Influence of Mesoscopic Parameters of Weakly Cemented Rocks on Macroscopic Mechanical Properties

Abstract

In weakly cemented rocks, the mesoscopic parameters have a great influence on the macroscopic mechanical properties. One example of a typical weakly cemented rock is the Cretaceous coarse sandstone in the Hongqinghe Coal Mine. In this study, rock samples were subjected to physical and mechanical experiments, from which a sample model was constructed based on particle flow theory. Uniaxial compression numerical simulation experiments and analyses were conducted, and sensitivity analyses of various microscopic parameters in relation to the macroscopic mechanical properties of the rock were performed via a control variable method. A response mechanism between the macroscopic and mesoscopic parameters was then inferred. On the microscopic scale, the rock is porous with a loose structure and extremely low average uniaxial compressive strength, indicating looseness and weakness. The mesoscopic parameters were then divided into three grades based on their degrees of influence from high to low on the peak strength, peak strain, and elastic modulus. Laboratory experiments revealed that the fracture form of weakly cemented coarse sandstone is typically due to single-section shear failure, whereas through simulation, cracks are caused mainly by tension failure. These two failure modes were inferred to be consistent with each other.

1. Introduction

In recent years, solving the mechanical problems of discontinuous media via the discrete element method has become a hot research direction for the numerical analysis of the properties of geotechnical materials. However, an important prerequisite to allow researchers to perform reliable numerical simulations is the determination of the mesoscopic parameters that correspond to the macroscopic properties of the materials. Compared with other rocks, weakly cemented rocks have several peculiar properties, such as low strength, poor cementation, and easy disintegration. Thus, the parameter assignments used in the simulation of other rocks may not be suitable for weakly cemented rocks. Therefore, it is necessary to conduct numerical simulation experiments specifically for the properties of weakly cemented rocks.

In line with this objective, Liu et al. [1] formulated a method for selecting a rock’s mesoscopic parameters according to the influence of these parameters on the stress–strain curve and the specimen failure style. Cong et al. [2] studied the close relationship between macroscopic mechanical characteristics and mesoscopic parameters and determined a quantitative correlation between the two. Cheng et al. [3] used the Particle Flow Code in 2 Dimensions (PFC^2D) software (PFC2D.5.00.25(64bit), Itasca, IL, USA) to analyze the sensitivities of mesoscopic parameters, such as parallel bond strength, the particle stiffness ratio, particle gradation, and the friction coefficient, to the mechanical properties of the filling body. Chen et al. [4] calibrated mesoscopic parameters to study the relationship between macroscopic and mesoscopic parameters, and they verified the feasibility of PFC simulation by comparing the results of laboratory experiments and simulations. Jiang et al. [5] simulated uniaxial tensile and uniaxial compression tests by calibrating different void ratios and uniformity coefficients. Furthermore, they reported on the shortcomings of the parallel bonding model from the perspective of a microscopic failure mechanism and proposed an improved method. Song and Ning [6] studied the relationship between the quantitative parameters of the meso-structure of weakly cemented rocks and their macroscopic mechanical behavior. Consequently, they inferred that the properties of cemented sandstones are affected by changes in the stiffness ratio among the mesoscopic parameters. Xu and Liu [7] investigated the influence of the cementation radius and the particle friction factor on the macro-mechanical characteristics of rock based on weakly cemented rock. They concluded that when the cementation radius is constant, variations in the particle friction factor produce corresponding variations in the macro-mechanical characteristics of the rock. Huang and Xia [8] studied the effect of the mesoscopic parameters of sandstone on its macroscopic mechanical properties. They determined that the elastic modulus E is affected mainly by the particle bond modulus Ec and the stiffness ratio kn/ks, whereas the uniaxial compressive strength σ is affected mainly by the normal bond strength σc, shear bond strength τc, and particle bond modulus Ec. Hou et al. [9] analyzed the mechanical properties of weakly cemented sandstone at different cementation strengths. They concluded that there is an evident linear correlation between the particle bond strength and the macroscopic uniaxial compressive strength of the rock; the strain at peak time also varies with cementation. Furthermore, the intensity increases and continues to grow. Liu and Xu [10] studied the influence of different particle sizes on the mechanical properties of weakly cemented sandstone, which were in line with the general law of rock mechanical response, and they concluded that the particle size distribution has a greater impact on the mechanical properties. As shown by the aforementioned studies, the use of numerical simulation to study the macro-mechanical properties of coal and rock masses is an effective research method that can reflect the macroscopic, mesoscopic, and mechanical property response mechanisms more intuitively.

