基于修正偶应力理论的Timoshenko微梁模型和尺寸效应研究

基于修正偶应力理论，将Timoshenko微梁的应力、偶应力、应变、曲率等基本变量，描述为位移分量偏导数的表达式.根据最小势能原理，推导了决定Timoshenko微梁位移场的位移场控微分方程.利用级数法求解了任意载荷作用下Timoshenko简支微梁的位移场控微分方程，得到了反映尺寸效应的挠度、转角及应力的偶应力理论解.通过对承受余弦分布载荷Timoshenko简支微梁的数值计算，研究了Timoshenko微梁的挠度、转角和应力的尺寸效应，分析了Poisson比对Timoshenko微梁力学行为及其尺寸效应的影响.结果表明:当截面高度与材料特征长度的比值小于5时，Timoshenko微梁的刚度和强度均随着截面高度的减小而显著提高，表现出明显的尺寸效应；当截面高度与材料特征长度的比值大于10时，Timoshenko微梁的刚度与强度均趋于稳定，尺寸效应可以忽略；材料Poisson比是影响Timoshenko微梁力学行为及尺寸效应的重要因素，Poisson比越大Timoshenko微梁刚度和强度的尺寸效应越显著.该文建立的Timoshenko微梁模型，能有效描述Timoshenko微梁的力学行为及尺寸效应，可为微电子机械系统（MEMS）中的微结构设计与分析提供理论基础和技术参考.

A Timoshenko Micro-Beam Model and Its Size Effects Based on the Modified Couple Stress Theory

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).