基于Cosserat理论的平面等参元

摘 要:为解释材料在微尺度下的尺度效应,基于Cosserat理论,从势能泛函驻值条件出发提出构造8节点Serendipity平面等参元,并建立平面有限元法. 每个节点拥有3个独立节点自由度,分别为2个方向的线位移和1个逆时针方向的角位移. 用该方法分析含中心小孔的无限平板在单轴拉伸情况下的应力集中问题. 数值计算结果与Cosserat理论的解析解非常符合,表明应力集中因数k受泊松比μ,常数c及a/l值的影响很大;由于偶应力的存在,小孔周围的应力分布明显小于经典弹性力学理论的预测. 通过对材料常数c的调节可以将该方法推广应用于基于Mindlin偶应力理论的数值分析中.

关键词:Cosserat理论; 偶应力; 平面有限元法; 等参元; 尺度效应; 应力集中

中图分类号:O241.82

文献标志码:A

Plane isoparametric element based on Cosserat theory

QI Lei, ZHANG Ruojing

(College of Aerospace Eng. & Mechanics, Tongji Univ., Shanghai 200092, China)

Abstract: To explain the scale effect of materials under the microscale, based on Cosserat theory and deduced from the stationary condition of potential energy functional, an eight-node Serendipity plane isoparametric element is proposed and a finite element method for plane is developed. There are three independent node freedoms at each node, including linear displacement in two directions and counterclockwise angular displacement. The stress concentration problem of a infinite flat plate with a central small hole is analyzed by the method in the case of uniaxial tension. The numerical calculation results are in good agreement with the analytical solution based on Cosserat theory, which indicate that the stress concentration factor k is strongly influenced by Poisson ratio μ, constant c and the value of a/l; the stress distribution around the little hole is significantly less than the prediction of classical elasticity theory due to the existence of couple stress. The method can be applied to the numerical analysis based on Mindlin couple stress theory by the regulation of constant c.

Key words: Cosserat theory; couple stress; finite element method for plane; isoparametetric element; scale effect; stress concentration

0 引 言

经典的连续介质力学理论已被广泛应用于许多工程领域,如航空航天、车船、机械和土木等.事实证明,这种理论适合应用于具有宏观结构力学性能的分析.在微尺度下,由于尺度效应的影响,材料的部分力学性能不同于经典连续介质力学理论的预测.

近年来的大量实验研究也表明,当金属试件的尺寸在微米或亚微米量级时,材料具有很强的尺度效应.如在细铜丝扭转实验中,FLECK等[1]发现,当铜丝直径从170 μm下降到12 μm时,无量纲的扭转硬化增加至3倍;在Ni材质的薄梁弯曲试验中,STOLKEN等[2]

观察到当梁的厚度从100 μm下降到12.5 μm时,无量纲的弯曲硬化显著增加,而在单轴拉伸情况下这种尺度效应不存在.在微米量级尺度下,微观硬度试验与颗粒增强金属基复合材料中也观察到尺度效应.当压痕深度从10 μm减至1 μm时, 金属硬度增加1倍[3-6]

;对于以碳化硅颗粒加强的Al-Si基复合材料,LLOYD[7]等观察到当保持颗粒体积比为15%的条件下, 将颗粒直径从16 μm减为7.5 μm时,复合材料的强度显著增加.