1. Introduction
Extensive studies on the geological disposal of high-level radioactive waste (HLW) have been carried out for several decades [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28]. Alxa area is one of the three candidate areas with a large volume of granitic intrusions in China [
22,
23,
25,
29]. Geomechanical properties of the host rock are of paramount importance for evaluating the stability of the repository during the periods of construction and operation.
Owing to the effects of far field in-situ stresses or redistributed stresses induced by excavation, the shearing load may play an important role in the stability of underground opening [
30,
31,
32,
33]. Dilation behavior is considered as an important factor for better modeling of the deformation behavior of the surrounding rock mass [
34,
35,
36]. Many previous researches have been carried out based on conventional tri-axial compression experiments on cylindrical rock samples. According to these experimental studies, different stages of stress-strain relations have been divided, and dilation behaviors have been studied associated with the process of crack propagation and acoustic emission events during the compression [
1,
35,
37,
38]. These studies are mainly on the phenomenological and mechanism researches. However, a quantified estimation of the rock dilation behavior is required. As a parameter controlling the plastic volume changes [
34], the dilation angle is always used in the plastic constitutive model for rock. For example, for the flow rule:
where
is the incremental plastic strain tensor,
is the stress tensor,
is a non-negative multiplier, and
is the plastic potential function. For rock and soil material, we always use the non-associated flow rule, and the plastic potential function
usually takes the form of
where
and
are the maximum and minimum principal stresses,
is a constant, and
is the dilation angle [
34].
The dilation angle of rock has been widely studied in many published references [
34,
35,
36]. In the early stage, the dilation angle was considered to be constant, and suggestions for determining the constant dilation angle for rock masses were also provided [
34,
39]. However, the researchers later found that the assumption of constant dilation angle should not be realistic, and different methods for describing the dilation angle have been provided considering the varying confining pressures and plastic parameters [
40,
41,
42]. More recently, Zhao and Cai [
35] proposed a mobilized dilation angle model dependent on confining pressures and plastic shear strain; thereafter, this model was put forward for analyses of rock masses [
43]. Walton and Diederichs [
36] argued that this model has too many parameters that are difficult for determination, and a new dilation angle was proposed with a piecewise style [
36,
44,
45].
These studies were mainly based on conventional tri-axial compression experiments with the stress state of
σ1 >
σ2 =
σ3 (
σ1,
σ2, and
σ3 are the maximum, intermediate, and minimum principal stress, respectively). By a series of deductions based on the theory of plasticity, the dilation angle
is determined as [
34]:
where,
and
is the volumetric and axial plastic strain increments, respectively. For the conventional tri-axial compression test, the lateral plastic strain increment (
) should be measured carefully to calculate
. In fact, the lateral deformation is not uniform during the post-peak period, so it is not easy to measure the lateral and volumetric strain accurately.
Comparatively, the dilation angle defined according to the direct shear experiment has a very clear physical meaning, which can be better understood by the sketch presented in
Figure 1a. A sketch of the direct shear experiment on a cubic sample is presented in
Figure 1b. Under the effect of shear loading, a shear zone may be formed with the evolution of the fractures. In
Figure 1c, an element is selected in the shear zone to illustrate the physical meaning of shear strain
and dilation angle
. The dashed line shows the original element, and solid line shows the sheared element. The shear strain
arises from the distortion of the element, and the normal strain
is determined the normal deformation Δ
y divided by the original
y-length of the element. By conducting the direct shear tests, the dilation angle
can be described as [
34]:
where,
and
are the plastic normal strain increment and plastic shear strain increment, respectively. The symbols
x and
y here follow the coordinate system shown in
Figure 1.
Consequently, it may be a considerable choice to study the dilation behavior of rock by carrying out a series of direct shear tests under different normal stresses. In addition to the stability analyses on underground excavation, this method can also be used to study the deformation behavior related to landslide and earthquake.
