With the increase in the population worldwide and people's living standards, the energy demand has continuously increased globally. The exploitation of natural resources is constantly advancing toward deep earth (Jiang & Ashworth, 2021; Lu, 2018; Ranjith et al., 2017; Wang, Ma, et al., 2018). Geothermal resources in deep hot dry rocks, a renewable and widely distributed green energy, have attracted more research attention (Cheng et al., 2022; Li et al., 2022; Lu and Wang, 2015; Qu et al., 2019; Wang, Li, et al., 2018). However, due to the high burial depth and extremely low natural porosity and permeability (Feng et al., 2022; Gao et al., 2022; Lu et al., 2020; National Energy Administration of the People's Republic of China, 2018; Su, Chen, et al., 2017; Xie et al., 2021; Zhuang et al., 2022), challenges still remain in effective extraction of geothermal resources in deep hot dry rocks. The key to solve this problem is the creation of a fracture network through hydraulic fracturing (Dobson et al., 2021; Gao et al., 2022; Jia et al., 2022). However, due to some reasons including high in situ stresses, the risk of induced seismicity in the deep formations remains high, and sometimes hazards are likely to occur (Ellsworth, 2013; Liu et al., 2022; Majer et al., 2007; McGarr, 2014). Experimental research is an important method for a more in-depth understanding of the development process of geothermal resources in hot dry rocks. Compared with material testing on small-sized specimens, tests of large-sized models can be more authentic to reflect the mechanical properties of rock masses conforming to practical engineering (Li et al., 2023; Mao, 2020; Sun et al., 2022; Zou et al., 2021). Therefore, tests of large-sized models are crucial in deep rock mechanics.

A series of triaxial devices that can conduct large-sized model tests have been developed for various purposes (Feng et al., 2015, 2018, 2020; FLOXLAB, 2023; Frash et al., 2014; GCTS Testing System, 2023; He et al., 2003; Li et al., 2019; Lu et al., 2020; Mao, 2020; Su, Chen, et al., 2017; Su, Feng, et al., 2017; Wille Geotechnik, 2023; Zhao et al., 2008; Zhou et al., 2005, 2016). The multi-functional true triaxial geophysical apparatus in Chongqing University has been used to conduct a series of mechanical tests of 200- or 100-mm cubic specimens (Lu et al., 2020). The test system can provide a loading force of 6000 kN in two directions and 4000 kN in another direction, respectively. Zhao et al. (2008) from the China University of Mining and Technology, have developed a 600°C, 20 MN servo-controlled high-temperature and high-pressure triaxial rock testing machine to conduct research in fields such as deep coal mining and geothermal mining (Feng et al., 2020). The tested specimens are cylinders with a diameter of 200 mm and a height of 400 mm. Both the axial and lateral loads of the machine can reach a maximum of 10 000 kN. The Geotechnical Consulting and Testing Systems (GCTS) in the United States can conduct tests of 300 mm cubic specimens (GCTS Testing System, 2023). The maximum loading stress is 103 MPa in three directions, and the temperature of the specimen can reach up to 200°C. The XPS-1000 servo-controlled triaxial system developed by Taiyuan University of Technology can perform experiments on cubic specimens with a maximum size of 1000 mm (Mao, 2020). The maximum axial loading capacity and horizontal loading capacity of the testing machine are 10 000 and 9000 kN, respectively. These triaxial testing systems for large-sized specimens usually adopt a mechanical loading scheme, which is achieved by the hydraulic cylinder connected to the loading indenter. The required loading capacity for a given stress is proportional to the square of the specimen size, which means that the loading framework of the triaxial test system for high stresses of large-sized specimens is not cost-effective. Therefore, the development of a low-cost loading system will be of great significance for large-sized model tests.

