On the vibrations of a mechanically-based non-local beam model

The vibration problem of a Timoshenko non-local beam is addressed. The beam model involves assuming that the equilibrium of each volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are taken as depending on the product of the interacting volume elements and on their relative displacement through a material-dependent distance-decaying function. To derive the motion equations and the related mechanical boundary conditions, the Hamilton's principle is applied