Liu and Dong [11] believed that the cementation properties play an influential role on the stresses of weakly cemented sandstone structures. The mesoscopic parameters related to the cementation properties in weakly cemented rocks have a great influence on the macroscopic properties. Because of their peculiar characteristics, weakly cemented rocks cannot be accurately simulated using the parameter assignments typically utilized in simulations of other types of rocks. Moreover, the previous research lacks systematic experiments on the mesoscopic parameters of weakly cemented rocks; the influence of mesoscopic parameters on macroscopic properties of weakly cemented rocks needs further analysis. In order to carry out accurate simulation experiments on weakly cemented rocks, this study conducted simulations on the relationship between the mesoscopic parameters of rock and the macroscopic properties based on a linear parallel bond contact model. Subsequently, based on an analysis of the macroscopic stress–strain eigenvalues of different mesoscopic parameters, the mesoscopic sensitivity parameters that affect the macroscopic eigenvalues of rocks were summarized. The specific technical route is shown in Figure 1.

2. Indoor Physical Mechanics Experiment

2.1. Experimental Sampling and Preparation

For this experiment, samples were obtained from the Cretaceous coarse sandstone in the Hongqinghe Coal Mine. The sandstone is a dark red, iron-cemented, typically weakly cemented rock.

The microstructures of the collected rocks were observed via scanning electron microscopy (SEM). The collected rock cores were then sampled and manually polished into cylindrical cores, each with a diameter of 50 mm and height of 100 mm, as shown in Figure 2, according to the engineering rock sample method standard. At the same time, Vaseline was smeared onto the upper and lower surfaces of the samples to serve as a coupling agent between the acoustic emission probe and the rock specimen. In addition, this strategy also reduces the influence of the outside world on the acoustic emission collection during the loading process. A good sound wave propagation path was thus created between the specimen and the sensor. The test equipment included a YA-600 test machine (YA-600, Changchun Kexin Test Instrument Co., Ltd.) and AE21C acoustic emission detector (AE21C, American Acoustics). The loading rate was 0.01 mm/s, a single-channel acquisition method was adopted for acoustic emission, a 15γ probe was used, and the threshold value was set to 30 dB.

2.2. Experimental Results and Analysis

2.2.1. Characteristics of Rock Mesostructure

The rock microstructures observed via SEM are shown in Figure 3. The weakly cemented sandstone is composed of particles as its basic skeleton. The skeleton structure has numerous pores and is relatively loose. The fine clay particles are unevenly arranged and randomly distributed. Adhesive particles are attached to the surfaces of the particles, mainly as thin films or on the contact points between the particles, which play a role in the bonding or contact mode between the particles.

2.2.2. Rock Stress–Strain Law

Five standard specimens were subjected to uniaxial compression tests, and the collected mechanical parameters and acoustic emission signal parameters were analyzed. The main mechanical parameters are shown in

Table 1. Mechanical parameters of weakly cemented sandstone samples.

Samples	Density ρ (kg·m−1)	Elastic Modulus E/GPa	Compressive Strength σc/MPa	Peak Strain ε/%
S-1	2440	1.75	13.5	0.63
S-2	2460	3.48	14.7	0.35
S-3	2380	2.65	12.0	0.37
S-4	2320	1.79	8.35	0.51
S-5	2420	2.03	10.6	0.47

.

Figure 4 shows the rock stress–strain curves. The peak strength of this rock is much lower than that of the same kind of rock from mines in the middle-eastern part of China. In the natural state, the rock peak strength ranges from 8.35 MPa to 14.7 MPa with an average of 11.83 MPa. The rock strength is equivalent to that of hard coal. According to the definition of soft rock in the code, this kind of rock is categorized as a soft rock.

According to the stress–strain curve of the whole process, the failure forms for this kind of sandstone are divided into two types: one is the normal strain softening stage, as exhibited by S-1, S-3, and S-4; and the other is the cementation ductility in the strain softening stage, as exhibited by S-2 and S-5.

