Journal of Mechanical Science and Technology 25 (9) (2011) 2365~2375 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0711-6 Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions † S. Sahmani * and R. Ansari Department of Mechanical Engineering, University of Guilan, P.O.Box 3756, Rasht, Iran (Manuscript Received September 15, 2010; Revised April 18, 2011; Accepted April 21, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract Buckling analysis of nanobeams is investigated using nonlocal continuum beam models of the different classical beam theories namely as Euler-Bernoulli beam theory (EBT), Timoshenko beam theory (TBT), and Levinson beam theory (LBT). To this end, Eringen’s equa- tions of nonlocal elasticity are incorporated into the classical beam theories for buckling of nanobeams with rectangular cross-section. In contrast to the classical theories, the nonlocal elastic beam models developed here have the capability to predict critical buckling loads that allowing for the inclusion of size effects. The values of critical buckling loads corresponding to four commonly used boundary con- ditions are obtained using state-space method. The results are presented for different geometric parameters, boundary conditions, and values of nonlocal parameter to show the effects of each of them in detail. Then the results are fitted with those of molecular dynamics simulations through a nonlinear least square fitting procedure to find the appropriate values of nonlocal parameter for the buckling analy- sis of nanobeams relevant to each type of nonlocal beam model and boundary conditions.analysis. Keywords: Nanomechanics; Nanobeams; Nonlocal elasticity parameter; Beam theory; State-space method high ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Due to exceptionally good physical, mechanical, and elec- trical properties [1-6], nano-sized structures have attracted much investment to develop innovatory applications in a wide range of disciplines. To accomplish the design of nanostruc- tures and systems, an essential study of their mechanical be- havior seems necessary. Nanomechanics is a branch of me- chanics in which the mechanical properties and behavior of structures at nanoscale are investigated. Modified continuum models have been the subject of much attention in nanomechanics due to their computational effi- ciency and the capability to produce accurate results which are comparable to those of atomistic models [7-11]. One approach for including nanoscale size-effects into the classical contin- uum mechanics is the use of modified continuum models based on the concept of nonlocal elasticity. Nonlocal contin- uum model has gained much popularity among the researchers because of its efficiency as well as simplicity to analyze the behavior of various nanostructures [11-21]. It has been ob- served that the mechanical properties of nanostrustures pre- dicted by nonlocal continuum models are different from those previously obtained by the classical continuum mechanics which shows the size-effects on the behavior of structures at nanoscale. Based on the above introduction, it seems that size-effects consideration in the analysis of nanobeams is necessary. In this work, different nonlocal beam models corresponding to the different classical beam theories [22-24] are presented on the basis of Eringen’s equations of nonlocal elasticity [25] to predict the buckling behavior of nanobeams with four com- monly used boundary conditions. State-space method is used to solve the governing differential equations for each type of nonlocal beam model with different boundary conditions. Various numerical results are given to show the influences of boundary conditions, aspect ratio, and values of nonlocal con- stant, separately. Then the results are matched with those of molecular dynamics simulations which are available in the literature to extract the correct values of nonlocal parameter corresponding to each type of nonlocal beam model and boundary conditions. 2. Overview of different beam theories 2.1 Introduction There are various types of beam theory to describe the be- havior of beams. Consider a straight uniform beam with the length L and rectangular cross-section of thickness h which is † This paper was recommended for publication in revised form by Editor Maenghyo Cho * Corresponding author. Tel.: +98 131 6690276, Fax.: +98 131 6690276 E-mail address: sasahmani@yahoo.com © KSME & Springer 2011