Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory

Abstract

In the present article, the total potential energy principle and the nonlocal integral elasticity theory have been used to develop a novel finite element method for studying the free vibration behavior of nano-scaled beams. The formulations are based on Euler-Bernoulli beam theory and this method is able to properly analyze the free vibration of beams with various boundary conditions. By implementing the variational statements, the eigenvalue problem of the free vibration is obtained. The validation investigation is pursued by comparing the results of the current study with those available in the literature. The effects of nonlocal parameter, geometry parameters and boundary conditions on the free vibration of the Euler-Bernoulli beam are then studied.

Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach

“…However, in order to compare the results with those of nonlocal differential elasticity, the kernel of equation (3) will be considered in the current analysis. In addition, the nonlocal constitutive equation has been assumed as a two-phase material (local and nonlocal elasticity), in line with some studies published in the literature (Polizzotto, 2001; Pisano et al., 2009; Naghinejad and Ovesy, 2017). For this purpose, the nonlocal kernel is expressed as follows where ζ1 and ζ2 are positive constants which refer to the local and nonlocal fractions of the material, respectively (ζ1+ζ2=1), δ is the Dirac delta function and λ0 is a normalization-factor expressed as …”

“…It is noted that the differential form can be achieved only for some particular kernels. Furthermore, equation (4) is valid when x is far enough from the boundaries or the limits of integration are infinity (Taghizadeh et al., 2016; Naghinejad and Ovesy, 2017). Apart from the latter conditions, this method would lead to some inaccurate results.…”

“…Apart from the latter conditions, this method would lead to some inaccurate results. Moreover, in the case of nano-plates, applying free and simply-supported boundary conditions may cause some ambiguity in considering the differential nonlocal equations, and just in the case of clamped boundary conditions these problems will not arise (Phadikar and Pradhan, 2010; Taghizadeh et al., 2016; Naghinejad and Ovesy, 2017). It is clear that working with the differential equations is simpler than the more general form equation (1).…”

“…Results were compared with those of nonlocal differential elasticity theory. Naghinejad and Ovesy (2017) developed a FE method considering the nonlocal integral elasticity theory and Euler–Bernoulli beam theory to investigate the vibration behavior of nano-scaled beams. They compared the results with those of nonlocal differential elasticity theory available in the literature.…”

Finite Element Model of Functionally Graded Nanobeam for Free Vibration Analysis

“…For this reason, some researchers have been used some higher order theories that take into account small-scale effect analysis of micro and nano structures [3][4][5]. Among higher order theories, nonlocal elasticity theory [6] have been widely studied recently [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Ebrahimi et al [2] presented the applicability of differential transformation method (DTM) in investigations on vibrational characteristics of FG size-dependent nanobeams.…”