#### Abstract

In underground mining engineering, the residual strength of surrounding rock has an important influence on the secondary stress distribution caused by excavation. Generally, the residual strength of coal can be measured by experiments, but for soft coal, due to its large postpeak deformation characteristics and the limitation of strain sensor range, it is difficult to measure residual strength. Therefore, it is of great practical significance to estimate the residual strength of soft coal with incomplete stress-strain curves. In this paper, a method for estimating residual strength of soft coal based on the ratio of equivalent residual strain to peak strain is proposed. Taking advantage of the characteristics that the strain-softening curve of soft coal decreases approximately linearly in the initial stage, keeping the drop modulus unchanged, the endpoint of the extension line can be determined based on the ratio of equivalent residual strain to peak strain, which is estimated residual strength. There are 342 groups of typical complete stress-strain curves analyzed to determine the value range of the ratio of equivalent residual strain to peak strain. It is concluded that for briquette composed of soft coal, the ratio of equivalent residual strain to peak strain is between 1 and 3.35 when the confining pressure is 0-6 MPa, and the maximum probability interval of the ratio value under different confining pressures is further calculated. When the confining pressure is higher than 6 MPa, the briquette composed of soft coal tends to ideal plastic state after peak. The feasibility of the residual strength estimation method is verified by comparing the numerical simulation results of triaxial compression of coal pillars with the experimental results.

#### 1. Introduction

Coal is one type of solid-state combustible organic rock, which is transformed from plant remains by complex biochemistry, physical chemistry, and geochemistry effects [1], and has certain heterogeneity. There are also a large number of pores, cracks, bedding, and other types of defects in the coal. These structural factors not only affect the strength properties of the coal mass but also affect the deformation properties [2]. Under the influence of mining disturbance, the stress of roadway surrounding rock produces secondary distribution, forming stress concentration area and pressure relief area. It brings difficulties to surrounding rock support and roadway construction and increases the engineering risk; at the same time, the unloading damage of coal seam is caused by surrounding rock pressure relief, promoting coal seam gas flow. The heterogeneous properties of soft coal lead to reservoir permeability with heterogeneous characteristics, in which the gas flow state also needs to be studied in detail [3, 4]. Therefore, the study of postpeak stress-strain state of soft coal is of great significance to the stability support of surrounding rock, the safe construction of roadways, and the improvement of permeability.

The mechanical parameters of soft coal are usually obtained by triaxial compression test. In the triaxial compression test, when the stress reaches the peak strength, the soft coal will be destroyed continuously and its strength will drop to a lower level rapidly as the deformation continues to increase, and this phenomenon of degradation of material properties due to deformation is called strain softening. With the further increase of deformation, penetrating macrofractures are formed inside the soft coal, the cohesion between the fracture surfaces is basically lost, and the bearing capacity is completely provided by the friction between the fracture surfaces and maintained at a stable value which is the residual strength. At present, researchers have made relevant progress in the study of strain-softening characteristics. When studying the structural stability during construction, Japanese scholar Toshikazu [5] first proposed a postpeak linear strain-softening model for the material. In order to study the mechanism of mechanical deformation and failure of soft rock, Lu et al. [6] established a subsequent yield surface model of soft rock characterized by two state parameters: generalized cohesion and generalized internal friction angle. When studying the postfailure behavior of rock mass and analyzing the stability of tunnel, Alejano et al. [7] believed that the actual mechanical behavior of soft rock () is very similar to the ideal elastic-plastic behavior. Wang et al. [8] adopted the idea of simplifying the stress-strain curve in the softening stage into a series of brittle plastic models when excavating a circular tunnel in isotropic strain-softening rock mass. In order to study the difference law of mechanical behavior of brittle-ductile transformation of rocks at different depths, Xie et al. [9] believed that the brittle-ductile transformation of rocks at different depths is not instantaneous and there is a stress range of brittle-ductile transition gradually. The residual strength is a mechanical strength characterization of the material into the ideal plastic state and predicts the transition from the strain-softening stage to the residual stage. With the increase of confining pressure, the residual strength of rock materials increases more than the peak strength, and the residual strength gradually becomes the main factor affecting the postpeak section of rock [10]. The residual strength of rock also plays an important role in maintaining the stability of roadway [11]. However, limited by the range of the testing machine, the testing method, or the rock materials, it is difficult to obtain complete stress-strain curves and directly measure the residual strength [12]. In the past, when the postpeak strain-softening characteristics of coal and rock mass were studied, the value of residual strength was often derived from references or determined based on experience [10, 13, 14], not from experimental results. Although Cai et al. [15] applied the GSI system to the residual strength estimation of jointed rock masses and concluded that the residual strength corresponds to strain approximately 5-10 times the strain corresponding to the peak strength, the mechanical properties of soft coal and rock materials differ significantly, leading to the inapplicability of the above method to soft coal.

