Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory

Abstract

In this paper, a novel size-dependent beam model made of functionally graded materials (FGMs) is developed based on the strain gradient elasticity theory and sinusoidal shear deformation theory. The material properties of the functionally graded (FG) microbeams are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. Governing equations and boundary conditions are derived simultaneously by using Hamilton’s principle. The new model contains three material length scale parameters and can consequently capture the size effect. In addition, the newly developed model degenerates to the modified couple stress sinusoidal beam model or the classical sinusoidal beam model by setting two or all material length scale parameters to zero. The Navier-type solution is developed for simply-supported boundary conditions. Numerical results are presented to investigate the influences the material length scale parameter, different material compositions, and shear deformation on the bending and free vibration behavior of FG microbeams. Some of the present results are compared with the previously published results to establish the validity of the present formulation. It is established that the present FG microbeams exhibit significant size-dependence when the thickness of the microbeam approaches to the material length scale parameter.

Introduction

Functionally graded materials (FGMs) are special composites that have continuous variation of material properties from one surface to another and have many advantages. By now, the application of FGMs has been spread in many engineering fields, such as thin films (Fu et al., 2003, Lü et al., 2009), atomic force microscopes (Rahaeifard, Kahrobaiyan, & Ahmadian, 2010), micro-and nano-electro-mechanical systems (MEMS and NEMS) (Witvrouw & Mehta, 2005). In these applications, the thickness of the microstructures such as microbeams and microplates is typically on the order of microns and sub-microns. The size-dependent deformation behavior of micro-scale structures has been experimentally observed by many investigators (Chong et al., 2001, Fleck et al., 1994, Liu et al., 2012, Lam et al., 2003, Stölken and Evans, 1998). Because of this, the size effect plays an important role on the mechanical behavior of microstructures at the micrometer scale. Since the wide applications of the FGMs, it is important to study size effects of the FG structures. However, conventional strain-based mechanical theories cannot compute such a size dependent phenomenon as a result of lacking of the material length scale parameter. Thus, needs exist for the development of size dependent beam models which account for these size effects. Recently, higher-order (non-local) continuum theories such as strain gradient elasticity theory (SGT) (Lam et al., 2003) and couple stress theory (CST) (Koiter, 1964; Mindlin & Tiersten, 1962; Toupin, 1962) which contain additional material constants have been developed to predict these size dependence phenomenon. Mindlin and Tiersten (1962) and Koiter (1964) elaborated a type of higher-order continuum theories named couple stress theory (CST) involving four material length scale parameters (two classical and two additional) which are used to compute the size effect. Yang, Chong, Lam, and Tong (2002) first modified the classical couple stress theories (CST) (Koiter, 1964; Mindlin & Tiersten, 1962; Toupin, 1962) and proposed a modified couple stress theory (MCST) in which the constitutive equations contain only one additional material length scale parameter which causes to create symmetric couple stress tensor and to use it more easily. Several size-dependent microbeam and microplate models made of homogeneous materials have been developed by using the modified couple stress theory (MCST) (Asghari, 2012, Chen et al., 2012, Fu and Zhang, 2010, Ma et al., 2008, Rahaeifard et al., 2011, Yin et al., 2010), etc. Meanwhile, based on the modified couple stress theory, several size-dependent FG microbeam and microplate models have been also developed. Asghari, Ahmadian, Kahrobaiyan, and Rahaeifard (2010) developed a FG Timoshenko beam model based on the modified couple stress theory. Ke and Wang (2011) examined the size effect on dynamic stability of FG Timoshenko microbeams and the differential quadrature method (DQM) is employed to solve the governing differential equations. Nateghi, Salamat-Talab, Rezapour, and Daneshian (2012) studied the buckling behaviors of size-dependent FG microbeams by using the modified couple stress theory and parabolic shear deformation beam theory. Reddy (2011) developed microstructure-dependent Euler–Bernoulli and Timoshenko beam theories for FG microbeams taking into account the von Kármán geometric nonlinearity. Ke, Wang, Yang, and Kitipornchai (2012) studied the nonlinear free vibration of size-dependent FG Timoshenko microbeams. Şimşek, Kocatürk and Akba s (2013) examined the static bending of FG microbeams based on the modified couple stress theory and Timoshenko beam theories. To avoid the use of the shear correction factor, Salamat-Talab, Nateghi, and Torabi (2012) utilized modified couple stress theory to model the static and free vibration behaviors of FG microbeams based on a third-order shear deformation theory. Thai and Kim (2013) developed a size-dependent model for bending and free vibration of FG linear Reddy plate. Thai and Vo (2013) further proposed a size-dependent FG sinusoidal plate model by using the principle of virtual displacements.

