## 2 Test Methodology
### Testing Setup and Method  
The tests were carried out by the INSTRON 1346 electrohydraulic servo universal testing machine at the Central South University, monitored by a 3D-DIC system. The overview and details of the test equipment are shown in Figure 4. The usage, basic principles, and calculation methods of 3D-DIC have been fully discussed in our previous studies and thus are not repeated here [21; 33; 34; 35]. After tunnel excavation in practical engineering, the stress in the upper part of the tunnel structure is transmitted to the surrounding rock below the inverted arch of the tunnel along the surrounding rock on both sides of the tunnel. To simulate the stress condition, the mold was designed to transfer the force from the testing machine to the local position of the sample. The loading concave mold and test method are shown in Figure 4. Different from the general biaxial test, the biaxial step-loading test has different stresses in the lateral direction. Although the forces in both directions of X are equal, the force applied area is different for the specimen. Therefore, the surface in contact with the mold is subjected to greater stress compared to the other side. The lateral stresses (5 MPa, 10 MPa, and 15 MPa) refer to the stress on the surface in contact with the mold, while the other side  
Figure 3: Prepared rock samples: (**a**) sample photos; (**b**) diagram of sample size. Figure 2: Physical model of the layered surrounding rock at the bottom of the tunnel: (**a**) the stress state of a tunnel engineering project; (**b**) the stress state of a novel designed rock sample. only provides a restraining effect. This can help simulate the stress concentration state of the tunnel arch foot. In addition, the concave mold provides a free surface and deformation space for the specimen. It is possible to clearly observe the heave phenomenon and record the corresponding displacement by using the DIC system. Underground excavation activities change the stress state surrounding the excavation from a three-dimensional stress state to a two-dimensional or one-dimensional stress state. In this study, a step-stress-loading scheme was developed. To simulate different stress states of rock samples, three lateral stress values were set. In addition, to reasonably formulate the step-loading gradient value of rock samples, the maximum uniaxial compression load of rock samples was tested. The loading schemes for the rock samples are listed in Table 3. The test methods were introduced as follows: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Loading Method** & **Lithology** & **Sample Shape** & **Sample No.** & **Lateral Stress (MPa)** & \(\Delta F\) \\ \hline \multirow{6}{*}{Sandstone} & & & SA-UI-1 & 0 & / \\  & & Intact sample & SA-UI-2 & 0 & / \\  & & & SA-UI-3 & 0 & / \\  & Sandstone & Sample with & SA-UP-1 & 0 & / \\  & & prefabricated & SA-UP-2 & 0 & / \\  & & invert arch & SA-UP-3 & 0 & / \\  & & & SC-UI-1 & 0 & / \\  & & & SC-UI-2 & 0 & / \\  & & & SC-UI-3 & 0 & / \\  & & & SC-UP-1 & 0 & / \\  & & & prefabricated & SC-UP-2 & 0 & / \\  & & invert arch & SC-UP-3 & 0 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Loading schemes for rock samples. Figure 4: Photos of test devices: (**a**) overview of the test system; (**b**) detail of biaxial loading mold; (**c**) schematic diagram of loading direction. Uniaxial compression test: The displacement control method was used to apply the load until the overall failure of the sample, and the maximum uniaxial compressive load was obtained. The samples were loaded at a displacement rate of 0.006 mm/s. The load-time curves were recorded by the data acquisition system of the testing machine, and the displacement, strain, and deformation information were recorded by the 3D-DIC system. In this study, the \"strain \(\varepsilon\)\" denotes the axial strain of the overall sample and was measured by using an extensometer. Biaxial step-loading test: First, the sample was loaded to the predetermined values at a loading rate of 0.42 kN/s in the X-direction.