https://orcid.org/0009-0004-8677-8442
Figures
Abstract
Rock can undergo shape deformation and damage due to the influence of joint fissures, and the range of damage caused by joints at different rock bridge angles varies. To study the influence of rock bridge angle on the size effect of rock secant modulus, this paper adopts the principle of regression analysis and combines numerical simulation to carry out relevant research. The research results indicate that: (1) As the rock bridge angle increases, the secant modulus gradually decreases, following a power function relationship. (2) As the rock size increases, the secant modulus shows a trend of first decreasing and then stabilizing, following a power function relationship. (3) As the rock bridge angle increases, the characteristic tangent modulus and characteristic size gradually decrease, following a power function relationship. On this basis, we obtained their relationship formulas separately.
Citation: Zhou J, Meng S, Hu G, Zha L (2024) Exploring the influence of rock bridge angle on the rock secant modulus. PLoS ONE 19(9): e0307709. https://doi.org/10.1371/journal.pone.0307709
Editor: Peitao Wang, University of Science and Technology Beijing, CHINA
Received: March 15, 2024; Accepted: July 9, 2024; Published: September 25, 2024
Copyright: © 2024 Zhou et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript.
Funding: This research is supported by the University-local government scientific and technical cooperation cultivation project of Ordos Institute-LNTU, (YJY-XD-2024-A-004). The Scientific Research Fund of Liaoning Provincial Education Department (No. LJ2020JCL037). The funder had important roles in the study design, data collection and analysis.
Competing interests: The authors have declared that no competing interests exist.
1.Introduction
Secant modulus (E50) refers to the ratio of stress to corresponding longitudinal strain in rock specimens under longitudinal stress. It is an indispensable parameter in rock mechanics research and plays an important role in rock mechanics theory. It is widely used in engineering design and rock stability analysis. The size of rocks and the geometric shape of internal cracks have a significant impact on their mechanical properties, which is reflected in the variation of E50. Secondly, the rock bridge angle is also one of the key indicators for evaluating rock stability, which also affects the E50. Therefore, in-depth research on the relationship between E50 and rock bridge angle is of great significance for accurately evaluating rock stability.
The E50, as an important parameter reflecting the deformation characteristics of rocks, has always been a hot topic in the field of rock mechanics’ research. Scholars have conducted in-depth research on the mechanical properties and deformation behavior of rocks within the nonlinear strain range through the study of E50. For example, Davarpanah et al. [1] explored the correlation between the critical mechanical properties of rocks through laboratory testing and statistical analysis and found a high correlation between the tangent value of E50 and Poisson’s ratio. Kang et al. [2] found that the decay rate of E50 decreases with increasing confining pressure. Jiang et al. [3] found that the E50 decreases with the axial stress based on the theory of continuous damage mechanics. Peng et al. [4] conducted a series of experiments on granite with different burial depths and found that the E50 and burial depth basically follows a quadratic function relationship. Peng et al. [5] found that as the strain rate increased, the E50 showed an increasing trend. Zong et al. [6] studied the effect of confining pressure and found that the E50 increases linearly with the confining pressure. Zhao [7] investigated the influence of prefabricated cracks on E50. Although these studies have explored the effects of confining pressure, burial depth, strain rate, and cracks on E50, they have rarely considered the influence of rock bridge angle and the impact of rock size changes. Therefore, to further understand the deformation characteristics of rocks, it is necessary to conduct in-depth research on the quantitative relationship between E50 and rock bridge angle.
Studying the relationship between E50 and size is an important topic in the field of rock mechanics. As research deepens, scholars have discovered some phenomena and patterns. For example, Liu et al. [8] studied the influence of joint length on rock mechanical properties using 3D printing technology and found that the E50 continued to decrease with the joint length. Zhu [9] explored the relationship between the E50 and size ratio of the sample based on the hardened soil model. Zhu et al. [10] investigated the effect of interface energy on E50 and found that the E50 is dependent on particle size. Koutou A et al. [11] found that the E50 is influenced by the material shape. Sun et al. [12] explored the influence of rock size on E50 using regression analysis. The above research mainly explores the effects of joint length, particle size, and particle shape on the E50, but there is little research on the influence of rock size on the E50, and the effect of rock bridge angle is also rarely considered.
