## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Forward Problem for the DR of the Cracked Plate
#### 3.1.1 Plate of the Uniform Thickness (UT)
(2017) [118] analyzed the nonlinear dynamic properties and stability of plates, revealing relationships between different parameters including temperature, crack length, and boundary conditions. [PERSON] et al. (2017) examined the free vibration of specific FG porous CP under nonlinear thermal loads. In 2018, their study used the FSDT to analyze the vibration of rotating nano/micro-CP made from FG materials [119; 120]. [PERSON] et al., in 2017 and 2018, utilized FSDT to investigate the vibration of various rotating CP, focusing on the relationship between variables such as angular velocity, size, and external electric voltage on the vibrational behavior of cracked plates [121; 122]. Research by [PERSON] et al. (2018) [123] and [PERSON] et al. (2018) [124] presented the Hilbert transform and Mindlin theory to detect and study the characteristics of cracks on plates, focusing on structural deterioration and the impact of load. [PERSON] et al. (2018) [125]investigated the damage location effects on the primary frequency of plates in thermal environments, emphasizing temperature and damage-related factors. From 2018 to 2020, [PERSON] et al. [126, 127, 128, 129, 130, 131] conducted a series of studies on cracked plates, focusing on vibration analysis and the influence of various factors such as temperature, immersion depth, and internal material length scale. Their research investigated the dynamics of different types of plates with cracks, providing analytical models for submerged, thermally influenced plates, and exploring the influence of crack length and microstructure on frequency. Meanwhile, [PERSON] et al. (2019) [132] explored the vibration characteristics of a uniquely cracked plate, offering new understandings of frequency responses related to crack angle. Their findings indicate an inverse relationship between crack angle and frequency response, as shown in Figure 3.  
Recently, [PERSON] and [PERSON] (2020) [133] introduced an analytical approach based on [PERSON]'s theory to investigate the natural vibrational characteristics of isotropic plates with surface or internal cracks, focusing on how various crack and boundary condition parameters affect natural frequencies. They concluded that the presence of internal cracks significantly altered the natural frequencies of a structure compared to surface cracks. Additionally, the longer the crack, the lower the natural frequencies. [PERSON] et al. (2020) [134] utilized a multi-phase field approach to address the initiation of intra-laminar cracking in long fiber-reinforced composites. However, these studies did not offer any models or methodologies for examining buckling and free vibration issues, which are commonly encountered in practical applications. To address this gap, [PERSON] et al. (2021) [135] applied phase-field theory to analyze multi-crack plates using Mindlin plate theory, where each layer contains a crack that extends through the layer's thickness according to the angle-ply orientation, but does not penetrate the entire plate thickness. In developing this theory, multiple phase-field variables were introduced, with each variable representing the crack in a specific layer. Furthermore, [PERSON] and [PERSON] (2021) [136] developed a mathematical model to explore the impact of through-thickness damage on thin plate frames of different configurations, highlighting how much the location of damage affects the strength of the structure. [PERSON] et al. initiated a two-phase investigation into breathing cracks in plate-like structures in 2021, using DR analysis and Hilbert transform to assess crack severity [137]. Their follow-up study in 2022 presented a crack identification technique based on instantaneous frequency ratio (IFR), highlighting the importance of different excitation types for damage indication and severity estimation. [PERSON] et al. (2022) [138] analyzed thin RP with cracks under moving masses ad found that the mass weight and velocity, crack dimensions, angular orientations, and plate aspect ratios significantly influenced the dynamic outcomes.