## 3 Dynamic-Static Conversion Relationship
### Relation Establishment
Blanco Alcora (BA): white crystalline limestone; Piedra de Colmenar (PdC): grey and white lacustrine fossiliferous limestone; Traverting Amarillo (TAM): porous layered limestone; Traverting Rojo (TR): porous layered limestone; Gris Macael (GM): grey calcite marble; Blanco Tranco (BT): white homogeneous calcite marble; Amarillo Triana (AT): yellow dolomite marble; Crema Valencia (CV): cream mritic limestone; Rojo Cehegi 'n (RC): micritic limestone; Marro 'n Emperador (ME): brown brecciated dolostone. \[E_{\text{s}}=\frac{E_{\text{d}}}{3.8\lambda^{-0.68}}\]  
[PERSON] et al. [33] proposed a new relationship that exists between the static and dynamic moduli of elasticity of eight distinct types of igneous, sedimentary, and metamorphic rocks. They used a linear model (Equation (28)) and a nonlinear model (Equation (29)) to associate the static modulus with the dynamic modulus. They fully considered the mineral composition, content, and porosity of the rock sample, but did not take into account factors such as cracks and weathering. Therefore, this model is not suitable for rocks with more complex microstructures (cracks, weathering, etc.). \[E_{\text{s}}=0.932E_{\text{d}}-3.421\quad\text{ with R}^{2}=0.97\]  
\[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]0  
Based on the summary and analysis of the above research literature, numerous investigators have conducted experimental comparative studies on the dynamic and static parameters of rocks (stones), resulting in several empirical formulas. This has provided a solid theoretical foundation and practical insights for the exploration of the correlation between dynamic and static mechanical parameters. These studies mainly focus on igneous rocks, metamorphic (such as marble, granite, etc. ), and sedimentary (such as sandstone, siltstone, sandy mudstone, and mudstone, etc. ), and there are also some innovations in this area, such as the establishing of the best prediction model for different types of rocks. \[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]1 Relationship between \(E_{\text{d}}\) and \(E_{\text{s}}\) in the studied specimens. The straight line corresponds to the ideal ratio \(E_{\text{d}}/E_{\text{s}}=1\)[32]. However, overall, there is a lack of research on the interrelationships between the parameters of empirical formulas. Few experiments consider the factors affecting temperature and pressure. There is also a lack of research on the micro-mechanisms underlying the discrepancy in dynamic and static elastic parameters; the obtained empirical formulas have limitations, some of which may apply to particular rock types; and most experiments select fewer samples, which cannot accurately demonstrate the validity of the experimental data and lack persuasiveness. This paper will provide an overview of the aforementioned issues based on previous research. Table 3 is obtained by organizing the empirical equations of linear correlation in \[E_{\rm s}=0.69E_{\rm d}+6.40\quad\quad\text{with R}^{2}=0.75\]2 There may be some connection between the constants of these empirical equations. A long time ago, [PERSON] [8] studied the correlation between the constants c and d in type II, and he analyzed 50 related correlation equations, which represent the correlation between static and dynamic deformation indices under different conditions and stresses. He believed there is a certain regularity between c and d and that c and d are also affected by the type of rock and the conditions of the static parameters test. His results also showed this by using the correlation of type II. When estimating static values using dynamic values, the accuracy was better than 20% for 70% of the available results. Later on, [PERSON] [9] conducted experimental studies on ten different types of rocks (the stresses considered were 10, 20, 30, and 40 MPa) and established a correlation for type III based on the obtained data. Then, the constants \(\alpha\) and \(\beta\) in the relationship were studied.