Hot dry rock at great depths is usually subjected to high stress and high temperature simultaneously. Therefore, hot dry rock tests are often accompanied by a heating process. The increase in temperature of the materials under constrained conditions also leads to the generation of thermal stress (Chen, 2015). The use of the thermal stress generated by temperature change is a potential approach to exert stress to the specimen, which differs from the traditional mechanical loading scheme. Completely or partially replacing the mechanical loading scheme of traditional triaxial systems with thermal loading is a possible way to reduce the cost of the testing system, which has not been studied in depth yet.

This paper aims to propose and validate a new method that utilizes thermal expansion to achieve stress loading of large-sized specimens. The paper is organized as follows: the theoretical knowledge as well as the basic configuration of the thermal stress triaxial loading system are presented in Section 2. A corresponding simulation model is established with numerical software (COMSOL Multiphysics®) to demonstrate the feasibility of the proposed solution in Section 9. Then, a series of numerical experiments considering practical situations are carried out in Section 10. The results and discussions are presented in Section 14. Although technical details still remain to be solved, the numerical results have confirmed the feasibility of the proposed approach.

2 PRINCIPLES AND DESIGN OF THERMAL STRESS LOADING

The traditional mechanical loading scheme faces the challenge of a cumbersome loading frame and difficulties in the high-pressure loading process in large-sized model tests. To address these issues, this paper proposes a thermal stress-loading method. This method utilizes the thermal stress generated by the temperature change of the thermal expansion material surrounding the specimen to achieve the stress loading and unloading process. This loading approach can achieve both isotropic loading and anisotropic loading implementing uniaxial loading, conventional triaxial loading, and true triaxial loading by adjusting the expansion material and temperature configuration. The details are described below.

2.2 Triaxial thermal stress loading system

According to the concept of thermal stress loading introduced above, the key to generating thermal stress is to constrain the free deformation of materials during temperature changes. Therefore, a triaxial test device based on thermal stress should be equipped with at least a rigid body section, a heating section, and an expansion loading section. Additionally, to monitor the loading effect and meet the requirements of experimentation, the triaxial testing device should also include a data monitoring section. Taking large-sized hot dry rock (cubic specimen with a side length of 400 mm) fracturing tests as an example, the following triaxial loading equipment is designed (Figures 1 and 2). The characteristics of each section are as follows:

               

               

2.2.1 Heating section

The heating section is composed of heating rods (Figure 1c), heating plates (Figure 1d), and heat insulation plates (Figure 1e). The heating rods are connected to the power supply and are responsible for heat generation. The heating plates are the containers/carriers of the heating rods, which have good thermal conductivity, but small thermal expansion coefficients, and can be regarded as rigid bodies. To minimize the heat loss from the heating plates to the box system of the tested specimen, several heat insulation plates are placed between the box system and the heating plates (Figure 1e).

2.2.2 Loading section

The loading section is the core part of the triaxial device. In the demonstrative device to be investigated in this paper, axial loading ( ${𝜎}_{𝑧}$ ) is accurately applied using a loading indenter with a hydraulic servo controller (Figure 1f). This configuration enables the adoption of stress paths in both displacement- and load-controlled modes for diverse testing purposes. By using thermal expansion plates with different or identical thermal expansion coefficients in different directions, the device can achieve conventional triaxial or true triaxial loading. The thermal expansion plates of the same color in Figure 1b form a matched pair to apply ${𝜎}_{𝑥}$ and ${𝜎}_{𝑦}$ on the tested specimen.

2.2.3 Rigid box section

The box section (Figure 1g) possesses high stiffness and can be regarded as a rigid body. Its primary function is to restrain the deformation of the loading section, thereby effectively transforming the thermal stress generated within the loading section into the surface pressure of the tested specimen.

2.2.4 Data measurement section

The data measurement section is for the measurement of load, strain, and temperature. Given the large size of the specimen, conventional measuring devices such as linear variable differential transformers are not suitable. Therefore, stress–strain gauges are recommended to monitor the stress–strain of the specimen. Additionally, several temperature probes are placed at different representative locations of the specimen to measure its temperature. The recorded data are then transferred to the controlling software for further analysis and processing.

