## 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)
In 2011 [83], they introduced a model toanalyze the vibration dynamics of damaged isotropic plates, with the results aligning well with other research in the field. In addition, [PERSON] et al. (2008) [99] examined the vibration behaviors of cracked cantilevered plates by considering changes in crack location and length, using FEM and the hybrid frequency/time (HFT) method. The study focused on nonlinearity due to crack closing effects, particularly near veerings. The research highlighted the effect of crack location and length on the first 15 natural frequencies, as shown in Figure 2. [PERSON] et al. (2011) [100] studied the vibration characteristics of cracked FGM plates by considering various parameters and observed a decrease in natural frequency with increasing temperature gradients, crack lengths, and gradient indices. [PERSON] et al. (2011) [101] used the RRM to analyze vibrations in plates with internal disruptions, introducing functions that accurately represented stress near crack tips and discontinuity across cracks. A 2018 study built on this presented a refined method for analyzing plates with through-cracks, concentrating on how crack parameters influence frequency variations, mode transition, and localization [102]. Soon after, [PERSON] and [PERSON] (2012) [84] explored how crack orientation influenced the base frequency of partially cracked isotropic plates using both analytical and experimental methods. [PERSON] et al. (2013) [103] conducted a comprehensive investigation into the impact of crack orientation and position on the natural frequency of cracked plates by employing the LSM. Their findings unveiled an inverse correlation between the natural frequency and crack length, independent of the plate's boundary constraints. Moreover, for SS or CS plates, an increase in crack angle up to 45 degrees induced a reduction in the natural frequency. Conversely, the natural frequency exhibited an upward trend with increasing crack eccentricity. In 2016, [PERSON] et al. [104] extended their research to include mobility analysis, which provided a deeper understanding of side cracks within the RP. In their analysis, the RRM has demonstrated its effectiveness in providing valuable insights into the dynamics of forced vibration, showing the connections between normalized mobility, crack length ratios, and crack angles. [PERSON] et al. (2014) [105] developed a bending and vibration analysis model for specific microplates, while [PERSON] et al. (2014) [106] examined the responses of cracked aluminum plates repaired with composite patches. [PERSON] et al. (2014) [107] conducted a study using the FEM to understand the impact of various crack parameters such as crack orientation, length, and position on plate vibrations. [PERSON] et al., working from 2014 to 2017 [108; 109; 110; 111; 112], extensively used the LSM to explore the effects of cracks on the vibration dynamics of various plate types, highlighting the decline in the base frequency with rising temperatures. Their results shed light on the vibrational changes caused by cracks and the comprehensive impacts of thermal moments and in-plane forces on plates of different materials including isotropic and FGM plates. They also explored the thermal effects on these plates using the modified couple stress theory in later studies [113]. Moreover, [PERSON] and [PERSON] (2015) [114] introduced a novel sinusoidal plate model that was founded on the modified strain gradient theory. Using [PERSON]'s principle, they derived the governing equations and their associated boundary conditions, enabling a more precise analysis of free vibrations in microplates. [PERSON] et al., in 2015 and 2016, developed a framework highlighting the influence of microstructure and fiber orientation on vibrational characteristics of different cracked plates [115; 116]. [PERSON] et al. (2017) [117] and [PERSON] et al.