This study addresses the deficiency of means for analysis of planar arbitrarily curved microbeams. More precisely, a formulation is developed for static analysis employing the modified couple stress theory and the Euler-Bernoulli beam model. Geometric and kinematic descriptions of a slender three-dimensional continuum body are consistently reduced to those of its beam axis. A systematic framework is presented to enable elegant determination of essential strain and stress measures. Then, the virtual work principle is employed to derive governing equations and boundary conditions. Some remarks are given on the numerical implementation with the isogeometric approach. In addition, to facilitate the verification of the derivation, the isogeometric approach is also applied to two-dimensional problems of the modified couple stress theory, and the implementation is detailed. Two comprehensive examples are used to investigate the size-dependent behavior of planar arbitrarily curved microbeams. Several rigorous tests are designed to examine the accuracy of the derived beam formulation and the validity of kinematic assumptions of the Euler-Bernoulli beam model. Finally, the robustness and efficiency of the isogeometric implementation for the proposed beam formulation are verified.

本研究解决了平面任意弯曲微束分析方法的不足。更准确地说，使用修正的耦合应力理论和 Euler-Bernoulli 梁模型开发了用于静态分析的公式。细长的三维连续体的几何和运动学描述始终简化为其光束轴的描述。提出了一个系统框架，以便能够优雅地确定基本应变和应力测量。然后，利用虚功原理推导出控制方程和边界条件。对等几何方法的数值实现给出了一些评论。此外，为了便于推导的验证，等几何方法也应用于修正偶应力理论的二维问题，并且实现很详细。两个综合示例用于研究平面任意弯曲微束的尺寸依赖性行为。设计了几个严格的测试来检查导出的梁公式的准确性和欧拉-伯努利梁模型的运动学假设的有效性。最后，验证了所提出的梁公式的等几何实现的鲁棒性和效率。