Nonlinear post‐buckling analysis of viscoelastic nano‐scaled beams by nonlocal integral finite element method

Abstract

The viscoelastic buckling and nonlinear post‐buckling behavior of nano‐scaled beams are analyzed using the nonlocal integral elasticity theory. Eringen's nonlocal theory is one of the well‐known and popular size‐dependent theories, which has been used by several researchers to study the mechanical behavior of, mostly, the elastic nanostructures. A finite element method is developed using Hamilton's principle based on the two‐phase nonlocal integral theory and taking into account the buckling related terms and viscoelastic effects. The corresponding formulations are derived by implementing the Euler–Bernoulli beam theory and Kelvin–Voigt viscoelastic model. Furthermore, for analyzing the post‐buckling and viscoelastic buckling behavior, the nonlinear strains, and initial displacement (imperfection) have been considered. By employing the variational relations, the governing equations are obtained and then solved numerically using the finite difference and Newton–Raphson methods. Because of the finite element nature of the current method, various boundary conditions can be properly implemented. The results of the current study are compared with those available in the literature and the effects of nonlocal parameter, viscoelastic parameter, axial compressive load and boundary conditions on the viscoelastic buckling, and postbuckling behavior have been investigated.

Nonlocal Elasticity for Nanostructures: A Review of Recent Achievements

“…To overcome the issues emerging from the application of the strain-driven model, a mixture model of elasticity was proposed by Eringen in [12] and then restored in [14,15]. Eringen's two-phase model was proposed in [46] for buckling analysis of slender beams, free vibration analyses of Euler-Bernoulli curved beams have been performed in [47], post-buckling of viscoelastic nanobeams has been analysed in [48], bending problem of two-phase elastic structures has been addressed in [49], mixture nonlocal integral theories have been adopted in [50] for functionally graded Timoshenko beams.…”

Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions

“…In this section of the paper, the solution approach applied to obtain the buckling loads of the FG nanostructure, whose buckling equation was presented in the previous section, will be presented. In the literature, it is possible to see many solution methods adopted by researchers [8,9,13,31,[63][64][65][66][67][68][69][70]. In this study, the adopted solution approach is based on the combination of Fourier sine series and Stokes' transform like in refs.…”

Size‐dependent Levinson beam theory for thermal vibration of a nanobeam with deformable boundary conditions

“…[25][26][27][28][29][30][31][32] modified couple stress theory have been considered to show the responses of scale-dependent nano/micro elements. Nonlocal elasticity theory has been used by researchers [33][34][35][36][37][38][39][40][41][42] to investigate the different effects on the nano/micro-scaled beams, rods, etc. Strain gradient elasticity has been adopted the buckling of nano/microbeams in refs.…”