Application of nonlocal continuum models to nanotechnology

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

“…The solution in Eq. ͑26͒ is the same as that obtained by Peddieson et al 4 The two solutions match, although a lower-order governing differential equation was employed by Peddieson et al, 4 because the boundary conditions are simply supported and all constants of integration vanish. There are cases where the constants of integration do not vanish and therefore the solutions are not the same.…”

“…There are cases where the constants of integration do not vanish and therefore the solutions are not the same. In some cases, the solutions of Peddieson et al 4 are less interpretable because the small-scale effect disappears.…”

“…Since the first publication of nonlocal elasticity theory by Eringen and his associate, 1-3 many articles have been published on the application of this model in nanomechanics, particularly in the early 21st century, such as Peddieson et al, 4 Sudak, 5 Zhang et al, 6 Wang, 7,8 Lu et al, 9 Xu, 10 Wang and Hu, 11 etc. Almost all articles presented a simplified nonlocal beam model by assuming that the beam midplane is governed by a second-order ordinary differential equation, while in the transverse direction the classical Euler-Bernoulli beam model or the Timoshenko beam model applies.…”

Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics

“…For such small beamlike structures, many researchers have applied the nonlocal elasticity concept for the bending, buckling and vibration analyses. 2,6,[8][9][10][11][12][13] Recently, Wang et al 7 derived the closed form solutions for the free vibration problem of nanobeams using the nonlocal Timoshenko beam theory. The solutions may be specialized for CNT by assuming that it is a hollow cylinder, i.e., the CNT is viewed as a thin-walled circular tube.…”

Free vibration of nanorings/arches based on nonlocal elasticity

“…In order to account for small length scale effect, Eringen 30 nonlocal elasticity theory is adopted. His nonlocal theory has been widely used to derive bending solutions, buckling loads, vibration frequencies, and phase velocities of micro-and nanobeams, rods and tubes ͑for example, see papers by Peddieson et al, 31 , Sudak, 32 Wang et al, 33 Zhang et al, 34 Wang and Varadan, 35 Wang and Liew, 36 Lu et al, 37,38 Lim et al, 39 Xu, 40 Wang et al, 41 Wang, 42 Duan et al, 43,44 Wang et al, 45 and Reddy 46 ͒. The effect of defects represented by a hinge and rotational restraint and elastic boundary conditions on vibrations of nanorings/arches are also considered.…”