Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory

Abstract

Size-dependent models for bending, buckling, and vibration of functionally graded Kirchhoff and Mindlin plates are developed using a modified couple stress theory. The present models contain one material length scale parameter and can capture the size effect, geometric nonlinearity, and two-constituent material variation through the plate thickness. The equations of motion are derived from Hamilton’s principle based on a modified couple stress theory, the von Karman nonlinear strains, and the power law variation of the material through the thickness of the plate. Analytical solutions for deflection, buckling load, and frequency of a simply supported plate are presented to bring out the effect of the material length scale parameter on the bending, buckling, and vibration responses of microplates.

Introduction

Functionally graded materials (FGMs) are a class of composites that have continuous variation of material properties from one surface to another and thus eliminate the stress concentration found in laminated composites. The FGMs which are often isotropic and nonhomogeneous, are made from a mixture of two materials to achieve a composition that provides a certain functionality. In recent years, the application of FGMs has broadly been spread in micro- and nano-scale devices and systems such as thin films [1], [2], atomic force microscopes [3], micro- and nano-electro-mechanical systems (MEMS and NEMS) [4], [5]. In such applications, size effects have been experimentally observed [6], [7], [8], [9], and conventional plate models based on classical continuum theories do no account for such size effects due to lack of material length scale parameters. Therefore, size-dependent plate models based on size-dependent continuum theories that contain additional material length scale parameters have been developed. Several size-dependent continuum theories have been developed to account for the size effects such as the classical couple stress theory [10], [11], [12] with two material length scale parameters, the nonlocal elasticity theory [13] with two material length scale parameters, and the and strain gradient theory [9] with three material length scale parameters. In view of the difficulties in determining material length scale parameters, the modified couple stress theory [14] takes an advantage over the aforementioned size-dependent continuum theories due to involving only one material length scale parameter.

The modified couple stress theory proposed by Yang et al. [14] results from the classical couple stress theory [10], [11], [12]. The two main advantages of the modified couple stress theory over the classical couple stress theory are the inclusion of asymmetric couple stress tensor and the involvement of only one material length scale parameter. Based on the modified couple stress theory, several size-dependent beam and plate models have been developed to capture the size effects in small scale structures. For example, Park and Gao [15] developed a Euler-Bernoulli beam model for bending analysis of nanobeams. This model was employed by Kong et al. [16] and Kahrobaiyan et al. [17] to study vibration of microbeams. Xia et al. [18] included the von Karman nonlinear strain in the Euler–Bernoulli beam model, but they neglected the coupling between axial and transverse displacements which can be significant in beams where the axial displacement is constrained. Ma et al. [19] developed a Timoshenko beam model to incorporate the effects of transverse shear deformation and rotary inertia. The Timoshenko beam model was adopted to study the buckling [20] and vibration [21], [22] of microtubes. Asghari et al. [23] includes the von Karman nonlinear strain in the Timoshenko beam model. Tsiatas [24] first developed a Kirchhoff plate model for static analysis of microplates. This model was used by Yin et al. [25] and Jomehzadeh et al. [26] to study the vibration of microplates. Recent papers by Asghari [27] and Asghari and Taati [28] separately deal with Kirchhoff plate theory of isotropic plates and FGM plates, with the first one including the geometric nonlinearity (but without FGM). A positive contribution of these papers is that they present the general form of boundary conditions considering the sharp corners. To account for the effects of transverse shear deformation and rotary inertia in moderately thick microplates, Ma et al. [29] and Ke et al. [30] developed a Mindlin plate model. It should be noted that the above mentioned studies dealt with the microbeams and microplates made of homogeneous materials only. Recently, the modified coupled stress theory is further used to develop functionally graded Euler–Bernoulli beam [31], functionally graded Timoshenko beam [32], [33], [34], [35], and laminated Timoshenko beam [36].

Although the size-dependent beam and plate models have been developed in the aforementioned studies based on the modified couple stress theory, no literature has been reported for the size-dependent plate models accounting for geometric nonlinearity and material variation through the thickness of the plate. In this paper, size-dependent functionally graded Kirchhoff and Mindlin plate models are developed to account for the size effect, geometric nonlinearity, and material variation through the thickness of the plate. The equations of motion are derived from Hamilton’s principle based on the modified couple stress theory, von Karman nonlinear strains, and power law variation of material through the thickness. Analytical solutions for the static bending, buckling, and free vibration problems are presented for a simply supported plate to bring out the effects of material length scale parameter on the deflection, buckling load, and frequency. Since most nano-scale devices involve plate-like elements that may be functionally graded and undergo moderately large rotations, the newly developed plate models can be used to capture the size effects in functionally graded nanoplates.