Isogeometric Vibration Analysis of Small-Scale Timoshenko Beams Based on the Most Comprehensive Size-Dependent Theory

By taking the nonlocal and strain gradient effects into account, the vibrational behavior of Timoshenko micro- and nano-beams is studied in this paper based on a novel size-dependent model. The nonlocal effects are captured using both differential and integral formulations of Eringen’s nonlocal elasticity theory. Moreover, the strain gradient influences are incorporated into the model according to the most general form of strain gradient theory which can be reduced to simpler strain gradient-based theories such as modified strain gradient and modified couple stress theories. Hamilton’s principle is employed to derive the variational form of governing equations. The isogeometric analysis (IGA) is then utilized for the solution approach. Comprehensive results for the effects of small scale and nonlocal parameters on the natural frequencies of beams under various types of boundary conditions are given and discussed. It is revealed that using the differential nonlocal strain gradient model for computing the fundamental frequency of cantilevers leads to paradoxical results, and one must recourse to the integral nonlocal strain gradient model to obtain consistent results.