## 1 Introduction  
It is well known that drilling efficiency is closely linked to project profitability. Compared to traditional extrusion-based rock-breaking methods, cutting techniques significantly enhance rock-breaking efficiency [1]. As a result, PDC (polycrystalline diamond compact), which utilizes cutting for rock breakage, has become widely adopted in drilling engineering. Since its introduction in the 1950s, the study of rock-breaking mechanisms has remained a key research focus in drilling engineering, serving as a theoretical foundation for improving drilling efficiency [2, 3, 4]. A typical Polycrystalline Diamond Compact (PDC) bit consists of multiple cutters, and its rock-breaking mechanism is complex [6, 7, 8]. To better understand this mechanism and optimize the rock-breaking parameters, single-tool testing was first employed in the 1970s and has been used ever since [9, 10]. Based on the relative motion between the tool and the rock, single-tooth cutting tests can be classified into three types: linear cutting, rotary cutting, and mixed cutting [8; 11; 12]. Due to its convenience and clarity, the linear cutting test is considered the most suitable for studying tool-rock interactions [13]. As a result, extensive research has been conducted on single-tooth cutting tests for rock breaking [6; 11; 14; 15; 16]. In the study of single-tool cutting, researchers have developed mechanical models for rock breaking, which are influenced by drilling operation parameters, the mechanical properties of the rock, and the geometric parameters of the cutting teeth [6; 17; 18; 19]. Subsequently, by incorporating the shape and distribution of the cutting teeth on the drill bit, a comprehensive drilling and rock-breaking model for the entire drill bit was derived using the superposition principle [6; 20]. In the drilling mechanics model, the geometric parameters of the tool can be directly measured, while the rotational speed, drilling speed, and drilling load can be accurately obtained through various functional sensors [11]. As a result, to derive the fundamental parameters for geotechnical engineering design and safety assessment, researchers have applied the drilling model to predict lithological parameters, based on drilling operations and the monitoring of mechanical response parameters [18; 21; 22; 23; 24; 25]. According to previous literature, the cutting tool does not make direct contact with the rock at the front during the cutting process; instead, an extrusion zone typically exists between them [26]. The primary function of this zone is to transfer the force and energy applied by the tool to the rock, thereby inducing and maintaining the continuous occurrence of rock-breaking activities [27; 28; 29]. Considering this characteristic, the presence of the extrusion zone has been incorporated into the study of the rock-cutting mechanism, and a corresponding theoretical model has been proposed [27]. Additionally, scholars have observed that as the rock failure process progresses in front of the tool, the cutting force fluctuates accordingly, indicating a close relationship between cutting force variations and the rock failure process [16; 17]. Moreover, achieving optimal rock-breaking efficiency has always been a key objective for engineers and researchers. To characterize the efficiency of rock cutting, the concept of mechanical specific energy, based on cutting force, was introduced [30; 31]. This concept represents the energy consumption per unit of rock failure and has led to several valuable conclusions [8; 10; 28; 32]. Furthermore, the numerical simulation method has been applied to the study of rock cutting due to its excellent capability in handling large deformations, yielding many significant results [14; 33; 34; 35]. Based on the analysis above, cutting force is one of the most valuable indicators in the rock-breaking process, as it contains a wealth of critical information related to the cutting behavior.