The small length scale effect for a non-local cantilever beam: a paradox solved

Abstract

Non-local continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. This paper presents some simplified non-local elastic beam models, for the bending analyses of small scale rods. Integral-type or gradient non-local models abandon the classical assumption of locality, and admit that stress depends not only on the strain value at that point but also on the strain values of all points on the body. There is a paradox still unresolved at this stage: some bending solutions of integral-based non-local elastic beams have been found to be identical to the classical (local) solution, i.e. the small scale effect is not present at all. One example is the Euler-Bernoulli cantilever nanobeam model with a point load which has application in microelectromechanical systems and nanoelectromechanical systems as an actuator. In this paper, it will be shown that this paradox may be overcome with a gradient elastic model as well as an integral non-local elastic model that is based on combining the local and the non-local curvatures in the constitutive elastic relation. The latter model comprises the classical gradient model and Eringen's integral model, and its application produces small length scale terms in the non-local elastic cantilever beam solution.

Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

“…Nonlocal structural models, including nonlocal shell, plate, and beam models, have developed rapidly and a detailed survey of this line of investigation is presented in the review papers by Askes and Aifantis (2011) and Arash and Wang (2012). The application of such a nonlocal theory to beam structures like CNTs is extensively investigated including bending, buckling, vibration, and wave propagation analysis (see, for instance, Peddieson et al (2003) Challamel and Wang (2008)). There are two nonlocal theories for the vibration of Timoshenko beams, i.e., (1) the Reddy theory (Reddy and Pang, 2008) that allows for length scale coefficient terms in both the normal stress-strain relation and the shear stress-strain relation and (2) the Wang theory that has the length scale coefficient term in the normal stress-strain relation only.…”

A new displacement-based framework for non-local Timoshenko beams

“…For instance, Eringen's integral theory has been used by Peddieson et al [30], Wang and Shindo [31], Civalek and Demir [32] to derive non-local EB beam models, by Wang and Liew [33], Wang and et al [34] to derive non-local TM beam models, the latter in conjunction with the principle of virtual work. A non-local EB beam model has been presented by Challamel and Wang [35] based on a gradient elastic model and a non-local integral elastic model, where the constitutive relation is expressed by combining local and nonlocal curvatures. A non-local EB model has been also presented by McFarland and Colton [13] based on a micropolar elasticity constitutive law.…”

Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials

“…Nonlocal elasticity theory for the first time was introduced by Eringen (1972). Recent literature shows that the nonlocal elasticity theory which includes small scale effect arising at nanoscale level is being increasingly used for reliable and quick analysis of nanostructures Wang 2005;Zhang et al 2005;Shen 2011;Lu et al 2006;Challamel and Wang 2008) like nanobeams, nanoplates, nanorings, carbon nanotubes, graphenes, nanoswitches and microtubules. Aydogdu (2009) proposed a general nonlocal beam theory to study bending, buckling and free vibration of nanobeams.…”