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.摘要 本文针对现有文献中基于梯度弹性理论（gradient elasticity theory）开展纳米梁弯曲分析的若干模型开展批判性评述研究。在纳米梁的非局部弹性理论（nonlocal elasticity theory）模型中，控制方程与边界条件未能恰当考虑尺寸效应。这意味着此类模型中，由于将尺寸效应替换为惯性梯度效应，纳米梁弯曲分析中对尺寸相关性的考量被忽略。因此，当梁的几何尺寸相较于其材料长度尺度参数增大时，基于梯度弹性理论（无论是非局部弹性理论还是应变梯度弹性理论（strain gradient elasticity theory））得到的解应当收敛于经典弹性解。故而，边界条件的满足是关键要点。本文基于两类梯度弹性理论（即非局部弹性理论与应变梯度弹性理论）推导得到控制方程与边界条件，同时采用量纲分析方法对应变梯度弹性理论中的边界条件进行了修正。分析结果表明，在弯曲分析中，相较于非局部弹性理论，应变梯度弹性理论对尺寸效应的捕捉更为灵敏；此外，经修正后的应变梯度弹性理论边界条件可确保大尺寸尺度下解收敛于经典弹性解。为验证纳米梁边界条件对结构力学行为具有直接影响，本文针对碳纳米管(CNT)的尺寸相关杨氏模量开展了研究，结果表明，采用修正后边界条件的应变梯度弹性理论预测结果与实验结果吻合良好。