Nonlinear Dyn (2012) 67:373–383 DOI 10.1007/s11071-011-9985-6 ORIGINAL PAPER Nonlinear finite element analysis for vibrations of double-walled carbon nanotubes R. Ansari · M. Hemmatnezhad Received: 9 June 2010 / Accepted: 10 February 2011 / Published online: 2 March 2011 © Springer Science+Business Media B.V. 2011 Abstract The large-amplitude free vibration analy- sis of double-walled carbon nanotubes embedded in an elastic medium is investigated by means of a fi- nite element formulation. A double-beam model is utilized in which the governing equations of layers are coupled with each other via the van der Waals interlayer forces. Von-Karman type nonlinear strain- displacement relationships are employed where the ends of the nanotube are constrained to move axially. The amplitude-frequency response curves for large- amplitude free vibrations of single-walled and double- walled carbon nanotubes with arbitrary boundary con- ditions are graphically illustrated. The effects of ma- terial constant of the surrounding elastic medium and the geometric parameters on the vibration characteris- tics are investigated. For a double-walled carbon nan- otube with different boundary conditions between in- ner and outer tubes, the nonlinear frequencies are ob- tained apparently for the first time. Comparison of the results with those from the open literature is made for the amplitude-frequency curves where possible. This comparison illustrates that the present scheme yields very accurate results in predicting the nonlinear fre- quencies. R. Ansari ( ) · M. Hemmatnezhad Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran e-mail: r_ansari@guilan.ac.ir Keywords Large-amplitude vibration · Carbon nanotubes · Frequency response · Boundary conditions 1 Introduction The field of nanotechnology is growing at a phenom- enal pace as reflected by increasing number of pub- lications devoted to synthesis, fundamentals, and ap- plications of nanostructured materials. This is because of unique physical (mechanical, electrical, and ther- mal) as well as chemical properties of nanomateri- als. Amongst the materials at the scale of nanome- ters, carbon nanotubes (CNTs) discovered by Iijima [1] in 1991 promise a large variety of new applica- tions in nanoelectronics, nanodevices, nanocompos- ites, and so on [2–9]. A comprehensive review on their properties and industrial applications can be found in [10]. They exhibit extraordinary strength which is measured up to 100 times that of steel at one-sixth of the weight [11], as well as superior electrical and thermal conductivities. Consequently, they are consid- ered as one of the most commonly mentioned building blocks of nanotechnology in the early twenty-first cen- tury. The buckling and bending problems of CNTs have been widely studied using experimental methods and molecular-dynamics (MD) simulations [12–18]. Al- though MD simulations have generated abundant re- sults for understanding the behavior of structures in