Direct shear experiments have been widely used for analyzing the mechanical properties of geomaterials. For soil and sand samples, a direct shear test is usually used for studying the shear strength and deformation behaviors [
46,
47,
48,
49,
50]. For rock samples, on the one hand, a direct shear test is often applied to research the strength and deformation behaviors of rock discontinuities [
51,
52,
53,
54,
55,
56,
57,
58,
59,
60,
61]; on the other hand, a direct shear test has been carried out for studying the fracturing patterns inside the rock [
31,
62]. There are still very few studies discussing the deformation behavior of rock, or associating the fracturing process with the stress-strain relations under direct shear experiments [
31]. The dilation behavior has been analyzed in some references by conducting direct shear tests considering various normal stresses; however, these studies are mainly focused on the descriptions based on observation, and a detailed, quantified analysis on the dilation angle is still required to be conducted.
In the Alxa candidate area, field investigations have been carried out in two sub-areas (TMS and NRG sub-areas), and four 600 m deep boreholes have been drilled in these two sub-areas. For more detailed information about the Alxa candidate area, readers are referred to [
22]. Rock structures have been studied, and cored samples have been tested in the laboratory for analyzing their strength and deformation properties, seepage behaviors, and thermal effect on mechanical characteristics [
22,
23,
25]. Nonetheless, the dilation behavior has not been studied. The mobilization of the dilation angle is still required to study for a reasonable plastic model of the underground repository.
Consequently, based on a series of direct shear experiments on the granite samples from Alxa candidate areas, the stress-strain relations will be studied in detail, and the dilation behavior will be investigated considering both the normal stress and the plastic shear strain. This paper is organized as follows: In
Section 2, the granite samples, as well as the experimental setup and methods, are introduced. Thereafter, the experimental results are provided in
Section 3, with a detailed characterization of both shear stress—shear strain and shear stress—normal strain relations.
Section 4 provides a systematic discussion on the mobilization of dilation angle dependent on the normal stress and plastic shear strain. With collected data from the experiments, a fitted empirical model of dilation angle will be proposed.
3. Experimental Results
A series of peak strength values can be obtained from the direct shear tests on the granite samples under different normal stresses (
Figure 4). It should be noted that only the tests under the normal stresses of 3, 9, and 15 MPa were repeated three times considering the limited numbers of the samples, and each of the tests under the other normal stresses was carried out on just one single sample. It is found that these strength values can be well fitted by the linear Coulomb criterion. The fitting result shows that the granite samples have a cohesion of 6.6 MPa and internal friction angle of 65.7°.
The obtained stress–strain curves are presented in
Figure 5. The shear stress–shear strain curves have been obtained for each of the tests under different normal stresses. In
Figure 5, only eight shear stress–shear strain curves under the eight different normal stresses (without the curves obtained from the repeated tests) are presented for clear observation. Unfortunately, the shear stress–normal strain curves have only been recorded successfully for the experiments under the normal stresses
σn = 7, 11, 13, 15, and 17 MPa. Nonetheless, these curves provided enough data for the analyses in this study. It should be noted that the negative value of the normal strain means the expansion of the specimen at the direction of normal stress, and the positive value of the shear strain means the decrease of the angle ∠
xoy under the effect of shear loading (according to the coordinate system, as shown in
Figure 1).
According to the shear stress–shear strain curves and shear stress–normal strain curves presented in
Figure 5, several characteristics can be observed as follows:
- (1)
The shear stress–shear strain curves and shear stress–normal strain curves can be divided into five different stages: (I) Crack closure stage. In this stage, the slope of the shear stress–shear strain curve keeps growing to reach a constant value, showing the closure of the pre-existing cracks inside the specimen. This process may also be mixed with the effect of seating and sample adjustment. It should be noted that the normal strain curves show a slight expansion instead of crack closure induced by compaction because the crack closure process at the normal direction has occurred during the preparation process of applying a normal load, which is prior to the shear loading, so it cannot be shown in this curve. (II) Linear elastic stage. In this stage, both the shear stress–shear strain curve and the shear stress–normal strain curve behave in an almost linear style. (III) Stable crack growth stage. Although the shear stress–shear strain curve still appears linear, the normal strain shows an apparent depart from linearity. Scattered distributed cracks begin to develop in a stable way, and the onset of crack initiation also means the beginning of dilation. (IV) Unstable crack propagation stage. Both the shear strain and normal strain increase in an apparently non-linear style, meaning that crack coalescence and the unstable crack propagation occurs. (V) Post-peak stage. The shear stress decreases with the increasing shear strain, meaning that the shear zone is forming, and the shear load cannot be maintained. The detailed division of the different stages is demonstrated in
Figure 6, with a typical stress-strain curve from this group of experiments. It should be noted that the information of the cracking process has not been monitored in this study, but the above-mentioned analyses are believed to be reasonable because they are based on the features of the stress-strain curves, as well as a comparison with the stage division of conventional tri-axial compression experiments [
1,
35,
37,
38].
