Simulation Study on the Size Effect of Secant Modulus of Rocks Containing Rough Joints

Abstract

The secant modulus reflects the ability of rocks to resist deformation, and it is mostly used to evaluate rock strength and deformation evolution. Due to the existence of rough joints in rocks, the secant modulus changes according to rock size. Therefore, it is very important to effectively obtain the secant modulus to evaluate rough-jointed rock deformation. In this paper, the regression analysis method is used, and 25 sets of simulation models are set up to discuss the influence of joint roughness and rock size on the rock secant modulus. The research shows that the secant modulus increases exponentially with the increase in rock size, and it increases as a power function with the increase in joint roughness. The characteristic size of the secant modulus increases exponentially with the increase in joint roughness, also as a power function. This paper gives the specific forms of these four relationships. The establishment of these relationships enables the prediction and calculation of the secant modulus and provides guidance for rock deformation analysis.

1. Introduction

Rock is a heterogeneous material that contains joints, which is the main reason for its deformation and damage. The secant modulus is one of the important parameters of rock resistance to deformation, reflecting the stiffness of the rock, and it possesses a size effect. Therefore, the in-depth study of rock deformation and failure laws is of great significance in maintaining the stability of underground engineering.

In the study of rock deformation and failure, scholars have investigated the effects of stress, strain rate, and confining pressure on the secant modulus. For stress, Dai et al. [1] proposed a new equation to characterize the variation in the secant modulus with axial stress. Jiang et al. [2] carried out conventional triaxial compression numerical experiments with rock under different deformation parameters and determined the law that as axial stress increases, the secant modulus decreases. Zhang et al. [3] studied the effect of stress on the compaction characteristics of rock mass and determined that the secant modulus increases with the increase in strain. For strain rate, Kang et al. [4] obtained the failure law of the stress–strain curve of rock by carrying out uniaxial compression tests at different strain rates. Peng et al. [5] determined that the rock secant modulus increases with increasing strain rate. For confining pressure, Zong et al. [6] determined through deformation tests of fractured sandstone that the secant modulus increases linearly with the increase in confining pressure. Kang et al. [7] determined through triaxial compression tests that confining pressure affects the secant modulus. The above studies discussed the factors affecting the secant modulus, and they obtained some rules, but they did not establish the mathematical relationships between them.

The joints and fissures existing in rock will have an impact on the secant modulus. Scholars have carried out research on the angle, length, diameter, and size of fissures. Zhu et al. [8] studied the influence of different arc fractures (γ) on the secant modulus. Yin et al. [9] determined that the reduction degree of the secant modulus is related to crack angle and length. Zhang et al. [10] obtained the variation law of the secant modulus with fracture size by studying the deformation of fractured rock. Joints also affect the secant modulus. Wu et al. [11] studied the effect of fractures’ geometric parameters on the deformation characteristics and failure mode of a rock mass. Liu et al. [12] studied the relationship between the rock secant modulus and joint dip. These studies did not consider the influence of joint roughness, nor did they establish a quantitative description between them.

In terms of the size effect of the secant modulus, scholars have carried out research from the perspectives of rock aspect ratio, particle size, and joint length. For example, Ban et al. [13] characterized mechanical properties with size effect based on the secant modulus method and the effective model reduction method. Sun et al. [14] found that the strain rate sensitivity of the secant modulus in the plastic segment of specimens with large aspect ratios was stronger than that of specimens with small aspect ratios. Zhu et al. [15] used PFC software to obtain the stress–strain curves of substitute materials at all levels and found that there was an exponential correlation between the ratio of the secant modulus and the maximum particle size (D) of the sample. Liu et al. [12] studied the effect of joint length on the secant modulus of rocks with rough joints. Hu et al. [16] established the size effect relationship between joint spacing and rock uniaxial compressive strength. The above research was carried out from different angles, and a size effect of the secant modulus of rocks was confirmed. These works show the size effect is related to many factors, but the exact relationship between size and the secant modulus has not yet been revealed. Therefore, it is particularly important in engineering to study the size effect of the secant modulus of rocks with rough joints, as well as obtain the relationship between the secant modulus and the rock size.

