A new nonlocal bending model for Euler–Bernoulli nanobeams

• A new variational formulation of nonlocal Euler–Bernoulli nanobeams is provided.
• Small-scale effects are described by two nonlocal parameters.
• The proposed nonlocal model can make a nanobeam stiffer or not.
• Eringen and gradient models are recovered as special cases.
• Closed-form solutions of a nanocantilever are given.

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.