Quantizing soliton-like phonon pulse and optical branch of lattice vibration at crack tip excited by crack propagation as shock waves

Abstract In the previous paper (Kobayashi, 2019) based on the proposed micro-crack evolution equation, a solitary pulse wave radiated by crack propagation as the shock wave was analyzed and the solitary character of the solitary pulse wave was examined due to its energy and acceleration. Subsequently, in this viewpoint, the solitary pulse wave was concluded to be a soliton. In this paper, the lattice vibration at the crack tip excited by the crack propagation as a shock wave is examined by considering the shock wave velocity at the crack tip, and then the lattice vibration excited at the crack tip is concluded to be optical branch (mode). Therefore, optical and/or electrical phenomena may be exited at the crack tip due to the crack propagation as shock waves. Based on the optical branch, the mechanism, that causes discontinuity, singularity, the non-locality and also the entanglement in the fracture progress process, is discussed. The correlation of parameters introduced in the previous paper with number and energy of phonons is examined in the viewpoint of quantizing a soliton-like phonon pulse. As a result, it is clarified that the amplitude of the soliton-like phonon pulse cannot be determined arbitrarily and, hence, is a discrete value. Here, the proposed soliton-like phonon pulse may be recognized as a topological soliton of the strain induced by a kink of the displacement occurred by explicit or spontaneous symmetry breaking in the (1+1) dimensional space due to the crack propagation. Therefore, it is suggested that fracture induces phase transformation. The long-period pulse waves that were observed in the seismograms of the 2016 Kumamoto Earthquake and the 2018 Hokkaido Eastern Iburi Earthquake at the neighborhood of the seismic centers are examined in detail in the viewpoint of the proposed quantization of a soliton-like phonon pulse. Then the long-period pulse waves observed in earthquakes are simulated using the value of the proposed parameter λIc determined by the earthquake energy calculated by its magnitude, and its fault area is estimated using the relation between the fracture area and the fracture energy derived in the previous paper (Kobayashi, 2017b). In this paper, the author suggests some examples in which micro-scale properties such as quantum character visibly reflect in macro-scale phenomena.