## 5 Discussion  
As shown in Figures 9 and 10, tensile stress concentrated on the top and bottom of the opening under unidirectional compression, whilst the sides of the opening were concentrated by compressive stress. The stress concentration factor was related to the number and configuration of openings. For the intact sample, it is ideal that tensile cracks parallel to the compression direction occur in the sample and lead to splitting failure. However, the end friction changed the stress state of the ends from one dimension to three dimensions, which resulted in the formation of shear plane (crack) on the diagonal. This is why the tensile-shear failure pattern of the intact sample appeared. With respect to the sample containing an inverted U-shaped opening, numerical results show that the maximum stress concentration factors (SCF) on the roof and floor of the opening were \(-0.87\) and \(-0.93\), respectively. Obviously, the elementary tensile crack \(1^{\mathrm{a}}\) grew first from the bottom. However, it was observed that no tensile crack occurred on the top, this might be because the radius of the semicircle was very small [26]. The crack \(1^{\mathrm{a}}\) developed slowly with the increasing stress. Nevertheless, as the distance from the opening rose, the concentrated tensile stress became smaller and smaller. Once it decreased to zero, the critical stress condition shifted to both sides of the elementary tensile crack and resulted in the occurrence of the posterior tensile cracks. During this period, the slabbing cracks emerged due to the increasing concentrated compressive stresses with a SCF of 2.52 \(p\) on the sides of the opening. It was also found that, as the posterior tensile crack expanded, the elementary tensile crack became shorter and shorter until it disappeared. This was caused by the lateral compression of  
Figure 10: Hoop stress concentration factors on the boundaries of openings under vertical compressive stress (**a**) H-model; (**b**) D-model and (**c**) V-model. the posterior tensile crack. When the constraint was lifted at the end of the test, the elementary tensile crack appeared again. Moreover, the end friction force increased with the rising load, leading to the emergence of the shear bands on the diagonal. After the maximum stress, the shear bands developed into macro cracks. Afterwards, the failure would take place if the shear cracks and the V-shaped notch or the posterior tensile crack were connected. In regard to the sample containing two horizontal inverted U-shaped openings, the SCFs at the centers of the roofs and floors were \(-0.81\) and \(-0.91\), respectively. By contrast, the maximum SCFS on the left sidewall of the left opening and the right sidewall of the right opening were both \(2.53\), and those on the right sidewall of the left opening and the left sidewall of the right opening were equal to \(2.49\). Therefore, the cracks \(1^{\mathrm{a}}\) and \(1^{\mathrm{b}}\) were formed firstly at the floors of the two openings, and then the slabbing cracks \(3^{\mathrm{a}}\)-\(3^{\mathrm{d}}\) occurred on their sides. With the growing of the exerted stress, the two adjacent V-shaped notches were gradually connected. Similar to the sample with an opening, after the elementary tensile cracks stopped growing, the posterior tensile cracks \(2^{\mathrm{a}}\)-\(2^{\mathrm{f}}\) would appear one after another due to the transferred critical stress condition. The shear bands on the diagonal would also grow into shear cracks after the peak point. When the shear crack intersected with the V-shaped notch of the left opening and the posterior tensile crack on its upper-right corner propagated to the upper surface of the sample, the failure happened. Concerning the appearance of the spalling cracks, they were caused by the local high compressive stress. By contrast, the mechanism of crack development in the samples containing two diagonal or vertical openings was similar, but the sequences of the initiated cracks were different.