic instability, accelerates the bifurcated propagation of the secondary crack, and significantly increases the energy required for the propagation of the secondary crack. Furthermore, the high-speed friction effect intensifies the thermodynamic coupling effect of the existing fracture surface, resulting in an increase in the energy dissipation efficiency of frictional slip.

3.3.3. Relationship Between Energy and Failure Mechanism

Figure 5 shows the relationship between the energy and the destruction mechanism. Because sandstone is a typical heterogeneous multiphase geological rock, its mechanical response is controlled by both mineral components and energy transfer paths. Zhang et al. [27] found that sandstone is mainly composed of minerals such as quartz crystals and clay substances. Among them, quartz crystals are connected through clay substances. The energy distribution characteristics of this dual-phase structure show significant differences under the effect of the strain rate, resulting in changes in the failure mode. At a low strain rate, the mechanical properties of clay substances dominate the initial damage evolution process. The longer stress action time provides sufficient conditions for the internal stress redistribution of the clay material, promoting the priority initiation of secondary cracks at the weak interface. With the increase in the loading stress, secondary cracks extend slowly along the clay–quartz interface by selectively choosing the dominant propagation path. During this process, the elastic strain energy is mainly released slowly in the form of frictional heat, eventually forming a through main crack extending along the boundary of the quartz crystal, and the rock sample undergoes intergranular failure, as shown in Figure 5a,b. At a high strain rate, the energy transfer mechanism undergoes a transformation. The high strain rate significantly inhibits the stress relaxation ability of the clay material, forcing the energy distribution to shift to the quartz crystal. At this point, the clay material is unable to regulate the local stress concentration, resulting in the secondary cracks being forced to enter a non-equilibrium propagation state. The tip of the secondary crack rapidly expands under the drive of high elastic strain energy. This energy pulse prompts the secondary crack to break through the energy failure threshold of the mineral interface and directly penetrate the internal lattice structure of the quartz crystal, thereby causing a transgranular failure of the rock sample, as shown in Figure 5c,d.

4. Discussion

Wang et al. [28] defined the rock damage variables based on the evolution characteristics of acoustic emission ringing count and constructed a constitutive model based on this damage characteristic. This paper draws on the model construction concept of Wang et al. [28] and constructs a constitutive model based on the evolution characteristics of dissipative strain energy.

4.1. Constructing Constitutive Model

The damage variable, as a parameter reflecting the degree of internal damage of rock materials, is closely related to the strain equivalence assumption [29]. Therefore, a damaging constitutive model is constructed based on damage variables and the assumption of strain equivalence [30]:

$$\sigma =(1-D)E\epsilon$$

(5)

where D is the damage variable.

The damage behavior of the microunits inside the rock is random. In this paper, the Weibull distribution function [31] is introduced to quantitatively describe the damage behavior of internal microunits in rock samples [32]:

$$f\left(\epsilon \right)=\lambda k{\left(\lambda \epsilon \right)}^{k-1}{\mathrm{e}}^{-{\left(\lambda \epsilon \right)}^k}$$

(6)

where λ, k are the parameters of the Weibull distribution.

Under different strain rates, the greater the loading stress, the more internal failure units there are in the rock sample, and the more severe the damage to the rock sample. Based on this, the damage variable can be expressed as [33]

$$D={\displaystyle \frac{N}{N_P}}$$

(7)

where N is the destruction of microunits; N_P is the total number of microunits.

In combination with Equations (6) and (7), the destruction of the microunit is further expressed as

$$N={\displaystyle {\int }_0^{\epsilon }{N}_{P}f\left(x\right)}dx$$

(8)

$$N=N_P\left[1-{\mathrm{e}}^{-{\left(\lambda \epsilon \right)}^k}\right]$$

(9)

Substitute Equation (9) into Equation (8), and the damage variable is further expressed as

$$D=1-{\mathrm{e}}^{-{\left(\lambda \epsilon \right)}^k}$$

(10)

Substituting Equation (10) into Equation (5), the damaging constitutive model is further expressed as [28]

$$\sigma =E\epsilon {\mathrm{e}}^{-{\left(\lambda \epsilon \right)}^k}$$

(11)

Dissipative strain energy refers to the energy that cannot be recovered during the deformation process of the rock sample. Under different strain rates, damage accumulation will release elastic strain energy and convert it into dissipated strain energy. Meanwhile, an increase in dissipated energy will accelerate the accumulation of damage. Therefore, the damage variable based on cumulative dissipative strain energy is expressed as

$$D={\displaystyle \frac{U}{U_P}}$$

(12)

where U is the cumulative dissipated strain energy at a certain moment; U_P is the cumulative dissipated strain energy at the peak stress.