Due to the existence of the rock size effect, the mechanical parameters of rock will gradually stabilize and reach a constant value at a certain size, which is defined as the representative elementary volume (REV). For this research, Chen et al. [17] analyzed the size effect of the degree of rock mass block and determined the REV size of the rock mass. Wei et al. [18] obtained the rock characteristic size of a particular engineering case, which was 4 m. Loyola et al. [19] evaluated the REV of fractured rocks based on the central limit theorem (CLT). Peng et al. [20] analyzed the size effect through PFC software and obtained an REV size of 16 m. Hu et al. [21] obtained the characteristic size expression of the elastic modulus by simulating the uniaxial compression of rocks with rough joints. Zhang et al. [22] obtained the REV of deformation and strength parameters through numerical tests of rock mass samples of different sizes. Wu et al. [23] determined the REV of a fractured rock mass based on the concept of scale effect. The above studies indicate that there are relatively few investigations of the secant modulus in terms of the REV of rock mechanical parameters, especially the secant moduli of rocks with rough joints.

Therefore, this paper explores the effect of joint roughness and rock size on the secant modulus. The relationships between secant modulus and rock size/joint roughness are described quantitatively. The relationships between the characteristic size of secant modulus, characteristic secant modulus, and joint roughness are also described.

2. Numerical Simulation Plans

Numerical simulations were carried out to consider two different aspects: (i) the influence of rock size on the secant modulus (Plans 1–5), with rock sizes set to 100, 200, 300, 400, and 500 mm. (ii) The effect of joint roughness on the secant modulus of the rock (Plans 6–10), where the roughness values were set to 1.6, 2.6, 3.6, 4.6, and 5.6, as shown in

Table 1. Summary of joint roughness coefficient and rock size.

Simulation Plans	JRC	a	b	c	d	e
		Size 100 mm	Size 200 mm	Size 300 mm	Size 400 mm	Size 500 mm
1	1.6	1.6 × 100	1.6 × 200	1.6 × 300	1.6 × 400	1.6 × 500
2	2.6	2.6 × 100	2.6 × 200	2.6 × 300	2.6 × 400	2.6 × 500
3	3.6	3.6 × 100	3.6 × 200	3.6 × 300	3.6 × 400	3.6 × 500
4	4.6	4.6 × 100	4.6 × 200	4.6 × 300	4.6 × 400	4.6 × 500
5	5.6	5.6 × 100	5.6 × 200	5.6 × 300	5.6 × 400	5.6 × 500

. A total of 10 simulation plans were set up, and 25 numerical models were established in this simulation. The rock mechanical parameters, boundary conditions, and loading conditions in these numerical simulations were obtained by referring to Ref. [21].

The simulation software used in this paper was RFPA2D, and the 2D model was used in the article. The joint roughness was obtained through the geological survey of the slope rock of Zhang’ao Mine, which is located in Zhang’ao Village, Shengzhou City, Zhejiang Province, China. The method to obtain the roughness was as follows: first, a structural surface profile sampler was used to draw the field structural surface to obtain the roughness profile curve. Then, scanning and programming methods were used to convert the roughness profile curve into a CAD data curve, which was imported into RFPA2D. Finally, the corresponding parameters were assigned to the roughness curve in RFPA2D.

For the rock used in the numerical simulation, the elastic modulus was 8000 MPa, Poisson’s ratio was 0.25, the compressive strength was 60 MPa, the cohesion was 1.2 MPa, the friction angle was 30°, and the density was 2600 g/cm³. For the joint used in the numerical simulation, the elastic modulus was 0.01 MPa, Poisson’s ratio was 0.25, the compressive strength was 0.01 MPa, and the friction angle was 10°.

The constraint condition used in the numerical simulation was that the two sides of the model were free surfaces without force, and the upper and lower surfaces of the model bore the load. The numerical simulation adopted the displacement loading method, where the displacement on both sides of the model was 0 mm, and that on the upper and lower surfaces of the model was 0.01 mm.

