# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 1 Introduction  
Studying the characteristics of Type I-II composite fractures in hard rocks enables theoretical predictions of the fracturing behavior of ore rocks and serves as a foundational research basis for establishing mechanisms of rock breaking under hydraulic fracturing conditions [1; 2; 3; 4; 5]. In the field of fracture mechanics, a variety of traditional composite fracture criteria have been proposed to predict the linear elastic fracture behavior of cracked materials [6]. These criteria include the maximum shear stress criterion proposed by [PERSON] et al. [7], the strain energy density factor theory proposed by [PERSON] [8], and the maximum energy release rate theory developed based on the Griffith theory [9]. They analyzed the fracture characteristics of the material by considering the singular term of the stress field at the crack tip. However, studies have shown that there are significant differences between the prediction results of the traditional composite fracture criterion and the experimental observation results, mainly due to the influence of the non-singular term (T stress) in the stress field of the crack tip [10; 11; 12; 13; 14; 15]. Therefore, to more accurately analyze the fracture characteristics of materials, scholars began to retain the non-singular term (T stress) in the stress field. This concept was first proposed by [PERSON] and [PERSON] [16] and led to the development of the generalized maximum shear stress criterion [17], the generalized minimum strain energy density factor criterion [18], and the generalized average strain energy density factor criterion [19]. The impact of T stress on material fracture characteristics is closely related to the critical fracture distance, and the greater the critical fracture distance, the greater the influence of T stress on material fracture characteristics [20]. Hard rocks, as brittle materials, are generally considered to have a very small critical fracture distance [21]. Currently, there has been a considerable amount of research on the influence of T stress on the fracture characteristics of brittle materials. For example, [PERSON] et al. [22] derived analytical formulas for T stress in edge-cracked Brazilian disc (ECBD) specimens under distributed loads and found that its influence should be considered for larger crack angles. [PERSON] et al. [23] modified the traditional G criterion, considering the influence of T stress on crack propagation. [PERSON] et al. considered the influence of T stress and established the maximum tangential stress (MTS) criterion that simultaneously considers rock properties, crack parameters, strength, and deformation parameters, predicting Type I-II composite stress intensity factors [24]. Under the combined influence of in situ stress and hydraulic pressure, hydraulic fractures often exhibit Type I-II composite fractures [25; 26; 27]. Existing theoretical research calculates Type I-II composite stress intensity factors at the tip of hydraulic fractures by modeling without considering the wellbore, simplifying it as a model with inclined cracks in an infinite plate [28; 29; 30]. Moreover, hydraulic fracturing rocks are often treated as brittle materials, and to simplify calculations, linear elastic fracture criteria are used to describe the rock-breaking process of hydraulic fractures [31; 32; 33; 34]. The effect of T stress on the fracture characteristics of brittle materials was investigated. Based on the generalized maximum tangential stress (GMTS) criterion, the effect of T stress on the fracture characteristics of hard rock is verified in the splitting test. In addition, we also analyze the influence of T stress on the fracture characteristics of hydraulic fracturing and derive the calculation formula for the critical fracture pressure of hydraulic fracturing.