Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory

Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory

“…This simplified constitutive model is similar to that derived by some gradient elasticity theories or the hybrid nonlocal beam model (see, e.g., Aifantis, 2011;Challamel 2013;Challamel, Rakotomanana, & Le Marrec, 2009;Güven, 2014;Wu, Li, & Cao, 2013). It was recently reported by Lim et al (2015) that, the wave propagation of carbon nanotubes modelled using the general constitutive Eq.…”

Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

“…Suppose e = e 0 = e 1 and the nonlocal functions α 0 (x, x , e 0 a) and α 1 (x, x , e 1 a) satisfy the conditions in Eringen [1], thus The general constitutive relation (5) is also similar to some constitutive relation through different methods (see, e.g., [35,37,[47][48][49]). It should be noted that the general constitutive relation (5) can be reduced to that of the nonlocal continuum theory [1] by setting l = 0 and that of strain gradient theory [50,51] by setting ea = 0.…”

Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory

“…(37) is identical to [78,89,92] and Eq. (38) is identical to that developed by Wu et al [81] based on a hybrid Rayleigh beam theory. Under such case, the asymptotic phase velocity for both the longitudinal and transverse waves can be reduced to…”

“…However, when β > 0.1 1/nm, different continuum theories produce different trends. The phenomenon is also detected the transverse waves in a CNT [81] and the longitudinal waves in an axial bar [92]. This is because that the phase velocity is not sensitive to nanostructural properties when the wavelength is large (low wave numbers).…”

“…This simplified constitutive model is similar to some gradient elasticity theories (see, e.g., [78][79][80][81][82]). It is worth mentioning that Eq.…”