# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 2 Materials and Methods
For composite materials with periodic curvature along the \(x_{1}\) direction, the stiffness coefficients \(A_{\bar{\mathfrak{g}}}\) (\(x_{1}\)) are expressed as functions of the curvature parameters [8; 10]. As described in Ref. [7], the material constants in Equation (11) can be expanded as  
\[A_{11}(x_{1})=A_{11}^{0}\varnothing_{1}^{4}(x_{1})+2A_{12}^{0} \varnothing_{1}^{2}(x_{1})\varnothing_{2}^{2}(x_{1})+\] \[+A_{22}^{0}\varnothing_{2}^{4}(x_{1})+4A_{66}^{0}\varnothing_{1} ^{2}(x_{1})\varnothing_{2}^{2}(x_{1}),\] \[A_{13}(x_{1})=A_{13}^{0}\varnothing_{1}^{2}(x_{1})+A_{23}^{0} \varnothing_{2}^{2}(x_{1}), \tag{12}\] \[A_{33}(x_{1})=A_{33}^{0},\] \[A_{55}(x_{1})=A_{55}^{0}\varnothing_{1}^{2}(x_{1})+A_{44}^{0} \varnothing_{2}^{2}(x_{1}).\]  
Related to Equation (12), the material constants \(A_{11}{}^{0},A_{33}{}^{0},A_{12}{}^{0},A_{23}{}^{0},A_{22}{}^{0},A_{66}{}^{0},A _{55}{}^{0}\), \(A_{13}{}^{0}\), and \(A_{44}{}^{0}\) correspond to the normalized material constants for the composite plate without periodic curvatures [7]:\[A_{11}^{0}=A_{33}^{0}=\mu_{1}\eta_{1}+\mu_{2}\eta_{2}+(\mu_{1}+\eta_{1})\eta_{1} \ +(\mu_{2}+\eta_{2})\eta_{2}-\ \frac{(\lambda_{1}-\lambda_{2})^{2}}{(\lambda_{1}+2\mu_{1})\eta_{2}+( \lambda_{2}+2\mu_{2})\eta_{1}}\,\]  
\[A_{12}^{0}=A_{23}^{0}=\lambda_{1}\eta_{1}+\lambda_{2}\eta_{2}-(\lambda_{1}- \lambda_{2})\eta_{1}\eta_{2}\frac{(\lambda_{1}+2\mu_{1})(\lambda_{2}+2\mu_{2}) }{(\lambda_{1}+2\mu_{1})\eta_{2}+(\lambda_{2}+2\mu_{2})\eta_{1}}\,\]  
\[A_{22}^{0}=(\lambda_{1}+2\mu_{1})\eta_{1}+(\lambda_{2}+2\mu_{2})\eta_{2}-\eta_ {1}\eta_{2}\frac{(\lambda_{1}+2\mu_{1})(-\lambda_{2}+2\mu_{2})^{2}}{(\lambda_ {1}+2\mu_{1})\eta_{2}+(\lambda_{2}+2\mu_{2})\eta_{1}},\]  
\[A_{66}^{0}=A_{55}^{0}=\frac{\mu_{1}\mu_{2}}{\mu_{1}\eta_{2}+\mu_{2}\eta_{1}}\]  
\[A_{13}^{0}=(\lambda_{1}+\mu_{1})\eta_{1}+(\lambda_{2}+\mu_{2})\eta_{2}-\eta_{ 1}\eta_{2}\frac{(\lambda_{1}-\lambda_{2})^{2}}{(\lambda_{1}+2\mu_{1})\eta_{2} +(\lambda_{2}+2\mu_{2})\eta_{1}}-\]  
\[+A_{22}^{0}\varnothing_{2}^{4}(x_{1})+4A_{66}^{0}\varnothing_{1} ^{2}(x_{1})\varnothing_{2}^{2}(x_{1}),\]0  
\[+A_{22}^{0}\varnothing_{2}^{4}(x_{1})+4A_{66}^{0}\varnothing_{1} ^{2}(x_{1})\varnothing_{2}^{2}(x_{1}),\]1  
\(\lambda_{\rm i}\) and \(\mu_{\rm i}\) denote the Lame constants and \(\eta_{\rm i}\) volume fractions of the matrix and reinforcement materials, respectively, where the index \(i=1\) refers to the matrix material and \(i=2\) to the reinforcement material. In Equation (13),  
\[ \begin{array}{c}\phi_{1}=\frac{1}{\sqrt{1+\frac{dF(x_{1})}{dx_{1}}^{2}}}\\ \\ \phi_{2}=\phi_{1}\frac{dF(x_{1})}{dx_{1}}\end{array}  \tag{14}\]  
where the function \(F\) (\(x_{1}\)) in Equation (14) can be expressed as shown in Equation (15). \[ \begin{array}{c}F(x_{1})=\varepsilon f(x_{1})\\ \\ \varepsilon=\frac{\overline{h}}{\Lambda}\\ \\ f(x_{1})=sin\big{(}\frac{\pi x_{1}}{\Lambda}+\delta\big{)}\end{array}  \tag{15}\]  
Here, \(\Lambda\), \(\bar{\rm h}\), and \(\delta\) represent the wavelength, amplitude, and phase difference of the periodic curvature, respectively, as also illustrated in Figures 1 and 2. According to the three-dimensional elasticity theory, the equilibrium equations can be expressed as given in Equation (16). \[\sigma_{\rm ij,j}+F_{\rm i}=0,\ \ \ \ \ \ \ \ \ \ \ \ i,j=1,2,3. \tag{16}\]  
Integrating the stresses \(\sigma_{\rm ij}\) through the thickness of the plate gives  
\[\mathrm{T}_{ij}=\int_{-h/2}^{h/2}\sigma_{ij}dx_{2}\ \ \ \ {\rm i,j}=1,2,3. \tag{17}\]  
Figure 2: Periodic curvature of the laminate, showing the curvature parameters \(\mathrm{x_{1}}\) and the parameters \(\Lambda\), \(\bar{\rm h}\) and \(\delta\). Similarly, by multiplying the stress components \(\sigma_{ij}\) by \(x_{2}\) and integrating through the plate thickness, the bending moments \(M_{ij}\) can be obtained, as expressed in Equation (18). These resultant quantities, namely \(T_{ij}\) and \(M_{ij}\), are expressed per unit width of the plate.