Accurately determining the characteristic size of rocks is crucial in rock mechanics’ research. Representative Elemental Volume (REV) refers to a representative rock sample size that can provide accurate and statistically significant measurements of mechanical parameters at the minimum scale of macroscopic mechanical behavior. Previous studies have extensively explored the concept and applications of REV. For example, Huang et al. [13] validated and obtained an indicator for estimating rock mass REV based on GSI. Liu et al. [14] developed a method for estimating the geometric feature size REV of rocks based on rock anisotropy. Niazmandi et al. [15] studied the effect of constraints on the size of REV using the DFN to synthesize fractured rock masses. Zhou et al. [16] used three-dimensional laser scanning technology to study the influencing factors of REV. Ma et al. [17] studied the influence of crack size on rock masses REV. Wang et al. [18] proposed a REV estimation method based on permeability. These studies are mainly based on different indicator factors, such as GSI, anisotropy, permeability, etc., to estimate the REV value, with a focus on the definition and calculation methods of REV. However, little consideration is given to the properties of rock itself, such as rock size and angle.
Therefore, this article studies the influence of rock bridge angle on the E50, acquires their relationship, and obtains their influence on the characteristic size.
2. Numerical simulation scheme and boundary conditions
The numerical simulation scheme is carried out from the following two aspects. (1) Explore the relationship between the rock bridge angle and E50, and set the rock bridge angle to 15°, 30°, 45°, 60°, and 75°, including schemes 1–7. (2) Explore the relationship between rock size and E50 and set rock sizes as 100 mm to 400 mm, including schemes 8 to 12. The plans are shown in Table 1.
In the simulation, the parameters of rock mechanics were obtained through geological surveys of rocks in the Haoba Mine in Shaoxing City. The parameters are shown in Table 2.
The loading model of this article is shown in Fig 1, where α is the rock bridge angle. The model adopts displacement loading, with an initial loading amount of 0 and a displacement amount of 0.01mm. The calculation models of different sizes are shown in Fig 2.
(a) 100 mm, (b) 150 mm, (c) 200 mm, (d) 250 mm, (e) 300 mm, (f) 350 mm, (g) 400 mm.
3. Result analysis
3.1 Analysis of stress-strain laws
In rock mechanics, the stress-strain curve of rocks provides the relationship between stress and strain, which can reveal the variation of the E50 of rocks. We conducted simulation studies on schemes 1 to 7 and obtained their stress-strain curves, as shown in Fig 2. We conducted simulation studies on schemes 8 to 12 and obtained their stress-strain curves, as shown in Fig 3.
(a)15°, (b) 30°, (c) 45°, (d) 60°, (e) 75°.
Based on Figs 2 and 3, we select points corresponding to 50% compressive strength, and calculate the E50 value, as shown in Table 3.
Analyzing Fig 2C, as the angle increases, the slope of the curve gradually decreases, and the calculated E50 also gradually decreases. When the angle varies between 15°, 30°, 45°, 60°, and 75°, the corresponding E50 values are 16.32 GPa, 14.72 GPa, 14.24 GPa, 13.86 GPa, and 13.53 GPa, respectively. Within the range of rock bridge angles from 15° to 75°, the E50 decreased by 17.09%. This phenomenon can be explained as the rock bridge angle causing a change in the stress distribution inside the rock. Normally, the E50 of rocks is controlled by the stress state inside the rock. When the rock bridge angle increases, the component of stress along the direction of the rock bridge decreases, while the normal component remains relatively stable. This will lead to a decrease in stress concentration, resulting in a decrease in the E50.
Analyzing Fig 3C, we found that the slope of the curve gradually decreases with increasing size. When the size increased from 100 mm to 300 mm, the E50 decreased from 30.95 GPa to 9.82 GPa, a decrease of 68.3%. After the size exceeds 300 mm, the E50 gradually stabilizes. This indicates that the E50 is significantly influenced by size. This is mainly due to the dominant characteristic factors of rock microstructure at smaller sample sizes. As the size increases, the dominant role of internal joint fractures in rocks strengthens, resulting in a gradual decrease in the E50. As the size increases, the proportion of internal joint cracks in the rock increases, and therefore the E50 gradually decreases.
