2.1 Establishment of the constitutive model

According to Lemaitre’s strain equivalent theory and the principle of effective stress, the damage constitutive relationship of coal under gas pressure is expressed as [18]:

(1)

where is the effective stress influenced by gas action; is the nominal stress; is the damage variable; is the Biot coefficient, for convenience of calculation, ; is the Kronecker symbol; is gas pressure.

Therefore, the effective stress is:

(2)

Assuming that the strength of rock micro-elements conforms to the Weibull random distribution, it can be represented by the following equation:

(3)

where is the strength distribution function of rock microe-lements, is the independent variable, is the shape parameter, and is the scale parameter.

The damage variable can be obtained [18, 19]:

(4)

Let , the rock micro-element strength criterion can be uniformly represented as:

(5)

where is the stress function; is material parameter.

The Hoek-Brown strength criterion is commonly employed to describe the strength of rock micro-elements. When expressed in terms of stress invariants, this criterion can be written as follows:

(6)

(7)

(8)

where is the peak effective stress of undamaged coal in the uniaxial compression test; and are the constants related to rock characteristics; is Rhode point; is the first invariant; is the second invariant; , and are the first, second and third effective principal stress, respectively.

In the conventional triaxial test, the first principal stress , the second principal stress , and the third principal stress satisfy and is 30°. By substituting Eq. (2) into Eq. (6), the Hoek-Brown strength criterion can be transformed into the following expression:

(9)

According to Hooke’s law, the stress-strain relationship in the principal direction can be expressed as:

(10)

where is the elastic modulus; is the first effective principal strain; is Poisson ratio.

As the gas permeates the pores within the coal body, it causes the coal to absorb gas and expand. Consequently, the stress-strain relationship in the principal stress direction can be described as follows:

(11)

where is expansion strain. The gas pressure is constant, so the adsorption expansion strain remains unchanged.

When evaluating the coordinated deformation of rock materials, it is possible to determine that , is the first principal strain. By substituting Eq. (2) into Eq. (10), researchers can then derive an equation that describes the connection between axial stress and strain when stress and gas pressure are both present:

(12)

By substituting Eq. (12) into Eq. (9), it becomes possible to derive the Hoek-Brown strength criterion that expresses a nominal stress. The resulting equation is as follows:

(13)

where is the peak stress of undamaged coal in the uniaxial compression test.

The deviatoric stress is expressed as:

(14)

The initial strain before applying the axial stress can be expressed as:

(15)

The first principal strain is the sum of the test measured strain and the initial strain , which is expressed as:

(16)

Applying Eqs. (4), (14), (15) and (16) to Eqs. (12) and (13) yields the statistical damage constitutive model under gas pressure, namely

(17)

where .

In order to more accurately represent the post-peak behavior of coal in the constitutive model, a damage correction factor has been defined based on the material’s deformation characteristics [19]:

(18)

where is the residual stress; is the peak stress.

The effective stress with the damage correction factor can be expressed as:

(19)

By incorporating the correction of residual stress, it is possible to establish a comprehensive statistical damage model for coal under gas pressure. The final form of the model is presented below:

(20)

(21)

2.2 Calculation of model parameters

The peak point method and curve fitting method are both commonly used for calculating model parameters in stress-strain analysis. The peak stress and peak strain of triaxial compression test are defined as and , respectively. When using the peak point method to calculate model parameters, two requirements must be met: first, the two sides of the model equation must be equal at the peak point, and second, the derivative of the constitutive model at the peak point must be 0.

By substituting the peak point into Eq. (20), the resulting outcome is:

(22)

where , .

Firstly, the derivative of Eq. (20) is calculated. Next, the values of and are substituted into the equation to make the derivative equal to 0, i.e., at and , the following holds:

(23)

By solving Eqs. (22) and (23) simultaneously, the following expressions are obtained:

, (24)

where , .

3.1 Testing equipment and procedure

Coal samples used in this study were taken from the Pingmei No. 8 Coal Mine in China. The samples were taken from depths ranging from 580 to 705 m. To conform to international rock test size standards, the coal was transformed into a cylinder standard sample with 50 mm in diameter and 100 mm in height.

