# Development of a Hollow Cylinder Torsional Apparatus for Rock

## Abstract

The mechanical characteristics of rock subjected to the changing of the principal stress magnitude and orientation caused by excavation are significant for the construction of larger and deeper underground engineering. However, there have been few experimental studies on rock mechanical characteristics under the changing principal stress orientation due to the lack of the test device. Hence, in this paper, a new rock mechanical experimental technique and device was developed to conduct the complex stress path with coupling variations of stress magnitude and orientation. The theoretical principle and apparatus composition were introduced in this work, and two test cases were conducted to verify its feasibility and reliability. This study has important practical significance and scientific value for promoting the technical level of rock mechanical test and enriching the theoretical frame of rock mechanics.

## Keywords

Rock mechanical test Hollow cylinder torsional apparatus Complex stress path Principal stress axis rotation## List of Symbols

*F*Axial force

*P*_{1}Inner confining pressure

*P*_{2}Outer confining pressure

*M*_{t}Torque

- \({\sigma _{\text{z}}}\)
Axial stress

- \({\sigma _{\text{r}}}\)
Radial stress

- \({\sigma _\theta }\)
Circumferential stress

- \({\tau _{{\text{z}}\theta }}\)
Shear stress

- \({\sigma _1}\)
Maximum principal stress

- \({\sigma _2}\)
Intermediate principal stress

- \({\sigma _3}\)
Minimum principal stress

- \(\alpha\)
Rotation angle of the \({\sigma _1}\) and \({\sigma _3}\) caused by

*M*_{t}*L*Length of the torque arm

*D*Diameter of the axial loading piston

*R*Radius of the piston in the torque hydraulic jack

## 1 Introduction

In large and deep underground excavations, surrounding rockmass is subject to a very complex stress field, and the stress variation caused by excavation has a great influence on the mechanical response of rockmass, possibly leading to large deformation and serious damage (Abel and Lee 1973; Abuov et al. 1988; Kielbassa and Duddeck 1991; Read et al. 1998; Germanovich and Dyskin 2000; Bobet 2010; Zhang et al. 2012; Yong et al. 2013). Practically, the stress variation consists of two aspects, i.e., stress magnitude variation and stress orientation variation. Various studies to evaluate the influence of the stress redistribution (stress magnitude and orientation) on the mechanical response of surrounding rock and the stability of the tunnel excavation, have shown that the stress change can influence the stability of underground openings. Eberhardt (2001) reported that the stress magnitude and orientation varied as the tunnel face advanced, and proposed that the rotation of the principal stress axis was a controlling factor in the direction of fracture propagation and the type of damage induced in the rock mass. Kaiser et al. (2001) discussed the mining-induced stress change and showed that the large stress rotations can contribute to the degradation of the rockmass. Diederichs et al. (2004) observed the mechanisms leading to in situ strength drop, which included the factor of influence of tunnel-induced stress rotation on crack propagation, interaction and ultimately coalescence and failure. Lee et al. (1999, 2002) performed a series of hollow cylindrical triaxial tests to investigate the effect of stress paths on the mechanical behaviour and the yield surface of the sandstone. Most of those studies are focused on the numerical calculation and lack the experimental research on the stress orientation change.

However, almost all the traditional mechanical test techniques and devices in the field of rock mechanics can only control the stress magnitude variation, while the simultaneous control of both stress magnitude variation and orientation is actually impossible (Bieniawski 1967; Eberhardt et al. 1999; Ganne and Vervoort 2006; Amann et al. 2011, 2012). Thus, it is important to develop a new test technique and device to conduct complex stress paths on intact rock specimens with coupling variations of stress magnitude and orientation.

In fact, the hollow cylinder torsional apparatus for soil has been successfully developed to study the influence of the principal stress axis rotation on the strength and deformation behaviours of soil (Hight et al. 1983; Ishihara and Towhata 1983; Vaid et al. 1990; Sayao and Vaid 1991; O’Kelly and Naughton 2005), and it was demonstrated that strength and deformation parameters of soil are significantly stress path dependent.

The above mentioned works provided inspiration for us to develop a new approach to simulate the complex stress paths on intact rock specimens. This paper illustrates the characteristic and capability of the new apparatus. The results obtained from two different stress paths were used to check the correct operation and accuracy of the new equipment.