In addition, the developed triaxial loading device can be equipped with other sections to meet further experimental needs, for example, a pore pressure section, including syringe pumps and pressure transducers, to carry out hydraulic fracturing tests. The pumps can operate in constant flow or constant pressure mode to impose fracturing pressure on the tested specimen, while the transducers are used to monitor the pressure of the injected fluid throughout the testing process. The basic configuration of triaxial equipment capable of conducting large-sized dry hot rock fracturing tests has been introduced above. However, the feasibility of the proposed solution still needs to be verified, which will be described in the following section.

3 VALIDATION BASED ON NUMERICAL SIMULATION

To validate the loading scheme, a model of a triaxial testing device based on the design described in Section 3 is established in COMSOL Multiphysics® (Figure 2a). For simplicity and efficiency of computation, specific details such as the rigid box system and the axial loading indenter are not included in the model. The model couples the modules of solid mechanics and heat transfer with thermal expansion. Additionally, the events (ev) in mathematics module are also implemented in the model to regulate the operation of the heating rods and control the working state. The temperatures at the center of the sides of the tested specimen are monitored (Figure 3) as indicators of the system's performance. During the heating process, the corresponding heating rods are activated and operated at the specified power level if the monitored temperature falls below the target value. Conversely, the heating rods are deactivated once the monitored temperature exceeds the target temperature. By defining appropriate “events,” the occurrence of temperature overload is effectively mitigated.

           

Assuming that the components of the testing platform are strongly interconnected, each component is considered uniform and isotropic, and the material parameters do not change with temperature, as outlined in Table 1. A basic experiment with an initial temperature of 293.15 K and a target temperature of 373.15 K is conducted. Both the average temperature and average stresses of the tested specimen gradually increase, as depicted in Figure 4. The values of stresses are negative, indicating that the specimen is subjected to compressive stress. It is also noteworthy that the ${𝜎}_{𝑥}$ and ${𝜎}_{𝑦}$ lines are perfectly aligned, that is, conventional triaxial loading, as the material parameters of thermal expansion plates in the x-direction and y-direction are the same (Table 1).

           

Figure 5 illustrates the change of temperature on the z = 0 mm section with time. Initially, the temperature at the center of the specimen is lower than that at the sides. However, the temperature across the specimen equalizes after a certain time, which agrees with the commonly recognized principles. After the temperature becomes stable, the stress field inside the specimen is observed and shown in Figure 6. It should be noted that stress concentration near the edges of the specimen is observed. In contrast, the distribution of stresses inside the specimen is relatively even. The stress concentration is caused by the mutual compression of the thermal expansion plates at the corners (Figure 7). Fortunately, this impact is localized.

           

           

           

Furthermore, the deformations of different parts during thermal loading are also investigated by monitoring the displacements of several key points (Figure 8). The displacements of the heat insulation plate, the thermal loading plate, and the thermal expansion plate are 0.002, −0.001, and 0.020 mm, respectively. Due to differences in material parameters, the thermal loading plates under the designed conditions are subjected to compression, resulting in a negative value of deformation. Besides, these displacements are significantly smaller than each plate's thickness (10, 15, and 20 mm, respectively). As the materials of the test platform are assumed to be ideal materials, including no structural damage during use, all the plates would return to their original shape after the thermal unloading process.

           

Different combinations of target temperatures and thermal expansion coefficients of thermal expansion plates result in varying effects of pressure loading (Figure 9). Setting the target temperature to 573.15 K and taking the thermal expansion coefficient of thermal expansion plates as 10 × 10−6/K, the horizontal stress of the specimen reaches up to 75 MPa (Figure 9), fully restoring the high-temperature and high-stress environment of the deep rock mass. The stress of the specimen shows a linear relationship with the target temperature. When the thermal expansion coefficients of thermal expansion plates are taken as 14 × 10−6/K, the stress increases by 3 MPa for every 10 K increase in temperature. Therefore, precise control of the temperature of the loading device based on thermal stress is crucial. Temperature exceding expectations should be avoided as much as possible. Even if the specimen can be cooled to the target temperature, the thermal damage is often irreversible and may have unpredictable effects on subsequent tests.

