Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory
Graphical abstract
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.
Section snippets
Functionally graded materials
Fig. 1 shows the three-dimensional configuration of an FG microbeam of length L, thickness h and width b mounted on a rotating rigid hub. The floating coordinate system o-xyz is fixed on the microbeam. The radius and rotational speed of the hub are z and Ω, respectively.
It is assumed that the microbeam is made of a two-constituent FG material along the thickness following the power law defined by [9], [10], [11], [35], [37] where P(z) denotes either the
The modified couple stress theory
The size-dependent governing equations of motion of the rotating FG microbeam can be derived by Hamilton's principle where T, U and W denote the kinetic energy, strain energy and external work, respectively. In this paper, it is assumed that no external force is applied, hence the external work W = 0.
Based on the modified couple stress theory [33], the strain energy is written by where εij and σij denote the components of the strain tensor and stress
Numerical results and discussion
In this section, the influences of the size-dependency in conjunction with other parameters, such as material gradient index, hub radius ratio and dimensionless rotational speed, on dynamics of rotating FG microbeams are presented. A complex modal analysis method is applied to solving the coupling vibration for the steady-state rotation (i.e., ).
Since the effect of aspect ratio (b/h) is not discussed herein, b/h is set to 1.0. Moreover, we assume that the FG microbeam is made of Al and Al2O3
Summary
In this study, a size-dependent three-dimensional model is presented to investigate the dynamic characteristics of rotating FG cantilevered Euler–Bernoulli microbeams using Hamilton's principle based on a modified couple stress theory and the von Kármán geometric nonlinearity. A two-constituent material varying through the thickness is considered following the power law while Poisson's ratio is assumed constant. The present model of the FG microbeam includes the coupling of the axial, chordwise
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
Acknowledgments
The authors are very grateful to the National Natural Science Foundation of China (Grant No. 11302096), the Natural Science Foundation of Nanjing Institute of Technology (Grant No. ZKJ201701) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20170759) for funding this work. The support from the Outstanding Scientific and Technological Innovation Team in Colleges and Universities of Jiangsu Province is also appreciated.
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