Eringen’s nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation. However, several recent studies have pointed out that some inconsistent results appear while applying it in the analysis of bounded structures, which indicates that it is necessary to adopt other suitable models. In this work, both the well-posed strain-driven and stress-driven two-phase local/nonlocal integral models are used to study the size effect in the free vibration of Euler–Bernoulli curved beams. The governing equations of motion and the associated boundary conditions are derived on the basis of Hamilton’s principle. The two-phase nonlocal integral relation is transformed into an equivalent differential law with two constitutive boundary conditions. Using the generalized differential quadrature method, the governing equation in terms of displacements is solved numerically. The vibration frequencies of the beam under different boundary conditions are obtained and validated by comparing with those existing results. For all boundary conditions, the nonlocal related parameters of the two types of two-phase nonlocal strategies show consistent softening and stiffening effects on vibration response, respectively. Moreover, the effect of the curvature radius of the beam is also investigated.

Eringen 的非局部弹性模型已被广泛应用于解决微/纳米结构的尺寸相关响应，这在实验测试和分子动力学模拟中观察到。然而，最近的几项研究指出，将其应用于有界结构的分析中会出现一些不一致的结果，这表明有必要采用其他合适的模型。在这项工作中，适定应变驱动和应力驱动的两相局部/非局部积分模型都用于研究欧拉-伯努利曲梁自由振动中的尺寸效应。运动的控制方程和相关的边界条件是根据哈密尔顿原理推导出来的。将两相非局部积分关系转化为具有两个本构边界条件的等效微分定律。使用广义微分求积法，对位移控制方程进行数值求解。通过与现有结果的比较，获得并验证了不同边界条件下梁的振动频率。对于所有边界条件，两种两相非局部策略的非局部相关参数分别对振动响应表现出一致的软化和硬化效应。此外，还研究了梁曲率半径的影响。通过与现有结果的比较，获得并验证了不同边界条件下梁的振动频率。对于所有边界条件，两种两相非局部策略的非局部相关参数分别对振动响应表现出一致的软化和硬化效应。此外，还研究了梁曲率半径的影响。通过与现有结果的比较，获得并验证了不同边界条件下梁的振动频率。对于所有边界条件，两种两相非局部策略的非局部相关参数分别对振动响应表现出一致的软化和硬化效应。此外，还研究了梁曲率半径的影响。