# Study of the Critical Speed of Ultra-High-Speed Diamond Drilling Technology in Marble
## 2 Rock Breaking Simulation Model  
### Basic Assumptions  
The impregnated diamond bit is the most commonly used drill bit type in deep hard rock geological drilling, meeting the drilling needs of highly abrasive formations with its unique self-sharpening process and rock-breaking method [7; 8; 9; 10; 11; 12; 13]. Its rock-breaking mechanism is similar to that of a grinding wheel working on a workpiece. It involves the grinding and fracturing of the rock at the bottom of the hole by a large number of small diamond particles on the lip surface of the drill bit, as illustrated in the rotary rock-breaking model of the impregnated diamond bit shown in Figure 2.  
The drilling response of the impregnated diamond bit is primarily described by the relationships among drilling pressure (W), torque (T), drilling speed (V), and speed (\(\Omega\)), or it can be described by the cutting depth per revolution of the drill bit. In this context, \(\mathrm{d_{s}}\) and \(\mathrm{d_{d}}\) represent the cutting depths of the impregnated block and individual diamond, respectively. It is assumed that the exposed diamonds on the impregnated block surface are sufficient to cover the entire surface, and the cutting depth of each diamond is considered fixed, i.e., \(\mathrm{d_{d}}=\mathrm{d_{s}}/\mathrm{n_{d}}\), where \(\mathrm{n_{d}}\) represents the quantity of diamonds on the same cutting path. Additional variables include the edge angle \(\beta\), which represents the ratio of the average exposed height (h) of the impregnated block surface to the overlapping length (\(\mathrm{l_{0}}\)) between adjacent diamonds. \(\mathrm{l_{s}}\) and \(\mathrm{l_{w}}\), respectively, denote the lengths of the impregnated block and the waterway [14]. As diamonds gradually wear and the matrix containing diamonds continues to wear, new diamonds continue to expose and contribute to the ongoing rock fracturing. The impregnated diamond bit possesses a self-sharpening characteristic, and under normal drilling conditions, its drilling speed should be constant, meaning the cutting depth per revolution for each diamond particle on the drill bit is fixed. Based on this nature and requirement, the first assumption is proposed as follows: the impregnated diamond bit should operate with a fixed cutting depth per revolution to achieve optimal drilling results and maintain a consistent rate of penetration (ROP), unaffected by the wear of diamonds and the matrix. Figure 2: Schematic diagram of ID bit rotary drilling technology. [PERSON] and [PERSON] conducted extensive research on the rotational drilling interface law, finding that, during the drilling process, the bit only undergoes rotation and vertical displacement [14; 15; 16; 17]. The relationships between W and T and ROP and \(\Omega\) are conjugate, and it is assumed that the interface relationship between the bit/rock is independent of the velocity. However, through numerous simulations and experimental studies, it was discovered that an increase in velocity has a significant impact on the rock-breaking efficiency of the bit. This paper discusses the interaction between diamond particles on impregnated diamond bits and rocks, considering velocity as the sole variable. The drilling process of impregnated diamond bits is a highly complex issue, involving the bonding effect of the matrix on diamonds, the particle size and strength of diamond particles, the tribo-system wear relationship between cuttings and the diamond matrix, and the erosive effect of drilling fluid on the matrix. The focus of the study on the rock-breaking mechanism of ultra-high-speed single-rotation drilling is on the variation of diamond particle stress during the drilling process and the fracture mechanism during rock fragmentation. The model only considers the interaction between diamonds and rocks, neglecting frictional wear between the matrix and rocks and the impact of cuttings on the drilling process.