Modeling of Non-Local Beam Theories for Vibratory and Buckling Problems of Nano-Tubes

Abstract

This article is concerned with the study of vibratory and buckling problems associated to micro and nano-beams modeled with first-order shear approaches in the context of non-local elasticity.
Normally this kind of studies is performed in the context of a single Timosheko beam model or even coupled with axial motions and also with extended Bernoulli-Euler beam models, among others. In the present study the slender structure is conceived as a thin-walled beam with motion in three axes, i.e. bending in two directions and the axial motion and twisting. Thus a model for both isotropic and functionally graded beams is developed. Then the model is employed to calculate buckling loads and vibration frequencies with or without the presence of initial stresses. The derivation procedure of the non-local beam model follows three steps. First: the differential equations of a local model are derived in terms of internal forces; second: the internal forces are re-deduced in terms of the non-local constitutive equations which are substituted, as a third step, in the previous differential equations giving the differential equations of the non-local beam model. The power series method is employed to offer an analytical response for basic buckling and vibration problems, as well as the finite element method in used to calculate the vibratory and buckling features for more complex configurations. A number of examples are presented in order to show the influence of the non-local formulation in the buckling response and the vibratory patters of nano/microbeams.