## 3 Mechanical Properties of Norite
### Deformation Characteristics  
[PERSON]'s modulus and Poisson's ratio are important for addressing geomechanical issues. However, the determination of the deformation modulus and Poisson's ratio of rock masses depends on the rock mass quality and various geological features [52]. The Poisson's ratio is an elastic constant that measures the lateral deformation of a material when subjected to uniaxial tension or compression [53]. The Poisson's ratios of various rocks vary significantly; for example, norite ranges from 0.2 to 0.25 are observed in Figure 7. For example, marble, which is relatively uniform in composition, has a Poisson's ratio range from 0.15 to 0.30. This wide variability in Poisson's ratios is largely due to the heterogeneities in rocks, influenced by their geological history, mineral content, crystallization, and depositional structure [2]. A small Poisson's ratio indicates minimal lateral deformation under axial strain, while a wide range suggests varying degrees of elasticity in materials.  
The elastic modulus is made up of two important mechanical parameters for linear elastic materials, which are the [PERSON]'s modulus (\(E\)) and [PERSON]'s ratio (\(v\)), and they are expressed as  
\[E=\frac{\sigma}{\in} \tag{2}\]  
\[v=\frac{\in_{lateral}}{\in_{axial}} \tag{3}\]  
where \(E\) is the Young's modulus, \(\sigma\) is stress, and \(\in\) is strain. \(v\) is the [PERSON]'s ratio, \(\in_{lateral}\) is lateral strain, and \(\in_{axial}\) is axial strain. The [PERSON]'s modulus quantifies a material's stiffness by comparing stress and strain in its elastic region, while the Poisson's ratio  
Figure 6: Split grips for dog-bone-shaped specimens are one type of direct stress testing method. (**a**) For cylindrical specimens, bonded end caps are used. (**b**) Uniform cylindrical specimen compressed under applied force (Fa), with no necking. (**c**) Testing for biaxial extension and (**d**) load converters that convert compression to tension. (Applied force (Fa), radius (R), diameter (D), area (A), and water pressure (P) are crucial variables in these experiments [50].)  
indicates the material's lateral deformation in response to axial stress. [PERSON] et al. [2] (2021) discovered that the [PERSON]'s modulus describes the strain (\(\epsilon\)) response to uniaxial stress in the force-applied direction, while the Poisson's ratio describes the response in perpendicular directions.  
According to the International Society for Rock Mechanics (ISRM), in the study by [PERSON] et al. [52] (2020), the stress-strain curve for axial deformation is the most effective method for determining rock elasticity and the Young's modulus (\(E\)). Tangent Young's modulus (\(E_{t}\)) is the slope of the curve at a specific percentage of ultimate strength, while the secant Young's modulus (\(E_{s}\)) is the slope of the line from the origin to a fixed percentage of ultimate strength, typically at 50% (Figure 8). The secant and tangent Young's modulus values in Figure 7 are nearly identical, showing a strong correlation with an R\({}^{2}\) value of 0.99 across all intact rock types, including Monzodiorite, sharing similarities with norite as both are coarse-grained, mafic, and intrusive igneous rocks.  
Figure 8: Relationship between secant [PERSON]’s modulus (E\({}_{t}\)) and tangent Young’s modulus (E\({}_{s}\)) ([PERSON] et al. [52], 2020.)  
Figure 7: Distribution of Poisson’s ratios for different types of rocks (after [54]).  
The elastic modulus of intact rock (such as norite) is crucial for rock engineering projects like tunnels, slopes, and foundations. This parameter has become a popular area of research due to the need for high-quality core samples, advanced testing equipment, and empirical models to estimate the roughness of a rock [55]. The Young's modulus measures the stiffness of a rock wherein higher values indicate less deformation under load, while lower values suggest greater flexibility and compression.