## 2 Hollow Cylinder Torsional Apparatus for Rock

### 2.1 Theoretical Principle

The theoretical principle of hollow cylinder torsional apparatus for rock is based on the thin-walled cylinder theory within the frame of isotropic linear-elasticity. As shown in Fig. 1 and Table 1, the stress of the specimen in cylindrical coordinate system, including the components of axial stress \({\sigma _{\text{z}}}\), radial stress \({\sigma _{\text{r}}}\), circumferential stress \({\sigma _\theta }\) and shear stress \({\tau _{{\text{z}}\theta }}\), is related to four applied loads on the hollow cylinder specimen including axial force *F*, the inner confining pressure *P*_{1}, the outer confining pressure *P*_{2} and the torque *M*_{t} (Height et al. 1983; Vaid et al. 1990; Sayao and Vaid 1991; O’Kelly and Naughton 2005). Furthermore, the principal stress (\({\sigma _1}\), \({\sigma _2}\), \({\sigma _3}\)) in rectangular coordinate system and the rotation angle *α* of \({\sigma _1}\) and \({\sigma _3}\) caused by *M*_{t} can be expressed by (\({\sigma _{\text{z}}}\), \({\sigma _r}\), \({\sigma _\theta }\), \({\tau _{{\text{z}}\theta }}\)). Therefore, the complex stress path with coupling variations of stress magnitude and orientation can be simulated by controlling four applied loads independently.

Stress expressions in two coordinate systems

Stress in cylindrical coordinate system | Stress in rectangular coordinate system |
---|---|

\({\sigma _{\text{z}}}=\frac{{F \cdot {\text{(}}{{{D^2}} \mathord{\left/ {\vphantom {{{D^2}} 4}} \right. \kern-0pt} 4} - {r_0}^{{\text{2}}}{\text{)}} - {P_1}r_{{\text{i}}}^{2}}}{{r_{0}^{2} - r_{i}^{2}}}\) | \({\sigma _1}={{\left( {{\sigma _{\text{z}}}+{\sigma _\theta }} \right)} \mathord{\left/ {\vphantom {{\left( {{\sigma _{\text{z}}}+{\sigma _\theta }} \right)} 2}} \right. \kern-0pt} 2}+\sqrt {{{{{\left( {{\sigma _{\text{z}}} - {\sigma _\theta }} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {{\sigma _{\text{z}}} - {\sigma _\theta }} \right)}^2}} 4}} \right. \kern-0pt} 4}+\tau _{{{\text{z}}\theta }}^{2}}\) |

\({\sigma _{\text{r}}}=\frac{{{P_2}{r_0}+{P_1}{r_{\text{i}}}}}{{{r_0}+{r_{\text{i}}}}}\) | \({\sigma _2}={\sigma _r}\) |

\({\sigma _\theta }=\frac{{{P_2}{r_0} - {P_1}{r_i}}}{{{r_0} - {r_i}}}\) | \({\sigma _3}={{\left( {{\sigma _{\text{z}}}+{\sigma _\theta }} \right)} \mathord{\left/ {\vphantom {{\left( {{\sigma _{\text{z}}}+{\sigma _\theta }} \right)} 2}} \right. \kern-0pt} 2} - \sqrt {{{{{\left( {{\sigma _{\text{z}}} - {\sigma _\theta }} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {{\sigma _{\text{z}}} - {\sigma _\theta }} \right)}^2}} 4}} \right. \kern-0pt} 4}+\tau _{{{\text{z}}\theta }}^{2}}\) |

\({\tau _{{\text{z}}\theta }}=\frac{{3{M_t}{R^2} \cdot L}}{{2\left( {r_{0}^{3} - r_{i}^{3}} \right)}}\) | \(\alpha =\frac{1}{2}{\text{arctan}}\frac{{2{\tau _{{\text{z}}\theta }}}}{{{\sigma _{\text{z}}} - {\sigma _\theta }}}\) |

### 2.2 Description of the Apparatus

As shown in Fig. 2, the hollow cylinder torsional apparatus for rock is composed of a triaxial cell, a torque-applying device, a hydraulic system, a strain measuring system and a data acquisition device. The triaxial cell and torque-applying device are the key components to independently apply *F, P*_{1}, *P*_{2} and *M*_{t} on the hollow cylinder specimen with the assistance of hydraulic system which is mainly composed of four independent high-pressure servo pumps.

