## 3 Crack Detection in Plates Using Dynamic Response (DR)
### Forward Problem for the DR of the Cracked Plate
#### 3.1.2 Plate of Thickness (VT)  
A few studies have focused on the oscillatory behavior of cracked plates of VT. [PERSON] et al. (2008) [158] examined the dynamics of viscoelastic RP with VT and the impact of cracks on their vibrational properties by applying the differential quadrature methodology. The study established differential equations to describe the motion of a viscoelastic plate whose thickness varies linearly and contains part-through cracks. These equations include terms for the additional rotation caused by the cracks. The study revealed that the dimensionless complex frequencies, which describe the vibrational properties of a system, are influenced by the material's non-dimensional delay time, the plate's aspect and thickness ratio, and the characteristics of any cracks present. The research shows that cracks impact vibration by changing these frequencies, particularly when they are closer to the plate's center, which decreases the rigidity of the plate. Moreover, a higher number of cracks leads to a significant decrease in frequency, whereas an increase in the plate's aspect and thickness ratios results in a considerable increase in frequency. [PERSON] et al. (2018) [27] combined classical plate theory with modified couple stress theory to develop an analytical model to understand the nonlinear vibration dynamics of cracked RP with LVT and PVT. In Table 3, their research showed that the fundamental frequency rose as the taper parameter increased in both one and two directions, with a more pronounced effect in the latter. Furthermore, the fundamental frequency increased with the taper parameter more significantly in PVT than in LVT. Additionally, an increase in crack length and gradient index resulted in a reduction in the fundamental frequency. The impact of boundary conditions on the fundamental frequency was also significant. Importantly, their results revealed similar vibrational behaviors between the UT and VT plates affected by cracks, though the magnitudes of influence differed. From 2019 to 2021, [PERSON] et al. [159; 160; 161; 162; 163; 164] conducted a series of studies on FGM plates with LVT and central cracks, investigating the effects of crack dimensions, material volume fraction exponents, and plate thickness on vibration frequencies. They used various shear deformation theories to analyze stability and vibrational parameters, considering factors like temperature, boundary conditions, and plate geometry. The study's findings indicated that several factors contributed to reduced vibration frequency in plates. An increase in crack length, length-to-width ratio, and power-law index all led to decreased plate stiffness and vibration frequency. Additionally, plates with a higher ratio of initial to final thickness (h\({}_{0}\)/ha) exhibited a more significant reduction in vibration frequency. In 2020, [PERSON] et al. [165] examined the free vibration of cracked plates of LVT, employing the peridynamics (PD) approach. They found that thicker plates had higher natural frequency values, while longer cracks led to lower frequencies. Interestingly, larger crack orientation resulted in higher frequencies, and plates with LVT showed mode shapes favouring their thinner ends. [PERSON] et al. (2021 and 2022) [87; 166] studied the nonlinear vibrations of cracked isotropic and orthotropic plates with LVT and PVT. They observed that the nonlinearity of the plate increased with the increasing or decre