基于Cosserat理论的平面等参元  

Plane isoparametric element based on Cosserat theory

作　　者：齐磊[1] 张若京[1] 

机构地区：[1]同济大学航空航天与力学学院,上海200092

出　　处：《计算机辅助工程》2010年第1期12-16,26,共6页Computer Aided Engineering

基　　金：国家自然科学基金(10772136);教育部博士点基金(20060247016);上海市重点学科建设项目(B302)

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.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.

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 

分 类 号：O33[理学—一般力学与力学基础] O241.82[理学—力学]