基于平面偶应力-Reissner/Mindlin板比拟的偶应力有限元
Finite Element Of Elasticity With Couple-Stress Using The Analogy Between Plane Couple-Stress And Reissner/Mindlin Plate Bending
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摘要: 偶应力理论的有限元列式面临本质性的C1连续性困难.平面偶应力理论和Reissner/Mindlin板弯曲理论之间的比拟关系表明这两个理论系统的有限元的同一性，而R/M板有限元并不存在C1连续性困难.因此，研究将R/M板单元转化为具有一般位移自由度的平面偶应力单元的一般方法.根据这一方法，将典型的8节点Serendipity型R/M板单元Q8S转化为一个4节点12自由度的四边形平面偶应力单元，数值结果表明该单元具有良好的精度和收敛性Abstract: In order to include the effect of microstructure, thetheory of elasticity with couple stress considers couple stress which doesnot appear in the classical elasticity theory. However, there exists acrucial C$^{1}$ continuity difficulty in the finite element formulation ofelasticity with couple stress.The analogy between plane elasticity with couple stress and Reissner/Mindlinplate bending provides an important way to avoid the C$^{1}$ continuitydifficulty. According to the analogy, the C$^{1}$ continuity difficulty canbe avoided naturally by the formulation in the space of stress functions,and the formulation can be analogous to the one of certain Reissner/Mindlinplate bending element in the space of transversal deflection and rotation.The unsettled problem is how to transform the finite element with stressfunctions as degree of freedom (DOF) into the one with usual planardisplacement and rotation as DOF. Using the analogy, the present workprovides an effective and rigorous method to deal with this problem. Thefinal finite element has two important characteristics. Firstly, theformulation in space of stress functions avoids C$^{1}$ continuitydifficulty. Secondly, the discrete unknown DOF are usual displacement androtation.As an application of the present method, a finite element of plane couplestress with 12 DOF is transformed from the eight nodes serendipityReissner/Mindlin plate bending element. Numerical results of typicalproblems show that the present element has satisfactory precision andconvergence.
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