## 2 Materials and Methods  
### Mindlin Plate Theory  
Initially, [PERSON] [34] proposed the static equilibrium equations of plates in which the effect of transverse shear was considered. Subsequently, [PERSON] [25] pursued a systematic investigation of the dynamics in the plate and developed a theory known as Mindlin plate theory. In this theory, the effects of transverse shear deformation and rotational inertia are taken into consideration to model flexural wave propagation in the plate.  
Consider a thin homogeneous, isotropic and linearly elastic plate having thickness \(h\) subjected to an excitation force \(P\), as shown in Figure 1a. The general configuration of the defect in the plate is shown in Figure 1b. It should be noted that the actual one-sided thickness defect is approximated by the symmetric two-sided thickness reduction. The internal forces and deformations in the plate are shown in Figure 1c.  
The three displacement components of the plate can be denoted as follows:  
\[u=-z\beta_{x}(x,y,t),v=-z\beta_{y}(x,y,t),w=\bar{w}(x,y,t) \tag{1}\]  
where \(u\), \(v\) are displacement components in the \(x\)- and \(y\)-direction and the \(w\) component is given as an average value of vertical displacement \(\bar{w}\) at mid-plane, respectively. The rotations are specified as  
\[\beta_{x}=\frac{\partial w}{\partial x}-\gamma_{zx},\beta_{y}=\frac{\partial w }{\partial y}-\gamma_{zy} \tag{2}\]  
where \(\beta_{x}\) and \(\beta_{y}\) are the rotations of the normal to the undeformed mid-plane surface in the \(x-z\) and \(y-z\) planes, respectively. The terms \(\gamma_{zx}\) and \(\gamma_{zy}\) are the shear angles of the cross-section in the \(x\)- and \(y\)-direction. The moments \(M_{x}\), \(M_{y}\), \(M_{xy}\) and shear forces \(Q_{x}\), \(Q_{y}\) are specified as  
\[\begin{split} M_{x}&=-D\Big{(}\frac{\partial\beta_ {x}}{\partial x}+v\frac{\partial\beta_{y}}{\partial y}\Big{)}\\ M_{y}&=-D\Big{(}\frac{\partial\beta_{y}}{\partial y }+v\frac{\partial\beta_{x}}{\partial x}\Big{)}\\ M_{xy}&=-(1-v)\frac{D}{2}\Big{(}\frac{\partial \beta_{x}}{\partial y}+\frac{\partial\beta_{y}}{\partial x}\Big{)}\\ Q_{x}&=KCh\Big{(}\frac{\partial w}{\partial x}- \beta_{x}\Big{)}\\ Q_{y}&=KCh\Big{(}\frac{\partial w}{\partial y}- \beta_{y}\Big{)},\end{split} \tag{3}\]  
where \(G\) is the shear modulus, \(K=5/(6-\
u)\) is the shear correction factor [35], \(D=Eh^{3}/12(1-\
u^{2})\) is the bending stiffness, with \(E\) and \(\
u\) being the [PERSON]'s modulus and Poisson ratio, respectively.  
Figure 1: (**a**) Geometry of the plate and wave propagation problem; (**b**) General configuration of the defect; (**c**) Internal forces and deformations in a plate.  
The plate equations of motion are specified as  
\[\left\{\begin{array}{l}\frac{\partial M_{x}}{\partial x}+\frac{\partial M_{yy}}{ \partial y}-Q_{x}=-\frac{\rho h^{3}}{12}\frac{\partial^{2}\beta_{y}}{\partial l ^{2}}\\ \frac{\partial M_{yy}}{\partial x}+\frac{\partial M_{y}}{\partial y}-Q_{y}=- \frac{\rho h^{3}}{12}\frac{\partial^{2}\beta_{y}}{\partial l^{2}}\\ \frac{\partial Q_{x}}{\partial x}+\frac{\partial Q_{y}}{\partial y}+P=\rho h \frac{\partial^{2}\psi}{\partial l^{2}},\end{array}\right. \tag{4}\]  
where \(\rho\) is the mass density of the material and \(P\) is the forcing function normal to the plane of the plate.