Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory

Applied Physics Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory Milad Hemmatnezhad 0 Reza Ansari 1 0 Faculty of Mechanical Engineering , Takestan Branch , Islamic Azad University , Takestan , Iran 1 Department of Mechanical Engineering, University of Guilan , P.O. Box 3756, Rasht , Iran The present paper is concerned with the free vibration analysis of double-walled carbon nanotubes embedded in an elastic medium and based on Eringen's nonlocal elasticity theory. The effects of the transverse shear deformation and rotary inertia are included according to the Timoshenko beam theory. The governing equations of motion which are coupled with each other via the van der Waals interlayer forces have been derived using Hamilton's principle. The thermal effect is also incorporated into the formulation. Using the statically exact beam element with displacement fields based on the first order shear deformation theory, the finite element method is employed to discretize the coupled governing equations which are then solved to find the natural frequencies. The effects of the small scale parameter, boundary conditions, thermal effect, changes in material constant of the surrounding elastic medium, and geometric parameters on the vibration characteristics are investigated. Furthermore, our analysis includes nonlocal double-walled carbon nanotubes with different boundary conditions between inner and outer tubes which seem to be scarcely considered in the literature, and the corresponding given results for this case can be considered as a benchmark for further studies. Comparison of the present numerical results with those from the open literature shows an excellent agreement. Vibration; Double-walled carbon nanotubes; Nonlocal elasticity; Timoshenko beam theory; Finite element method - Introduction Carbon nanotubes (CNTs) have occupied the chief topic of research in nanotechnology since they were first discovered by Iijima [1] in 1991. During the past two decades, research on CNTs increased as reflected by extensive number of publications devoted to synthesis, fundamentals, and applications of these nanostructured materials. Their unique physical (mechanical, electrical, and thermal) as well as chemical properties enable them for a large variety of new applications in nanoelectronics, nanodevices, nanocomposites, and so on [2-7]. They possess extraordinary strength which is measured up to 100 times that of steel at one-sixth of the weight [8], as well as superior electrical and thermal conductivities. Until now, a wide range of studies have been conducted on the mechanical behavior of CNTs such as buckling and bending problems using experimental methods and molecular-dynamics (MD) simulations, but performing experiments at the scale of nanometers is very difficult and needs high expenses. Also, the atomistic methods such as MD simulations are dependent to the small-scale modeling. Therefore, developing continuum models which may overcome these restrictions are expected to be the dominant tool for modeling structures at the scale of nanometers and performing analytical analysis of CNTs of larger scales. Recently, many elastic continuum models have been successfully used for studying the bending, buckling, and vibrational behaviors of CNTs including cylindrical shell models [9-12] and beam models [13-21]. The beam models implemented are often developed on the basis of the Euler-Bernoulli theory [13-17] and the Timoshenko beam theory [18-21] which takes the effects of shear deformation and rotary inertia into the consideration. Fu et al. [22] studied the nonlinear vibrations of embedded multiwalled nanotubes, with the inclusion of intertube radial displacements and the internal van der Waals (vdW) forces, using the incremental harmonic balanced method. They only considered CNTs with simply supported end conditions. Xu et al. [23] studied free linear vibrations of double-walled nanotubes (DWNTs) modeled as elastic beams due to different boundary conditions between inner and outer tubes. Related to the work done by Fu et al. [22], Ansari et al. [24] and Ansari and Hemmatnezhad [25] investigated the nonlinear vibrations of single-, double- and triple-walled CNTs on the basis of a multiple-beam model and found the nonlinear frequencies using the homotopy perturbation method and the variational iteration method Recently, they proposed a general finite element formulation for investigating the nonlinear oscillations of DWNTs with different boundary conditions [26]. They also extended the work done by Xu et al. [23] to the large-amplitude vibrations of DWNTs with different boundary conditions between inner and outer tubes. However, the continuum models proposed in all of these works, so-called the classical continuum models, are scale-independent and their application in smallscale nanomaterials are of some concern. In fact, the si (...truncated)