Determination of the Appropriate Gradient Elasticity Theory for Bending Analysis of Nano-beams by Considering Boundary Conditions Effect

Abstract In the present paper, a critique study on some models available in the literature for bending analysis of nano-beams using the gradient elasticity theory is accomplished. In nonlocal elasticity models of nano-beams, the size effect has not been properly considered in governing equations and boundary conditions. It means that in these models, because of replacing of the size effect with the inertia gradient effect, the size dependency has been ignored in bending analysis of nano-beams. Therefore, as the beam dimensions increase in comparison to its material length scale parameter, the obtained solution based on the gradient elasticity theory (either in the nonlocal elasticity theory or the strain gradient elasticity theory) should converge to the classical elasticity solution. Hence, satisfying of boundary conditions is a crucial point. In this paper, governing equations and boundary conditions are presented based on two gradient elasticity theories (i.e., nonlocal elasticity and strain gradient elasticity theories). Also, boundary conditions in strain gradient elasticity theory are modified based on a dimensional analysis approach. The results indicate that the strain gradient elasticity theory captures the size effect more sensitive in comparison with the nonlocal elasticity theory in bending analysis. In addition, modified boundary conditions in strain gradient elasticity theory can lead to converge the classical solution at large scales. To prove that the boundary conditions of nano-beam have the direct effect on mechanical behavior of structure, the size-dependent Young modulus of carbon nanotube (CNT) is investigated and the results show that the prediction of strain gradient elasticity theory with modified boundary conditions is in a good agreement with experimental results