Testing was conducted using the MTS815 rock mechanics test system and gas input system, with acoustic emission detection carried out using the PCI-2 acoustic emission test system. Coal samples were loaded into the system and subjected to a confining pressure of 10 MPa at a rate of 3 MPa/min. The coal samples and natural gas pipelines were evacuated to below 50 kPa while maintaining the confinement pressure, after which methane gas was introduced to reach the desired pressure. Once the pressure equilibrium was reached and the samples were saturated, a constant axial pressure was applied at a rate of 10 kN/min. Four groups of compression failure tests were conducted using two coal samples in each group. According to different gas pressures, tests were divided into 1, 2, 3 and 5 MPa. Figure 1 shows the schematic diagram of the equipment test.

Figure 1全屏查看

Schematic diagram of equipment test

3.2 Model validation

The test curves and theoretical curves of the statistical damage constitutive model under different gas pressures were compared using triaxial compression test data. Figure 2 shows the contrast between the test curves and theoretical curves at various gas pressures. The comparison reveals that the four theoretical curves fit well with the test curves in the pre-peak stage, but the theoretical curves obtained from the peak point method fit best at the peak point. The curve modified model exhibits the best fitting degree with the test curves, accurately capturing the stress drop phenomenon and softening characteristics of the coal. Conversely, the curve obtained from the unmodified model has a poor fitting degree with the test curves, failing to reflect the residual stress of the coal. Overall, the curves obtained from the modified model accurately represent the real stress characteristics of coal, with the modified curve fitting method proving to be the best approach to fit the test curve.

Figure 2全屏查看

Comparison between theoretical curves and test curves at various gas pressures: (a) 1 MPa; (b) 2 MPa; (c) 3 MPa; (d) 5 MPa

It is noteworthy that while the fitting performance of the modified curve fitting method is optimal, the alignment between the theoretical curve and the test curve remains somewhat unsatisfactory in the post-peak stage. As elucidated through the analysis in Figure 2, the proposed statistical damage constitutive model in this study demonstrates accurate predictions of the stress-strain curve for gas-bearing coal in the pre-peak stage. However, it falls short in accurately forecasting the stress-strain trend in the post-peak stage. This limitation arises from the deformation and failure process of coal involving the initiation, propagation, and coalescence of micro-cracks. By the post-peak stage, these micro-cracks have interconnected to form macroscopic fractures. Consequently, analyzing the evolution of micro-cracks post-macroscopic failure of coal based on the statistical damage constitutive model no longer holds theoretical significance. Nevertheless, with modifications to the statistical damage constitutive model, the theoretical curve in the post-peak stage can still to some extent reflect the stress reduction characteristics and residual stress of gas-bearing coal.

Given the high heterogeneity of coal, the average value of model parameters at different gas pressure levels is taken for analysis. As gas pressure increases, the elastic modulus, peak stress, and residual stress of coal gradually decrease, while the Poisson ratio gradually increases. As the gas pressure escalates from 1 to 5 MPa, the elastic modulus, peak stress, and residual stress of gas-bearing coal decrease by 42.66%, 19.99% and 11.45%, respectively, with the Poisson ratio witnessing a rise of 13.11%. Gas pressure notably weakens the stress of coal, playing a crucial role in its deformation and failure. From a microscopic perspective, this phenomenon stems from the gas adsorbed on the surface of coal particles weakening the cohesive forces within the coal structure, thereby facilitating the development of micro-cracks and significantly diminishing the mechanical properties of coal. Furthermore, damage correction factor is approximately 0.52. As the gas pressure increases from 1 to 5 MPa, the model parameters and decrease by 22.18% and 60.45%, respectively.

The established damage constitutive model accurately characterizes the deformation behavior of gas-bearing coal and effectively reflects the stress drop phenomenon and softening characteristics of coal during the post-peak stage. Additionally, comparing the stress-strain curves of coal under different gas pressures reveals that the peak stress of coal decreases gradually with increasing gas pressure, and the rate of stress drop decreases significantly. Typically, a larger stress drop rate during the post-peak stage indicates a higher brittleness of the coal [20]. Therefore, the reduced rate of stress drop during the post-peak stage of gas-bearing coal implies a gradual decrease in its brittleness with increasing gas pressure.