2.2.3. Acoustic Emission Characteristics of Rocks

To study the acoustic emission of this kind of rock, two typical rock samples, S-4 and S-5, were selected for analysis.

Figure 5 shows the acoustic emission curve of S-4. As shown in the figure, the strength of the specimen was low, and acoustic emission occurred at relatively low pressures. Cracks in the specimen gradually developed in the compaction, elasticity, and yield stages. Finally, the ringing count would increase sharply, and energy would be released in the failure stage, which is a typical rock mechanical failure characteristic [12].

For comparison, Figure 6 shows the acoustic emission curve of S-5. As shown in the figure, ringing count and energy were generated at the initial loading stage of the specimen. This is because there were pores and cementation forces between particles in the specimen. The strength of the weakly cemented rock was low. A decrease in the number of pores broke the original state between particles, resulting in acoustic emission. At the same time, the sample was categorized as a weakly cemented sandstone, which is prone to weathering, hydrolysis, and weak cementation, and is characterized by low strength. Before reaching the strain softening stage, the sandstone would be destroyed, indicating continuous strain growth and constant stress.

From these experimental results, the weakly cemented rock had low strength and poor cementation performance. Furthermore, it had the characteristics of rock mechanical failure and exhibited a cementation ductility stage. Therefore, in simulations of weakly cemented rocks, it is necessary to focus on the assignment of parameters that control strength and cementation performance to obtain similar results between experiments and simulations.

3. Numerical Simulation Experiment Scheme

3.1. Bonding Model Selection

In the software simulation, the cementation of rock is based mainly on a bonding model and has great influence on the macroscopic mechanical properties of the rock. In the particle flow method, a bonding effect similar to cementation in actual tests can be imparted at the contact points of unit particles. The bonding model includes a contact bonding model and a parallel bonding model. These two models are very different from each other. The scope of contact bonding exists only within a small range at the contact point, in which the point contact form is applicable. Conversely, for the parallel bonding model, the scope of action is in a limited rectangular or circular area, in which the surface contact property is applicable. As shown in Figure 7, the parallel bonding model also includes two contact interfaces: the first is an infinitely small linear elastic interface, which does not bear tension, can bear friction, and can only transmit force; the second is a linear elastic bonding interface with specific dimensions, which can transmit forces and moments. Therefore, the first model is equivalent to a linear model and cannot withstand torsional forces, and sliding is achieved via the provision of a coulomb-limited shear force. By contrast, the second model is for parallel bonding. When such bonding occurs, it resists torque and behaves in a linearly elastic manner until the force exceeds its strength limit and the bond model fails. Conversely, when no such bonding occurs, the load cannot be transmitted. This unbonded linear parallel model is also equivalent to a linear model [13,14].

As shown in Figure 3, the weakly cemented rock has large spacings between particles and the particles are filled with cement, which plays a role in cementation. Therefore, the parallel bond model used in this study will be able to reflect the macroscopic mechanical properties of weakly cemented rocks more realistically and accurately [15,16].

3.2. Model Construction

The numerical model of weakly cemented rock was established using PFC^2D for subsequent uniaxial compression experiments. The relationship between meso-parameters, macro-parameters, and rock failure laws were studied. The basic mechanical parameters of the rock, as shown in Table 1, were obtained through laboratory tests. Based on these parameters, the coarse sandstone is characterized by low strength and poor cementation.

A standard numerical model ϕ50 × 100 mm was then constructed, and five simulation processes, i.e., sample forming, pre-pressing, cementing, unloading, and loading, were written using PFC command flow and Fish language. This model contains a total of 6339 particles. The particles are connected by parallel bonding properties. A total of 15,741 contacts are generated in the rock, and the parallel bonding properties of the contacts between the particles are activated. The model is shown in Figure 8.

Based on the macro-parameters of the S-2 specimen, as shown in Table 1, the “trial-and-error method” was used to continuously adjust the meso-parameters until the simulation results, such as elastic modulus, compressive strength, and peak strain, were similar to the properties obtained from rock laboratory experiments. In this way, the meso-parameters were finally determined [17]. The basic meso-parameters of the rock that were matched by the model are shown in

Table 2. Rock mesoscopic parameters.

