## 1 Introduction  
Rock failure is very common in energy development engineering. The failure behavior of rocks affects the efficiency of the engineering project and may even be directly related to its success or failure. In some engineering applications, crack-formation is desirable, for instance in deep geothermal energy development where crack-formation in the artificial storage layer will increase the heat exchange area [1]. In such contexts, rock failure is conducive to improving the efficiency of energy extraction. In other engineering contexts, ensuring the stability and integrity of rock is essential to the smooth completion of the project. An example of this is in the rock surrounding underground coal gasification [2] or coal mining roadways [3]. Rock failure occurs through the initiation, development, and penetration of one or more cracks until macroscopic cracks are formed [4, 5]. Cracks play a key role in rock failure, so the problem of rock failure can be analyzed by way of fracture mechanics. Rock fracture mechanics can be used to identify and predict the imminent failure of rock mass structures, thereby providing vital information for improving the stability and safety of these structures [6]. Fracture toughness is an important parameter in fracture mechanics, as it reflects the ability of rock to resist fracturing [7; 8; 9]. Fracture toughness has been receiving a large amount of attention in theoretical research and engineering applications such as rock cutting, drilling, hydraulic fracturing, blasting, tunnel excavation, underground reservoir construction, and in-situ modified mining reservoir infiltration. This property has therefore become an important research object. However, testing fracture toughness in an underground engineering context is practically difficult. Rocks deep in the formation can only be sampled by drilling, which requires high-tech drilling techniques and incurs high costs. Direct fracture toughness testing requires a large number of samples. The cores obtained by drilling are not only extremely small; they also have poor integrity. The sample is easily damaged during processing, and it is difficult to cut artificial notches of uniform size with precision [10]. Therefore, a simple, reasonable, economical, and efficient method for determining the fracture toughness of rock would be extremely helpful. One proposed solution was that it should be possible to study the fracture behavior of rock through the relationships between fracture parameters and index test results [11]. This allows the fracture toughness value to be estimated on the basis of conventional mechanical parameters, thereby reducing core consumption. The relationship between fracture toughness and density has been studied [12]. There is a remarkably good match between normalized P-wave compressional velocity and the decreasing trend in normalized fracture toughness (\(K_{IC}\)) [13]. In addition, a method has been developed for quickly estimating the mode-I fracture toughness using point load testing [10]. Mode-I fracture is characterized by tensile stress perpendicular to the crack surface and relatively separated upper and lower surfaces. The study found that the fracture toughness of rock can be related to uniaxial compressive strength, uniaxial tensile strength, and point load strength index as well as rock density. The mode-I fracture toughness of several types of rock and soil has been found to be directly proportional to their tensile strength [14; 15]. [PERSON] and [PERSON] [16] similarly investigated the relationship between tensile strength and mode-I fracture toughness for a wide variety of rocks and soils such as Melbourne mudstone, Johnstone, oil shale, granite, marble, micrite, basalt, and syenite. Their results showed a reasonably strong correlation between these two parameters. [PERSON] et al. [17] showed a strong empirical relationship between mode-I fracture toughness and the tensile strength of soil using a ring test. Both the results of [PERSON] and [PERSON] [16] and [PERSON] et al.