基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

Abstract

Based on the modified couple stress theory, the basic variables of the Timoshenko microbeam, such as the stress, couple stress, strain and curvature, were described as the functions of partial derivatives of displacement components. According to the principle of minimum total potential energy, the governing differential equation was derived to determine the displacement field of the Timoshenko microbeam. The series method was utilized to solve the governing differential equation for the simply supported Timoshenko microbeam under arbitrary load, and the theoretical couple stress solutions of the deflection, rotation angle and stress, which can reflect the size effects, were obtained. The size effects of the deflection, rotation angle and stress of the Timoshenko microbeam subjected to a cosine distribution load were investigated in detail, and the influence of Poisson’s ratio on the mechanical behaviors of the Timoshenko microbeam and the size effects were analyzed. The results show that, both the stiffness and the strength of the Timoshenko microbeam improve clearly with the decreased cross section height and their size effects are obvious when the ratio of the cross section height to the material characteristic length is less than 5. However, both the stiffness and the strength of the Timoshenko microbeam tend to be stable and their size effects can be neglected when the ratio of the cross section height to the material characteristic length is greater than 10. Poisson’s ratio is an important factor influencing the mechanical behaviors of the Timoshenko microbeam and the size effects. The smaller Poisson’s ratio is, the more significant the size effects of the stiffness and the strength will be. The developed model can effectively describe the mechanical behaviors of Timoshenko microbeams and their size effects, and makes a theoretical basis and technical reference for the design and analysis of microstructures in the micro electro mechanical systems (MEMS).