Free vibrations of Timoshenko nanoscale beams based on a hybrid integral strain- and stress-driven nonlocal model

Several non-classical elasticity theories are used for considering the size-dependent behavior of structures at small scales. The nonlocal theory is widely used to reflect the softening behavior of material at small scales, and theories like the strain gradient theory are employed to reflect the hardening behavior. In this article, the most general form of integral strain- and stress-driven nonlocal models with two nonlocal parameters is developed which is able to consider both hardening and softening influences simultaneously. To this end, it is considered that the stress field at the entire points of the domain is a function of strain field of the entire points of the domain. The free vibration problem of first-order shear deformable beams is solved herein. The integral form of governing equations and associated boundary conditions are obtained first, and then directly solved in a numerical approach. Through developing an efficient matrix formulation and using differential and integral matrix operators, the discretized governing equations are obtained. The simultaneous effects of strain- and stress-driven nonlocal parameters on the natural frequencies of fully clamped, fully simply-supported, and clamped-free nanobeams are investigated. The results indicate that the paradox related to the behavior of clamped-free nanobeams is resolved using the presented integral nonlocal formulation. Also, it is revealed that it is possible to find some specific values of nonlocal parameters at which the prediction of hybrid nonlocal model coincides with that of classical elasticity theory.