# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 2 Formulation of the P-FGM Model
### Displacement Fields and Strain-Displacement Relations  
The displacement field of the plate/shell domain is based on [PERSON]'s third-order shear deformation theory [10,11]:  
\[\begin{array}{l}u(x,y,z,r)=u_{0}(x,y,r)-z\,\theta_{y}(x,y,r)+z^{3}c_{1}\Big{[} \theta_{y}(x,y,r)-\frac{\partial w_{0}}{\partial x}\Big{]}\\ v(x,y,z,r)=v_{0}(x,y,r)+z\,\theta_{x}(x,y,r)+z^{3}c_{1}\Big{[}-\theta_{x}(x,y, )-\frac{\partial w_{0}}{\partial y}\Big{]}\end{array} \tag{3}\]  
\[\begin{array}{l}w(x,y,z,r)=w_{0}(x,y,r)\end{array}\]  
Figure 1: Cross-section representation of an FGM plate.  
where \(u_{0},\ v_{0},\ w_{0}\) are displacements of a generic point in the middle plane referred to the local axes--\(x,y,z\) directions; \(\theta_{x},\ \theta_{y}\) are the rotations of the normal to the middle plane about the \(x\) axis (clockwise) and \(y\) axis (anticlockwise); \(\frac{\partial w_{0}}{\partial x},\ \frac{\partial w_{0}}{\partial y}\) are the slopes of the tangents of the deformed mid-surface in the \(x,y\) directions; \(c_{1}=4/3\,h^{2}\), with \(h\) denoting the total thickness of the plate/shell structure; and \(r\) is the time.  
For the geometrically nonlinear behavior analysis, the strain tensor is conveniently represented in terms of the linear and nonlinear parts of the strain tensor as follows:  
\[\varepsilon=\left\{\ \varepsilon^{L}+\varepsilon^{NL}\right\} \tag{4}\]  
The linear strain components associated with the displacement fields defined above can be written in the following form:  
\[\varepsilon^{L}=\left\{\begin{array}{c}\varepsilon^{m}+z\varepsilon^{b}+z^{ 3}\ \varepsilon^{sb}\\ \varepsilon^{s}+z^{2}\ \varepsilon^{ss}\end{array}\right\} \tag{5}\]  
Therefore, we can write in the detailed form:  
\[\begin{array}{c}\varepsilon_{xx}=\frac{\partial\ u_{0}}{\partial\ x}-z\ \frac{\partial\theta_{y}}{\partial\ x}+z^{3}c_{1}\Big{(}\frac{\partial\theta_ {y}}{\partial x}-\frac{\partial^{2}w_{0}}{\partial x^{2}}\Big{)}+\frac{1}{2} \bigg{[}\bigg{(}\frac{\partial u_{0}}{\partial x}\bigg{)}^{2}+\Big{(}\frac{ \partial w_{0}}{\partial x}\Big{)}^{2}+\Big{(}\frac{\partial w_{0}}{\partial x }\Big{)}^{2}\bigg{]}\\ \varepsilon_{yy}=\frac{\partial\ v_{0}}{\partial\ y}+z\frac{\partial\theta_{x }}{\partial\ y}+z^{3}c_{1}\Big{(}-\frac{\partial\theta_{y}}{\partial y}-\frac {\partial^{2}w_{0}}{\partial y^{2}}\Big{)}+\frac{1}{2}\bigg{[}\bigg{(}\frac{ \partial u_{0}}{\partial y}\bigg{)}^{2}+\Big{(}\frac{\partial v_{0}}{\partial y }\bigg{)}^{2}+\Big{(}\frac{\partial w_{0}}{\partial y}\bigg{)}^{2}\bigg{]}\\ \gamma_{xy}=\Big{(}\frac{\partial\ u_{0}}{\partial\ y}+\frac{\partial\ v_{0}}{ \partial\ x}\Big{)}+z\ \Big{(}-\frac{\partial\ g_{y}}{\partial\ y}+\frac{\partial\ g_{x }}{\partial\ x}\Big{)}+z^{3}c_{1}\Big{(}\ \frac{\partial\theta_{y}}{\partial y}-\frac{\partial\ g_{x }}{\partial x}-2\frac{\partial^{2}w_{0}}{\partial x\ \By integrating through the thickness, the linear elastic constitutive equation for each virtual layer, \(k\), can be written as follows:  
\[\hat{\sigma}_{k}=\hat{D}_{k}\,\varepsilon_{k}=\begin{cases}N\\ M\\ M^{*}\\ Q\\ Q^{*}\end{cases}=\begin{bmatrix}A&B&D&0&0\\ B&C&E&0&0\\ D&E&G&0&0\\ 0&0&0&A_{s}&C_{s}\\ 0&0&0&C_{s}&E_{s}\end{bmatrix}\begin{split}\varepsilon^{m}\\ \varepsilon^{b}\\ \varepsilon^{sb}\\ \varepsilon^{ssc}\end{bmatrix}_{k} \tag{10}\]  
where \(\hat{\
u}_{k}\) is the resultant forces and moments, \(\hat{D}_{k}\) is the constitutive matrix, and the submatrices of \(\hat{D}\) are given by the following:  
\[\begin{split}\left(A,B,C,D,E,G\right)_{k}=\overline{Q}_{k}\,\int _{z_{k-1}}^{z_{k}}\left(1,z,z^{2},z^{3},z^{4},z^{6}\right)\,dz\\ \left(A_{s},C_{s},E_{s}\right)_{k}=\overline{Q}_{k}\,\int_{z_{k-1}}^{z_{k}} \left(1,z^{2},z^{4}\right)\,dz\end{split} \tag{11}\]