In summary, the E50 decreases with the increase of rock bridge angle and rock size.
3.2 The effect of rock bridge angle on E50
The E50 decreases with the increase of the rock bridge angle, but their relationship still needs further exploration. Plot the curve of the change of E50 with the angle in Fig 4.
Fig 4 shows that each curve has two poles, namely the E50 at rock bridge angles of 15° and 75°. Analyzing the curve with a rock size of 350 mm, the E50 was the maximum value of the rock at a rock bridge angle of 15°, reaching 9.75 GPa. When the angle is 75°, its E50 is the minimum value, which is 7.26 GPa. During the process of increasing the angle from 15° to 75°, the E50 decreased by 25.54%. This indicates that as the angle increases, the E50 gradually decreases. Fit the relationship between them, as shown in Table 4.
By analyzing the function types of these formulas, we found that they are all power functions. Based on this discovery, the following relationship between the rock bridge angle and E50 was proposed:
(1)
In the formula, E50(α) is the E50, units: GPa; a, b are constant.
a and b are constants closely related to rock size. According to Table 4, the values of a and b were calculated, as shown in Table 5. Scatter plots of a and b with changes in rock size were plotted, and the curves were fitted in Fig 5.
(a) a, (b) b.
Fig 5 shows that parameter a follows a power function relationship with rock size, and parameter b also follows a power function relationship with rock size, as follows:
(2)
(3)
From Eqs (1)-(3), we can obtain a special relationship for the E50 of rocks as follows:
(4)
Formula (4) is applicable for solving the E50 values of rocks of specific sizes. In this formula, the rock size is a known quantity. For a rock mass on site, when the rock bridge angle changes, we can calculate the corresponding E50 value.
3.3 The effect of rock size on E50
Due to the size effect of rocks, changes in size will inevitably affect the E50. To obtain this effect, the relationship curves between E50 and rock size are plotted in Fig 6.
Analyzing Fig 6, as the size increases, the E50 decreases. When the rock size exceeds 350 mm, the E50 gradually tends to stabilize. This reflects the variation pattern of the pore structure inside the rock. At the beginning stage, as the rock size increases, the pore structure gradually increases. When the size reaches a certain threshold, the pore structure gradually tends to stabilize and no longer undergoes significant changes. Solve the fitting relationship for each curve in Table 6.
The formulas in Table 6 shows the relationships between E50 and rock size. Analyzing the function types of these formulas, we found that they all exhibit the characteristics of power functions. Based on this discovery, the following relationship was proposed:
(5)
In the formula, E50(I) is the E50, units: GPa.
c and d are constants closely related to the rock bridge angle. According to Table 6, the values of c and d under 5 rock bridge angles were calculated, as shown in Table 7. Scatter plots of c and d with the change of angle were plotted, and the curves were fitted in Fig 7.
(a) c, (b) d.
Fig 7 shows that c and the angle follow a power function relationship, while d and the angle follow an exponential function relationship, as follows:
(6)
(7)
A special relationship for the E50 of rocks can be obtained from Eqs (5)-(7):
(8)
Formula (8) provides a method for solving the E50, in which the angle is a known quantity, and the rock size is a variable. For a rock, we can calculate the E50 value as it varies in size.
3.4 Fluctuation coefficient
Fluctuation coefficient reflecting the predictability of data and playing an important role in determining the engineering properties of rocks. Formula (9) provides the calculation formula for the fluctuation coefficient of the E50 [19]:
(9)
Among them, Al is the fluctuation coefficient; E50I is the E50; is the average value of the E50
We calculated the fluctuation coefficient of E50 according to formula (9), as shown in Table 8, and plotted the relationship curve in Fig 8.
Variation law of E50 fluctuation coefficient (a) angle, (b) size.
Analyzing Fig 8A, taking the rock size of 250 mm as an example, when the rock bridge angle increases from 15° to 75°, the fluctuation coefficient of E50 increases from 22.14 to 28.02, with an increase of up to 26.56%. This phenomenon indicates that as the angle increases, the fluctuation coefficient will increase.
