## 6 Conclusions  
In this research, to increase the understanding of fracture behavior around openings, a host of sandstone specimens, including three groups of specimens containing two inverted U-shaped openings (H, D and V), one group of intact specimens (C) and one group of specimens containing one single inverted U-shaped opening (S), were loaded in uniaxial compression combining the DIC and AE experimental techniques. Moreover, a numerical investigation was further conducted to simulate the stress distributions around the openings. Based on the experimental and numerical results, several conclusions could be drawn as follows.  
1. Both the number and configuration of openings noticeably affected the strength and deformation properties of rock specimens. The uniaxial compressive strength of these groups could be ordered from large to small as: C \(>\) V \(>\) S \(>\) H \(>\) D, and the degree of attenuation ranged from \(15.80\%\)to 29.94%. For the Young's modulus, Group C had the largest value, followed by Groups H, V, D and S, with a decrease extent of 14.76%-24.93%. Besides, the peak strains of the pre-holed specimens also showed a different degree of reduction from the intact specimen.
2. According to the distribution of strain localization and the variation of AE signals during loading, the crack development around the openings under increasing loads was clarified. Five sorts of cracks appeared around the openings, i.e., elementary tensile crack, posterior tensile crack, slabbing crack, shear crack and spalling crack. For specimens with two openings, three categories of hole coalescence, namely slabbing coalescence for Group H, shear coalescence for Group D and tensile coalescence for Group V, were formed. The failure mode of Groups S and D was shear failure, whilst that of the others was tensile-shear failure. With regard to the threshold stresses in the fracture process, the initiation stress of the intact specimen was 44.8 MPa, while that of the specimens with one single or two openings were 21.2 MPa, 22.9 MPa, 18.8 MPa and 24.6 MPa, respectively. Moreover, the damage stress of the intact specimen was 91% of the peak stress, while that of the specimens with openings was (0.65-0.78) times their peak stresses.
3. Under unidirectional compressive loads, tensile and compressive stresses concentrate on the roof-floor and both sides of the opening, respectively. The hoop SCFs on the boundaries of the two diagonal openings was the largest, while those of the two vertical openings were the smallest. On the whole, the stress distribution at the periphery of the opening could effectively interpret the formation, initiation location and sequence of the cracks. The order of the crack initiation stresses of the specimens with openings could also be verified on the basis of the magnitude of the SCF on the roofs or floors of the openings.  
[PERSON] conducted the laboratory tests and numerical simulation, analyzed the results and wrote the paper. [PERSON] designed the experimental scheme and revised the paper. [PERSON] helped to perform stress analysis, collect literature and process data.  
This research was funded by National Natural Science Foundation of China (51774321), National Key Research and Development Program of China (2018 YC0604606) and Fundamental Research Funds for the Central Universities of Central South University (2018 zzts215).  
Special thanks should be given to [PERSON] for his strong support on the laboratory experiments.  
The authors declare no conflict of interest.