The unknown parameters k and λ in Equation (10) can be determined by fitting the energy damage variable. Introduce the control parameter (ω), which is expressed as [28]

$$W=A{\epsilon }^k$$

(13)

$$\rho ={\displaystyle \frac{1}{{\epsilon }_n}}$$

(14)

where ε_n is peak strain corresponding to peak stress.

In combination with Equations (10), (13), and (14), the constitutive model of energy damage is expressed as

$$\sigma =E\epsilon \cdot {\mathrm{e}}^{-{\left(\rho \lambda \epsilon \right)}^k}$$

(15)

To accurately characterize the compaction and hardening characteristics of rock samples, Liu et al. [34] proposed the compaction coefficient. The compaction coefficient (η) is expressed as

$$\eta ={\log }_{m_1}\left[{\displaystyle \frac{\left(m_2-1\right)\epsilon }{{\epsilon }_m}}+1\right]$$

(16)

where ε_m is the yield strain; m₁ and m₂ are the experimental coefficients.

Substituting Equation (16) into Equation (15), the constitutive model of energy damage can be obtained:

$$\sigma =\mu {\log }_{m_1}\left[{\displaystyle \frac{\left(m_2-1\right)\epsilon }{{\epsilon }_m}}+1\right]E\epsilon \cdot {\mathrm{e}}^{-{\left(\rho \lambda \epsilon \right)}^k}$$

(17)

where μ is the correction coefficient.

4.2. Constitutive Model Rationality

Figure 6 shows the comparison between the experimental curve and the model curve. The model curve has a relatively high degree of fit with the experimental curve. This model can not only accurately characterize the nonlinear hardening characteristics of the rock sample during the compaction stage under different strain rates, but it can also respond well to the strength and deformation laws of the rock sample.

To quantitatively evaluate the rationality of the model, the coefficient of determination (R²) and the root mean square error (RMSE) are introduced for judgment [35].

$$R^2=1-{\displaystyle \frac{\left(n-1\right){\displaystyle {\sum }_{t=1}^n{\left({\sigma }_t-{\sigma }_t^{\prime }\right)}^2}}{\left(n-2\right){\displaystyle {\sum }_{t=1}^n{\left({\sigma }_t-\overline{\sigma }\right)}^2}}}$$

(18)

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{t=1}^n{\left({\sigma }_t-{\sigma }_t^{\prime }\right)}^2}}{n}}}$$

(19)

where n is the total amount of data; σ_t and ${\sigma }_t^{\prime }$ are, respectively, the measured and theoretical values of the stress corresponding to the t-th data point; $\overline{\sigma }$ is the average measured stress of the total data quantity.

Figure 7 shows the rationality analysis of the model. With the increase in the strain rate, the fitting degree among rock samples with different strain rates gradually increases, and all are greater than 0.908. Furthermore, as the strain rate increases, the standard deviations of the rock samples with different strain rates first increase and then decrease. The standard deviations among the rock samples fluctuated greatly but were all less than 4. This indicates that the model constructed in this paper is reasonable. This model establishes a cross-scale correlation equation between the macroscopic mechanical response and the mesoscopic damage evolution, providing a quantifiable assessment basis for the prevention and control of engineering dynamic disasters in deep rock masses.

4.3. Constitutive Model Prediction

Figure 8 shows the relationship between the damage amount and the strain. The damage evolution process of the rock samples under different strain rates shows significant three-stage characteristics, and there is a significant correspondence between the damage variables and the strain. By analyzing the damage evolution law of the rock samples during the loading process, the damage evolution process can be divided into the unchanged stage, the slow growth stage, and the accelerated growth stage. At 1 mm/min, the thresholds are a 0.38% strain between the unchanged and slow growth and a 0.72% strain between the slow growth and fast growth. At 3 mm/min, they are 0.32% and 0.54%. At 5 mm/min, they are 0.27% and 0.43%. At 7 mm/min, they are 0.26% and 0.40%, respectively. During the invariant stage, the internal structure of the rock sample is mainly characterized by the closure of the primary fractures. At this stage, dominated by the rearrangement of mineral particles and the compression of fractures, no secondary cracks occurred inside the rock samples, and the damage variables remained basically unchanged. After entering the slow growth stage, the microcracks stably expand along the mineral interface, and the damage variable grows slowly at a constant rate. When the strain increases to the yield strain, the secondary crack mesh shows bifurcation and fusion phenomena. The secondary crack propagation mode changes from steady state to unsteady state. The stress concentration effect at the tip of the secondary crack is significant, and the damage variable shows an accelerating growth trend. Furthermore, it is not difficult to see that the greater the strain rate, the greater the increase rate of the rock sample damage variable, and the more severe the damage and destruction. The research results provide an important theoretical basis for the assessment of surrounding rock damage under different pressure rates of the mine roof.

