Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions

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. 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.