Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates

TL;DR: In this article, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium with oriented, straight carbon nanotubes (CNTs).

Abstract: Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.