## 1 Introduction  
Natural rock contains microcracks, micropores, and other defects. The initial stress release and redistribution during the construction of underground structures lead to the further development of original microcracks and micropores in rock. As a result, the strength and stability of surrounding rock are reduced [1, 2]. Therefore, a good understanding of the mechanical characteristics of rock mass with different initial damage levels is essential for stability analysis of the space underground [3]. The surrounding rock masses of the underground engineering structures, such as tunnel, champers, deep foundations, and structures [4, 5, 6, 7], will have a certain degree of rupture and damage due to the release of the in-situ stress [8, 9]. [PERSON] first defined the damage variable by using the ratio of the damaged area of the material section to the initial non-damaged area, opening the way for the development of damage mechanics [10, 11, 12]. Over the past decades, comprehensive studies on the measurement means of rock damage have been conducted using various laboratory tests. Many scholars have since chosen parameters that are related to the rock mechanics in order to characterize the initial damage, such as the number, spacing, direction of microcracks [13, 14, 15, 16], crack volumetric strain [17], X-ray, computed tomographic scan number [18; 19], ratio of initial compression and yield strength [20; 21], the acoustic emission signals [22; 23; 24; 25], and the rock's acoustic wave speed [26; 27; 28; 29; 30; 31]. Among them, the direct methods are preferred for measuring the P-wave velocities of rocks, and the test process produces no secondary damage to the rock test sample. Hence, the P-wave velocity test provides an accurate estimation of the total material damage. Acoustic wave speed has been widely applied in rock damage evaluation research in recent years due to its ability to accurately represent rock deformation and the damage information. Acoustic wave technology is the procedure of longitudinal wave reception after propagation in a crack-containing rock. The longitudinal wave velocity was calculated using the transmission travel time of an acoustic pulse along the axial direction of the samples [29]. The internal damage behaviors in the compression fracture tests have been introduced and are of great concern [25; 32; 33; 34]. [PERSON] et al. [35], [PERSON] et al. [36], [PERSON] et al. [37] and [PERSON] et al. [38] observed that the crack development and rock strength were influenced by damage development in rock creep tests. [PERSON] et al. [39] conducted a rock failure pattern under irreversible strain and damage accumulation in a uniaxial experiment of porous sandstone. [PERSON] et al. [40] measured the elastic wave velocities of granite and marble under triaxial compression tests and determined the location of damaged cracks. [PERSON] et al. [41] developed a damage threshold of brittle rock which describes the crack damage of brittle rock. [PERSON] et al. [42] compared pre-damaged and intact granite results under a high strain rate loading test. [PERSON] [43] constructed model parameters for the strength damage of granite after freezing and thawing. Overall, compared to the accumulated damage of compression tests, the studies on the initial damage of brittle rock are insufficient. In addition, the relevant experimental data regarding initial damage are quite scarce, especially those in terms of high strain rate. These above investigations have provided insights into the compression static and dynamic characteristics of limestone with different initial levels of damage. Hence, in this paper a series of compression tests were conducted on block type limestone by employing a large-scale rock multi-function shear device subjected to 2 \(\times\) 10\({}^{-4}\) s\({}^{-1}\) strain rates and a Hopkinson rod device subjected to a 100 s\({}^{-1}\) strain rate.