## 2 Experimental Materials and Methods
### Experimental Procedure  
Mechanical tests were performed using a TAJW-2000 rock triaxial testing system form Changchun, China, with a maximum axial force of 2000 kN and maximum confining pressure of 80 MPa. Axial and radial displacement were measured using displacement sensors. The real-time development of internal cracks was monitored using a DS5 AE system from Beijing Softland Times company, Beijing, China, a sampling frequency of 1.5 MHz, a maximum signal amplitude of 100 dB, and a dynamic range > 40 dB. Figure 2 is a schematic of the experimental setup used.  
Figure 1: (**a**) Schematic diagram of shale sampling; (**b**) shale specimens; (**c**) definition of bedding inclination angle \(\theta\).  
Figure 2: Schematic diagram of the test system.  
The successful implementation of triaxial incremental cyclic loading and unloading tests is dependent on the compressive strength (\(\sigma_{\rm p}\)) of the shale under triaxial monotonic loading. To determine this, a triaxial monotonic loading test was conducted using five \(\theta\) and three \(\sigma_{3}\) (10, 20, 30 MPa), with four pre-peak unloading points at \(0.2\sigma_{\rm p}\). \(0.4\sigma_{\rm p}\). \(0.6\sigma_{\rm p}\), and \(0.8\sigma_{\rm P}\). Triaxial incremental cyclic loading and unloading tests were then performed under the same conditions. When the specimen begins to be loaded formally, AE monitoring is performed simultaneously. The main steps used were as follows:  
(1) Shale samples were sealed and fixed in place within a triaxial pressure chamber under an axial load of 2 kN. Good contact between the sample and the pressure-measuring element was ensured.  
(2) Confining pressure was applied to the designed value at a rate of 0.05 MPa/s using the force control mode. The obtained confining pressure was then maintained for the entire test period.  
(3) The axial pressure was increased to \(0.2\sigma_{\rm p}\) using the displacement control mode (with a loading rate of 0.05 mm/min). The axial pressure was then unloaded to 5 MPa at the same rate to complete the first loading and unloading cycle, followed by sequential unloading points to \(0.4\sigma_{\rm p}\), \(0.6\sigma_{\rm p}\), and \(0.8\sigma_{\rm p}\) for the second, third, and fourth cycles.  
(4) The axial pressure continued until the specimen failed.  
Because of the high constant confining pressure that was applied during the experiment, rock samples were subjected to relatively small circumferential displacement, resulting in lower circumferential strain energy. However, as stated in [23], changes in the circumferential energy of a rock sample can be disregarded. Figure 3 displays the strain and energy indicators for shale during a single loading and unloading cycle, including the total strain (\(\varepsilon^{\rm tot}\)), elastic strain (\(\varepsilon^{\rm e}\)), and plastic strain (\(\varepsilon^{\rm irr}\)). The total energy density absorbed by the shale (external input energy and work performed by the testing machine) is denoted as \(U_{0}\), the elastic energy density accumulated inside the shale is denoted as \(U_{\rm e}\), and the dissipated energy density due to the compaction of pores or microcracks and the connection and expansion of microcracks is denoted as \(U_{\rm d}\). \(U_{0}\) and \(U_{\rm e}\) represent the areas under the loading and unloading curves for each loading and unloading cycle, respectively, and \(U_{\rm d}\) is the difference between \(U_{0}\) and \(U_{\rm e}\).  
Basic Mechanical Behaviors of Anisotropic Shale under Incremental Triaxial Cyclic Loading and Unloading