Size Effect on the Axisymmetric Vibrational Response of ‎Functionally Graded Circular Nano-Plate Based on the Nonlocal Stress-Driven Method

Abstract

In this work, the axisymmetric-vibrational behavior of a size-dependent circular nano-plate with functionally graded material with different types of boundary conditions was investigated. The analysis was performed based on the Stress-driven model (SDM) and Strain-gradient theory (SGT) in conjunction with classical plate theory. The governing equations of motion and their corresponding equations for boundary conditions were obtained based on Hamilton’s principle and solved using the generalized differential quadrature rule. Results show that this method is applicable to the vibrational analysis of such structures with a fast convergence rate; as N approaches 6 for the first mode, and 10 for the second as well as the third and fourth modes, regardless of the type of boundary condition. In both models, the influences of various parameters such as size-effect parameter Lc, material heterogeneity index n, and types of boundary conditions were obtained on the first four modes and compared with each other. Results indicate that the natural frequencies in these modes increase with an increase in the heterogeneity index n, and size-effect parameter Lc. Additionally, these parameters appear to have a stiffening effect on the nano-plate vibrational behavior. However, for a nano-plate resting on a knife or simply supported edge, in the first mode, the SDM shows a more stiffening effect on the plate behavior as compared with the SGT. Nonetheless, for the clamped and free edge boundary conditions, both models predicted the same behavior. The SGT showed a higher-stiffening effect only in the fourth mode, for all types of considered boundary conditions.

Keywords

• vibrational response
• functionally graded material
• circular nano-plate
• stress-driven model
• strain gradient ‎theory‎