Free vibrations of elastic beams by modified nonlocal strain gradient theory

Size-dependent axial and flexural free vibrations of Bernoulli-Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018). The corresponding elastodynamic problem, with the natural constitutive boundary conditions, is solved by an effective analytical methodology. Axial and flexural fundamental frequencies are determined for cantilever and fully-clamped nano-beams. Effects of nonlocal and gradient scale parameters on fundamental frequencies are examined and compared with those obtained by Eringen's local/nonlocal mixture model. New benchmarks are found for vibrations of beams. The adopted nonlocal strain gradient model, with the appropriate constitutive boundary conditions, is capable of capturing both softening and stiffening dynamical responses. Accordingly, it provides an advantageous approach for design and optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).