# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 1 Introduction
Numerical results obtained via three-dimensional finite element modeling revealed that the local maxima of the thickness-direction normal stress significantly increased due to the structural curvature, emphasizing the necessity of considering these effects when predicting the mechanical performance, such as the adhesion strength, of composite plates with two-directional periodic curvature. [PERSON] [16] investigated the effect of periodic structural curvature on the stress distribution of thick rectangular composite plates subjected to forced vibration, using the exact three-dimensional theory of elasticity. The study revealed that even low-frequency dynamic loading leads to significantly higher stress concentrations due to the curvature effects. In a related study, [PERSON] [17] analyzed the natural vibration behavior of rectangular composite plates with a periodically curved structure using the three-dimensional elasticity theory. His findings further highlighted the significant influence of structural curvature on the vibration characteristics of such plates. [PERSON] and [PERSON] [18] investigated the stress distribution in multilayered composite plates exhibiting local structural damage in the form of curved fiber layers. By employing three-dimensional elasticity theory and a semi-analytical finite element method, they demonstrated that such local curvatures, which may arise during manufacturing, significantly affect the internal stress fields and must be accounted for through accurate constitutive models based on continuum mechanics. [PERSON], [PERSON], [PERSON], and [PERSON] [19] investigated crack problems in composite materials with periodically curved layers by employing the exact three-dimensional elasticity theory within the framework of a piecewise-homogeneous body model. Their study highlights a specific fracture mechanism observed in unidirectional composites under uniaxial loading, where the material splits along the loading direction, an effect not seen in homogeneous materials. The authors emphasize that even slight curvatures in the reinforcing elements can significantly influence the internal stress distribution and act as a trigger for crack formation. To address this, they developed a modeling approach that incorporates the presence of periodically curved layers and allows for a more accurate investigation of fracture behavior. The proposed method considers cracks as pre-existing along the most vulnerable regions of the matrix, parallel to the external load, offering deeper insight into the mechanical response of such complex composite systems. While the modeling in [19] assumes periodically curved reinforcing layers, it has been observed that, in practical applications, the curvature of the reinforcing elements may not always follow a periodic pattern. Instead, local curvatures can arise due to manufacturing imperfections or design irregularities. In this context, [PERSON] [20] extended the existing approach to analyze the fracture behavior of composites with locally curved reinforcing layers, using the same piecewise-homogeneous body model and exact elasticity theory. His findings highlight the importance of incorporating such irregularities in fracture modeling, especially when standard macroscopic criteria fail to capture the underlying failure mechanisms. [PERSON], [PERSON], and [PERSON] [21] proposed a two-scale asymptotic homogenization method for composite Kirchhoff plates with in-plane periodicity. By simplifying a 3D periodic plate problem into a fourth-order elliptic PDE, they developed an efficient approach to capture both macro- and microscale behaviors. Their method, validated through static and dynamic analyses, demonstrated high accuracy in predicting the mechanical response of thin periodic plates. [PERSON] [22] presented the first known natural frequencies for laminated composite angle-ply triangular and trapezoidal plates with completely free boundaries using the Ritz method.