## 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

${\dot{\epsilon}}_{ij}^{p}$ is the incremental plastic strain tensor,

${\sigma}_{ij}$ is the stress tensor,

$\dot{\lambda}$ is a non-negative multiplier, and

$g$ is the plastic potential function. For rock and soil material, we always use the non-associated flow rule, and the plastic potential function

$g$ usually takes the form of

where

${\sigma}_{1}$ and

${\sigma}_{3}$ are the maximum and minimum principal stresses,

$c$ is a constant, and

$\psi $ 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

σ_{1} >

σ_{2} =

σ_{3} (

σ_{1},

σ_{2}, and

σ_{3} are the maximum, intermediate, and minimum principal stress, respectively). By a series of deductions based on the theory of plasticity, the dilation angle

$\psi $ is determined as [

34]:

where,

${\dot{\epsilon}}_{\mathrm{v}}^{\mathrm{p}}$ and

${\dot{\epsilon}}_{1}^{\mathrm{p}}$ is the volumetric and axial plastic strain increments, respectively. For the conventional tri-axial compression test, the lateral plastic strain increment (

${\dot{\epsilon}}_{2}^{\mathrm{p}}={\dot{\epsilon}}_{3}^{\mathrm{p}}$) should be measured carefully to calculate

${\dot{\epsilon}}_{\mathrm{v}}^{\mathrm{p}}={\dot{\epsilon}}_{1}^{\mathrm{p}}-2{\dot{\epsilon}}_{2}^{\mathrm{p}}$. 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

${\gamma}_{xy}$ and dilation angle

$\psi $. The dashed line shows the original element, and solid line shows the sheared element. The shear strain

${\gamma}_{xy}$ arises from the distortion of the element, and the normal strain

${\epsilon}_{yy}$ 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

$\psi $ can be described as [

34]:

where,

${\dot{\epsilon}}_{yy}^{\mathrm{p}}$ and

${\dot{\gamma}}_{xy}^{\mathrm{p}}$ 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.

## 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.