The higher order mixture nonlocal gradient theory of elasticity is conceived via consistent unification of the higher order stress- and strain-driven mixture nonlocal elasticity and the higher order strain gradient theory. The integro-differential constitutive law is established applying an abstract variational approach and appropriately replaced with the equivalent differential condition subject to nonclassical boundary conditions. The introduced higher order elasticity theory provides, as special cases, a variety of generalized elasticity theories adopted in nanomechanics to assess size effects in continua with nanostructural features. The well-posed higher order mixture nonlocal gradient theory is elucidated and invoked to examine the flexural wave propagation. The closed-form wave propagation relation between the phase velocity and the wave number is analytically derived. The determined wave propagation response and ensuing results are compared and calibrated with the pertinent molecular dynamic simulations. The demonstrated results of the phase velocity of the flexural wave propagation detect new benchmarks for numerical analyses. The proposed higher order size-dependent elasticity approach can be profitably employed in rigorous analysis of pioneering nanotechnological devices.

中文翻译：

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波传播的高阶混合非局部梯度理论

高阶混合非局部弹性梯度理论是通过高阶应力应变驱动的混合非局部弹性和高阶应变梯度理论的一致统一而构想出来的。积分微分本构律是应用抽象变分方法建立的，并适当地替换为服从非经典边界条件的等效微分条件。引入的高阶弹性理论作为特例提供了纳米力学中采用的各种广义弹性理论，以评估具有纳米结构特征的连续体的尺寸效应。阐明并引用适定的高阶混合非局部梯度理论来检查弯曲波的传播。解析推导了相速度与波数之间的闭合形式的波传播关系。确定的波传播响应和随后的结果与相关的分子动力学模拟进行比较和校准。弯曲波传播的相速度的证明结果为数值分析提供了新的基准。所提出的高阶尺寸依赖弹性方法可以有利地用于对开创性纳米技术设备的严格分析。