3. Numerical Simulation Results and Analysis

3.1. Stress–Strain Curve Analysis

The stress–strain curves of rock failure under different sizes in research content (i) are shown in Figure 1, where Figure 1a–e represent different roughnesses. The stress–strain curves of rock failure corresponding to joint roughness in research content (ii) are shown in Figure 2, where Figure 2a–e represent different rock sizes.

Figure 1 shows that when the axial strain was in the range of 0–0.001, the rock with rough joints was in the linear elastic stage. In this stage, the rocks experienced two conditions: tight compaction and micro-crack expansion. As the axial stress increased gradually, the deformation increased, and the microcracks expanded.

Figure 1 also shows that when the rock stress reached its peak strength, the rock began to undergo brittle failure, and the strength decreased rapidly. The peak strength decreased gradually with the increase in rock size, but it would not decrease when the strength decreased to a certain value, indicating that the peak strength has an obvious size effect. The elastic stage of the stress–strain curves of the rock was mostly linear, and its slope was characterized by the elastic modulus. When the rock size increased, the slope of the stress–strain curves gradually decreased, indicating that the rock size affects the elastic modulus, and that they are negatively correlated.

The laws of stress–strain curves for rock failure in Figure 2a–e are basically similar. Taking Figure 2e as an example, the rock failure law when the joint roughness changes was analyzed. When the joint roughness increased from 1.6 to 5.6, the peak strength gradually increased accordingly, which indicates that the joint roughness has a certain influence on the rock strength, and the two are positively correlated.

The slope of the curve in Figure 2e decreased gradually with the increase in joint roughness, which indicates that the elastic modulus of the rock is affected by the joint roughness, as the former decreased when the latter increased.

3.2. Influence of the Size Effect of Secant Modulus

To study the size effect of the rock secant modulus, first, according to Figure 1, the secant modulus of the rock was determined by selecting the slope of the 50% stress peak point and the origin according to the curves of the linear elastic part of the stress–strain curves. The secant modulus values are shown in

Table 2. Secant modulus values.

Simulation Plans	JRC	Secant Modulus (GPa)
		a	b	c	d	e
		Size 100 mm	Size 200 mm	Size 300 mm	Size 400 mm	Size 500 mm
1	1.6	21.67	14.38	7.92	6.79	5.45
2	2.6	23.62	17.79	9.22	8.35	5.59
3	3.6	29.55	17.88	11.51	8.88	6.38
4	4.6	30.68	22.31	16.37	9.98	9.10
5	5.6	31.88	23.88	16.98	11.99	9.41

.

Table 2 shows that changes in rock size influence the secant modulus. From Table 2, a scatter diagram of secant modulus and rock size was drawn, and the corresponding curve was fitted, as shown in Figure 3.

Combined with Figure 3 and Table 2, the curve and data for a joint roughness of 1.6 were analyzed. When the rock size increased from 100 mm to 500 mm, the secant modulus of the rock decreased from 21.67 GPa to 5.45 GPa, which indicated that as the rock size increases, the secant modulus decreases, and they are negatively correlated. As the rock size increased further, the secant modulus changed less, and the curve gradually tended to be stable, showing an exponential decay trend.

To better illustrate this relationship, the formulas of the fitting curves are listed in

Table 3. Fitting formulas between secant modulus and rock sizes.

Roughness	Fitting Formula	R2
1.6	$E_{50}(l)=5.51+37.52{\mathrm{e}}^{-l/120.17}$	0.978
2.6	$E_{50}(l)=5.2+38.91{\mathrm{e}}^{-l/146.54}$	0.942
3.6	$E_{50}(l)=4.95+45.02{\mathrm{e}}^{-l/171.18}$	0.998
4.6	$E_{50}(l)=4.75+45.5{\mathrm{e}}^{-l/210.34}$	0.981
5.6	$E_{50}(l)=3.19+46.07{\mathrm{e}}^{-l/265.83}$	0.996

. The coefficients of determination were all greater than 0.9, which proves that the secant modulus has a good correlation with rock size.

