## 2 The TPB-IC Specim for Mixed-Mode I/III Fracture Study
### Experimental Investigation of TPB-IC Specimens  
Sandstone was selected as the material for the 3D mixed-mode fracture experiments using the TPB-IC specimen because of its low price, convenient processing, wide distribution and application in various engineering projects and experimental research. All the specimens in this experiment were taken from one large homogeneous piece of sandstone. The stone was cut into 20 mm thin sections, and then the thin sections were cut into long rectangular specimens of sizes (120 mm long and 40 mm wide). The cracks with different horizontal and vertical rotation angles were cut out using a circular slicer with a thickness of only 0.8 mm. The crack tip was sawn again with a 0.3 mm thick saw line, which made the crack height ratio a/W reach 0.5. The specimens were tested four times for each angle, and the experiments were carried out on the same universal compression testing machine (Figure 9). This consistent manufacturing and experimental process eliminated part of the error sources. The specimens were loaded at a constant rate of 0.05 mm per minute until a fracture occurred. The fractured TPB-IC specimens (\(\alpha\) = 30\({}^{\circ}\)\(\beta\) = 30\({}^{\circ}\)) are shown in Figure 10. When the loading began, the crack tip was under tensile effect and the upper and lower crack surfaces opened, which resulted in a mode I deformation. Due to the existence of a horizontal deflection angle, the deformation direction of the front-to-back crack was opposite, and the crack surface rotated and resulted in a mode III deformation. The relative displacement of the upper and lower crack surfaces occurred because of the vertical deflection angle, so the crack surfaces sheared, and the mode II deformation occurred. The crack propagation of the specimens started from the crack tip and then propagated  
Figure 8: Variation of biaxiality ratio B with respect to horizontal deflection angle \(\alpha\) and vertical deflection angle \(\beta\). (**a**) The relationship between B and \(\alpha\). (**b**) The relationship between B and \(\beta\). Figure 9: (**a**) The full view of experimental machine. (**b**) The loading situation. to the loading site (Figure 10a,b). Although the end of the crack propagation path was far away from the loading point and stopped at a small displacement from the loading point (see Figure 10a), it was believed that when the specimen height was large enough, the final end of the crack propagation path would reach the loading point, as in previous experimental results for large thickness on I/III mixed-mode fractures [48]. The kinks and deflections of the crack propagation path appeared during the I/II/III mixed-mode fracture seen Figure 10c. Although kinks were common in composite fractures [49], the same phenomenon occurred in a fracture experiment using sandstone when the displacement gradient of the crack propagation of sandstone was large, which led a displacement discontinuity of the crack surface [50]. Figure 10c shows the obvious deflection of the crack during crack propagation, which indicates that K\({}_{\rm II}\) and K\({}_{\rm III}\) have a significant influence on the mixed-mode fracture behavior. Nine groups of load-displacement curves are shown in Figure 11. The load-displacement curves are concave curves with low loads and are short. This is because there were some microparticles on the surface of sandstone and microcracks in the interior, and the specimen was not compacted at the beginning of loading. However, it can be seen from the subsequent curve that there was a linear relationship between the displacement and load before the peak load. So, sandstone can be assumed to be a linear elastic material. The same phenomenon was found in the investigation on the correlation of mode II fracture toughness of sandstone with tensile strength [51]. In the case of three-point bending, the inclined crack needed more fracture load and fracture energy than the straight crack, which made the fracture load of the inclined crack specimen increase [32].