3.3 Model parameter analysis

To investigate the impact of the damage correction factor on the stress-strain curve of gas-bearing coal, the test data of coal sample with a gas pressure of 5 MPa is selected. Keeping all other parameters constant, the damage correction factor is taken as the variable, which is gradually increased from 0.5 to 1. The effect of the damage correction factor on the whole stress-strain curve of gas-bearing coal is depicted in Figure 3.

Figure 3全屏查看

Effect of damage correction factor on the whole stress-strain curve of gas-bearing coal

The findings show that the damage correction factor has an important effect on the post-peak behavior of coal, with the residual stress decreasing steadily as this factor increases. Notably, no observable impact on the pre-peak stage was observed. When the damage correction factor reaches a value of 1, the model fails to reflect the residual stress of the coal. Overall, the use of a damage correction factor can enhance the accuracy and applicability of the constitutive model for gas-bearing coal.

To further analyze the influence of model parameters on the mechanical behavior of gas-bearing coal, the test data from coal sample at a gas pressure of 5 MPa is selected. Taking the shape parameter and the scale parameter as variables, is gradually increased from 10 to 18, and is increased from 14.5 to 16.5 MPa. Figure 4 shows the impact of the shape parameter and the scale parameter on the stress-strain curve of gas-bearing coal.

Figure 4全屏查看

Effect of model parameters on the stress-strain curve of gas-bearing coal: (a) Shape parameter ; (b) Scale parameter

The results indicate that the shape parameter primarily reflects the brittleness and distribution of micro-element strength in coal materials, while the scale parameter reflects the macro statistical average strength. Figure 4(a) demonstrates that the stress reduction rate of gas-bearing coal after the peak-point increases with the rising value of , indicating a greater degree of brittleness in the coal. By contrast, Figure 4(b) shows that the strength of gas-bearing coal increases as parameter rises. Further analysis, as presented in Figure 2, reveals that gas pressure has a significant impact on the strength and brittleness of gas-bearing coal. In particular, higher gas pressures increase the likelihood of ductile failure rather than brittle failure, thereby diminishing the strength and brittleness of the coal.

3.4 Analysis of brittleness based on damage variable

The process of coal deformation and failure is actually the accumulation of internal damage in coal. Comparing the stress-strain curve with the damage evolution curve of coal can intuitively and effectively reflect the crack propagation and evolution process of coal.

Figure 5 shows the stress-strain and damage evolution curves of the entire process of coal failure. The damage variable can be obtained through the statistical damage constitutive model. By combining the stress-strain curve with the evolution curve of the damage variable, the fracture failure process of coal can be divided into four stages. Stage I is the crack closure stage, during which micro-cracks close, and the stress-strain curve exhibits a slight upward concave feature. Next is the elastic deformation stage (Stage II), where the stress-strain curve approximates a straight line, and the coal undergoes elastic deformation. In the first two stages, the coal experiences little to no damage, with the damage variable remaining at 0. This is followed by the crack growth stage (Stage III), where cracks begin to propagate and the damage variable gradually increases from 0. As the stress reaches its peak, the coal enters the failure stage (Stage IV), marked by a significant decrease in stress and a rapid increase in the damage variable. The maximum damage rate reflects the rate of stress drop in the coal. Finally, with the failure of the coal, the damage variable reaches its maximum value of 1.

Figure 5全屏查看

Stress-strain and damage evolution curve of the entire process of coal failure

At present, when evaluating the brittleness characteristics of coal based on stress-strain curves, the brittleness of coal is mainly evaluated based on the amplitude and rate of post-peak stress drop. Therefore, it is feasible to evaluate the brittle characteristics of coal based on its damage evolution. In addition, the maximum damage rate of the damage variable can reflect the maximum growth rate of damage during the post peak stress drop process of the coal. The higher the maximum damage rate, the higher the post peak stress drop rate of coal, indicating a stronger brittleness characteristic of coal. Based on this characteristic, the maximum damage rate is chosen as an indicator to reflect the brittleness characteristics of coal. The damage rate of coal is defined as the partial derivative of the damage variable to strain, which can be expressed as:

(25)

