A new method for estimating crack initiation stress of rock under compression based on chaotic characteristic of self-vibration period

The crack initiation stress threshold (${\sigma }_{ci}/{\sigma }_p$) is an important parameter in the process of brittle failure of rocks, and it plays a significant role in assessing the macroscopic deformation and micro-fracture characteristics of rocks. In this study, the rock system during loading stage is equivalently modeled as a viscoelastic system undergoing light damping vibration, and the bifurcation characteristics of the vibration period are analyzed based on chaos theory. The stress corresponding to first bifurcation of the iterative growth factor of self-vibration period is chosen as the crack initiation stress (${\sigma }_{ci}$), and the analytical solution is ${\sigma }_{ci}=\frac{4}{9}{\sigma }_p+\frac{5}{9}\nu \left({\sigma }_2+{\sigma }_3\right)$. The existing experimental results were employed to validate the feasibility and efficacy of the proposed method. Additionally, the proposed method was compared with results obtained by energy-based method, strain-based methods and the Acoustic Emission (AE) method. Compared to energy-based method, the proposed method is more consistent with existing results. Compared to strain-based methods and the AE method, the proposed method is effective and rapid in determining the crack initiation stress ${\sigma }_{ci}$, and it also reduces the subjectivity associated with parameter determination.