On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory

In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that, for a beam, a meaningful energy bounded solution of the purely nonlocal theory may still be defined as the limit solution of the two-phase nonlocal theory. For this, we consider the problem of free vibrations of a flexural beam under the two-phase theory of nonlocal elasticity with an exponential kernel, in the presence of rotational inertia. After recasting the integro-differential governing equation and the boundary conditions into purely differential form, a singularly perturbed problem is met that is associated with a pair of end boundary layers. A multi-parametric asymptotic solution in terms of size-effect and local fraction is presented for the eigenfrequencies as well as for the eigenforms for a variety of boundary conditions. It is found that, for simply supported end, the weakest boundary layer is formed and, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free end conditions bring a strong boundary layer and eigenfrequencies are heavily affected by rotational inertia, even for the lowest mode, in a manner opposite to that brought by nonlocality. Remarkably, all asymptotic solutions admit a well defined and energy bounded limit as the local fraction vanishes and the purely nonlocal model is retrieved. Therefore, we may define this limiting case as the proper solution of the purely nonlocal model for a beam. Finally, numerical results support the accuracy of the proposed asymptotic approach.