## 2 Fundamental Assumptions and Formulations
### Spectral Displacement Formulation  
For high-precision analyses of thick FGM plates, as well as for the stress analyses of thin FGM plates, refined shear deformation theories are necessary [4]. In this paper, the displacement fields are expanded into the following spectral series:  
\[\left\{ \begin{array}{l}u=u_{0}+\sum_{i=1}^{M}(u_{i}-w_{i-1,x})\kappa_{i}(z)+ \sum_{i=M+1}^{N}u_{i}\kappa_{i}(z)\\ v=v_{0}+\sum_{i=1}^{M}(v_{i}-w_{i-1,y})\kappa_{i}(z)+\sum_{i=M+1}^{N}v_{i} \kappa_{i}(z)\\ w=\sum_{i=1}^{M}w_{i-1}\kappa_{i}^{\prime}(z)\end{array} \right. \tag{3}\]  
where \(u\), \(v\), and \(w\) are displacements in the \(x\), \(y\), and \(z\) directions; \(\{u_{i}\}_{i=0}^{N}\)\(\{v_{i}\}_{i=0}^{N}\), and \(\{w_{i}\}_{i=0}^{M-1}\) are basic unknowns. \(\{\kappa_{i}(z)\}_{i=1}^{N}\) are the spectral bases with the first constant term omitted. \((\ )^{\prime}\) represents the first-order derivative, while \((\ )_{,x}\) and \((\ )_{,y}\)_ indicate first-order partial derivatives with respect to \(x\) and \(y\), respectively. \(M\) and \(N\) are the truncation numbers for the transverse and the in-plane displacement fields, \(M\leq N\). Equation (3) can be rearranged in the following vector form:  
\[ \begin{array}{l}\mathbf{u}=\mathbf{u}_{0}+\sum_{i=1}^{M}\mathbf{A}_{i}(z) \mathbf{u}_{i}+\sum_{i=M+1}^{N}\mathbf{A}_{i}(z)\mathbf{u}_{i}\\ \mathbf{A}_{i}(z)=\text{diag}\big{[}\kappa_{i}(z),\kappa_{i}(z),\kappa_{i}^{ \prime}(z)\big{]}\end{array}  \tag{4}\]  
where \(\mathbf{u}=[u,v,w]^{\text{T}}\), \(\{\mathbf{u}_{i}\}_{i=0}^{N}\) are the generalized displacements defined as follows:  
\[\mathbf{u}_{i}=\left\{ \begin{array}{l}[u_{i}-w_{i-1,x},v_{i}-w_{i-1,y},w_{i- 1}]^{\text{T}}\ \,\ \ 1\leq i\leq M\\ [u_{i},v_{i},0]^{\text{T}}\ \,\ \ i=0\text{ and }M+1\leq i\leq N\end{array} \right. \tag{5}\]  
Figure 1: Functionally graded material plate. The infinitesimal strain fields can be expressed as [40]  
\[ \begin{array}{l}\epsilon=\epsilon_{0}+\sum_{i=1}^{M}\mathbf{B}_{i}(z)\epsilon_{i }+\sum_{i=M+1}^{N}\mathbf{B}_{i}(z)\epsilon_{i}\\ \mathbf{B}_{i}(z)=\text{diag}[\kappa_{i}(z),\kappa_{i}(z),\kappa_{i}^{\prime }(z),\kappa_{i}^{\prime}(z),\kappa_{i}^{\prime}(z),\kappa_{i}(z)]\\ \end{array}  \tag{6}\]  
where \(\epsilon=\left[\epsilon_{xx},\epsilon_{yy},\epsilon_{zz},\gamma_{yz},\gamma_ {xz},\gamma_{xy}\right]^{\text{T}}\), \(\left\{\epsilon_{i}\right\}_{i=0}^{N}\) are generalized strains in the following form:  
\[\epsilon_{i}=\left\{ \begin{array}{l}\left[u_{i,x}-w_{i-1,xx},\;v_{i,y}-w_{i -1,yy},w_{i-1},\right.\\ v_{i,u},u_{i,y}+v_{i,x}-2w_{i-1,xy}\right]^{\text{T}}\\ \left[u_{i,x},v_{i,y},0,v_{i},u_{i,y}+v_{i,x}\right]^{\text{T}},\;\;\;i=0\text { and }M+1\leq i\leq N\\ \end{array} \right. \tag{7}\]  
Applying the 3D elastic constitutive relations [40], the stress fields are expressed as  
\[ \begin{array}{l}\boldsymbol{\sigma}=\mathbf{D}\boldsymbol{\epsilon}= \mathbf{D}\boldsymbol{\epsilon}_{0}+\sum_{i=1}^{M}\mathbf{DB}_{i}(z) \boldsymbol{\epsilon}_{i}+\sum_{i=M+1}^{N}\mathbf{DB}_{i}(z)\boldsymbol{ \epsilon}_{i}\\ \mathbf{D}=\left[\begin{array}{l}\mathbf{D}_{\mathbf{n}}\\ 0_{3\times 3}\end{array}\right],\;\;\;\left(\mathbf{D}_{\mathbf{n}}\right)_{ij}= \lambda+2\mu\delta_{ij},\;\;\;\left(\mathbf{D}_{\mathbf{s}}\right)_{ij}=\mu \delta_{ij}\end{array}  \tag{8}\]  
where \(\boldsymbol{\sigma}=\left[\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{yz}, \sigma_{xz},\sigma_{xy}\right]^{\text{T}}\), \(\delta_{ij}\) is the Kronecker symbol. \(\lambda=Ev/(1+v)/(1-2v)\) and \(\mu=E/(1+v)/2\) are Lame parameters. In this paper, the Chebyshev polynomials of the first kind are taken as the spectral bases whose first-order derivatives are Chebyshev polynomials of the second kind. Specifically, Chebyshev polynomials [41] of the first kind on \([-h/2,h/2]\) are defined as [41]  
\[ \begin{array}{l}T_{0}(z)=1,\;\;\;T_{1}(z)=\frac{2z}{h}\\ T_{i}(z)=\frac{4z}{h}T_{i-1}(z)-T_{i-2}(z),\;\;\;i\geq 2\end{array}  \tag{9}\]  
and Chebyshev polynomials of the second kind are defined as [41]  
\[