The phase-field method in damage mechanics is attracting increasing popularity in rock engineering, which bridges the gap between microcracking fracture mechanics and quantified damage mechanics. The quasi-static microcrack propagation can be well captured in FEM via the phase-field damage method. However, the determination and calibration of the length scale parameter (ℓ) in the phase-field method are still unclear since the influence width of microcracks cannot be directly observed. In this research, the phase-field damage method is applied to simulate rock behavior under triaxial tests. The length scale parameter is calibrated via ultrasonic wave velocity measurements. The results show that the phase-field damage can be calibrated via ultrasonic wave velocity measurement, which empowers the application of phase-field damage in engineering applications.


The microcracks, joints and faults are always challenging geotechnical issues for underground engineering (X. Li et al., 2022; Si et al., 2020). A novel approach to capture microcracking activities is to use the phase-field damage mechanics (Francfort & Marigo, 1998; You et al., 2021). The phase-field damage mechanics enables to simulate crack propagation and coalescence without introducing complicated remeshing in FEM, plausible problem-specific functions (e.g. in XFEM) or any other thermodynamically unfriendly auxiliary equations or mesh manipulations.

The usage of the phase-field damage model involves two additional parameters, namely the critical fracture energy per unit of area (Gc) and length scale parameter (). Gc describes the energy consumed to form the unit length of fractures, which is related to fracture toughness (Kc). The fracture toughness can be easily captured by ISRM suggested SBC test. However, the length scale parameter represents the influencing width of microcracks, which is hard to be directly measured. Some previous studies casually selected the minimum size of the mesh as the length scale parameter, which may influence the accuracy of the entire phase-field damage model (Duda et al., 2015). In this paper, we proposed an approach to calibrate parameters in the phase-field damage model via ultrasonic wave velocity measurement. The bridge between traditional damage mechanics and phase-field damage mechanics has been established. The proposed model shows great consistency with laboratory results.

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