Abstract

Two frameworks of the nonlocal integral elasticity and the modified strain gradient theory are consistently merged to conceive the nonlocal modified gradient theory. The established augmented continuum theory is applied to a Timoshenko–Ehrenfest beam model. Nanoscopic effects of the dilatation, the deviatoric stretch, and the symmetric rotation gradients together with the nonlocality are suitably accommodated. The integral convolutions of the constitutive law are restored with the equivalent differential model subject to the nonclassical boundary conditions. Both the elastostatic and elastodynamic flexural responses of the nano-sized beam are rigorously investigated and the well posedness of the nonlocal modified gradient problems on bounded structural domains is confirmed. The analytical solution of the phase velocity of flexural waves and the deflection and the rotation fields of the nano-beam is detected and numerically illustrated. The transverse wave propagation in carbon nanotubes is furthermore reconstructed and validated by the molecular dynamics simulation data. Being accomplished in revealing both the stiffening and softening structural responses at nano-scale, the proposed nonlocal modified gradient theory can be beneficially implemented for nanoscopic examination of the static and dynamic behaviors of stubby nano-sized elastic beams.

中文翻译：

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非局部修饰梯度纳米束的弯曲力学

摘要

非局部积分弹性和修正应变梯度理论的两个框架被一致地合并以构想非局部修正梯度理论。已建立的增强连续谱理论被应用于Timoshenko-Ehrenfest光束模型。适当地适应了扩张，偏向拉伸和对称旋转梯度以及非局部性的纳米效应。服从非经典边界条件的等效微分模型恢复了本构律的积分卷积。严格研究了纳米梁的弹性静力和弹性动力弯曲响应，并确认了非局部修正梯度问题在有界结构域上的适定性。检测并数值表示了弯曲波的相速度以及纳米束的偏转和旋转场的解析解。碳纳米管中的横向波传播还可以通过分子动力学模拟数据进行重建和验证。通过揭示纳米级结构的刚度和软化结构的响应，提出的非局部修正梯度理论可以有益地用于对粗短纳米级弹性梁的静态和动态行为进行纳米级检查。