Calibration of Eringen's small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model

In this paper, we calibrate Eringen's small length scale coefficient e0 for an initially stressed vibrating nonlocal Euler beam via a microstructured beam modelled by some repetitive cells comprising finite rigid segments and elastic rotational springs. By adopting the pseudo-differential operator and Pade's approximation, an analytical solution for the vibration frequency in terms of initial stress may be developed for the microstructured beam model. When comparing this analytical solution with the established exact vibration solution from the nonlocal beam theory, one finds that the calibrated Eringen's small length scale coefficient e0 is given by where σ0 is the initial stress and is the mth mode buckling stress of the corresponding local Euler beam. It is shown that e0 varies with respect to the initial axial stress, from at the buckling compressive stress to when the axial stress is zero and it monotonically increases with increasing initial tensile stress. The small length scale coefficient e0, however, does not depend on the vibration/buckling mode considered.