Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method

• Published in: Meccanica (Milan) 2015-09, Vol.50 (9), p.2391-2408
• Main Authors: Zare, Mojtaba, Nazemnezhad, Reza, Hosseini-Hashemi, Shahrokh
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Civil Engineering; Classical Mechanics; Mechanical Engineering; Physics; Physics and Astronomy
• ISSN: 0025-6455; 1572-9648
• DOI: 10.1007/s11012-015-0161-9

In this paper, the natural frequencies of a functionally graded nanoplate are analyzed for different combinations of boundary conditions. Application of new materials and specially the functionally graded materials in the micro- and nano-scale devices and systems is increasingly spread. Therefore, the study of the natural frequencies of functionally graded materials for different boundary conditions seems to be necessary in the micro/nano-structures. The article presented here covers broad types of common boundary conditions for the free vibration of functionally graded rectangular nanoplates for the first time. The analytical solution method used here is new for this subject and solves the governing equations with no approximation. The size dependency is considered according to Eringen’s differential form of nonlocal elasticity theory. The elasticity modulus and mass density of the plate are varied along the thickness of the plate according to a power-law distribution of the constituents’ volume fractions. As the in-plane and out-of-plane displacement variables are coupled in the equations of motion, a new exact solution method is introduced to solve the displacement fields analytically. The method is capable of dealing with new combinations of boundary conditions which have not been studied before in the literature. The validity and accuracy of the present method is investigated by comparing some of the present results to their counterparts reported in the literature. The results presented here are new and discussed for the first time in this subject. As a novelty a detailed study is carried out to examine the effects of power-law distribution, the characteristic internal length, the plate aspect ratio and the mode number on the natural frequencies of functionally graded rectangular plates for different boundary conditions. It’s shown that the type of boundary condition affects considerably on the values of natural frequency but the behavior of the frequency variations is in the same manner for different combinations of boundary conditions. These results can be used as benchmark for future studies.