## 4 Discussion  
This section offers an in-depth analysis of plate theories, with a focus on modeling cracks and thickness variation in governing equations of plates and exploring damage detection methods in plate structure. The discussion aims to methodically evaluate the benefits, constraints, and complexities inherent in these areas. Plate theories provide a fundamental framework for understanding the behavior of thin plates under various loading conditions and boundary constraints. These theories are essential for designing safe and efficient plate structures in a wide range of engineering applications. However, these theories have both advantages and disadvantages that need to be carefully considered when selecting the most appropriate theory for a specific application. Classical plate theories, such as the Euler-Bernoulli, Lagrange, Navier, and Kirchhoff theories, provide valuable tools for analyzing thin plates under static loading, but they have limitations. These theories cannot accurately predict the behavior of thick plates, plates with large deflections, or plates made of nonlinear or anisotropic materials. Additionally, they are not suitable for analyzing plates subjected to shear loads, dynamic loads, or three-dimensional effects. Additionally, the Rayleigh plate theory further restricts its applicability to thin plates with small deflections and linear elastic, homogeneous, and isotropic materials. To address these limitations, more sophisticated plate theories have been developed. The MR plate theory relaxes the assumption of negligible shear strains, allowing for the modeling of transverse shear deformation and stresses. This theory provides more accurate predictions for thick plates or plates with high aspect ratios. However, its ability to handle nonlinear materials or dynamic loads is still limited. For even more complex behaviors, such as three-dimensional effects and nonlinear material properties, HSDT incorporates additional terms in the governing equations. These theories provide the most accurate predictions but are computationally more demanding and may not be necessary for all applications. Generally, the selection of plate theory depends on the specific characteristics of the plate, the loading conditions, and the required level of accuracy. Classical plate theories are suitable for simple plate geometries and moderate loading conditions. For more complex problems or situations where high accuracy is required, more advanced plate theories, such as the MR or HSDT, should be used. The governing equations differ for damaged and undamaged plates in accounting for the presence of damage. In an undamaged plate, the material properties are assumed to be uniform throughout the plate. This means that the governing equations can be expressed in a relatively simple form. However, in a damaged plate, the material properties are no longer uniform. This is because the damage has caused a change in the stiffness, mass, and damping properties of the plate. As a result, the governing equations for a damaged plate must be more complex in order to account for these changes. Modeling the behavior of damaged plates requires modifying the governing equations to account for the changes in material properties caused by the damage. Two main approaches exist, explicit damage modeling, which directly incorporates the damage characteristics into the equations [207; 208; 209; 210], and implicit damage modeling, which utilizes a damage parameter to represent the overall impact on the plate's behavior [212; 213; 214; 215]. The selection of the approach depends on the damage complexity and desired accuracy. Explicit modeling is suitable for simple damage types, while implicit modeling is more appropriate for complex damage scenarios. Generally, governing equations for damaged plates are more intricate and computationally demanding due to the introduction of additional unknowns, necessitating the use of numerical methods for solutions.