Vibration and buckling of first-order shear deformable circular cylindrical micro-/nano-shells based on Mindlin's strain gradient elasticity theory
Introduction
In recent years, micro-structures whose characteristic size is on the order of microns or sub-microns have been widely used in micro-sized systems such as sensors, actuators, atomic force microscopes and micro-/nano-electromechanical systems (MEMS and NEMS) (Fu et al., 2004, Fu et al., 2011). In all of such applications, the role of size effect is crucial in studying the behaviors of such small scale structures. The scale effects of micro-structures are as a result of impurities, crystal lattice mismatch and micro cracks at micro scales (Jung et al., 2014).
Experimental studies have shown that the mechanical behavior of micro-structures is size-dependent (Stölken and Evans, 1998, Fleck et al., 1994, Lam et al., 2003). Since the conventional continuum models fail to predict the size-dependent response of micro-structures due to the lack of inherent length scales, developing non-classical continuum theories which can capture the size effects is essential. Up to now, different types of higher-order theories such as couple stress theory (CST) (Fathalilou et al., 2014, Shaat and Mohamed, 2014, Zeighampour and Beni, 2014), strain gradient theory (SGT) (Aifantis, 1999, Setoodeh and Afrahim, 2014, Li et al., 2014), micropolar theory (Eringen, 1967), nonlocal elasticity theory (Eringen, 1972, Zang et al., 2014) and surface elasticity theory (Gurtin et al., 1998, Ansari et al., 2014a, Ansari et al., 2014b) have been introduced to develop size-dependent continuum models. Among these non-classical theories, the CST and SGT include some higher-order stress components in addition to the classical stresses.
Mindlin and Tiersten (1962) and Koiter (1964) developed the CST in which four material length scale parameters including two classical and two additional are used to capture the size effect of micro-structure. In CST, the gradient stress of higher-order rotation is taken into consideration in deformation relations. Numerous studies have been undertaken in which size-dependent continuum beam models are developed on the basis of CST (Eringen and Suhubi, 1964a, Eringen and Suhubi, 1964b, Toupin, 1964, Mindlin, 1964). Yang et al. (2002) first proposed the modified couple stress theory (MCST). In this theory, one additional equilibrium equation, known as the couple of couples, is assumed to capture the material properties. Thereafter, using the principle of minimum potential energy, Park and Gao (2008) presented a variational formulation for MCST. The successful application of MCST in buckling and vibration behaviors of beams, plates and shells has been reported by many research workers. For instance, employing the MCST along with classical, first- and third-order shear deformation beam theories, Nateghi et al. (2012) investigated the size-dependent buckling of functionally graded (FGM) microbeams. Park and Gao (2006) using a variational formulation of the MCST introduced a new model for the bending of Bernoulli-Euler beam. Ma et al. (2008) analyzed the vibration and static bending behavior of Timoshenko beams on the basis of MCST and demonstrated the considerable effects of Poisson's coefficient and size parameter on beam deflection and beam vibration. Asghari et al. (2010) using MCST determined the nonlinear equations of Timoshenko beam and compared the linear and nonlinear vibrations and bending of the beam with classical theory (CT). Based on the MCST, Tsiatas (2009) developed a new Kirchhoff plate model for the static analysis of isotropic microplates. By means of MCST, Sahmani et al. (2013) presented a study on the dynamic stability of FG higher-order shear deformable microshells.
