## 2 Theoretical Definition of the Problem  
### Poroelastic Modeling of Plates  
Let us consider a rectangular FG porous plate, with in-plane dimensions \(a\) and \(b\), and thickness \(h\), as depicted in Figure 1, along with three different porosity patterns throughout the thickness direction (\(0\leq z\leq h\)), namely, a non-symmetric nonlinear porous distribution (PNND), a symmetric nonlinear porous distribution (PNSD) and a uniform porous distribution (PUD). Except for uniform porosities, the mechanical properties of the material in terms of shear modulus, Young's modulus and mass density, for a PNND and PNSD, are defined as in Equation (1) [36; 37; 38; 39; 40].  
Figure 1: Geometrical scheme and loading conditions for a FG-saturated porous plate.  
\[E =E_{1}[1-e_{0}\,Q] \tag{1}\] \[G =G_{1}[1-e_{0}\,Q]\] \[\rho =\rho_{1}[1-e_{m}\,Q]\]  
In which  
\[Q(z)=\left\{\begin{array}{ll}(a)PNND&cos\Big{(}\dfrac{\pi z}{2h}\Big{)}\\ (\mathrm{b})-\mathrm{PNSD}&cos\Big{(}\dfrac{\pi}{2}-\dfrac{\pi z}{h}\Big{)} \end{array}\right. \tag{2}\]  
and \(0\leq e_{0}\leq 1\) is the porosity coefficient. Moreover, \(E_{1}\), \(G_{1}\) and \(\rho_{1}\) denote the Young's modulus, the shear modulus and the mass density at \(z=h\) (for a PNND) and at \(z=0\) (for a PNSD), whereby \(E_{j}=2G_{j}(1+v)\), \(j=0,1\) and the [PERSON]'s ratio, \(v\), is assumed to be constant in the \(z\)-direction. The constitutive equations of FG-saturated porous rectangular plates are derived from [PERSON]'s theory, which accounts for the displacements field of the solid, the pore fluid movement as well as their interactions owing to the applied loads [41]. Based on [PERSON]'s theory, the constitutive law is thus written as [42]:  
\[\sigma_{ij}=2G\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij}-p\alpha\delta_{ij} \tag{3}\]  
where  
\[p=\overline{M}(\Psi-\alpha\epsilon_{kk})\]  
\[\overline{M}=\dfrac{2G(v_{u}-v)}{\alpha^{2}(1-2v_{u})(1-2v)}\]  
\[v_{u}=\dfrac{v+\alpha\beta(1-2v)/3}{1-\alpha\beta(1-2v)/3}\]  
\[v_{=}\dfrac{\epsilon_{ij}}{\epsilon_{ii}}|_{\sigma_{ii}=0},p=0,i\
eq\mathrm{j}\]  
\[v_{u} =\dfrac{\epsilon_{ij}}{\epsilon_{ii}}|_{\sigma_{ii}=0},\Psi=0,i\
eq \mathrm{j}\]  
\[\alpha=1-\dfrac{K}{K_{S}}\]  
\[K =\dfrac{2(1+v)}{3(1-2v)}G\] \[K_{u} =\dfrac{2(1+v_{u})}{3(1-2v_{u})}G\]  
Note that \(p\) is the pore fluid pressure, such that, for \(p=0\), [PERSON]'s law reverts to the classical [PERSON]'s law (or drained condition). In addition, \(\lambda\) denotes the Lame constant, \(\delta_{ij}\) is the Kronecker delta and \(\alpha\) is the Biot's effective stress coefficient (with \(0<\alpha<1\)). This parameter accounts for the effect of porosity on the structural behavior and resistance of porous materials in the absence of an internal fluid. At the same time, \(\overline{M}\), \(G\), \(\
u_{u}\), \(\epsilon_{kk}\), \(\Psi\), \(K_{s}\) and \(\beta\) stand for the Biot's modulus, shear modulus, undrained Poisson's ratio (\(v<\
u_{u}<0.5\)), volumetric strain, variation of fluid volume content, bulk modulus of a homogeneous material and the Skempton coefficient, which introduces the pore fluid property, respectively. This last coefficient, \(\beta\), in particular, denotes a dimensionless parameter to include the impact of a fluid within cavities on the overall response of a porous material in undrained condition (\(\Psi=0\)), and it is described as the ratio of the cavity pressure to the total body stress, namely,  
\[\beta=\dfrac{dp}{d\sigma}\bigg{|}_{\Psi=0}=\dfrac{1}{1+e_{0}\dfrac{C_{P}}{C_{ S}}}=\dfrac{K_{u}-K}{\alpha K_{u}} \tag{4}\]In which \(K_{u}\), \(K\) refer to the bulk modulus in undrained and drained conditions respectively, and \(C_{p}\) and \(C_{S}\) stand for the fluid and solid compressibility in pores. Thus, the Skempton coefficient defines the effect of fluid compressibility on the elastic modulus and compressibility of the whole porous material [42].