Table 3 shows that the fitting relationship between secant modulus and rock size conforms to an exponential function, and their relationship is proposed as follows:

$$E_{50}(l)=a+b{\mathrm{e}}^{-l/c}$$

(1)

where $E_{50}(l)$ is the secant modulus, unit: GPa; l is the rock size, unit: mm; a, b, and c are parameters.

The parameters a, b, and c are summarized in

Table 4. Parameters a, b, and c.

	Roughness	1.6	2.6	3.6	4.6	5.6
Parameter
a		5.51	5.2	4.95	4.75	3.19
b		37.52	38.91	45.02	45.5	46.07
c		120.17	146.54	171.18	210.34	265.83

, and the relationships between the parameters and joint roughness are plotted in Figure 4.

According to Figure 4, the relationships between parameters and joint roughness are expressed as follows:

$$a=5.38-0.01e^{JRC/0.96}$$

(2)

$$b=34.08JR{C}^{0.18}$$

(3)

$$c=63.43+35.68e^{JRC/3.23}$$

(4)

Subsequently, a special relationship between the secant modulus and rock size is obtained:

$$E_{50}(l)=6.73-0.78e^{JRC/3.69}+(34.08JR{C}^{0.18}){\mathrm{e}}^{-l/(63.43+{35.68}^{JRC/3.23})}$$

(5)

where $E_{50}(l)$ is the secant modulus, unit: GPa; l is the rock size, unit: mm; JRC is the joint roughness.

Equation (5) quantitatively gives the relationship between rock size and the secant modulus. At construction sites, when the joint roughness coefficient is measured, the corresponding secant modulus under any rock size can be obtained, which is beneficial for engineering applications.

3.3. Size Effect of Joint Roughness on Rock Secant Modulus

Combined with the analysis of the stress–strain curves in Figure 2, the rock secant modulus values under each working condition are given in

Table 5. Secant modulus under different rock roughnesses.

Numerical Plans	Size/mm	Secant Modulus/GPa
		Plan 1	Plan 2	Plan 3	Plan4	Plan 5
		JRC = 1.6	JRC = 2.6	JRC = 3.6	JRC = 4.6	JRC = 5.6
a	100	21.67	23.62	29.55	30.68	31.88
b	200	14.38	17.79	17.88	22.31	23.88
c	300	7.92	9.22	11.51	16.37	16.98
d	400	6.79	8.35	8.88	9.98	11.99
e	500	5.45	5.59	6.38	9.10	9.41

.

Table 5 shows that the joint roughness affects the secant modulus. From Table 5, a scatter diagram of secant modulus and joint roughness was drawn, and the corresponding curve was fitted, as shown in Figure 5.

Combining Table 5 and Figure 5, the effect of joint roughness on the secant modulus of rock was analyzed. Under a rock size of 500 mm, when the joint roughness was 1.6, the secant modulus was 5.45 GPa, and when the joint roughness increased to 5.6, the rock secant modulus increased to 9.41 GPa. Therefore, the secant modulus increases with the increase in joint roughness, and they are positively correlated. The larger the rock size, the greater the variation in the secant modulus, showing an increasing trend. The formulas of the regression curves under each condition are listed in

Table 6. Fitting formulas between secant modulus and joint roughness coefficient.

Size l/mm	Fitting Formula	R2
100	$E_{50}(JRC)=17JR{C}^{0.38}$	0.917
200	$E_{50}(JRC)=12.73JR{C}^{0.48}$	0.908
300	$E_{50}(JRC)=5.93JR{C}^{0.6}$	0.912
400	$E_{50}(JRC)=5.35JR{C}^{0.51}$	0.918
500	$E_{50}(JRC)=4.37JR{C}^{0.41}$	0.800

.

Table 6 shows that the secant modulus and joint roughness conform to a power function relationship, which is proposed as follows:

$$E_{50}(JRC)={dJRC}^f$$

(6)

where $E_{50}(JRC)$ is the secant modulus, unit: GPa; JRC is the joint roughness; d and f are parameters.

The parameters d and f in Table 6 are summarized in

Table 7. Values of parameters d and f.

