Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

中文翻译：

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混合物统一梯度纳米束弯曲的非标准界面条件


工程科学中应用兴趣的结构方案经常遇到复杂的不连续现象。本研究旨在通过采用抽象变分方案来解决弹性纳米梁弯曲的不连续性。引用弹性的混合统一梯度理论来实现超小尺度的尺寸效应。精心阐述了源于已建立的平稳变分原理的界面条件的一致形式。适当封闭平衡边值问题，求解弹性纳米梁横向位移场的解析解。作为一种替代方法，本征函数展开方法也用于检查所提出的变分公式在解决不连续弹性纳米梁弯曲方面的功效。对具有不同运动学约束的混合统一梯度梁的弯曲特性进行了数值说明和深入讨论。特征长度尺度参数的预期纳米级特征得到证实。所展示的数值结果可以有利地作为开创性超灵敏纳米传感器的分析和设计的基准。所建立的变化一致的尺寸相关框架为纳米力学的发展铺平了道路，并激发了对超小尺度弹性梁断裂力学的进一步研究。