#### 2.2.1 Triaxial Cell

As shown in Fig. 3, the triaxial cell is mainly composed of an axial loading piston, a sealing cap, top and bottom flange plates, top cap and base pedestal, a steel cylinder and eight bolts. The upper end of the axial loading piston is rigidly connected to the torque-applying device, and the lower end is connected to the top cap by a tenon structure. Both ends of the hollow cylinder specimen are, respectively, bonded with the top cap and base pedestal using the epoxy sealant, which is implemented with the assistance of a self-developed tooling device in this work (see Fig. 4). After that, the base pedestal is fixed on the bottom flange plate through bolt connection. Finally, the steel cylinder, top and bottom flange plates are connected by eight bolts.

As shown in Fig. 3, the axial force *F*, the inner confining pressure *P*_{1} and the outer confining pressure *P*_{2} can be applied on the specimen by controlling the hydraulic circuits (*F*_{in} and *F*_{out}, *P*_{1,in} and *P*_{1,out}, *P*_{2,in} and *P*_{2,out}), respectively. And the torque *M*_{t} can be applied through the axial loading piston which is driven to rotate by the torque-applying device.

It is important to note that the o-rings (between the steel cylinder and the flange plates, the base pedestal and the bottom flange plate) and epoxy sealant (between the specimen, top cap and base pedestal) must be adopted to ensure the absolute independence of the outer and inner confining pressures. RTV silicone rubber is daubed with the thickness of 3 mm on both outer and inner walls of hollow cylinder specimen to separate the specimen from the hydraulic oil. After test, the epoxy sealant can be removed with special solution to enable reuse of the top cap and base pedestal.

#### 2.2.2 Torque-Applying Device

As shown in Fig. 5, the torque-applying device is composed of a torque arm, a torque hydraulic jack and a torque reaction frame, which are connected to each other with pin connection. The torque reaction frame is bolted with the top flange plate, and the torque arm is fixed to the upper end of axial loading piston with gear-occlude connection. It is very important that the torque arm must be perpendicular to the axial loading piston, and the torque hydraulic jack perpendicular to the torque arm.

#### 2.2.3 Strain Measuring System

To characterize the mechanical response of the specimen, axial strain, radial strain, circumferential strain and shear strain are measured. As shown in Fig. 6, two linear variable differential transformers (LVDTs) with the accuracy of 0.001 mm and the range of 10 mm are used in this work to measure the axial strain. The outer and inner strain-rings are developed by the authors to measure the deformation of outer and inner walls of the specimen, which can be used to calculate the radial and circumferential strains. To measure the shear strain, a vertical steel sheet is adopted and fixed to the bottom flange plate, then the shear strain can be measured by a torsional strain gauge which is stuck to the vertical steel sheet.

An Ethernet-based data acquisition device (DS-NET) is chosen in this work to automatically record the test data (see Fig. 2). As shown in Figs. 5 and 6, data wires from LVDTs, torsional strain gauge and outer strain-ring are connected to data acquisition device through outlet no. 2, and the test data from inner strain-ring is transmitted to data acquisition device through outlet no. 1 successively.

### 2.3 Specimen Preparation

The size of hollow cylinder rock specimen is an important issue for this test technique. Comprehensively considering the strength, deformation and failure characteristics of rock and the principles of the previous research results (Hight et al. 1983; Sayao and Vaid 1991), a hollow cylinder specimen with the outer diameter of 50 mm, inner diameter of 30 mm, and height of 120 mm is selected in this work. To prepare the hollow cylinder specimen more precisely and efficiently, a double-drill device was developed as shown in Fig. 7.

## 3 Test Cases

In this part, two test cases were conducted to verify the feasibility and reliability of the technique and apparatus proposed in this work.

### 3.1 Stress Variation Under a Constant Rotation Angle

Considering the test repeatability, an aluminium alloy sample was firstly selected, and the expected stress path was set as shown in Fig. 8a. First, *P*_{1} and *P*_{2} were applied simultaneously from point O to A, and the stress state is \({\sigma _1}>{\sigma _2}={\sigma _3}\) at point A according to the equations listed in Table 1. Then, *F* was successively applied to the value calculated by Eq. (1) while *P*_{1} and *P*_{2} were kept constant from A to B/C, and the stress state is \({\sigma _1}={\sigma _2}>{\sigma _3}\) at point B/C. *M*_{t} was applied after point C and the stress varied under a constant rotation angle. Note that the equations \({\sigma _{\text{z}}}={\sigma _\theta }\) and \(\alpha ={45^ \circ }\) remained constant as the consequences of the applied loads during the test.