Rock Category	Minimum Particle Radius Rmin/mm	Ratio of Particle Radius Rmax/Rmin	Friction Coefficient	Particle Density ρ (kg·m−1)	Particle Effective Modulus Ec*/GPa	Particle Stiffness Ratio kn/ks
Weakly cemented coarse sandstone	0.35	1:1.66	0.3	2460	3.1	3.0
Parallel Bond Effective Modulus ${{\overline{E}}_c}^{*}$/GPa	Parallel Bond Stiffness Ratio	Parallel Bond Cohesion pb-coh/MPa	Parallel Bond Tensile Strength pb-ten/MPa	Parallel Bond Friction Angle pb-fa (°)		Parallel Bond Friction Coefficient
3.1	3.0	6.4	8	40		0.3

. These parameters were used as the original data for controlling a single variable in the simulation experiment.

Figure 9 shows the uniaxial compression stress–strain curves of weakly cemented coarse sandstone as obtained from laboratory tests and simulations. The figure shows that the characteristics of the two curves are basically similar, with only a small error. Thus, the numerical simulation results can reflect the deformation law of rock under load [18].

3.3. Test Plan

The properties of weakly cemented rocks are affected mainly by inter-particle contact and inter-particle cementation performance. On this basis, to study the influences of different parameters on the test results in different gradients, this experiment was performed on eight parameters: In the linear group, emod and kratio, and in the parallel bonding group, pb_emod, pb_kratio, pb_coh, pb_ten, pb_fa, and pb_fric, were assigned to the simulation. Emod and pb_emod as well as kratio and pb_kratio have similar properties, and thus, their values were similarly varied within 1–21 and 0–30, respectively. To reflect the proportional relationship between pb_10 and pb_coh, the values were varied within a relatively large range of 1–110 MPa. The values of pb_fric were within the range 0–3, whereas the values of pb_fa were within the range 0–90°. The aforementioned parameters were all within reasonable ranges and were non-uniformly selected. Thus, based on the scheme designed in

Table 3. Contact attribute values.

Group		Mesoscopic Parameter Name	Value
Parallel Bond Material Group	Linear Group	emod/GPa	1 3 6 10 15 21
		Kratio	0.1 0.3 1 3 10 30
	Parallel Bond Group	pb_emod/GPa	1 3 6 10 15 21
		pb_kratio	0.1 0.3 1 3 10 30
		pb-coh/MPa	1.4 6.4 16.4 56.4 106.4
		pb-ten/MPa	3 8 18 58 108
		pb_fric	0 0.06 0.3 1 3
		pb_fa (°)	0 15 40 65 80

, experimental simulations based on the method of controlling a single variable were conducted on each parameter for linear contact and parallel bonding one at a time [19].

4. Correlation Analysis of Rock Meso-Parameters and Macro-Mechanical Properties

Weakly cemented rocks have discontinuous defects within them, such as particle interfaces, cementing materials, micro-cracks, and pores, which cause these rocks to act as special anisotropic and heterogeneous materials [20]. Most conventional rock fracture studies have been based on homogeneous, continuous media mechanics models, which are difficult to apply to the study of fracture evolution and damage mechanisms in weakly cemented rocks. Weakly cemented rocks, which consist of large numbers of particles and cementing materials, are structurally complex and produce discrete properties in terms of mechanical properties [21,22,23].

4.1. Deformation Law of Rock under Uniaxial Compression

Figure 10 shows the deformation curves of rock at different compression stages under uniaxial compression. The fracture of weakly cemented rock had evident zoning deformation characteristics, which can be divided into four typical stress–strain stages: (1) Compaction stage (OA). At this stage, the skeleton structure of the weakly cemented rock can be regarded as a continuously solid structure. Regarding its deformation, the process is dominated by linear elasticity, whereas the porous part undergoes mainly nonlinear elastic changes. The porous part of the rock is closed under the action of external force, the macroscopic deformation mechanical ability of the rock is strengthened, and the rock’s resistance to external force is realized mainly by the cementation force between particles. (2) Elastic stage (AB). As the load of the rock sample continues to increase, the compaction stage ends and it enters the elastic stage, indicated herein as the AB interval. In this stage, both the rock particles and the cement jointly bear the external load. At this time, if the rock sample is unloaded, its strain can be recovered. (3) Plastic stage (BC). In this stage, the external load is greater than the yield strength of the composite structure of rock particles and cement, the number of cracks in the rock gradually increases, and the strain is irreversible. (4) Post-peak stage (CD). In this stage, the rock enters the failure stage. The combined structural system of rock particles and cement follows the principle of minimum energy: the cement material will be destroyed first, and rock cementation will be lost. The rock relies mainly on the friction between particles to resist external influences [24,25,26,27,28], namely, the carrying capacity of the residual after the peak.