However, Shu and Fleck (1998) have pointed out that couple stress theory (Koiter, 1964; Mindlin & Tiersten, 1962; Toupin, 1962), which is a general form of the modified couple stress theory (Yang et al., 2002), usually underestimates the size effect because it only employs rotation gradient and neglects other gradients. Therefore, to account for size effect more effectively, a general strain gradient theory, incorporating not only rotation gradient but also stretch gradient and other gradients, should be introduced. For example, strain gradient elasticity theory (SGT) proposed by Lam et al. (2003) introduces three material length scale parameters to characterize dilatation gradient, deviatoric stretch gradient and symmetric rotation gradient. The higher-order stress work-conjugate to the new higher-order deformation metrics and the corresponding constitutive relations are defined. By using strain gradient elasticity theory (SGT), Kong, Zhou, Nie, and Wang (2009) studied the static and dynamic responses of Euler–Bernoulli microbeams. Wang, Zhao, and Zhou (2010) presented strain gradient Timoshenko microbeams formulations. Kahrobaiyan, Asghari, Rahaeifard, and Ahmadian (2011) developed a nonlinear Euler–Bernoulli beam model based on strain gradient elasticity theory. Ramezani (2012) developed a nonlinear Timoshenko microbeam model based on a general form of strain gradient elasticity theory. Akgöz and Civalek (2011) employed strain gradient elasticity theory to analyze microbeams with various boundary conditions. Ansari, Gholami and Sahmani (2011) studied the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient and Timoshenko beam theories. They also investigated the nonlinear free vibration behaviors of FG Timoshenko microbeams based on the strain gradient elasticity theory and von Karman geometric nonlinearity (Ansari, Gholami, & Sahmani, 2012). Akgöz and Civalek (2013) studied the bending and vibration of microbeams made of homogeneous materials based on the strain gradient elasticity theory and the sinusoidal shear deformation theory. Kahrobaiyan, Rahaeifard, Tajalli, and Ahmadian (2012) developed a FG Euler–Bernoulli beam model by using strain gradient elastic theory. Sahmani and Ansari (2013) investigated the free vibration behaviors of FG microplates based on strain gradient elasticity and higher-order shear deformable plate theory. Ansari, Faghih Shojaei, Gholami, Mohammadi, and Darabi (2013) also investigated the postbuckling behavior of FG microbeams subjected to thermal loads. Rahaeifard, Kahrobaiyan, Ahmadian, and Firoozbakhsh (2013) studied the size-dependent static and dynamic behavior of nonlinear Euler–Bernoulli beams made of functionally graded materials (FGMs) on the basis of the strain gradient theory.

This paper studies the static bending and free vibration of FG microbeams based on the strain gradient elasticity theory (SGT) and sinusoidal shear deformation theory. This model involves three material length scale parameters which can capture the size effect of the FG microbeams. The material properties of the FG microbeams are assumed to vary in the thickness direction, which are estimated though the Mori–Tanaka homogenization technique. The governing equations and the related boundary conditions are derived by using Hamilton’s principle. Analytical solutions are obtained by using Navier procedure for the size-dependent sinusoidal microbeam with simply supported boundary conditions. Some of the present results are compared with the previously published results to establish the validity of the present beam model. The effects of the material length scale parameters, material property gradient index, each type of strain gradient on the static bending and free vibration behaviors of FG sinusoidal microbeams are examined in detail.