Spatial arbitrarily curved microbeams with the modified couple stress theory: Formulation of equations of motion

Abstract

This study commences the application of the modified couple stress theory in the analysis of spatial arbitrarily curved microbeams. The kinematic assumptions of the Timoshenko-Ehrenfest beam theory are employed. The moving trihedron of the beam axis are used to form a local Cartesian coordinate system. Displacements of the beam axis and cross-sectional rotations in the local coordinate system are considered as unknowns in the kinematic descriptions. The principle of virtual work is used to derive the equations of motion. The outputs, including the deformation measures, constitutive relations, cross-sectional stress resultants, equations of motion, and boundary conditions, are explicitly expressed in terms of the kinematic unknowns. The established general formulation is further specialized to the case of planar arbitrarily curved microbeams which cannot be found in the literature. The accuracy of the derived equations of motion is tested by a comparison with existing studies of planar circular and straight microbeams. Since this is the first study for the use of the modified couple stress theory in the setting of spatial arbitrarily curved microbeams, the outputs of the present study enable further in-deep analysis to comprehend these structures.