In this paper, a method for estimating residual strength of soft coal based on the ratio of equivalent residual strain to peak strain is proposed. A large number of complete stress-strain curves obtained from previous coal and rock mass experiments were collected and counted, from which the ratio data of equivalent residual strain to peak strain were extracted, the value range of the ratio of equivalent residual strain to peak strain is summarized, and the maximum probability interval for the ratio of equivalent residual strain to peak strain under different confining pressures is calculated. Finally, COMSOL Multiphysics is used to simulate the triaxial compression of coal pillar and error analysis.

#### 2. Mechanical Properties of Postpeak Strain Softening of Soft Coal

According to the different postpeak behaviors of rock, Hoek and Brown [16] simplified the postpeak stress-strain relationship into three stages based on GSI system: elastic brittleness, strain softening, and ideal elastic-plastic. Therefore, the stress-strain curve of soft coal can be shown in Figure 1; for the prepeak stage (OP segment), the classical linear elastic model is adopted. After point P (including PA, P-B-C, and PD), PA is brittle model, P-B-C is strain-softening model, and the stress-strain relationship in the strain-softening stage is nonlinear [8]. PD is an ideal plastic model.

In the softening stage, the bearing capacity of soft coal gradually transitions from peak strength to residual strength. The transition is controlled by softening parameter , and the postpeak failure obeys the yield function. The failure criterion is defined as

Combined with failure criteria, assumptions and simplifications are made for the postpeak strain-softening characteristics as shown in Figure 2. The drop modulus of softening stage is expressed by . The Mohr-Coulomb criterion is used as the postpeak failure criterion, and the expression composed of principal stress is [17] where is the maximum principal stress, is the minimum principal stress, is cohesion, is internal friction angle, and is a function of the internal friction angle.

When studying the softening parameter , it is generally believed that the softening parameter is more closely related to the experimental type than the characteristics of the material itself [18]. is often defined by the maximum and minimum principal plastic strains [19]: where is plastic shear strain and and are the maximum and minimum principal plastic strains.

The principal strain is composed of elastic strain () and plastic strain () [20]. The strain-softening characteristics are studied according to the cyclic loading and unloading principle, and the elastic modulus remains unchanged [21]; the maximum and minimum principal plastic strains are where is the postpeak axial stress value and is the modulus of elasticity in the linear elastic stage.

It is known that the softening parameters at the peak point. The strain relationship at the peak point is given by .

The final expression for the softening parameter can be obtained as where is the maximum principal strain, is the peak strain, and is Poisson’s ratio.

The strength parameters are considered as a function of softening parameter [22], as follows: where is strength parameter, is initial strength parameter, is residual strength parameter, and represents the transition value during the transformation from strain-softening stage to residual stage.

The variation of strength parameters is described by the bilinear function of softening parameter, as shown in Figure 3.

From the peak point to the residual stage, the loss or decrease ratio of strength parameters of materials is similar [24]. The following is the compound expression of postpeak stress-strain relationship:

Cohesion and internal friction angle can be defined as where is the initial cohesion and is the residual cohesion. where is the initial internal friction angle and is the residual internal friction angle.

