Variationally consistent dynamics of nonlocal gradient elastic beams

The variational static formulation contributed in [International Journal of Engineering Science 143, 73-91 (2019)] is generalized in the present paper to model axial and flexural dynamic behaviors of elastic nano-beams by nonlocal strain and stress gradient approaches. Appropriate forms of non-standard boundary conditions are detected and properly prescribed. Equivalence between differential laws and integral convolutions is elucidated and invoked to analytically evaluate size-dependent axial and flexural fundamental frequencies of cantilever and fully-clamped beams which significantly characterize new-generation nano-actuators. The proposed methodology and ensuing results are tested by pertinent outcomes in literature. Advantageously, in comparison with available nonlocal gradient models, the developed formulation of elasticity leads to well-posed dynamic structural problems of nano-mechanics. Outcomes obtained by the well-established strain-driven and stress-driven nonlocal and gradient theories of Engineering Science are recovered under special ad hoc assumptions. The present study offers a simple and effective strategy to predict peculiar stiffening and softening dynamic responses of nano-scaled components of advanced technological devices, such as Nano-Electro-Mechanical-Systems (NEMS), and modern composite nano-structures.