## 1 Introduction  
The mechanical properties of rock masses exhibit significant variations depending on burial depth, dynamic disturbances, and loading paths [1]. Dynamic loading intensities can be categorized into six regimes based on strain rate levels [2, 3], as shown in Figure 1. Under both creep (\(<\)\(10^{-7}\) s\({}^{-1}\)) and static loading (\(10^{-7}\)-\(10^{-4}\) s\({}^{-1}\)) conditions, the compressive strength of rocks typically exhibits strain rate independence, with engineering fields encompassing fault slip, slope stability, and long-term stability of surrounding rock in underground caverns. In quasi-static regimes (\(10^{-4}\)-\(10^{-1}\) s\({}^{-1}\)), there is a gradual increase in compressive strength with strain rate increases (distinct from inertia-dominated mechanisms), correlating with engineering scenarios such as seismic loads (\(10^{-3}\)-\(10^{-1}\) s\({}^{-1}\)) [4], rockburst(\(1.5\times 10^{-3}\)-\(1.7\times 10^{-1}\) s\({}^{-1}\)) [5], and mining tremors (\(1.6\times 10^{-3}\)-\(1.6\times 10^{-1}\) s\({}^{-1}\)) [5]. When the strain rate exceeds \(10^{-1}\) s\({}^{-1}\), transitional rate dependency governed by inertial effects becomes prominent in rock compressive strength, observed in blasting engineering, impact loading, and explosions. This analysis reveals that the compressive strength of rocks under quasi-static loading exhibits distinct deviations from both conventional static loading conditions and dynamic loading regimes (e.g., explosive/shock wave scenarios), highlighting their strain rate dependency [6]. Nevertheless, current engineering practices predominantly employ static mechanical parameters or empirically adjusted values (typically increased by 30%) based on early foreign experimental data to determine dynamic parameters [7]. This oversimplified approach, neglecting project-specific dynamic loading environments, substantially elevates the risk of rock mass instability. The strain rate effects on rock mechanical properties under quasi-static loading have been extensively validated through experimental studies. For instance, [PERSON] et al. [8] performed triaxial compression tests on granite at strain rates of \(10^{-4}\)-\(10^{0}\) s\({}^{-1}\) and confining pressures of 20-170 MPa, revealing that peak strength increased synchronously with both the strain rate and confining pressure, whereas the elastic modulus and Poisson's ratio remained rate-insensitive. Similarly, [PERSON] et al. [6,9] identified \(5\times 10^{-4}\) s\({}^{-1}\) as the threshold strain rate: strength remained constant below this value, while both strength and the elastic modulus increased above it. In contrast, [PERSON] et al. [10] conducted dynamic compression tests at confining pressures up to 225 MPa and reported that the strain rate sensitivity of strength is influenced by the confinement at low pressures. [PERSON] [11] demonstrated strain rate dependence in basalt/granite peak strength: under quasi-static conditions, brittle fracture persisted, while strain rate effects mirrored temperature-induced strengthening, supporting the thermal activation theory. [PERSON] et al. [12] and [PERSON] et al. [13] found that the total absorbed energy, elastic strain energy, and the dissipated energy of rock specimens at the peak strength point increased as the strain rate increased under uniaxial compression or true triaxial compression. [PERSON] et al. [14] reported that the maximum energy and cumulative energy of acoustic emission increase continuously with the strain rate. In addition, the strain rate effect on compressive strength of rock was investigated across lithologies: shale [15] showed lower strength growth rates compared to marble [12], while sandstone [13,16] and coal [17] exhibited different rate dependencies. Characteristic stress, mainly including crack initiation stress (\(\sigma_{\text{cl}}\)), crack damage stress (\(\sigma_{\text{cd}}\)), and peak stress (\(\sigma_{\text{p}}\)), is a mechanical parameter used to describe the failure process of rock. [PERSON] et al. [18] and [PERSON] et al.