A comprehensive investigation of the bending and vibration behavior of size- dependent functionally graded nanoplates via an enhanced first order shear deformation theory and nonlocal finite element analysis

Abstract This research paper conducts a comprehensive investigation into the bending and free vibration of size-dependent functionally graded (FG) nanoplates, using an improved first-order shear deformation theory (IFSDT). The IFSDT, proposed in this study, offers an enhanced representation and precise computation of normal and shear stresses across the thickness of the nanoplate. Notably, it not only ensures compliance with free conditions on both upper and lower surfaces but also eliminates the need for a conventional correction factor commonly employed in FSDT. The material properties of the FG nanoplate undergo continuous grading throughout the thickness direction using a power-law function. To transcend the assumptions of classical continuum mechanics and address the impacts of small sizes in discrete nanoplates, Eringen's nonlocal elasticity theory is applied. The formulation of the governing equation for bending and free vibration analyses of the FG nanoplate is achieved through the application of Hamilton’s principle. The proposed IFSDT is implemented with a computationally efficient C0-continuous quadrilateral element, tackling large-scale discrete numerical problems. The model's performance is showcased through a comparative evaluation against literature predictions, highlighting its high accuracy and rapid convergence. Additionally, the research scrutinizes various parameters such as plate thickness, boundary conditions, aspect ratio, nonlocal parameter, different material compositions, and power-law index. The thorough examination and discussion of these parameters provide insights into their influence on the deflection, stresses, and natural frequency of FG nanoplates. The results underscore the significant impact of size-dependent effects on the bending and vibration behaviors of nanoplates, emphasizing the necessity of incorporating these effects in the design and analysis of FG nanoplates. Ultimately, the developed nonlocal finite element model serves as a valuable predictive tool for understanding the bending and vibration behavior of size-dependent functionally graded nanoplates.