The failure characteristics of coal and its damage and brittle evolution under different gas pressures exhibit significant differences. To confirm whether the maximum damage rate can specifically reflect the brittle characteristics of gas-bearing coal, the damage evolution curves of coal under varying gas pressures are compared and analyzed, as shown in Figure 6. The slope of the curve represents the damage rate of coal. During the elastic deformation stage, the degree of damage to coal is almost 0. As the strain continues to increase, the destruction of coal gradually accelerates until it reaches a critical point, and then undergoes rapid and widespread degradation. Moreover, it is evident that the maximum damage rate of coal varies with changes in gas pressure. Under gas pressures of 1, 2, 3 and 5 MPa, the maximum damage rates of coal are 14.611×10², 8.192×10², 3.463×10² and 1.758×10², respectively. As the gas pressure increases, the maximum damage rate of coal decreases. This indicates that the stress reduction rate of coal in the post-peak stage decreases, the process of failure and deformation slows down, and the brittleness gradually decreases. Therefore, the maximum damage rate of coal provides a valuable indicator for evaluating its brittleness characteristics.

Figure 6全屏查看

Damage variable-strain curve of coal at various gas pressures

4.1 Energy and damage evolution

The process of coal damage development is inherently linked to energy conversion mechanisms. Indeed, energy conversion, involving energy accumulation, dissipation, and release, are frequently present during coal damage progression [21]. There are three forms of strain energy: total absorbed energy , elastic energy , and dissipated energy . The calculation expressions of is shown as follows:

(26)

During the deformation and fracturing of coal, the pore pressure of coal can have a profound influence. Notably, the action of gas pressure adds a certain amount of energy to coal, resulting in its deformation. As such, the impact of pore pressure on coal energy accumulation and dissipation cannot be overlooked.

The input and dissipation of energy for the triaxial test are composed of many parts. By introducing the principle of effective stress, the gas pressure and confining pressure in the tests are both fixed values. Based on the definitions of elastic energy, the following formulas can be used to determine the values of and [22]:

(27)

(28)

where , and are the first, second and third principal strain, respectively; , and are the first, second, and third principal elastic strains, respectively; is the volumetric strain, and . During coal expansion, the pore pressure performs positive work. To simplify the analysis, an idealized model is used where the pore pressure is treated as a uniform internal pressure acting on the volume of the coal samples.

The evolution of dissipated energy is related to the damage of coal. To quantify this relationship, an energy damage index is defined as follows:

(29)

where is the dissipated energy, and is the dissipated energy of coal at peak stress. Notably, when , the material is in an undamaged state before being subjected to any loading. Conversely, when , it indicates that the coal has been subjected to stresses that have brought it to peak stress and incurred damage.

Figure 7 displays the energy development curves of coal under various gas pressures. During the compression deformation and linear elastic deformation stages, the majority of absorbed energy is stored within coal samples as elastic energy, indicating energy accumulation within coal. As internal cracks develop, the rate of dissipated energy increases rapidly. As coal stress approaches its apex, the elastic energy progressively reaches maximum amount of energy it can store. Once the peak value is achieved, coal experiences macroscopic destruction, causing the stress curve to plummet quickly while the elastic energy curve mirrors this decline. In this stage, coal absorbs additional energy, most of which is used as dissipated energy. Under low gas pressure, the elastic energy decrease after coal damage is more pronounced, leading to diminished residual elastic energy and displaying more apparent brittle characteristics of coal.

Figure 7全屏查看

Energy evolution curve of coal samples at various gas pressures: (a) 1 MPa; (b) 2 MPa; (c) 3 MPa; (d) 5 MPa

As the gas pressure increases, the maximum capacity of elastic energy storage in coal gradually decreases. For example, at gas pressures of 1, 2, 3 and 5 MPa, the maximum elastic energy storage values of coal are 287.8, 178.6, 113.3 and 106.6 kJ/m³, respectively. Compared to the elastic energy storage at 1 MPa, the values at 2, 3 and 5 MPa decrease by 38%, 61% and 63%, respectively. This decline can be attributed to the progressive expansion of micro-cracks within the coal during the pre-peak stage, as the axial deviatoric stress intensifies and the pore gas pressure fosters micro-crack propagation [23]. The presence of pore pressure and adsorbed gas can significantly impair the load-bearing capacity of coal. Under the influence of internal pore gas pressure, micro-cracks within coal samples tend to propagate, thereby diminishing the mechanical properties of coal. Additionally, gas adsorption on the surface of coal particles weakens the cohesive forces within the coal’s internal structure, leading to a certain degree of softening and a reduction in its brittle fracture characteristics. As gas pressure increases, coal develops more irregular seepage channels internally, consequently resulting in a substantial decrease in energy storage levels and cumulative elastic properties. Therefore, the brittle characteristics of coal are weakened. Furthermore, to mitigate the influence of coal sample dispersion, these findings were validated by comparing ( is the elastic energy of coal at peak stress). As the gas pressure rises, this ratio decreases, indicating a diminishing proportion of elastic energy and a gradual attenuation of the brittle characteristics of coal.