#### 3. Residual Strength Estimation Method Based on Equivalent Residual Strain Ratio

For soft coal, the postpeak part of stress-strain curve is usually composed of a linear segment in the early stage of failure and a nonlinear segment in the middle and late stage of failure. Based on the above theoretical analysis, it is necessary to construct a residual strength estimation method, which can be divided into the following steps: (1)Assumptions and simplifications: as shown in Figure 4, when only the O-A-B segment stress-strain curve can be obtained in the experiment, according to the constitutive model, the stress-strain curve generally evolves as the A-B-D segment curve, where point A is the peak strength point and point D is the real residual strength point. The strain-softening stage is assumed and simplified as A-B-C-D segment, in which point C is the equivalent residual strength point(2)Numerical solution: the residual strength is theoretically deduced according to the peak point, the drop modulus, and the equivalent residual strain ratio. Parameters include peak strength , peak strain, and drop modulus , which are obtained from experimental data directly. The equivalent residual strain ratio is expressed as , and the equivalent residual strain value () is obtained by the equivalent residual strain ratio and peak strain, as shown inwhere is the estimated residual strength, is the peak strength, is the drop modulus, and is the equivalent residual strain (3)Combined with failure criteria: the proposed residual strength estimation method is combined with the failure criterion to quantitatively describe strain-softening characteristics. The softening parameter is expressed as a function of the equivalent residual strain ratio , as shown in

The cohesion is shown in

The internal friction angle is shown in

Limited to many situations, it is difficult to select the equivalent residual strain ratio directly. However, with the increase of the number of experiments and the development of advanced testing machines [25], a number of complete stress-strain curves can be collected, in which the existence of residual strength makes it possible to obtain the equivalent residual strain ratio. By studying the strain-softening relationship of rock from different sources, Tutluoğlu et al. [26] believed that corresponding data extraction methods should be adopted for stress-strain curves of different rock. Therefore, a large number of complete stress-strain curves in previously published literature were collected. Typical data extraction cases are shown in Figures 5(a) and 5(b). The peak strength point is obtained directly from the experimental curves, the real residual strength point is obtained from the point where the drop modulus of the postpeak part of the stress-strain curve of soft coal becomes horizontal, and the equivalent residual strength point is obtained by intersecting the postpeak linear part with the real residual strength line. The scope includes raw coal, briquette composed of soft coal, and soft rock, and a total of 342 groups of data are collected. See tables for details (raw coal data (Tables 1 and 2), briquette composed of soft coal data (Tables 3 and 4), and soft rock data (Tables 5 and 6)).

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#### 4. Data Analysis of the Equivalent Residual Strain Ratio

##### 4.1. Value Range of the Equivalent Residual Strain Ratio

Many scholars have studied the factors affecting strain-softening behavior. In [27], the stress-strain curves obtained from Tennessee marble at different confining pressures were analyzed, showing that the stress-strain curve is sensitive to the confining pressure. Fang and Harrison [28] believed that the microscopic mechanism of rock fracture is affected by many factors, among which confining pressure is one of the most important. Therefore, confining pressure will be an important consideration in this paper.

The peak strength and the residual strength , the corresponding peak strain , and the real residual strain of coal and rock mass in the published literature are analyzed (the data are from these literatures [6, 29–34]). Calculating the variation trend of strength ratio and real strain ratio with confining pressure, the results are shown in Figure 6.

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According to Figure 6(a), as the confining pressure increases, the strength ratio tends to 1. The strain ratio of coal and rock mass, as shown in Figure 6(b), is not 5-10 times proposed by Cai et al. [15], but between 1 and 4.5 times.

The variation characteristics of peak strength and residual strength of coal and rock mass with confining pressure are statistically analyzed, as shown in Figure 7. With the increase of confining pressure, the peak strength of raw coal, briquette composed of soft coal, and soft rock all show an increasing trend, and the change of residual strength is basically consistent with the peak strength. In addition, the increase of residual strength of coal and rock mass with confining pressure is greater than that of peak strength, indicating that confining pressure has a significant effect on the residual strength.

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The variation law of strength ratio with confining pressure is quantitatively analyzed, as shown in Figure 8. With the increase of confining pressure, the strength ratio tends to 1. The postpeak strength degrades significantly at low confining pressure and exhibits more ductile mechanical properties at high confining pressure.