5. Conclusions

This paper takes sandstone rock samples as the research object and conducts mechanical experiments on the rock samples at different strain rates. By analyzing the strength characteristics, deformation characteristics, failure characteristics, energy characteristics and damage characteristics of the rock samples, the damage and failure mechanisms of the rock samples under different strain rates were further revealed. The conclusions obtained are as follows:

(1)

The strain rate has a significant influence on the mechanical behavior of rock samples. Both the peak stress and the elastic modulus show a monotonically increasing trend with the increase in the strain rate. When the strain rate increased from 1 mm/min to 3 mm/min, the growth ratios of the peak stress and the elastic modulus were the largest, increasing by 71.1% and 147.4%, respectively. The peak strain, however, shows a monotonically decreasing law with the increase in the strain rate, presenting an evolution trend opposite to the peak stress and elastic modulus. The particle size distribution of the destroyed rock blocks varies under different strain rates. Under a low strain rate, large-sized complete rock blocks of the rock samples dominate, while rock samples with a high strain rate retain more small-sized fragmented rock blocks. This difference stems from the variation of secondary crack propagation characteristics.

(2)

Rocks exhibit phased energy transformation characteristics during the loading process. In the early stage of loading, the axial stress prompts the closure of the primary fissures. The increase in dissipated strain energy with strain mainly results from the frictional slip of the fissures and the fragmentation of minerals. Entering the middle stage of loading, the residual fissures are compacting and maintaining the continuous increase in the dissipated strain energy, but the energy-dominated mechanism undergoes a transformation. The accelerated bifurcation propagation of secondary cracks in the later stage of loading exacerbated the accumulation of the dissipated strain energy, but the elastic energy storage still dominated. When the secondary crack expands to the critical length, the elastic strain energy is suddenly released, causing macroscopic brittle failure of the rock sample. The energy is dissipated in the forms of crack propagation, clastic kinetic energy, and acoustic emission.

(3)

With the increase in the loading stress, secondary cracks preferentially select the propagation path and slowly expand along the clay–quartz interface. The elastic strain energy is slowly released in the form of frictional heat, eventually forming a penetrating main crack that extends along the boundary of the quartz crystal, resulting in the failure of the rock sample along the crystal. At a high strain rate, the energy transfer mechanism changes, the stress relaxation ability of the clay is suppressed, and the energy is forced to transfer to the quartz crystal. Driven by the high elastic strain energy, the tip of the secondary crack rapidly breaks through the energy threshold of the mineral interface and directly penetrates the internal lattice structure of the quartz crystal, resulting in the transgranular failure of the rock sample.

(4)

Judging from R² and RMSE, the constitutive model of energy damage constructed in this paper is reasonable. With the increase in the strain rate, the fitting degree among rock samples with different strain rates gradually increases, and all are greater than 0.908. This model can not only accurately characterize the nonlinear hardening characteristics of the rock sample in the compaction stage under different strain rates, but also respond well to the strength and deformation laws of the rock sample. Based on the evolution characteristics of the damage variables of this model, its evolution process can be divided into the unchanged stage, the slow growth stage, and the accelerated growth stage. The characteristics of this stage reveal the sudden change mechanism from the dissipation of elastic strain energy of rock samples to the unstable propagation of secondary cracks, clarify the influence of strain rate on damage accumulation, and provide a theoretical basis for the dynamic assessment of surrounding rock damage and disaster early warning when the mine roof is crushed.

6. Further Research Plan

First, we intend to utilize CT scanning and SEM to elucidate the clay–quartz interface properties governing secondary crack path selection and energy transfer, specifically clarifying the transgranular failure mechanism where crack tips breach mineral energy thresholds and penetrate quartz lattices under high strain rates, with SEM providing direct microstructural evidence of microscale damage evolution. Second, we intended to rigorously verify the energy damage constitutive model’s applicability under complex stress paths and varied lithologies by integrating on-site monitoring data and optimizing damage variable evolution stage criteria to enhance accuracy. Third, we intend to expand the manuscript with a section comparatively evaluating failure prediction accuracy across diverse theories using our experimental dataset, conducting simulations to quantify predictive performance under different loading conditions, identifying applicability limits. Fourth, our comparative analysis of multiple constitutive models will enhance the rigor of the method and provide deeper insights into the mechanisms. Fifth, we will conduct triaxial compression tests on the specimens at different loading rates. Finally, characterization of detailed sample composition and porosity analysis and specific extraction site refinement is explicitly slated for subsequent work.