# Feasibility Study and Prospects of Rock Fragmentation Using Ultrasonic Vibration Excitation
## 3 The Theoretical Feasibility of Ultrasonic Rock Breakage
### Ultrasonic Vibration Excitation Mechanisms for Rock Breakage  
(1) Ultrasonic mechanical rock breakage  
When the ultrasonic impact load reaches the fatigue failure threshold of the rock under static stress, each impact will further reduce its strength. Due to the high frequency of the ultrasonic wave, the fatigue failure threshold of the rock decreases rapidly in a short period of time, thereby optimising the rock fragmentation. In the process of ultrasonic excitation of rock, the P wave is in the same direction of micro particle vibration; therefore, the particles are under compressive stress and tensile stress that alternately change at a high frequency, accordingly leading to tension and compression deformation. The propagation of the S wave induces shear stress on the rock micro particles, and therefore accounts for the shear deformation. The adjacent rock micro-particles are continually subjected to tensile stress, compressive stress, and shearing stress, and the contacts between particles are constantly under the effects of friction, deformation, and damage, leading to the initiation of microcracks. Under the continuous excitation of the ultrasonic wave, the microcracks continue to propagate, expand, and pierce the rock to form large cracks. With the increase of ultrasonic excitation time, the large cracks inside the rock mass continue to develop, penetrate, and form the fracture network for the rock breakage. The effective ultrasonic sound pressure is described in Equation (1) [21]:  
\[p_{e}=\frac{\sqrt{2}UI}{\omega AS}, \tag{1}\]  
where \(U\) is the working voltage of the ultrasonic generator, \(I\) is the working current of the ultrasonic generator, \(\omega\) is the ultrasonic vibration frequency, \(A\) is the vibration amplitude, and \(S\) is the area of the ultrasonic contact action. Considering the rock strength \(\sigma_{0}\), when \(p_{e}\geq\sigma_{0}\), the micro-cracks in the rock mass initiate, propagate, and penetrate, leading to the rock damage, that is as described in Equation (2):  
\[\frac{\sqrt{2}UI}{\omega AS}\geq\sigma_{0}. \tag{2}\]  
The equation above can be used to derive the ultrasonic frequency range for the formation of microcracks as in Equation (3):  
\[\omega\leq\frac{\sqrt{2}UI}{\sigma_{0}AS}. \tag{3}\]  
Outside this frequency range, the ultrasonic excitation will not cause damage to the rock and does not contribute to the crack propagation inside the rock. All the ultrasonic energy is dissipated in the form of elastic waves and the rock is hardly damaged by fatigue loading. Within this frequency range, part of the ultrasonic energy is dissipated in the form of elastic waves, and the remainder is used for the propagation of cracks and formation of the new surface area. Each action of ultrasonic excitation can cause cracks within the rock mass. Under these conditions, cyclic loading with constant amplitude can lead to the accumulation of rock damage until reaching fatigue failure [9]. Within the effective ultrasonic frequency range, the higher the frequency of ultrasonic excitation is, the faster the fatigue damage occurs to the rock; and the longer the ultrasonic excitation time is, the higher the damage degree of the rock. Therefore, from the perspective of rock breakage using impact loading, the ultrasonic method is applicable, and its high frequency makes the method very competitive. (2) Ultrasonic resonance  
When the impact frequency generated by ultrasonic excitation reaches the natural frequency of the rock, the rock will resonate. When the single impact load is the same under the resonance effect,the effective energy of the impact load acting on the rock will be greater per unit time, which contributes to the rapid development of fracture propagation within the rock mass.