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##### 4.2. Determination of Value Range of the Equivalent Residual Strain Ratio

Figure 9 shows the relationship between the equivalent residual strain ratio with confining pressure. Compared with raw coal, briquette composed of soft coal has a larger strain ratio range. The equivalent residual strain ratio of soft coal needs to be further analyzed as follows.

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The equivalent residual strain ratio is quantitatively analyzed, as shown in Figure 10. For raw coal, the value range of equivalent residual strain ratio is 1-2.28. For briquette composed of soft coal, the value range of equivalent residual strain ratio is 1-3.35.

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##### 4.3. Probability Analysis of the Value Range of the Equivalent Residual Strain Ratio

In order to determine the occurrence probability of a specific equivalent residual strain ratio interval under a certain confining pressure, the probability method is used to characterize the proportion of different equivalent residual strain ratio intervals, as shown in Figure 11.

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As shown in Figure 11(a), when the confining pressure is 0-2 MPa (including 2 MPa), the proportion of equivalent residual strain ratio of raw coal and briquette composed of soft coal between 1 and 1.75 is 0.924 and 0.915, respectively, which is greater than 90%. As shown in Figure 11(b), when the confining pressure is 2-4 MPa (including 4 MPa), the proportion difference between the equivalent residual strain ratio of raw coal in the range of 1-1.5 and 1-1.75 is not significant, which are 0.826 and 0.869, respectively; the probability is close to 90%. The proportion of equivalent residual strain ratio of briquette composed of soft coal is 1 in the range of 1-2.3. As shown in Figure 11(c), when the confining pressure is 4-6 MPa (including 6 MPa), the proportion of raw coal equivalent residual strain ratio is 0.95 in the range of 1-1.75, and briquette composed of soft coal is 0.938 in the range of 1-2. As shown in Figure 11(d), when the confining pressure is above 6 MPa, raw coal can be divided into 6-8 MPa (including 8 MPa), 8-15 MPa (including 15 MPa), and 15-25 MPa (including 25 MPa). In the case of 6-8 MPa (including 8 MPa) and 8-15 MPa (including 15 MPa), the proportion of raw coal equivalent residual strain ratio is 1 and 0.88 in the range of 1-1.75. In the case of 15-25 MPa (including 25 MPa), the proportion of raw coal equivalent residual strain ratio distribution in the range of 1-1.5 has a probability of 0.947.

#### 5. Numerical Simulation Verification

##### 5.1. Numerical Model

In this paper, COMSOL Multiphysics is used to simulate and verify the conventional triaxial test data of raw coal and briquette composed of soft coal in reference [35]. The model settings are as follows: firstly, the three-dimensional compression process is simplified in the model wizard part, which is set to two-dimensional axisymmetric form, and the solid mechanics module in the physical field interface of structural mechanics is selected for the study, and the Mohr-Coulomb criterion is modified by introducing the equivalent residual strain ratio to characterize the strength parameters as a function of the softening parameter. The modified Mohr-Coulomb criterion is finally used as the governing equations for numerical simulations, and the D-P criterion is matched to ensure that the yield surface in the principal stress space is a smooth cone, with the aim of improving computational efficiency. Secondly, the geometry is chosen as a rectangle, and the node summation of the boundary is applied in the form of an integral, using custom softening parameters and an evolution equation for the strength parameters. Thirdly, the whole loading process adopts the control mode of axial displacement, which is controlled as 1 mm/min; the geometric model and the experimental coal sample have the same size of mm. Finally, in the boundary settings, the left and lower boundaries are roller support, the right boundary is subjected to boundary load, the upper axial applies specified displacement, and the structured mesh is divided and solved, as shown in Figure 12.

In the setting of simulation parameters, initial cohesion, initial internal friction angle, residual cohesion, and residual internal friction angle are defined, respectively. Since they are indirectly measurable parameters, they can be obtained by means of regression analysis through triaxial tests under different confining pressures [36]. According to the Mohr-Coulomb failure criterion, when the coal body is damaged, the relationship between maximum and minimum principal stress is as follows:

The results of the linear fitting of strength values are shown in Figure 13.

