# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 2 Materials and Methods  
The geometric boundaries of the plate in the \(Ox_{1}x_{2}x_{3}\) coordinate system are defined as:  
\[-\frac{h}{2}\leq x_{2}\leq\frac{h}{2}\hskip 56.905512 pt0\leq x_{1}\leq l_{1} \hskip 56.905512 pt0\leq x_{3}\leq l_{3} \tag{1}\]  
As illustrated in Figure 1.  
The equations of displacements, according to the Kirchhoff-Love plate theory, are expressed in Equations (2)-(4). This theory is based on the assumption that the normals to the mid-surface remain straight and perpendicular after deformation. Therefore, it is applicable only to thin plates where transverse shear effects are negligible [30]. \[\mathrm{u}_{1}(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=u(\mathrm{x}_{1}, \mathrm{x}_{3})-x_{2}w_{,1} \tag{2}\]  
\[u_{2}(x_{1},x_{2},\mathrm{x}_{3})=w(x_{1},\mathrm{x}_{3}) \tag{3}\]  
\[u_{3}(x_{1},x_{2},\mathrm{x}_{3})=v(x_{1},\mathrm{x}_{3})-x_{2}w_{,3} \tag{4}\]  
where \(u_{1}\), \(u_{2}\), and \(u_{3}\) are the displacements in the \(x_{1}\), \(x_{2}\), and \(x_{3}\) directions, respectively. Similarly, \(u\), \(w\), and \(v\) represent the displacements of the mid-surface (or mid-plane) of the plate in the same \(x_{1}\), \(x_{2}\), and \(x_{3}\) directions. The notation used herein is expressed in Equation (5), the stress-strain relationships are given by Equation (6), and the strain-displacement expressions are presented in Equation (7). \[(\dots)_{i}=\frac{\partial(\dots)}{\partial x_{i}} i=1,3. \tag{5}\]  
\[\sigma_{\mathrm{ij}}=C_{\mathrm{ijkl}}\epsilon_{\mathrm{kl}} i,j,k,l=1,2,3. \tag{6}\]  
\[\epsilon_{\mathrm{ij}}=\frac{1}{2}\left(\frac{\partial u_{\mathrm{i}}}{x_{ \mathrm{j}}}+\frac{\partial u_{j}}{x_{\mathrm{i}}}\right) i,j=1,2,3. \tag{7}\]  
For a composite plate, Equation (6) can be rewritten in the form given in Equation (8). \[\sigma_{\mathrm{i}}=A_{\mathrm{ij}}\epsilon_{\mathrm{j}} i,j=1,2,3,4,5,6. \tag{8}\]  
Figure 1: Plate Geometry. The expanded form of Equation (8) is given as follows:  
\[ \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix} = \begin{bmatrix}A_{11}&A_{12}&A_{13}&A_{14}&A_{15}&A_{16 }\\ A_{21}&A_{22}&A_{23}&A_{24}&A_{25}&A_{26}\\ A_{31}&A_{32}&A_{33}&A_{34}&A_{35}&A_{36}\\ A_{41}&A_{42}&A_{43}&A_{44}&A_{45}&A_{46}\\ A_{51}&A_{52}&A_{53}&A_{54}&A_{55}&A_{56}\\ A_{61}&A_{62}&A_{63}&A_{64}&A_{65}&A_{66}\end{bmatrix}\begin{bmatrix}\epsilon_{ 1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\end{bmatrix}  \tag{9}\]  
The following notation is expressed in Equation (9), as shown in Equation (10):  
\[ \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix} = \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{13}\\ \sigma_{12}\end{bmatrix} ,.... \begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\end{bmatrix} = \begin{bmatrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{13}\\ \epsilon_{12}\end{bmatrix}  \tag{10}\]  
In the present study, \(A_{15}\) and \(A_{35}\) are taken as zero due to the presence of periodic curvature only in the \(x_{1}\) direction [3]. Substituting Equations (2)-(4) into Equation (7) and Equation (9)  
\[ \begin{bmatrix}\sigma_{1}\\ \sigma_{3}\\ \sigma_{5}\end{bmatrix} = \begin{bmatrix}A_{11}&A_{13}&0\\ A_{31}&A_{33}&0\\ 0&0&A_{55}\end{bmatrix} . \begin{bmatrix}\epsilon_{11}^{0}-x_{2}w_{,11}\\ \epsilon_{33}^{0}-x_{2}w_{,33}\\ \epsilon_{13}^{0}-2x_{2}w_{,13}\end{bmatrix}  \tag{11}\]  
It is obtained where \(\epsilon_{ij}^{0}\) denotes the strains at the mid-surface of the plate. It should be noted that \(\epsilon_{12}=~{}\epsilon_{23}=~{}\epsilon_{22}=0\), and the periodic curvature is introduced only along the \(x_{1}\) direction [7]. This assumption is based on the classical Kirchhoff-Love plate theory, which adopts the plane stress condition for thin plates and neglects transverse shear deformations [30].