1. Introduction
In the natural environment, under the long-term action of complex geological conditions, large numbers of joints, cracks and faults have accumulated in rocks. One of the most common is joints. The existence of parallel joints greatly weakens the strength of a rock, resulting in great challenges regarding the stability of rock engineering [
1]. The size effect is a phenomenon in which the mechanical properties of the rock show a certain change law with the size change, and it is also a universal property of the rock [
2].
Compressive strength is one of the mechanical properties of rocks, and the compressive strength has a size effect. Hudson et al. [
3] obtained the law wherein the strength of a rock decreases with the increase in the length–diameter ratio through laboratory experiments, and proposed that a rock has a size effect. Fu [
4] obtained the size effect of rock compressive strength through uniaxial compression tests with different height-to-diameter ratios. Lu et al. [
5] performed uniaxial compression tests on rocks with different height-to-diameter ratios and obtained the size effect of rock strength as well. Zhao et al. [
6] studied the size effect of rock strength from the perspective of dynamics by conducting indoor rock burst tests on granites of different heights. The above studies showed that the compressive strength of a rock has a size effect. With the expanding research of scholars, the size effect of rocks has been transformed from qualitative description to quantitative analysis. Liu et al. [
7] proposed an empirical formula to describe the size effect of rocks, which was verified in uniaxial compression tests on seven types of rock. The above studies are based on theory or laboratory experiments. The size of the model is mostly below 0.4 m, which limits the study of larger-sized rocks. With the development and application of numerical simulation technology, numerical methods to solve geotechnical engineering problems have gradually become a trend. The most representative one is the Rock Failure Process Analysis System (RFPA), developed by Professor Tang [
8]. Numerical simulations are used to study the size effect of the rock, which overcomes the size limitation of the traditional indoor test model. For example, Luo et al. [
9], Wu et al. [
10], and Wang et al. [
11] all used numerical simulations to study the size effect of rock strength, overcoming the size limit of the model. Although scholars have performed a number of studies on the size effect of rock strength, there are relatively few studies on the effect of the number of joints on the size effect of rock strength.
The number of joints affects the compressive strength of the rock. Both Wang et al. [
12] and Wang et al. [
13] used indoor uniaxial compression tests to study the influence of the number of joints on the compressive strength. Zhao et al. [
14] used uniaxial compression experiments to study rock mechanics. Yang et al. [
15], Meng et al. [
16], Gao et al. [
17], and Liu et al. [
18] studied the influence of the number of joints on rock strength based on numerical simulations. Patel et al. [
19] used compression experiments and numerical simulations to study the influence of stress paths on rock failure. Liu et al. [
20] used RFPA2D to simulate the influence of the number of joints on the blasting effect of rock mass. Cao et al. [
21] studied the influence of joint density on rock mass by a combination of indoor experiments and numerical simulations. Pan et al. [
22] used numerical simulation and a uniaxial compression test to discuss the influence of joint density on rock burst. Whether from the perspective of laboratory tests or numerical simulations, scholars have carried out many studies on the influence of the number of joints on the compressive strength of rocks, but few of them involve the element of rock size effect.
The quantitative description of the size effect of rock strength is characterized by the characteristic size or REV, and the strength of the rock under the characteristic size is called the characteristic strength [
23]. The number of joints has an important influence on the characteristic size and strength. Through case analysis, Gasmi et al. [
24] proposed two representative characteristic sizes—geometric REV and mechanical REV, and classified REV. Some scholars have adopted different methods to evaluate the characteristic size. For example, Ying et al. [
25], based on data collected on-site at the Magi dam site, proposed a method to estimate rock mass REV based on volumetric fracture strength (P32) and statistical tests. Loyola et al. [
26] used the Central Limit Theorem (CLT) to evaluate the REV of fractured rock. Scholars have obtained the characteristic size mostly based on field data, theoretical models, and probability statistics. Cui et al. [
27] estimated the characteristic size of the rock mass REV to be 50–60 m
2, taking the Danba Hydropower Station as an example. Based on the size effect of a jointed rock mass, Chong et al. [
28] estimated the REV size of the rock mass as 10 m × 10 m. Ma et al. [
29] studied the size effect of rock mass strength by using 3DEC, and obtained the characteristic size of 4 × 4 × 8 m
3. Liang et al. [
30] studied the size effect of the uniaxial compressive strength (UCS) of a rock mass, and obtained a characteristic size of 14 m × 14 m. The acquisition of these characteristic sizes has particularity. Scholars have theoretically analyzed the factors that affect the characteristic size. For example, Niazmandi et al. [
31] studied the impact of different constraints on REV based on the discrete fracture network–discrete element method (DFN-DEM). Chen et al. [
32] studied the influence of rock mass (Bz) on REV. Although the above studies have obtained the characteristic sizes of rocks from different methods and theories, they have not studied them from the perspective of mathematical relationships. Few scholars have established mathematical models of the characteristic size and number of parallel joints (NPJ), and few scholars have established mathematical models of the characteristic strength and NPJs.
