Vol.:(0123456789) 1 3 Appl. Phys. A (2017) 123:260 DOI 10.1007/s00339-017-0811-y Vibration analysis of single-walled carbon peapods based on nonlocal Timoshenko beam theory Majid Ghadiri 1  · Hamid Hajbarati 1  · Mohsen Sai 1   Received: 8 November 2016 / Accepted: 25 January 2017 © Springer-Verlag Berlin Heidelberg 2017 single-walled carbon nanotube (SWCNT) has surrounded a cylindrical empty space, and thus, it is an interesting idea to attempt to insert molecules into this cavity. Many applica- tions of SWNTs rely on the ability to use or controllably modify their natural properties by manipulating or choos- ing their structures, which have received many experimen- tal and theoretical attentions in the past years [7–13]. When size of a vibrational system becomes compara- ble to the dimension of its material structure, size efect is seen. As experiments and molecular dynamics simulations at micro/nanoscale are costly to operate and the classi- cal continuum theory cannot consider size efect. Conse- quently, various higher order continuum models, such as strain gradient theory, modiied couple stress theory, nonlo- cal theory, and etc, are proposed. These models employ the intrinsic parameters in macro-structure constitutive equa- tions to an accounting of size dependent. The couple stress theory has been developed by Toupin [14] and Mindlin and Tiersten [15] which contains two higher order material con- stants besides the Lame constants. The two additional con- stants are related to the material structures. As it is diicult to specify the microstructural material length scale param- eters, Yang et al. [16] originated modiied couple stress the- ory. Based on the modiied couple stress theory, the strain energy is considered as a quadratic function of the strain tensor and symmetric part of the curvature tensor. In this theory, only a higher order material constant is considered. The couple stress theory is a class of the strain gradient theory. The strain gradient theory was irst introduced by Mindin [17, 18] in which the strain energy density has been exhibited to be a function of the symmetric strain tensor, the dilatation gradient vector, the deviatoric stretch gradi- ent tensor, and the symmetric rotation gradient tensor. This theory includes ive higher order material constants. The nonlocal elasticity theory of Eringen [19, 20] that contains Abstract In this article, vibration behavior of single- walled carbon nanotube encapsulating C 60 molecules is studied using the Eringen’s nonlocal elasticity theory within the frame work of Timoshenko beam theory. The governing equation and boundary conditions are derived using Hamilton’s principle. It is considered that the nan- opeapod is embedded in an elastic medium and the C 60 molecules are modeled as lumped masses attached to the nanobeam. The Galerkin’s method is applied to determine the natural frequency of the nanobeam with clamped– clamped boundary conditions. Efects of nonlocality, foun- dation stifness, and ratio of the fullerenes’ mass to the nanotube’s mass on the natural frequencies are investigated. In addition, by vanishing efects of shear deformation and rotary inertia, the results based on Euler–Bernoulli beam theory are presented. 1 Introduction In recent years, researchers have focused on nanotechnol- ogy and nanoscience with the discovery of carbon nano- tubes (CNTs) [1] and fullerenes [2]. Fullerenes are a cat- egory of molecules made by carbon atoms. C60 is a known fullerene, and it is the simplest member of this class. Many investigations have been concentrated on the unusual thermal, electrical, and mechanical properties of CNTs and their usages [3–6]. The combination of fullerene and CNT gives carbon nanopeapods (NPPs). An individual * Majid Ghadiri ghadiri@eng.ikiu.ac.ir 1 Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin 3414916818, Iran