Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates

Abstract

The small- and large-amplitude vibrations are presented for a functionally graded rectangular plate resting on a two-parameter (Pasternak-type) elastic foundation in thermal environments. Two kinds of micromechanics models, namely, Voigt (V) model and Mori–Tanaka (M–T) model, are considered. The motion equations are based on a higher order shear deformation plate theory that includes plate-foundation interaction. The thermal effects are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. Two cases of the in-plane boundary conditions are considered. Initial stresses caused by thermal loads or in-plane edge loads are introduced. The accuracy of Voigt and Mori–Tanaka models for the vibration analysis of FGM plates is investigated. The comparison studies reveal that the difference between these two models is much less compared to the difference caused by different solution methodologies and plate theories. The results show that the difference of the fundamental frequencies between M–T and V solutions is very small, and the difference of the nonlinear to linear frequency ratios between M–T and V solutions may be negligible.

Introduction

The demand for improved structural efficiency in aerospace structures has resulted a new class of materials, called functionally graded materials (FGMs). FGMs are designed so that material properties vary smoothly and continuously through the thickness from the surface of a ceramic exposed to high temperature to that of a metal on the other surface. They are now developed for the general use as structural components in high temperature environments. Consequently, investigations on linear vibration characteristics of FGM plates are identified as an interesting field of study in recent years [1], [2], [3].

Many nonlinear vibration analyses of FGM plates subjected to mechanical and thermal loading are available in the literature. Among those, Yang et al. [4] studied the large amplitude vibration of an initially stressed FGM plate with surface-bonded piezoelectric layers. Huang and Shen [5], [6] presented nonlinear free and forced vibration analyses of FGM plates with or without piezoelectric layers in thermal environments. This work was then extended to the case of nonlinear vibration of FGM plates with piezoelectric fiber reinforced composite (PFRC) actuators [7]. Similar work for nonlinear free and forced vibration analyses of FGM plates with piezoelectric layers in thermal environments was carried out by Fakhari et al. [8] using the finite element method (FEM). On the other hand, Sundararajan et al. [9] calculated frequencies for nonlinear free flexural vibration of FGM rectangular and skew plates under thermal environments using the FEM. Chen and his co-authors [10], [11], [12], [13], [14], [15] studied the nonlinear vibration and stability of initially stressed FGM plates with or without initial geometric imperfections using the Galerkin method. Mao and Fu [16] presented the nonlinear dynamic response and active vibration control for piezoelectric functionally graded plates using the finite difference method. Recently, Talha and Singh [17] re-examined the large amplitude free flexural vibration of shear deformable FGM plates using the nonlinear FEM. In the aforementioned studies, however, the effective material properties of FGMs are usually assumed to obey from the Voigt model except for [9] in which the Mori–Tanaka model was adopted. Although Voigt model was adopted in most analyses of FGM structures [18], someone still has doubts on the accuracy of applying the Voigt model in bending, buckling and vibration analyses of structures made of FGMs. Recently, Huang et al. [19] compared the frequencies of cantilevered square functionally graded Al/Al₂O₃ plates having horizontal side cracks using Mori–Tanaka and Voigt models. They found that the largest difference of frequencies based on these two models is around 6% in the case of very thick plate (width-to-thickness ratio b/h = 5) with volume fraction index N = 1. However, in their analysis, the Poisson’s ratio is taken to be a constant for Voigt model, whereas it is assumed to be a function of Z (direction in the plate thickness) for Mori–Tanaka model. Hence, to clarify the accuracy of applying Mori–Tanaka and Voigt models will play an important role in the structural analysis of FGM structures.

In the present work, we focus our attention on the nonlinear free vibration for FGM plates resting on an elastic foundation of Pasternak-type in thermal environments. Two kinds of micromechanics models, namely, Voigt model and Mori–Tanaka model, are adopted. The motion equations are based on Reddy’s higher order shear deformation plate theory [20] and general von Kármán-type equations [21]. The plate-foundation interaction and thermal effects are also included, and the material properties are assumed to be temperature-dependent. All four edges of the plate are assumed to be simply supported and two cases of in-plane boundary conditions are considered. Initial stresses caused by thermal loads or in-plane edge loads are introduced. Assessments are presented through numerical comparisons.