In this paper, numerical simulation methods are used to establish a numerical simulation program with different numbers of parallel joints and rock sizes. The paper studies the influence of the NPJs and rock size on UCS, and establishes a mathematical model of UCS and the NPJs and rock size. The paper establishes mathematical models of the characteristic size and strength of the rock and the NPJs. The paper obtains a method of solving the parameters under each mathematical model, which can provide a theoretical reference for engineering applications.
4. Discussion
The NPJ and rock size have an impact on the UCS, characteristic size, and characteristic strength of the rock; however, this relationship is yet to be obtained. This paper establishes four mathematical relationships: (i) UCS and NPJ, (ii) UCS and rock size, (iii) rock characteristic size and NPJ, (iv) rock characteristic strength and NPJ. Through the establishment of four mathematical models, the relationships between the UCS and rock size and NPJ were quantified. The relationships between the characteristic size, characteristic strength, and the NPJ were quantified. The results obtained have important scientific significance in engineering. When the NPJ or rock size at the engineering site are obtained, the UCS, characteristic size, and characteristic strength of the rock can be quickly obtained, which has significance for engineering applications. Below, we list the differences between them and consider the existing research from the perspective of mathematical relationships.
- (i)
The relationship between UCS and NPJ
First, the general form of the mathematical relationship is given by considering different rock sizes and NPJs. Then, combined with the size of the rock, the specific form of the mathematical relationship is given. In existing research, scholars rarely consider the role of size in discussing the influence of NPJ on UCS. The relationship among UCS, NPJ, and rock size has not yet been established.
- (ii)
The relationship between UCS and rock size
First, the general form of the mathematical relationship is given by considering the NPJs and rock sizes. Then, combined with the NPJs, the specific form of the mathematical relationship is given. In the existing research on the size effect of UCS, few scholars consider the influence of NPJ on it. The relationship among UCS, rock size, and NPJ has not yet been established.
- (iii)
The relationship between rock characteristic size and NPJ
The establishment of the relationship is based on relationship (i). In the existing research, no scholar has established the relationship between rock characteristic size and NPJ.
- (iv)
The relationship between rock characteristic strength and NPJ
The establishment of this relationship is based on relationships (i) and (iii). In the existing research, no scholar has established the relationship between rock characteristic strength and NPJ.
The relationship obtained in this paper applies to a specific rock type. The obtained mathematical formulas have a good effect in analyzing the relationships between the UCS, rock size, and NPJ of similar rocks. At the same time, this article also provides an analysis method. When the content of mineral petrology changes, the properties of rocks will change. According to the research ideas and methods presented in this article, any rock can be analyzed and the mathematical relationships between them can be obtained.
The rationality of the research is discussed below.
In RFPA, the rock medium model is discretized into a numerical model composed of meso-elements. It is assumed that the mechanical properties of the discretized meso-elements obey a certain statistical distribution law (Weibull distribution). This establishes the relationship between mesoscopic and macroscopic media mechanical properties. Based on this idea, the RFPA software realizes the characterization of rock heterogeneity. Whether it is 2D or 3D, the mechanical properties of the meso-primitives in the numerical model established by RFPA are different, which well reflects the non-uniformity of the rock.
Structural surfaces such as joints and cracks in the rock, as well as geological conditions such as pores, constitute the anisotropic characteristics of the rock. The mechanical properties of rocks in different directions and angles are different. In the process of numerical simulation modeling, the influence of some minor factors is often ignored, and only the major factors are considered for research. In this paper, the influence of the NPJ and rock size on the UCS is studied. Therefore, in the modeling process, the changes in these two are mainly considered. In the process of numerical simulation, the selection of 2D and 3D models is a crucial matter. Due to the non-uniformity and anisotropy of the rock, it is a very good choice to use a 3D model to build the real internal structure of the rock without any simplification, which is also where the 3D model is superior to the 2D model. However, when the influence of some insignificant factors is ignored, the advantages of the 3D model over the 2D model will become less obvious. In the research, the stress–strain curves of different sizes of rocks after compression were analyzed. The 2D model can perfectly reflect the deformation and failure characteristics of the rock. On the contrary, if a 3D model is used, in the analysis process, it may be necessary to intercept the corresponding 2D section for analysis, which will complicate simple problems. In consideration of these issues, we chose a 2D model to study.
In this article, numerical simulation is used to verify the anisotropic characteristics of rocks. The parallel joints cannot always be horizontal. When the angle of the parallel joints changes, the article verifies that the obtained relational formula is still applicable. This verification proves that the relationship obtained in the article is reasonable and applicable to anisotropic rocks.
However, this article still has some shortcomings. Because this paper simplifies the secondary factors in rocks when studying the number and size of parallel joints in the rock, the relationship obtained in this article is suitable for analyzing on-site rocks with parallel joints. At the same time, this article gives less consideration to indoor tests. In follow-up studies, comparative analyses including indoor tests will be performed.