Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory

Highlights

• A three-dimensional model of rotating functionally graded microbeams is established.

• Modified couple stress theory and Hamilton's principle are used to derive the model.

• The size-dependent model contains the von Kármán geometric nonlinearity.

• The influence of size effect on dynamics is studied combined with other parameters.

Abstract

A size-dependent three-dimensional dynamic model of rotating functionally graded (FG) microbeams is developed based on the Euler–Bernoulli beam theory. A two-constituent material varying along the thickness is considered following the power law. In addition, Poisson's ratio is assumed constant in the present model. Hamilton's principle is adopted in conjunction with a modified couple stress theory to derive the governing equations with the von Kármán geometric nonlinearity incorporated. The Galerkin method is utilized to solve these equations in which the coupling of the axial, chordwise and flapwise deformations is included. The centrifugal stiffening effect of the rotating FG microbeam is captured by the nonlinear coupling deformations. The convergence, accuracy and validity of the present method are confirmed by several examples. The influence of the size-dependency on modal characteristics is investigated combined with the material gradient index, dimensionless rotational speed and hub radius ratio. Finally, the achieved results of dynamic responses indicate that the deformations of rotating FG microbeams are greatly affected by the size-dependency and material gradient variation.

Introduction

Functionally graded materials (FGMs) were first developed by material scientists in Japan as heat-resistant composite materials [1], which are made of two or even more different materials, usually ceramics and metals, with continuously and smoothly variable properties from one surface to another in desired directions. The continuity not only avoids the high interlaminar stresses that habitually occur in conventional laminated composites, but also satisfies some special features in engineering design. Since the FGMs have many advantages over ordinary laminates composites and homogeneous materials, the applications of FGMs have widely been spread in aerospace, defense and civil industries or even in biomedical sectors, electronics and nuclear reactions. Therefore, the study on the static [2], [3], [4], [5], dynamic [6], [7], [8], [9], [10], [11] and buckling [12], [13], [14] properties of macro-scaled structural members (e.g., beams, plates and shells) made of FGMs has attracted the attention of numerous researchers. It should be pointed out that all of the reviewed literatures were based on the classical continuum theory.

Nowadays, FGMs have been broadly applied in micro- and nano-electro-mechanical system (MEMS and NEMS) and even in atomic force microscopes (AFMs). As the key structural members, microbeams are common in such systems. The experimental results shown that the mechanical behavior of the microstructures has obvious size effect which cannot be captured by the classical continuum theory [15], [16], [17], [18], [19]. In order to describe the size-dependent behaviors of microstructures, several nonclassical continuum theories, e.g. couple stress [20], [21], [22], strain gradient [18], [23], [24], [25], [26], [27], [28] and nonlocal micropolar [29], [30], [31], have been proposed with incorporating size-dependent material length scale parameters.

The couple stress theory was originally proposed by Cosserat brothers in 1909 [32], also known as Cosserat theory, however, did not attract much attention for a long time after that. Until 1960s, a general (classical) couple stress theory was developed by Toupin [20] and Mindlin and Tiersten [21] containing two material length scale parameters and two Lamé constants. Due to the difficulties of determining material length scale parameters, Yang et al. [33] introduced a new additional equilibrium of the moment of couples in conjunction with traditional equilibrium relations of forces and moments of forces to develop a modified couple stress theory that includes only one material length scale parameter. Subsequently, the modified couple stress theory was widely used to interpret the size effect of FG microstructures [34], [35], [36], [37], [38], [39], [40], [41]. Recently, Dehrouyeh-Semnani et al. [42] explored the longitudinal-transverse coupling vibrations of FG microbeams with geometric imperfection. Shafiei et al. [43] applied the generalized differential quadrature method (GDQM) to solve the size-dependent nonlinear vibrations of tapered axially functionally graded (AFG) microbeams including the von-Kármán nonlinearity. Şimşek and Aydın [44] solved the static bending and forced vibration of an imperfect FG Mindlin microplate using the Newmark's method. Attia [45] developed nonlocal-couple stress-surface elasticity model to study size-dependent bending, buckling and free vibration responses of FG nanobeams. Then, Shanab et al. [46] presented the nonlinear bending response of FG nanoscale beams including the surface energy. Babaei et al. [47] investigated the temperature-dependent free vibration of FG Euler–Bernoulli microbeams. Based on the hyperbolic shear deformation beam model, Akgöz and Civalek [48] determined the more accurate natural frequencies of FG thick microbeams. Shojaeefard et al. [49] studied free vibration and thermal buckling of FG porous circular microplates subjected to a nonlinear thermal load using both classical and the first-order shear deformation theories.

Rotating beam-type structures are broadly applied in blades of turbo-engine and turbine, spinning space structures and other engineering fields. Hence, numerous investigations on mechanical behavior of rotating beams have been published using different analytical or/and numerical methods on the basis of the classical continuum theory [50], [51], [52], [53], [54], [55]. However, the available literature for rotating microstructures based on the modified coupled stress theory is very limited. Dehrouyeh-Semnani [56] developed the nonclassical Euler–Bernoulli and Timoshenko beam elements to study the influences of size-dependency on flapwise vibrations of rotating microbeams. Then, Dehrouyeh-Semnani et al. [57] investigated the size-dependent lead-lag vibration of rotating cantilevered microbeams considering the coupling of the transverse bending and axial stretching motions. Ghadiri and Shafiei [58] analyzed the thermal vibration of rotating FG Timoshenko microbeams in the environments with high temperature changes. Shafiei et al. studied the transverse vibration of rotating non-uniform AFG Euler–Bernoulli microbeams [59], and then, they extended the work to rotating tapered AFG Timoshenko microbeams [60]. Meanwhile, Shafiei and his collaborators [61] also presented the size-dependent free vibration behaviors of rotating non-uniform FG microbeams. More recently, Mahinzare and his collaborators studied the free vibrations of a rotary smart two directional functionally graded piezoelectric material (2-FGPM) circular nanoplate [62] and a spinning bi-dimensional functionally graded (2-FGM) micro plate subjected to thermal load [63]. As far as authors know, no work has been done on the size-dependent three-dimensional dynamics of rotating FG microbeams in the existing literatures.

The objective of this study is to develop a three-dimensional dynamic model of a rotating FG microbeam including the size effect on the basis of the modified couple stress theory, and investigate its dynamics. The physical model of the FG microbeam and the distribution law of FGMs are introduced in Section 2. In Section 3, the Hamilton's principle and the modified couple stress theory are employed to derive the size-dependent three-dimensional governing equations of the rotating FG microbeam via the Euler–Bernoulli beam theory, finally the Galerkin method is utilized to solve the dynamics of the FG microbeam. In Section 4, convergence and comparative examples are given to verify the effectiveness of the current method, and then the influences of size-dependency on the vibrations and responses of rotating FG microbeams are investigated in detail combined with other parameters. The summary is given in Section 5.