# Static and Dynamic Response of FG-CNT-Reinforced Rhombic Laminates
## 3 Mathematical Formulation  
### Displacement Fields and Strains  
The displacement field for the FG-CNT-reinforced rhombic plate is considered to derive the mathematical model based on the third-order shear deformation theory [39]:  
\[u(\text{x},\text{y},\text{z}) = u_{0}(\text{x},\text{y})+x\theta_{\text{x}}(\text{x},\text{y})+x^ {2}\xi_{\text{x}}(\text{x},\text{y})+x^{3}\xi_{\text{x}}(\text{x},\text{y})\] \[v(\text{x},\text{y},\text{z}) = v_{0}(\text{x},\text{y})+z\theta_{\text{y}}(\text{x},\text{y})+x ^{2}\xi_{\text{y}}(\text{x},\text{y})+x^{3}\zeta_{\text{y}}(\text{x},\text{y} )\;\;, \tag{7}\] \[w(\text{x},\text{y},\text{z}) = w_{0}(\text{x},\text{y})\]  
where \(u\), \(v\) and \(w\) are the displacements of any generic point in the plate geometry, (\(u_{0}\), \(v_{0}\) and \(w_{0}\)) are displacements at the mid-plane and \(\theta_{\text{x}}\), \(\theta_{\text{y}}\) are the bending rotations defined at the midplane about the y and x-axes respectively. \(\xi_{\text{x}}\), \(\xi_{\text{y}}\), \(\xi_{\text{x}}\) and \(\zeta_{\text{y}}\) are higher order terms of [PERSON]'s series expansion. The function \(\xi_{\text{x}}\), \(\xi_{\text{y}}\), \(\zeta_{\text{x}}\) and \(\zeta_{\text{y}}\) will be calculated by vanishing shear stress at top and bottom of the plate. By applying the boundary conditions \(\gamma_{\text{xz}}(\text{x},\text{y},\text{z}\text{h}/2)=\gamma_{\text{yz}}( \text{x},\text{y},\text{z}\text{h}/2)=0\) at the upper layer and lower layer of the plate in Equation (7) and rearranging the terms that appear in the displacement field (\(u\) and \(v\)), we obtained  
\[\xi_{\text{x}}=\xi_{\text{y}}=0 \tag{8}\]  
\[\xi_{\text{x}}=-\frac{4}{3h^{2}}\left(\theta_{\text{x}}+\frac{\partial w}{ \partial\text{x}}\right),\text{ }\zeta_{\text{y}}=-\frac{4}{3h^{2}}\left(\theta_{\text{y}}+\frac{\partial w}{ \partial\text{y}}\right). \tag{9}\]  
By substituting Equations (8) and (9) into Equation (7), we obtain  
\[u(\text{x},\  
first-order derivatives of transverse displacement. By applying \(C^{0}\) continuity to the present problem, the out of plane derivatives are exchanged by the following relations in Equation (10):  
\[\psi_{\mathrm{x}}=\frac{\partial w}{\partial\mathrm{x}},\ \psi_{\mathrm{y}}=\frac{ \partial w}{\partial\mathrm{y}}. \tag{11}\]  
The final form of higher order theory possessing \(C^{0}\) continuity may be presented in the following manner:  
\[ \begin{array}{l}u(\mathrm{x},\mathrm{y},\mathrm{z})=u_{0}+\left(\mathrm{z}- \frac{4\mathrm{z}^{3}}{3\mathrm{h}^{2}}\right)\theta_{\mathrm{x}}-\frac{4 \mathrm{z}^{3}}{3\mathrm{h}^{2}}\psi_{\mathrm{x}}\\ v(\mathrm{x},\mathrm{y},\mathrm{z})=v_{0}+\left(\mathrm{z}-\frac{4\mathrm{z}^ {3}}{3\mathrm{h}^{2}}\right)\theta_{y}-\frac{4\mathrm{z}^{3}}{3\mathrm{h}^{2} }\psi_{\mathrm{y}}\\ w(\mathrm{x},\mathrm{y},\mathrm{z})=w_{0}\end{array} . \tag{12}\]  
Hence, the basic field variables interpreted in the present investigation with the assumption of constant transverse displacement component are \(u_{0}\), \(\
u_{0}\), \(w_{0}\), \(\theta_{\mathrm{x}}\), \(\theta_{\mathrm{y}}\), \(\psi_{\mathrm{x}}\), and \(\psi_{\mathrm{y}}\) for each node. Mathematically, the nodal displacement vector \(\{\delta\}\) corresponding to displacement field in Equation (12) may be represented as  
\[\{\delta\}=\left[u_{0}\ v_{0}\ w_{0}\ \theta_{\mathrm{x}}\ \theta_{\mathrm{y}}\ \psi_{\mathrm{x}}\psi_{\mathrm{y}}\right]^{T} \tag{13}\]  
From the displacement field presented above in Equation (12), the strain can be written as  
\[\{\varepsilon\}=\big{\{}\varepsilon_{\mathrm{x}},\ \varepsilon_{\mathrm{y}},\ \gamma_{ \mathrm{xy}},\ \gamma_{\mathrm{xz}},\ \gamma_{\mathrm{yz}}\big{\}}^{T}. \tag{14}\]  
Furthermore, the expression of strain vector \(\{\varepsilon\}\) can be correlated with the displacement vector \(\{\delta\}\) by means of the following relationship:  
\[\{\varepsilon\}=[B]\{\delta\}, \tag{15}\]  
where \([B]\) is known as a strain-displacement matrix and involves the derivatives of shape function terms.