	Size/mm	100	200	300	400	500
Parameter
d		17	12.73	5.93	5.35	4.37
f		0.38	0.48	0.6	0.51	0.41

. Then, the relationships between parameters and rock size are fitted, as shown in Figure 6.

According to Figure 6, the relationships between parameters d and f and rock size are obtained as follows:

$$d=713.24l^{-0.8},$$

(7)

$$f=0.34+(6.5E\text{-}4)l,$$

(8)

where d and f are parameters; l is the rock size, unit: mm.

Subsequently, a special relationship between the secant modulus and joint roughness is obtained:

$$E_{50}(JRC)=(713.24l^{-0.8}){JRC}^{0.34+(6.5E\text{-}4)l},$$

(9)

where E₅₀(JRC) is the secant modulus under different joint roughnesses, unit: GPa; JRC is the joint roughness; l is the rock size, unit: mm.

Equation (9) quantitatively describes the relationship between the secant modulus and joint roughness. When the rock size is determined, the corresponding secant modulus under any kind of joint roughness can be obtained, which provides certain theoretical support for the evaluation of rock engineering safety and stability.

3.4. Mathematical Model of the Characteristic Size of Secant Modulus and Characteristic Secant Modulus

3.4.1. Mathematical Model of Secant Modulus Characteristic Size

In practical engineering, the rock size is usually relatively large, and the REV can generally be reached. The mechanical parameters of the rock tend to gradually become stable with the increase in rock size. Therefore, when the rock size reaches the REV, the deformation parameters obtained by the test are more conducive to meeting the needs of practical engineering or theoretical research. The characteristic size of the secant modulus can be used to characterize the size effect of the secant modulus. Liang [24] provided a quantitative calculation method for the characteristic size of the secant modulus; the formulas are as follows:

$$\left|k\right|=\left|\frac{b{e}^{(-l/\mathrm{c})}}{\mathrm{c}}\right|,$$

(10)

$$\left|k\right|\le r,$$

(11)

$$l\ge [\ln (\frac{b}{c})-\ln r]c,$$

(12)

where r is the absolute value of the acceptable slope.

3.4.2. Relationship between Characteristic Size of the Secant Modulus and Joint Roughness

The characteristic size of the secant modulus under joint roughnesses of 1.6, 2.6, 3.6, 4.6, and 5.6 is solved and summarized in

Table 8. Characteristic size of the secant modulus and joint roughness coefficient.

Roughness	1.6	2.6	3.6	4.6	5.6
Characteristic size/mm	413.52	480.52	559.68	646.62	758.27

; the two parameters are plotted and fitted in Figure 7.

Figure 7 shows that when the joint roughness increases, the characteristic size of the secant modulus increases gradually; the fitting relationship is an exponential function. Therefore, the following special relation is obtained:

$$L=46.53+282.43e^{JRC/6.07},$$

(13)

where L is the characteristic size of the secant modulus, unit: mm; JRC is the joint roughness.

3.4.3. Mathematical Model of Characteristic Secant Modulus and Joint Roughness

To calculate the characteristic secant modulus, we substituted the characteristic size of the secant modulus in Table 8 in Section 3.4.2 into Equation (5), and the characteristic secant moduli under different characteristic sizes could be obtained. The calculation results are listed in

Table 9. Joint roughness coefficient and characteristic secant modulus.

Roughness	1.6	2.6	3.6	4.6	5.6
Characteristic secant modulus (GPa)	6.78	6.57	6.32	6.13	5.84

. The obtained results were fitted, and the curve of the rock’s characteristic secant modulus and joint roughness is shown in Figure 8.

Figure 8 shows that that the characteristic secant modulus increases with the decreases i joint roughness, and the two parameters have a power function relationship. Therefore, the following special relation is obtained:

$$E(JRC)=7.22JR{C}^{-0.11},$$

(14)

where E(JRC) is the characteristic secant modulus, unit: GPa; JRC is the joint roughness.