4.2 Damage stress threshold

The failure process of coal is linked to the transformation of different forms of energy. The effect of energy on coal can be divided into the energy hardening effect (energy storage) and the energy softening effect (energy release) [24]. In order to explore the evolution trend of energy, the elastic energy proportion and dissipated energy proportion were calculated.

Figure 8 shows the energy proportion and energy damage index of coal under gas pressure of 2 MPa. The initially decreases, then increases, and eventually experiences a steady drop, while the demonstrates the opposite trend. Both curves exhibit an S-shaped evolution law. Additionally, two damage stress thresholds and can be gained, and the energy development process can be separated into three stages:

Figure 8全屏查看

Energy proportion and energy damage index of coal under gas pressure of 2 MPa

(I) Initial compression stage : The imperfections such as micro-cracks, pores, and holes in the coal are compressed and closed. This results in an increase in the proportion of dissipated energy. The damage caused by energy within the interior of the coal can be neglected during this stage.

(II) Energy hardening stage : increases while decreases as the stress of the coal exceeds . This phenomenon, known as energy hardening, can be observed in the evolution of energy damage. serves as the starting point for energy hardening. Coal mainly absorbs elastic energy during this stage, and the energy damage index shows that the damage growth rate of coal is relatively slow.

(III) Energy softening stage and failure stage : The energy development process enters the energy softening stage and failure stage once the stress of the coal exceeds . During this stage, decreases rapidly while increases quickly. Simultaneously, grows rapidly, indicating a sharp increase in internal damage to the coal. Most of the damage of coal occurs during this stage. As continues to increase, the coal stress reaches its peak, and the coal undergoes macro-fracture. The damage stress threshold serves as the starting point for the energy softening stage.

To investigate the differences in the damage development process of coal at various gas pressures, the energy damage parameters of gas-bearing coal were calculated. Figure 9 shows the energy proportion and energy damage index of coal under different gas pressures. The declines as the coal stress decreases, and the decreasing trend is consistent. When reaching the residual stage, and also tend to be stable. In addition, the damage of coal starts to increase sharply at , while the increasing rate of damage to the coal is smaller than that before .

Figure 9全屏查看

Energy proportion and energy damage index of coal under different gas pressures: (a) 1 MPa; (b) 2 MPa; (c) 3 MPa; (d) 5 MPa

The result shows that damage growth is more pronounced at low gas pressures, while it is slower at high gas pressures. At a gas pressure of 1 MPa, the coal sample exhibits 94% of its damage produced in only 18% of strain increments during the stage. In comparison, at a gas pressure of 5 MPa, the coal sample shows that 98% of its damage is produced in 46% of strain increments during the stage. The significant damage and relatively smaller strain increments observed at low gas pressure suggest that coal damage is rapid and intense, exhibiting more brittle characteristics.

5.1 Establishment of brittleness index

After conducting thorough research and analysis, it has been observed that the stress-strain curve effectively represents the brittle characteristics of coal. Furthermore, the statistical damage constitutive model excels in simulating the complete stress-strain curve of coal. Leveraging this advantage, the damage variable is selected within the statistical damage constitutive model to accurately reflect the brittle nature of coal. Additionally, an energy evolution analysis of gas-bearing coal is performed to investigate the mechanisms of energy dissipation and release during coal fracturing. This analysis involves monitoring various parameters throughout the deformation process, including changes in coal elastic energy , dissipated energy , peak elastic energy storage limit , and the ratio of peak elastic energy to dissipated energy. By integrating the energy evolution mechanism with the statistical damage constitutive model, a comprehensive understanding of the brittle behavior exhibited by gas-bearing coal is achieved.