           

In addition, the true triaxial loading of the tested specimen can be achieved by adjusting the thermal expansion coefficient of the lateral thermal expansion plates (Figure 10). The simulation results (Figures 6 and 10) demonstrate that the thermal stresses generated in the loading plates are effectively transformed into the lateral loads of the tested specimen. Therefore, the thermal stress loading scheme is feasible.

           

4 FURTHER EVALUATION CONSIDERING PRACTICAL FACTORS

In the previous sections, a simulation model was established by COMSOL Multiphysics®. The model demonstrates that the proposed loading scheme is feasible and effective. However, it is essential to note that the model is idealized. To address these concerns, additional research considering some practical conditions is conducted and the following section outlines the findings.

4.1 Effect of heterogeneity and anisotropy of materials

Anisotropy and heterogeneity of specimens and thermal expansion plates might be important factors affecting the loading result of a specimen. The heterogeneity of materials has many aspects. In this section, we investigate the influence of the elastic modulus and Poisson's ratio of the tested specimen (Figure 11) on the loading result. Upon comparing the obtained results of stress loading on the inhomogeneous specimen (Figure 12) with Figure 10, it is evident that the heterogeneity of material plays a notable role.

               

               

Subsequently, an investigation is conducted to examine the impact of the specimen's anisotropy on the loading result. According to the parameters presented in Table     1, the elastic matrix of the above-mentioned isotropic specimen is as follows:

      $$\mathit{𝐷}=\left[\begin{array}{cccccc}3.6& 1.2& 1.2& 0& 0& 0\\ 1.2& 3.6& 1.2& 0& 0& 0\\ 1.2& 1.2& 3.6& 0& 0& 0\\ 0& 0& 0& 1.2& 0& 0\\ 0& 0& 0& 0& 1.2& 0\\ 0& 0& 0& 0& 0& 1.2\end{array}\right]\times {10}^{10}\text{Pa}.$$     (2)    

For comparison, the anisotropic elastic matrix adopted in this section is determined as follows:

      $$\mathit{𝐷}=\left[\begin{array}{cccccc}3.0& 1.0& 1.5& 0& 0& 0\\ 1.3& 3.6& 1.1& 0& 0& 0\\ 1.2& 1.0& 3.3& 0& 0& 0\\ 0& 0& 0& 1.0& 0& 0\\ 0& 0& 0& 0& 1.1& 0\\ 0& 0& 0& 0& 0& 1.3\end{array}\right]\times {10}^{10}\text{Pa}.$$     (3)    

The remaining parameters are the same as those in Table 1. The stresses along the z-axis are shown in Figure 13. For the anisotropic specimen, ${𝜎}_{𝑥}$ , ${𝜎}_{𝑦}$ , and ${𝜎}_{𝑧}$ are recorded as −17.04, −18.07, and −16.71 MPa, respectively, indicating a true triaxial loading condition. This does not imply that a conventional triaxial loading setup for an anisotropic specimen is impossible. A series of tests of physical properties on the specimen are required before loading to obtain the anisotropic parameters, thus achieving better adjustment of the experimental temperature and selection of material for the thermal expansion plates.

               

4.2 Influence of air gaps between the equipment and the specimen

Due to factors such as the processing accuracy of specimens or the loading equipment itself, there might be air gaps between the thermal expansion plates and the specimen in practical tests. It is important to note that the generation of thermal stress in the thermal expansion plates can only occur when the temperature reaches a certain threshold. At this point, the deformations of the plates and the tested specimen compensate for the gaps, resulting in close contact among all components. Any errors in the size of an actual specimen in the longitudinal direction can be automatically corrected. Therefore, in this section, only the impact of the accuracy of the specimen's horizontal size on the effectiveness of stress loading is discussed.