4.2. Parameter Sensitivity Analysis

The sensitivities of the rock parameters to macroscopic deformation were analyzed based on the normalization of each fine parameter after simulation. The peak strength, peak strain, and elastic modulus of the rock were used as macroscopic sensitivity indicators, and the degrees of sensitivity of the fine parameters were classified according to changes in the indicators. After analysis of the experimental data, the parameters that exhibited increases of ≥200% or decreases of ≥70% were defined as the most sensitive parameters (grade I); the parameters that exhibited increases of 50–200% or decreases of 30–70% were defined as the sensitive parameters (grade II); and the parameters that exhibited increases of <50% or decreases <30% were defined as insensitive parameters (grade III) [29].

4.2.1. Sensitivity Analysis of Peak Intensity

Figure 11 shows the variation curves for the peak intensities of the fine view parameters, such as particle effective modulus, particle stiffness ratio, parallel cohesive effective modulus, and parallel cohesive stiffness ratio, at different parameter assignments. According to

Table 4. Table of changes in peak intensity.

Mesoscopic Parameters	Extent of Peak Increase/%	Extent of Peak Decrease/%	Grade
pb-coh	454.1	97.8	I
pb_fa	412.5	97.6	I
pb-ten	400.3	80.0	I
pb_kratio	72.14	41.9	II
pb_emod	27.5	21.6	III
pb_fric	21.5	17.7	III
kratio	13.12	11.6	III
emod	4.2	4.1	III

, the degrees of influence on the peak strength can be divided into three grades, from high to low: I, II, and III. The most sensitive parameters (grade I) were the parallel bond cohesion, parallel bond tensile strength, and parallel bond friction angle. As the bond cohesion and parallel bond tensile strength were increased, the peak strength first increased and then tended to be stable within 0–18 MPa and 3–16.5 MPa, respectively. As the friction angle was increased from 0 to 15°, the peak strength first increased, and then as the angle was increased from 15 to 80°, the peak strength gradually decreased. The peak strength variation range was 0.36–15.2 MPa, and the results for friction angles greater than 60° had no research significance, and thus are not analyzed below. Conversely, the sensitive parameter (grade II) was the parallel bond stiffness ratio. Increasing the parallel bond stiffness ratio resulted in two change trends for the peak strength. When the parallel bond stiffness ratio was within 0.1–3, the peak strength exhibited an increasing trend; when the ratio was within 3–30, the peak intensity continued to decrease. These two processes caused the peak intensity to fluctuate in the range 6–14.6 MPa. Lastly, the insensitive parameters (grade III) were the parallel bond effective modulus, parallel bond friction coefficient, particle stiffness ratio, and particles effective modulus; the influence of the four parameters on peak strength was not evident. The maximum and minimum amplitude differences were 2.7 MPa and 0.8 MPa, respectively.

The strength of the weakly cemented rock is low. The uniaxial compressive strengths of rock samples collected from the Hongqinghe Coal Mine were approximately 11.56–37.89 MPa, and the average compressive strength of the Cretaceous coarse sandstone was 13.48 MPa. In different parameter simulations, the peak strengths were within 0–18.1 MPa, which is consistent with the characteristics of weakly cemented rocks.

The relationship between the mesoscopic parameters in grade I and the peak intensity fitting is shown in Figure 12, and expressed mathematically as follows:

$$\sigma =-42.3e^{\frac{-x_1}{2.56}}+16.4, {\mathrm{R}}^2=0.98,$$

(1)

$$\sigma =-0.145e^{\frac{x_3}{17.3}}+15.1, {\mathrm{R}}^2=0.97,$$

(2)

$$\sigma =-27.86e^{\frac{-x_2}{3.1}}+18.1, {\mathrm{R}}^2=0.98,$$

(3)

where:

• σ—peak intensity (MPa);
• $x_1$—pb_coh (MPa);
• $x_2$—pb_fa (MPa); and
• $x_3$—pb_ten (°).

Figure 12. Fitting curve of peak intensity sensitivity parameter.