Analyzing Fig 8B, as the rock size increases, the fluctuation coefficient of E50 decreases from 101.94 to 7.20, with a decrease of 92.94%. When the rock size reaches 350 mm, even if the rock bridge angle changes, the fluctuation coefficient is less than 10%, indicating that the E50 has stabilized.
3.5 Relationship between CSM and rock bridge angle
The characteristic size of rock secant modulus (CSSM) refers to the minimum size required for the determination of rock mechanics parameters, which has an impact on the study of rock mechanical properties. Reference [20] provides a calculation method for characteristic size.
In the formula: γ is the absolute value of the slope. can be considered as a formula for solving CSSM.
3.5.1 CSSM and rock bridge angle.
According to formula (12), we calculated the CSSM for different angles, as shown in Table 9. To clearly observe the trend and pattern of rock CSSM changes under different angles, their scatter plots and fitting curves were plotted in Fig 9.
By analyzing Fig 9, the following conclusion can be drawn: as the angle increases, the CSSM shows a gradually decreasing trend. We conducted data fitting analysis on the collected data and found that they have a power function form, as follows:
(13)
In the formula: D(α) is the CSSM, units: mm.
Formula (13) is used to calculate the CSSM, which is related to the rock bridge angle. But only applicable to rock bridges with two rough joints. The acquisition of this formula makes it convenient and fast to solve the CSSM on site.
3.5.2 CSM and rock bridge angle.
Based on the relevant data in Table 9, the characteristic secant modulus (CSM) values in Table 10 were obtained by formula (5), and the relationship was fitted in Fig 10.
According to Fig 10, as the rock bridge angle increases, the CSM gradually decreases. This decreasing trend can be described by a power function, the regression formula is derived:
(14)
In the formula, Ew(α) is the CSM, units: GPa.
Formula (14) can be used to solve the CSM of rocks, which is related to the rock bridge angle. But it is only applicable for calculating rock bridges with two rough joints.
3.6 Validation analysis
To verify the accuracy of formulas (1) and (5), this section cites stress-strain curve data for different sizes and angles of references [19,21], as shown in Fig 11. Based on Fig 11, the E50 value is calculated, as shown in Tables 11 and 12.
The relationship curve between size and E50, as well as angle and E50, were fitted and plotted in Fig 12.
Through linear regression analysis of Fig 12, the relationship between size and E50, as well as angle and E50, were obtained as follows:
(15)
(16)
The function types of the formulas proposed in formula (15) and formula (5) are consistent, as well as formula (16) and formula (1), which proves the accuracy of formulas (5) and (1) and verifies the accuracy of the analysis method in this article.
4. Discussion
This article establishes three relationships:
- The relationship between E50 and angle. scholars mostly explore the effects of confining pressure [6], burial depth [4], and strain rate [5] on the E50, but rarely discuss the relationship between angle and E50. Therefore, the relationship between angle and E50 obtained in this article provides important help for us to deeply understand the influence of rock bridge angle on E50.
- The relationship between E50 and rock size. In existing studies, scholars have mostly studied the effects of joint length [8], particle size ratio [9], and particle shape [11] on the E50, with little consideration given to the influence of angle on size changes. The relationship obtained in this article provides us with assistance in gaining a deeper understanding of the size effect of E50.
- The relationship between the CSM and the angle. This relationship is established based on obtaining the CSSM. In existing research, there has been relatively little research on the CSSM and CSM, and there has been little discussion on the relationship between CSM and angle. The relationship obtained in this article compensates for this deficiency.
5. Conclusion
The size effect exists in the E50 and the presence of rock bridge angle can also affect the results of size effect. This study investigated this issue through numerical simulation and drew the following conclusions:
- (1) The relationship between the rock bridge angle and E50 is:
By solving the a and b, a specific formula was obtained as follows:
- (2) The relationship between rock size and E50 is:
The special relationship was obtained by solving the c and d as follows:
- (3) The CSSM is related to the rock bridge angle, and the following specific forms are given based on the fitting:
- (4) The CSM is related to the rock bridge angle, and the following specific forms are given based on fitting:
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