According to the suggestion provided by the International Society for Rock Mechanics and Rock Engineering (ISRM) (Ulusay, 2015), the acceptable height error for a specimen with a size of 400 mm in each direction should not exceed 1.2 mm, which is significantly greater than the displacement observed in the previous study (Figure 8). In this section, the thermal expansion coefficients of the thermal expansion plates are assumed to be 8 × 10−5/K. The results of loading tests with different thicknesses of air gaps are illustrated in Figure 14. As the thickness increases, the stresses on the specimen decrease significantly, indicating that the presence of air gaps can weaken the loading capacity of the triaxial device significantly. For instance, an air gap with a thickness of 0.2 mm led to a decrease in horizontal stress from −450.16 to −291.48 MPa, representing a reduction of 35.25%. This emphasizes the importance of accurately controlling the tight fit between each part to achieve the desired loading result.

               

4.3 Effect of friction on contact interfaces

Several previous studies have highlighted the significant impact of the friction between the force loading indenter and the specimen on the loading effect of rock specimens (Frash et al., 2014; Zhang et al., 2017). The influence of friction between the thermal expansion plates and the specimen on the loading result is also investigated. The simulation model is further simplified, as shown in Figure 15. The boundary conditions of the model are as follows: the displacement of the top surface of the specimen is fixed, while the bottom surface is supported by rollers. The plate on the left side of the specimen is taken as an example to explain the boundary conditions of plates briefly. As depicted in Figure 15b, the displacement of the plates in its thickness direction in the specified direction is constrained. Additionally, the friction coefficient of the contact surface between the plate and the specimen is specified. In the case of adopting a thermal stress loading scheme, the blue surface represents the temperature boundary and constrains the deformation of the plate in the thickness direction. However, if the traditional mechanical loading scheme with indenters is used, the blue side only serves as the temperature boundary. Simultaneously, a specified pressure is applied to the face. Some material parameters adopted in the model are shown in Table 2, while the remaining parameters are the same as those listed in Table 1.

               

Table 2. Some parameters adopted in the model to study the influence of friction between side plates and the specimen on the loading result while the remaining parameters are the same as those listed in Table     1.

	Loading scheme	Coefficient of thermal expansion (10−6/K)	Elastic moduli (GPa)
Plate-x	Thermal stress loading	8	280
	Force loading indenter	0
Plate-y	Thermal stress loading	12
	Force loading indenter	0

An ideal model, that is, a friction-free model, is first studied. Assuming that a rock specimen is subjected to heating from an initial temperature of 293.15 K to a final temperature of 373.15 K, the resulting average values of ${𝜎}_{𝑥}$ and ${𝜎}_{𝑦}$ within the specimen are −20.66 MPa and −21.92 MPa, respectively. These values serve as the loading stresses in the traditional loading scheme. Figures     16 and     17 show the loading effects observed when different loading methods and friction coefficients between the contact faces are used. It can be seen that,

• 1.

  The friction between indenters and specimens significantly influences the loading results of the traditional loading scheme. As the friction coefficient increases, the non-uniformity of internal stresses in the specimen intensifies progressively. The curve of ${𝜎}_{𝑧}$ along the z-direction in Figure 16c appears smooth, which is attributed to the application of the roller support constraint on the bottom surface and the fixed displacement constraint on the top surfaces of the specimen. That is, no contact friction is considered in the z-direction.

• 2.

  The average stresses of the specimen with the traditional loading scheme decrease by about 9.8% (Figure 17) with the increase of the friction coefficient. However, when the loading scheme based on thermal stress is adopted, the average stresses of the specimen decrease slightly (less than 0.3%; Figure 17) with the increase of the friction coefficient, which is within an acceptable range.

               

               

Moreover, although there are uneven stresses near the specimen surface, the distribution of stresses inside the specimen is relatively uniform. In summary, the thermal stress loading scheme proposed in this paper demonstrates clear superiority over the traditional scheme when considering the friction effect.

5 DISCUSSION

6 CONCLUSIONS

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.