Figure 8b, c showed the load paths and principal stress paths in the test respectively. During test, *P*_{1} was loaded to 5 MPa and *P*_{2} was 10 MPa simultaneously. Then, *F* was loaded to 2.82 MPa calculated by Eq. (1), and *M*_{t} was successively applied under the constant values of *P*_{1}, *P*_{2} and *F*. Great consistency between the expected stress path and principal stress paths in the test is shown by comparing Fig. 8a, c.

### 3.2 Stress Variations of Both Magnitude and Orientation

A marble specimen drilled by the double-drill device was selected for testing and the expected stress path is shown in Fig. 9a. First, *P*_{1} and *P*_{2} were applied simultaneously from point O to A, and the stress state is \({\sigma _1}={\sigma _2}={\sigma _3}\) at point A according to the equations listed in Table 1. Then, *F* was successively applied while *P*_{1} and *P*_{2} were kept constant from A to B/C, and the stress state is \({\sigma _1}>{\sigma _2}={\sigma _3}\) at point B/C. *M*_{t} was applied after point C and the stress magnitude and orientation both varied. Note that, as a consequence of the applied loads during the test, the relational expression \({\sigma _{\text{z}}}>{\sigma _\theta }\) remained true, while the magnitudes and orientations of the principal stresses kept changing.

Figure 9b, c showed the load paths and principal stress paths in the test respectively. During the test, *P*_{1}, *P*_{2} and *F* were kept constant and equal to the selected target value. *M*_{t} was continuously increased until failure. Figure 9c reveals good agreement with the expected stress path shown in Fig. 9a. It can be seen that great fluctuation appeared in the load paths of *M*_{t} and the principal stress paths of \({\sigma _1}\) and \({\sigma _3}\), which might be a consequence of the progressive failure process and the heterogeneousness of the specimen by analyzing the failure characteristics. Hence, the local damage expanded gradually as the loads were continuously applied until complete failure of the specimen occurred. Figure 10 shows the failure pattern of the marble sample. There are main crack penetrated the sample and several minor cracks distributed on the surface under variation of the stress magnitude and orientation. While the failure surface of the marble sample in the conventional triaxial test is mainly composed of a through diagonal crack. Compared with the conventional triaxial test, the failure surfaces showed individual characteristics. Further study is required to reveal the underlying damage mechanism.

The stress paths mentioned above innovatively considered the influence of stress variations containing both magnitude and orientation on the mechanical characteristics of rock. Moreover, the test results confirmed the feasibility of the independent application of the four loads and the stability control of the test system, making the complex stress path realizable in the laboratory.

## 4 Conclusions

In this work, a new rock mechanical test technique and apparatus for simulating the complex stress path were developed to investigate the rock mechanical characteristics. With the independence of four loads on the hollow cylinder specimen, the simulation of complex stress paths, including variations of principal stress magnitude and orientation, become realizable. Two test cases were conducted to verify the feasibility and reliability of the technique and apparatus proposed in this work.

Further study will focus on the following aspects. First, by using the new device, the influence of stress orientation variation on rock failure can be investigated, with the assistance of the acoustic emission (AE) technique, the scanning electron microscope (SEM) technique and the computed tomography (CT) technique. Second, different types of rocks and samples with original cracks will be adopted to study the different mechanical responses and anisotropy. This study has important practical significance and scientific value for promoting the technical level of rock mechanical test and enriching the theoretical frame of rock mechanics.

## Notes

### Acknowledgements

The authors would like to thank the financial supports provided by China National Key Basic Research Program under Grant no. 2014CB046902, the Scientific Instrument Developing Project of the Chinese Academy of Sciences (YZ201553), National Natural Science Foundation of China (NSFC) (51427803, 51404240, 51709257, and 51704097) and Youth Innovation Promotion Association CAS. Besides, the authors are also grateful to the anonymous reviewers for their careful reading of our manuscript and their many helpful comments.

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