Figure 12. Fitting curve of peak intensity sensitivity parameter.

For weakly cemented rocks, the peak strength is affected mainly by the cement. Herein, the bond properties between particles are indicated by the parameters in the parallel bond model. Parallel bond cohesive force and parallel bond tensile strength are similar to macroscopic cohesive force and tensile strength, and their values affect the characteristics of cement between the particles. These two parameters, as shown in the figure, exhibit a positive correlation with the peak strength. For simulations of weakly cemented rock, the values of these two are assumed to be relatively small. There is also a constraint relationship between the two. Zhao et al. [30] analyzed the influence of the pb_ten and pb_coh ratio on compressive strength. When 0 < pb_ten/pb_coh < 2, the compressive strength was affected more significantly by pb_ten. By contrast, when pb_ten /pb_coh ≥ 2, the compressive strength is affected more significantly by pb_coh. This change is also reflected in the microscopic parameter simulation of weakly cemented rock. According to the aforementioned research and analysis, in weakly cemented rock simulations, pb_ten and pb_coh should be kept between 5–18 MPa and 2–20 MPa, respectively, and the ratio between the two should be kept at about 2. According to Cullen’s law:

$$\tau =\sigma \tan \phi +c,$$

(4)

where:

• $\tau$—ultimate shear stress under normal stress (MPa);
• $c$—cohesion of rock (MPa); and
• $\phi$—internal friction angle of rock (°).

The numerical simulation also generated a similar Mohr–Coulomb curve, which was an oblique straight line. Of the parameters, pb_coh is the intercept, similar to the macroscopic cohesion, whereas pb_fa is the inclination angle, similar to the macroscopic internal friction angle [3]. Under the control of a single variable, the internal friction angle is negatively correlated with the compressive strength, which is basically consistent with the trend of decreasing peak strength with respect to increasing pb_fa.

4.2.2. Peak Strain Sensitivity Analysis

Figure 13 shows the variation curves of peak strain for different assignments of mesoscopic parameters, such as particle effective modulus, particle stiffness ratio, parallel bond effective modulus, and parallel bond stiffness ratio. According to the analysis in

Table 5. Table of changes in strain.

Mesoscopic Parameters	Extent of Strain Increase/%	Extent of Strain Decrease/%	Grade
pb_emod	707.2	87.6	I
pb-ten	241.6	70.7	I
pb_kratio	233.3	70	I
emod	152.9	60.47	II
pb_fa	68.2	40.5	II
pb-coh	57.53	36.5	II
pb_fric	21.9	18	III
kratio	2.85	2.78	III

, the degrees of influence on the peak strain can be divided into three grades, from high to low: I, II, and III. The most sensitive parameters (grade I) were the parallel bond effective modulus, parallel bond tensile strength, and parallel bond stiffness ratio; changes to the parallel bond effective modulus caused the peak strain to exhibit a decreasing trend, with the extent of decrease continuously decreasing, whereas the change trends in peak strain with respect to parallel bond tensile strength and parallel bond stiffness ratio were consistent with each other: the strain first increased and then tended to be stable. Conversely, the sensitive parameters (grade II) were the particle effective modulus, parallel bond cohesion, and parallel bond friction angle. Changes in the particle effective modulus tended to decrease the peak strain, although the peak strain changes caused by the parameter changes were relatively uniform. Parallel bond cohesion appeared to be critical at 16.4 MPa; the peak strain increased in the early stage and then tended to be stable. Meanwhile, as the friction angle was increased from 0 to 15°, the strain increased and changed, after which, as the angle was increased from 15 to 80°, the strain decreased monotonically. Lastly, the insensitive parameters (grade III) were the particle stiffness ratio and parallel bond friction coefficient; the effect of particle stiffness ratio on strain was stable at approximately 0.365%. The influence of the friction coefficient on strain was 0.07%.

The weakly cemented rock has poor cementation and is easily disintegrated. It easily ruptures when subjected to an external load. Generally, the deformation amount is small; for example, among the collected rock samples, the peak strain was less than 0.65%. From each variation curve of peak strain versus the rock mesoscopic parameters, as shown in Figure 13, the peak strain varies below 0.7%, which is consistent with the deformation characteristics of weakly cemented rocks.

