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

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.