Static, free vibration and stability analysis of three-dimensional nano-beams by atomistic refined models accounting for surface free energy effect

This work presents several higher-order atomistic-refined models for the static, free vibration and stability analysis of three-dimensional nano-beams. Stemming from a one-dimensional approach and thanks to a compact notation for the a priori kinematic field approximation over the beam cross-section, the model derivation is made general regardless the approximation order. This latter is a free parameter of the formulation. Several higher-order beam theories can be obtained straightforwardly. Classical beam models, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. The assumed constitutive equations for orthotropic materials account for the surface free energy effect as well as the third-order elastic constants. The resulting stiffness coefficients depend upon the cross-section side length. The governing equations and boundary conditions are variationally obtained through the Principle of Virtual Displacements. A Navier-type, strong form solution is adopted. Simply supported beams are, therefore, investigated. Static, free vibration and buckling analyses are carried out in order to investigate the effect of the cross-section side as well as the crystallographic plane orientation on the mechanical response. Beams with different values of the length-to-thickness ratio are considered. Results are validated in terms of accuracy and computational costs towards three-dimensional FEM solutions. Numerical investigations show the advantages of refined beam models over the classical ones demonstrating that accurate results can be obtained with reduced computational costs.