Static analysis of planar arbitrarily curved microbeams with the modified couple stress theory and Euler-Bernoulli beam model

Abstract

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.