Nonlocal integral approach to the dynamical response of nanobeams

Nonlocal continuum theories have been formulated and evolved in our era to explain size effect phenomena in micro– and nano– structures. The differential approach of nonlocal Euler-Bernoulli beam theory (NEBBT) has widely used to simulate the static and dynamical response of carbon nanotubes (CNTs) and nanobeams. However, this approach often gives rises to paradoxes, such as the calculation of the fundamental eigenfrequency for the case of a cantilever beam. Another disadvantage is that the nonlocal differential beam models are not capable of leading to the formulation of quadratic energy functionals. On the other hand, recent studies attest to the integral approach of NEBBT overcomes the aforementioned disadvantage for the static case. This work revolves around the dynamical response of nanobeams by employing the nonlocal integral form for the first time. In particular, we formulate the quadratic energy functional and then deduce the nonlocal integral Euler-Bernoulli equation of motion by using Hamilton's principle. Our overall research objective is to investigate the free vibration problem for three engineering benchmark cases (a cantilever, a simply supported and a clamped-clamped nanobeam, respectively). Carrying out finite element method (FEM) to our problems, the eigenfrequencies of the nonlocal integral model take smaller values than eigenfrequencies of classic-local and the nonlocal differential model, respectively, which implies that the behavior of the nonlocal integral model appears to be more softening than the behavior of the two other models. It is crucial that the nonlocal integral model does not give rise to paradoxes as the nonlocal differential model does. Our results are significant and capable of triggering the study of nanostructures, such as CNTs, biomaterials, micro- and nano- electro mechanical systems (MEMS and NEMS). •We formulate the nonlocal integral EB beam (NIEBB) model for dynamical problems.•The finite element method is employed to solve the dynamical problems.•The NIEBB model is more softening than classic and nonlocal differential model.•The NIEBB model does not give rise to paradoxes for the case of a cantilever beam.•The use of the NIEBB model becomes a viable option for engineering applications.