Fleck and Hutchinson (1993) through reformulating and extending the Mindlin's theory introduced a new type of continuum theory named as SGT. In this theory, the second-order deformation tensor is divided into stretch gradient tensor and rotation gradient tensor leading to additional higher-order stress components in comparison with CST. Altan and Aifantis (1997) then developed a simplified form of SGT containing a single length scale parameter. Utilizing the principle of minimum total potential energy, Gao and Park (2007) presented the variational formulation of simplified SGT and developed the model for the analysis of pressurized thick-walled cylinders. Later, Lam et al. (2003) proposed modified strain gradient theory (MSGT) which includes three material length scale parameters related to dilatation gradient, deviatoric gradient and symmetric rotation gradient tensors. The MSGT can be reduced to MCST if two of its length scale parameters are set equal to zero. In recent years, the application of MSGT has received considerable attention in the study of mechanical responses of microstructures. In this regard, Wang et al. (2011) proposed a size-dependent Kirchhoff microplate formulations on the basis of MSGT. In another study, Wang et al. (2010) using MSGT studied the static bending and free vibration problem of a simply-supported (SS) micro-scale Timoshenko beam. Kong et al. (2009) employed MSGT to explore the static and dynamic responses of Euler-Bernoulli micro-beams and examined the effect of thickness to the material length scale parameter ratio of the micro-beams on their static deformation and vibrational characteristics. Using MSGT, Ansari and his elaborates (Ansari et al., 2013) conducted an investigation into size-dependent bending, buckling and free vibration of FG Timoshenko microbeams. They also made a comparison between the results generated based on various beam models such as MCST, MSGT and CT. More recently, Gholami et al. (2014) developed a size-dependent first-order shear deformable shell model based on the MSGT for the axial buckling response of FG cylindrical microshells. Their results revealed that the small scale effect is more prominent for lower values of dimensionless length scale parameter.
The objective of this work is to develop a general size-dependent micro-/nano-scale shell model based on the most general form of Mindlin's strain gradient elasticity theory which includes shear deformation and rotary inertia effects and can be reduced into the models based on the simple forms of the strain gradient theory for specific values of length scale parameters. Hamilton's principle is utilized to derive the size-dependent governing equations and associated boundary conditions. Then, as two case studies, the size-dependent free vibration and axial buckling of circular cylindrical microshells are investigated. The non-dimensional natural frequencies and buckling loads of microshells subjected to SS boundary conditions are analytically determined by using a Navier-type solution. The effects of non-dimensional material length scale parameter, length-to-radius ratio and circumferential mode number on the dimensionless natural frequencies and buckling loads of microshells are fully investigated. Also, the dimensionless natural frequencies and axial buckling loads obtained from MSGT are compared with ones predicted by MCST, SGT and CT.
Section snippets
Mathematical formulation
Depicted in Fig. 1 is a circular cylindrical micro-/nano-shell of uniform thickness h, mid-surface radius R and length L. As shown, a curvilinear coordinate system whose origin is located in the middle surface of microshell is used where parameters x, y and z respectively denote the coordinates of a typical point in the axial, circumferential and radial directions. Based on the first-order shear deformation shell theory (FSDT), the dynamic displacement field can be expressed as follows (
Closed-form solution for axial buckling and free vibration of cylindrical microshells
In this section, as a case study, the axial buckling and free vibration of a circular cylindrical microshell with SS boundary condition at edges x = 0 and x = L are investigated.
According to Eq. (19a), (19b), (19c), (19d), (19e), (19f), (19g), (19h), (19i), (19j), the mathematical expressions corresponding to different sets of boundary conditions for micro- and nano-shells can be selected. In the current study, to show the direct application of the developed size-dependent shell model, a
Results and discussion
In this section, selected numerical results are presented for free vibration and axial buckling of circular cylindrical microshells subjected to SS end conditions and the effects of dimensionless material length scale parameters, length-to-radius ratio and circumferential mode number on the non-dimensional natural frequencies and buckling loads are thoroughly examined.
First of all, to verify the validity of developed formulation, the non-dimensional natural frequencies of a
Conclusion
In this study, a most general form of size-dependent first order shear deformation micro-/nano-scale shell model based on Mindlin's strain gradient elasticity theory was developed. Unlike the classical continuum theory, the proposed model has the capability of accommodating the microstructural size effects through using additional internal length scale parameters. The size-dependent governing equations and corresponding boundary conditions were derived using Hamilton's principle. For specific
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