3.5. Experimental Comparison and Verification Analysis

To verify the general applicability of Equation (6), Shen [25] were cited, and the rock size selected for the laboratory test was 80 mm, as shown in Figure 9. According to Figure 9, the peak strength of the stress–strain curve was taken as the compressive strength of the rock, and the slope of the curve connecting 50% of the stress peak to the origin was taken as the secant modulus of the rock, as shown in

Table 10. Secant modulus with different joint roughness coefficients.

Joint roughness	7	12	17
Compressive strength/KPa	88.16	107.9	122.27
Secant modulus/KPa	72.93	85.64	92.23

.

The data in Table 10 are combined with the scatter plot of secant modulus and joint roughness, and their fitted curves are drawn, as shown in Figure 10.

The relationship between secant modulus and joint roughness is obtained in Figure 10 as follows:

$$E_{50}(JRC)=43.87JR{C}^{0.26},$$

(15)

where E₅₀(JRC) is the secant modulus of the rock, unit: MPa; JRC is the joint roughness.

The functional type of Equation (15) conforms to the mathematical model proposed in Equation (6). Therefore, the numerical simulations are consistent with the experimental conclusion. The verification proves the reliability of the mathematical model proposed in Section 3.3 (Equation (6)) to determine the corresponding secant modulus under different joint roughnesses.

4. Discussion

This work established the following four relationships: (1) the secant modulus and rock size; (2) the secant modulus and joint roughness; (3) the characteristic size of the secant modulus and joint roughness; (4) the characteristic secant modulus and joint roughness.

(1) Relationship between secant modulus and rock size

Rock size affects the secant modulus of rock. Here, the changes in the secant modulus under different rock sizes were analyzed, and a general mathematical formula for the relationship between secant modulus and rock size was put forward. The parameter value in the general formula is related to the change in joint roughness.

In the existing research on the influence of the secant modulus and rock size, most consider the aspect ratio effect [14] and changes in fracture size [10], where the main discussion is on the effect of an increase in the aspect ratio or a change in the fracture size on the secant modulus. However, these papers rarely establish the relationship between the secant modulus and rock size, and they rarely consider the effect of roughness changes on the size effect of the secant modulus.

(2) Relationship between secant modulus and joint roughness

Here, the changes in the secant modulus with different joint roughnesses were analyzed, and a general formula for the functional relationship between secant modulus and joint roughness was proposed. The parameter value in the general formula is related to the change of rock size. Some scholars have carried out research on the inclination angle of rough joints [12] and the roughness coefficient [26]. For example, Ref. [12] obtained the trend of an increasing secant modulus with the increase in roughness. This conclusion is consistent with the results shown in Figure 6 of this article. However, Su [26] did not establish the relationship between the secant modulus and roughness.

At the same time, according to the relationship between the rock secant modulus and roughness obtained in this paper, a verification study was also performed based on laboratory test results [25]. The accuracy of the formula derivation and the applicability of the formula in this paper were proved, and a solid verification foundation was provided for the popularization and application of the subsequent formula.

(3) The relationship between characteristic size of the secant modulus, characteristic secant modulus, and joint roughness

Based on Equation (1), the relationships between the characteristic size of the secant modulus, characteristic secant modulus, and joint roughness were established. According to the current research, no scholars have studied the relationships between these factors.

The four relationships established in this study reveal the size effect of the secant modulus, which has important engineering application value; to a certain extent, it enriches the theoretical basis of the rock size effect.

5. Conclusions

Based on the numerical simulations and regression analysis, this paper studied the influence of joint roughness and rock size on the rock secant modulus and the characteristic secant modulus. The results are as follows:

(1) With the increase in rock size, the secant modulus of rock increases exponentially.

(2) With the increase in joint roughness, the secant modulus of rock increases as a power function.

(3) With the increase in joint roughness, the characteristic size of the secant modulus increases exponentially.

(4) With the increase in joint roughness, the characteristic secant modulus of rock increases as a power function.

We also established the relationships between the secant modulus and rock size, joint roughness, and the relationships between the characteristic size of the secant modulus, the characteristic secant modulus, and joint roughness. These relationships take into account the influence of the rock size effect and joint roughness, and they realize the prediction and calculation of the rock secant modulus, providing guidance for rock deformation analyses.