## 2 Functionally Graded Porous Plates  
Figure 1a represents a full ceramic-metal FG porous plate with length \(a\), width \(b\) and thickness \(h\), where the plate mid-surface is labeled by \(\varpi\in\Re^{2}\). A Cartesian coordinate system \((O;x,y,z)\) sits at the upper left corner of \(\varpi\) so that the plate top and bottom surfaces are positioned at \(z=\pm h/2\). The composition of ceramic and metal particles varies through the thickness according to their volume fractions. Letting \(V_{c}(z)\) and \(V_{m}(z)\) be the volume fractions of ceramic and metal, both fractions satisfy the following constraint, given by  
\[V_{c}(z)+V_{m}(z)=1,\ -h/2\leq z\leq h/2 \tag{1}\]  
By virtue of this constraint, one can choose only one volume fraction to present two volume fractions within the plate. In the current study, the ceramic volume fraction \(V_{c}(z)\) is selected and defined by  
\[V_{c}(z)=(0.5+z/h)^{n},\ -h/2\leq z\leq h/2 \tag{2}\]  
with \(n(n\geq 0)\) being the ceramic index. Figure 1b represents the thickness-wise ceramic volume fractions for different values of \(n\), which were plotted using Excel. The plate becomes ceramic-rich as \(n\) approaches zero and metal-rich as \(n\) approaches infinity. Figure 2 represents four different porosity distributions taken for the current study in which the porosity density changes in the \(z-\)direction only. Uniform and center-biased distributions are symmetric while lower- and upper-biased distributions are non-symmetric. In the current study, uniform distribution is called even, center-biased distribution is called  
Figure 1: A ceramic–metal FG plate: (**a**) geometry and dimensions, (**b**) thickness-wise distributions of ceramic volume fraction \(V_{c}\). uneven or sym, and lower- and upper-biased distributions are called unsym-1 and unsym-2, respectively. First, the porosity \(\psi(z)\) in even and uneven distributions is expressed by  
\[\text{Even}:\ \psi(z)=e \tag{3}\]  
\[\text{Uneven}:\ \psi(z)=e\bigg{(}1-\frac{2|z|}{h}\bigg{)} \tag{4}\]  
Second, referring to a paper by [PERSON] et al. [19], the porosity \(\psi(z)\) in sym, unsym-1 and unsym-2 distributions is expressed by  
\[\text{Sym}:\ \psi(z)=e\cos\Big{[}\pi\Big{(}\frac{z}{h}\Big{)}\Big{]} \tag{5}\]  
\[\text{Unsym}-1:\ \psi(z)=e\cos\Big{[}\frac{\pi}{2}\Big{(}\frac{z}{h}-0.5 \Big{)}\Big{]} \tag{6}\]  
\[\text{Unsym}-2:\ \psi(z)=e\cos\Big{[}\frac{\pi}{2}\Big{(}\frac{z}{h}+0.5 \Big{)}\Big{]} \tag{7}\]  
using cosine functions. Here, \(e(0\leq e\leq 1)\) is the porosity parameter, and all the five porosity distributions have the same porous volume for a given value of \(e\). According to the linear rule of mixtures, the equivalent material properties \(\wp_{eff}(z)\) at any point within the FG porous plate is calculated by  
\[\wp_{eff}(z)=[V_{c}(z)\wp_{c}+(1-V_{c}(z))\wp_{m}](1-\psi(z)) \tag{8}\]  
in terms of two base material properties \(\wp_{c}\) and \(\wp_{m}\) of ceramic and metal, the volume fraction \(V_{c}(z)\) and the porosity \(\psi(z)\). The mid-surface surface \(\varpi\in\Re^{2}\) with its boundary \(\partial\varpi\) can define a dimension-reduced 2D hierarchical model in which the displacement field \(\mathbf{u}(x,y,z)\) is decomposed into \(\mathbf{\Theta}(x,y)\cdot Q(z)\). Here, \(\mathbf{\Theta}(x,y)\) and \(Q(z)\) indicate the in-plane triple vector function and the assumed thickness monomial, respectively. The classical plate and shell theories and the hierarchical models [29; 30; 31] were developed according to this dimension reduction technique. Referring to reference [31], the (3,3,2) hierarchical model, which can be considered third-order SDPT, is expressed by  
\[ \begin{cases}u_{x}\\ u_{y}\\ u_{z}\end{cases} _{(x,y)}= \begin{cases}u_{x}^{0}\\ u_{y}^{0}\\ u_{z}^{0}\end{cases} _{(x,y)}+ \begin{cases}u_{x}^{1}\\ u_{y}^{1}\\ u_{z}^{1}\end{cases} _{(x,y)}\times\bigg{(}\frac{2z}{h}\bigg{)}+ \begin{cases}u_{ x}^{2}\\ u_{y}^{2}\\ u_{z}^{2}\end{cases} _{(x,y)}\times\bigg{(}\frac{2z}{h}\bigg{)}^{2}+ \begin{cases}u_ {y}^{3}\\ u_{y}^{3}\\ 0\end{cases}