- (2)
When the normal stress is relatively low (e.g.,
σn = 3 and 5 MPa), the shear stress drops in a gradual manner after the failure of the specimen, showing relatively ductile post-peak behavior; with increasing normal stress, the post-peak behavior turns to be more brittle. Under the normal stress
σn = 13 MPa to 17 MPa, it seems that the granite samples are prone to fail before reaching the residual shear strength. It appears that the direct shear test results show a ductile–brittle transition behavior, instead of the brittle–ductile transition behavior usually found in the conventional tri-axial compression experiments [
35]. This study tries to provide an explanation as follows: For the cases under lower normal stresses, the peak shear strength is lower, so the strain energy stored in the test system is also lower. Consequently, lower strain energy will be released when failure occurs; therefore, the shear stress drops in a more stable manner. On the other hand, the higher normal stress may result in higher peak shear strength, and higher strain energy will be released during the post-peak stage, leading to a more violent failure of the specimen. It should be noted that the post-peak behavior should also be closely related to the fracturing patterns under different normal stresses, which requires a more systematic research.
- (3)
When the failure occurs, the normal strain is apparently higher under relatively lower normal stresses (e.g., σn = 7 MPa and 11 MPa) than that under higher stresses (e.g., σn = 13, 15, and 17 MPa). It is interesting to find that the normal strain may decrease during the post-failure stage. This decrease is more obvious for the samples under lower normal stresses (e.g., σn = 7 and 11 MPa). This phenomenon is related to the dilation behavior and will be discussed in detail in the next section.
5. Conclusions
Based on a series of direct shear experiments on the granite samples from the Alxa candidate area in China for HLW disposal, this paper supplied a systematic analysis of the shear stress–shear strain and shear stress–normal strain relations. The dilation behaviors of the granite samples were especially studied in detail, and an empirical model on the mobilization of dilation angle dependent on the normal stress and plastic shear strain was proposed. The main contributions are as follows:
- (1)
The shear stress–shear strain curves and shear stress–normal strain curves are divided into five typical stages, which are associated with the deformation and fracturing process. The typical stress thresholds were proposed to divide the different stages.
- (2)
It is found that the increasing normal stress may reduce the maximum dilation angle. When the normal stress is lower, the negative dilation angle may occur; however, this phenomenon has not been observed in the cases under higher normal stresses.
- (3)
An empirical model of the mobilized dilation angle dependent on normal stress and plastic shear strain is proposed based on the fitting of the data collected from the direct shear tests. This model can be used in further studies on the constitutive modeling of the host rock.
This study provided a method to analyze the mobilization angle of rock under direct shear test. The proposed model has well understood physical meanings, and it is easy to determine the values of the parameters. This study can be used for better modeling on the stability of the repository for HLW disposal, and this method can also be put forward to analyze the stability of other geomechanical problems, including the deformation behaviors related to landslides, earthquakes, and so on.
It should also be noted that the effect of temperature induced by the nuclear waste cannot be ignored when considering the mechanical behavior of the host rock, so more studies are required considering the effect of heat on the dilation behavior of the granite samples. In addition, more systematic experimental studies on more types of rock associated with the monitoring of the fracturing process will be conducted to extend the applicability of the dilation angle model supplied in this study.