The relationship between the mesoscopic parameters of grade I and the peak strain fitting is shown in Figure 14 and expressed mathematically as follows:

$$\epsilon =0.8e^{\frac{-x_1}{2.75}}+0.107, {\mathrm{R}}^2=0.99,$$

(5)

$$\epsilon =-0.75e^{\frac{-x_2}{3.13}}+0.4, {\mathrm{R}}^2=0.99,$$

(6)

$$\epsilon =-0.28e^{\frac{-x_3}{1.14}}+0.375, {\mathrm{R}}^2=0.98,$$

(7)

where:

• $\epsilon$—peak strain (%);
• $x_1$—pb_emod (MPa);
• $x_2$—pb_ten (MPa); and
• $x_3$—pb_kratio.

Figure 14. Fitting curve of peak strain sensitivity parameters.

Figure 14. Fitting curve of peak strain sensitivity parameters.

Chen et al. [31] determined that the parallel bonding effect of the effective modulus on the strain is significant, exhibiting a positive correlation trend. Herein, the simulation results for weakly cemented rocks revealed similar changes. The parallel bond tensile strength, which is similar to the macroscopic tensile strength, was the main property acting on the cement. As the parallel bond tensile strength is increased, the yield stage of the rock continues to move backward, and the deformation amount increases under the loading of the force. Song and Ning [6] and Liu et al. [32] observed that as kn/ks decreases, the tangential stiffness of particle contact increases, and the weakly cemented sandstone becomes more brittle and easier to break after being loaded. The rock also exhibits brittle characteristics, and thus, the deformation quantity decreases as the parallel bond stiffness ratio decreases.

4.2.3. Sensitivity Analysis of Elastic Modulus

Figure 15 shows the influence of meso-parameters on the elastic modulus. The elastic modulus varied within 0–12.4 GPa as the different parameters are changed. According to

Table 6. Table of changes in modulus of elasticity.

Mesoscopic Parameters	Extent of Elastic Modulus Increase/%	Extent of Elastic Modulus Decrease/%	Grade
pb-coh	639.1	98.45	I
pb_emod	475.8	82.6	I
pb_fa	352	97.2	I
emod	139.7	58.3	II
pb_kratio	98.8	49.7	II
pb-ten	91.4	47.75	II
kratio	8	7.45	III
pb_fric	4.2	4	III

, the elastic modulus amplitude of variation, i.e., the degrees of influence of the parameters on the elastic modulus, can be divided into three grades: I, II, and III, from high to low. The most sensitive parameters (grade I) were the parallel bond cohesion, parallel bond effective modulus, and parallel bond friction angle. The parallel bond cohesion had the most significant influence on the variation range of the elastic modulus. The variation trend of the elastic modulus was to increase to 4.18 GPa and then stabilize at approximately 4 GPa; the influence of parallel bond effective modulus on the elastic modulus is demonstrated when the elastic modulus increases as the modulus value is increased. The variation range of the elastic modulus was 2.11–2.4 GPa. As the friction angle was increased from 0 to 40°, the elastic modulus remained stable, after which, as the angle was increased from 40 to 60°, the elastic modulus began to exhibit a decreasing trend. Conversely, the sensitive parameters (grade II) were the particle effective modulus, parallel bond stiffness ratio, and parallel bond tensile strength. There was an approximately proportional relationship between the particle effective modulus and the elastic modulus, and increasing the parameter changed the elastic modulus in roughly the same way. Meanwhile, as the parallel bond stiffness ratio was increased, the elastic modulus decreased [33], and the range of decrease also gradually decreased; as the parallel bond tensile strength was increased, the elastic modulus first increased and then tended to be stable. Lastly, the insensitive parameters (grade III) were the particle stiffness ratio and parallel bond friction coefficient, and the difference of the influence of the particle stiffness ratio on the elastic modulus was 0.25 GPa. The change of the parallel friction coefficient stabilized the elastic modulus between 4.1–4.2 GPa. The elastic moduli of typical weakly cemented rocks in China are roughly in the range 0–17 GPa. By comparison, in the numerical simulation of weakly cemented meso-parameters, the elastic modulus was in the range 0–13 GPa.

