Element-free Galerkin model of nano-beams considering strain gradient elasticity

This paper deals with the investigation of the static bending response of nano-beams upon application of a mechanical load, considering the effect of gradient elasticity. The gradient elasticity is modelled using the strain gradient elasticity model with three independent material length constants. For the numerical modelling of the structural response, the higher-order displacement theory is used. The element-free Galerkin method is employed to interpolate the displacement field in the continuum in terms of the nodal displacements. Thereafter, using the variational principle, the governing equations of motion in terms of the nodal displacements are derived in the algebraic form. The model developed in this paper is validated by comparison with the exact solutions available in the literature, for the specific case of a simply supported beam under a sinusoidally distributed load. Following this, the numerical model is solved for various other boundary conditions and loading functions. The effect of the strain gradient elasticity is studied for different aspect ratios, and the results are compared. The reduction of the bending response due to the strain gradient effect is compared for each case. A comparison of the effects of each of the material length scales, corresponding to different forms of the higher-order gradients, over the structural response is performed. The present study reveals that the strain gradient elasticity must be considered for an accurate analysis of nanostructures.