Free vibration of nonlocal beams: Boundary value problem and a calibration method

Abstract

The paradox of abnormal frequencies exists for free vibration of nonlocal cantilever beams. This work is concerned with solving this paradox within the framework of nonlocal strain gradient theory. The inconsistences of the reported boundary conditions are updated by using the weighted residual method. Then, the closed-form solutions for free vibration of cantilever beams is obtained in terms of the inconsistences of the reported boundary value problems (BVPs) and the reformulated BVPs. The softening phenomenon is captured by the updated numerical results. A method for calibrating the size-effect parameters is firstly proposed in this work. The numerical results show that the present model is capable for capturing the size-dependent mechanical properties of materials, which exhibit either the stiffness-hardening effect or the stiffness-softening effect, depending upon the relative values of the two size-effect parameters.