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The numerical simulation parameters are shown in Table 7.

##### 5.2. Numerical Simulation Results and Error Analysis

Numerical simulations are carried out for raw coal at 2 MPa, 4 MPa, and 6 MPa confining pressure, respectively, and numerical simulations are carried out for briquette composed of soft coal at 2 MPa, 3 MPa, 4 MPa, and 6 MPa confining pressure, respectively, as shown in Figure 14.

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When studying the possible errors of selecting different equivalent residual strain ratios, different equivalent residual strain ratios are selected for briquette composed of soft coal at different confining pressure for analysis, respectively. The results are shown in Figure 15.

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The quantitative analysis of the errors is shown in Figure 16. At 2 MPa confining pressure, when the equivalent residual strain ratio is 1.25, 1.5, and 1.75, the estimated residual strength is 1.2153, 1.0433, and 0.8714 times of the real residual strength, respectively. At 3 MPa confining pressure, when the equivalent residual strain ratio is 1.25, 1.5, and 1.75, the estimated residual strength is 1.3108, 1.1105, and 0.9103 times of the real residual strength, respectively. At 4 MPa confining pressure, when the equivalent residual strain ratio is 1.25, 1.3, and 1.5, the estimated residual strength is 1.0126, 0.9674, and 0.7865 times of the real residual strength, respectively. At 6 MPa confining pressure, when the equivalent residual strain ratio is 1.25, 1.5, and 1.75, the estimated residual strength is 1.2305, 0.9963, and 0.7620 times of the real residual strength, respectively.

#### 6. Conclusions

In this paper, based on the equivalent residual strain ratio, a method for estimating the residual strength of soft coal is proposed. The behavior of the softening parameter composed of the equivalent residual strain ratio, together with the Mohr-Coulomb criterion to jointly control the postpeak stress drop, is discussed. Then, the extracted stress-strain data are summarized, the variation laws of strength ratio and strain ratio with confining pressure are analyzed in detail, and the value range and maximum interval probability of equivalent residual strain ratio are obtained. The residual strength estimation method is simulated and the errors are analyzed. The flowchart of the new method for residual strength estimation is shown in Figure 17.

In conclusion, the following conclusions can be drawn from this study: (1)The study on the strength ratio of coal and rock mass shows that the strength ratio tends to 1 with the increase of confining pressure, which is relatively small under low confining pressure(2)From the analysis of the variation law of the equivalent residual strain ratio of raw coal and briquette composed of soft coal with confining pressure, it can be concluded that for raw coal, the equivalent residual strain ratio ranges from 1 to 2.28, under the confining pressure of 0-25 MPa (including 25 MPa), and the probability of equivalent residual strain ratio of raw coal in the range of 1-1.75 is about 90%. For briquette composed of soft coal, the value range of equivalent residual strain ratio is 1-3.35, under the confining pressure of 0-2 MPa (including 2 MPa), 2-4 MPa (including 4 MPa), and 4-6 MPa (including 6 MPa), and the equivalent residual strain ratio intervals with a probability of occurrence greater than 90% are 1-1.75, 1-2.3, and 1-2, respectively. When the confining pressure is greater than 6 MPa, the briquette composed of soft coal tends to be ideal plastic(3)The reliability of the residual strength estimation method is verified by triaxial compression simulation. This shows that the evolution mode of the postpeak numerical simulation curve of soft coal is basically consistent with that of the triaxial test curve, indicating that the model can better describe the postpeak strain-softening characteristics of soft coal and accurately obtain the residual strength estimation

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that the research was conducted in the absence of any commercial of financial relationships that could be construed as a potential conflict of interest.

#### Acknowledgments

The authors are grateful for the financial support from projects funded by the National Natural Science Foundation of China (No. 52274238 and No. 42211540389), Fundamental Research Funds for the Central Universities (No. 2021YCPY0110 and No. 2020ZDPY0224), and China Postdoctoral Science Foundation (No. 2021M692733).