Recently, it has been proved that the common nonlocal strain gradient theory has inconsistence behaviors. The order of the differential nonlocal strain gradient governing equations is less than the number of all mandatory boundary conditions, and therefore, there is no solution for these differential equations. Given these, for the first time, transverse vibrations of nanobeams are analyzed within the framework of the two-phase local/nonlocal strain gradient (LNSG) theory, and to this aim, the exact solution as well as finite-element model are presented. To achieve the exact solution, the governing differential equations of LNSG nanobeams are derived by transformation of the basic integral form of the LNSG to its equal differential form. Furthermore, on the basis of the integral LNSG, a shear-locking-free finite-element (FE) model of the LNSG Timoshenko beams is constructed by introducing a new efficient higher order beam element with simple shape functions which can consider the influence of strains gradient as well as maintain the shear-locking-free property. Agreement between the exact results obtained from the differential LNSG and those of the FE model, integral LNSG, reveals that the LNSG is consistent and can be utilized instead of the common nonlocal strain gradient elasticity theory.

中文翻译：

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用一致的两相非局部应变梯度理论研究纳米梁的振动：精确解和积分非局部有限元模型

最近，证明了常见的非局部应变梯度理论存在不一致行为。微分非局部应变梯度控制方程的阶数小于所有强制边界条件的数量，因此，这些微分方程没有解。鉴于此，首次在两相局部/非局部应变梯度 (LNSG) 理论的框架内分析了纳米梁的横向振动，并为此目的，提出了精确解和有限元模型。为了获得精确解，LNSG 纳米梁的控制微分方程是通过将 LNSG 的基本积分形式转换为其等微分形式而导出的。此外，在积分 LNSG 的基础上，LNSG Timoshenko 梁的无剪切锁定有限元 (FE) 模型是通过引入具有简单形状函数的新的高效高阶梁单元构建的，该单元可以考虑应变梯度的影响并保持剪切锁定 -免费财产。从微分 LNSG 和有限元模型（积分 LNSG）获得的精确结果之间的一致性表明，LNSG 是一致的，可以用来代替常见的非局部应变梯度弹性理论。