Vibration Examination of Higher-Order Trigonometric Auxetic Meta-Structure Laminated Plates Exposed to Lorentz Magnetic Load in Hygro-Thermal Environment

This study presents an analytical approach to determine the natural frequencies of sinusoidal shear deformable laminated plates, emphasizing the significance of examining the mechanical performance of engineering structures constructed from novel advanced materials under various multi-physical loadings and realistic environmental conditions. The inner layer of the structure comprises auxetic metamaterials exhibiting a negative Poisson’s ratio, while the outside layers consist of polymeric materials reinforced with carbon nanotubes. The plate’s performance is influenced by temperature and moisture fluctuations, and a longitudinal magnetic load based on Lorentz criteria is also applied. Additionally, a bi-axial mechanical preload is exerted on the plate. The entire edifice is situated on the Pasternak foundation. Exact values of natural frequencies under various situations are acquired and reported by deriving and solving the motion differential equations and validating them against pertinent literature. This work introduces unique elements and presents precise result values, enabling their usage as benchmarks for future research in this domain.