# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 1 Introduction  
Due to their widespread use in industries such as aerospace, shipbuilding, aviation, automotive, and medicine, composite materials have become a subject of interest in various scientific disciplines, including chemistry and mechanics. To enhance their strength or to meet specific functional requirements, these materials can be manufactured with periodic curvature. In fiber-reinforced and laminated composite materials, structural curvatures may arise either during the design stage or as a result of manufacturing processes. These geometric imperfections can significantly affect the mechanical properties of the material, including stiffness, strength, and stability [1, 2, 3, 4, 5]. As a result, the deflection, buckling, and vibration behavior of such materials under various static and dynamic loads has become a significant topic of research [1, 2, 3, 4, 5, 6]. [PERSON] and [PERSON]' [1; 6; 7] derived the continuity equations for composite materials with curvature and presented both theoretical and numerical results for different types of matrix and fiber materials. [PERSON] [8], using the Continuum Theory developed by [PERSON] and [PERSON]', investigated two- and three-dimensional boundary value problems of curved composite materials through the finite element method and provided numerical examples. [PERSON] [9; 10] conducted analyses on the natural vibration and stability behavior of composite beams and plates with curvature based on the Continuum Theory of Akbarov and Guz' and employed the third-order shear deformation plate theory. [PERSON] [11] carried out a three-dimensional stress analysis of multilayered composite plates with small-scale periodic curvatures by employing the Continuum Theory of Akbarov and Guz' and the semi-analytical finite element method. The results highlighted the significant impact of structural curving on the stress distribution in such plates. [PERSON] and [PERSON]' [1] investigated the problems of stability loss, nonlinear vibrations, and failure under compressive loads in composite structures with material curvature. They emphasized that the influence of curvature on stress distribution constitutes a significant area for future research. [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] [12] investigated the parametric nonlinear vibration behavior of three-layer symmetric laminated plates using the harmonic balance method and the Galerkin approach. They analyzed the frequency response curves for various materials and geometric configurations of the core layer. [PERSON] and [PERSON] [13] performed the static bending and free vibration analysis of cross-ply laminated composite spherical shells by employing an equivalent single-layer shell displacement model. For this purpose, they developed a new hyperbolic shear deformation theory (RHSDT) based on the equivalent single-layer shell displacement model. The governing equations were derived using [PERSON]'s principle and validated through the Navier solution method. Additionally, the study examined the impact of geometric parameters on the bending and free vibration responses of supported laminated spherical shells. Studies on the dynamic behavior of laminated sandwich composite shells indicate that there is a need for advanced analytical approaches in this field. [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] [14] investigated the free vibration behavior of laminated sandwich composite shells using both analytical methods and the finite element method, highlighting the importance of combining theoretical and numerical approaches in this area. [PERSON] and [PERSON] [15] investigated the natural and forced vibration behavior of composite plates with periodic curvature in two directions within the framework of the three-dimensional theory of elasticity. The plates were clamped on all edges and subjected to uniformly distributed time-varying normal loads on their upper surfaces.