In order to assess the brittle evolution characteristics of coal more accurately and comprehensively, two damage variables, denoted as and , are proposed from the perspectives of statistical damage constitution model and energy evolution mechanism. These two indices of brittleness mutually corroborate and complement each other, collectively reflecting the brittle characteristics of coal. In establishing the brittleness indices, the balanced evaluation of each indicator is considered. Through the multiplication synthesis method based on and multiple energy parameters (, , , , ), a novel brittleness index is constructed. Specifically, the brittleness of coal encompasses several key traits: 1) The stress-strain curve ideally exhibits a vertical trend after reaching its peak. This article addresses this characteristic by incorporating damage variable into the statistical damage constitutive model. Notably, a higher maximum damage rate signifies stronger brittle characteristics; 2) Coal should possess a greater capacity for storing elastic energy prior to failure, and a larger elastic energy storage capacity implies enhanced brittleness; 3) During the post-peak stage, the release of elastic energy should occur rapidly, accompanied by swift damage progression. A more pronounced manifestation of these factors indicates increased brittleness. Based on these observations, a new definition for the brittleness index is proposed:

(30)

where is the maximum damage rate, and are the elastic energy and dissipated energy of coal at peak stress, respectively, is the elastic energy proportion at , is the energy damage index at , is peak strain, is the strain at the , and is peak elastic energy storage limit.

5.2 Validation of the brittleness index

To evaluate the effectiveness of , several commonly used brittleness indexes based on energy and stress-strain are compared with the proposed in this paper. The commonly used brittleness indexes are as follows [10-14]:

(31)

(32)

(33)

(34)

(35)

where is the peak stress, is the residual stress, is peak strain, is residual strain, is the slope of post-peak stress drop, is elastic energy released after the peak, and is post-peak absorbed energy.

It is noteworthy that the formation of coal is influenced by various factors, including the original plant species, depositional environment, and coalification processes. These factors result in variations in composition and structure within the same coal seam, thereby influencing its mechanical properties. Therefore, in quantitatively assessing the brittle characteristics of coal, the average value is used to investigate the evolution of coal brittleness under different gas pressure conditions.

Table 1 presents the brittleness index of coal at various gas pressures, while Figure 10 displays the comparison of the calculated brittleness index values using different evaluation methods. Among them, the brittleness indices , , , , and show a positive correlation, signifying that higher values correspond to greater coal brittleness. In contrast, the indicates negative correlation, implying that lower values indicate a higher degree of coal brittleness. The results indicate that BIE, B₁, B₂ and B₃ all indicate that the brittleness of coal gradually decreases with increasing gas pressure. The brittleness of coal characterized by them is consistent with the brittleness reflected by the stress-strain when the coal is damaged. However, and can only reflect the overall trend of brittleness change, they cannot show the amplitude features of coal brittleness with different gas pressures. Moreover, the decreasing rate of tends to level off after the gas pressure reaches 2 MPa, which cannot fully capture the brittleness reduction characteristic of coal with increasing gas pressure. Because only considers the stress-strain characteristics of the coal but does not consider the effect of energy evolution. is not fully decreasing with increasing gas pressure, indicating that this index is not accurate in assessing the brittleness of gas-bearing coal. It only considers the energy change of coal in the post-peak stage.

Table 1全屏查看

Brittleness index of coal at various gas pressures

Gas pressure/MPa
1	0.325	0.377	0.047	37.279	0.344	29.227
2	0.283	0.456	0.032	13.927	0.244	17.772
3	0.263	0.615	0.026	13.872	0.339	8.517
5	0.244	1.177	0.028	13.759	0.297	0.54

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Figure 10全屏查看

Comparison of the calculated brittleness index values with different evaluation methods

Figure 11 shows the of coal at various gas pressures. The link between and gas pressure for gas-bearing coal may be determined by the data fitting as follows:

Figure 11全屏查看

Brittleness index of coal at various gas pressures

(36)

The results show that the brittleness of coal steadily reduces as gas pressure increases, which is compatible with test analysis results. This finding confirms that can objectively reflect the brittleness characteristics of coal and can be utilized to assess the damage evolution process of gas-bearing coal.