## 1 Introduction  
The demand for resources is escalating annually. The depletion of shallow reserves has compelled a gradual transition in the exploitation of mineral resources towards deeper layers [1]. The intricate stress environment of deep formations exacerbates the challenges associated with fracturing the inner hard rock. For instance, the primary rock-breaking methods employed in underground coal mining encompass drilling and blasting, as well as mechanical approaches. However, as the mining depth surpasses 1000 m, the conventional drilling and blasting technique becomes increasingly ineffective. Conversely, the mechanical rock-breaking method generally encounters issues such as elevated wear rate and low efficiency, hindering the promotion of industrial development. Consequently, there is an urgent need to unearth a novel, efficient rock-breaking method. The utilization of ultrasonic high-frequency vibrations for rock fragmentation represents a novel approach in this field. Compared to conventional rock fragmentation methods, the advantages of ultrasonic vibration technology are multifaceted: Firstly, rocks exposed to high-frequency impact loads in excess of 20,000 times per second rapidly develop fatigue damage, thereby reducing the challenges associated with fragmentation. Secondly, when the natural frequency of the rock aligns with the vibrational excitation frequency, resonance effects occur, increasing the likelihood of rock fragmentation. Lastly, the implementation of ultrasonic vibrational rock fragmentation is relatively straightforward mechanistically. Overall, ultrasonic vibrational rock fragmentation effectively compensates for the shortcomings of traditional methods and holds significant potential for application in deep underground rock fragmentation engineering. Ultrasonic vibrational fracture technology has been successfully implemented in space drilling sampling [2; 3; 4] and oil exploration [5] due to its high efficiency, offering a novel approach for the efficient fracturing of hard rock formations. [PERSON] was the first to integrate ultrasonic high-frequency axial vibrations with conventional rock breaking, significantly enhancing drilling rates [6]. Utilizing numerical simulations, [PERSON] discovered that the wear rate of the drill bit and rock decreases under high-frequency ultrasonic vibrations [7]. [PERSON] demonstrated that the drill bit drilling speed under high-frequency ultrasonic vibrations surpasses that of conventional drilling methods by constructing mechanical models and conducting test analyses [8; 9]. The aforementioned studies validate the feasibility of the ultrasonic vibration-enabled rock-breaking method. In recent years, an abundance of research has been conducted on the damage and fracture behavior of rocks under cyclic loading. However, the majority of these studies have been limited to frequencies below 100 Hz, with only a scarce number of investigations focusing on high-frequency ultrasonic vibrational rock fracture. To fill this gap, some researchers have developed specialized testing platforms. Laboratory experiments, theoretical analyses, and numerical computations have been employed to explore the failure properties and fracture laws of hard rocks subjected to high-frequency ultrasonic vibrations. Using electron microscopy scanning techniques, [PERSON] elucidated that the mechanism governing the propagation of granite microcracks under ultrasonic vibration excitation is tensile failure [10]. [PERSON] employed an infrared thermal imager to monitor the changes in the infrared radiation of the rock under ultrasonic vibration excitation, thereby categorizing the rock into fracture zones [11]. [PERSON] examined the properties of granite damage at varying amplitudes and discovered that stress waves generated by ultrasound vibrations attenuate with the increasing depth of the rock, and that augmenting the amplitude value results in amplified stress in the region [12].