Application of nonlocal strain gradient theory for the analysis of bandgap formation in metamaterial nanobeams

Abstract

The aim of this paper is to investigate bandgap formation in an Euler-Bernoulli nanobeam using the nonlocal strain gradient theory. Such a model enables the capture of both softening and stiffening size-dependent behavior of the nanobeams using two length-scale parameters, namely, a nonlocal parameter and a strain gradient parameter. Hamilton's principle is used to develop the normalized equation of motion yielding a sixth order differential equation. The nanobeam is assumed to be infinitely long and the dispersion of an elastic wave in a single representative periodic unit cell of the structure is used to estimate the bandgap edge frequencies. To this end, two unit cell approaches are employed to obtain the dispersion relations, namely, the homogenization and transfer matrix approaches. Both methods yield similar dispersion curves and showed good agreement with classical cases from the literature. A parametric study was carried out to investigate the effect of several parameters on the bandgap formation in the metamaterial nanobeams. These parameters include, the nonlocal parameter, the strain gradient parameter, the length of the unit cell and the resonator mass to unit cell mass ratio. The study conducted using a purely strain gradient model showed a comparable response to that of the classical macro problem with the usual hardening when increasing the value of the strain gradient length scale. Nonlocal and nonlocal strain gradient models introduced coupling between parameters of the study with the occasional introduction of non-monotonic dispersion curves. Such non-monotonic behavior is believed to be characteristic of large values of the nonlocal length-scale.