Obtaining Eringen׳s length scale coefficient for vibrating nonlocal beams via continualization method

Eringen׳s length scale coefficients e0e0 are presented herein for initially stressed vibrating nonlocal beams with various boundary conditions. The coefficients are obtained by applying the continualization method to the discrete equations of microstructured beam models. When compared to another continualized approach that is based on the Padé approximant, the proposed continualization method (which is based on assuming continuous exponential displacement functions for the discrete displacement field) is a more straightforward approach for solution. Unlike the Padé approximant approach, this latter method needs no sophisticated mathematical manipulations for obtaining the continuous equation. Moreover, the continualization method allows one to directly solve the discrete governing equations for vibration of microstructured beams in the discrete domain rather than solving an approximated continuous equation obtained via the Padé approximant. By using this method, it is found that e0=1/12 for the buckling case of beams and e0=1/6 for the vibration case for any combination of end conditions. However, the coefficients are not the same in the presence of an initial stress for vibrating beams with different boundary conditions.