The relationship between the mesoscopic parameters in grade I and the elastic modulus fitting is shown in Figure 16, and quantitatively expressed as follows:

$$E=-350.9e^{\frac{-x_1}{0.31}}+4.0, {\mathrm{R}}^2=0.99,$$

(8)

$$E=0.5x_2+2.53, {\mathrm{R}}^2=0.97,$$

(9)

$$E=-2.33e^{\frac{x_3}{0.31}}+4.2, {\mathrm{R}}^2=0.99,$$

(10)

where:

• $E$—Young’s modulus (GPa);
• $x_1$—pb_coh (MPa);
• $x_2$—pb_emod (MPa); and
• $x_3$—pb_fa (°).

Figure 16. Fitting curve of elastic modulus sensitivity parameter.

Figure 16. Fitting curve of elastic modulus sensitivity parameter.

Shi et al. [13] determined that the parallel bond effective modulus approximately corresponds to the elastic modulus, and that the parallel bond effective modulus is proportional to the elastic modulus. The elastic modulus is a measure of the ability of an object to resist elastic deformation. In the elastic stage, particles and cements are subjected to external loads, static friction between particles, and the combined effect of cementation. According to the previously presented analysis, the parallel bond cohesion and parallel bond friction angle are related to the macroscopic cohesion and internal friction angle. Cohesion plays a key role in the bonding between particles, whereas the internal friction angle reflects friction between the particles. The parallel bond effective modulus and parallel bond friction angle have significant effects in the elastic stage. The weakly cemented rock exhibits poor cementation performance, and thus the values of pb_coh and pb_fa should not be excessively large; pb_coh should be 2–20 MPa, whereas pb_fa should be 0–60°.

4.3. Analysis of Breakage of Specimen

After uniaxial compression tests on rock samples collected from the Hongqinghe Coal Mine, the failure form of the rock sample, shown in Figure 17a, was obtained. The fracture form of the Cretaceous coarse sandstone is typically a single-section shear failure, with an evident main failure section. After the main failure section is penetrated, the rock sample enters the post-peak stage.

Figure 17b shows a numerical simulation of the failure crack of the Cretaceous coarse sandstone. In the figure, the red crack is that generated by shear failure, whereas the green crack is that generated by tension failure. The simulation results were practically the same as the laboratory experiment results: there was a main penetrating surface for diagonal rupture with an angle of approximately 60°. This crack was mainly generated by tension cracks, with shear cracks randomly distributed around the main failure surface [34]. The overall analysis is consistent with the indoor results.

5. Conclusions

Herein, PFC was used for the numerical analysis part of the study because it can simulate material migration and stress transfer within a specimen. It can also simulate crack formation, propagation, and extension processes. Furthermore, it can be used to monitor the positions and numbers of micro-cracks generated in the failure process of the specimen, the stress and strain inside the specimen, the morphology of crack coalescence, and the strength of the specimen to enable observation of the mechanical testing of rock and soil materials simulated at the microscale.

• Typical weakly cemented coarse sandstone, which was collected onsite, was subjected to meso-structure and uniaxial compression experiments. The results show that the mesoscopic structure of the rock is characterized by numerous pores and that the distribution of fine particles is loose; that the uniaxial compressive strength of the rock ranges from 8.35 to 14.7 MPa, with an average value of 11.83 MPa; that the strength of the rock is extremely low; and that the rock is evidently characterized by looseness and weakness.
• The degrees of influence of mesoscopic parameters on the peak strength of weakly cemented rock can be divided into three grades from high to low: grade I includes the parallel bond cohesion, parallel bond tensile strength, and parallel bond friction angle; grade II includes the parallel bond stiffness ratio, parallel bond friction coefficient, and parallel bond effective modulus; and grade III includes the particle effective modulus and particle stiffness ratio.
• The degrees of influence of mesoscopic parameters on peak strain can similarly be divided into three grades from high to low: grade I includes the parallel bond effective modulus, parallel bond tensile strength, and parallel bond stiffness ratio; grade II includes the particle effective modulus, parallel bond cohesion, and parallel bond friction angle; and grade III includes the particle stiffness ratio and parallel bond friction coefficient.
• The degrees of influence of mesoscopic parameters on the elastic modulus can also be divided into three grades from high to low: grade I includes the parallel bond cohesion, parallel bond effective modulus, and parallel bond friction angle; grade II includes the particles effective modulus, parallel bond stiffness ratio, and parallel bond tensile strength; and grade III includes the particle stiffness ratio and parallel bond friction coefficient.