Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory

Abstract In this work the problem of the in-plane free vibrations (axial and bending) of a Bernoulli–Euler nanobeam using the mixed local/nonlocal Eringen elasticity theory is studied. The natural frequencies of vibration have been analytically obtained solving two uncoupled integro-differential eigenvalue problems, which are properly transformed in differential eigenvalue problems. Different kinds of end supports have been considered, and the influence of both mixture parameter and length scale has been analysed. The results show the softening effect of the Eringen’s nonlocality, which is more pronounced as the local phase fraction decreases. A large number of papers devoted to the dynamics of Bernoulli–Euler beams considering the fully nonlocal Eringen elasticity theory has been previously published. However, as recently stated by Romano, Barretta, Diaco and de Sciarra (2017), the problem is ill-posed in general, and the existence of a solution is an exception, the rule being non-existence. Nevertheless, the presence of a local term in the constitutive equation, leading to the two-phase formulation, renders the problem well-posed. To the best knowledge of the authors, this is the first time an exact solution is presented for a dynamic problem involving structures with constitutive equations corresponding to nonlocal integral Eringen’s elasticity.