## 6 Conclusions  
In this study, in order to evaluate the static and dynamic mechanical properties of limestone, longitudinal wave velocity tests were conducted. The influence of the initial damage on the compression strength, failure strain, modulus of elasticity, and Poisson's ratio is illustrated. Based on the analysis and interpretation of experimental results, the conclusions are summarized:  
1. With the increasing degree of longitudinal wave speed, the initial damage decreases nonlinearly and remains constant when entering 6000 m/s, indicating the correlation between the initial damage degree of limestone and the location depth. 2. The compression strength and modulus of elasticity decrease nonlinearly with increasing initial damage and increase nonlinearly with an increasing strain rate. Conversely, the failure strain increases nonlinearly with increasing initial damage and strain rate. The Poisson's ratio changes slightly with the increase in initial damage and strain rates. Moreover, an empirical formula for describing the nonlinear relationship between initial damage and strength is presented and verified by an engineering example. The threshold of the damage variable of the limestone under the initial damage state is proposed, indicating that the initial damage value corresponds to the stress peak during the compression deformation of the nondestructive limestone. The initial strain threshold at a high strain rate of \(\epsilon=100\) s\({}^{-1}\) is approximately D\({}_{thr}^{1}=0.2\), which is 0.02 higher than that at a low strain rate of \(\epsilon=2\times 10^{-4}\) s\({}^{-1}\). When the threshold of initial damage of limestone is less than 0.18, the compressive strength of the limestone is found to meet the engineering requirements. Figure 21: Comparison of longitudinal wave velocity before and after blasting. (**a**) Radius of influence; (**b**) Vibration acceleration. Figure 20: Comparison of longitudinal wave velocity before and after blasting. As the initial damage increases, the strain softening modulus of the limestone decreases; thus, the fragility of the limestone decreases after the peak stress. With different initial degrees of damage of the rock, the strain softening modulus is much larger than the corresponding initial damage of the elastic modulus. The descending section of the stress-strain curve is steeper than the rising section, so the conclusions obtained by the strain softening analysis reflect the brittle characteristics of the rock. * The initial damage modulus defined based on the longitudinal wave velocity can be applied to understand the compression strength properties of rock, and it also has a certain reference value for the study of damage constitutive theory. It should be noted that while the definition of initial damage proposed in this paper has provided useful insights for the construction of the rock damage constitutive model, the determination of damage degree requires accurate analysis and evaluation on the number, spacing, angle and other factors of cracks. The above conclusions are only valid for the qualitative analysis of initial damage, or simple semi-quantitative analysis, and without sufficient field verification. Further numerical model analysis is suggested to determine a more perfect initial damage model to reasonably determine the long-term damage characteristics of rock for engineering applications. Conceptualization, Y.W. and X.W. ; methodology, J.Z. and B.Y. ; software, J.C.; validation, Y.W., and X.W. ; formal analysis, Y.W. ; investigation, X.W. ; resources, J.Z. ; data curation, W.Z. ; writing--original draft preparation, Y.W. ; writing--review and editing, Y.W. ; visualization, X.W. ; supervision and project administration, J.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Science Foundation of China under grant number CNNSF51608533.